Identification of Regression Models with Misclassified and Endogenous Binary Regressor

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1 Identification of Regression Models with Misclassified and Endogenous Binary Regressor A Celebration of Peter Phillips Fourty Years at Yale Conference Hiroyuki Kasahara 1 Katsumi Shimotsu 2 1 Department of Economics University of British Columbia 2 Faculty of Economics University of Tokyo 1

2 Identification In 1970 s, Peter had already recognized the importance of (non-)identification: It seems important that we should understand the implications of identification failure for statistical inference. Yet, this is a subject that seems to be virtually untouched in the literature (Phillips, 1973). Peter s work on identification includes Phillips, P. C. B. (1972), The Structural Estimation of a Stochastic Differential Equation System, Econometrica. Phillips, P. C. B. (1973), The Problem of Identification in Finite Parameter Continuous Time Models, Journal of Econometrics. Phillips, P. C. B. (1989), Partially Identified Econometric Models, Econometric Theory. Choi, I. and Phillips, P. C. B. (1992), Asymptotic and Finite Sample Distribution Theory for IV Estimators and Tests in Partially Identified Structural Equations, Journal of Econometrics. Shi, X. and Phillips, P. C. B. (2012), Nonlinear Cointegrating Regression under Weak Identification, Econometric Theory. Su, L., Shi, Z., and Phillips, P. C. B. (2016), Identifying Latent Structures in Panel Data, Econometrica. 2

3 Misclassified and endogenous binary regressor The model Variables Y = g(x, T ) + ε = α(x) + β(x)t + ε, E[ε X] = 0. Y : outcome (for example, log-wage) X: exogenous controls T : unobservable binary regressor (for example, true educational qualification). T may be endogenous = correlated with ε. T : observable misclassified measurement of T (for example, self-reported schooling). 3

4 Misclassified and endogenous binary regressor Mismeasured binary regressors are prevalent. Academic degree, self-reported drug use, job training reported by worker,... When a regressor is binary, its measurement error is necessarily nonclassical (correlated with the true value). Mahajan (2006) shows identification when a binary instrument variable Z satisfies T Z conditionally on (T, X), E[ε X, Z ] = 0, and additional assumptions. The assumptions in Mahajan (2006) imply that T is exogenous (DiTraglia and García-Jimeno, 2017). 4

5 Brief literature review: exogenous T Mahajan (2006) shows identification of g(x, T ) under one binary instrument Z. Hu (2008) generalizes Mahajan (2006) to discrete T. Battistin et al. (2014) use two measurements T 1 and T 2 that are independent to each other conditionally on T. 5

6 Brief literature review: endogenous T Lewbel (2007) shows E(Y X, T = 1) E(Y X, T = 0) can be identified when Z takes at least three values. DiTraglia and García-Jimeno (2017) show identification under additional assumption E[ε 2 X, Z ] = E[ε 2 X], E[ε 3 X, Z ] = E[ε 3 X], which can be sensitive to measurement error / outlier in Y. Identification of LATE: Yanagi (2018), Ura (2018), Calvi et al. (2018) 6

7 Our contribution We show identification when there exists V such that E[ε X, Z, V ] = 0, V T T, X, Z V may enter the outcome equation for Y. So, V can be a covariate. V may be binary. V must be correlated with T. Example of V : gender dummy. Z must be correlated with T but does not need to be mean independent of T conditionally on (T, X). We use only the first moment of ε. 7

8 Our contribution Many papers use a single instrument Z and assume (T, ε) Z conditionally on T (Mahajan, 2006; Lewbel, 2007; Hu, 2008; DiTraglia and García-Jimeno, 2017). As noted in Mahajan (2006), this is a strong assumption. If T is one s true educational attainment and Z is college proximity, then Z may affect one s incentive to lie about T. We allow T to depend on Z conditionally on T. We need T V conditionally on (T, Z ). 8

9 Model and Assumptions Henceforth, we suppress X. The model is Assumption 1 Y = α(v ) + β(v )T + ε, E[ε V, Z ] = 0. (a) Non-differential measurement error E[ε T, T, Z, V ] = E[ε T, Z, V ]. (b) Relevance of Z Pr(T = 1 Z = 0, V ) Pr(T = 1 Z = 1, V ). (c) Exclusion restriction on Z from the outcome equation E[ε T, Z, V ] = E[ε T, V ]. 9

10 Model and Assumptions Model Assumption 1 Y = α(v ) + β(v )T + ε, E[ε V, Z ] = 0. (d) Relevance of V Pr(T = 1 Z, V = 0) Pr(T = 1 Z, V = 1). (e) Exclusion restriction on V from misclassification probability Pr(T = 1 T, Z, V ) = Pr(T = 1 T, Z ). (f) Pr(T = 1 T = 1, Z ) > Pr(T = 1 T = 0, Z ) for all Z. (g) β(v ) 0. 10

11 Model and Assumptions Assumption 2 Pr(T = 0 Z = 0, V = 0) Pr(T = 0 Z = 1, V = 1) Pr(T = 0 Z = 1, V = 0) Pr(T = 0 Z = 1, V = 0) Pr(T = 1 Z = 0, V = 0) Pr(T = 1 Z = 1, V = 1) Pr(T = 1 Z = 1, V = 0) Pr(T = 1 Z = 1, V = 0). This assumption is likely to hold when both Z and V are relevant for T. 11

12 Identification Proposition Under Assumptions 1 and 2, α(v ), β(v ), Pr(T = 1 T, Z ), and Pr(T = 1 Z, V ) are identified for all (T, Z, V ). By using two variables Z and V, we achieve identification under weaker requirements on these variables. Each variable needs to satisfy only one exclusion restriction. The proof is similar to those in Hu (2008), Kasahara and Shimotsu (2009), and Carroll et al. (2010). 12

13 Intuition E[Y Z, V ] = E[Y T = 0, Z, V ] Pr(T = 0 Z, V ) + E[Y T = 1, Z, V ](1 Pr(T = 0 Z, V )). Because Y Z T, V, the right hand side equals E[Y T = 0, V ] Pr(T = 0 Z, V ) + E[Y T = 1, V ](1 Pr(T = 0 Z, V )). Similarly, because T V T, Z, E[T Z, V ] = E[T T = 0, Z ] Pr(T = 0 Z, V ) Because Y T T, V, Z, E[YT Z, V ] + E[T T = 1, Z ](1 Pr(T = 0 Z, V )). = E[Y T = 0, V ]E[T T = 0, Z ] Pr(T = 0 Z, V ) + E[Y T = 1, V ]E[T T = 1, Z ](1 Pr(T = 0 Z, V )). 13

14 Identification a model with heterogenous effect Y = α(x, V, U ) + β(x, V, U )T + ε, E[ε X, V, Z ] = 0. T is unobservable and binary. U is unobservable and has a finite support. U may be correlated with T. Example: ability Similar to the models analyzed by Vytlacil and Heckman and their co-authors. Correlation bet. α(x, V, U ) and T = sorting on the level Correlation bet. β(x, V, U ) and T = sorting on the gain 14

15 Identification a model with heterogenous effect Y = α(x, V, U ) + β(x, V, U )T + ε, E[ε X, V, Z ] = 0. T is observable measurement for T. Z is an instrument-like variable for T. V is an instrument or covariate that moves Pr(T = 1 X, Z ). Let U be an observable proxy for U, for example, AFQT score. 15

16 Model and Assumptions Henceforth, we suppress X. Y = α(v, U ) + β(v, U )T + ε, E[ε V, Z ] = 0. U has a finite support, and there is an observable proxy U. This model has a similar structure as before except that T is replaced with S = (T, U ). As before, we assume S V conditionally on (S, X, Z ). 16

17 Model and Assumptions Model Y = α(v, U ) + β(v, U )T + ε, ε Z T, U, V, X. Assumption 3 (S=(T,U)) (a) The support of U is {1, 2,..., K }. (b) ε S conditional on (S, V ). (non-differential measurement error) (c) V and Z are relevant for T : Pr(T = 1, U Z, V ) is in (0, 1) and strictly increasing in Z and V for any U. (d) ε Z (S, V ). (Z cannot affect Y given (S, V )) 17

18 Model and Assumptions Model Y = α(v, U ) + β(v, U )T + ε, ε Z U, T, V, X. Assumption 3 (S=(T,U)) (e) S V (S, V ). (V cannot affect S given (S, Z )). (f) P(S = s S = s) > P(S = s S = s) for any s s. (g) Y takes at least 2K (= S ) different values. (h) α(v, U ) and β(v, U ) are strictly increasing in U. (i) α(v, U = 1) + β(v, U = 1) > α(v, U = K ). 18

19 Identification Assumption 4 { λ j } K S j=1 take all distinct values across j = 1,..., 2K, where λ j := Pr(S = s j Z = 0, V = 0) Pr(S = s j Z = 1, V = 1) Pr(S = s j Z = 1, V = 0) Pr(S = s j Z = 0V = 1). Proposition Under Assumptions 3 and 4, Pr(S Z, V ), Pr(Y S, V ), and Pr(S S, Z ) are identified for all (S, Z, V ). 19

20 Bibliography i References Battistin, E., De Nadai, M., and Sianesi, B. (2014), Misreported Schooling, Multiple Measures and Returns to Educational Qualifications, Journal of Econometrics, 181, Carroll, R. J., Chen, X., and Hu, Y. (2010), Identification and Estimation of Nonlinear Models Using Two Samples with Nonclassical Measurement Errors, Journal of Nonparametric Statistics, 22, DiTraglia, F. J. and García-Jimeno, C. (2017), Mis-Classified, Binary, Endogenous Regressors: Identification and Inference, Tech. rep., NBER NBER Working Paper

21 Bibliography ii Hu, Y. (2008), Identification and Estimation of Nonlinear Models with Misclassification Error Using Instrumental Variables: A General Solution, Journal of Econometrics, 144, Kasahara, H. and Shimotsu, K. (2009), Nonparametric Identification of Finite Mixture Models of Dynamic Discrete Choices, Econometrica, 77, Lewbel, A. (2007), Estimation of Average Treatment Effects with Misclassification, Econometrica, 75, Mahajan, A. (2006), Identification and Estimation of Regression Models with Misclassification, Econometrica, 74, Ura, T. (2018), Heterogeneous Treatment Effects with Mismeasured Endogenous Treatment, Quantitative Economics, Forthcoming. Yanagi, T. (2018), Inference on Local Average Treatment Effects for Misclassified Treatment, Hitotsubashi University. 21

A Note on the Closed-form Identi cation of Regression Models with a Mismeasured Binary Regressor

A Note on the Closed-form Identi cation of Regression Models with a Mismeasured Binary Regressor Xiaohong Chen Yale University Yingyao Hu y Johns Hopkins University Arthur Lewbel z Boston College First