# 250 15 introduction measurement error simultaneous

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Unformatted text preview: (Z , v ) − β2 Cov (Z , w ) + Cov (w , v ) − β2 Cov (w , w ) = −β2 Cov (w , w ) = 2 −β2 σw (9) (10) (11) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Consequences Measurement Error Consequences Variance of regressor with measurement error: Var (X ) = Var (Z + w ) (12) = Var (Z ) + Var (w ) + 2Cov (Z , w ) (13) 2 2 = σZ + σw (14) Probability limit of OLS estimator b2 : plim(b2 ) = β2 + cov (X , u ) σ2 = β2 − β2 2 w 2 Var (X ) σZ + σw → OLS estimator b2 inconsistent → Graphical intuition for inconsistency: P.250 (15) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Consequences Measurement Error Consequences To give more insight: 2 σw 2 2 σZ + σw 1 ) = β2 (1 − σ2 Z 2 +1 σ plim(b2 ) = β2 − β2 w Attenuation of OLS estimate towards smaller absolute number 2 2 Effect smaller if σZ larger compared to σw Why? Then relative importance of measurement error smaller (16) (17) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Consequences Measurement Error Consequences Note: s.e.’s invalid, t- and F-statistics invalid Contemporaneous correlation between X and u implies B7 does not hold: unbiasedness proof for OLS with stochastic regressors also does not go through. We made lots of assumptions on the way on covariances - think about variations to the basic textbook case. Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Alternative Case Contrast: Measurement Error In Dependent Variable True model: Qi = β1 + β2 Xi + vi Observe dependent variable with with measurement error: Yi = Qi + ri Yi − ri = β1 + β2 Xi + vi (18) Yi = β1 + β2 Xi + vi + ri (19) Yi = β1 + β2 Xi + ui (20) Consequence: Under textbook assumptions (!), B7 result carries through s.e.’s remain valid But: disturbance variance larger and lower precision of estimators. Discussion along the lines of multicollinearity i.e. OLS still most efﬁcient but only the variance is larger than what it would be if there were no meas. error in Y . Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Remedy Instrumental Variable Regression Suppose we have instrument Ii such that: 1 cov (I , u ) = 0 2 cov (I , X ) = 0 3 I itself is not a regressor. Consistency of IV estimator based on β2 : IV b2 = = n i =1 (Ii n i =1 (Ii n i =1 (Ii = β2 + 1 n 1 n ¯ − ¯)(Yi − Y ) I ¯ − ¯)(Xi − X ) I ¯ ¯ − ¯)(β2 (Xi − X ) + (ui − u )) I n ¯ (Ii − ¯)(Xi − X ) I i =1 n ¯ i =1 (Ii − I )(ui n ¯ i =1 (Ii − I )(Xi ¯ − u) ¯ − X) (21) (22) (23) Introduction Measurement Error Simultaneous Equations Past Exam Practice Question Remedy Instrumental Variable Regression IV estimator consistent despite measurement error in regressor: IV plim(b2 ) = β2 + We can also show: 2 σbIV = 2 Cov (I , u ) = β2 Cov (I , X ) 2 σu n i =1 (Xi − 1 ¯ )2 r 2 X XI Note: we have not shown IV unbiased in smal...
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