The Simple Regression Model

The Simple Regression Model
DEFINITION OF THE SIMPLE REGRESSION MODEL DERIVING THE ORDINARY LEAST SQUARES(OLS) ESTIMATES MECHANICS OF OLS UNITS OF MEASUREMENT AND FUNCTIONAL FORM EXPECTED VALUES AND VARIANCES OF THE OLS ESTIMATORS REGRESSION THROUGH THE ORIGIN Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

The Simple Regression Model
Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

DEFINITION OF THE SIMPLE REGRESSION MODEL
In the simple linear regression model, where y = b0 + b1x + u, we typically refer to y as the Dependent Variable, or Left-Hand Side Variable, or Explained Variable, or Regressand（回归子） Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

In the simple linear regression of y on x, we typically refer to x as the
Independent Variable, or Right-Hand Side Variable, or Explanatory Variable, or Regressor（回归元）, or Covariate（协变量）, or Control Variables Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

A Simple Assumption The variable u, called the error term or disturbance in the relationship, represents factors other than x that affect y. If the other factors in u are held fixed, so that the change in u is zero, then x has a linear effect on y, and get 2.2 The average value of u, the error term, in the population is 0. That is, E(u) = 0 This is not a restrictive assumption, since we can always use b0 to normalize E(u) to 0 Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Zero Conditional Mean Assumption
We need to make a crucial assumption about how u and x are related We want it to be the case that knowing something about x does not give us any information about u, so that they are completely unrelated. That is, that E(u|x) = E(u) = 0, which implies E(y|x) = b0 + b1x ——population regression function（总体回归函数） Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

E(y|x) as a linear function of x, where for any x
the distribution of y is centered about E(y|x) Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Deriving the Ordinary Least Squares Estimate
Basic idea of regression is to estimate the population parameters from a sample Let {(xi,yi): i=1, …,n} denote a random sample of size n from the population For each observation in this sample, it will be the case that yi = b0 + b1xi + ui Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

. . . . Population regression line, sample data points
and the associated error terms y E(y|x) = b0 + b1x . y4 u4 { . y3 u3 } . y2 u2 { u1 . y1 } x1 x2 x3 x4 x Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Deriving OLS Estimates
To derive the OLS estimates we need to realize that our main assumption of E(u|x) = E(u) = 0 also implies that Cov(x,u) = E(xu) = 0 Why? Remember from basic probability that Cov(X,Y) = E(XY) – E(X)E(Y) Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Deriving OLS continued
We can write our 2 restrictions just in terms of x, y, b0 and b1 , since u = y – b0 – b1x E(y – b0 – b1x) = 0 E[x(y – b0 – b1x)] = 0 These are called moment restrictions（矩条件）. Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Deriving OLS using M.O.M. The method of moments approach to estimation implies imposing the population moment restrictions on the sample moments What does this mean? Recall that for E(X), the mean of a population distribution, a sample estimator of E(X) is simply the arithmetic mean（算术平均值） of the sample. Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

More Derivation of OLS We want to choose values of the parameters that will ensure that the sample versions of our moment restrictions are true The sample versions are as follows: Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

So the OLS estimated slope is
The sample covariance between x and y the sample variance of x Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Summary of OLS slope estimate
The slope estimate is the sample covariance between x and y divided by the sample variance of x If x and y are positively correlated, the slope will be positive If x and y are negatively correlated, the slope will be negative Only need x to vary in our sample Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

More OLS Intuitively, OLS is fitting a line through the sample points such that the sum of squared residuals is as small as possible, hence the term least squares The residual, û, is an estimate of the error term, u, and is the difference between the fitted line (sample regression function) and the sample point Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

. . . . { Sample regression line, sample data points
and the associated estimated error terms y . y4 { û4 . y3 } û3 . y2 û2 { û1 . } y1 x1 x2 x3 x4 x Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Alternate approach to derivation
Given the intuitive idea of fitting a line, we can set up a formal minimization problem That is, we want to choose our parameters such that we minimize the following: Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Alternate approach, continued
If one uses calculus to solve the minimization problem for the two parameters you obtain the following first order conditions, which are the same as we obtained before, multiplied by n Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Algebraic Properties of OLS
The sum of the OLS residuals is zero Thus, the sample average of the OLS residuals is zero as well The sample covariance between the regressors and the OLS residuals is zero The OLS regression line always goes through the mean of the sample Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Algebraic Properties (precise)

Example 2.3 CEO Salary and Return on Equity

Goodness-of-Fit How do we think about how well our sample regression line fits our sample data? Can compute the fraction of the total sum of squares (SST) that is explained by the model, call this the R-squared of regression R2 = SSE/SST = 1 – SSR/SST Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Units of Measurement and Functional Form

Units of Measurement and Functional Form (cont)
The goodness-of-fit of the model should not depend on the units of measurement of our variables. Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Incorporating Nonlinearities in Sample Regression

Example 2.11 CEO Salary and Firm Sales

Units of Measurement and Functional Form (cont)

Expected Values and Variances of the OLS Estimators
Unbiasedness of OLS Variances of the OLS Estimators Estimating the Error Variance Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Unbiasedness of OLS: four assumptions
Assumption SLR.1: the population model is linear in parameters as y = b0 + b1x + u Assumption SLR.2: we can use a random sample of size n, {(xi, yi): i=1, 2, …, n}, from the population model. Thus we can write the sample model yi = b0 + b1xi + ui Assumption SLR.3: E(u|x) = 0 Assumption SLR.4: there is variation in the xi Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Unbiasedness of OLS: Four Assumptions

Unbiasedness of OLS (cont)
In order to think about unbiasedness, we need to rewrite our estimator in terms of the population parameter Start with a simple rewrite of the formula as Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Unbiasedness of OLS (cont)

Unbiasedness of OLS (cont):Proof

Unbiasedness Summary The OLS estimates of b1 and b0 are unbiased if
Proof of unbiasedness depends on our 4 assumptions – if any assumption fails, then OLS is not necessarily unbiased Remember unbiasedness is a description of the estimator – in a given sample we may be “near” or “far” from the true parameter Using simple regression when u contains factors affecting y that are also correlated with x can result in spurious correlation. Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Example 2.12 Student Math Performance and the School Lunch Program
Let math10 denote the percentage of tenth graders at a high school receiving A passing score on a standardized mathematics exam, and lnchprg denote the Percentage of students who are eligible for the lunch program. Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Variance of the OLS Estimators
Now we know that the sampling distribution of our estimate is centered around the true parameter Want to think about how spread out this distribution is important. Much easier to think about this variance under an additional assumption, so Assumption SLR.5: Var(u|x) = s2 (Homoskedasticity) Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Variance of OLS (cont) Var(u|x) = E(u2|x)-[E(u|x)]2
E(u|x) = 0, so s2 = E(u2|x) = E(u2) = Var(u) Thus s2 is also the unconditional variance, called the error variance s, the square root of the error variance is called the standard deviation of the error Can say: E(y|x)=b0 + b1x and Var(y|x) = s2 In other words, the conditional expectation of y given x is linear in x, but the variance of y given x is constant. Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Example:Heteroskedasticity in a Wage Equation
When Var(u|x) depends on x, the error term is said to exhibit heteroskedasticity (or onconstant varianve). Since Var(u|x)=Var(y|x), heteroskedasticity is present whereever Var(y|x) is a function of x. Example:Heteroskedasticity in a Wage Equation Wage variability depends on the level of education: more wage variability at higher levels education, vice versa. Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Variance of OLS Summary
The larger the error variance, s2, the larger the variance of the slope estimate The larger the variability in the xi, the smaller the variance of the slope estimate As a result, a larger sample size should decrease the variance of the slope estimate Problem that the error variance is unknown Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Estimating the Error Variance
We don’t know what the error variance, s2, is, because we don’t observe the errors, ui What we observe are the residuals, ûi We can use the residuals to form an estimate of the error variance Copyright © 2007 Thomson Asia Pte. Ltd. All rights reserved.

Error Variance Estimate (cont)

The Simple Regression Model

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