Using Matching, Instrumental Variables and Control Functions to Estimate Economic Choice Models

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1 Using Matching, Instrumental Variables and Control Functions to Estimate Economic Choice Models James J. Heckman and Salvador Navarro The University of Chicago Review of Economics and Statistics 86(1) (2004) This draft for Nu eld College, Oxford August 7,

2 1 Introduction 1. We orient the discussion of the selection of alternative estimators around the economic theory of choice. We compare the di erent roles that the propensity score plays in three widely used econometric methods, and the implicit economic assumptions that underlie applications of these methods. 2

3 2. Conventional matching methods do not distinguish between excluded and included variables.we show that matching breaks down when there are variables that predict the choice of treatment perfectly whereas control function methods take advantage of exclusion restrictions and use the information available from perfect prediction to obtain identification. Matching assumes away the possibility of perfect prediction while selection models rely on this property in limit sets. 3

4 3. We define the concepts of relevant information and minimal relevant information, and distinguish agent and analyst information sets. We state clearly what information is required to identify di erent treatment parameters. In particular we show that when the analyst does not have access to the minimal relevant information, matching estimators of di erent treatment parameters are biased. 4

5 Having more information, but not all minimal relevant information, can increase bias compared to having less information. Enlarging the analyst s information set with variables that do not belong in the relevant information set may either increase or decrease bias from matching. Because the method of control functions explicitly models omitted relevant variables, rather than assuming that there are none, it is more robust to omitted conditioning variables. 5

6 4. The method of matching o ers no guidance as to which variables to include or exclude in conditioning sets. Such choices can greatly a ect inference. There is no support for the commonly used rules of selecting matching variables by choosing the set of variables that maximizes the probability of successful prediction into treatment or by including variables in conditioning sets that are statistically significant in choice equations. This weakness is shared by many econometric procedures but is not fully appreciated in recent applications of matching which apply these selection rules when choosing conditioning sets. 6

7 2 A Prototypical Model of Economic Choice Two potential outcomes ( 0 1 ). =1if 1 is selected. =0if 0 is selected. Let be utility. = ( ) =1( 0) (1) : observed factors determining choices, unobserved 1 = 1 ( 1 ) 0 = 0 ( 0 ) (2a) (2b) 0 1 are (absolutely) continuous 7

8 = 1 0 Additively Separable Case: For Familiarity. Not essential. = ( )+ ( )=0 (1 0 ) 1 = 1 ( )+ 1 ( 1 )=0 (2 0 ) 0 = 0 ( )+ 0 ( 0 )=0 (2 0 ) 8

9 3 Parameters Discussed Today : ( 1 0 ) (Average Treatment E ect) : ( 1 0 =1)(Treatment on the Treated) : ( 1 0 =0)(Marginal Treatment E ect) These are familiar but by no means the only parameters we could consider From MTE, can identify many other parameters (Recall IV Lectures) 9

10 4 The Selection Problem Samples generated by choices: Data: ( =1)= ( 1 =1) ( =0)= ( 0 =0) Pr ( =1 ) ( 1 =1) and ( 0 =0) From raw means, we get biases. Can form ( 1 =1) ( 0 =0) 10

11 In General This Produces BIASES : Bias = [ ( =1) ( =0)] ( 1 0 =1) = [ ( 0 =1) ( 0 =0)] Under Additive Separability Bias = ( 0 =1) ( 0 =0) : Bias = ( =1) ( =0) ( 1 0 ) 11

12 Under Additive Separability Bias = [ ( 1 =1) ( 1 )] [ ( 0 =0) ( 0 )] : Bias = ( =1) ( =0) ( 1 0 =0) Under Additive Separability Bias = ( 1 =1) ( 1 =0) [ ( 0 =0) ( 0 = 0)] 12

13 5 How Di erent Methods Solve the Bias Problem 5.1 Matching =( ) ( 1 0 ) " denotes independence given (M-1) 0 Pr( =1 )= ( ) 1 (M-2) Rosenbaum and Rubin (1983) show (M-1) and (M-2) imply ( 1 0 ) ( ) (M-3) 13

14 ( 1 =0 ( )) = ( 1 =1 ( )) = ( 1 ( )) ( 0 =1 ( )) = ( 0 =0 ( )) = ( 0 ( )) Dependence between and ( 1 0 ) is eliminated by conditioning on : ( 1 0 ) Selection on Observables If ( )=1or ( )=0 method breaks down for those values. 14

15 Extensions (Heckman, Ichimura, Smith and Todd) Distinction between and Introducing allows one to solve the breakdown problem arising from ( ) =1or ( ) =0 Thus if outcomes are defined in terms of and ( ) = ( ) If we can find another value 0 such that Pr( 0 ) 6= 1 can match using this (IV assumption) 15

16 Require only weaker mean independence assumptions ( 1 =1) = ( 1 ) ( 0 =0) = ( 0 ) CanbeusedforMeans. 16

17 Matching is for free (Gill and Robins (2001)): ( 0 =1 ) is not observed. Can just as well replace it by ( 0 =1 )= ( 0 =0 ) However, the implied economic restrictions are not for free. Imposes that, conditional on and, the marginal person is thesameastheaverageperson. ThisisthesameasaflatMTE( ) in (MTE does not depend on ) 17

18 5.2 Control Functions Additively Separable Case We observe left-hand sides of ( 1 =1)= 1 ( )+ ( 1 =1) ( 0 =0)= 0 ( )+ ( 0 =0) If ( 1 ) ( 1 =1)= ( 1 ( ) )= 1 ( ( )) If ( 1 ) ( 0 =0)= ( 0 ( ) )= 0 ( ( )) 18

19 So, key assumption ( 1 0 ) ( ) Under this condition ( 1 =1)= 1 ( )+ 1 ( ( )) ( 0 =0)= 0 ( )+ 0 ( ( )) Need Limit Set Results lim 1 1 ( )=0and lim 0 0 ( )=0 19

20 If there are limit sets Z 0 and Z 1 such that lim ( ) = Z 0 0 and lim ( ) =1, then we can identify the constants. Z 1 There are semiparametric versions of these estimators. Use polynomials in ; Local Linear Regression in 20

21 From this model can obviously identify = 1 ( ) 0 ( ) (Aswehaveseen) Plus, = 1 ( ) 0 ( )+ ( 1 0 =1) = 1 ( ) 0 ( )+ 1 ( ( )) μ ( ( )) 21

22 = 1 ( ) 0 ( ) + [ ( 1 0 =1) ( )] ( ) = 1 ( ) 0 ( ) + ( ) 1 ( ( )+ 1 0 ( ( ) ª ( ) Marginal and Average are allowed to be di erent. 22

23 Both Matching and Control functions are defined only over Support ( =1) Support ( =0) Method of control functions does not require ( 0 1 ) ( ) But Matching does. 23

24 Matching is a special case of control functions in the additively separable case. Additive separability and control functions assumptions are central to this claim. ( 1 =1) = ( 1 ) = ( 1 ( )) ( 0 =0) = ( 0 ) = ( 0 ( )) If then 1 ( )= ( 1 ) and 0 ( )= ( 0 ) ( 1 ( )) = 0 and ( 0 ( )) = 0 24

25 In the method of control functions, If ( ) ( 0 1 ) ( ) = ( 1 =1) + ( 0 =0)(1 ) = 0 ( )+( 1 ( ) 0 ( )) + ( 1 =1) + ( 0 ( ) =0)(1 ) = 0 ( )+( 1 ( ) 0 ( )) + ( 1 ( ) =1) + ( 0 ( ) =0)(1 ) = 0 ( ) +[ 1 ( ) 0 ( )+ 1 ( ( )) 0 ( ( ))] + 0 ( ( )) 25

26 To identify must isolate it from and 1 ( ) 0 ( ) 1 ( ( ) 0 ( ( ) 26

27 Under matching assumptions, ( ( ) ) = 0 ( ( )) + ( 1 ( ( )) 0 ( ( ))) +[ ( 1 ( )) ( 0 ( ))] + ( 0 ( )) ( ( ) )= 0 ( ( )) + ( 1 ( ( )) 0 ( ( ))) since ( 1 ( )) = ( 0 ( )) = 0 27

28 We can get identification without exclusion, conditional on ( 1 0 ) 28

29 Tables 1 and 2 present sensitivity analysis for the case of so ( 1 0 ) 0 (0 ) ( ) = ( ) = 2 ; = {0 1} Bias ( ( ) = ) = 0 0 ( ) Bias ( ( ) = ) = ( )[ 1 1 (1 )+ 0 0 ] where ( ) = ( 1 (1 )) (1 ) 29

30 Even if the correlation between the observables and the unobservables is small, so that one might think that selection on unobservables is relatively unimportant, we still get substantial biases if we do not control for relevant omitted conditioning variables. These examples also demonstrate that sensitivity analyses can be conducted for control function models even when they are not fully identified. 30

31 31

32 32

33 5.3 Instrumental Variables Matching and Method of control functions work with ( ) and Pr ( =1 ). = 1 +(1 ) 0 = 0 ( )+( 1 ( ) 0 ( )+ 1 0 ) + 0 = 0 ( )+ ( ) + 0 If 1 = 0 ( 0 ( ) )= ( 0 ) (IV-1) Pr ( =1 ) is a nontrivial function of for each (IV-2) 33

34 When 1 6= 0 but ( 1 0 ) or alternatively ( 1 0 ) we have = ( 1 0 ) = ( ( ) ) = ( 1 0 =1)= ( 1 0 ) = 34

35 Analytically More Interesting Case 1 6= 0 and 6 ( 1 0 ) For : ( 0 + ( 1 0 ) ( ) )= ( 0 + ( 1 0 ) ) (IV-3) For : ( 0 + ( 1 0 ) ( 0 + ( 1 0 ) ) ( ) ) = ( 0 + ( 1 0 ) ( 0 + ( 1 0 ) ) ) For we can rewrite: ( 0 ( ) )+ ( 1 0 =1 ( ) ) ( ) = ( 0 )+ ( 1 0 =1 ) ( ) Conditions Satisfied If 1 = 0,or( 1 0 ) ( ) 35

36 1. Method of Control Functions Models Dependence between ( 1 0 ) and 2. Matching assumes ( 1 0 ) 3. Linear IV requires that 0 + ( 1 0 ) be mean independent of (or ( )) givenx. 36

37 Local Instrumental Variables (LIV) require that ( ) be a non-degenerate random variable given (existence of an exclusion restriction) (LIV-1) ( 0 1 ) (LIV-2) 0 Pr ( ) 1 (LIV-3) Support ( ( )) = [0 1] (LIV-4) Under these conditions, ( ( )) ( ( )) = ( ( ) =0) 37

38 Proof: ( ( )) = ( 1 =1 ( )) ( ) = + ( 0 =0 ( )) (1 ( )) Z + where = ( ) Thus ( ( )) ( ) Z ( ) Z Z ( ) 1 ( 1 ) 1 0 ( 0 ) 0 = ( 1 0 = ( )) = 38

39 39

40 6 The bias from Matching: Information requirements Fundamental Problem: Information of the Analyst often less than that of the Agent. Definition 1 We say that ( ) is a relevant information set if its associated random variable,, satisfies (M-1) so ( 1 0 ) 40

41 Definition 2 We say that ( ) is a minimal relevant information set if it is the intersection of all sets ( ) and ( 1 0 ). The associated random variable is the minimum amount of information that guarantees that (M-1) is satisfied. Intersection may be empty. May not be a unique minimal information set. Definition 3 The agent s information set, ( ),isdefined by the information used by the agent when choosing among treatments. Accordingly, we call the agent s information. 41

42 Definition 4 The econometrician s full information set, ( ), is defined by all the information available to the econometrician, Definition 5 The econometrician s information set, ( ) is defined by the information used by the econometrician when analyzing the agent s choice of treatment,. 42

43 Obvious Inclusions: ( ) ( ) ( ) ( ) ( ) ( ) and This assumes exists. Matching implies ( ) ( ) 43

44 Generalized Roy Examples (Assume Factor Structure for error terms) = + = = 1 if 0 =0otherwise 1 = = = = ( ) mean zero random variables, mutually independent of each other and 44

45 The minimal relevant information set when factor loadings are not zero: = { 1 2 } Agent information sets may include di erent variables. If shocks to the outcomes not known, but other terms are: = { 1 2 } Under perfect certainty, = { }. (ThisistheoriginalRoymodel) Construct examples using: ( ) (0 ) ( ) =

46 6.1 The economist uses the minimal relevant information: ( ) ( ) Suppose = { 1 2 } ( 1 =1 ) ( 0 =0 ) = 1 0 +( ) 1 +( ) 2 Knowledge of ( 1 2 ) and knowledge of ( 1 2 ) equivalent μ ( ) = Pr μ = 1 = 46

47 ( 1 =1 ( )= ) ( 0 =0 ( )= ) = ( 1 =1 ( )= ) ( 0 =0 ( )= ) μ = (1 ) μ 0 1 (1 ) =

48 This is equal to all the treatment parameters (MTE=ATE=LATE=TT) μ 1 1 (1 ) μ 0 1 (1 ) = ( 1 ) 1 ( ) = ( 0 ) 0 ( ) where 1 ( )= ( 1 (1 )) because and 0 ( )= ( 1 (1 )) 1 ( )= ( )=0 48 =0 1

49 6.2 The Economist does not use all of the Minimal Relevant Information: = { } q S 1 (1 ) Bias ( ( ) = ) = 0 ( ) ( )= 1 ( ) 0 ( ) Bias ( ( ) = ) = ( )[ 1 (1 )+ 0 ] 49

50 where Bias ( ( ) = ) = ( )[ 1 (1 )+ 0 ] ( ) = ( 1 (1 )) (1 ) 1 (1 )[ 1 0 ] 1 = q = q

51 6.3 Adding information to the econometrician s information set : using Some, but not all, Information from the Minimal Relevant Information Set 0 = { 2 } May raise or lower the bias. Comparable to 1 and 0 above, we can define 0 1 = 0 0 = q q

52 Condition 1 The bias produced by using matching to estimate is smaller in absolute value for any given when the new information set ( 0 ) is used if Condition 2 The bias produced by using matching to estimate is smaller in absolute value for any given when the new information set ( 0 ) is used if 1 (1 ) (1 )

53 Condition 3 The bias produced by using matching to estimate is smaller in absolute value for any given when the new information set ( ) 0 is used if ( )[ 1 (1 )+ 0 ] 1 (1 )[ 1 0 ] ( )[ 0 1 (1 )+ 0 0 ] 1 (1 )[ ] 53

54 Proof of Condition 1 Suppose ³ ( 2 2) 0 = When q {z } when 2 is in information set ³ 02 2 = q {z } when 1 is not = 0 0 ³ 0 = q

55 Thereissomecriticalvalue 02 beyond which Assume 01 = 1 = 2 =1 02 = 12 =1 11 = 2 55

56 Figure 1.--Bias for Treatment on the Treated Special case:adding relevant information f2 increases the bias Matching on P(Z) Matching on P(Z,f2) Bias Average Bias = Average Bias = P 56

57 Figure 2.--Bias for Average Treatment Effect Special case: Adding relevant information f2 increases the bias 4 Matching on P(Z) Matching on P(Z,f2) Bias Average Bias = Average Bias = P 57

58 Figure 3.--Bias for Marginal Treatment Effect Special case: Adding relevant information f2 increases the bias 4 Matching on P(Z) Matching on P(Z,f2) Bias Average Bias = Average Bias = P 58

59 Control Function Method Models the Bias In control function method, adding 2 we simply change the control function. We go from to 1 ( ( ) = ) = 1 1 ( ) 0 ( ( ) = ) = 0 0 ( ) 1 0 ( ( 2 )= ) = ( ) 0 0 ( ( 2 )= ) = ( ) where 1 ( ) = ( 1 (1 )) and 0 ( ) = ( 1 (1 )) 1 This protects us against misspecification. 59

60 6.4 Adding information to the econometrician s information set: using proxies for the relevant information Suppose we do not know 2,justproxyitby e e = n o e Suppose = e Suppose further that e 0 2 ³ e 2 = and e ( ) 60

61 Expressions corresponding to 0 and 1 : e 1 = (1 2 ) 2 2 q (1 2 )+ 2 e 0 = (1 2 ) 2 2 q (1 2 )+ 2 61

62 By substituting 0 for e and 0 for e ( =0 1) into Conditions (1), (2) and (3) we obtain equivalent results for this case. Whether e will be bias reducing depends on how well it spans and the signs of the terms in the absolute values. In this case, there is another parameter ( =0). 62

63 The bias generated when the econometrician s information is e can also be smaller than when it is 0.Itcan be the case that knowing variable e is than knowing the actual variable 2. Take treatment on the treated as the parameter. The bias is reduced when e is used instead of 2 if (1 2 ) 2 q (1 2 )+ 2 q

64 Figure 4.--Bias for Treatment on the Treated ~ ~ Special case: Adding irrelevant information Z increases the bias correlation(z,f2)= Matching on P(Z) ~ Matching on P(Z,Z) Bias Average Bias = Average Bias = P 64

65 Figure 5.--Bias for Average Treatment Effect ~ ~ Special case: Adding irrelevant information Z increases the bias correlation(z,f2)= Matching on P(Z) ~ Matching on P(Z,Z) Bias Average Bias = Average Bias = P 65

66 3.2 3 Figure 6.--Bias for Marginal Treatment Effect ~ ~ Special case: Adding irrelevant information Z increases the bias correlation(z,f2)=0.5 Matching on P(Z) ~ Matching on P(Z,Z) Bias Average Bias = Average Bias = P 66

67 Figure 7.--Bias for Treatment on the Treated ~ Using proxy Z for f2 increases the bias ~ correlation (Z,f2)= Matching on P(Z,f2) ~ Matching on P(Z,Z) Bias Average Bias = Average Bias = P 67

68 4 Figure 8.--Bias for the Average Treatment Effect ~ Using proxy Z for f2 increases the bias ~ correlation (Z,f2)=0.5 Matching on P(Z,f2) ~ Matching on P(Z,Z) Bias Average Bias = Average Bias = P 68

69 Figure 9.--Bias for the Marginal Treatment Effect ~ Using proxy Z for f2 increases the bias ~ correlation (Z,f2)=0.5 Matching on P(Z,f2) ~ Matching on P(Z,Z) Bias Average Bias = Average Bias = P 69

70 6.5 The case of a discrete treatment = + = , =0 1 = 1 if 0 =0otherwise, People receive treatment according to the rule = + = = 1 if 0 =0otherwise; ( 1 2 ) ( 0 1 ) 70

71 The e ect of treatment is given by: 1 ( )= Pr ( 1 =1 =1 ) Pr ( 0 =1 =1 ) A second definition works with odds ratios: 2 ( )= Pr( 1 =1 =1 ) Pr( 1 =0 =1 ) Pr( 0 =1 =1 ) Pr( 0 =0 =1 ) 71

72 Onecouldalsoworkwithlog Under the null hypothesis of no e ect of treatment 1 = 2 =1 b 1 ( )= Pr ( 1 =1 =1 ) Pr ( 0 =1 =0 ) The denominator replaces the desired probability Pr ( 0 =1 =1 ) by Pr ( 0 =1 =0 ) 72

73 Under the Null Hypothesis of no real e ect of treatment 1 = 0 = 1 = 0 = can be generated by 11 = 01 = 1 12 = 02 = 2 1 = 0 = 73

74 Assume initially that = { 1 2 } b 1 ( )= Pr ( 1 =1 =1 1 2 ) Pr ( 0 =1 =0 1 2 ) In general: 1 ( )= Pr ( 1 =1 =1 ) Pr ( 0 =1 =1 ) = Pr ( 1 =1 1 2 ) Pr ( 0 =1 1 2 ) = 1( ) 6= Pr ( 1 =1 =1 ) Pr ( 0 =1 =0 ) = b 1 ( ) 74

75 16 14 Figure 10.--Estimated Effect of Treatment under Different Information Sets No Effect of Treatment and v2 =1 ^ 1 I E ={f 1,f 2 } I E ={f 2 } < f 2 =1(V>0) 75

76 6 Figure 11.--Estimated Effect of Treatment under Different Information Sets No Effect of Treatment and v2 =-1 ^ 1 I E ={f 1,f 2 } I E ={f 2 } 5 4 < f2 76

77 14 Figure 12.--Estimated Effect of Treatment under Different Information Sets No Effect of Treatment and v2 =1 ^ 2 I E ={f 1,f 2 } I E ={f 2 } < f 2 77

78 25 Figure 13.--Estimated Effect of Treatment under Different Information Sets No Effect of Treatment and v2 =-1 ^ 2 I E ={f 1,f 2 } I E ={f 2 } < f 2 78

79 3 2.5 Figure 14.--Estimated Effects of Treatment under Different Information Sets No Effect of Treatment and v2 =1 ^ 3 IE={f1,f2} I E ={f 2 } < f 2 79

80 2 Figure 15.--Estimated Effects of Treatment under Different Information Sets No Effect of Treatment and v2 =-1 ^ 3 I E ={f 1,f 2 } I E ={f 2 } 1 0 < f 2 80

81 3 Figure 16.--Estimated Effect of Treatment under Different Infomation Sets No Effect of Treatment and v2 =1 4 ^ I E ={f 1,f 2 } I E ={f 2 } < f

82 3.5 3 Figure 17.--Estimated Effects of Treatment under Different Information Sets No Effect of Treatment and v2 =-1 ^ 4 I E ={f 1,f 2 } I E ={f 2 } < f 2 82

83 6.6 On the use of model selection criteria to choose matching variables Adding more variables to the Information Set may increase Bias. How to choose the relevant variables? Standard methods on model selection criteria fail. An implicit assumption underlying such procedures is that the added conditioning variables are exogenous in the following sense ( 0 1 ) (M-4) ( is the list of initial variables used as conditioning variables.) 83

84 Sometimes procedures suggested Add variables when ratios big in propensity score Improve Fit Such procedures can raise the bias. Consider the following example: where e e = { } = ( ) might be an elicitation from a questionaire. 84

85 Same expressions for the biases using e ( =0 1) instead of where: e 1 = q e 0 = q = q Bias can be in- In general, these expressions are not zero. creased or decreased. 85

86 When 2 =0we can perfectly predict. (Will pass a goodnessof-fit criterion) Thus for It follows that lim Pr ( =1 + ) 0 = 1 for Pr ( =1 + ) = 0 for lim 0 Assumption (M-2) is violated and matching breaks down Making 2 arbitrarily small, we can predict arbitrarily well. Can improve over the fit with ( 1 2 ) in the set which produces no bias. 86

87 87

88 A More General Example Considers use of a proxy regressor = ( 1 2 ); ( 1 2 ) has mean zero 1 2, and ( 1 2 ) ( ); in the latent variable gener- possibly dependent on ating the treatment choice is measurement error. 88

89 For di erent levels of dependence between and,and di erent weights on 1 2 and on the scale of measurement error, can be a better predictor of than 1 2 or even 1 2. However, in general, ( 1 0 ) Á because is an imperfect proxy for the combinations of 1 and 2 entering 1 and 0. Thus conditioning on can produce a better fit for but greater bias for the treatment parameters. 89

90 Consider the following example where is an outcome and is an index = + = + where ( ) Obviously. 90

91 Suppose instead that we have a candidate conditioning variable = + +. Suppose that all variables are normal with zero mean and are mutually independent. Then we may write where = = It is assumed that is independent of all other error components on the right-hand sides of the equations for and. 91

92 From normal regression theory we know that conditioning is equivalent to residualizing. Constructing the residuals we obtain = (1 )+ ( + ) By a parallel argument = (1 )+ ( + ) requires that and be uncorrelated, which in general does not happen. Letting the dependence between and get large, and setting to suitable values, we can predict better (in the sense of 2 )with than with. 92

93 Letting =1( 0) produces a simple version of the example because better prediction of produces better prediction of. 93

94 Appendix Consider a general model of the form: 1 = = = ( )+ = 1 if 0 =0otherwise = 1 +(1 ) 0 94

95 where ( 1 0 ) 0 (0 ) ( ) = 2 ( ) = =0; =1 ( 1 ) = 1 ( 0 ) = 0 Let ( ) and ( ) be the pdf and the cdf of a standard normal random variable. Then, the propensity score for this model is given by: Pr ( 0 ( )) = ( ( )) = Pr ( ( )) = μ ( ) = 1 = 95

96 so Since the event ( ) = 1 (1 ) ³ S 0 ( ( )) = can be written as S ( ) S 1 (1 ) we can write the conditional expectations required to get the biases as a function of. 96

97 For 1 : ( 1 0 ( ( )) = ) = μ 1 = 1 ( ) μ ( ( )) = 1 (1 ) = 1 1 ( ) 97

98 ( 1 =0 ( ( )) = ) = 1 = 1 μ μ = ( ) ( ( )) = = 1 (1 ) ( ( )) = = 1 1 (1 ) where Similarly for 0 : 1 = 1 ( 0 0 ( )= ) = 0 1 ( ) ( 0 0 ( )= ) = 0 0 ( ) ( 0 =0 ( )= ) = 0 1 (1 ) 98

99 where and 0 = 0 1 ( ) = ( 1 (1 )) 0 ( ) = ( 1 (1 )) (1 ) are inverse Mills ratio terms. Substituting these into the expressions for the biases Bias ( ) = 0 1 ( ) 0 0 ( ) = 0 ( ) 99

100 Bias ( ) = 1 1 ( ) 0 0 ( ) = ( )( 1 (1 )+ 0 ) Bias = 1 1 ( ) 0 0 ( ) 1 1 (1 )+ 0 1 (1 ) = ( )( 1 (1 )+ 0 ) 1 (1 )[ 1 0 ] where ( ) = 1 ( ) 0 ( ) = ( 1 (1 )) (1 ) 100

Using Matching, Instrumental Variables and Control Functions to Estimate Economic Choice Models

DISCUSSION PAPER SERIES IZA DP No. 768 Using Matching, Instrumental Variables and Control Functions to Estimate Economic Choice Models James J. Heckman Salvador Navarro-Lozano April 2003 Forschungsinstitut