斯托克、沃森着《计量经济学》第六章

Chapter 6. Linear Regression with Multiple Regressors 6.1 Omitted Variable Bias（遗漏变量偏差）OLS estimate of the Test Score/STR relation:nTestScore= 698.9 – 2.28×STR, R2 = .05, SER = 18.6(10.4) (0.52)Is this a credible estimate of the causal effect on test scores of a change in the student-teacher ratio?1No: there are omitted confounding factors (family income; whether the students are native English speakers) that bias the OLS estimator: STR could be “picking up” the effect of these confounding factors.2Omitted Variable BiasThe bias in the OLS estimator that occurs as a result of an omitted factor is called omitted variable bias. For omitted variable bias to occur, the omitted factor “Z” must be:1.a determinant of Y; and2.correlated with the regressor X.3Both conditions must hold for the omission of Z to result in omitted variable bias.Example #1: In the test score example:1.English language ability (whether the student hasEnglish as a second language) plausibly affectsstandardized test scores: Z is a determinant of Y.42.Immigrant communities tend to be less affluent and thushave smaller school budgets – and higher STR: Z iscorrelated with X.ˆβ is biased•Accordingly,1•What is the direction of this bias?•What does common sense suggest?•If common sense fails you, there is a formula…5Example #2: Time of day of the test.Time of day of the test could affect test scores, but is uncorrelated with the student-teacher ratio (STR).Example #3: Parking lot space per pupil.Schools with more teachers per pupil probably have more teacher parking space, but parking lot space has no direct6effect on learning (learning takes place in the classroom, not parking lot).Example #4: The Mozart EffectA study published in Nature in 1993 suggested that listening to Mozart for 10-15 minutes could temporarily raise your IQ by 8 or 9 points.A review of dozens of studies found that students who take7optional music or arts courses in high school do in fact have higher English or math test scores than those who don’t.Problem: The academically better students might have more time to take optional music courses or more interest in doing so, or those schools with a deeper music curriculum might just be better schools across the board.Evidence from randomized controlled experiments: Many89 controlled experiments on the Mozart effect fail to show that listening to Mozart improves IQ or general test performance.A formula for omitted variable bias : Recall the equation,1ˆβ – β1 = 121()()n i i i n i i X X u X X ==−−∑∑ =1211nii Xv n n s n =−⎛⎞⎜⎟⎝⎠∑where v i = (X i –X )u i ≅ (X i – μX )u i .10 Under Least Squares Assumption #1,E[(X i – μX )u i ] = cov(X i , u i ) = 0.But what if E[(X i – μX )u i ] = cov(X i , u i ) = σXu ≠ 0? Then1ˆβ = β1 + 121()()n i i i n i i X X u X X ==−−∑∑ = β1 +121()1n ii i XX X u n n s n =−−⎛⎞⎜⎟⎝⎠∑11 其中，22p X Xs σ⎯⎯→，11n n−→， []11()E ()(0)cov(,)n p i i i X i i i Xu X u i X X u X u X u n μρσσ=−⎯⎯→−−==∑，where ρXu = corr(X , u ) （X 与u 的相关系数）因此，有如下Omitted variable bias formula:1ˆβp ⎯⎯→ β1 + u Xu X σρσ⎛⎞⎜⎟⎝⎠If an omitted factor Z is both :(1) a determinant of Y (that is, it is contained in u); and(2) correlated with X,ˆβ is biased.then ρXu≠ 0 and the OLS estimator1The math makes precise the idea that districts with few ESL students (1) do better on standardized tests and (2) have smaller classes (bigger budgets), so ignoring the ESL factor results in overstating the class size effect.12Is this actually going on in the CA data?13•Districts with fewer English Learners have higher test scores•Districts with lower percent EL (PctEL) have smaller classes•Among districts with comparable PctEL, the effect of class size is small (recall overall “test score gap” = 7.4)14Three ways to overcome omitted variable bias：1.Run a randomized controlled experiment in whichtreatment (STR) is randomly assigned: then PctEL is stilla determinant of TestScore, but PctEL is uncorrelated withSTR. (But this is unrealistic in practice.)2.Adopt the “cross tabulation” approach, with finergradations of STR and PctEL (But soon we will run out of15data, and what about other determinants like familyincome and parental education?)e a method in which the omitted variable (PctEL) is nolonger omitted: include PctEL as an additional regressor ina multiple regression.166.2 The Multiple Regression Model（多元回归模型）Consider the case of two regressors:Y i = β0 + β1X1i + β2X2i + u i, i = 1,…,n•X1, X2 are the two independent variables (regressors) •(Y i, X1i, X2i) denote the i th observation on Y, X1, and X2. •β0 = unknown population intercept17•β1 = effect on Y of a change in X1, holding X2 constant•β2 = effect on Y of a change in X2, holding X1 constant•u i = “error term” (omitted factors)Interpretation of multiple regression coefficientsY i = β0 + β1X1i + β2X2i + u i, i = 1,…,n Consider changing X1 by ΔX1 while holding X2 constant:18Population regression line before the change:Y = β0 + β1X1 + β2X2Population regression line, after the change:Y + ΔY = β0 + β1(X1 + ΔX1) + β2X21920 Before : Y = β0 + β1(X 1 + ΔX 1) + β2X 2After : Y + ΔY = β0 + β1(X 1 + ΔX 1) + β2X 2Difference : ΔY = β1ΔX 1 That is, β1 = 1Y X ΔΔ, holding X 2 constant also,21β2 = 2YX ΔΔ, holding X 1 constantandβ0 = predicted value of Y when X 1 = X 2 = 0.22With two regressors, the OLS estimator solves:[]01221122,,1min()nii i b b b i Y bb X b X =−++∑• The OLS estimator minimizes the average squared difference between the actual values of Y i and the prediction (predicted value) based on the estimated line.•This minimization problem is solved using calculus23•The result is the OLS estimators of β0, β1, and β2.更一般的多元回归模型：Y i = β0 + β1X1i + β2X2i + …+βk X ki+ u i, i = 1,…,n如果包括常数项，则共有k + 1个解释变量。