Text Books Text book #1: Using Econometrics: A Practical Guide

Professor of Statistics
The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences Course: Econometrics- MDEC 6301 Semester: Spring 2015 Instructor: Prof. Dr. Samir Safi Professor of Statistics Note: The original Power point files are designed by the publisher “Prentice Hall” 1-0

Text Books Text book #1: Using Econometrics: A Practical Guide
Author: Studenmund, A.H. Edition: Sixth edition, 2011 Publisher: Prentice Hall , New Jersey, USA. Text book #2: Introduction to Econometrics Author: Stock, J. H. and Watson, M. W. Edition: Third edition, 2011 1-1

Text Books - Continued 3. الكتاب الثالث
اسم الكتاب: مقدمة في تحليل نماذج الانحدار باستخدام EViews اسم المؤلف: أ.د. سمير خالد صافي الطبعة: الأولى 2015 دار النشر: مكتبة آفاق – غزة، فلسطين. 4. الكتاب الرابع اسم الكتاب: الحديث في الاقتصاد القياسي بين النظرية والتطبيق اسم المؤلف: أ.د. عبد القادر محمد عطية الطبعة: 2005 دار النشر: الدار الجامعية، الإسكندرية، مصر. 1-2

An Overview of Regression
Chapter 1 An Overview of Regression © 2011 Pearson Addison-Wesley. All rights reserved. 1-3

What is Econometrics? Econometrics literally means “economic measurement” It is the quantitative measurement and analysis of actual economic and business phenomena—and so involves: economic theory Statistics Math observation/data collection © 2011 Pearson Addison-Wesley. All rights reserved. 1-4 4

What is Econometrics? (cont.)
Three major uses of econometrics: Describing economic reality Testing hypotheses about economic theory Forecasting future economic activity So econometrics is all about questions: the researcher (YOU!) first asks questions and then uses econometrics to answer them © 2011 Pearson Addison-Wesley. All rights reserved. 5

Example Consider the general and purely theoretical relationship:
Q = f(P, Ps, Yd) (1.1) Econometrics allows this general and purely theoretical relationship to become explicit: Q = 27.7 – 0.11P Ps Yd (1.2) © 2011 Pearson Addison-Wesley. All rights reserved. 6

What is Regression Analysis?
Economic theory can give us the direction of a change, e.g. the change in the demand for dvd’s following a price decrease (or price increase) But what if we want to know not just “how?” but also “how much?” Then we need: A sample of data A way to estimate such a relationship one of the most frequently ones used is regression analysis © 2011 Pearson Addison-Wesley. All rights reserved. 7

What is Regression Analysis? (cont.)
Formally, regression analysis is a statistical technique that attempts to “explain” movements in one variable, the dependent variable, as a function of movements in a set of other variables, the independent (or explanatory) variables, through the quantification of a single equation © 2011 Pearson Addison-Wesley. All rights reserved. 8

Example Return to the example from before:
Q = f(P, Ps, Yd) (1.1) Here, Q is the dependent variable and P, Ps, Yd are the independent variables Don’t be deceived by the words dependent and independent, however A statistically significant regression result does not necessarily imply causality We also need: Economic theory Common sense © 2011 Pearson Addison-Wesley. All rights reserved. 9

Single-Equation Linear Models
The simplest example is: Y = X (1.3) The are denoted “coefficients” is the “constant” or “intercept” term is the “slope coefficient”: the amount that Y will change when X increases by one unit; for a linear model, is constant over the entire function © 2011 Pearson Addison-Wesley. All rights reserved. 10

Figure 1.1 Graphical Representation of the Coefficients of the Regression Line
© 2011 Pearson Addison-Wesley. All rights reserved. 11

Single-Equation Linear Models (cont.)
Application of linear regression techniques requires that the equation be linear—such as (1.3) By contrast, the equation Y = X2 (1.4) is not linear What to do? First define Z = X (1.5) Substituting into (1.4) yields: Y = Z (1.6) This redefined equation is now linear (in the coefficients and in the variables Y and Z) © 2011 Pearson Addison-Wesley. All rights reserved. 12

Is (1.3) a complete description of origins of variation in Y? No, at least four sources of variation in Y other than the variation in the included Xs: Other potentially important explanatory variables may be missing (e.g., X2 and X3) Measurement error Incorrect functional form Purely random and totally unpredictable occurrences Inclusion of a “stochastic error term” (ε) effectively “takes care” of all these other sources of variation in Y that are NOT captured by X, so that (1.3) becomes: Y = β0 + β1X + ε (1.7) © 2011 Pearson Addison-Wesley. All rights reserved. 13

Two components in (1.7): deterministic component (β0 + β1X) stochastic/random component (ε) Why “deterministic”? Indicates the value of Y that is determined by a given value of X (which is assumed to be non-stochastic) Alternatively, the det. comp. can be thought of as the expected value of Y given X—namely E(Y|X)—i.e. the mean (or average) value of the Ys associated with a particular value of X This is also denoted the conditional expectation (that is, expectation of Y conditional on X) © 2011 Pearson Addison-Wesley. All rights reserved. 14

Example: Aggregate Consumption Function
Aggregate consumption as a function of aggregate income may be lower (or higher) than it would otherwise have been due to: consumer uncertainty—hard (impossible?) to measure, i.e. is an omitted variable Observed consumption may be different from actual consumption due to measurement error The “true” consumption function may be nonlinear but a linear one is estimated (see Figure 1.2 for a graphical illustration) Human behavior always contains some element(s) of pure chance; unpredictable, i.e. random events may increase or decrease consumption at any given time Whenever one or more of these factors are at play, the observed Y will differ from the Y predicted from the deterministic part, β0 + β1X © 2011 Pearson Addison-Wesley. All rights reserved. 15

Figure 1.2 Errors Caused by Using a Linear Functional Form to Model a Nonlinear Relationship

Extending the Notation
Include reference to the number of observations Single-equation linear case: Yi = β0 + β1Xi + εi (i = 1,2,…,N) (1.10) So there are really N equations, one for each observation the coefficients, β0 and β1, are the same the values of Y, X, and ε differ across observations © 2011 Pearson Addison-Wesley. All rights reserved. 17

Extending the Notation (cont.)
The general case: multivariate regression Yi = β0 + β1X1i + β2X2i + β3X3i + εi (i = 1,2,…,N) (1.11) Each of the slope coefficients gives the impact of a one-unit increase in the corresponding X variable on Y, holding the other included independent variables constant (i.e., ceteris paribus) As an (implicit) consequence of this, the impact of variables that are not included in the regression are not held constant (we return to this in Ch. 6) © 2011 Pearson Addison-Wesley. All rights reserved. 18

Example: Wage Regression
Let wages (WAGE) depend on: years of work experience (EXP) years of education (EDU) gender of the worker (GEND: 1 if male, 0 if female) Substituting into equation (1.11) yields: WAGEi = β0 + β1EXPi + β2EDUi + β3GENDi + εi (1.12) © 2011 Pearson Addison-Wesley. All rights reserved. 19

Indexing Conventions Subscript “i” for data on individuals (so called “cross section” data) Subscript “t” for time series data (e.g., series of years, months, or days—daily exchange rates, for example ) Subscript “it” when we have both (for example, “panel data”) © 2011 Pearson Addison-Wesley. All rights reserved. 20

The Estimated Regression Equation
The regression equation considered so far is the “true”—but unknown—theoretical regression equation Instead of “true,” might think about this as the population regression vs. the sample/estimated regression How do we obtain the empirical counterpart of the theoretical regression model (1.14)? It has to be estimated The empirical counterpart to (1.14) is: (1.16) The signs on top of the estimates are denoted “hat,” so that we have “Y-hat,” for example © 2011 Pearson Addison-Wesley. All rights reserved. 21

The Estimated Regression Equation (cont.)
For each sample we get a different set of estimated regression coefficients Y is the estimated value of Yi (i.e. the dependent variable for observation i); similarly it is the prediction of E(Yi|Xi) from the regression equation The closer Y is to the observed value of Yi, the better is the “fit” of the equation Similarly, the smaller is the estimated error term, ei, often denoted the “residual,” the better is the fit © 2011 Pearson Addison-Wesley. All rights reserved. 22

The Estimated Regression Equation (cont.)
This can also be seen from the fact that (1.17) Note difference with the error term, εi, given as (1.18) This all comes together in Figure 1.3 © 2011 Pearson Addison-Wesley. All rights reserved. 23

Figure 1.3 True and Estimated Regression Lines

Example: Using Regression to Explain Housing prices
Houses are not homogenous products, like corn or gold, that have generally known market prices So, how to appraise a house against a given asking price? Yes, it’s true: many real estate appraisers actually use regression analysis for this! Consider specific case: Suppose the asking price was $230,000 © 2011 Pearson Addison-Wesley. All rights reserved. 25

Example: Using Regression to Explain Housing prices (cont.)
Is this fair / too much /too little? Depends on size of house (higher size, higher price) So, collect cross-sectional data on prices (in thousands of $) and sizes (in square feet) for, say, 43 houses Then say this yields the following estimated regression line: (1.23) © 2011 Pearson Addison-Wesley. All rights reserved. 26

Figure 1.5 A Cross-Sectional Model of Housing Prices

Note that the interpretation of the intercept term is problematic in this case (we’ll get back to this later, in Section 7.1.2) The literal interpretation of the intercept here is the price of a house with a size of zero square feet… © 2011 Pearson Addison-Wesley. All rights reserved. 28

How to use the estimated regression line / estimated regression coefficients to answer the question? Just plug the particular size of the house, you are interested in (here, 1,600 square feet) into (1.23) Alternatively, read off the estimated price using Figure 1.5 Either way, we get an estimated price of $260.8 (thousand, remember!) So, in terms of our original question, it’s a good deal—go ahead and purchase!! Note that we simplified a lot in this example by assuming that only size matters for housing prices © 2011 Pearson Addison-Wesley. All rights reserved. 29

Ordinary Least Squares (OLS)
Chapter 2 Ordinary Least Squares (OLS) © 2011 Pearson Addison-Wesley. All rights reserved. 1-30

Estimating Single-Independent-Variable Models with OLS
Recall that the objective of regression analysis is to start from: (2.1) And, through the use of data, to get to: (2.2) Recall that equation 2.1 is purely theoretical, while equation (2.2) is it empirical counterpart How to move from (2.1) to (2.2)? © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-31 31 31

Estimating Single-Independent-Variable Models with OLS (cont.)
One of the most widely used methods is Ordinary Least Squares (OLS) OLS minimizes (i = 1, 2, …., N) (2.3) Or, the sum of squared deviations of the vertical distance between the residuals (i.e. the estimated error terms) and the estimated regression line We also denote this term the “Residual Sum of Squares” (RSS) © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-32 32 32

Similarly, OLS minimizes: Why use OLS? Relatively easy to use The goal of minimizing RSS is intuitively / theoretically appealing This basically says we want the estimated regression equation to be as close as possible to the observed data OLS estimates have a number of useful characteristics © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-33 33 33

OLS estimates have at least two useful characteristics: The sum of the residuals is exactly zero OLS can be shown to be the “best” estimator when certain specific conditions hold (we’ll get back to this in Chapter 4) Ordinary Least Squares (OLS) is an estimator A given produced by OLS is an estimate © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-34 34 34

How does OLS work? First recall from (2.3) that OLS minimizes the sum of the squared residuals Next, it can be shown (see Exercise 12) that the coefficients that ensure that for the case of just one independent variable are: (2.4) (2.5) © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-35 35 35

Estimating Multivariate Regression Models with OLS
In the “real world” one explanatory variable is not enough The general multivariate regression model with K independent variables is: Yi = β0 + β1X1i + β2X2i βKXKi + εi (i = 1,2,…,N) (1.13) Biggest difference with single-explanatory variable regression model is in the interpretation of the slope coefficients Now a slope coefficient indicates the change in the dependent variable associated with a one-unit increase in the explanatory variable holding the other explanatory variables constant © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-36 36 36

Estimating Multivariate Regression Models with OLS (cont.)
Omitted (and relevant!) variables are therefore not held constant The intercept term, β0, is the value of Y when all the Xs and the error term equal zero Nevertheless, the underlying principle of minimizing the summed squared residuals remains the same © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-37 37 37

Example: financial aid awards at a liberal arts college
Dependent variable: FINAIDi: financial aid (measured in dollars of grant) awarded to the ith applicant © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-38 38 38

Example: financial aid awards at a liberal arts college
Theoretical Model: (2.9) (2.10) where: PARENTi: The amount (in dollars) that the parents of the ith student are judged able to contribute to college expenses HSRANKi: The ith student’s GPA rank in high school, measured as a percentage (i.e. between 0 and 100) © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-39 39 39

Example: financial aid awards at a liberal arts college (cont.)
Estimate model using the data in Table 2.2 to get: (2.11) Interpretation of the slope coefficients? Graphical interpretation in Figures 2.1 and 2.2 © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-40 40 40

Figure 2.1 Financial Aid as a Function of Parents’ Ability to Pay
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Figure 2.2 Financial Aid as a Function of High School Rank

Total, Explained, and Residual Sums of Squares
(2.12) (2.13) TSS = ESS + RSS This is usually called the decomposition of variance © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-43 43 43

Figure 2.3 Decomposition of the Variance in Y

Evaluating the Quality of a Regression Equation
Checkpoints here include the following: 1. Is the equation supported by sound theory? 2. How well does the estimated regression fit the data? 3. Is the data set reasonably large and accurate? 4. Is OLS the best estimator to be used for this equation? 5. How well do the estimated coefficients correspond to the expectations developed by the researcher before the data were collected? 6. Are all the obviously important variables included in the equation? 7. Has the most theoretically logical functional form been used? 8. Does the regression appear to be free of major econometric problems? *These numbers roughly correspond to the relevant chapters in the book © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-45 45 45

Describing the Overall Fit of the Estimated Model
The simplest commonly used measure of overall fit is the coefficient of determination, R2: (2.14) Since OLS selects the coefficient estimates that minimizes RSS, OLS provides the largest possible R2 (within the class of linear models) © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-46 46 46

Figure 2.4 Illustration of Case Where R2 = 0

Figure 2.5 Illustration of Case Where R2 = .95

Figure 2.6 Illustration of Case Where R2 = 1

The Simple Correlation Coefficient, r
This is a measure related to R2 r measures the strength and direction of the linear relationship between two variables: r = +1: the two variables are perfectly positively correlated r = –1: the two variables are perfectly negatively correlated r = 0: the two variables are totally uncorrelated © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-50 50 50

The adjusted coefficient of determination
A major problem with R2 is that it can never decrease if another independent variable is added An alternative to R2 that addresses this issue is the adjusted R2 or R2: (2.15) Where N – K – 1 = degrees of freedom © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-51 51 51

The adjusted coefficient of determination (cont.)
So, R2 measures the share of the variation of Y around its mean that is explained by the regression equation, adjusted for degrees of freedom R2 can be used to compare the fits of regressions with the same dependent variable and different numbers of independent variables As a result, most researchers automatically use instead of R2 when evaluating the fit of their estimated regressions equations © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 2-52 52 52

Learning to Use Regression Analysis
Chapter 3 Learning to Use Regression Analysis © 2011 Pearson Addison-Wesley. All rights reserved.

Steps in Applied Regression Analysis
The first step is choosing the dependent variable – this step is determined by the purpose of the research (see Chapter 11 for details) After choosing the dependent variable, it’s logical to follow the following sequence: 1. Review the literature and develop the theoretical model 2. Specify the model: Select the independent variables and the functional form 3. Hypothesize the expected signs of the coefficients 4. Collect the data. Inspect and clean the data 5. Estimate and evaluate the equation 6. Document the results © 2011 Pearson Addison-Wesley. All rights reserved. 54

Step 1: Review the Literature and Develop the Theoretical Model
Perhaps counter intuitively, a strong theoretical foundation is the best start for any empirical project Reason: main econometric decisions are determined by the underlying theoretical model Useful starting points: Journal of Economic Literature or a business oriented publication of abstracts Internet search, including Google Scholar EconLit, an electronic bibliography of economics literature (for more details, go to © 2011 Pearson Addison-Wesley. All rights reserved. 55

Step 2: Specify the Model: Independent Variables and Functional Form
After selecting the dependent variable, the specification of a model involves choosing the following components: the independent variables and how they should be measured, the functional (mathematical) form of the variables, and the properties of the stochastic error term © 2011 Pearson Addison-Wesley. All rights reserved. 56

Step 2: Specify the Model: Independent Variables and Functional Form (cont.)
A mistake in any of the three elements results in a specification error For example, only theoretically relevant explanatory variables should be included Even so, researchers frequently have to make choices –also denoted imposing their priors Example: when estimating a demand equation, theory informs us that prices of complements and substitutes of the good in question are important explanatory variables But which complements—and which substitutes? © 2011 Pearson Addison-Wesley. All rights reserved. 57

Step 3: Hypothesize the Expected Signs of the Coefficients
Once the variables are selected, it’s important to hypothesize the expected signs of the regression coefficients Example: demand equation for a final consumption good First, state the demand equation as a general function: (3.2) The signs above the variables indicate the hypothesized sign of the respective regression coefficient in a linear model © 2011 Pearson Addison-Wesley. All rights reserved. 58

Step 4: Collect the Data & Inspect and Clean the Data
A general rule regarding sample size is “the more observations the better” as long as the observations are from the same general population! The reason for this goes back to notion of degrees of freedom (mentioned first in Section 2.4) When there are more degrees of freedom: Every positive error is likely to be balanced by a negative error (see Figure 3.2) The estimated regression coefficients are estimated with a greater deal of precision © 2011 Pearson Addison-Wesley. All rights reserved. 59

Figure 3.1 Mathematical Fit of a Line to Two Points

Figure 3.2 Statistical Fit of a Line to Three Points

Step 4: Collect the Data & Inspect and Clean the Data (cont.)
Estimate model using the data in Table 2.2 to get: Inspecting the data—obtain a printout or plot (graph) of the data Reason: to look for outliers An outlier is an observation that lies outside the range of the rest of the observations Examples: Does a student have a 7.0 GPA on a 4.0 scale? Is consumption negative? © 2011 Pearson Addison-Wesley. All rights reserved. 62

Step 5: Estimate and Evaluate the Equation
Once steps 1–4 have been completed, the estimation part is quick using Eviews or Stata to estimate an OLS regression takes less than a second! The evaluation part is more tricky, however, involving answering the following questions: How well did the equation fit the data? Were the signs and magnitudes of the estimated coefficients as expected? Afterwards may add sensitivity analysis (see Section 6.4 for details) © 2011 Pearson Addison-Wesley. All rights reserved. 63

Step 6: Document the Results
A standard format usually is used to present estimated regression results: (3.3) The number in parentheses under the estimated coefficient is the estimated standard error of the estimated coefficient, and the t-value is the one used to test the hypothesis that the true value of the coefficient is different from zero (more on this later!) © 2011 Pearson Addison-Wesley. All rights reserved. 64

Case Study: Using Regression Analysis to Pick Restaurant Locations
Background: You have been hired to determine the best location for the next Woody’s restaurant (a moderately priced, 24-hour, family restaurant chain) Objective: How to decide location using the six basic steps of applied regression analysis, discussed earlier? © 2011 Pearson Addison-Wesley. All rights reserved. 65

Step 1: Review the Literature and Develop the Theoretical Model
Background reading about the restaurant industry Talking to various experts within the firm All the chain’s restaurants are identical and located in suburban, retail, or residential environments So, lack of variation in potential explanatory variables to help determine location Number of customers most important for locational decision  Dependent variable: number of customers (measured by the number of checks or bills) © 2011 Pearson Addison-Wesley. All rights reserved. 66

Step 2: Specify the Model: Independent Variables and Functional Form
More discussions with in-house experts reveal three major determinants of sales: Number of people living near the location General income level of the location Number of direct competitors near the location © 2011 Pearson Addison-Wesley. All rights reserved. 67

Step 2: Specify the Model: Independent Variables and Functional Form (cont.)
Based on this, the exact definitions of the independent variables you decide to include are: N = Competition: the number of direct competitors within a two-mile radius of the Woody’s location P = Population: the number of people living within a three-mile radius of the location I = Income: the average household income of the population measured in variable P With no reason to suspect anything other than linear functional form and a typical stochastic error term, that’s what you decide to use © 2011 Pearson Addison-Wesley. All rights reserved. 68

Step 3: Hypothesize the Expected Signs of the Coefficients
After talking some more with the in-house experts and thinking some more, you come up with the following: (3.4) © 2011 Pearson Addison-Wesley. All rights reserved. 69

Step 4: Collect the Data & Inspect and Clean the Data
You manage to obtain data on the dependent and independent variables for all 33 Woody’s restaurants Next, you inspect the data The data quality is judged as excellent because: Each manager measures each variable identically All restaurants are included in the sample All information is from the same year The resulting data is as given in Tables 3.1 and 3.3 in the book (using Eviews and Stata, respectively) © 2011 Pearson Addison-Wesley. All rights reserved. 70

Step 5: Estimate and Evaluate the Equation
You take the data set and enter it into the computer You then run an OLS regression (after thinking the model over one last time!) The resulting model is: Estimated coefficients are as expected and the fit is reasonable Values for N, P, and I for each potential new location are then obtained and plugged into (3.5) to predict Y (3.5) © 2011 Pearson Addison-Wesley. All rights reserved. 71

Step 6: Document the Results
The results summarized in Equation 3.5 meet our documentation requirements Hence, you decide that there’s no need to take this step any further © 2011 Pearson Addison-Wesley. All rights reserved. 72

Chapter 4 The Classical Model
© 2011 Pearson Addison-Wesley. All rights reserved.

The Classical Assumptions
The classical assumptions must be met in order for OLS estimators to be the best available The seven classical assumptions are: I. The regression model is linear, is correctly specified, and has an additive error term II. The error term has a zero population mean III. All explanatory variables are uncorrelated with the error term IV. Observations of the error term are uncorrelated with each other (no serial correlation) V. The error term has a constant variance (no heteroskedasticity) VI. No explanatory variable is a perfect linear function of any other explanatory variable(s) (no perfect multicollinearity) VII. The error term is normally distributed (this assumption is optional but usually is invoked) © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-74 74 74

I: linear, correctly specified, additive error term
Consider the following regression model: Yi = β0 + β1X1i + β2X2i βKXKi + εi (4.1) This model: is linear (in the coefficients) has an additive error term If we also assume that all the relevant explanatory variables are included in (4.1) then the model is also correctly specified © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-75 75 75

II: Error term has a zero population mean
As was pointed out in Section 1.2, econometricians add a stochastic (random) error term to regression equations Reason: to account for variation in the dependent variable that is not explained by the model The specific value of the error term for each observation is determined purely by chance This can be illustrated by Figure 4.1 © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-76 76 76

Figure 4.1 An Error Term Distribution with a Mean of Zero

III: All explanatory variables are uncorrelated with the error term
If not, the OLS estimates would be likely to attribute to the X some of the variation in Y that actually came from the error term For example, if the error term and X were positively correlated then the estimated coefficient would probably be higher than it would otherwise have been (biased upward) This assumption is violated most frequently when a researcher omits an important independent variable from an equation © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-78 78 78

IV: No serial correlation of error term
If a systematic correlation does exist between one observation of the error term and another, then it will be more difficult for OLS to get accurate estimates of the standard errors of the coefficients This assumption is most likely to be violated in time-series models: An increase in the error term in one time period (a random shock, for example) is likely to be followed by an increase in the next period, also Example: Hurricane Katrina If, over all the observations of the sample εt+1 is correlated with εt then the error term is said to be serially correlated (or auto-correlated), and Assumption IV is violated Violations of this assumption are considered in more detail in Chapter 9 © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-79 79 79

V: Constant variance / No heteroskedasticity in error term
The error term must have a constant variance That is, the variance of the error term cannot change for each observation or range of observations If it does, there is heteroskedasticity present in the error term An example of this can bee seen from Figure 4.2 © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-80 80 80

Figure 4.2 An Error Term Whose Variance Increases as Z Increases (Heteroskedasticity)

VI: No perfect multicollinearity
Perfect collinearity between two independent variables implies that: they are really the same variable, or one is a multiple of the other, and/or that a constant has been added to one of the variables Example: Including both annual sales (in dollars) and the annual sales tax paid in a regression at the level of an individual store, all in the same city Since the stores are all in the same city, there is no variation in the percentage sales tax © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-82 82 82

VII: The error term is normally distributed
Basically implies that the error term follows a bell-shape (see Figure 4.3) Strictly speaking not required for OLS estimation (related to the Gauss-Markov Theorem: more on this in Section 4.3) Its major application is in hypothesis testing, which uses the estimated regression coefficient to investigate hypotheses about economic behavior (see Chapter 5) © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-83 83 83

Figure 4.3 Normal Distributions

The Sampling Distribution of
We saw earlier that the error term follows a probability distribution (Classical Assumption VII) But so do the estimates of β! The probability distribution of these values across different samples is called the sampling distribution of We will now look at the properties of the mean, the variance, and the standard error of this sampling distribution © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-85 85 85

Properties of the Mean A desirable property of a distribution of estimates in that its mean equals the true mean of the variables being estimated Formally, an estimator is an unbiased estimator if its sampling distribution has as its expected value the true value of . We also write this as follows: (4.9) Similarly, if this is not the case, we say that the estimator is biased © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-86 86 86

Properties of the Variance
Just as we wanted the mean of the sampling distribution to be centered around the true population , so too it is desirable for the sampling distribution to be as narrow (or precise) as possible. Centering around “the truth” but with high variability might be of very little use. One way of narrowing the sampling distribution is to increase the sampling size (which therefore also increases the degrees of freedom) These points are illustrated in Figures 4.4 and 4.5 © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-87 87 87

Figure 4.4 Distributions of

Figure 4.5 Sampling Distribution of for Various Observations (N)

Properties of the Standard Error
The standard error of the estimated coefficient, SE( ), is the square root of the estimated variance of the estimated coefficients. Hence, it is similarly affected by the sample size and the other factors discussed previously For example, an increase in the sample size will decrease the standard error Similarly, the larger the sample, the more precise the coefficient estimates will be © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-90 90 90

The Gauss-Markov Theorem and the Properties of OLS Estimators
The Gauss-Markov Theorem states that: Given Classical Assumptions I through VI (Assumption VII, normality, is not needed for this theorem), the Ordinary Least Squares estimator of –k is the minimum variance estimator from among the set of all linear unbiased estimators of –k, for k = 0, 1, 2, …, K We also say that “OLS is BLUE”: “Best (meaning minimum variance) Linear Unbiased Estimator” © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-91 91 91

The Gauss-Markov Theorem and the Properties of OLS Estimators (cont.)
The Gauss-Markov Theorem only requires the first six classical assumptions If we add the seventh condition, normality, the OLS coefficient estimators can be shown to have the following properties: Unbiased: the OLS estimates coefficients are centered around the true population values Minimum variance: no other unbiased estimator has a lower variance for each estimated coefficient than OLS Consistent: as the sample size gets larger, the variance gets smaller, and each estimate approaches the true value of the coefficient being estimated Normally distributed: when the error term is normally distributed, so are the estimated coefficients—which enables various statistical tests requiring normality to be applied (we’ll get back to this in Chapter 5) © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 4-92 92 92

Hypothesis Testing Chapter 5

What Is Hypothesis Testing?
Hypothesis testing is used in a variety of settings The Food and Drug Administration (FDA), for example, tests new products before allowing their sale If the sample of people exposed to the new product shows some side effect significantly more frequently than would be expected to occur by chance, the FDA is likely to withhold approval of marketing that product Similarly, economists have been statistically testing various relationships, for example that between consumption and income Note here that while we cannot prove a given hypothesis (for example the existence of a given relationship), we often can reject a given hypothesis (again, for example, rejecting the existence of a given relationship) © 2011 Pearson Addison-Wesley. All rights reserved. 94

Classical Null and Alternative Hypotheses
The researcher first states the hypotheses to be tested Here, we distinguish between the null and the alternative hypothesis: Null hypothesis (“H0”): the outcome that the researcher does not expect (almost always includes an equality sign) Alternative hypothesis (“HA”): the outcome the researcher does expect Example: H0: β ≤ 0 (the values you do not expect) HA: β > 0 (the values you do expect) © 2011 Pearson Addison-Wesley. All rights reserved. 95

Type I and Type II Errors
Two types of errors possible in hypothesis testing: Type I: Rejecting a true null hypothesis Type II: Not rejecting a false null hypothesis Example: Suppose we have the following null and alternative hypotheses: H0: β ≤ 0 HA: β > 0 Even if the true β really is not positive, in any one sample we might still observe an estimate of β that is sufficiently positive to lead to the rejection of the null hypothesis This can be illustrated by Figure 5.1 © 2011 Pearson Addison-Wesley. All rights reserved. 96

Figure 5.1 Rejecting a True Null Hypothesis Is a Type I Error

Type I and Type II Errors (cont.)
Alternatively, it’s possible to obtain an estimate of β that is close enough to zero (or negative) to be considered “not significantly positive” Such a result may lead the researcher to “accept” the null hypothesis that β ≤ 0 when in truth β > 0 This is a Type II Error; we have failed to reject a false null hypothesis! This can be illustrated by Figure 5.2 © 2011 Pearson Addison-Wesley. All rights reserved. 98

Figure 5.2 Failure to Reject a False Null Hypothesis Is a Type II Error

Decision Rules of Hypothesis Testing
To test a hypothesis, we calculate a sample statistic that determines when the null hypothesis can be rejected depending on the magnitude of that sample statistic relative to a preselected critical value (which is found in a statistical table) This procedure is referred to as a decision rule The decision rule is formulated before regression estimates are obtained The range of possible values of the estimates is divided into two regions, an “acceptance” (really, non-rejection) region and a rejection region The critical value effectively separates the “acceptance”/non-rejection region from the rejection region when testing a null hypothesis Graphs of these “acceptance” and rejection regions are given in Figures 5.3 and 5.4 © 2011 Pearson Addison-Wesley. All rights reserved. 100

Figure 5.3 “Acceptance” and Rejection Regions for a One-Sided Test of β

Figure 5.4 “Acceptance” and Rejection Regions for a Two-Sided Test of β

The t-Test The t-test is the test that econometricians usually use to test hypotheses about individual regression slope coefficients Tests of more than one coefficient at a time (joint hypotheses) are typically done with the F-test, presented in Section 5.6 The appropriate test to use when the stochastic error term is normally distributed and when the variance of that distribution must be estimated Since these usually are the case, the use of the t-test for hypothesis testing has become standard practice in econometrics © 2011 Pearson Addison-Wesley. All rights reserved. 103

The t-Statistic For a typical multiple regression equation: (5.1)
we can calculate t-values for each of the estimated coefficients Usually these are only calculated for the slope coefficients, though (see Section 7.1) Specifically, the t-statistic for the kth coefficient is: (5.2) © 2011 Pearson Addison-Wesley. All rights reserved. 104

The Critical t-Value and the t-Test Decision Rule
To decide whether to reject or not to reject a null hypothesis based on a calculated t-value, we use a critical t-value A critical t-value is the value that distinguishes the “acceptance” region from the rejection region The critical t-value, tc, is selected from a t-table (see Statistical Table B-1 in the back of the book) depending on: whether the test is one-sided or two-sided, the level of Type I Error specified and the degrees of freedom (defined as the number of observations minus the number of coefficients estimated (including the constant) or N – K – 1) © 2011 Pearson Addison-Wesley. All rights reserved. 105

The Critical t-Value and the t-Test Decision Rule (cont.)
The rule to apply when testing a single regression coefficient ends up being that you should: Reject H0 if |tk| > tc and if tk also has the sign implied by HA Do not reject H0 otherwise © 2011 Pearson Addison-Wesley. All rights reserved. 106

Note that this decision rule works both for calculated t-values and critical t-values for one-sided hypotheses around zero (or another hypothesized value, S): H0: βk ≤ 0 H0: βk ≤ S HA: βk > 0 HA: βk > S H0: βk ≥ 0 H0: βk ≥ S HA: βk < 0 HA: βk < S © 2011 Pearson Addison-Wesley. All rights reserved. 107

As well as for two-sided hypotheses around zero (or another hypothesized value, S): H0: βk = 0 H0: βk = S HA: βk ≠ 0 HA: βk ≠ S From Statistical Table B-1 the critical t-value for a one-tailed test at a given level of significance is exactly equal to the critical t-value for a two-tailed test at twice the level of significance of the one-tailed test—as also illustrated by Figure 5.5 © 2011 Pearson Addison-Wesley. All rights reserved. 108

Figure 5.5 One-Sided and Two-Sided t-Tests

Choosing a Level of Significance
The level of significance must be chosen before a critical value can be found, using Statistical Table B The level of significance indicates the probability of observing an estimated t-value greater than the critical t-value if the null hypothesis were correct It also measures the amount of Type I Error implied by a particular critical t-value Which level of significance is chosen? – 5 percent is recommended, unless you know something unusual about the relative costs of making Type I and Type II Errors © 2011 Pearson Addison-Wesley. All rights reserved. 110

Confidence Intervals A confidence interval is a range that contains the true value of an item a specified percentage of the time It is calculated using the estimated regression coefficient, the two-sided critical t-value and the standard error of the estimated coefficient as follows: (5.5) What’s the relationship between confidence intervals and two-sided hypothesis testing? If a hypothesized value fall within the confidence interval, then we cannot reject the null hypothesis © 2011 Pearson Addison-Wesley. All rights reserved. 111

p-Values This is an alternative to the t-test
A p-value, or marginal significance level, is the probability of observing a t-score that size or larger (in absolute value) if the null hypothesis were true Graphically, it’s two times the area under the curve of the t-distribution between the absolute value of the actual t-score and infinity. In theory, we could find this by combing through pages and pages of statistical tables But we don’t have to, since we have EViews and Stata: these (and other) statistical software packages automatically give the p-values as part of the standard output! In light of all this, the p-value decision rule therefore is: Reject H0 if p-valueK < the level of significance and if has the sign implied by HA © 2011 Pearson Addison-Wesley. All rights reserved. 112

Examples of t-Tests: One-Sided
The most common use of the one-sided t-test is to determine whether a regression coefficient is significantly different from zero (in the direction predicted by theory!) This involves four steps: 1. Set up the null and alternative hypothesis 2. Choose a level of significance and therefore a critical t-value 3. Run the regression and obtain an estimated t-value (or t-score) 4. Apply the decision rule by comparing calculated t-value with the critical t-value in order to reject or not reject the null hypothesis Let’s look at each step in more detail for a specific example: © 2011 Pearson Addison-Wesley. All rights reserved. 113

Examples of t-Tests: One-Sided (cont.)
Consider the following simple model of the aggregate retail sales of new cars: (5.6) Where: Y = sales of new cars X1 = real disposable income X2 = average retail price of a new car adjusted by the consumer price index X3 = number of sports utility vehicles sold The four steps for this example then are as follows: © 2011 Pearson Addison-Wesley. All rights reserved. 114

Step 1: Set up the null and alternative hypotheses
From equation 5.6, the one-sided hypotheses are set up as: H0: β1 ≤ 0 HA: β1 > 0 H0: β2 ≥ 0 HA: β2 < 0 H0: β3 ≥ 0 HA: β3 < 0 Remember that a t-test typically is not run on the estimate of the constant term β0 © 2011 Pearson Addison-Wesley. All rights reserved. 115

Step 2: Choose a level of significance and therefore a critical t-value
Assume that you have considered the various costs involved in making Type I and Type II Errors and have chosen 5 percent as the level of significance There are 10 observations in the data set, and so there are 10 – 3 – 1 = 6 degrees of freedom At a 5-percent level of significance, the critical t-value, tc, can be found in Statistical Table B-1 to be 1.943 © 2011 Pearson Addison-Wesley. All rights reserved. 116

Step 3: Run the regression and obtain an estimated t-value
Use the data (annual from 2000 to 2009) to run the regression on your OLS computer package Again, most statistical software packages automatically report the t-values Assume that in this case the t-values were 2.1, 5.6, and –0.1 for β1, β2, and β3, respectively © 2011 Pearson Addison-Wesley. All rights reserved. 117

Step 4: Apply the t–test decision rule
As stated in Section 5.2, the decision rule for the t-test is to: Reject H0 if |tk| > tc and if tk also has the sign implied by HA In this example, this amounts to the following three conditions: For β1: Reject H0 if |2.1| > and if 2.1 is positive. For β2: Reject H0 if |5.6| > and if 5.6 is positive. For β3: Reject H0 if |–0.1| > and if –0.1 is positive. Figure 5.6 illustrates all three of these outcomes © 2011 Pearson Addison-Wesley. All rights reserved. 118

Figure 5.6a One-Sided t-Tests of the Coefficients of the New Car Sales Model

Figure 5.6b One-Sided t-Tests of the Coefficients of the New Car Sales Model

Examples of t-Tests: Two-Sided
The two-sided test is used when the hypotheses should be rejected if estimated coefficients are significantly different from zero, or a specific nonzero value, in either direction So, there are two cases: Two-sided tests of whether an estimated coefficient is significantly different from zero, and Two-sided tests of whether an estimated coefficient is significantly different from a specific nonzero value Let’s take an example to illustrate the first of these (the second case is merely a generalized case of this, see the textbook for details), using the Woody’s restaurant example in Chapter 3: © 2011 Pearson Addison-Wesley. All rights reserved. 121

Examples of t-Tests: Two-Sided (cont.)
Again, in the Woody’s restaurant equation of Section 3.2, the impace of the average income of an area on the expected number of Woody’s customer’s in that area is ambiguous: A high-income neighborhood might have more total customers going out to dinner (positive sign), but those customers might decide to eat at a more formal restaurant that Woody’s (negative sign) The appropriate (two-sided) t-test therefore is: © 2011 Pearson Addison-Wesley. All rights reserved. 122

Figure 5.7 Two-Sided t-Test of the Coefficient of Income in the Woody’s Model

Examples of t-Tests: Two-Sided (cont.)
• The four steps are the same as in the one-sided case: Set up the null and alternative hypothesis H0: βk = 0 HA: βk ≠ 0 2. Choose a level of significance and therefore a critical t-value Keep the level at significance at 5 percent but this now must be distributed between two rejection regions for 29 degrees of freedom hence the correct critical t-value is (found in Statistical Table B-1 for 29 degrees of freedom and a 5-percent, two-sided test) 3. Run the regression and obtain an estimated t-value: The t-value remains at 2.37 (from Equation 5.4) Apply the decision rule: For the two-sided case, this simplifies to: Reject H0 if |2.37| > 2.045; so, reject H0 © 2011 Pearson Addison-Wesley. All rights reserved. 124

The F-Test of Overall Significance
We can test for the predictive power of the entire model using the F statistic Generally these compare two sources of variation F = V1/V2 and has two df parameters Here V1 = ESS/K has K df And V2 = RSS/(n-K-1) has n-K-1 df © 2011 Pearson Addison-Wesley. All rights reserved. 125

F Tables Numerator d.f. denom. d.f. Value of F at a
Usually will see several pages of these; one or two pages at each specific level of significance (.10, .05, .01). Numerator d.f. denom. d.f. Value of F at a specific significance level

F Test Hypotheses H0: 1 = 2 = …= K = (None of the Xs help explain Y) Ha: Not all s are (At least one X is useful) H0: R2 = is an equivalent hypothesis Reject H if F≥Fc Do Not Reject H if F<Fc The critical F-value, Fc, is determined from Statistical Tables B-2 or B3 depending on a level of significance, α, and degrees of freedom, df1=K , (K, the number of the independent variables) and df2=n-k-1

Example: The Woody's restaurant
Since there are 3 independent variables, the null and alternative hypotheses are: H0: N = P = I = 0 Ha: Not all s are 0 From E-Views output, F=15.65, Fc(0.05;3,29)=2.93 Fc is well below the calculated F-value of 15.65, so we can reject the null hypothesis and conclude that the Woody's equation does indeed have a significance of overall fit.

Model Specification: Choosing the Independent Variables
Chapter 6 Model Specification: Choosing the Independent Variables © 2011 Pearson Addison-Wesley. All rights reserved.

Specifying an Econometric Equation and Specification Error
Before any equation can be estimated, it must be completely specified Specifying an econometric equation consists of three parts, namely choosing the correct: independent variables functional form form of the stochastic error term Again, this is part of the first classical assumption from Chapter 4 A specification error results when one of these choices is made incorrectly This chapter will deal with the first of these choices (the two other choices will be discussed in subsequent chapters) © 2011 Pearson Addison-Wesley. All rights reserved. 130

Omitted Variables Two reasons why an important explanatory variable might have been left out: we forgot… it is not available in the dataset, we are examining Either way, this may lead to omitted variable bias (or, more generally, specification bias) The reason for this is that when a variable is not included, it cannot be held constant Omitting a relevant variable usually is evidence that the entire equation is a suspect, because of the likely bias of the coefficients. © 2011 Pearson Addison-Wesley. All rights reserved. 131

The Consequences of an Omitted Variable
Suppose the true regression model is: (6.1) Where is a classical error term If X2 is omitted, the equation becomes instead: (6.2) Where: (6.3) Hence, the explanatory variables in the estimated regression (6.2) are not independent of the error term (unless the omitted variable is uncorrelated with all the included variables—something which is very unlikely) But this violates Classical Assumption III! © 2011 Pearson Addison-Wesley. All rights reserved. 132

The Consequences of an Omitted Variable (cont.)
What happens if we estimate Equation 6.2 when Equation 6.1 is the truth? We get bias! What this means is that: (6.4) The amount of bias is a function of the impact of the omitted variable on the dependent variable times a function of the correlation between the included and the omitted variable Or, more formally: (6.7) So, the bias exists unless: the true coefficient equals zero, or the included and omitted variables are uncorrelated © 2011 Pearson Addison-Wesley. All rights reserved. 133

Correcting for an Omitted Variable
In theory, the solution to a problem of specification bias seems easy: add the omitted variable to the equation! Unfortunately, that’s easier said than done, for a couple of reasons Omitted variable bias is hard to detect: the amount of bias introduced can be small and not immediately detectable Even if it has been decided that a given equation is suffering from omitted variable bias, how to decide exactly which variable to include? Note here that dropping a variable is not a viable strategy to help cure omitted variable bias: If anything you’ll just generate even more omitted variable bias on the remaining coefficients! © 2011 Pearson Addison-Wesley. All rights reserved. 134

Correcting for an Omitted Variable (cont.)
What if: – You have an unexpected result, which leads you to believe that you have an omitted variable – You have two or more theoretically sound explanatory variables as potential “candidates” for inclusion as the omitted variable to the equation is to use How do you choose between these variables? One possibility is expected bias analysis – Expected bias: the likely bias that omitting a particular variable would have caused in the estimated coefficient of one of the included variables © 2011 Pearson Addison-Wesley. All rights reserved. 135

Correcting for an Omitted Variable (cont.)
Expected bias can be estimated with Equation 6.7: (6.7) When do we have a viable candidate? When the sign of the expected bias is the same as the sign of the unexpected result Similarly, when these signs differ, the variable is extremely unlikely to have caused the unexpected result © 2011 Pearson Addison-Wesley. All rights reserved. 136

Irrelevant Variables This refers to the case of including a variable in an equation when it does not belong there This is the opposite of the omitted variables case—and so the impact can be illustrated using the same model Assume that the true regression specification is: (6.10) But the researcher for some reason includes an extra variable: (6.11) The misspecified equation’s error term then becomes: (6.12) © 2011 Pearson Addison-Wesley. All rights reserved. 137

Irrelevant Variables (cont.)
So, the inclusion of an irrelevant variable will not cause bias (since the true coefficient of the irrelevant variable is zero, and so the second term will drop out of Equation 6.12) However, the inclusion of an irrelevant variable will: Increase the variance of the estimated coefficients, and this increased variance will tend to decrease the absolute magnitude of their t-scores Decrease the R2 (but not the R2) Table 6.1 summarizes the consequences of the omitted variable and the included irrelevant variable cases (unless r12 = 0) © 2011 Pearson Addison-Wesley. All rights reserved. 138

Table 6.1 Effect of Omitted Variables and Irrelevant Variables on the Coefficient Estimates

Four Important Specification Criteria
We can summarize the previous discussion into four criteria to help decide whether a given variable belongs in the equation: 1. Theory: Is the variable’s place in the equation unambiguous and theoretically sound? 2. t-Test: Is the variable’s estimated coefficient significant in the expected direction? 3. R2: Does the overall fit of the equation (adjusted for degrees of freedom) improve when the variable is added to the equation? 4. Bias: Do other variables’ coefficients change significantly when the variable is added to the equation? If all these conditions hold, the variable belongs in the equation If none of them hold, it does not belong The tricky part is the intermediate cases: use sound judgment! © 2011 Pearson Addison-Wesley. All rights reserved. 140

Specification Searches
Almost any result can be obtained from a given dataset, by simply specifying different regressions until estimates with the desired properties are obtained Hence, the integrity of all empirical work is open to question To counter this, the following three points of Best Practices in Specification Searches are suggested: Rely on theory rather than statistical fit as much as possible when choosing variables, functional forms, and the like 2. Minimize the number of equations estimated (except for sensitivity analysis, to be discussed later in this section) 3. Reveal, in a footnote or appendix, all alternative specifications estimated © 2011 Pearson Addison-Wesley. All rights reserved. 141

Sequential Specification Searches
The sequential specification search technique allows a researcher to: Estimate an undisclosed number of regressions Subsequently present a final choice (which is based upon an unspecified set of expectations about the signs and significance of the coefficients) as if it were only a specification Such a method misstates the statistical validity of the regression results for two reasons: 1. The statistical significance of the results is overestimated because the estimations of the previous regressions are ignored 2. The expectations used by the researcher to choose between various regression results rarely, if ever, are disclosed © 2011 Pearson Addison-Wesley. All rights reserved. 142

Bias Caused by Relying on the t-Test to Choose Variables
Dropping variables solely based on low t-statistics may lead to two different types of errors: 1. An irrelevant explanatory variable may sometimes be included in the equation (i.e., when it does not belong there) 2. A relevant explanatory variables may sometimes be dropped from the equation (i.e., when it does belong) In the first case, there is no bias but in the second case there is bias Hence, the estimated coefficients will be biased every time an excluded variable belongs in the equation, and that excluded variable will be left out every time its estimated coefficient is not statistically significantly different from zero So, we will have systematic bias in our equation! © 2011 Pearson Addison-Wesley. All rights reserved. 143

Model Specification: Choosing a Functional Form
Chapter 7 Model Specification: Choosing a Functional Form © 2011 Pearson Addison-Wesley. All rights reserved.

The Use and Interpretation of the Constant Term
An estimate of β0 has at least three components: 1. the true β0 2. the constant impact of any specification errors (an omitted variable, for example) 3. the mean of ε for the correctly specified equation (if not equal to zero) Unfortunately, these components can’t be distinguished from one another because we can observe only β0, the sum of the three components As a result of this, we usually don’t interpret the constant term On the other hand, we should not suppress the constant term, either, as illustrated by Figure 7.1 © 2011 Pearson Addison-Wesley. All rights reserved. 145

Figure 7.1 The Harmful Effect of Suppressing the Constant Term

Alternative Functional Forms
An equation is linear in the variables if plotting the function in terms of X and Y generates a straight line For example, Equation 7.1: Y = β0 + β1X + ε (7.1) is linear in the variables but Equation 7.2: Y = β0 + β1X2 + ε (7.2) is not linear in the variables Similarly, an equation is linear in the coefficients only if the coefficients appear in their simplest form—they: are not raised to any powers (other than one) are not multiplied or divided by other coefficients do not themselves include some sort of function (like logs or exponents) © 2011 Pearson Addison-Wesley. All rights reserved. 147

Alternative Functional Forms (cont.)
For example, Equations 7.1 and 7.2 are linear in the coefficients, while Equation 7:3: (7.3) is not linear in the coefficients In fact, of all possible equations for a single explanatory variable, only functions of the general form: (7.4) are linear in the coefficients β0 and β1 © 2011 Pearson Addison-Wesley. All rights reserved. 148

Linear Form This is based on the assumption that the slope of the relationship between the independent variable and the dependent variable is constant: For the linear case, the elasticity of Y with respect to X (the percentage change in the dependent variable caused by a 1-percent increase in the independent variable, holding the other variables in the equation constant) is: © 2011 Pearson Addison-Wesley. All rights reserved. 149

What Is a Log? If e (a constant equal to ) to the “bth power” produces x, then b is the log of x: b is the log of x to the base e if: eb = x Thus, a log (or logarithm) is the exponent to which a given base must be taken in order to produce a specific number While logs come in more than one variety, we’ll use only natural logs (logs to the base e) in this text The symbol for a natural log is “ln,” so ln(x) = b means that ( ) b = x or, more simply, ln(x) = b means that eb = x For example, since e2 = ( ) 2 = 7.389, we can state that: ln(7.389) = 2 Thus, the natural log of is 2! Again, why? Two is the power of e that produces 7.389 © 2011 Pearson Addison-Wesley. All rights reserved. 150

What Is a Log? (cont.) Let’s look at some other natural log calculations: ln(100) = 4.605 ln(1000) = 6.908 ln(10000) = 9.210 ln( ) = n(100000) = Note that as a number goes from 100 to 1,000,000, its natural log goes from to only ! As a result, logs can be used in econometrics if a researcher wants to reduce the absolute size of the numbers associated with the same actual meaning One useful property of natural logs in econometrics is that they make it easier to figure out impacts in percentage terms (we’ll see this when we get to the double-log specification) © 2011 Pearson Addison-Wesley. All rights reserved. 151

Double-Log Form Here, the natural log of Y is the dependent variable and the natural log of X is the independent variable: (7.5) In a double-log equation, an individual regression coefficient can be interpreted as an elasticity because: (7.6) Note that the elasticities of the model are constant and the slopes are not This is in contrast to the linear model, in which the slopes are constant but the elasticities are not © 2011 Pearson Addison-Wesley. All rights reserved. 152

Figure 7.2 Double-Log Functions

Semilog Form The semilog functional form is a variant of the double-log equation in which some but not all of the variables (dependent and independent) are expressed in terms of their natural logs. It can be on the right-hand side, as in: Yi = β0 + β1lnX1i + β2X2i + εi (7.7) Or it can be on the left-hand side, as in: lnY = β0 + β1X1 + β2X2 + ε (7.9) Figure 7.3 illustrates these two different cases © 2011 Pearson Addison-Wesley. All rights reserved. 154

Figure 7.3 Semilog Functions

Polynomial Form Polynomial functional forms express Y as a function of independent variables, some of which are raised to powers other than 1 For example, in a second-degree polynomial (also called a quadratic) equation, at least one independent variable is squared: Yi = β0 + β1X1i + β2(X1i)2 + β3X2i + εi (7.10) The slope of Y with respect to X1 in Equation 7.10 is: (7.11) Note that the slope depends on the level of X1 © 2011 Pearson Addison-Wesley. All rights reserved. 156

Figure 7.4 Polynomial Functions

Inverse Form The inverse functional form expresses Y as a function of the reciprocal (or inverse) of one or more of the independent variables (in this case, X1): Yi = β0 + β1(1/X1i) + β2X2i + εi (7.13) So X1 cannot equal zero This functional form is relevant when the impact of a particular independent variable is expected to approach zero as that independent variable approaches infinity The slope with respect to X1 is: (7.14) The slopes for X1 fall into two categories, depending on the sign of β1 (illustrated in Figure 7.5) © 2011 Pearson Addison-Wesley. All rights reserved. 158

Figure 7.5 Inverse Functions

Table 7.1 Summary of Alternative Functional Forms

Lagged Independent Variables
Virtually all the regressions we’ve studied so far have been “instantaneous” in nature In other words, they have included independent and dependent variables from the same time period, as in: Yt = β0 + β1X1t + β2X2t + εt (7.15) Many econometric equations include one or more lagged independent variables like X1t-1 where “t–1” indicates that the observation of X1 is from the time period previous to time period t, as in the following equation: Yt = β0 + β1X1t-1 + β2X2t + εt (7.16) © 2011 Pearson Addison-Wesley. All rights reserved. 161

Using Dummy Variables A dummy variable is a variable that takes on the values of 0 or 1, depending on whether a condition for a qualitative attribute (such as gender) is met These conditions take the general form: (7.18) This is an example of an intercept dummy (as opposed to a slope dummy, which is discussed in Section 7.5) Figure 7.6 illustrates the consequences of including an intercept dummy in a linear regression model © 2011 Pearson Addison-Wesley. All rights reserved. 162

Figure 7.6 An Intercept Dummy

Slope Dummy Variables Contrary to the intercept dummy, which changed only the intercept (and not the slope), the slope dummy changes both the intercept and the slope The general form of a slope dummy equation is: Yi = β0 + β1Xi + β2Di + β3XiDi + εi (7.20) The slope depends on the value of D: When D = 0, ΔY/ΔX = β1 When D = 1, ΔY/ΔX = (β1 + β3) Graphical illustration of how this works in Figure 7.7 © 2011 Pearson Addison-Wesley. All rights reserved. 164

Figure 7.7 Slope and Intercept Dummies

Problems with Incorrect Functional Forms
If functional forms are similar, and if theory does not specify exactly which form to use, there are at least two reasons why we should avoid using goodness of fit over the sample to determine which equation to use: 1. Fits are difficult to compare if the dependent variable is transformed 2. An incorrect function form may provide a reasonable fit within the sample but have the potential to make large forecast errors when used outside the range of the sample The first of these is essentially due to the fact that when the dependent variable is transformed, the total sum of squares (TSS) changes as well The second is essentially die to the fact that using an incorrect functional amounts to a specification error similar to the omitted variables bias discussed in Section 6.1 This second case is illustrated in Figure 7.8 © 2011 Pearson Addison-Wesley. All rights reserved. 166

Figure 7.8a Incorrect Functional Forms Outside the Sample Range

Figure 7.8b Incorrect Functional Forms Outside the Sample Range

Multicollinearity Chapter 8

Introduction and Overview
The next three chapters deal with violations of the Classical Assumptions and remedies for those violations This chapter addresses multicollinearity; the next two chapters are on serial correlation and heteroskedasticity For each of these three problems, we will attempt to answer the following questions: 1. What is the nature of the problem? 2. What are the consequences of the problem? 3. How is the problem diagnosed? 4. What remedies for the problem are available? © 2011 Pearson Addison-Wesley. All rights reserved. 170

Perfect Multicollinearity
Perfect multicollinearity violates Classical Assumption VI, which specifies that no explanatory variable is a perfect linear function of any other explanatory variables The word perfect in this context implies that the variation in one explanatory variable can be completely explained by movements in another explanatory variable A special case is that of a dominant variable: an explanatory variable is definitionally related to the dependent variable An example would be (Notice: no error term!): X1i = α0 + α1X2i (8.1) where the αs are constants and the Xs are independent variables in: Yi = β0 + β1X1i + β2X2i + εi (8.2) Figure 8.1 illustrates this case © 2011 Pearson Addison-Wesley. All rights reserved. 171

Figure 8.1 Perfect Multicollinearity

Perfect Multicollinearity (cont.)
What happens to the estimation of an econometric equation where there is perfect multicollinearity? OLS is incapable of generating estimates of the regression coefficients most OLS computer programs will print out an error message in such a situation What is going on? Essentially, perfect multicollinearity ruins our ability to estimate the coefficients because the perfectly collinear variables cannot be distinguished from each other: You cannot “hold all the other independent variables in the equation constant” if every time one variable changes, another changes in an identical manner! Solution: one of the collinear variables must be dropped (they are essentially identical, anyway) © 2011 Pearson Addison-Wesley. All rights reserved. 173

Imperfect Multicollinearity
Imperfect multicollinearity occurs when two (or more) explanatory variables are imperfectly linearly related, as in: X1i = α0 + α1X2i + ui (8.7) Compare Equation 8.7 to Equation 8.1 Notice that Equation 8.7 includes ui, a stochastic error term This case is illustrated in Figure 8.2 © 2011 Pearson Addison-Wesley. All rights reserved. 174

Figure 8.2 Imperfect Multicollinearity

The Consequences of Multicollinearity
There are five major consequences of multicollinearity: 1. Estimates will remain unbiased 2. The variances and standard errors of the estimates will increase: a. Harder to distinguish the effect of one variable from the effect of another, so much more likely to make large errors in estimating the βs than without multicollinearity b. As a result, the estimated coefficients, although still unbiased, now come from distributions with much larger variances and, therefore, larger standard errors (this point is illustrated in Figure 8.3) © 2011 Pearson Addison-Wesley. All rights reserved. 176

Figure 8.3 Severe Multicollinearity Increases the Variances of the s

The Consequences of Multicollinearity (cont.)
3. The computed t-scores will fall: a. Recalling Equation 5.2, this is a direct consequence of 2. above 4. Estimates will become very sensitive to changes in specification: a. The addition or deletion of an explanatory variable or of a few observations will often cause major changes in the values of the s when significant multicollinearity exists b. For example, if you drop a variable, even one that appears to be statistically insignificant, the coefficients of the remaining variables in the equation sometimes will change dramatically c. This is again because with multicollinearity, it is much harder to distinguish the effect of one variable from the effect of another 5. The overall fit of the equation and the estimation of the coefficients of nonmulticollinear variables will be largely unaffected © 2011 Pearson Addison-Wesley. All rights reserved. 178

The Detection of Multicollinearity
First realize that that some multicollinearity exists in every equation: all variables are correlated to some degree (even if completely at random) So it’s really a question of how much multicollinearity exists in an equation, rather than whether any multicollinearity exists There are basically two characteristics that help detect the degree of multicollinearity for a given application: 1. High simple correlation coefficients 2. High Variance Inflation Factors (VIFs) We will now go through each of these in turn: © 2011 Pearson Addison-Wesley. All rights reserved. 179

High Simple Correlation Coefficients
If a simple correlation coefficient, r, between any two explanatory variables is high in absolute value, these two particular Xs are highly correlated and multicollinearity is a potential problem How high is high? Some researchers pick an arbitrary number, such as 0.80 A better answer might be that r is high if it causes unacceptably large variances in the coefficient estimates in which we’re interested. Caution in case of more than two explanatory variables: Groups of independent variables, acting together, may cause multicollinearity without any single simple correlation coefficient being high enough to indicate that multicollinearity is present As a result, simple correlation coefficients must be considered to be sufficient but not necessary tests for multicollinearity © 2011 Pearson Addison-Wesley. All rights reserved. 180

High Variance Inflation Factors (VIFs)
The variance inflation factor (VIF) is calculated from two steps: 1. Run an OLS regression that has Xi as a function of all the other explanatory variables in the equation—For i = 1, this equation would be: X1 = α1 + α2X2 + α3X3 + … + αKXK + v (8.15) where v is a classical stochastic error term Calculate the variance inflation factor for : (8.16) where is the unadjusted from step one © 2011 Pearson Addison-Wesley. All rights reserved. 181

High Variance Inflation Factors (VIFs) (cont.)
From Equation 8.16, the higher the VIF, the more severe the effects of mulitcollinearity How high is high? While there is no table of formal critical VIF values, a common rule of thumb is that if a given VIF is greater than 5, the multicollinearity is severe As the number of independent variables increases, it makes sense to increase this number slightly Note that the authors replace the VIF with its reciprocal, , called tolerance, or TOL Problems with VIF: No hard and fast VIF decision rule There can still be severe multicollinearity even with small VIFs VIF is a sufficient, not necessary, test for multicollinearity © 2011 Pearson Addison-Wesley. All rights reserved. 182

Remedies for Multicollinearity
Essentially three remedies for multicollinearity: Do nothing: a. Multicollinearity will not necessarily reduce the t-scores enough to make them statistically insignificant and/or change the estimated coefficients to make them differ from expectations b. the deletion of a multicollinear variable that belongs in an equation will cause specification bias 2. Drop a redundant variable: a. Viable strategy when two variables measure essentially the same thing b. Always use theory as the basis for this decision! © 2011 Pearson Addison-Wesley. All rights reserved. 183

Remedies for Multicollinearity (cont.)
Increase the sample size: This is frequently impossible but a useful alternative to be considered if feasible The idea is that the larger sample normally will reduce the variance of the estimated coefficients, diminishing the impact of the multicollinearity © 2011 Pearson Addison-Wesley. All rights reserved. 184

Serial Correlation Chapter 9

Pure Serial Correlation
Pure serial correlation occurs when Classical Assumption IV, which assumes uncorrelated observations of the error term, is violated (in a correctly specified equation!) The most commonly assumed kind of serial correlation is first-order serial correlation, in which the current value of the error term is a function of the previous value of the error term: εt = ρεt–1 + ut (9.1) where: ε = the error term of the equation in question ρ = the first-order autocorrelation coefficient u = a classical (not serially correlated) error term © 2011 Pearson Addison-Wesley. All rights reserved. 186

Pure Serial Correlation (cont.)
The magnitude of ρ indicates the strength of the serial correlation: If ρ is zero, there is no serial correlation As ρ approaches one in absolute value, the previous observation of the error term becomes more important in determining the current value of εt and a high degree of serial correlation exists For ρ to exceed one is unreasonable, since the error term effectively would “explode” As a result of this, we can state that: –1 < ρ < +1 (9.2) © 2011 Pearson Addison-Wesley. All rights reserved. 187

Pure Serial Correlation (cont.)
The sign of ρ indicates the nature of the serial correlation in an equation: Positive: implies that the error term tends to have the same sign from one time period to the next this is called positive serial correlation Negative: implies that the error term has a tendency to switch signs from negative to positive and back again in consecutive observations this is called negative serial correlation Figures 9.1–9.3 illustrate several different scenarios © 2011 Pearson Addison-Wesley. All rights reserved. 188

Figure 9.1a Positive Serial Correlation

Figure 9.1b Positive Serial Correlation

Figure 9.2 No Serial Correlation

Figure 9.3a Negative Serial Correlation

Figure 9.3b Negative Serial Correlation

Impure Serial Correlation
Impure serial correlation is serial correlation that is caused by a specification error such as: an omitted variable and/or an incorrect functional form How does this happen? As an example, suppose that the true equation is: (9.3) where εt is a classical error term. As shown in Section 6.1, if X2 is accidentally omitted from the equation (or if data for X2 are unavailable), then: (9.4) The error term is therefore not a classical error term © 2011 Pearson Addison-Wesley. All rights reserved. 194

Impure Serial Correlation (cont.)
Instead, the error term is also a function of one of the explanatory variables, X2 As a result, the new error term, ε* , can be serially correlated even if the true error term ε, is not In particular, the new error term will tend to be serially correlated when: X2 itself is serially correlated (this is quite likely in a time series) and the size of ε is small compared to the size of Figure 9.4 illustrates 1., for the case of U.S. disposable income © 2011 Pearson Addison-Wesley. All rights reserved. 195

Figure 9.4 U.S. Disposable Income as a Function of Time

Impure Serial Correlation (cont.)
Turn now to the case of impure serial correlation caused by an incorrect functional form Suppose that the true equation is polynomial in nature: (9.7) but that instead a linear regression is run: (9. 8) The new error term ε* is now a function of the true error term and of the differences between the linear and the polynomial functional forms Figure 9.5 illustrates how these differences often follow fairly autoregressive patterns © 2011 Pearson Addison-Wesley. All rights reserved. 197

Figure 9.5a Incorrect Functional Form as a Source of Impure Serial Correlation

Figure 9.5b Incorrect Functional Form as a Source of Impure Serial Correlation

The Consequences of Serial Correlation
The existence of serial correlation in the error term of an equation violates Classical Assumption IV, and the estimation of the equation with OLS has at least three consequences: 1. Pure serial correlation does not cause bias in the coefficient estimates 2. Serial correlation causes OLS to no longer be the minimum variance estimator (of all the linear unbiased estimators) 3. Serial correlation causes the OLS estimates of the SE to be biased, leading to unreliable hypothesis testing. Typically the bias in the SE estimate is negative, meaning that OLS underestimates the standard errors of the coefficients (and thus overestimates the t-scores) © 2011 Pearson Addison-Wesley. All rights reserved. 200

The Durbin–Watson d Test
Two main ways to detect serial correlation: Informal: observing a pattern in the residuals like that in Figure 9.1 Formal: testing for serial correlation using the Durbin–Watson d test We will now go through the second of these in detail First, it is important to note that the Durbin–Watson d test is only applicable if the following three assumptions are met: 1. The regression model includes an intercept term 2. The serial correlation is first-order in nature: εt = ρεt–1 + ut where ρ is the autocorrelation coefficient and u is a classical (normally distributed) error term 3. The regression model does not include a lagged dependent variable (discussed in Chapter 12) as an independent variable © 2011 Pearson Addison-Wesley. All rights reserved. 201

The Durbin–Watson d Test (cont.)
The equation for the Durbin–Watson d statistic for T observations is: (9.10) where the ets are the OLS residuals There are three main cases: 1. Extreme positive serial correlation: d = 0 2. Extreme negative serial correlation: d ≈ 4 3. No serial correlation: d ≈ 2 © 2011 Pearson Addison-Wesley. All rights reserved. 202

To test for positive (note that we rarely, if ever, test for negative!) serial correlation, the following steps are required: 1. Obtain the OLS residuals from the equation to be tested and calculate the d statistic by using Equation 9.10 2. Determine the sample size and the number of explanatory variables and then consult Statistical Tables B-4, B-5, or B-6 in Appendix B to find the upper critical d value, dU, and the lower critical d value, dL, respectively (instructions for the use of these tables are also in that appendix) © 2011 Pearson Addison-Wesley. All rights reserved. 203

3. Set up the test hypotheses and decision rule: H0: ρ ≤ 0 (no positive serial correlation) HA: ρ > 0 (positive serial correlation) if d < dL Reject H0 if d > dU Do not reject H0 if dL ≤ d ≤ dU Inconclusive In rare circumstances, perhaps first differenced equations, a two-sided d test might be appropriate In such a case, steps 1 and 2 are still used, but step 3 is now: © 2011 Pearson Addison-Wesley. All rights reserved. 204

3. Set up the test hypotheses and decision rule: H0: ρ = 0 (no serial correlation) HA: ρ ≠ 0 (serial correlation) if d < dL Reject H0 if d > 4 – dL Reject H0 if 4 – dU > d > dU Do Not Reject H0 Otherwise Inconclusive Figure 9.6 gives an example of a one-sided Durbin Watson d test © 2011 Pearson Addison-Wesley. All rights reserved. 205

Figure 9.6 An Example of a One-Sided Durbin–Watson d Test

Remedies for Serial Correlation
The place to start in correcting a serial correlation problem is to look carefully at the specification of the equation for possible errors that might be causing impure serial correlation: Is the functional form correct? Are you sure that there are no omitted variables? Only after the specification of the equation has bee reviewed carefully should the possibility of an adjustment for pure serial correlation be considered There are two main remedies for pure serial correlation: 1. Generalized Least Squares 2. Newey-West standard errors We will no discuss each of these in turn © 2011 Pearson Addison-Wesley. All rights reserved. 207

Generalized Least Squares
Start with an equation that has first-order serial correlation: (9.15) Which, if εt = ρεt–1 + ut (due to pure serial correlation), also equals: (9.16) Multiply Equation 9.15 by ρ and then lag the new equation by one period, obtaining: (9.17) © 2011 Pearson Addison-Wesley. All rights reserved. 208

Generalized Least Squares (cont.)
Next, subtract Equation from Equation 9.16, obtaining: (9.18) Finally, rewrite equation 9.18 as: (9.19) (9.20) © 2011 Pearson Addison-Wesley. All rights reserved. 209

Equation 9.19 is called a Generalized Least Squares (or “quasi-differenced”) version of Equation 9.16. Notice that: The error term is not serially correlated a. As a result, OLS estimation of Equation 9.19 will be minimum variance b. This is true if we know ρ or if we accurately estimate ρ) 2. The slope coefficient β1 is the same as the slope coefficient of the original serially correlated equation, Equation Thus coefficients estimated with GLS have the same meaning as those estimated with OLS. © 2011 Pearson Addison-Wesley. All rights reserved. 210

3. The dependent variable has changed compared to that in Equation This means that the GLS is not directly comparable to the OLS. 4. To forecast with GLS, adjustments like those discussed in Section 15.2 are required Unfortunately, we cannot use OLS to estimate a GLS model because GLS equations are inherently nonlinear in the coefficients Fortunately, there are at least two other methods available: © 2011 Pearson Addison-Wesley. All rights reserved. 211

The Cochrane–Orcutt Method
Perhaps the best known GLS method This is a two-step iterative technique that first produces an estimate of ρ and then estimates the GLS equation using that estimate. The two steps are: Estimate ρ by running a regression based on the residuals of the equation suspected of having serial correlation: et = ρet–1 + ut (9.21) where the ets are the OLS residuals from the equation suspected of having pure serial correlation and ut is a classical error term 2. Use this to estimate the GLS equation by substituting into Equation 9.18 and using OLS to estimate Equation 9.18 with the adjusted data These two steps are repeated (iterated) until further iteration results in little change in Once has converged (usually in just a few iterations), the last estimate of step 2 is used as a final estimate of Equation 9.18 © 2011 Pearson Addison-Wesley. All rights reserved. 212

The AR(1) Method Perhaps a better alternative than Cochrane–Orcutt for GLS models The AR(1) method estimates a GLS equation like Equation 9.18 by estimating β0, β1 and ρ simultaneously with iterative nonlinear regression techniques (that are well beyond the scope of this chapter!) The AR(1) method tends to produce the same coefficient estimates as Cochrane–Orcutt However, the estimated standard errors are smaller This is why the AR(1) approach is recommended as long as your software can support such nonlinear regression © 2011 Pearson Addison-Wesley. All rights reserved. 213

Newey–West Standard Errors
Again, not all corrections for pure serial correlation involve Generalized Least Squares Newey–West standard errors take account of serial correlation by correcting the standard errors without changing the estimated coefficients The logic begin Newey–West standard errors is powerful: If serial correlation does not cause bias in the estimated coefficients but does impact the standard errors, then it makes sense to adjust the estimated equation in a way that changes the standard errors but not the coefficients © 2011 Pearson Addison-Wesley. All rights reserved. 214

Newey–West Standard Errors (cont.)
The Newey–West SEs are biased but generally more accurate than uncorrected standard errors for large samples in the face of serial correlation As a result, Newey–West standard errors can be used for t-tests and other hypothesis tests in most samples without the errors of inference potentially caused by serial correlation Typically, Newey–West SEs are larger than OLS SEs, thus producing lower t-scores © 2011 Pearson Addison-Wesley. All rights reserved. 215

Heteroskedasticity Chapter 10

Pure Heteroskedasticity
Pure heteroskedasticity occurs when Classical Assumption V, which assumes constant variance of the error term, is violated (in a correctly specified equation!) Classical Assumption V assumes that: (10.1) With heteroskedasticity, this error term variance is not constant © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-217 217 217

Pure Heteroskedasticity (cont.)
Instead, the variance of the distribution of the error term depends on exactly which observation is being discussed: (10.2) The simplest case is that of discrete heteroskedasticity, where the observations of the error term can be grouped into just two different distributions, “wide” and “narrow” This case is illustrated in Figure 10.1 © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-218 218 218

Figure 10.1a Homoskedasticity versus Discrete Heteroskedasticity

Figure 10.1b Homoskedasticity versus Discrete Heteroskedasticity

Pure Heteroskedasticity (cont.)
Heteroskedasticity takes on many more complex forms, however, than the discrete heteroskedasticity case Perhaps the most frequently specified model of pure heteroskedasticity relates the variance of the error term to an exogenous variable Zi as follows: (10.3) (10.4) where Z, the “proportionality factor,” may or may not be in the equation This is illustrated in Figures 10.2 and 10.3 © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-221 221 221

Figure 10.2 A Homoskedastic Error Term with Respect to Zi

Figure 10.3 A Heteroskedastic Error Term with Respect to Zi

Impure Heteroskedasticity
Similar to impure serial correlation, impure heteroskedasticity is heteroskedasticity that is caused by a specification error Contrary to that case, however, impure heteroskedasticity almost always originates from an omitted variable (rather than an incorrect functional form) How does this happen? The portion of the omitted effect not represented by one of the included explanatory variables must be absorbed by the error term. So, if this effect has a heteroskedastic component, the error term of the misspecified equation might be heteroskedastic even if the error term of the true equation is not! This highlights, again, the importance of first checking that the specification is correct before trying to “fix” things… © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-224 224 224

The Consequences of Heteroskedasticity
The existence of heteroskedasticity in the error term of an equation violates Classical Assumption V, and the estimation of the equation with OLS has at least three consequences: Pure heteroskedasticity does not cause bias in the coefficient estimates Heteroskedasticity typically causes OLS to no longer be the minimum variance estimator (of all the linear unbiased estimators) Heteroskedasticity causes the OLS estimates of the SE to be biased, leading to unreliable hypothesis testing. Typically the bias in the SE estimate is negative, meaning that OLS underestimates the standard errors (and thus overestimates the t-scores) © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-225 225 225

Testing for Heteroskedasticity
Econometricians do not all use the same test for heteroskedasticity because heteroskedasticity takes a number of different forms, and its precise manifestation in a given equation is almost never known Before using any test for heteroskedasticity, however, ask the following: 1. Are there any obvious specification errors? – Fix those before testing! 2. Is the subject of the research likely to be afflicted with heteroskedasticity? – Not only are cross-sectional studies the most frequent source of heteroskedasticity, but cross-sectional studies with large variations in the size of the dependent variable are particularly susceptible to heteroskedasticity 3. Does a graph of the residuals show any evidence of heteroskedasticity? – Specifically, plot the residuals against a potential Z proportionality factor – In such cases, the graph alone can often show that heteroskedasticity is or is not likely – Figure 10.4 shows an example of what to look for: an expanding (or contracting) range of the residuals © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-226 226 226

Figure 10.4 Eyeballing Residuals for Possible Heteroskedasticity

The Park Test The Park test has three basic steps:
1. Obtain the residuals of the estimated regression equation: (10.6) 2. Use these residuals to form the dependent variable in a second regression: (10.7) where: ei = the residual from the ith observation from Equation 10.6 Zi = your best choice as to the possible proportionality factor (Z) ui = a classical (homoskedastic) error term © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-228 228 228

The Park Test 3. Test the significance of the coefficient of Z in Equation 10.7 with a t-test: If the coefficient of Z is statistically significantly different from zero, this is evidence of heteroskedastic patterns in the residuals with respect to Z Potential issue: How do we choose Z in the first place? © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-229 229 229

The White Test The White test also has three basic steps:
Obtain the residuals of the estimated regression equation: – This is identical to the first step in the Park test Use these residuals (squared) as the dependent variable in a second equation that includes as explanatory variables each X from the original equation, the square of each X, and the product of each X times every other X—for example, in the case of three explanatory variables: (10.9) © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-230 230 230

The White Test (cont.) 3. Test the overall significance of Equation 10.9 with the chi-square test – The appropriate test statistic here is NR2, or the sample size (N) times the coefficient of determination (the unadjusted R2) of Equation 10.9 – This test statistic has a chi-square distribution with degrees of freedom equal to the number of slope coefficients in Equation 10.9 – If NR2 is larger than the critical chi-square value found in Statistical Table B-8, then we reject the null hypothesis and conclude that it's likely that we have heteroskedasticity – If NR2 is less than the critical chi-square value, then we cannot reject the null hypothesis of homoskedasticity © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-231 231 231

Remedies for Heteroskedasticity
The place to start in correcting a heteroskedasticity problem is to look carefully at the specification of the equation for possible errors that might be causing impure heteroskedasticity : Are you sure that there are no omitted variables? Only after the specification of the equation has been reviewed carefully should the possibility of an adjustment for pure heteroskedasticity be considered There are two main remedies for pure heteroskedasticit1 Heteroskedasticity-corrected standard errors Redefining the variables We will now discuss each of these in turn: © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-232 232 232

Heteroskedasticity-Corrected Standard Errors
Heteroskedasticity-corrected errors take account of heteroskedasticity correcting the standard errors without changing the estimated coefficients The logic behind heteroskedasticity-corrected standard errors is power If heteroskedasticity does not cause bias in the estimated coefficients but does impact the standard errors, then it makes sense to adjust the estimated equation in a way that changes the standard errors but not the coefficients © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-233 233 233

Heteroskedasticity-Corrected Standard Errors (cont.)
The heteroskedasticity-corrected SEs are biased but generally more accurate than uncorrected standard errors for large samples in the face of heteroskedasticity As a result, heteroskedasticity-corrected standard errors can be used for t-tests and other hypothesis tests in most samples without the errors of inference potentially caused by heteroskedasticity Typically heteroskedasticity-corrected SEs are larger than OLS SEs, thus producing lower t-scores © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-234 234 234

Redefining the Variables
Sometimes it’s possible to redefine the variables in a way that avoids heteroskedasticity Be careful, however: Redefining your variables is a functional form specification change that can dramatically change your equation! In some cases, the only redefinition that's needed to rid an equation of heteroskedasticity is to switch from a linear functional form to a double-log functional form: The double-log form has inherently less variation than the linear form, so it's less likely to encounter heteroskedasticity © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-235 235 235

Redefining the Variables (cont.)
In other situations, it might be necessary to completely rethink the research project in terms of its underlying theory For example, a cross-sectional model of the total expenditures by the governments of different cities may generate heteroskedasticity by containing both large and small cities in the estimation sample Why? Because of the proportionality factor (Z) the size of the cities © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-236 236 236

Redefining the Variables (cont.)
This is illustrated in Figure 10.5 In this case, per capita expenditures would be a logical dependent variable Such a transformation is shown in Figure 10.6 Aside: Note that Weighted Least Squares (WLS), that some authors suggest as a remedy for heteroskedasticity, has some serious potential drawbacks and can therefore generally is not be recommended (see Footnote 14, p. 355, for details) © 2011 Pearson Addison-Wesley. All rights reserved. © 2011 Pearson Addison-Wesley. All rights reserved. 10-237 237 237

Figure 10.5 An Aggregate City Expenditures Function

Figure 10.6 A Per Capita City Expenditures Function

Choosing Your Topic There are at least three keys to choosing a topic:
Try to pick a field that you find interesting and/or that you know something about Make sure that data are readily available with a reasonable sample (we suggest at least 25 observations) Make sure that there is some substance to your topic – Avoid topics that are purely descriptive or virtually tautological in nature – Instead, look for topics that address an inherently interesting economic or behavioral question or choice © 2011 Pearson Addison-Wesley. All rights reserved. 241

Choosing Your Topic (cont.)
Places to look: your textbooks and notes from previous economics classes economics journals For example, Table 11.1 contains a list of the journals cited so far in this textbook (in order of the frequency of citation) © 2011 Pearson Addison-Wesley. All rights reserved. 242

Table 11.1a Sources of Potential Topic Ideas

Table 11.1b Sources of Potential Topic Ideas

Collecting Your Data Before any quantitative analysis can be done, the data must be: collected organized entered into a computer Usually, this is a time-consuming and frustrating task because of: the difficulty of finding data the existence of definitional differences between theoretical variables and their empirical counterparts and the high probability of data entry errors or data transmission errors But time spent thinking about and collecting the data is well spent, since a researcher who knows the data sources and definitions is much less likely to make mistakes using or interpreting regressions run on that data We will now discuss three data collection issues in a bit more detail © 2011 Pearson Addison-Wesley. All rights reserved. 245

What Data to Look For Checking for data availability means deciding what specific variables you want to study: dependent variable all relevant independent variables At least 5 issues to consider here: 1. Time periods: If the dependent variable is measured annually, the explanatory variables should also be measured annually and not, say, monthly 2. Measuring quantity: If the market and/or quality of a given variable has changed over time, it makes little sense to use quantity in units Example: TVs have changed so much over time that it makes more sense to use quantity in terms of monetary equivalent: more comparable across time © 2011 Pearson Addison-Wesley. All rights reserved. 246

What Data to Look For (cont.)
3. Nominal or real terms? Depends on theory – essentially: do we want to “clean” for inflation? TVs, again: probably use real terms 4. Appropriate variable definitions depend on whether data are cross-sectional or time-series TVs, again: national advertising would be a good candidate for an explanatory variable in a time-series model, while advertising in or near each state (or city) would make sense in a cross-sectional model 5. Be careful when reading (and creating!) descriptions of data: Where did the data originate? Are prices and/or income measured in nominal or real terms? Are prices retail or wholesale? © 2011 Pearson Addison-Wesley. All rights reserved. 247

Where to Look for Economic Data
Although some researchers generate their own data through surveys or other techniques (see Section 11.3), the vast majority of regressions are run on publicly available data Good sources here include: 1. Government publications: Statistical Abstract of the U.S. the annual Economic Report of the President the Handbook of Labor Statistics Historical Statistics of the U.S. (published in 1975) Census Catalog and Guide © 2011 Pearson Addison-Wesley. All rights reserved. 248

Where to Look for Economic Data (cont.)
2. International data sources: U.N. Statistical Yearbook U.N. Yearbook of National Account Statistics 3. Internet resources: “Resources for Economists on the Internet” Economagic WebEC EconLit ( “Dialog” Links to these sites and other good sources of data are on the text’s Web site: © 2011 Pearson Addison-Wesley. All rights reserved. 249

Missing Data Suppose the data aren’t there? A few observations:
What happens if you choose the perfect variable and look in all the right sources and can’t find the data? The answer to this question depends on how much data is missing: A few observations: in a cross-section study: Can usually afford to drop these observations from the sample in a time-series study: May interpolate value (taking the mean of adjacent values) © 2011 Pearson Addison-Wesley. All rights reserved.

Missing Data (cont.) 2. No data at all available (for a theoretically relevant variable!): From Chapter 6, we know that this is likely to cause omitted variables bias A possible solution here is to use a proxy variable For example, the value of net investment is a variable that is not measured directly in a number of countries Instead, might use the value of gross investment as a proxy, the assumption being that the value of gross investment is directly proportional to the value of net investment © 2011 Pearson Addison-Wesley. All rights reserved.

Advanced Data Sources So far, all the data sets have been:
1. cross-sectional or time-series in nature 2. been collected by observing the world around us, instead being created It turns out, however, that: 1. time-series and cross-sectional data can be pooled to form panel data 2. data can be generated through surveys We will now briefly introduce these more advanced data sources and explain why it probably doesn't make sense to use these data sources on your first regression project: © 2011 Pearson Addison-Wesley. All rights reserved.

Surveys Surveys are everywhere in our society and are used for many different purposes—examples include: marketing firms using surveys to learn more about products and competition political candidates using surveys to finetune their campaign advertising or strategies governments using surveys for all sorts of purposes, including keeping track of their citizens with instruments like the U.S. Census © 2011 Pearson Addison-Wesley. All rights reserved.

Surveys (cont.) While running your own survey might be tempting as a way of obtaining data for your own project, running a survey is not as easy as it might seem surveys: must be carefully thought through; it’s virtually impossible to go back to the respondents and add another question later must be worded precisely (and pretested) to avoid confusing the respondent or "leading" the respondent to a particular answer must have samples that are random and avoid the selection, survivor, and nonresponse biases explained in Section 17.2 As a result, we don't encourage beginning researchers to run their own surveys... © 2011 Pearson Addison-Wesley. All rights reserved.

Panel Data Again, panel data are formed when cross-sectional and time-series data sets are pooled to create a single data set Two main reasons for using panel data: To increase the sample size To provide an insight into an analytical question that can't be obtained by using time-series or cross-sectional data alone © 2011 Pearson Addison-Wesley. All rights reserved.

Panel Data (cont.) Example: suppose we’re interested in the relationship between budget deficits and interest rates but only have 10 years’ of annual data to study But ten observations is too small a sample for a reasonable regression! However, if we can find time-series data on the same economic variables-interest rates and budget deficits—for the same ten years for six different countries, we’ll end up with a sample of 10*6 = 60 observations, which is more than enough The result is a pooled cross-section time-series data set—a panel data set! Panel data estimation methods are treated in Chapter 16 © 2011 Pearson Addison-Wesley. All rights reserved.

Practical Advice for Your Project
We now move to a discussion of practical advice about actually doing applied econometric work This discussion is structured in three parts: 1. The 10 Commandments of Applied Econometrics (by Peter Kennedy) What to check if you get an unexpected sign 3. A collection of a dozen practical tips, brought together from other sections of this text that are worth reiterating specifically in the context of actually doing applied econometric work © 2011 Pearson Addison-Wesley. All rights reserved.

The 10 Commandments of Applied Econometrics
Use common sense and economic theory: Example: match per capita variables with per capita variables, use real exchange rates to explain real imports or exports, etc Ask the right questions: Ask plenty of, perhaps, seemingly silly questions to ensure that you fully understand the goal of the research Know the context: Be sure to be familiar with the history, institutions, operating constraints, measurement peculiarities, cultural customs, etc, underlying the object under study 4. Inspect the data: a. This includes calculating summary statistics, graphs, and data cleaning (including checking filters) b. The objective is to get to know the data well © 2011 Pearson Addison-Wesley. All rights reserved. 259

The 10 Commandments of Applied Econometrics (cont.)
5. Keep it sensibly simple: a. Begin with a simple model and only complicate it if it fails b. This both goes for the specifications, functional forms, etc and for the estimation method 6. Look long and hard at your results: a. Check that the results make sense, including signs and magnitudes b. Apply the “laugh test” 7. Understand the costs and benefits of data mining: a. “Bad” data mining: deliberately searching for a specification that “works” (i.e. “torturing” the data) b. “Good” data mining: experimenting with the data to discover empirical regularities that can inform economic theory and be tested on a second data set © 2011 Pearson Addison-Wesley. All rights reserved. 260

8. Be prepared to compromise: a. The Classical Assumptions are only rarely are satisfied b. Applied econometricians are therefore forced to compromise and adopt suboptimal solutions, the characteristics and consequences of which are not always known c. Applied econometrics is necessarily ad hoc: we develop our analysis, including responses to potential problems, as we go along… 9. Do not confuse statistical significance with meaningful magnitude: a. If the sample size is large enough, any (two-sided) hypothesis can be rejected (when large enough to make the SEs small enough) b. Substantive significance—i.e. “how large?”—is also important, not just statistical significance © 2011 Pearson Addison-Wesley. All rights reserved. 261

10. Report a sensitivity analysis: a. Dimensions to examine: i. sample period ii. the functional form iii. the set of explanatory variables iv. the choice of proxies b. If results are not robust across the examined dimensions, then this casts doubt on the conclusions of the research © 2011 Pearson Addison-Wesley. All rights reserved. 262

What to Check If You Get an Unexpected Sign
1. Recheck the expected sign Were dummy variables computed “upside down,” for example? 2. Check your data for input errors and/or outliers 3. Check for an omitted variable The most frequent source of significant unexpected signs 4. Check for an irrelevant variable Frequent source of insignificant unexpected signs 5. Check for multicollinearity Multicollinearity increases the variances and standard errors of the estimated coefficients, increasing the chance that a coefficient could have an unexpected sign © 2011 Pearson Addison-Wesley. All rights reserved. 263

What to Check If You Get an Unexpected Sign
6. Check for sample selection bias An unexpected sign sometimes can be due to the fact that the observations included in the data were not obtained randomly 7. Check your sample size The smaller the sample size, the higher the variance on SEs 8. Check your theory If nothing else is apparently wrong, only two possibilities remain: the theory is wrong or the data is bad © 2011 Pearson Addison-Wesley. All rights reserved. 264

A Dozen Practical Tips Worth Reiterating
1. Don’t attempt to maximize (Chapter 2) 2. Always review the literature and hypothesize the signs of your coefficients before estimating a model (Chapter 3) 3. Inspect and clean your data before estimating a model. Know that outliers should not be automatically omitted; instead, they should be investigated to make sure that they belong in the sample (Chapter 3) 4. Know the Classical Assumptions cold! (Chapter 4) 5. In general, use a one-sided t-test unless the expected sign of the coefficient actually is in doubt (Chapter 5) © 2011 Pearson Addison-Wesley. All rights reserved. 265

A Dozen Practical Tips Worth Reiterating (cont.)
6. Don’t automatically discard a variable with an insignificant t-score. In general, be willing to live with a variable with a t-score lower than the critical value in order to decrease the chance of omitting a relevant variable (Chapter 6) 7. Know how to analyze the size and direction of the bias caused by an omitted variable (Chapter 6) 8. Understand all the different functional form options and their common uses, and remember to choose your functional form primarily on the basis of theory, not fit (Chapter 7) © 2011 Pearson Addison-Wesley. All rights reserved. 266

9. Multicollinearity doesn’t create bias; the estimated variances are large, but the estimated coefficients themselves are unbiased: So, the most-used remedy for multicollinearity is to do nothing (Chapter 8) 10. If you get a significant Durbin–Watson, Park, or White test, remember to consider the possibility that a specification error might be causing impure serial correlation or heteroskedasticity. Don’t change your estimation technique from OLS to GLS or use adjusted standard errors until you have the best possible specification. (Chapters 9 and 10) © 2011 Pearson Addison-Wesley. All rights reserved. 267

11. Adjusted standard errors like Newey–West standard errors or HC standard errors use the OLS coefficient estimates. It’s the standard errors of the estimated coefficients that change, not the estimated coefficients themselves. (Chapters 9 and 10) 12. Finally, if in doubt, rely on common sense and economic theory, not on statistical tests © 2011 Pearson Addison-Wesley. All rights reserved.

The Ethical Econometrician
We think that there are two reasonable goals for econometricians when estimating models: 1. Run as few different specifications as possible while still attempting to avoid the major econometric problems The only exception is sensitivity analysis, described in Section 6.4 2. Report honestly the number and type of different specifications estimated so that readers of the research can evaluate how much weight to give to your results © 2011 Pearson Addison-Wesley. All rights reserved.

Writing Your Research Report
Most good research reports have a number of elements in common: A brief introduction that defines the dependent variable and states the goals of the research A short review of relevant previous literature and research An explanation of the specification of the equation (model): Independent variables functional forms expected signs of (or other hypotheses about) the slope coefficients A description of the data: generated variables data sources data irregularities (if any) © 2011 Pearson Addison-Wesley. All rights reserved.

Writing Your Research Report (cont.)
A presentation of each estimated specification, using our standard documentation format If you estimate more than one specification, be sure to explain which one is best (and why!) A careful analysis of the regression results: discussion of any econometric problems encountered complete documentation of all: equations estimated tests run A short summary/conclusion that includes any policy recommendations or suggestions for further research A bibliography An appendix that includes all data, all regression runs, and all relevant computer output © 2011 Pearson Addison-Wesley. All rights reserved.

Table 11.2a Regression User’s Checklist

Table 11.2b Regression User’s Checklist

Table 11.2c Regression User’s Checklist

Table 11.2d Regression User’s Checklist

Table 11.3a Regression User’s Guide

Table 11.3b Regression User’s Guide

Table 11.3c Regression User’s Guide

Key Terms from Chapter 11 Choosing a research topic Data collection
Missing data Surveys Panel data The 10 Commandments of Applied Econometrics What to Check If You Get An Unexpected Sign A Dozen Practical Tips Worth Reiterating The Ethical Econometrician Writing your research report A Regression User’s Checklist A Regression User’s Guide © 2011 Pearson Addison-Wesley. All rights reserved. 279

Time-Series Models Chapter 12

Dynamic Models: Distributed Lag Models
An (ad hoc) distributed lag model explains the current value of Y as a function of current and past values of X, thus “distributing” the impact of X over a number of time periods For example, we might be interested in the impact of a change in the money supply (X) on GDP (Y) and model this as: Yt = α0 + β0Xt + β1Xt–1 + β2Xt– βpXt–p + εt (12.2) Potential issues from estimating Equation 12.2 with OLS: 1. The various lagged values of X are likely to be severely multicollinear, making coefficient estimates imprecise © 2011 Pearson Addison-Wesley. All rights reserved. 281

Dynamic Models: Distributed Lag Models (cont.)
2. In large part because of this multicollinearity, there is no guarantee that the estimated coefficients will follow the smoothly declining pattern that economic theory would suggest Instead, it’s quite typical to get something like: 3. The degrees of freedom tend to decrease, sometimes substantially, since we have to: estimate a coefficient for each lagged X, thus increasing K and lowering the degrees of freedom (N – K – 1) b. decrease the sample size by one for each lagged X, thus lowering the number of observations, N, and therefore the degrees of freedom (unless data for lagged Xs outside the sample are available) © 2011 Pearson Addison-Wesley. All rights reserved. 282

What Is a Dynamic Model? The simplest dynamic model is: (12.3)
Note that Y is on the left-hand side as Yt, and on the right-hand side as Yt–1 It’s this difference in time period that makes the equation dynamic Note that there is an important connection between a dynamic model such as the Equation 12.3 and a distributed lag model such as Equation 12.2 © 2011 Pearson Addison-Wesley. All rights reserved. 283

What Is a Dynamic Model? (cont.)
Yt = α0 + β0Xt + β1Xt–1 + β2Xt– βpXt–p + εt (12.2) where: β1 = λβ0 (12.8) β2 = λ2β0 β3 = λ3β0 . βp = λPβ0 As long as λ is between 0 and 1, these coefficients will indeed smoothly decline, as shown in Figure 12.1 © 2011 Pearson Addison-Wesley. All rights reserved. 289

Figure 12.1 Geometric Weighting Schemes for Various Dynamic Models

Serial Correlation and Dynamic Models
The consequences of serial correlation depend crucially on the type of model in question: 1. Ad hoc distributed lag models: serial correlation has the effects outlined in Section 9.2: causes no bias in the OLS coefficients themselves causes OLS to no longer be the minimum variance unbiased estimator causes the standard errors to be biased 2. Dynamic models: Now serial correlation causes bias in the coefficients produced by OLS Compounding all this this is the fact that the consequences, detection, and remedies for serial correlation that we discussed in Chapter 9 are all either incorrect or need to be modified in the presence of a lagged dependent variable We will now discuss the issues of testing and correcting for serial correlation in dynamic models in a bit more detail © 2011 Pearson Addison-Wesley. All rights reserved. 293

Testing Koyck Lag Models for Serial Correlation

Testing for Serial Correlation in Dynamic Models
Using the Lagrange Multiplier to test for serial correlation for a typical dynamic model involves three steps: Obtain the residuals of the estimated equation: 2. Use these residuals as the dependent variable in an auxiliary regression that includes as independent variables all those on the right-hand side of the original equation as well as the lagged residuals: © 2011 Pearson Addison-Wesley. All rights reserved. 297

Testing for Serial Correlation in Dynamic Models (cont.)
3. Estimate Equation using OLS and then test the null hypothesis that a3 = 0 with the following test statistic: LM = N*R2 (12.19) where: N = the sample size R2 is the unadjusted coefficient of determination both of the auxiliary equation, Equation For large samples, LM has a chi-square distribution with degrees of freedom equal to the number of restrictions in the null hypothesis (in this case, one). If LM is greater than the critical chi-square value from Statistical Table B-8, then we reject the null hypothesis that a3 = 0 and conclude that there is indeed serial correlation in the original equation © 2011 Pearson Addison-Wesley. All rights reserved. 298

Correcting for Serial Correlation in Dynamic Models
There are essentially three strategies for attempting to rid a dynamic model of serial correlation: improving the specification: Only relevant if the serial correlation is impure instrumental variables: substituting an “instrument” (a variable that is highly correlated with YM but is uncorrelated with ut) for Yt: in the original equation effectively eliminates the correlation between Ytl and ut Problem: good instruments are hard to come by (also see Section 14.3) modified GLS: Technique similar to the GLS procedure outlined in Section 9.4 Potential issues: sample must be large and the standard © 2011 Pearson Addison-Wesley. All rights reserved. 299

Granger Causality Granger causality, or precedence, is a circumstance in which one time series variable consistently and predictably changes before another variable A word of caution: even if one variable precedes (“Granger causes”) another, this does not mean that the first variable “causes” the other to change There are several tests for Granger causality They all involve distributed lag models in one form or another, however We’ll discuss an expanded version of a test originally developed by Granger © 2011 Pearson Addison-Wesley. All rights reserved. 300

Granger Causality (cont.)
Granger suggested that to see if A Granger-caused Y, we should run: Yt = β0 + β1Yt– βpYt–p + α1At– αpAt–p + εt (12.20) and test the null hypothesis that the coefficients of the lagged As (the αs) jointly equal zero If we can reject this null hypothesis using the F-test, then we have evidence that A Granger-causes Y Note that if p = 1, Equation is similar to the dynamic model, Equation 12.3 Applications of this test involve running two Granger tests, one in each direction © 2011 Pearson Addison-Wesley. All rights reserved. 301

Granger Causality (cont.)
That is, run Equation and also run: At = β0 + β1At– βpAt–p + α1Yt– αpYt–p + εt (12.21) testing for Granger causality in both directions by testing the null hypothesis that the coefficients of the lagged Ys (again, the αs) jointly equal zero If the F-test is significant for Equation but not for Equation 12.21, then we can conclude that A Granger-causes Y © 2011 Pearson Addison-Wesley. All rights reserved. 302

Spurious Correlation and Nonstationarity
Independent variables can appear to be more significant than they actually are if they have the same underlying trend as the dependent variable Example: In a country with rampant inflation almost any nominal variable will appear to be highly correlated with all other nominal variables Why? Nominal variables are unadjusted for inflation, so every nominal variable will have a powerful inflationary component Such a problem is an example of spurious correlation: a strong relationship between two or more variables that is not caused by a real underlying causal relationship If you run a regression in which the dependent variable and one or more independent variables are spuriously correlated, the result is a spurious regression, and the t-scores and overall fit of such spurious regressions are likely to be overstated and untrustworthy © 2011 Pearson Addison-Wesley. All rights reserved. 303

Stationary and Nonstationary Time Series
a time-series variable, Xt, is stationary if: 1. the mean of Xt is constant over time, 2. the variance of Xt is constant over time, and 3. the simple correlation coefficient between Xt and Xt–k depends on the length of the lag (k) but on no other variable (for all k) If one or more of these properties is not met, then Xt is nonstationary If a series is nonstationary, that problem is often referred to as nonstationarity © 2011 Pearson Addison-Wesley. All rights reserved. 304

Stationary and Nonstationary Time Series (cont.)
To get a better understanding of these issues, consider the case where Yt is generated by an equation that includes only past values of itself (an autoregressive equation): Yt = γYt–1 + vt (12.22) where vt is a classical error term Can you see that if | γ | < 1, then the expected value of Yt will eventually approach 0 (and therefore be stationary) as the sample size gets bigger and bigger? (Remember, since vt is a classical error term, its expected value = 0) Similarly, can you see that if | γ | > 1, then the expected value of Yt will continuously increase, making Yt nonstationary? This is nonstationarity due to a trend, but it still can cause spurious regression results © 2011 Pearson Addison-Wesley. All rights reserved. 305

Stationary and Nonstationary Time Series (cont.)
Most importantly, what about if |γ| = 1? In this case: Yt = Yt–1 + vt (12.23) This is a random walk: the expected value of Yt does not converge on any value, meaning that it is nonstationary This circumstance, where γ = 1 in Equation (or similar equations), is called a unit root If a variable has a unit root, then Equation holds, and the variable follows a random walk and is nonstationary © 2011 Pearson Addison-Wesley. All rights reserved. 306

The Dickey–Fuller Test
From the previous discussion of stationarity and unit roots, it makes sense to estimate Equation 12.22: Yt = γYt–1 + vt (12.22) and then determine if |γ| < 1 to see if Y is stationary This is almost exactly how the Dickey-Fuller test works: 1. Subtract Yt–1 from both sides of Equation 12.22, yielding: (Yt – Yt–1) = (γ – 1)Yt–1 + vt (12.26) © 2011 Pearson Addison-Wesley. All rights reserved. 307

The Dickey–Fuller Test (cont.)
If we define ΔYt = Yt – Yt–1 then we have the simplest form of the Dickey–Fuller test: ΔYt = β1Yt–1 + vt (12.27) where β1 = γ – 1 Note: alternative Dickey-Fuller tests additionally include a constant and/or a constant and a trend term 2. Set up the test hypotheses: H0: β1 = 0 (unit root) HA: β1 < 0 (stationary) © 2011 Pearson Addison-Wesley. All rights reserved. 308

The Dickey–Fuller Test (cont.)
3. Set up the decision rule: If is statistically significantly less than 0, then we can reject the null hypothesis of nonstationarity If is not statistically significantly less than 0, then we cannot reject the null hypothesis of nonstationarity Note that the standard t-table does not apply to Dickey–Fuller tests For the case of no constant and no trend (Equation 12.27) the large-sample values for tc are listed in Table 12.1 © 2011 Pearson Addison-Wesley. All rights reserved. 309

Table 12.1 Large-Sample Critical Values for the Dickey–Fuller Test

Cointegration If the Dickey–Fuller test reveals nonstationarity, what should we do? The traditional approach has been to take first differences (ΔY = Yt – Yt–1 and ΔX = Xt – Xt–1) and use them in place of Yt and Xt in the regressions Issue: the first-differencing basically ”throws away information” about the possible equilibrium relationships between the variables Alternatively, one might want to test whether the time-series are cointegrated, which means that even though individual variables might be nonstationary, it’s possible for linear combinations of nonstationary variables to be stationary © 2011 Pearson Addison-Wesley. All rights reserved. 311

Cointegration (cont.) To see how this works, consider Equation 12.24:
(12.24) Assume that both Yt and Xt have a unit root Solving Equation for ut, we get: (12.30) In Equation 12.24, u t is a function of two nonstationary variables, so u t might be expected also to be nonstationary Cointegration refers to the case where this is not the case: Yt and Xt are both non-stationary, yet a linear combination of them, as given by Equation 12.24, is stationary How does this happen? This could happen if economic theory supports Equation as an equilibrium © 2011 Pearson Addison-Wesley. All rights reserved. 312

Cointegration (cont.) We thus see that if Xt and Yt are cointegrated then OLS estimation of the coefficients in Equation can avoid spurious results To determine if Xt and Yt are cointegrated, we begin with OLS estimation of Equation and calculate the OLS residuals: (12.31) Next, perform a Dickey-Fuller test on the residuals Remember to use the critical values from the Dickey-Fuller Table! If we are able to reject the null hypothesis of a unit root in the residuals, we can conclude that Xt and Yt are cointegrated and our OLS estimates are not spurious © 2011 Pearson Addison-Wesley. All rights reserved. 313

A Standard Sequence of Steps for Dealing with Nonstationary Time Series
1. Specify the model (lags vs. no lags, etc) 2. Test all variables for nonstationarity (technically unit roots) using the appropriate version of the Dickey–Fuller test 3. If the variables don’t have unit roots, estimate the equation in its original units (Y and X) 4. If the variables have unit roots, test the residuals of the equation for cointegration using the Dickey–Fuller test 5. If the variables have unit roots but are not cointegrated, then change the functional form of the model to first differences (∆X and ∆Y) and estimate the equation 6. If the variables have unit roots and also are cointegrated, then estimate the equation in its original units © 2011 Pearson Addison-Wesley. All rights reserved. 314

The Linear Probability Model
The linear probability model is simply running OLS for a regression, where the dependent variable is a dummy (i.e. binary) variable: (13.1) where Di is a dummy variable, and the Xs, βs, and ε are typical independent variables, regression coefficients, and an error term, respectively The term linear probability model comes from the fact that the right side of the equation is linear while the expected value of the left side measures the probability that Di = 1 © 2011 Pearson Addison-Wesley. All rights reserved. 316

Problems with the Linear Probability Model
1. R2 is not an accurate measure of overall fit: Di can equal only 1 or 0, but must move in a continuous fashion from one extreme to the other (as also illustrated in Figure 13.1) Hence, is likely to be quite different from Di for some range of Xi Thus, R2 is likely to be much lower than 1 even if the model actually does an exceptional job of explaining the choices involved As an alternative, one can instead use , a measure based on the percentage of the observations in the sample that a particular estimated equation explains correctly To use this approach, consider a > .5 to predict that Di = 1 and a < .5 to predict that Di = 0 and then simply compare these predictions with the actual Di is not bounded by 0 and 1: The alternative binomial logit model, presented in Section 13.2, will address this issue © 2011 Pearson Addison-Wesley. All rights reserved. 317

Figure 13.1 A Linear Probability Model

The Binomial Logit Model
The binomial logit is an estimation technique for equations with dummy dependent variables that avoids the unboundedness problem of the linear probability model It does so by using a variant of the cumulative logistic function: (13.7) Logits cannot be estimated using OLS but are instead estimated by maximum likelihood (ML), an iterative estimation technique that is especially useful for equations that are nonlinear in the coefficients Again, for the logit model is bounded by 1 and 0 This is illustrated by Figure 13.2 © 2011 Pearson Addison-Wesley. All rights reserved. 319

Figure 13.2 Is Bounded by 0 and 1 in a Binomial Logit Model

Interpreting Estimated Logit Coefficients
The signs of the coefficients in the logit model have the same meaning as in the linear probability (i.e. OLS) model The interpretation of the magnitude of the coefficients differs, though, the dependent variable has changed dramatically. That the “marginal effects” are not constant can be seen from Figure 13.2: the slope (i.e. the change in probability) of the graph of the logit changes as moves from 0 to 1! We’ll consider three ways for helping to interpret logit coeffcients meaningfully: © 2011 Pearson Addison-Wesley. All rights reserved. 321

Interpreting Estimated Logit Coefficients (cont.)
1. Change an average observation: Create an “average” observation by plugging the means of all the independent variables into the estimated logit equation and then calculating an “average” Then increase the independent variable of interest by one unit and recalculate the The difference between the two s then gives the marginal effect 2. Use a partial derivative: Taking a derivative of the logit yields the result that the change in the expected value of caused by a one unit increase in holding constant the other independent variables in the equation equals To use this formula, simply plug in your estimates of and Di From this, again, the marginal impact of X does indeed depend on the value of 3. Use a rough estimate of 0.25: Plugging in into the previous equation, we get the (more handy!) result that multiplying a logit coefficient by 0.25 (or dividing by 4) yields an equivalent linear probability model coefficient © 2011 Pearson Addison-Wesley. All rights reserved. 322

Other Dummy Dependent Variable Techniques
The Binomial Probit Model: Similar to the logit model this an estimation technique for equations with dummy dependent variables that avoids the unboundedness problem of the linear probability model However, rather than the logistic function, this model uses a variant of the cumulative normal distribution The Multinomial Logit Model: Sometimes there are more than two qualitative choices available The sequential binary model estimates such choices as a series of binary decisions If the choice is made simultaneously, however, this is not appropriate The multinomial logit is developed specifically for the case with more than two qualitative choices and the choice is made simultaneously © 2011 Pearson Addison-Wesley. All rights reserved. 323

The Nature of Simultaneous Equations Systems
In a typical econometric equation: Yt = β0 + β1X1t + β2X2t + εt (14.1) a simultaneous system is one in which Y has an effect on at least one of the Xs in addition to the effect that the Xs have on Y Jargon here involves feedback effects, dual causality as well as X and Y being jointly determined Such systems are usually modeled by distinguishing between variables that are simultaneously determined (the Ys, called endogenous variables) and those that are not (the Xs, called exogenous variables): Y1t = α0 + α1Y2t + α2X1t + α3X2t + ε1t (14.2) Y2t = β0 + β1Y1t + β2X3t + β3X2t + ε2t (14.3) © 2011 Pearson Addison-Wesley. All rights reserved. 325

The Nature of Simultaneous Equations Systems (cont.)
Equations 14.2 and 14.3 are examples of structural equations Structural equations characterize the underlying economic theory behind each endogenous variable by expressing it in terms of both endogenous and exogenous variables For example, Equations 14.2 and 14.3 could be a demand and a supply equation, respectively © 2011 Pearson Addison-Wesley. All rights reserved. 326

The Nature of Simultaneous Equations Systems (cont.)
The term predetermined variable includes all exogenous variables and lagged endogenous variables “Predetermined” implies that exogenous and lagged endogenous variables are determined outside the system of specified equations or prior to the current period The main problem with simultaneous systems is that they violate Classical Assumption III (the error term and each explanatory variable should be uncorrelated) © 2011 Pearson Addison-Wesley. All rights reserved. 327

Reduced-Form Equations
An alternative way of expressing a simultaneous equations system is through the use of reduced-form equations Reduced-form equations express a particular endogenous variable solely in terms of an error term and all the predetermined (exogenous plus lagged endogenous) variables in the simultaneous system © 2011 Pearson Addison-Wesley. All rights reserved. 328

Reduced-Form Equations (cont.)
The reduced-form equations for the structural Equations 14.2 and 14.3 would thus be: Y1t = π0 + π1X1t + π2X2t + π3X3t + v1t (14.6) Y2t = π4 + π5X1t + π6X2t + π7X3t + v2t (14.7) where the vs are stochastic error terms and the πs are called reduced-form coefficients © 2011 Pearson Addison-Wesley. All rights reserved. 329

Reduced-Form Equations (cont.)
There are at least three reasons for using reduced-form equations: Since the reduced-form equations have no inherent simultaneity, they do not violate Classical Assumption III – Therefore, they can be estimated with OLS without encountering the problems discussed in this chapter The interpretation of the reduced-form coefficients as impact multipliers means that they have economic meaning and useful applications of their own Reduced-form equations play a crucial role in Two-Stage Least Squares, the estimation technique most frequently used for simultaneous equations (discussed in Section 14.3) © 2011 Pearson Addison-Wesley. All rights reserved. 330

The Bias of Ordinary Least Squares (OLS)
Simultaneity bias refers to the fact that in a simultaneous system, the expected values of the OLS-estimated structural coefficients are not equal to the true βs, that is: (14.10) The reason for this is that the two error terms of Equation and are correlated with the endogenous variables when they appear as explanatory variables As an example of how the application of OLS to simultaneous equations estimation causes bias, a Monte Carlo experiment was conducted for a supply and demand model As Figure 14.2 illustrates, the sampling distributions differed greatly from the “true” distributions defined in the Monte Carlo experiment © 2011 Pearson Addison-Wesley. All rights reserved. 331

Figure 14.2 Sampling Distributions Showing Simultaneity Bias of OLS Estimates

What Is Two-Stage Least Squares?
Two-Stage Least Squares (2SLS) helps mitigate simultaneity bias in simultaneous equation systems 2SLS requires a variable that is: 1. a good proxy for the endogenous variable 2. uncorrelated with the error term Such a variable is called an instrumental variable 2SLS essentially consist of the following two steps: © 2011 Pearson Addison-Wesley. All rights reserved. 333

What Is Two-Stage Least Squares?
STAGE ONE: Run OLS on the reduced-form equations for each of the endogenous variables that appear as explanatory variables in the structural equations in the system That is, estimate (using OLS): (14.18) (14.19) © 2011 Pearson Addison-Wesley. All rights reserved. 334

What Is Two-Stage Least Squares? (cont.)
STAGE TWO: Substitute the Ys from the reduced form for the Ys that appear on the right side (only) of the structural equations, and then estimate these revised structural equations with OLS That is, estimate (using OLS): (14.20) (14.21) © 2011 Pearson Addison-Wesley. All rights reserved. 335

The Properties of Two-Stage Least Squares
1. 2SLS estimates are still biased in small samples But consistent in large samples (get closer to true βs as N increases) 2. Bias in 2SLS for small samples typically is of the opposite sign of the bias in OLS 3. If the fit of the reduced-form equation is poor, then 2SLS will not rid the equation of bias even in a large sample 4. 2SLS estimates have increased variances and standard errors relative to OLS Note that Two-Stage Least Squares cannot be applied to an equation unless that equation is identified, however We therefore now turn to the issue of identification © 2011 Pearson Addison-Wesley. All rights reserved. 336

What Is the Identification Problem?
Identification is a precondition for the application of 2SLS to equations in simultaneous systems A structural equation is identified only when enough of the system’s predetermined variables are omitted from the equation in question to allow that equation to be distinguished from all the others in the system Note that one equation in a simultaneous system might be identified and another might not Most simultaneous systems are fairly complicated, so econometricians need a general method by which to determine whether equations are identified The method typically used is the order condition of identification, to which we now turn © 2011 Pearson Addison-Wesley. All rights reserved. 337

The Order Condition of Identification
Is a systematic method of determining whether a particular equation in a simultaneous system has the potential to be identified If an equation can meet the order condition, then it is almost always identified We thus say that the order condition is a necessary but not sufficient condition of identification © 2011 Pearson Addison-Wesley. All rights reserved. 338

The Order Condition of Identification (cont.)
A necessary condition for an equation to be identified is that the number of predetermined (exogenous plus lagged endogenous) variables in the system be greater than or equal to the number of slope coefficients in the equation of interest Or, in equation form, a structural equation meets the order condition if: # predetermined variables ≥ # slope coefficients (in the simultaneous system) (in the equation) © 2011 Pearson Addison-Wesley. All rights reserved. 339

Figure 14.1 Supply and Demand Simultaneous Equations

Figure 14.3 A Shifting Supply Curve

Figure 14.4 When Both Curves Shift

Table 14.1a Data for a Small Macromodel

Table 14.1b Data for a Small Macromodel

Key Terms from Chapter 14 Endogenous variable Predetermined variable
Structural equation Reduced-form equation Simultaneity bias Two-Stage Least Squares Identification Order condition for identification © 2011 Pearson Addison-Wesley. All rights reserved. 345

What Is Forecasting? In general, forecasting is the act of predicting the future In econometrics, forecasting is the estimation of the expected value of a dependent variable for observations that are not part of the same data set In most forecasts, the values being predicted are for time periods in the future, but cross-sectional predictions of values for countries or people not in the sample are also common To simplify terminology, the words prediction and forecast will be used interchangeably in this chapter Some authors limit the use of the word forecast to out-of-sample prediction for a time series © 2011 Pearson Addison-Wesley. All rights reserved. 347

What Is Forecasting? (cont.)
Econometric forecasting generally uses a single linear equation to predict or forecast Our use of such an equation to make a forecast can be summarized into two steps: Specify and estimate an equation that has as its dependent variable the item that we wish to forecast: (15.2) © 2011 Pearson Addison-Wesley. All rights reserved. 348

What Is Forecasting? (cont.)
2. Obtain values for each of the independent variables for the observations for which we want a forecast and substitute them into our forecasting equation: (15.3) Figure 15.1 illustrates two examples © 2011 Pearson Addison-Wesley. All rights reserved. 349

Figure 15.1a Forecasting Examples

Figure 15.1b Forecasting Examples

More Complex Forecasting Problems
The forecasts generated in the previous section are quite simple, however, and most actual forecasting involves one or more additional questions—for example: 1. Unknown Xs: It is unrealistic to expect to know the values for the independent variables outside the sample What happens when we don’t know the values of the independent variables for the forecast period? 2. Serial Correlation: If there is serial correlation involved, the forecasting equation may be estimated with GLS How should predictions be adjusted when forecasting equations are estimated with GLS? © 2011 Pearson Addison-Wesley. All rights reserved. 352

More Complex Forecasting Problems (cont.)
3. Confidence Intervals: All the previous forecasts were single values, but such single values are almost never exactly right, so maybe it would be more helpful if we forecasted a confidence interval instead How can we develop these confidence intervals? 4. Simultaneous Equations Models: As we saw in Chapter 14, many economic and business equations are part of simultaneous models How can we use an independent variable to forecast a dependent variable when we know that a change in value of the dependent variable will change, in turn, the value of the independent variable that we used to make the forecast? © 2011 Pearson Addison-Wesley. All rights reserved. 353

Conditional Forecasting (Unknown X Values for the Forecast Period)
Unconditional forecast: all values of the independent variables are known with certainty This is rare in practice Conditional forecast: actual values of one or more of the independent variables are not known This is the more common type of forecast © 2011 Pearson Addison-Wesley. All rights reserved. 354

Conditional Forecasting (Unknown X Values for the Forecast Period) (cont.)
The careful selection of independent variables can sometimes help avoid the need for conditional forecasting This opportunity can arise when the dependent variable can be expressed as a function of leading indicators: A leading indicator is an independent variable the movements of which anticipate movements in the dependent variable The best known leading indicator, the Index of Leading Economic Indicators, is produced each month © 2011 Pearson Addison-Wesley. All rights reserved. 355

Forecasting with Serially Correlated Error Terms
Recall from Chapter 9 that when serial correlation is severe, one remedy is to run Generalized Least Squares (GLS) as noted in Equation 9.18: (9.18) If Equation 9.18 is estimated, the dependent variable will be: (15.7) Thus, if a GLS equation is used for forecasting, it will produce predictions of Y*T + 1 rather than of YT+1 Such predictions thus will be of the wrong variable! © 2011 Pearson Addison-Wesley. All rights reserved. 356

Forecasting with Serially Correlated Error Terms (cont.)
If forecasts are to be made with a GLS equation, Equation 9.18 should first be solved for YT before forecasting is attempted: (15.8) Next, substitute T+1 for t (to forecast time period T+1) and insert estimates for the coefficients, ρs and Xs into the equation to get: (15.9) Equation 15.9 thus should be used for forecasting when an equation has been estimated with GLS to correct for serial correlation © 2011 Pearson Addison-Wesley. All rights reserved. 357

Forecasting Confidence Intervals
The techniques we use to test hypotheses can also be adapted to create forecasting confidence intervals Given a point forecast, all we need to generate a confidence interval around that forecast are tc, the critical t-value (for the desired level of confidence), and SF, the estimated standard error of the forecast: (15.11) The critical t-value, tc, can be found in Statistical Table B-1 (for a two-tailed test with T-K-1 degrees of freedom) © 2011 Pearson Addison-Wesley. All rights reserved. 358

Forecasting Confidence Intervals (cont.)
Lastly, the standard error of the forecast, SF, for an equation with just one independent variable, equals the square root of the forecast error variance: (15.13) where: s2 = the estimated variance of the error term T = the number of observations in the sample XT+1 = the forecasted value of the single independent variable = the arithmetic mean of the observed Xs in the sample Figure 15.2 illustrates an example of a forecast confidence interval © 2011 Pearson Addison-Wesley. All rights reserved. 359

Figure 15.2 A Confidence Interval for

Forecasting with Simultaneous Equations Systems
How should forecasting be done in the context of a simultaneous model? There are two approaches to answering this question, depending on whether there are lagged endogenous variables on the right-hand side of any of the equations in the system: © 2011 Pearson Addison-Wesley. All rights reserved. 361

Forecasting with Simultaneous Equations Systems (cont.)
1. No lagged endogenous variables in the system: the reduced-form equation for the particular endogenous variable can be used for forecasting because it represents the simultaneous solution of the system for the endogenous variable being forecasted 2. Lagged endogenous variables in the system: then the approach must be altered to take into account the dynamic interaction caused by the lagged endogenous variables For simple models, this sometimes can be done by substituting for the lagged endogenous variables where they appear in the reduced-form equations If such a manipulation is difficult, however, then a technique called simulation analysis can be used © 2011 Pearson Addison-Wesley. All rights reserved. 362

ARIMA Models ARIMA is a highly refined curve-fitting device that uses current and past values of the dependent variable to produce often accurate short-term forecasts of that variable Examples of such forecasts are stock market price predictions created by brokerage analysts (called “chartists” or “technicians”) based entirely on past patterns of movement of the stock prices If ARIMA models thus essentially ignores economic theory (by ignoring “traditional” explanatory variables), why use them? The use of ARIMA is appropriate when: little or nothing is known about the dependent variable being forecasted, the independent variables known to be important cannot be forecasted effectively all that is needed is a one or two-period forecast © 2011 Pearson Addison-Wesley. All rights reserved. 363

ARIMA Models (cont.) The ARIMA approach combines two different specifications (called processes) into one equation: 1. An autoregressive process (AR): expresses a dependent variable as a function of past values of the dependent variable This is similar to the serial correlation error term function of Chapter 9 and to the dynamic model of Chapter 12 2. a moving average process (MA): expresses a dependent variable as a function of past values of the error term Such a function is a moving average of past error term observations that can be added to the mean of Y to obtain a moving average of past values of Y © 2011 Pearson Addison-Wesley. All rights reserved. 364

ARIMA Models (cont.) To create an ARIMA model, we begin with an econometric equation with no independent variables: and then add to it both the autoregressive and moving-average processes: (15.17) where the θs and the φs are the coefficients of the autoregressive and moving-average processes, respectively, and p and q are the number of past values used of Y and ε, respectively © 2011 Pearson Addison-Wesley. All rights reserved. 365

ARIMA Models (cont.) Before this equation can be applied to a time series, however, it must be ensured that the time series is stationary, as defined in Section 12.4 For example, a non-stationary series can often be converted into a stationary one by taking the first difference: (15.18) If the first differences do not produce a stationary series, then first differences of this first-differenced series can be taken—i.e. a second-difference transformation: (15.19) © 2011 Pearson Addison-Wesley. All rights reserved. 366

ARIMA Models (cont.) If a forecast of Y* or Y** is made, then it must be converted back into Y terms For example, if d = 1 (where d is the number of differences taken to make Y stationary), then: (15.20) This conversion process is similar to integration in mathematics, so the “I” in ARIMA stands for “integrated” ARIMA thus stands for Auto-Regressive Integrated Moving Average An ARIMA model with p, d, and q specified is usually denoted as ARIMA (p,d,q) with the specific integers chosen inserted for p, d, and q If the original series is stationary and d therefore equals 0, this is sometimes shortened to ARMA © 2011 Pearson Addison-Wesley. All rights reserved. 367

Key Terms from Chapter 15 Unconditional forecast Conditional forecast
Leading indicator Confidence interval (of forecast) Autoregressive process Moving-average process ARIMA(p,d,q) © 2011 Pearson Addison-Wesley. All rights reserved. 368

Random Assignment Experiments
When medical researchers want to examine the effect of a new drug, they use an experimental design called an random assignment experiment In such experiments, two groups are chosen randomly: 1. Treatment group: receives the treatment (a specific medicine, say) 2. Control group: receives a harmless, ineffective placebo The resulting equation is: OUTCOMEi = β0 + β1TREATMENTi + εi (16.1) where: OUTCOMEi = a measure of the desired outcome in the ith individual TREATMENTi = a dummy variable equal to 1 for individuals in the treatment group and 0 for individuals in the control group © 2011 Pearson Addison-Wesley. All rights reserved. 370

Random Assignment Experiments (cont.)
But random assignment can’t always control for all possible other factors—though sometimes we may be able to identify some of these factors and add them to our equation Let’s say that the treatment is job training: Suppose that random assignment, by chance, results in one group having more males and being slightly older than the other group If gender and age matter in determining earnings, then we can control for the different composition of the two groups by including gender and age in our regression equation: OUTCOMEi = β0 + β1TREATMENTi + β2X1i + β3X2i + εi (16.2) where: X1 = dummy variable for the individual’s gender X2 = the individual’s age © 2011 Pearson Addison-Wesley. All rights reserved. 371

Unfortunately, random assignment experiments are not common in economics because they are subject to problems that typically do not plague medical experiments—e.g.: 1. Non-Random Samples: Most subjects in economic experiments are volunteers, and samples of volunteers often aren’t random and therefore may not be representative of the overall population As a result, our conclusions may not apply to everyone 2. Unobservable Heterogeneity: In Equation 16.2, we added observable factors to the equation to avoid omitted variable bias, but not all omitted factors in economics are observable This “unobservable omitted variable” problem is called unobserved heterogeneity © 2011 Pearson Addison-Wesley. All rights reserved. 372

3. The Hawthorne Effect: Human subjects typically know that they’re being studied, and they usually know whether they’re in the treatment group or the control group The fact that human subjects know that they’re being observed sometimes can change their behavior, and this change in behavior could clearly change the results of the experiment 4. Impossible Experiments: It’s often impossible (or unethical) to run a random assignment experiment in economics Think about how difficult it would be to use a random assignment experiment to study the impact of marriage on earnings! © 2011 Pearson Addison-Wesley. All rights reserved. 373

Natural Experiments Natural experiments (or quasi-experiments) are similar to random assignment experiments, except: observations fall into treatment and control groups “naturally” (because of an exogenous event) instead of being randomly assigned by the researcher By “exogenous event” is meant that the natural event must not be under the control of either of the two groups © 2011 Pearson Addison-Wesley. All rights reserved. 374

Natural Experiments (cont.)
The appropriate regression equation for such a natural experiment is: ΔOUTCOMEi = β0 + β1TREATMENTi + β2X1i + β3X2i + εi (16.3) where: ΔOUTCOMEi is defined as the outcome after the treatment minus the outcome before the treatment for the ith observation β1 is called the difference-in-differences estimator, and it measures the difference between the change in the treatment group and the change in the control group, holding constant X1 and X2 Figure 16.1 illustrates an example of a natural experiment © 2011 Pearson Addison-Wesley. All rights reserved. 375

Figure 16.1 Treatment and Control Groups for Los Angeles

What Are Panel Data? Panel (or longitudinal) data combine time-series and cross-sectional data such that observations on the same variables from the same cross sectional sample are followed over two or more different time periods Why use panel data? At least three reasons—using panel data: 1. certainly will increase sample sizes! 2. can help provide insights into analytical questions that can’t be answered by using time-series or cross-sectional data alone: Allows determining whether the same people are unemployed year after year or whether different individuals are unemployed in different years 3. often allow researchers to avoid omitted variable problems that otherwise would cause bias in cross-sectional studies © 2011 Pearson Addison-Wesley. All rights reserved. 377

What Are Panel Data? (cont.)
There are four different kinds of variables that we encounter when we use panel data: 1. Variables that can differ between individuals but don’t change over time: e.g., gender, ethnicity, and race 2. Variables that change over time but are the same for all individuals in a given time period: e.g., the retail price index and the national unemployment rate 3. Variables that vary both over time and between individuals: e.g., income and marital status 4. Trend variables that vary in predictable ways: e.g., an individual’s age © 2011 Pearson Addison-Wesley. All rights reserved. 378

The Fixed Effects Model
There are several alternative panel data estimation procedures Most researchers use the fixed effects model, which allows each cross-sectional unit to have a different intercept: Yit = β0 + β1Xit + β2D2i βNDNi + vit (16.4) where: D2 = intercept dummy equal to 1 for the second cross-sectional entity and 0 otherwise DN = intercept dummy equal to 1 for the Nth cross-sectional entity and 0 otherwise Note that Y, X, and v have two subscripts! © 2011 Pearson Addison-Wesley. All rights reserved. 379

The Fixed Effects Model (cont.)
One major advantage of the fixed effects model is that it avoids bias due to omitted variables that don’t change over time e.g., race or gender Such time-invariant omitted variables often are referred to as unobserved heterogeneity or a fixed effect To understand how this works, consider what Equation 16.4 would look like with only two years worth of data: Yit = β0 + β1Xit + β2D2i + vit (16.5) Let’s decompose the error term, vit, into two components, a classical error term (εit) and the unobserved impact of the time-invariant omitted variables (ai): vit = εit + ai (16.6) © 2011 Pearson Addison-Wesley. All rights reserved. 380

If we substitute Equation 16.6 into Equation 16.5, we get: Yit = β0 + β1Xit + β2D2i + εit + ai (16.7) Next, average Equation 16.7 over time for each observation i, thus producing: Yi = β0 + β1Xi + β2D2i + εi + ai (16.8) where the bar over a variable indicates the mean of that variable across time Note that ai, β2D2i, and β0 don’t have bars over them because they’re constant over time © 2011 Pearson Addison-Wesley. All rights reserved. 381

If we now subtract Equation 16.8 from Equation 16.7, we get: Note that ai, β2, D2i, and β0 are subtracted out because they’re in both equations We’ve therefore shown that estimating panel data with the fixed effects model does indeed drop the ai out of the equation Hence, the fixed effects model will not experience bias due to time-invariant omitted variables! Example: The death penalty and the murder rate: Figures 16.2 and 16.3 illustrates the importance of the fixed-effects model: the unlikely (positive) result from the cross-section model is reversed by the fixed effects model! © 2011 Pearson Addison-Wesley. All rights reserved. 382

Figure 16.3 In a Panel Data Model, the Murder Rate Decreases with Executions

The Random Effects Model
Recall that the fixed effects model is based on the assumption that each cross-sectional unit has its own intercept The random effects model instead is based on the assumption that the intercept for each cross-sectional unit is drawn from a distribution (that is centered around a mean intercept) Thus each intercept is a random draw from an “intercept distribution” and therefore is independent of the error term for any particular observation Hence the term random effects model © 2011 Pearson Addison-Wesley. All rights reserved. 385

The Random Effects Model (cont.)
Advantages of the random effects model: 1. more degrees of freedom than a fixed effects model This is because rather than estimating an intercept for virtually every cross-sectional unit, all we need to do is to estimate the parameters that describe the distribution of the intercepts. 2. Can now also estimate time-invariant explanatory variables (like race or gender). Disadvantages of the random effects model: 1. Most importantly, the random effects estimator requires us to assume that ai is uncorrelated with the independent variables, the Xs, if we’re going to avoid omitted variable bias This may be an overly strong assumption in many cases © 2011 Pearson Addison-Wesley. All rights reserved. 386

Choosing Between Fixed and Random Effects
One key is the nature of the relationship between ai and the Xs: If they’re likely to be correlated, then it makes sense to use the fixed effects model If not, then it makes sense to use the random effects model Can also use the Hausman test to examine whether there is correlation between ai and X Essentially, this procedure tests to see whether the regression coefficients under the fixed effects and random effects models are statistically different from each other If they are different, then the fixed effects model is preferred If the they are not different, then the random effects model is preferred (or estimates of both the fixed effects and random effects models are provided) © 2011 Pearson Addison-Wesley. All rights reserved. 387

Key Terms from Chapter 16 Treatment group Control group
Differences estimator Difference in differences Unobserved heterogeneity The Hawthorne effect Panel data The fixed effects model The random effects model Hausman test © 2011 Pearson Addison-Wesley. All rights reserved. 393

Probability A random variable X is a variable whose numerical value is determined by chance, the outcome of a random phenomenon A discrete random variable has a countable number of possible values, such as 0, 1, and 2 A continuous random variable, such as time and distance, can take on any value in an interval A probability distribution P[Xi] for a discrete random variable X assigns probabilities to the possible values X1, X2, and so on For example, when a fair six-sided die is rolled, there are six equally likely outcomes, each with a 1/6 probability of occurring Figure 17.1 shows this probability distribution © 2011 Pearson Addison-Wesley. All rights reserved. 395

Figure 17.1 Probability Distribution for a Six-Sided Die

Mean, Variance, and Standard Deviation
The expected value (or mean) of a discrete random variable X is a weighted average of all possible values of X, using the probability of each X value as weights: (17.1) the variance of a discrete random variable X is a weighted average, for all possible values of X, of the squared difference between X and its expected value, using the probability of each X value as weights: (17.2) The standard deviation σ is the square root of the variance © 2011 Pearson Addison-Wesley. All rights reserved. 397

Continuous Random Variables
Our examples to this point have involved discrete random variables, for which we can count the number of possible outcomes: The coin can be heads or tails; the die can be 1, 2, 3, 4, 5, or 6 For continuous random variables, however, the outcome can be any value in a given interval For example, Figure 17.2 shows a spinner for randomly selecting a point on a circle A continuous probability density curve shows the probability that the outcome is in a specified interval as the corresponding area under the curve This is illustrated for the case of the spinner in Figure 17.3 © 2011 Pearson Addison-Wesley. All rights reserved. 398

Figure 17.2 Pick a Number, Any Number

Figure 17.3 A Continuous Probability Distribution for the Spinner

Standardized Variables
To standardize a random variable X, we subtract its mean and then divide by its standard deviation : (17.3) No matter what the initial units of X, the standardized random variable Z has a mean of 0 and a standard deviation of 1 The standardized variable Z measures how many standard deviations X is above or below its mean: If X is equal to its mean, Z is equal to 0 If X is one standard deviation above its mean, Z is equal to 1 If X is two standard deviations below its mean, Z is equal to –2 Figures 17.4 and 17.5 illustrates this for the case of dice and fair coin flips, respectively © 2011 Pearson Addison-Wesley. All rights reserved. 401

Figure 17.4a Probability Distribution for Six-Sided Dice, Using Standardized Z

Figure 17.4b Probability Distribution for Six-Sided Dice, Using Standardized Z

Figure 17.4c Probability Distribution for Six-Sided Dice, Using Standardized Z

Figure 17.5a Probability Distribution for Fair Coin Flips, Using Standardized Z

Figure 17.5b Probability Distribution for Fair Coin Flips, Using Standardized Z

Figure 17.5c Probability Distribution for Fair Coin Flips, Using Standardized Z

The Normal Distribution
The density curve for the normal distribution is graphed in Figure 17.6 The probability that the value of Z will be in a specified interval is given by the corresponding area under this curve These areas can be determined by consulting statistical software or a table, such as Table B-7 in Appendix B Many things follow the normal distribution (at least approximately): the weights of humans, dogs, and tomatoes The lengths of thumbs, widths of shoulders, and breadths of skulls Scores on IQ, SAT, and GRE tests The number of kernels on ears of corn, ridges on scallop shells, hairs on cats, and leaves on trees © 2011 Pearson Addison-Wesley. All rights reserved. 408

Figure 17.6 The Normal Distribution

The Normal Distribution (cont.)
The central limit theorem is a very strong result for empirical analysis that builds on the normal distribution The central limit theorem states that: if Z is a standardized sum of N independent, identically distributed (discrete or continuous) random variables with a finite, nonzero standard deviation, then the probability distribution of Z approaches the normal distribution as N increases © 2011 Pearson Addison-Wesley. All rights reserved. 410

Sampling First, let’s define some key terms:
Population: the entire group of items that interests us Sample: the part of this population that we actually observe Statistical inference involves using the sample to draw conclusions about the characteristics of the population from which the sample came © 2011 Pearson Addison-Wesley. All rights reserved. 411

Selection Bias Any sample that differs systematically from the population that it is intended to represent is called a biased sample One of the most common causes of biased samples is selection bias, which occurs when the selection of the sample systematically excludes or underrepresents certain groups Selection bias often happens when we use a convenience sample consisting of data that are readily available Self-selection bias can occur when we examine data for a group of people who have chosen to be in that group © 2011 Pearson Addison-Wesley. All rights reserved. 412

Survivor and Nonresponse Bias
A retrospective study looks at past data for a contemporaneously selected sample for example, an examination of the lifetime medical records of 65-year-olds A prospective study, in contrast, selects a sample and then tracks the members over time By its very design, retrospective studies suffer from survivor bias: we necessarily exclude members of the past population who are no longer around! Nonresponse bias: The systematic refusal of some groups to participate in an experiment or to respond to a poll © 2011 Pearson Addison-Wesley. All rights reserved. 413

The Power of Random Selection
In a simple random sample of size N from a given population: each member of the population is equally likely to be included in the sample every possible sample of size N from this population has an equal chance of being selected How do we actually make random selections? We would like a procedure that is equivalent to the following: put the name of each member of the population on its own slip of paper drop these slips into a box mix thoroughly pick members out randomly In practice, random sampling is usually done through some sort of numerical identification combined with a computerized random selection of numbers © 2011 Pearson Addison-Wesley. All rights reserved. 414

Estimation First, some terminology:
Parameter: a characteristic of the population whose value is unknown, but can be estimated Estimator: a sample statistic that will be used to estimate the value of the population parameter Estimate: the specific value of the estimator that is obtained in one particular sample Sampling variation: the notion that because samples are chosen randomly, the sample average will vary from sample to sample, sometimes being larger than the population mean and sometimes lower © 2011 Pearson Addison-Wesley. All rights reserved. 415

Sampling Distributions
The sampling distribution of a statistic is the probability distribution or density curve that describes the population of all possible values of this statistic For example, it can be shown mathematically that if the individual observations are drawn from a normal distribution, then the sampling distribution for the sample mean is also normal Even if the population does not have a normal distribution, the sampling distribution of the sample mean will approach a normal distribution as the sample size increases It can be shown mathematically that the sampling distribution for the sample mean has the following mean and standard deviation: (17.5) © 2011 Pearson Addison-Wesley. All rights reserved. 416

The Mean of the Sampling Distribution
A sample statistic is an unbiased estimator of a population parameter if the mean of the sampling distribution of this statistic is equal to the value of the population parameter Because the mean of the sampling distribution of X is μ, X is an unbiased estimator of μ © 2011 Pearson Addison-Wesley. All rights reserved. 417

The Standard Deviation of the Sampling Distribution
One way of gauging the accuracy of an estimator is with its standard deviation: If an estimator has a large standard deviation, there is a substantial probability that an estimate will be far from its mean If an estimator has a small standard deviation, there is a high probability that an estimate will be close to its mean © 2011 Pearson Addison-Wesley. All rights reserved. 418

The t-Distribution When the mean of a sample from a normal distribution is standardized by subtracting the mean of its sampling distribution and dividing by the standard deviation of its sampling distribution, the resulting Z variable has a normal distribution W.S. Gosset determined (in 1908) the sampling distribution of the variable that is created when the mean of a sample from a normal distribution is standardized by subtracting and dividing by its standard error (≡ the standard deviation of an estimator): © 2011 Pearson Addison-Wesley. All rights reserved. 419

The t-Distribution (cont.)
The exact distribution of t depends on the sample size, as the sample size increases, we are increasingly confident of the accuracy of the estimated standard deviation Table B-1 at the end of the textbook shows some probabilities for various t-distributions that are identified by the number of degrees of freedom: degrees of freedom = # observations - # estimated parameters © 2011 Pearson Addison-Wesley. All rights reserved. 420

Confidence Intervals A confidence interval measures the reliability of a given statistic such as X The general procedure for determining a confidence interval for a population mean can be summarized as: 1. Calculate the sample average X 2. Calculate the standard error of X by dividing the sample standard deviation s by the square root of the sample size N 3. Select a confidence level (such as 95 percent) and look in Table B-1 with N-1 degrees of freedom to determine the t-value that corresponds to this probability 4. A confidence interval for the population mean is then given by: © 2011 Pearson Addison-Wesley. All rights reserved. 421

Sampling from Finite Populations
Notably, a confidence interval does not depend on the size of the population This may first seem surprising: if we are trying to estimate a characteristic of a large population, then wouldn’t we also need a large sample? The reason why the size of the population doesn’t matter is that the chances that the luck of the draw will yield a sample whose mean differs substantially from the population mean depends on the size of the sample and the chances of selecting items that are far from the population mean That is, not on how many items there are in the population © 2011 Pearson Addison-Wesley. All rights reserved. 422

Key Terms from Chapter 17 Random variable Probability distribution
Expected Value Mean Variance Standard deviation Standardized random variable Population Sample Selection, survivor, and nonresponse bias Sampling distribution Population mean Sample mean Population standard deviation Sample standard deviation Degrees of freedom Confidence interval © 2011 Pearson Addison-Wesley. All rights reserved. 423

Text Books Text book #1: Using Econometrics: A Practical Guide

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