Smaller system that determines x and y one way to see

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Unformatted text preview: ation of instrument to w , so estimation more precise (37) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question 2005 Question 5 Past Exam 2005 Question 5 Model: Yt = α1 + α2 Xt + ut (38) Xt = β1 + β2 Yt + vt (39) Zt = γ1 + γ2 Yt + γ3 Xt + γ4 Qt + wt (40) u,v,w have fixed distribution, independent of Q and each other, not autocorrelated Parameters positive & α2 β2 < 1 Introduction Measurement Error Simultaneous Equations 2005 Question 5 5)a)i) Endogenous & Exogenous Variables Determined within system (endogenous): Xt , Yt , Zt Determined outside the system (exogenous): Qt Past Exam Practice Question Introduction Measurement Error Simultaneous Equations Past Exam Practice Question 2005 Question 5 5)a)ii) Simultaneous Equations Bias Equations 1 & 2 form smaller system that determines X and Y. One way to see this: Can determine Y and X reduced forms just from first two equations β1 + α1 β2 + vt + β2 ut 1 − α2 β2 α1 + α2 β1 + ut + α2 vt Yt = 1 − α2 β2 Xt = So here we can ignore third equation and get expression for SEB as in textbook! (41) (42) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question 2005 Question 5 5)a)ii) Simultaneous Equations Bias As always, to determine properties of estimator substitute dependent variable (and this is X for this equation!) b2 = = = n ¯ ¯ t =1 (Xt − X )(Yt − Y ) n ¯ )2 t =1 (Yt − Y n ¯ ¯ t =1 (β2 (Yt − Y ) + (vt − v ))(Yt n ¯ )2 t =1 (Yt − Y n ¯ ¯ (vt − v )(Yt − Y ) β2 + t =1 n ¯2 t =1 (Yt − Y ) (43) ¯ − Y) (44) (45) (46) Can’t get closed form expressions for expectations, so look at plim: plim(b2 ) = β2 + n ¯ ¯ t =1 (vt − v )(Yt − Y )) n 1 ¯ plim( n t =1 (Yt − Y )2 ) 1 plim( n (47) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question 2005 Question 5 5)a)ii) Simultaneous Equations Bias plim(b2 ) = β2 + Cov (v , Y ) Var (Y ) (48) As above, use RF for Y to get covariance expression: plim(b2 ) = β2 + To sign “bias”: variances positive α2 > 0 1 − α2 β2 > 0 → “bias” positive 2 α2 σv 2 1 − α2 β2 σY (49) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question 2005 Question 5 5)a)iii) OLS on third equation consistent? Key: is there correlation between regressors and disturbance term w in third equation? Y and X reduced forms depend only on u and v (not w) β1 + α1 β2 + vt + β2 ut 1 − α2 β2 α1 + α2 β1 + ut + α2 vt Yt = 1 − α2 β2 Xt = u and v not correlated with w (by assumption). So Cov (Y , w ) = Cov (X , w ) = 0. Qt exogenous, uncorrelated with disturbance terms. So OLS consistent! (50) (51) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question 2005 Question 5 5)a)iv) Is Qt valid instrument to fit second SF equation? Consider if it satisfies conditions for valid instrument: 1 Qt not itself regressor → Yes! 2 cov (Q , u ) = 0 → Yes! 3 Correlated with regressor Yt → No! cov (Q , Y ) = cov (Q...
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