Prof. Dan Levy Harvard University API-209

1 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University API-209 pick up handout #18 take your seat as soon as possible put your name card on your desk you don t want to appear on the film, please sit in the left wing of the room (facing the front) Poll 1 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University API-209 Final Exercise posted o3 options: Targeting Assessment Sierra Leone Poverty Profile and Targeting Honduras Currency Unions oStart early oTeamwork Problem Set 8 oQuestion on Final Exercise 2 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University HANDOUT 18 Multiple regression III Various Topics of Fit Standard Error of OLS Estimators Source : Wooldridge (Ch 3), Hughes-Hallett (Math camp handouts) Today we study 2 broad topics related to estimation in the context of multiple regression : oGoodness of fit (the famous R 2 ) oVariance of OLS estimators OF FIT Consider the following terms: 3 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof.

2 Dan LevyHarvard University Total Sum of Squares = SS(Y)T= i Y2 Explained sum of squares=SS(Y)E= i Y2 Residual sum of squares=SSuR= i2 It turns out that TSS=ESS+RSS. (See Wooldridge for proof) 4 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University The R-squared is defined to be R2=ESSTSS R2= (Y Y) i2 (Y Y)i2=1 RSSTSS=1 u i2 (Y Y)i2 By definition R 2 is a number between zero and one (because TSS = ESS + RSS, ESS 0 and RSS 0). 5 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University Interpretation of R 2 : proportion of the sample variation in y that is explained by the OLS regression line. R 2 can also be shown to equal the squared correlation coefficient between the actual Yi and the fitted values.

3 This is where theY i term R-squared comes from. 6 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University Example Smoking and Lung Cancer . regress lcd cigs, robust regression with robust standard errors Number of obs = 5 F( 1, 3) = Prob > F = R-squared = Root MSE = ---------------------------------------- -------------------------------------- | Robust lcd | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+-------------------------- -------------------------------------- cigs |.

4 3445158 .072487 .1138297 .5752019 _cons | ---------------------------------------- -------------------------------------- QUESTION: How do we interpret the R 2 in this particular example? 7 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University QUESTION : What happens to R 2 when an explanatory variable is added to a regression ? must increase increases or stays the same must decrease decreases or stays the same enough information provided 8 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University Adjusted R 2 : Penalizes you for using irrelevant explanatory variables R 2 provides a measure of how well the OLS line fits the data oAn R 2 =1 means all the points lie on the same line, OLS provides a perfect fit to the data oAn R 2 close to zero means a poor fit of the OLS line 9 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof.

5 Dan LevyHarvard University QUESTION : The larger the R 2 , the lower the likelihood that our regression suffers from omitted variable bias (OVB) don t know 10 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University STANDARD ERROR OF OLS ESTIMATORS Idea : The discussion of unbiasedness gives us an assessment of the central tendencies of . j Now we would like to have a measure of the spread in the sampling distribution of . j Key idea : All else equal, we would like an estimator of that has a low standard error. j Why? 11 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University We first add an assumption to our model called homoskedasticity . We do so for two reasons: (1)The formulas for the standard error of j are simplified, which allows us to develop more easily the intuition behind the determinants of the standard error (2)OLS has important efficiency properties under the homoskedasticity assumption (see below) 12 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof.

6 Dan LevyHarvard University ASSUMPTION [HOMOSKEDASTICITY] ar[u|X,,,] 2 If this assumption fails, then the model exhibits heteroskedasticity. See Appendix #3 for details. Assumptions through are collectively known as the Gauss-Markov assumptions (for cross-sectional regression ) 13 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University Efficiency of OLS: The Gauss-Markov Theorem Under assumptions through , are the Best Linear Unbiased, ,.., 0 1 k Estimators (BLUEs) of , ,.., 0 1 k respectively. B est : lowest variance L inear : Can be expressed as a linear function of the data on the dependent variable U nbiased : ( )E j= j E stimator : Rule/Method/Formula that can be applied to any sample to produce an estimate Key idea : The importance of the Gauss-Markov Theorem is that, when the standard set of assumptions holds, we need not look for alternative linear unbiased estimators: none will be better than OLS.

7 Terminology For the purposes of the next section, it will be helpful to think about various R 2 s, which we define here. Consider the following regression : 14 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University XXXY= 0+ 11+ 22+ 33+u The following R 2 s can be defined: Name R 2 computed from the following regression : R2 XXX Y= 0+ 11+ 22+ 33+u R21 XX X1= 0+ 12+ 23+v R22 XX X2= 0+ 11+ 23+ R23 XX X3= 0+ 11+ 22+ More generally, is the R-squared fromRj2 regressing on all other explanatoryXj variables (and including an intercept). 15 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University QUESTION: When would you expect to be Rj2 large? 16 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof.

8 Dan LevyHarvard University THEOREM [Sampling variances of the OLS slope estimators] Under assumptions through , conditional on the sample values of the explanatory variables, td. Error( ) ( ) S j= 2 TSS(1 R)jj2 for j=1,2,..,k , where isSSTj= ni=1(X)ij Xj2 the total sample variation in , and is theXjRj2 R-squared from regressing on all otherXj explanatory variables (and including an intercept). Note: The proof of theorem can be found in Wooldridge. 17 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University FORMULA FOR STANDARD ERROR TD. ERROR( ) S J= 2 TSS(1 R)JJ2 EXAMPLE Determinant of Standard Error Analysis (1) The variance of the error term () 2 (2) The Total Sample Variation in (): XjSS Tj SSTj= ni=1(X)ij Xj2 (3) The Linear Relationships Among the Explanatory Variables ()Rj2 18 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof.

9 Dan LevyHarvard University THE COMPONENTS OF THE STANDARD ERROR OF OLS ESTIMATORS Eq. ( ) shows that the standard error of j depends on three factors: , , and . 2 SSTjRj2 We now consider each of these factors separately. (1) The variance of the error term () 2 Key : is a feature of the population; it has 2 nothing to do with sample size. 19 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University (2) The Total Sample Variation in ():XjSSTj SSTj= ni=1(X)ij Xj2 Everything else equal, for estimating , we j prefer to have as much variation in as Xj possible. When sampling randomly from the population, increases with sample 20 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University (3) The Linear Relationships Among the Explanatory Variables ()Rj2 It is important to see that this R-squared is distinct from the R-squared in the regression of Y on X 1 , X 2.

10 X k . Extreme cases : Rj2=0[smallest Var () for a given j 2 and ]SSTj Rj2=1 [violates assumption ] Key case : When is close to 1, Rj2ar( )V j might become too large. High (but not perfect) correlation between two or more of the independent variables is called multicollinearity . 21 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University Key idea #1 : Worrying about high degrees of correlation among the independent variables in the sample is really no different from worrying about a small sample size: both work to increase .ar( )V j Example : Estimating the effect of school expenditure categories on student performance. 22 advanced Quantitative methods ( API-209 ) Harvard Kennedy School Prof. Dan LevyHarvard University Key idea #2 : A high degree of correlation between certain explanatory variables can be irrelevant as to how well we can estimate other parameters in the model.