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MULTIPLE REGRESSION ANALYSIS: SPECIFICATION AND DATA ISSUES Chapter 9 1

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I. Introduction 2 Failure of zero conditional mean assumption Correlation between error, u, and one or more explanatory variables. Why variables can be endogenous Possible remedies Functional Form Misspecification If omitted variable is a function of an explanatory variable in the model, the model suffers from functional for misspecification Using proxy variables to address omitted variable bias Measurement error Not all variables are measured accurately.

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II. Functional Form 3 Regression model can suffer from misspecification when it doesn’t account for relationship between dependent and explanatory variables. wage = 0 + 1 educ + 2 exper + u Omit exper 2 or exper*educ Omitting variable can lead to biased estimates of all regressors Use wage rather than log(wage) (latter satisfies GM) using wrong variable to relate LHS and RHS can lead to biased estimates of all regressors.

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II. Functional Form 4 We can change linear relationship by: using logs on RHS, LHS or both using quadratic forms of x’s Using interactions of x’s How do we know if we’ve gotten the right functional form for our model? Use F-test for joint exclusion restrictions to detect misspecification

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II. Functional Form Ex: Model of Crime Quadratics or not? Each of sq terms is individually and jointly signficant (F=31.37, df=3; 2,713 Adding squares makes interpretation more difficult: Before, intuitive (–) sign on pcnv suggested conviction rate has deterrence on crime. Now, level is positive, quadratic is negative: for low levels conviction has no deterrent effect, only effective for large levels. Note: Don’t square qemp86, because it’s a discrete variable taking only few values. 5

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II. Functional Form 6 How do you know what to try? Use economic theory to guide you Think about the interpretation Does it make more sense for x to affect y in percentage (use logs) or absolute terms? Does it make more sense for the derivative of x 1 to vary with x 1 (quadratic) or with x 2 (interactions) or to be fixed?

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II. Ramsey’s RESET 7 Know how to test joint exclusion restrictions for higher order terms or interactions. Can be tedious to add and test extra terms May find a square term matters when really using logs would be even better A test of functional form is Ramsey’s regression specification error test (RESET) Intuition: If specification okay, no nonlinear functions of the independent variables should be significant when put in original equation. Cost: Degrees of freedom

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II. Ramsey’s RESET 8 RESET relies on a trick similar to the special form of the White test Instead of adding functions of the x’s directly, we add and test functions of ŷ y = 0 + 1 x 1 + … + k x k + 1 ŷ 2 + 1 ŷ 3 +error Don’t look at above for parameter estimates, just to test inclusion of extra terms H 0 : 1 = 0, 2 = 0 using F~F 2,n-k-3 Significant F-stat suggests there’s some sort of functional for problem

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II. Ramsey’s RESET 9 Ex: Housing Price Equation (n=88) price = 0 + 1 lotsize + 2 sqrft + 3 bdrms +u RESET statistic (up to yhat 3 )=4.67 F 2,82 and p-value.012 Evidence of functional form misspecification lprice = 0 + 1 llotsize + 2 lsqrft + 3 bdrms +u RESET statistic (up to yhat 3 )=2.56 F 2,82 and p-value.84. No evidence of functional form misspecification On basis of RESET, log equation is preferred. But just because loq equation “passed” RESET, does that mean it’s the right specification? Should still use economic theory to determine if functional form makes sense.

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III. Proxy Variables 10 Previously, assumed could resolve functional form misspecification because you had the relevant data. What if model is misspecified because no data is available on an important x variable? Log(wage) = 0 + 1 educ + 2 exper + 3 abil + u Would like to hold ability fixed, but have no measure of it. Exclusion causes parameter estimates to be biased. Potential solution: Obtain proxy variable for omitted variable

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III. Proxy Variables 11 A proxy variable is something that is related to the unobserved variable that we’d like to control for in our analysis-but can’t. Ex: IQ as proxy for ability x 3 * = 0 + 3 x 3 + v 3, where * implies unobserved v 3 signals that x 3 and x 3 * are not directly related 0 allows different scales to be compared (i.e. IQ scale may not be how ability measured) just substitute x 3 for x 3 * in y= 0 + 1 x 1 + 2 x 2 + 3 x 3 * + u

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III. Proxy Variables 12 What do we need for this solution to give us unbiased estimates of 1 and 2 ? Need assumptions on u and v 3 1.) u uncorrelated with x 1, x 2, x 3 * (standard) Also suggests u uncorrelated with x 3… once x 1, x 2, x 3 * included, x 3 is irrelevant (i.e. x 3 doesn’t directly affect y other than through x 3 *) 2.) v 3 is uncorrelated with x 1, x 2, x 3. For v 3 to be uncorrelated with x 1, x 2 that means x 3 * must be good proxy for x 3 Formally, this means E(x 3 * | x 1, x 2, x 3 ) = E(x 3 * | x 3 ) = 0 + 3 x 3 Once x 3 controlled for, x 3 * does not depend on x 1, x 2

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III. Proxy Variables 13 E(abil|educ,exper,IQ)=E(abil|IQ)= 0 + 3 IQ Implies ability only changes with IQ, and not with educ and epxer (once include IQ). So are really running: y = ( 0 + 3 0 ) + 1 x 1 + 2 x 2 + 3 3 x 3 + (u + 3 v 3 ) redefined intercept, error term, x 3 coefficient Can rewrite as: y = 0 + 1 x 1 + 2 x 2 + 3 x 3 + e Unbiased estimates of 0, 1 = 1 2 = 2, 3 Won’t get original 0 or 3.

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III. Proxy Variables IQ as proxy for ability Want to estimate return to education 6.5% when run regression w/o ability proxy 5.4% when include IQ Interact educ*IQ, allows for possibility that returns to education differ across different ability levels. See that interaction not significant though. 14

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III. Proxy Variables 15 Proxy variable can still lead to bias if assumptions are not satisfied Say x 3 * = 0 + 1 x 1 + 2 x 2 + 3 x 3 + v 3 (violation) Then running: y = ( 0 + 3 0 ) + ( 1 + 3 1 ) x 1 + ( 2 + 3 2 ) x 2 + 3 3 x 3 + (u + 3 v 3 ) Bias will depend on signs of 3 and j Can safely assume 1 >0 and 3 >0, so that return to education is upward biased even when using proxy variable. This bias may be smaller than omitted variable bias, though (if x 3 * and x 1 correlated less than x 3 and x 1 )

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III. Lagged Dependent Variables 16 What if there are unobserved variables, and you can’t find reasonable proxy variables? Can include a lagged dependent variable to account for omitted variables that contribute to both past and current levels of y must think past and current y are related for this to make sense allows you to account for historical factors that cause current differences in dependent variables

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III. Lagged Dependent Variables Ex: Model of Crime: Effect of expenditure on crime crime= 0 + 1 unem + 2 expend +u Concerned that cities which have lots of crime react by spending more on crime…biased estimates Coeff on unem and expend are not intuitive crime= 0 + 1 unem + 2 expend+ 3 crime -1 + u Lagged value controls for fact that cities with high historical crime rates may spend more on crime prevention Coefficient estimates now more intuitive 17

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IV. Properties of OLS under Measurement Error 18 Sometimes we have the variable we want, but we think it is measured with error how many hours did you work last year, how many weeks you used child care when your child was young When use imprecise measure of variable in our regression, then model contains measurement error. Consequences of M.E. Model is similar to that of omitted variable bias Often variable with measurement error is the one we’re interested in measuring There are some conditions under which we still get unbiased results Measurement error in y different from measurement error in x

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IV. Measurement Error in a Dependent Variable 19 Let y* denote variable we’d like to explain, like annual savings. Model: y* = 0 + 1 x 1 + …+ k x k + u Most often, respondents are not perfect in their reporting, and so reported savings is denoted y Define measurement error as observed-actual: e 0 = y – y* Thus, really estimating: y = 0 + 1 x 1 + …+ k x k + u + e 0

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IV. Measurement Error in a Dependent Variable 20 When will OLS produce unbiased results? Have assumed u has zero mean and that x j and u are uncorrelated Need to assume e 0 also has zero mean (otherwise just biases 0 ) but more importantly e 0 and x j are uncorrelated. That is, the measurement error in y is statistically independent of each explanatory variable. As result, estimates are unbiased. Generally find Var(u+ e 0 )= u 2 + e0 2 > u 2 When have m.e. in LHS variable, get larger variances for OLS estimators.

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IV. Measurement Error in a Dependent Variable Savings Function sav* = 0 + 1 inc + 2 size+ 3 educ+ 4 age + u e 0= sav-sav* Is m.e. correlated with RHS variables? May think families with higher incomes or more education more likely to report savings accurately. Never know if that’s true, so assume there is no systematic relationship: i.e. wealthy or more educated just as likely to mis-report as non-wealthy, uneducated Scrap Rates Log(scrap*) = 0 + 1 grant + u Error assumed to be multiplicative: y=(y*)*a 0 where e 0 =log(a 0 ) log(scrap)=log(scrap*)+e 0 Log(scrap) = 0 + 1 grant + u + e 0 It’s possible that measurement error more likely to at firms that receive grant underreport scrap rate to make grant look more effective-so get more in future. Can’t verify whether true, so assume no relationship: i.e. measurement error not correlated with grant. 21

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IV. Measurement Error in an Explanatory Variable 22 More complicated when measurement error occurs in the explanatory variable(s) Model: y = 0 + 1 x 1 * + u x 1 * is not observed, instead only observe x 1 define m.e. as e 1 =observed-actual = x 1 – x 1 * Assume E(e 1 ) = 0 (not strong assumption) E(y| x 1 *, x 1 ) = E(y| x 1 *)…means x 1 doesn’t affect y after control for x 1 *…means u uncorrelated with x 1 and x 1 *….similar to proxy variable assumption. Now are estimating y = 0 + 1 x 1 + (u – 1 e 1 )

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IV. Measurement Error in an Explanatory Variable 23 What kind of results will OLS give us? depends on our assumption about the correlation between e 1 and x 1 Suppose Cov(x 1, e 1 ) = 0 OLS remains unbiased Variances larger ( since Var(u- 1 e 1 )= u 2 + 1 2 e1 2 ) Assumption that Cov(x 1, e 1 ) is analogous to the proxy variable assumption.

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IV. Measurement Error in an Explanatory Variable What if that’s not the case? Suppose only that Cov(x 1 *, e 1 ) = 0 Called classical errors-in-variables assumption More realistic assumption than assuming Cov(x 1, e 1 ) =0 This means: Cov(x 1, e 1 ) = E(x 1 e 1 )-E(x 1 )E(e 1 ) =E[(x 1 * +e 1 )(e 1 )]= E(x 1 * e 1 ) + E(e 1 2 ) = 0 + e 2 ≠0. This means x 1 is correlated with the error so estimate is biased and inconsistent 24

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IV. Measurement Error in an Explanatory Variable Economics 20 - Prof. Anderson 25 Notice that the multiplicative portion Var(x 1 *)/Var(x 1 )< 1 Means the estimate is biased toward zero – called attenuation bias True regardless of if 1 is (+) or (-) Larger Var(x 1 *)/Var(x 1 ) suggests inconsistency with OLS is small, because variation in “noise” (a.k.a. m.e.) is small relative to variation in true value. It’s more complicated with a multiple regression, but can still expect attenuation bias when assume classical errors in variables.

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IV. Measurement Error in an Explanatory Variable Economics 20 - Prof. Anderson 26 y = 0 + 1 x* 1 + 2 x 2 + 3 x 3 +u Assume u uncorrelated with x* 1, x 1, x 2, x 3 If assume e 1 uncorrelated with x 1, x 2, x 3 then get y = 0 + 1 x 1 + 2 x 2 + 3 x 3 +u - 1 e 1 get consistent estimates But, if e 1 uncorrelated with x 2, x 3 but not necessarily x 1, get If x* 1 uncorrelated with x 2, x 3 get consistent estimates of 2, 3 If this doesn’t hold, then other estimates will be inconsistent (size and direction are indeterminate)

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IV. Measurement Error in an Explanatory Variable Economics 20 - Prof. Anderson 27 Ex: GPA with measurement error colGPA = 0 + 1 faminc* + 2 hsGPA+ 3 SAT + 4 smoke + u faminc* is actual annual family income faminc=faminc*+ e 1 Assuming CEV holds, get OLS estimator of 1 that is attenuated (biased toward zero). colGPA = 0 + 1 faminc + 2 hsGPA+ 3 SAT + 4 smoke* + u smoke=smoke*+ e 1 CEV unlikely to hold, because those who don’t smoke are really unlikely to mis-report. Those that do smoke can mis-report, such that error and actual number of times smoked (smoked*) are correlated. Deriving the implications of measurement error when CEV doesn’t hold is difficult and out of scope of text.

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V. Missing Data, Nonrandom Samples, Outlying Observations Economics 20 - Prof. Anderson 28 Introduction into data problems that can violated MLR.2 of G-M assumptions Cases when data problems have no effect on OLS estimates Other cases when get biased estimates Missing Data Generally collect data from random sample of observations (people, schools, firms) Discover that information from these observations on key variables are missing

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V. Missing Data – Is it a Problem? Economics 20 - Prof. Anderson 29 Consequences If any observation is missing data on one of the variables in the model, it can’t be used Data missing at Random If data is missing at random, using a sample restricted to observations with no missing values will be fine Simply reduces sample size, thus reducing precision of estimates

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V. Missing Data – Is it a Problem? Economics 20 - Prof. Anderson 30 Data not missing at random A problem can arise if the data is missing systematically High income individuals refuse to provide income data Low education people generally don’t report education People with high IQ more likely to report IQ When missing data does not lead to bias Sample chosen on basis of independent variables Ex: Savings, income, age, size for population of people 35 years and older No bias because E(savings|income, age, size) is same for any subset of population described by income, age, size in this data.

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V. Nonrandom Samples Economics 20 - Prof. Anderson 31 When missing data leads to bias If the sample is chosen on the basis of the y variable, then we have sample selection bias Ex: estimating wealth based on education, experience, and age. Only those with wealth below 250k included OLS gives biased estimates because E(wealth|educ, exper, age) not same as expected value conditional on wealth being less than 250k.

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V. Outliers /Influential Observations Economics 20 - Prof. Anderson 32 Sometimes an individual observation can be very different from the others “Influential” for estimates if dropping that observation(s) from the analysis changes the key OLS estimates by a lot Particularly important with small data sets OLS susceptible to outliers because by definition, minimizes sum of squared residual, and this outlier will have “large” residual. Causes of outliers errors in data entry – one reason why looking at summary statistics is important sometimes the observation will just truly be very different from the others

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V. Outliers /Influential Observations Example: R& D Intensity & Firm Size Sales more than triples, and now statistically significant. 33 Economics 20 - Prof. Anderson

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V. Outliers Economics 20 - Prof. Anderson 34 Not unreasonable to fix observations where it’s clear there was just an extra zero entered or left off, etc. Not unreasonable to drop observations that appear to be extreme outliers, although readers may prefer to see estimates with and without the outliers Can use Stata to investigate outliers graphicall

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