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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.
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:
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
1 − α 2 β2
(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
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
1 − α2 β2 up
plim(b2 ) = β2 +
1 − α2 β2 Var (w )
= → OLS inconsistent (31)
(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
Regress w on U and x → save ﬁtted values w 2 TSLS estimator
i =1 (wi − w )(pi
i =1 (wi − w ¯
− w) ˆ
Intuition: w maximizes correl...
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- Spring '13