Econometrics - Instrumental Variables

EconometricsChengyuan YinSchool ofMathematics

Econometrics Chengyuan Yin School of Mathematics

Econometrics13.Instrumental Variables

Econometrics 13. Instrumental Variables

InstrumentalVariablesFramework:y = xβ + ε, K variables in X.口口There exists a set of K variables, z such thatplim(z'X/n) + 0 but plim(Z's/n) = 0The variables in z are called instrumental variables.An alternative (to least squares)estimator of β is口biv = (z'X)-iZ'yWe consider thefollowing:口Why usethisestimator?What are its properties compared to least squares?We will also examine an important application口

Instrumental Variables  Framework: y = X + , K variables in X.  There exists a set of K variables, Z such that plim(Z’X/n)  0 but plim(Z’/n) = 0 The variables in Z are called instrumental variables.  An alternative (to least squares) estimator of  is bIV = (Z’X) -1Z’y  We consider the following: ◼ Why use this estimator? ◼ What are its properties compared to least squares?  We will also examine an important application

IVEstimatorsConsistentbrv = (z'X)-1z'y= (z'X/n)-1 (z'X/n)β+ (Z'X/n)-1Z'ε/n= β+ (Z'X/n)-1Z'e/n > βAsymptotically normal (same approach to proof asfor OLS)Inefficient - to be shown

IV Estimators Consistent bIV = (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 → β Asymptotically normal (same approach to proof as for OLS) Inefficient – to be shown

LSasanIVEstimatorThe least squares estimator is(X x)-1X'y = ((X X)-1Z;X;Yi= β + (XX)-12;X;8If plim(X'X/n) = Q nonzeroplim(X'e/n) = 0Under the usual assumptions LS is an IV estimatorX is its own instrument

LS as an IV Estimator The least squares estimator is (X X) -1Xy = (X X) -1ixiyi =  + (X X) -1ixiεi If plim(X’X/n) = Q nonzero plim(X’ε/n) = 0 Under the usual assumptions LS is an IV estimator X is its own instrument