Panel Data Analysis Part III — Modern Moment Estimation Arellano and Honoré (2000) James J. Heckman University of Chicago Econ 312 This draft, May 8, 2006
� � � � � � �� = � � �� + � � + � �� � = 1 � ��� I � �� = � � + � �� � = 1 � ���� � � �� is strictly exogenous if � � ( � �� | � � � ) = 0 � = ( � � 1 � ���� � �� ) �� OLS identifies �� 1
Panel Data Setting: Strictly exogenous given � � if � � ( � �� | � � � ) = 0 � = 1 � ���� � for all � � � but not necessarily for � �� 2
Consequence: First di � erence eliminates e � ects: � � ( � �� � � ��� � 1 | � � � ) = 0 . Multivariate regression with cross equation restrictions. (This is essentially all the information). Partial Adjustment Model With Strictly Exogenous Variable � �� = � � �� ( � � 1) + � 0 � ��� + � 1 � ��� � 1 + � � + � �� . Assume � � ( � �� | � � � ) = 0 � � = 2 � ���� �� Does not restrict the serial correlation in � �� . 3
Restriction: � � ( � � �� | � � � ) = 0 � Model identified for � � 3 � � = 3 case we acquire orthogonality restrictions � ( � �� ( � � � 3 � � � � � 2 � � 0 � � � 3 � � 1 � � � 2 ) = 0 = � � ( � �� ( � � � 3 )) = 0 � � = 1 � 2 � 3 Use these in GMM to identify model. 3 equations in 3 un- knowns and we acquire exact identification. Note: Strict exogeneity enables us to identify dynamic e � ect of � on � with arbitrary serial correlation in the errors; Price: Assumes � not influenced by past values of � and � . 4
� � � � � Definition: � is predetermined if � � � � � 1 (*) � � ( � �� | � � ) = 0 � � = 2 � ���� � � ) � � � � 1 � � ���� � � � 1 � = ( � 1 = ( � 1 ) . � � ���� � � Current shocks are uncorrelated with past values of � and cur- rent and past values of � . Feedback from lagged dependent variables to future � not ruled out. Eg . Euler equations. (Information set of agents uncorrelated with current and future idiosyncratic shocks but not past shocks). 5
� � � � � � Example: Euler Equation: ¸ � ( � � ) � � � | I � � 1 = 1 � � � 1 ( � � � 1 ) � � Power Utility: � � = ( � � ) 1 � � � 1 1 � � � � = ( � � ) � � "μ � � # ¶ � � � | I � � 1 = 0 � � � 1 6
� � � � � � � � � [ � � � ( � � ) � � � | I � � 1 ] = 0 £ ¤ � | I � � 1 = 0 � � � Instruments that are e � cient: £ ¤ = 0 � � � � 1 � � � � � � 1 is in the information set. Crucial that instruments don’t include variables that cause the innovation. 7
� � � � � � Implication: � � ( � ��� � � ��� � 1 | � � � 1 � � � � 2 ) = 0 , � = 3 � ���� � For � = 3 � we acquire � � 1 0 = � ( � � � 3 � � � � � 2 � � 0 � � � 3 � � 1 � � � 2 ) � � 1 � � 2 This condition is not the same as that in strictly exogenous models: We acquire 3 moments only 2 in common with last (across strictly exogenous and these models). Standard errors are consistent with arbitrary serial correlation. 8
� � � If we ruled out arbitrary serial correlation, by 1 � � � � 1 � � ( � � �� | � � ) = 0 � = 2 � ���� � we acquire superset of all conditions. (*) and previous ones. (*) = � � ( � � �� � � � ��� � � ) = 0 � � 1 because we have that the covariances are zero ��� ( � ��� � � � 1 ) = 0 �� ��� ( � �� � � ��� � 1 ) = 0 generically. 9
Observe that in the predetermined case we can have special cases of serial correlation. e.g. � = 4 � ( � � ��� � � ��� � � ) = 0 � � 2 first order MA . Valid orthogonality conditions: � � �� 3 � � � � �� 2 � � 0 � � �� 3 � � 1 � � �� 2 = � �� 3 � � �� 2 � � �� 4 � � � � �� 3 � � 0 � � �� 4 � � 1 � � �� 3 = � �� 4 � � �� 3 10
Orthogonality conditions: � ( � � 1 � � �� 4 ) = 0 � ( � � 1 � � �� 4 ) = 0 � ( � � 2 � � �� 4 ) = 0 � Other orthogonality conditions. � �� 4 = �� �� 3 + � 0 � �� 4 + � 1 � �� 3 + � � + � �� 4 � �� 3 = �� �� 2 + � 0 � �� 2 + � 1 � �� 1 + � � + � �� 3 � � �� 4 = � ( � �� 3 � � �� 2 ) + � 0 ( � �� 4 � � �� 3 ) + � 1 ( � � 32 � � �� 2 ) + ( � �� 4 � � �� 3 ) 11
Suppose Uncorrelated E � ects (Some � �� uncorrelated with � � ) � [ � � � ( � � 2 � �� � 1 � � 0 � � 2 � � 1 � � 1 )] = 0 � orthogonality conditions for each regressor. Predetermined variables could be uncorrelated with fixed e � ects � �� = �� �� ( � � 1) + �� �� ( � � 1) + �� � + � ��� if � = 0 � � would be uncorrelated with �� 12
Adds more orthogonality restrictions: � ( � � 1 ( � � 2 � �� � 1 � � 0 � � 2 � � 1 � � 1 )) = 0 � ( � �� ( � �� � �� ��� � 1 � � 0 � �� � � 1 � ��� � 1 )) = 0 � � = 2 � ���� �� Only identified when � � 3 . 13
� � � � � � � � � � Statistical Definitions: Strict Exogeneity: � � ( � �� | � � � � � � ) = � � ( � �� | � � � � � � ) � � ( � ��� +1 | � � � � � � ) = � � ( � �� ( � +1) | � � ( y does not Granger cause � ) � 14
� � � � � � � � � � � � Let � ( � +1) � = ( � ��� +1 � ���� � ��� ) if � � ( � +1) � � � � � ) = � 0 � + � 0 � � ( � �� | � � + � � � � and � � � � ) = � 0 � � � + � 0 � � ( � � ( � +1) | � � � + � � � � � � = 0 � � � � = 0 . 15
� � � � AR-1 Models Balestra - Nerlove Problem // Nickell � �� = �� ��� � 1 + � � + � ��� � = 1 � ���� I ; � = 2 � ���� � (A-1) � � ( � �� | � � � 1 ) = 0 � = 2 � ���� � � ( � � ) = � , � ( � 2 �� ) = � 2 � �� ( � � ) = � 2 � � and � �� freely correlated �� | � � � 1 � ( � 2 ) need not coincide with � 2 � . 16
� We get ( � � 1)( � � 2) � 2 moment restrictions: � ( � � � 2 ( � � �� � � � � ��� � 1 )) = 0 etc., etc. Using minimum discrepancy (CMD) methods take � �� = �� ��� � 1 + � � + � �� . For � � �� we obtain: � ��� � ��� = �� ��� � 1 � � ��� + � � � ��� + � ��� � ��� � ( � �� � �� ) = �� ( � ��� � 1 � ��� ) + � ( � � � ��� ) + � ( � ��� � ��� ) =0 � ( � �� � �� ) = � �� [ � ( � ��� � 1 � �� ) = � � � 1 �� ] � ( � �� � ��� ) = � � ] 17
� μ � + 1 ¶ We take � × distinct elements of 2 � = � ( � � � 0 � ) . For � = 3 , we obtain � 31 = �� 21 + � 1 � 21 = �� 11 + � 1 = � ( � 21 � � 11 ) � = � 31 � � 21 ( � 21 � � 11 ) � 21 � � 11 � 1 = � 31 � �� 21 � 2 = � 32 � �� 22 �� model just identified. Fit discrepancies between the population moments and fitted moments. 18
� Other Restrictions Lack of correlation between e � ects and errors � � ( � �� | � � � 1 � � � ) = 0 � � = 2 � ���� � 0 = � [( � �� � �� ��� � 1 ) [ � � ��� � 1 � � � � ��� � 2 ]] quadratic (in � ) restrictions: because � ( � � � � ��� � 1 ) = 0 . 19
When � � � � � �� (Ahn-Schmidt). Multiply � �� = �� ��� � 1 + � � + � �� by � � � � � �� = �� � � ��� � 1 + � 2 � + � � � �� � � = �� � � 1 + � 2 � . For � = 3 , imposes no further restrictions. � 2 � = ( � 32 � � 21 ) � � ( � 22 � � 11 ) � 20
Other Restrictions: Homoscedasticity: � ( � 2 �� ) = � 2 � = 2 � ��� � � ( � 2 �� � � 2 ��� 0 ) = 0 , etc. Time Series Homoscedasticity: � ( � 2 �� ) = � 2 � + � 2 + 2 �� � � 1 � �� = � 2 � ( � � 1)( � � 1) + � 2 q 0 21
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