Econometrics-I-13 计量经济分析（第六版英文）ppt教学教案.pptx

Applied EconometricsWilliam Greene Department of Economics Stern School of Business Applied Econometrics13. Instrumental Variables Instrumental VariablesFramework: y = X + , K variables in X. There exists a set of K variables, Z such thatplim(Z’X/n) 0 but plim(Z’ /n) = 0The variables in Z are called instrumental variables. An alternative (to least squares) estimator of isbIV= (Z’X)-1Z’yWe consider the following:Why use this estimator?What are its properties compared to least squares?We will also examine an important application IV EstimatorsConsistentbIV = (Z’X)-1Z’y= (Z’X/n)-1 (Z’X/n)β+ (Z’X/n)-1Z’ε/n= β+ (Z’X/n)-1Z’ε/n βAsymptotically normal (same approach to proof as for OLS)Inefficient – to be shown. LS as an IV EstimatorThe least squares estimator is(X X)-1X y = (X X)-1ixiyi= + (X X)-1ixiεiIf plim(X’X/n) = Q nonzero plim(X’ε/n) = 0Under the usual assumptions LS is an IV estimator X is its own instrument. The General ResultBy construction, the IV estimator is consistent. So, we have an estimator that is consistent when least squares is not. Asymptotic Covariance Matrix of bIV Asymptotic EfficiencyAsymptotic efficiency of the IV estimator. The variance is larger than that of LS. (A large sample type of Gauss-Markov result is at work.)It’s a moot point. LS is inconsistent.Mean squared error is uncertain:MSE[estimator|β]=Variance + square of bias. IV may be better or worse. Depends on the data Two Stage Least SquaresHow to use an “excess” of instrumental variables X is K variables. Some (at least one) of the K variables in X are correlated with ε. Z is M > K variables. Some of the variables in Z are also in X, some are not. None of the variables in Z are correlated with ε. Which K variables to use to compute Z’X and Z’y? Choosing the InstrumentsChoose K randomly?Choose the included Xs and the remainder randomly? Use all of them? How?A theorem: (Brundy and Jorgenson, ca. 1972) There isa most efficient way to construct the IV estimator from this subset: (1) For each