Yt 1 1 2 xt 1 ut 1 through equation 1 so yt 1 depends

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Unformatted text preview: , =0 → Clearly not valid instrument α1 + α2 β1 + u + α2 v ) 1 − α2 β2 (52) (53) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question 2005 Question 5 5)a)iv) Is Qt valid instrument to fit second SF equation? What goes wrong in IV consistency proof? IV plim(b2 ) = β2 + Cov (Q , v ) = 0 BUT: Cov (Q , Y ) = 0 as well The limiting value not defined! Cov (Q , v ) Cov (Q , Y ) (54) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question 2005 Question 5 5)b) Change specification for second equation Assume X depends on the lag of Y. The system becomes: Yt = α1 + α2 Xt + ut (55) Xt = β1 + β2 Yt −1 + vt (56) Zt = γ1 + γ2 Yt + γ3 Xt + γ4 Qt + wt (57) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question 2005 Question 5 5)b)i) Fit 2 with OLS and 1 with IV using Yt −1 as instrument for Xt Part 1: Can we fit equation 2 with OLS now? For consistency: Is Yt −1 uncorrelated with vt ? Yt −1 = α1 + α2 Xt −1 + ut −1 through equation 1 so Yt −1 depends only on past disturbance terms so Corr (Yt −1 , vt ) = 0 since no autocorrelation so OLS consistent for equation 2 now! Introduction Measurement Error Simultaneous Equations Past Exam Practice Question 2005 Question 5 5)b)i) Fit 2 with OLS and 1 with IV using Yt −1 as instrument for Xt Part 2: Can we fit equation 1 using Yt −1 as instrument for Xt ? For consistency: Is Yt −1 a valid instrument for Xt ? 1 Correlation instrument with regressor → Yes! 2 No correlation instrument with disturbance term → Yes! Xt = β1 + β2 Yt −1 + vt through equation 2 Since Yt −1 depends only on past realizations of disturbance terms Cov (Yt −1 , ut ) = 0 since no autocorrelation in disturbance terms 3 Instrument not in equation in its own right → Yes! Introduction Measurement Error Simultaneous Equations Past Exam Practice Question 2005 Question 5 5)b)ii) Fit both equations with OLS We have shown OLS estimator equation 2 consistent Is OLS consistent for equation 1? Boils down to: Is Cov (Xt , ut ) = 0? Cov (Xt , ut ) = Cov (β1 + β2 Yt −1 + vt , ut ) (58) = β2 Cov (Yt −1 , ut ) + Cov (vt , ut ) (59) =0 (60) since no correlation between disturbance terms since Cov (Yt −1 , ut ) = 0 so OLS also consistent for equation 1 now! Generally OLS more efficient when both OLS and IV consistent Introduction Time Series and OLS Two Dynamic Models Autocorrelation EC220 Review Lectures Lecture 5 Bonsoo Koo May 10, 2010 Bonsoo Koo EC220 Review Lectures Introduction Time Series and OLS Two Dynamic Models Autocorrelation Today Today: Time Series, Dynamic Models & Autocorrelation OLS Two Dymamic Models Autocorrelation 1 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Time Series and OLS What are the likely problems when estimating the following with OLS? Yt = β1 + β2 Xt + β3 Xt −1 + ut Assume that all the assumptions for model C hold. X could be highly autocorrelated ⇒ problems with multicollinearity i.e. high standard...
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