Issues large sample test low power especially if

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Unformatted text preview: l samples weak instrument if not highly correlated with regressor X if only weak instruments, you may want to trade-off some bias for lower variance and stick with OLS anyways (24) (25) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Detection Durbin-Wu-Hausman Specification Test H0 : OLS and IV estimators consistent H1 : Only IV consistent Test statistic based on differences between OLS and IV estimators. Reject H0 when test statistic large. Note: If cannot reject H0 , use OLS since more efficient. Intuition: if both consistent, difference between them small, test statistic small and do not reject H0 . Issues: large sample test & low power (especially if instruments weak) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Definition Simultaneous Equations Estimation System of equation determines set of variables jointly. Example: Structural Form: p = β1 + β2 w + u p (26) w = α1 + α2 p + α3 U + uw (27) Endogenous variables (p,w): “whose values are determined by the interaction ... in the model” Exogenous variables (U): “whose values are determined externally” Reduced form: “expressing the endogenous variables in terms of the exogenous variables and disturbance terms” β1 + α1 β2 + α3 β2 U + up + β2 uw 1 − α2 β2 α1 + α2 β1 + α3 U + uw + α2 up w= 1 − α 2 β2 p= (28) (29) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Definition Disease 5: Simultaneous Equations Bias Consider estimating β2 through OLS regression p on w . Adapting notation, we can take our expression for plim(b2 ) from before (insert structural form in equation): plim(b2 ) = β2 + cov (w , up ) Var (w ) (30) Can we ignore the fact that w is determined endogenously? Is OLS still consistent? In other words: If w is determined as part of system, does this mean generally cov (w , up ) = 0? Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Consequences SEB Consequences Use reduced form for wi to evaluate covariance: cov (w , up ) = cov (up , α1 + α2 β1 + α3 U + uw + α2 up ) 1 − α2 β2 1 (α3 cov (up , U ) + cov (up , uw ) + α2 cov (up , up )) 1 − α2 β2 α2 = σ2 1 − α2 β2 up 2 σup α2 plim(b2 ) = β2 + 1 − α2 β2 Var (w ) = → OLS inconsistent (31) (32) (33) (34) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Remedy & Detection SEB Remedy & Detection In a way similar problem to measurement error → similar solution Use IV: We need variable that is correlated with w , not correlated with up and is not itself a regressor in the p-equation p = β1 + β2 w + u p (35) w = α1 + α2 p + α3 U + uw (36) Use U as instrument! Under-identification, exact identification, over-identification of equation (in EC220) depends roughly on number of exogenous instruments available Detection of simultaneous equation bias: use DWH test Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Remedy & Detection Simultaneous Equations & TSLS Equation may be over-identified and we have more than one instrument (assume above besides U, we have x) Use two-stage least squares (TSLS) to make efficient use of all instruments Procedure: 1 ˆ Regress w on U and x → save fitted values w 2 TSLS estimator TSLS b2 = n ¯ ˆ ˆ i =1 (wi − w )(pi n ¯ )(wi ˆ ˆ i =1 (wi − w ¯ − p) ¯ − w) ˆ Intuition: w maximizes correl...
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