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Labor Supply and the Two-Step Estimator James J. Heckman - PowerPoint PPT Presentation

Labor Supply and the Two-Step Estimator James J. Heckman University of Chicago Econ 312 This draft, April 7, 2006 In this lecture, we look at a labor supply model and discuss various approaches to identify the key parameters of


  1. Labor Supply and the Two-Step Estimator James J. Heckman University of Chicago Econ 312 This draft, April 7, 2006

  2. � � � In this lecture, we look at a labor supply model and discuss various approaches to identify the key parameters of the model, including the two-step estimator. 1 A one period model Consider a simple one period model of the labor supply choice (with total time normalized to 1): μ � � � 1 ¶ μ � � � 1 ¶ max { ��� } � ( �� � ) = + � such that � + �� � � + � , where � is consumption, � is leisure, � is the wage rate, and � is non wage income. 1

  3. � � � The Euler equation is � = �� � � 1 � � � 1 and the reservation wage is given by: ¸ � �� � � 1 = = � � � 1 � � � 1 � =1 �� = � = � ln � � = ln � + (1 � � ) ln � � ( �� �� � ) , � � � (0 � � 2 Assume ln � = �� + � , � � � ) . 2

  4. Now we will consider three cases: 1. Wages are observed for everyone, i.e. for those participat- ing ( � � 1 ) and also for those not participating ( � = 1 ); 2. Wages are not known for everyone, but we know the func- tional form for wages; and 3. Wages are observed for workers only. We discuss the commonly used methodologies to identify the key parameters of the model in each of these cases below. 3

  5. � � � � � � � 2 Observe wages for everyone Pr( Person works | �� � ) = Pr(ln � � � ln � | �� � ) μ � ¶ � ln � � �� � (1 � � ) ln � = Pr = � ( � ) � where � = ln � � �� � (1 � � ) ln � 4

  6. � � � � � � � � � � � 2.1 Grouped data estimator Each cell has common values of � � � � � � � � . For each cell, obtain: ˆ � � ( � � = 1 | � � � � � � � � ) = cell proportion working = � (ˆ � � ) � � � = � � 1 ( ˆ Then calculate ˆ � � ) . Regress ˆ � � on ln � � �� � (1 � � ) ln � and obtain estimates of � � � �� � . Note that here, instead of standard normal ( � ), one could use a standard logistic model ( � ( � ) = 1 + � � ) or a linear probability model: � ( � ) = � � � � � � � � � � 5

  7. � � � � � 2.2 Justifying the grouped data estimator Suppose � � = 1 if agent i works, 0 if not. For each cell, get ˆ � ( � = 1 | �� � ) . By WLLN and Slutsky: plim � � 1 (ˆ � � 1 ( plim ˆ � ( � = 1 | �� � )) = � ( � = 1 | �� � )) � � 1 ( � ( � = 1 | �� � )) = ln � � �� � (1 � � ) ln � = Set up the regression function: £ ¤ � � 1 (ˆ � � 1 ( � ) + � � 1 (ˆ � ) � � � 1 ( � ) � ) = ln � � �� � (1 � � ) ln � = + �� 6

  8. �� � � � � � We need to characterize � = � � 1 (ˆ � ) � � � 1 ( � ) . By the delta method, assuming � is continuously di � erentiable, we get: ¯ ¯ ¯ � ( � (ˆ � ) � � ( � )) = �� � (ˆ � � � ) � ¯ where min(ˆ �� � ) � � � � max(ˆ �� � ) . Now assume � � � ) � � (0 � � 2 � (ˆ � ) � Applying this to our regression function, we get (using � � = � � 1 ( � � ) ) p p 1 � � ( � � 1 (ˆ � � ) � � � 1 ( � � )) = � � (ˆ � � � � � ) � cells �� � ( � � � ) 7

  9. � � � � � � � � � � � � � � � � � � Suppose { � � } � � and errors are uncorrelated asymptoti- cally. Then à ! ¸ 2 � � (1 � � � ) 1 0 � � ( � � � ) where � � (1 � � � ) is the variance of the binary random variable � � . We obtain the feasible GLS estimator by regressing: ln � � �� � (1 � � ) ln � � � 1 (ˆ � � ) on � ( � � � ) � ( � � � ) r r � � (1 � � � ) � � (1 � � � ) 8

  10. � � � We can show that the estimates are asymptotically e � cient and satisfy the orthogonality condition μ ln � � �� � (1 � � ) ln � ¶ ¡ X 1 ¢ � � 1 (ˆ � � ) � � � 1 ( � � ) plim � ( � � � ) � =1 = 0 � 9

  11. � 2.3 Microdata analogue Y Y Y � = � ( � � ) � ( � � � ) = � ( � � [2 � � � 1]) � � =1 � � =0 MLE gives consistent and asymptotically normal estimates of � � � �� � . 10

  12. � �� �� �� 3 Do not observe wages, but wages follow specific functional form Here we do not observe wages for anyone, but do know that the wages have the following functional form: ln � = �� + �� where � � � (0 � � 2 � ) � ( � � � ) � and ( � � � ) � � (0 � � 2 � + � 2 � � 2 � �� ) = � (0 � � � 2 ) � where ( � � ) 2 = ( � 2 � + � 2 � � 2 � �� ) . 11

  13. � � � � � Then: μ � � � ¶ � �� � �� � (1 � � ) ln � Pr( � works ) = Pr Note: if ( �� � ) are Extreme Value (Type I), then ( � � � ) is logistic. 12

  14. � � � � � � 3.1 Identification (2 cases) μ � ¶ � � � 1 � � 1. If � , � distinct, then can estimate , but � � � � can’t estimate � � . 2. If � , � have elements in common ( � � = � � � � � � � � ), then can estimate only: μ � � ¶ � (1 � � ) � � � � � � � � � � � � � and again not � � . 13

  15. � 4 Observe wages for workers only Here we have: ln � = �� + � ln � � = �� + (1 � � ) ln � + � ( �� � ) � ( �� �� � ) � � = ln � � ln � � = �� � �� � (1 � � ) ln � + � � �� Agent � works if � � � 0 . This is a the Roy model with 2 sectors: market ( � ) and nonmarket ( � ). 14

  16. � � � � � � � � � Following the derivations in the lecture ‘Empirical Content of Roy Model,’ we get the expression for the expected wages in the market sector, conditional on participation and the � and � variables as: � (ln � | ln � � ln � � � �� �� � ) μ �� � �� + (1 � � ) ln � ¶ = �� + � �� � � �� = �� + � �� � � �� � ( � � ) � where � = �� � �� � (1 � � ) ln � . 15

  17. � � � � � � � � � � � 4.1 2-step Estimator • Step 1: Run probit on LFP (labor force participation) decision (as we did in section 2.3) : ˆ X ( ˆ � � � ˆ � � ) = argmax ln � [ � � (2 � � � 1)] Form: � ( � ˆ � � ) � ( � ˆ � � ) = 1 � � ( � ˆ � � ) � � � � � ˆ � � ˆ � � (1 � ˆ � ) ln � � where ˆ = ˆ 16

  18. �� � � � � \ • Step 2: Estimate ( ˆ ) via OLS on � �� � � �� ln � � = � � � + � �� � � �� � ( � ˆ � � ) + � � using sample of workers only (refer to expression for con- ditional expectation of market wages derived above). 17

  19. 4.2 Identification 4.3 With one exclusion restriction With one exclusion restriction (1 variable, call it � 1 , in � not in � or ln � ; let all other � be common with � ), we can now identify everything: ( �� �� �� � � � � �� � � �� � � �� ) � We describe below how we recover all the relevant parameters: 18

  20. � � � � � � μ ( � � � ) � ¶ � 1 � � (i) Step 1 of 2-step gives � 1 � � as well as . μ ¶ Step 2 gives � 1 as well as . Solve for � � . � � � � �� � � �� Use � � to solve for ( � �� � � �� � �� � ) . 19

  21. � � � � � �� (ii) Look at residuals from step 2: ln � � = � � � + � �� � � �� � ( � ˆ � � ) + � � Then following results in the earlier lecture, we have: ¡ � 2 £ ¤ £ 1 � � 2 ¤¢ 1 � � 2 ( � � � ) � � ( � � � ) � � � ( � 2 � ) = + � �� � � �� � 2 £ ¤ � 2 = � + � � � � 2 � on ˆ � + ˆ � 2 Regression of ˆ � � ˆ � � gives consistent estimates of ( � �� � � ) . Solve for � �� . Use � � = � �� + � �� � 2 � �� to solve for � �� . 20

  22. � �� � � � 4.4 Without any exclusion restrictions on � Without any exclusion restrictions on � , we can only identify ( �� � �� ) : We cannot uniquely identify � �� or � �� . μ ¶ (i) Step 2 of 2-step gives �� � �� � � �� (ii) Obtain ( � �� � � ) from residual regression as above. (iii) To solve for ( � �� � � �� ) either normalize � �� = 1 or � �� = 0 . If we assume � �� = 1 , then from step 2 we obtain , from which we can obtain � �� . If we as- � �� � � �� 1 + � �� � 2 � �� sume � �� = 0 , then from step 2 we get and can � �� + � �� solve for � �� . 21

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