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

Transcription

1 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 version November 006; Revised December 007. Abstract This note considers the identi cation of a nonparametric regression model with an unobserved 0- dichotomous regressor. The sample consists of a dependent variable and a 0- dichotomous proxy of the unobserved regressor. We obtain nonparametric identi cation of every element in the model as a closed-form function of the observed moments or densities. Our identi cation strategy does not require any additional sample information, such as instrumental variables or a secondary sample. The closedform solution may be used to construct estimators of the unknowns. Keywords Errors-In-Variables (EIV), Identi cation, Dichotomous variable. Department of Economics, Yale University, Box 088, New Haven, CT , USA. Tel xiaohong.chen@yale.edu. y Department of Economics, Johns Hopkins University, 440 Mergenthaler Hall, 3400 N. Charles Street, Baltimore, MD 8, USA. Tel yhu@jhu.edu. z Department of Economics, Boston College, 40 Commonwealth Avenue, Chestnut Hill, MA 0467 USA. Tel lewbel@bc.edu.

2 INTRODUCTION Binary variables are widely used in statistical studies. Just drawing from economics, some well known examples include employment status, union status, and education level (diploma or not). When a model containing such variables is estimated, one major concern is that these variables may be subject to reporting errors. For example, self-reported smoking behavior may not be accurate because an individual may not want others to know that he or she smokes. Ignoring such reporting errors in regressors generally leads to inconsistent model estimates. A well known strategy to deal with such misreporting or misclassi cation errors in binary variables is to use a secondary measurement or an instrumental variable. See Aigner (973), or more recently Mahajan (006) and Lewbel (007). Such additional sample information can yield parametric or nonparametric identi cation of the latent model. Without additional sample information, one usually can only identify bounds on features of the model, as in Klepper (988) and Bollinger (996). In contrast, we provide full nonparametric identi cation without using additional sample information. We consider the nonparametric regression model Y = m (X ) + ; E[jX ] = 0 (.) where Y is a scalar dependent variable, X is a 0- dichotomous regressor, and is the regression error. The variables X and are not observed. We observe a random sample of Y and a 0- dichotomous scalar X, where X is a proxy of the unobserved X.

3 De ne m j = m(j) for j = 0;. Note that since X is binary, identifying the function m (X ) is equivalent to identifying the constants m 0 and m. We could alternatively de ne a = m 0 and b = m a and without loss of generality rewrite the model as Y = a + bx +. In addition to identifying m 0 and m or equivalently a and b, we also identify the conditional distributions of Y (and hence ) conditional on X and the probability mass function of X given X. As we note later, our results readily extend to the case of Y = m (X ; W ) + where W is a vector of additional regressors that are observed without error. Our identi cation relies on some assumptions regarding the regression model instead of on additional sample information. The key assumption is that the rst three moments of the regression error are independent of the latent regressor. We show that the latent regression function is nonparametrically identi ed as a known function of observed moments. Our identi cation is constructive in the sense that it can directly lead to a consistent estimator. Other examples of obtaining identi cation in measurement error models without additional sample information include exploiting model restrictions as in Huwang and Hwang (00) or the use of higher moment error restrictions as in Lewbel (997) and Erickson and Whited (00). This note is organized as follows section provides the main identi cation results and section 3 summarizes the note and discusses extensions. All the proofs are in the appendix. NONPARAMETRIC IDENTIFICATION We now show how to obtain identi cation of the regression model.. We rst assume Assumption. X? jx. 3

4 This assumption implies that the measurement error X X is independent of the dependent variable Y conditional on the true value X. De ne m j = m(j) for j = 0;. Assumption. implies that the relationship between the observed density and the latent ones becomes f Y jx (yjj) = f X jx (0jj) f jx (y m 0 j0) + f X jx (jj) f jx (y m j) for j = 0; (.) This equation implies that the observed density f Y jx (yjj) is a mixture of two conditional densities f jx (y m 0 j0) and f jx (y m j). Note that we are using f to denote either a probability density function or a probability mass function, so since X and X are discrete, f X jx (j0) is equivalent to Pr (X = jx = 0) for example. Given that E[jX ] = 0, we then obtain the ordering of m j from that of observed j E(Y jx = j) under the following assumption Assumption. (i) > 0 ; (ii) f X jx (j0) + f X jx (0j) < Assumption.(i) is not restrictive because one can always rede ne X as X if needed. Assumption.(ii) reveals the ordering of m and m 0, by making it the same as that of and 0 because f X jx (j0) f X jx (0j) = 0 m m 0 ; so m > 0 m 0. Assumption.(ii) says that the sum of misclassi cation probabilities is less than one, meaning that, on average, the observations X are more accurate predictions of X than pure guesses. See Lewbel (007) for further discussion of this assumption. Assumption.3 E k jx = E k for k = ; 3. 4

5 For identi cation we only require restrictions on two moments of in this assumption, because we only need to solve for two unknowns, m 0 and m. A su cient condition for assumption.3 is that be independent of X, which is stronger than necessary because it makes the assumption hold for all k. For k =, assumption.3 says that the model errors are homoskedastic, and for k = 3 the assumption is that the error distributions conditional on X have the same skewness. A su cient condition for k = 3 in assumption.3 is symmetry of jx, which would make E k jx = 0 for all odd k. Properties like homoskedasticity and symmetry, or more generally independence naturally arise in some contexts, for example, these are common assumptions regarding measurement errors (in this case, would be interpreted as measurement error in Y ), or they may arise when errors are unobserved factors that are unrelated to the dichotomy given by X. Identi cation could also be obtained using alternative restrictions on jx including possible restrictions such as quantiles or modes instead of moments. For example, one of the moments in assumption.3 might be replaced with assuming that the density f jx =0 has zero median. Equation (.) would then imply that 05 = m 0 0 Z m0 f Y jx=0 (y)dy + m Z m0 f Y jx= (y)dy which may uniquely identify m 0 under some testable assumptions. An advantage of Assumption.3 over alternative restrictions on jx is that we obtain a closed-form solution for m 0 and m (see Chen, Hu, and Lewbel (007) for the general case). 5

6 h De ne v j E h Y j jx = j i, s j E Y j 3 jx = j i, C (v + ) (v 0 + 0) ; C 0 ( 0 ) + 3 v v 0 s s We leave the detailed proof to the appendix and present the results as follows Theorem. Suppose that equation (.), assumptions.,., and.3 hold. Then the density f Y jx uniquely determines f Y jx and f X jx. To be speci c, we have m 0 = C r C ; m = r C + C ; C f X jx (j0) = 0 p C ; f X jx (0j) = C p C ; and m j f Y jx =j(y) = f Y jx=0 (y) + m j 0 f Y jx= (y) Summary and Extensions This note provides a closed-form identi cation solution for every element in a regression model with a mismeasured dichotomous regressor. Our identi cation does not use any additional sample information. The key identi cation assumption is that the rst three moments of the regression error are independent of the latent regressor. When such a restriction on the latent model is reasonable in an application, our results suggest that one does not need a secondary measurement or an instrumental variable to identify the latent model. The closed form solution may directly lead to a consistent estimator. 6

7 As noted in the introduction, with a binary X our model is equivalent to Y = a+bx +. It may be possible to extend our method of identi cation to this linear model, or to other models such as polynomials, with more general distributions of X. Our assumptions would not su ce for identi cation of a linear model with arbitrarily distributed with X (Reiersol 950 provides a counterexample), but related identi cation results for linear and polynomial models with continuous regressors based on error moment restrictions exist in the literature, e.g., Lewbel (997) and Erickson and Whited (00). Although we only consider the case where these is a single regressor X, the extension to Y = m (X ; W ) + ; E[jX ; W ] = 0 where W is an additional vector of observed error-free covariates is immediate because our assumptions and identi cation results for model (.) can be all restated as conditional upon W. APPENDIX. MATHEMATICAL PROOFS Proof. (Theorem.) First, we introduce notations as follows for j = 0; h v j = E Y m j = m (j) ; j = E(Y jx = j); i h j jx = j, s j = E Y j 3 jx = j i, p = f X jx (j0) ; q = f X jx (0j) ; f Y jx=j (y) = f Y jx (yjj) We start the proof with equation (.), which is equivalent to fy jx (yj0) fx = jx (0j0) f X jx (j0) fjx =0(y m 0 ) f Y jx (yj) f X jx (0j) f X jx (j) f jx =(y m ) Using the notations above, we have fy jx=0 (y) p p = f Y jx= (y) q q 7 fjx =0(y m 0 ) f jx =(y m ) (A.)

8 Since E[jX ] = 0, we have We may solve for p and q as follows 0 = ( p) m 0 + pm and = qm 0 + ( q) m p = 0 m 0 m m 0 and q = m m m 0 (A.) We also have p q = Assumption. implies that m m m m 0 m > 0 m 0 = 0 m m 0 and fjx =0(y m 0 ) f jx =(y m ) = q p p q q p Plug-in the expression of p and q in equation (A.), we have fy jx=0 (y) f Y jx= (y) p p q = m ; q p q = m 0 ; and p p q = q p q ; q p q = p p q ; fjx =0(y m 0 ) f jx =(y m ) = = m 0 0 m m m 0 0 m 0 m m m ! fy jx=0 (y) f Y jx= (y)! fy jx=0 (y) f Y jx= (y) In other words, we have for j = 0; m j f jx =j(y) = f Y jx=0 (y + m j ) + m j 0 f Y jx= (y + m j ) (A.3) 0 0 In summary, f X jx (or p and q) and f jx are identi ed if we can identify m 0 and m. Next, we show that m 0 and m are indeed identi ed. By assumption.3, we have E k jx = E k for k = ; 3 For k =, we consider v = E (m(x ) ) jx = + E = E m(x ) jx = + E = qm 0 + ( q) m + E Similarly, we have v 0 = ( p) m 0 + pm 0 + E 8

9 We eliminate E ( ) to obtain, That is We have shown that ( p) m 0 + pm v = qm 0 + ( q) m v + v + v = ( p q) m m 0 ; p q = 0 m m 0 Thus, m and m 0 satisfy the following linear equation m + m 0 = (v + ) (v 0 + 0) 0 C This means we need one more restriction to identify m and m 0.We consider and s = E (Y ) 3 jx = = E (m (X ) ) 3 jx = + E 3 = q (m 0 ) 3 + ( q) (m ) 3 + E 3 s 0 = ( p) (m 0 0 ) 3 + p (m 0 ) 3 + E 3 We eliminate E ( 3 ) in the two equations above to obtain, ( p) (m 0 0 ) 3 + p (m 0 ) 3 s 0 = q (m 0 ) 3 + ( q) (m ) 3 s Plug in the expression of p and q in equation (A.), we have (m 0 ) (m 0 0 ) (m + m 0 0 ) s 0 = (m ) (m 0 ) (m + m 0 ) s ; Since m + m 0 = C ; we have (C m 0 0 ) (m 0 0 ) (C 0 ) + s 0 = (C m 0 ) (m 0 ) (C ) + s ; that is, Moreover, we have m 0 0 (C 0 ) + (m 0 0 ) C (C 0 ) + s 0 = m 0 (C ) + (m 0 ) C (C ) + s m 0 ( 0 ) + C ( 0 ) m 0 = (C ) 0 (C 0 ) C (C ) + 0 C (C 0 ) + s s 0 = 0 C ( 0 ) C + 0 C + s s 0 9

10 Since ( 0 ) > 0, we have m 0 + C m 0 = 3 ( + 0 ) C C + s s 0 0 Finally, we have where m 0 C + C = 0 C = 3 C 3 ( + 0 ) C s s = 3 [C ( + 0 )] 3 ( + 0 ) s s = 3 [C ( + 0 )] + ( 0 ) s s 0 0 v v 0 s s 0 = ( 0 ) h s j = E Y j 3 jx = j i = E Y 3 jx = j 3E Y jx = j j j 3 j = E Y 3 jx = j 3E Y jx = j j + 3 j j 3 j j + 3 j, C = 3 C 3 ( + 0 ) C s s 0 0 = 3 C 3 ( + 0 ) C ( ) 0 = 3 C 3 ( + 0 ) C 3 ( ) 0 = 3 C 3 ( + 0 ) = Notice that we also have m C + C = 0; which implies that m and m 0 are two roots of this quadratic equation. Since m > m 0, we 0

11 have m 0 = C r C ; m = r C + C After we have identi ed m 0 and m, p and q (or f X jx) are identi ed from equation (A.), and the density f (or f Y jx ) is also identi ed from equation (A.3). Thus, we have identi ed the latent densities f Y jx and f X jx from the observed density f Y jx. REFERENCES Aigner, D. J. 973, "Regression With a Binary Independent Variable Subject to Errors of Obervation," Journal of Econometrics,, Bollinger, C. R. 996, "Bounding Mean Regressions When a Binary Regressor is Mismeasured," Journal of Econometrics, 73, Chen, X., Y. Hu, and A. Lewbel, 007, Nonparametric identi cation and estimation of nonclassical errors-in-variables models without additional information, Cemmap Working Papers CWP8/07. (Centre for Microdata Methods and Practice) Erickson T. and T. M. Whited, 00, Two-Step GMM Estimation of the Errors-in-Variables Model Using High-order Moments, Econometric Theory,8, Huwang, L and J.T.G. Hwang, 00, "Prediction and con dence intervals for nonlinear measurement error models without identi ability information," Statistics and Probability Letters, 58, Klepper, S., 988, "Bounding the E ects of Measurement Error in Regressions Involving Dichotomous Variables," Journal of Econometrics, 37, Lewbel, A. 997, "Constructing Instruments for Regressions With Measurement Error When No Additional Data are Available, With an Application to Patents and R&D," Econometrica, 65, 0-3. Lewbel, A., 007, Estimation of average treatment e ects with misclassi cation, Econometrica, 75, Mahajan, A. 006, Identi cation and estimation of regression models with misclassi cation, Econometrica, 74, Reiersol, O. 950, Identi ability of a Linear Relation between Variables Which Are Subject to Error, Econometrica, 8,