Identi cation and Inference of Nonlinear Models Using Two Samples with Arbitrary Measurement Errors

Transcription

1 Ienti cation an Inference of Nonlinear Moels Using Two Samples with Arbitrary Measurement Errors Xiaohong Chen y New York University Yingyao Hu z University of Texas at Austin First version: February This version: November 2006 Abstract This paper consiers ienti cation an inference of a general latent nonlinear moel using two samples, in which a covariate contains arbitrary measurement errors in both samples, an neither sample contains an accurate measurement of the corresponing true variable. The primary sample consists of some epenent variables, some errorfree covariates, an an error-rien covariate, in which the measurement error has unknown istribution an coul be arbitrarily correlate with the latent true values. The auxiliary sample consists of another noisy measurement of the mismeasure covariate an some error-free covariates. We rst show that a general latent nonlinear moel is nonparametrically ienti e using the two samples when both coul have nonclassical errors, with no requirement of instrumental variables nor inepenence between the two samples. When the two samples are inepenent an the latent nonlinear moel is parameterize, we propose sieve quasi maximum likelihoo estimation (MLE) for the parameter of interest, an establish its root-n consistency an asymptotic normality uner possible misspeci cation, an its semiparametric e ciency uner correct speci- cation. We also provie a sieve likelihoo ratio moel selection test to compare two possibly misspeci e parametric latent moels. A small Monte Carlo simulation an an empirical example are presente. JEL classi cation: C0, C4. Keywors: Data combination, nonlinear errors-in-variables moel, nonclassical measurement error, nonparametric ienti cation, misspeci e parametric latent moel, sieve likelihoo estimation an inference. The authors woul like to thank R. Blunell, R. Carroll, P. Cross, S. Donal, B. Fitzenberger, J. Heckman, S. Hoerlein, A. Lewbel, E. Mammen, L. Nesheim, S. Schennach, M. Stinchcombe, C. Taber, an conference participants at the 2006 North American Summer Meeting of the Econometric Society an 2006 Southern Economic Association annual meeting for valuable suggestions. Chen acknowleges partial support from the National Science Founation. y Department of Economics, New York University, 269 Mercer Street, New York, NY 0003, tel: , xiaohong.chen@nyu.eu. z Department of Economics, University of Texas at Austin, University Station C300, Austin, TX 7872, tel: , hu@eco.utexas.eu.

2 Introuction The Measurement error problems are frequently encountere by researchers conucting empirical stuies in economics an other els in the social an natural sciences. A measurement error is calle classical if it is inepenent of the latent true values; otherwise, it is calle nonclassical. In econometrics an statistics there have been many stuies on ienti- cation an estimation of linear, nonlinear, an even nonparametric moels with classical measurement errors. See, e.g., Fricsh (934), Amemiya (985), Hsiao (989), Chesher (99), Hausman, Ichimura, Newey an Powell (99), Fan (99), Wansbeek an Meijer (2000), Newey (200), Taupin (200), Li (2002), Schennach (2004a, b), an Carroll et al. (2004), to name only a few. However, as reviewe in Boun, Brown, an Mathiowetz (200), valiation stuies in economic survey ata sets inicate that the errors in self-reporte variables, such as earnings, are typically correlate with the true values, an hence, are nonclassical. In fact, in many survey situations, a rational agent has an incentive to purposely report wrong values conitioning on his/her truth. Ample empirical eviences of nonclassical measurement errors have rawn growing attentions from theoretical research on econometric moels with nonclassical measurement errors. In the meantime, given that linear moels with measurement errors have been stuie thoroughly, recent research activities have been focusing on nonlinear (an/or nonparametric) Errors-In-Variables (EIV) moels. However, the ienti cation an estimation of general nonlinear (an/or nonparametric) moels with nonclassical errors is notoriously i cult. In this paper, we provie one solution to the nonlinear (an nonparametric) EIV problem by combining a primary sample an an auxiliary sample, where each sample only contains one measurement of the error-rien variable, an the measurement errors in both samples may be nonclassical. Our ienti cation strategy oes not require the existence of instrumental variables for the nonlinear moel of interest, nor oes it require an auxiliary sample containing the true values or inepenence between the two samples. It is well known that, without aitional information or restrictions, a general nonlinear moel cannot be ienti e in the presence of measurement errors. When point ienti cation is not feasible uner weak assumptions, some research activities have focuse on partial ienti cation an boun analyses. See, e.g., Chesher (99), Horowitz an Manski (995), Manski an Tamer (2003), Molinari (2004), an others. One approach to regaining ienti cation an consistent estimation of nonlinear EIV moels with classical errors is to impose parametric restrictions on error istributions; see, e.g., Fan (99), Buzas an Stefanski (996), Taupin (200), Hong an Tamer (2003), an others. However, it might be i cult to impose a correct parametric speci cation on error istributions for a nonlinear EIV moel with nonclassical errors. Another popular approach to ienti cation an estimation of EIV moels is to assume

3 the existence of Instrumental Variables (IVs). See Mahajan (2006), Lewbel (2006), an Hu (2006) for the use of the IV approach to ientify an consistently estimate nonlinear moels with misclassi cation errors in iscrete explanatory variables. Much important work has been one on using the IV approach to solve linear, nonlinear, an/or nonparametric EIV moels with classical errors in continuous explanatory variables. See, e.g., Amemiya (985), Amemiya an Fuller (988), Carroll an Stefanski (990), Hausman, Ichimura, Newey, an Powell (99), Hausman, Newey, an Powell (995), Wang an Hsiao (995), Li an Vuong (998), Newey (200), Li (2002), Schennach (2004a, b), an Carroll et al. (2004), to name only a few. Most recently, Hu an Schennach (2006) establish ienti cation an estimation of nonlinear EIV moels with nonclassical errors in continuous explanatory variables using IVs, where the IVs are exclue from the nonlinear moel of interest an are inepenent of the measurement errors. Although the IV approach is powerful, it might be i cult to n a vali IV for a general nonlinear EIV moel with nonclassical errors in applications. The alternative popular approach to ientifying nonlinear EIV moels with nonclassical errors is to combine two samples. See Carroll, Ruppert, an Stefanski (995) an Rier an Mo tt (2006) for a etaile survey of this approach. The avantages of this approach are that the primary sample coul contain arbitrary measurement errors an might not require the existence of IVs. However, the earlier papers using this approach typically assume the existence of a true valiation sample (i.e., an i.i.. sample that is from the same population as the primary sample an contains an accurate measurement of the true values). See, e.g., Boun, Brown, Duncan, an Rogers (989), Hausman, Ichimura, Newey, an Powell (99), Carroll an Wan (99), an Lee an Sepanski (995), to name only a few. Recent stuies using the two-sample approach have relaxe the true valiation sample requirement. For example, Hu an Rier (2006) show that the marginal istribution of the latent true values from an inepenent auxiliary sample is enough to ientify nonlinear EIV moels with a classical error. Chen, Hong, an Tamer (2005), Chen, Hong an Tarozzi (2005), an Ichimura an Martinez-Sanchis (2006) ientify an estimate nonlinear EIV moels with nonclassical errors using an auxiliary sample, which coul be obtaine as a strati e sample of the primary sample. Their approach oes not require the auxiliary sample to be a true valiation sample nor oes it require the auxiliary sample to have the same marginal istributions as those of the primary sample. Nevertheless, the approach still requires that the auxiliary sample contain an accurate measurement of the true values; such a sample might be i cult to n in some applications. In this paper, we provie nonparametric ienti cation of a nonlinear EIV moel with measurement errors in covariates by combining a primary sample an an auxiliary sample, in which each sample contains only one measurement of the error-rien explanatory variable, an the errors in both samples may be nonclassical. Our approach i ers from the IV approach in that we o not require an IV exclue from the moel of interest, an all 2

4 the variables in our samples may be inclue in the moel. Our approach is closer to the existing two-sample approach, since we also require an auxiliary sample an allow for nonclassical measurement errors in both samples. However, our ienti cation strategy i ers from the existing two-sample approach in that neither of our samples contains an accurate measurement of the true values. We assume that the primary sample consists of some epenent variables, some error-free covariates, an an error-rien covariate, in which the measurement error has unknown istribution an is allowe to be arbitrarily correlate with the latent true values. The auxiliary sample consists of some error-free covariates an another measurement of the mismeasure covariate. Even uner the assumption that the measurement error in the primary sample is inepenent of other variables conitional on the latent true values, it is clear that a general nonlinear EIV moel is not ienti e using the primary sample only, let alone using the auxiliary sample only. The ienti cation is mae possible by combining the two samples. We assume there are contrasting subsamples in the primary an the auxiliary samples. These subsamples may be i erent geographic areas, i erent age groups, or subpopulations with i erent observe emographic characteristics. We use the i erence between the marginal istributions of the latent true values in the contrasting subsamples of both the primary sample an the auxiliary sample to show that the error istribution is ienti e. To be speci c, assuming that the istributions of the common error-free covariates conitional on the latent true values are the same in the two samples, we may ientify the relationship between the measurement error istribution in the auxiliary sample an the ratio of the marginal istribution of latent true values in the subsamples. In fact, the ratio of the marginal istributions plays the role of an eigenvalue of an observe linear operator, while the measurement error istribution in the auxiliary sample is the corresponing eigenfunction. Therefore, the measurement error istribution may be ienti e through a iagonal ecomposition of an observe linear operator uner the normalization conition that the measurement error istribution in the auxiliary sample has zero moe (or zero meian or mean). The nonlinear moel of interest, e ne here as the joint istribution of the epenent variables, all the error-free covariates, an the latent true covariate in the primary sample, may then be nonparametrically ienti e. In this paper, we rst illustrate our ienti cation strategy using a nonlinear EIV moel with nonclassical errors in iscrete covariates of two samples. We then focus on nonparametric ienti cation of a general latent nonlinear moel with arbitrary measurement errors in continuous covariates. Our ienti cation result allows for fully nonparametric EIV moels an also allows for two correlate samples. But, in most empirical applications, the latent moels of interest are parametric nonlinear moels, an the two samples are regare as inepenent. Within this framework, we propose a sieve quasi maximum likelihoo estimation (MLE) for the latent nonlinear moel of interest using two samples with nonclassical measurement errors. Uner 3

5 possible misspeci cation of the latent parametric moel, we establish root-n consistency an asymptotic normality of the sieve quasi MLE of the nite imensional parameter of interest, as well as its semiparametric e ciency uner correct speci cation. However, i erent economic moels typically imply i erent parametrically speci e structural econometric moels, an parametric nonlinear moels coul be all misspeci e. We then provie a sieve likelihoo ratio moel selection test to compare two possibly misspeci e parametric nonlinear EIV moels using two inepenent samples with arbitrary errors. These results are extensions of those in White (982) an Vuong (989) to possibly misspeci e latent parametric nonlinear structural moels, an are also applicable to other possibly misspeci e semiparametric moels involving unobserve heterogeneity an/or nonparametric enogeneity. For example, one coul apply these results to erive vali inference without imposing the correct speci cation of the parametric structural moel in the famous mixture moel of Heckman an Singer (984). Finally, we present a small Monte Carlo simulation an an empirical illustration. We rst use simulate ata to estimate a probit moel with i erent nonclassical measurement errors. The Monte Carlo simulations show that the new two-sample sieve MLE performs well with the simulate ata. Secon, we apply our new estimator to a probit moel to estimate the e ect of earnings on voting behavior. It is well known that self-reporte earnings contain nonclassical errors. The primary sample is from the Current Population Survey (CPS) in November 2004, an the auxiliary sample is from the Survey of Income an Program Participation (SIPP). We use i erent marital status an gener as contrasting subsamples to ientify the error istributions in the auxiliary sample. This empirical illustration shows that our new estimator performs well with real ata. The rest of the paper is organize as follows. Section 2 establishes the nonparametric ienti cation of a general nonlinear EIV moel with (possibly) nonclassical errors using two samples. Section 3 presents the two-sample sieve quasi MLE an the sieve likelihoo ratio moel selection test uner possibly misspeci e parametric latent moels. Section 4 applies the two-sample sieve MLE to a latent probit moel with simulate ata an real ata. Section 5 brie y conclues, an the Appenix contains the proofs of the large sample properties of the sieve quasi MLEs. 2 Nonparametric Ienti cation 2. The ichotomous case: An illustration We rst illustrate our ienti cation strategy by escribing a special case in which the key variables in the moel are 0- ichotomous. Suppose that we are intereste in the e ect of 4

6 the true college eucation level X on the labor supply Y with the marital status W u an the gener W v as covariates. This e ect woul be ienti e if we coul ientify the joint ensity f X ;W u ;W v ;Y. In this example, we assume X ; W u, an W v are all 0- ichotomous. The true eucation level X is unobserve an is subject to measurement errors. (W u ; W v ) are accurately measure an observe in both the primary sample an the auxiliary sample, an Y is only observe in the primary sample. The primary sample is a ranom sample from (X; W u ; W v ; Y ), where X is a mismeasure X. In the auxiliary sample, we observe (X a ; W u a ; W v a ), in which the observe X a is a proxy of a latent eucation level X a, W u a is the marital status, an W v a is the gener. In this illustration subsection, we use italic letters to highlight all the assumptions impose for the nonparametric ienti cation of f X ;W u ;W v ;Y, while etaile iscussions of the assumptions are postpone to subsection 2.2. We assume that the measurement error in X is inepenent of all other variables in the moel conitional on the true value X, i.e., f XjX ;W u ;W v ;Y = f XjX. In our simple example, this assumption implies that all the people with the same eucation level have the same pattern of misreporting the latent true eucation level. This assumption can be relaxe if there are more common covariates in the two samples. Uner this assumption, the probability istribution of the observables equals f X;W u ;W v ;Y (x; u; v; y) = X x =0; f XjX (xjx )f X ;W u ;W v ;Y (x ; u; v; y) for all x; u; v; y: (2.) We e ne the matrix representations of f XjX as follows: 0 L XjX f XjX (0j0) f XjX (j0) f XjX (0j) f XjX (j) A : Notice that the matrix L XjX contains the same information as the conitional ensity f XjX. Equation (2.) then implies for all u; v; y f X;W u ;W v ;Y (0; u; v; y) f X;W u ;W v ;Y (; u; v; y) A = L XjX f X ;W u ;W v ;Y (0; u; v; y) f X ;W u ;W v ;Y (; u; v; y) A : (2.2) Equation (2.2) implies that the ensity f X ;W u ;W v ;Y woul be ienti e provie that L XjX woul be ienti able an invertible. Moreover, equation (2.) implies, for the subsamples of 5

7 males (W v = ) an of females (W v = 0) f X;W u jw v =j(x; u) = X x =0; = X x =0; f XjX ;W u ;W v =j (xjx ; u) f W u jx ;W v =j(ujx )f X jw v =j(x ): f XjX (xjx ) f W u jx ;W v =j(ujx )f X jw v =j(x ); (2.3) in which f X;W u jw v =j(x; u) f X;W u jw v(x; ujj) an j = 0;. By counting the numbers of knowns an unknowns in equation (2.3), one can see that the unknown ensity f XjX together with other unknowns cannot be ienti e using the primary sample alone. In the auxiliary sample, we assume that the measurement error in X a satis es the same conitional inepenence assumption as that in X, i.e., f XajX a ;W a u;w a v = f X ajxa. Furthermore, we link the two samples by a stable assumption that the istribution of the marital status conitional on the true eucation level an gener is the same in the two samples, i.e., f W u a jxa;w a v =j(ujx ) = f W u jx ;W v =j(ujx ) for all u; j; x. Therefore, we have for the subsamples of males (Wa v = ) an of females (Wa v = 0): f Xa;W u a jw v a =j (x; u) = X x =0; = X x =0; f XajX a ;W u a ;W v a =j (xjx ; u) f W u a jx a ;W v a =j (ujx )f X a jw v a =j (x ) f XajX a (xjx ) f W u jx ;W v =j(ujx )f X a jw v a =j (x ): (2.4) We e ne the matrix representations of relevant ensities for the subsamples of males (W v = ) an of females (W v = 0) in the primary sample as follows: for j = 0;, L X;W u jw v =j = L W u jx ;W v =j = L X jw v =j = f X;W u jw v =j(0; 0) f X;W u jw v =j(0; ) A f X;W u jw v =j(; 0) f X;W u jw v =j(; ) f W u jx ;W v =j(0j0) f W u jx ;W v =j(0j) A f W u jx ;W v =j(j0) 0 f W u jx ;W v f X jw v =j(0) 0 A ; 0 f X jw v =j() T where the superscript T stans for the transpose of a matrix. We similarly e ne the matrix representations L Xa;W u a jw v a =j, L XajX a, L W u a jx a ;W v a =j, an L X a jw v a =j of the corresponing ensities f Xa;W u a jw v a =j, f XajX a, f W u a jx a;w v a =j an f X a jw v a =j in the auxiliary sample. We note 6

8 that equation (2.3) implies for j = 0; ; L XjX L X jw v =jl W u jx ;W v =j 0 0 = L f X jw v =j(0) 0 f W u jx ;W v =j(0j0) f W u jx ;W v =j(0j) 0 0 f X jw v =j() f W u jx ;W v =j(j0) f W u jx ;W v =j(j) = L f W u ;X jw v =j(0; 0) f W u ;X jw v =j(; 0) A 0 f W u ;X jw v =j(0; ) f W u ;X jw v =j(; ) 0 f XjX (0j0) f XjX (0j) f W u ;X jw v =j(0; 0) f W u ;X jw v =j(; 0) A f XjX (j0) 0 f XjX (j) f W u ;X jw v =j(0; ) f W u ;X jw v =j(; ) f X;W u jw v =j(0; 0) f X;W u jw v =j(0; ) A f X;W u jw v =j(; 0) f X;W u jw v =j(; ) = L X;W u jw v =j, A T that is L X;W u jw v =j = L XjX L X jw v =jl W u jx ;W v =j: (2.5) Similarly, equation (2.4) implies that L Xa;W u a jw v a =j = L XajX a L X a jw v a =j L W u jx ;W v =j: (2.6) We assume that the observable matrices L Xa;W a ujw a v=j an L X;W u jw v =j are invertible, that the iagonal matrices L X jw v =j an L X a jwa v =j are invertible, an that L XajX a is invertible. Then equations (2.5) an (2.6) imply that L XjX an L W u jx ;W v =j are invertible, an we can then eliminate L W u jx ;W v =j, to have for j = 0; L Xa;W u a jw v a =jl X;W u jw v =j = L X ajx a L X ajw v a =jl X jw v =j L XjX : 7

9 Since this equation hols for j = 0;, we may then eliminate L XjX, to have L Xa;X a L Xa;W a ujw a v= L X;W u jw v = L Xa;W a ujw a v=0 L X;W u jw v =0 = L XajX a 0 L X a jwa v =L X jw v = L X jw v =0L XajW a v =0 0 L X f X ajxa (0j0) f X ajxa (0j) k X (0) a 0 A (2.7) f XajX a (j0) f X ajxa (j) 0 0 k X a f X ajxa (0j0) f X ajxa (0j) A : f XajX a (j0) f X ajxa (j) with k X a (x ) = f X ajw v a = (x ) f X jw v =0 (x ) f X jw v = (x ) f X a jw v a =0 (x ) : L X a jw v a =L X jw v = L X jw v =0L Notice that the matrix is iagonal because L X jw v =j an L X a jwa v=j are iagonal matrices. The equation (2.7) provies an eigenvalue-eigenvector ecomposition of an observe matrix L Xa;Xa on the left-han sie. If such a ecomposition is unique, then we may ientify L XajX a, i.e., f X ajxa, from the observe matrix L X a;x a. We assume that k X a (0) 6= k X a (); i.e., the eigenvalues are istinctive. This assumption requires that the istributions of the latent eucation level of males or females in the primary sample are i erent from those in the auxiliary sample, an that the istribution of the latent eucation level of males is i erent from that of females in one of the two samples. Notice that each eigenvector is a column in L XajX a, which is a conitional ensity. That means each eigenvector is automatically normalize. Therefore, for an observe L Xa;Xa, we may have an eigenvalue-eigenvector ecomposition as follows: L Xa;X a = X a jw v a =0 f X ajxa (0jx ) 0 f XajX a (0jx 2) k Xa (x ) 0 A (2.8) f XajX a (jx ) f XajX a (jx 2) 0 0 k X a (x f X ajxa (0jx ) f XajX a (0jx 2) A : f XajX a (jx ) f XajX a (jx 2) The value of each entry on the right-han sie of equation (2.8) can be irectly compute from the observe matrix L Xa;Xa. The only ambiguity left in equation (2.8) is the value of the inices x an x 2, or the inexing of the eigenvalues an eigenvectors. In other wors, the ienti cation of f XajX a boils own to ning a -to- mapping between the following two 8

10 sets of inices of the eigenvalues an eigenvectors: fx ; x 2g () f0; g : Next, we make a normalization assumption that people with (or without) college eucation in the auxiliary sample are more likely to report that they have (or o not have) college eucation; i.e., f XajX a (x jx ) > 0:5 for x = 0;. (This assumption also implies the invertibility of L XajX a.) Since the values of f X ajx a (0jx ) an f XajX a (jx ) are known in equation (2.8), this assumption pins own the inex x as follows: 8 < 0 if f x XajX a = (0jx ) > 0:5 : if f XajX a (jx ) > 0:5 : The value of x 2 may be foun in the same way. In summary, we have ienti e L XajX a, i.e., f XajX a ; from the ecomposition of the observe matrix L X a;x a. After ientifying L XajX a, we can ientify L W u jx ;W v =j or f W u jx ;W v =j from equation (2.6) as follows: L X a jw v a =j L W u jx ;W v =j = L X ajx a L X a;w u a jw v a =j ; in which two matrices L X a jw v a =j an L W u jx ;W v =j can be ienti e through their prouct on the left-han sie. Moreover, the ensity f XjX or the matrix L XjX is ienti e from equation (2.5) as follows: L XjX L X jw v =j = L X;W u jw v =jl W u jx ;W v =j ; in which we may ientify two matrices L XjX an L X jw v =j from their prouct on the lefthan sie. Finally, the ensity of interest f X ;W u ;W v ;Y is ienti e from equation (2.2). This simple example with ichotomous variables emonstrates that we can nonparametrically ientify the moel of interest using the similarity of the error structures an the i erence in the latent istributions between the two samples. We next show that such a nonparametric ienti cation strategy is in fact generally applicable. 2.2 The general case We are intereste in a moel containing variables X ; W; an Y. ienti e if we can ientify the joint probability ensity of X ; W; Y : We say the moel is f X ;W;Y (x ; w; y); (2.9) 9

11 in which X is an unobserve scalar covariate subject to measurement errors, W is a vector of accurately measure covariates that are observe in both the primary sample an the auxiliary sample, an Y is a vector of other variables, incluing epenent variables an other covariates, which are observe in the primary sample only. The primary sample is a ranom sample from (X; W T ; Y T ), in which X is a mismeasure X. Suppose the supports of X; W; Y; an X are X R, W R w, Y R y, an X R, respectively. Let f XjX an f X jx enote the conitional ensities of X given X ; an of X given X; respectively. Let f X an f X enote the marginal ensities of X an X ; respectively. We assume that the measurement error in X satis es Assumption 2. f XjX ;W;Y (xjx ; w; y) = f XjX (xjx ) for all x 2 X, x 2 X, w 2 W, an y 2 Y. Assumption 2. implies that the measurement error in X is inepenent of all other variables in the moel conitional on the true value X. The measurement error in X may still be correlate with the true value X in an arbitrary way, an hence is nonclassical. We realize that assumption 2. might be restrictive in some applications. But it is reasonable to believe that the latent true value X is a more important factor in the reporte value X than any other variables W an Y. Assumption 2. allows for a very general nonclassical error an captures the major concern regaring misreporting errors. Uner assumption 2., the probability ensity of the observe vectors equals Z f X;W;Y (x; w; y) = f XjX (xjx )f X ;W;Y (x ; w; y)x X for all x; w; y: (2.0) Let L p (X ), p < enote the space of functions with R X jh(x)jp x <, an let L (X ) be the space of functions with sup x2x jh(x)j <. Then it is clear that for any xe w 2 W, y 2 Y, f X;W;Y (; w; y) 2 L p (X ) ; an f X ;W;Y (; w; y) 2 L p (X ) for all p. Let H X L 2 (X ) an H X L 2 (X ) : We e ne the integral operator L XjX : H X! H X as follows: Z fl XjX hg (x) = f XjX (xjx ) h (x ) x X for any h 2 H X, x 2 X. Therefore, equation (2.0) becomes: f X;W;Y (x; w; y) = fl XjX f X ;W;Y (; w; y)g (x). Then the latent ensity f X ;W;Y coul be ienti e from the observe ensity f X;W;Y, provie that the operator L XjX is ienti able an invertible. We will show that f XjX can be ienti e by combining the information from the primary sample with that from an auxiliary sample. Suppose that we observe an auxiliary sample, which is a ranom sample from (X a ; Wa T ), in which X a is a mismeasure Xa. In orer to combine them, the two samples shoul have 0

12 something in common. We consier a series of mutually exclusive subsets V ; V 2 ; :::; V J W in the two samples. For example, the two samples may contain subpopulations with i erent emographic characteristics, such as race, gener, profession, an geographic location. Suppose W = (W u ; W v ) T an W a = (Wa u ; Wa v ) T, in which W u an Wa u are scalar covariates with support W u R, an W v an Wa v are iscrete variables with the same support W v = fv ; v 2 ; :::; v J g inicating the characteristics above. We will iscuss a case in which there exist extra common covariates i.e., (W u ; W v ) W an (Wa u ; Wa v ) W a later, in Remark 2.5. We may let V j = fv j g. Let X a R enote the support of X a. We assume Assumption 2.2 (i) Xa, Wa u, an Wa v have the same supports as X, W u, an W v, respectively; (ii) f XajX a ;W a u;w a v(xjx ; u; v) = f XajX a (xjx ) for all x 2 X a, x 2 X, u 2 W u, an v 2 W v. Assumption 2.2 implies that the istribution of measurement error in X a is inepenent of (W u a ; W v a ), conitional on the true value X a. This assumption is consistent with assumption 2. impose on the primary sample. Assumption 2.3 f W u a jx a ;V j (ujx ) = f W u jx ;V j (ujx ) for all u 2 W u R an x 2 X. Assumption 2.3 implies the conitional istribution of the scalar covariate W u given the true value X is the same in each subsample corresponing to V j in the two samples. If the conitional ensities escribe an unknown economic relationship between the two variables, assumption 2.3 requires that such a relationship be stable across the two samples. If such a common covariate oes not exist in either of the two samples, there is basically no common information to link them. In the case in which V j correspons to each possible value of W v, assumption 2.3 can be written as f W u a jxa;w a v = f W u jx ;W v. We note that, uner assumption 2.3, the marginal istributions of the true value X an the vector of covariates W in the primary sample may still be i erent from those of Xa an W a in the auxiliary sample. For each subsample V j, assumption 2. implies that f X;W u jv j (x; u) = = Z Z f XjX ;W u ;V j (xjx ; u) f W u jx ;V j (ujx )f X jv j (x )x f XjX (xjx ) f W u jx ;V j (ujx )f X jv j (x )x (2.) in the primary sample. Similarly, assumptions 2.2 an 2.3 imply that f Xa;W a ujv (x; u) = f j XajX a ;W a u;v j (xjx ; u) f W u a jxa ;V j (ujx )f X a jv j (x )x Z Z = f XajX a (xjx ) f W u jx ;V j (ujx )f X a jv j (x )x (2.2)

13 in the auxiliary sample. We e ne the following operators for the primary sample L X;W u jv j h Z (x) = f X;W u jv j (x; u)h (u) u; L W u jx;v j h Z (x) = f W u jx;v j (ujx)h (u) u; L W u jx ;V j h Z (x ) = f W u jx ;V j (ujx )h (u) u; L X jv j h (x ) = f X jv j (x )h (x ) : We also e ne the operators L XajX a, L X a;w u a jv j, L W u a jxa;v j, an L X a jv j for the auxiliary sample in the same way as their counterparts for the primary sample. Notice that operators L X jv j an L X a jv j are iagonal operators. By equation (2.) an the e nition of the operators, we have, for any function h, L X;W u jv j h (x) Z = f X;W u jv j (x; u)h (u) u Z Z = f XjX (xjx ) f W u jx ;V j (ujx )f X jv j (x )x h (u) u Z Z = f XjX (xjx ) f X jv j (x ) f W u jx ;V j (ujx )h (u) u x Z = f XjX (xjx ) f X jv j (x ) L W u jx ;V j h (x ) x Z = f XjX (xjx ) L X jv j L W u jx ;V j h (x ) x = L XjX L X jv j L W u jx ;V j h (x) : This means we have the operator equivalence L X;W u jv j = L XjX L X jv j L W u jx ;V j (2.3) in the primary sample. Similarly, equation (2.2) an the e nition of the operators imply L Xa;W u a jv j = L XajX a L X ajv j L W u jx ;V j (2.4) in the auxiliary sample. While the left-han sies of equations (2.3) an (2.4) are observe, the right-han sies contain unknown operators corresponing to the error istributions (L XjX an L XajX a ), the marginal istributions of the latent true values (L X jv j an L X a jv j ), an the conitional istribution of the scalar common covariate (L W u jx ;V j ). Equations (2.3) an (2.4) imply that one cannot apply the ienti cation results in Hu an Schennach (2006, the IV approach) to the primary sample (or the auxiliary sample) to ientify the error istribution f XjX (or f XajX a ). Although the epenent variable in their 2

14 paper may not be a variable of interest, the IV approach still requires that, conitional on the latent true values, the epenent variable be inepenent of the mismeasure values an the IV, an that the epenent variable vary with the latent true values. Intuitively, this requirement in the IV approach implies that the epenent variable still contains information on the latent true variable even conitional on all other observe variables, an that the epenent variable cannot be generate from the observe variables by researchers. Therefore, the inicator of subsample V j in our approach cannot play the role of epenent variable in their IV approach because V j is generate from the observe W (or W a ); hence, in general, W u (or W u a ) is not inepenent of V j conitional on X (or X a). In other wors, the common variable W u (or W u a ) cannot play the role of the IV because it is generally correlate with the inicator of subsample V j. In orer to ientify the unknown operators in equations (2.3) an (2.4), we assume Assumption 2.4 L XajX a : H Xa! H X a is injective, i.e., the set fh 2 H X a : L XajX a h = 0g = f0g. Assumption 2.4 implies that the inverse of the linear operator L XajX a exists. Recall that the conitional expectation operator of Xa given X a, E X a jx a : L 2 (X )! L 2 (X a ), is e ne as Z fe X a jx a h 0 g(x) = f X a jx a (x jx) h 0 (x )x = E[h 0 (Xa) jx a = x] for any h 0 2 L 2 (X ), x 2 X a. X We have fl XajX a hg (x) = R X f XajX a (xjx ) h (x ) x = E H X a, x 2 X a. Assumption 2.4 is equivalent to E h fxa (x)h(x a) f X a (X a ) h h (X a) f Xa (Xa) f X a (X a) jx a i jx a = x i for any h 2 = 0 implies h = 0. If 0 < f X a (x ) < over int(x ) an 0 < f Xa (x) < over int(x a ) (which are very minor restrictions), then assumption 2.4 is the same as the ienti cation conition impose in Newey an Powell (2003), Darolles, Florens, an Renault (2005), an Carrasco, Florens, an Renault (2006), among others. Moreover, as shown in Newey an Powell (2003), this conition is implie by the completeness of the conitional ensity f X a jx a, which is satis e when f X a jx a belongs to an exponential family. In fact, if we are willing to assume sup x ;w f X a ;W a (x ; w) c <, then a su cient conition for assumption 2.4 is the boune completeness of the conitional ensity f X a jx a ; see, e.g., Blunell, Chen, an Kristensen (2004) an Chernozhukov, Imbens, an Newey (2006). When X a an Xa are iscrete, assumption 2.4 requires that the support of X a is not smaller than that of Xa. Assumption 2.5 (i) f X jv j > 0 an f X a jv j > 0; (ii) L W u jx;v j is injective an f XjVj > 0; (iii) L W u a jx a;v j is injective an f XajV j > 0. By the e nition of L X;W u jv j, we have L W u jx;v j h (x) = f XjVj L (x) X;W u jv j h (x). Assumption 2.5(ii) then implies that L X;W u jv j is invertible. Similarly, assumption 2.5(iii) implies 3

15 that L Xa;W a u jv j is invertible. Therefore, assumptions 2.4 an 2.5 imply that all the operators involve in equations (2.3) an (2.4) are invertible. More precisely, assumptions 2.4, 2.5(i), an 2.5(iii), an equation (2.4) imply that the operator L W u jx ;V j is injective. Next, the injectivity of L W u jx ;V j, assumptions 2.5(i) an 2.5(ii), an equation (2.3) imply that L XjX is injective. Remark 2. Uner equations (2.3) an (2.4) an assumption 2.5 (i), the invertibility (or injectivity) of any three operators from L X;W u jv j, L XjX, L W u jx ;V j, L Xa;W a ujv an L j X ajxa implies the invertibility (or injectivity) of the remaining two operators. Therefore, assumptions 2.4 an 2.5 (ii)(iii) coul be replace by alternative conitions that still imply the invertibility of all ve operators. We ecie to impose assumptions 2.5 (ii) an (iii) since they are conitions on observables only an can be veri e from ata. Nevertheless, we coul replace assumption 2.4 with the assumption that the operator L XjX : H X! H X is injective. In this paper, we impose assumption 2.4 an allow the conitional istribution f XjX in the primary sample to be very exible. Uner assumptions 2.4 an 2.5, for any given V j we can eliminate L W u jx ;V j (2.3) an (2.4) to obtain in equations L Xa;W u a jv j L X;W u jv j = L XajX a L X a jv j L X jv j L XjX : (2.5) This equation hols for all V i an V j. We may then eliminate L XjX to have L ij X a;x a L Xa;W a u jv j L X;W u jv j L Xa;W a u jv i L = L XajX a L X a jv j L X jv j L X jv i L Xa jv L i X ajxa X;W u jv i L XajX a Lij X a L X ajx a : (2.6) The operator L ij X a;x a on the left-han sie is observe for all i an j. An important observation is that the operator L ij X = L a X a jv j L X jv j L X jv i L XajV i is a iagonal operator e ne as L ijxa h (x ) = k ij Xa (x ) h (x ) (2.7) with k ij X a (x ) f X ajv j (x ) f X jv i (x ) f X jv j (x ) f X a jv i (x ) : Equation (2.6) implies a iagonalization of an observe operator L ij X a;x a. An eigenvalue of L ij X a;x a equals k ij X a (x ) for a value of x, which correspons to an eigenfunction f XajX a (jx ). 4

16 The structure of equation (2.6) is similar to that of equation 8 in Hu an Schennach (2006, the IV approach), in the sense that both equations provie a iagonal ecomposition of observe operators whose eigenfunctions correspon to measurement error istributions. Therefore, the same technique of operator iagonalization is use for the ienti cation of f XajX a. On the one han, these results imply that the technique of operator iagonalization is a very powerful tool in the ienti cation of nonclassical measurement error moels. On the other han, the i erence between our equation (2.6) an their equation 8 also shows how the ienti cation strategy in our paper i ers from the IV approach in Hu an Schennach (2006). An eigenvalue in their IV approach is a value of the latent ensity of interest, while an eigenvalue in our paper is a value of the ratio of marginal istributions of the latent true values in i erent subpopulations. Moreover, the eigenvalues in our paper o not egenerate to those in the IV approach, or vice versa. Therefore, although both papers use the operator ecomposition technique, our ienti cation strategy is very i erent from that of the IV approach. Remark 2.2 We may also eliminate L XajX a in equation (2.5) for i erent V i an V j to obtain L Xa;W a ujv L i X;W u jv i L Xa;W a ujv L j X;W u jv j = L XjX L X jv i L Xa jv L X i a jv L j X jv j L XjX : This equation also provies a iagonalization of an observe operator on the left-han sie. If we impose the same restriction on f XjX as those that will be introuce on f XajX a, the same ienti cation proceure of f XajX a also applies to f XjX. In this paper, we impose the restrictions on the error istribution f XajX a in the auxiliary sample so that we may consier more general measurement errors in the primary sample. We now show the ienti cation of f XajX a an kij Xa (x ). First, we require the operator L ij X a;x a to be boune so that the iagonal ecomposition may be unique; see, e.g., Dunfor an Schwartz (97). Equation (2.6) implies that the operator L ij X a;x a has the same spectrum as the iagonal operator L ij X. Since an operator is boune by the largest element of a its spectrum, it is su cient to assume Assumption 2.6 k ij X a (x ) < for all i; j 2 f; 2; :::; Jg for all x. Notice that the subsets V ; V 2 ; :::; V J W o not nee to be collectively exhaustive. We may only consier those subsets in W in which these assumptions are satis e. Secon, although it implies a iagonalization of the operator L ij X a;x a, equation (2.6) oes not guarantee istinctive eigenvalues. If there exist uplicate eigenvalues, there exist two linearly inepenent eigenfunctions corresponing to the same eigenvalue. A linear combination of the two eigenfunctions is also an eigenfunction corresponing to the same eigenvalue. 5

17 Therefore, the eigenfunctions may not be ienti e in each ecomposition corresponing to a pair of i an j. However, such ambiguity can be eliminate by noting, importantly, that the observe operators L ij X a;x a for all i; j share the same eigenfunctions f XajX a (jx ). In orer to istinguish each linearly inepenent eigenfunction, we assume Assumption 2.7 For any x 6= x 2, there exist i; j 2 f; 2; :::; Jg, such that k ij X a (x ) 6= k ij X a (x 2). Assumption 2.7 implies that, for any two i erent eigenfunctions f XajX a (jx ) an f XajX a (jx 2), one can always n two subsets V j an V i such that the two i erent eigenfunctions correspon to two i erent eigenvalues k ij Xa (x ) an k ij Xa (x 2) an, therefore, are ienti e. Although there may exist uplicate eigenvalues in each ecomposition corresponing to a pair of i an j, this assumption guarantees that each eigenfunction f XajX a (jx ) is uniquely etermine by combining all the information from a series of ecompositions of L ij X a;x a for i; j 2 f; 2; :::; Jg. Remark 2.3 () Assumption 2.7 oes not hol if f X jv j (x ) = f X a jv j (x ) for all V j an all x 2 X. This assumption requires that the two samples be from i erent populations; one cannot use two subsets of a ranom sample as the primary sample an the auxiliary sample for ienti cation, which is i erent from the setup in Chen, Hong, an Tamer (2005). Given assumption 2.3 an the invertibility of the operator L W u jx ;V j, one coul check assumption 2.7 from the observe ensities f W u jv j an f W u a jv j. In particular, if f W u jv j (u) = f W u a jv j (u) for all V j an all u 2 W u, then assumption 2.7 is not satis e. (2) Assumption 2.7 oes not hol if f X jv j (x ) = f X jv i (x ) an f X a jv j (x ) = f X a jv i (x ) for all V j 6= V i an all x 2 X. This means that the marginal istribution of X or Xa shoul be i erent in the subsamples corresponing to i erent V j in at least one of the two samples. For example, if X or Xa are earnings an V j correspons to gener, then assumption 2.7 requires that the earning istribution of males be i erent from that of females in one of the samples (either the primary or the auxiliary). Given the invertibility of the operators L XjX an L XajX a, one coul check assumption 2.7 from the observe ensities f XjVj an f XajVj. In particular, if f XjVj (x) = f XjVi (x) for all V j 6= V i, an all x 2 X, then assumption 2.7 requires the existence of an auxiliary sample such that f XajVj (X a ) 6= f XajVi (X a ) with positive probability for some V j 6= V i. We now provie an example of the marginal istribution of X to illustrate that assumptions 2.6 an 2.7 are easily satis e. Suppose that the istribution of X in the primary sample is the stanar normal, i.e., f X jv j (x ) = (x ) for j = ; 2; 3, where is the probability ensity function of the stanar normal, an the istribution of X a in the auxiliary 6

18 sample is for 0 < < an 6= 0 8 >< f X a jv j (x ) = >: (x ) for j = ( x ) for j = 2 (x ) for j = 3 : (2.8) It is obvious that assumption 2.6 is satis e with 8 < k ij Xa (x ) = : exp 2 2 (x ) 2 for i =, j = 2 (x ) (x ) for i = ; j = 3 : (2.9) For i =,j = 2, any two eigenvalues k 2 X a (x ) an k 2 X a (x 2) of L 2 X a;x a may be the same if an only if x = x 2. In other wors, we cannot istinguish the eigenfunctions f XajX a (jx ) an f XajX a (jx 2) in the ecomposition of L 2 X a;x a if an only if x = x 2. Since k ij Xa (x ) for i = ; j = 3 is not symmetric aroun zero, the eigenvalues kx 3 a (x ) an kx 3 a (x 2) of L 3 X a;x a are i erent for any x = x 2. Notice that the operators L 2 X a;x a an L 3 X a;x a share the same eigenfunctions f XajX a (jx ) an f XajX a (jx 2). Therefore, we may istinguish the eigenfunctions f XajX a (jx ) an f XajX a (jx 2) with x = x 2 in the ecomposition of L 3 X a;x a. By combining the information obtaine from the ecompositions of L 2 X a;x a an L 3 X a;x a, we can istinguish the eigenfunctions corresponing to any two i erent values of x. The thir ambiguity is that, for a given value of x, an eigenfunction f XajX a (jx ) times a constant is still an eigenfunction corresponing to x. To eliminate this ambiguity, we nee to normalize each eigenfunction. Notice that f XajX a (jx ) is a conitional probability ensity for each x ; hence, R f XajX a (xjx ) x = for all x. This property of conitional ensity provies a perfect normalization conition. Fourth, in orer to fully ientify each eigenfunction, i.e., f XajX a, we nee to ientify the exact value of x in each eigenfunction f XajX a (jx ). Notice that the eigenfunction f XajX a (jx ) is ienti e up to the value of x. In other wors, we have ienti e a probability ensity of X a conitional on Xa = x with the value of x unknown. An intuitive normalization assumption is that the value of x is the mean of this ienti e probability ensity, i.e., x = R xf XajX a (xjx ) x; this assumption implies that the measurement error in the auxiliary sample has zero mean conitional on the latent true values. An alternative normalization assumption is that the value of x is the moe of this ienti e probability ensity, i.e., x = arg maxf XajX a (xjx ); this assumption implies that the error x istribution conitional on the latent true values has zero moe. The intuition behin this assumption is that people are more willing to report some values close to the latent true values than they are to report those far from the truth. Another normalization assump- 7

19 tion mayn be that the value of x is the meian of the ienti e probability ensity, i.e., x = inf x : R o x f X ajxa (xjx ) x ; this assumption implies that the error istribution 2 conitional on the latent true values has zero meian, an that people have the same probability of overreporting as that of unerreporting. Obviously, the zero meian conition can be generalize to an assumption that the error istribution conitional on the latent true values has a zero quantile. In summary, we use the following general normalizing conition to ientify the exact value of x for each eigenfunction f XajX a (jx ): Assumption 2.8 There exists a known functional M such that M f XajX a (jx ) = x for all x 2 X. Assumption 2.8 requires that the support of X a not be smaller than that of Xa. Recall that, in the ichotomous case, assumption 2.8 with zero moe also implies the invertibility of L XajX a (i.e., assumption 2.4). However, this is not true in the general iscrete case. For the general iscrete case, a comparable su cient conition for the invertibility of L XajX a is strictly iagonal ominance (i.e., the iagonal entries of L XajX a are all larger than 0.5), but assumption 2.8 with zero moe only requires that the iagonal entries of L XajX a be the largest in each row, which cannot guarantee the invertibility of L XajX a when the support of Xa contains more than 2 values. After fully ientifying the ensity function f XajX a, we now show that the ensity of interest f X ;W;Y an f XjX are also ienti e. By equation (2.2), we have f Xa;W a u jv j = L XajX a f Wa u;x a jv. By the injectivity of operator L j X ajxa, the joint ensity f Wa u;x a jv may be j ienti e as follows: f W u a ;Xa jv = L j X f ajxa X a;wa ujv : j Assumption 2.3 implies that f W u a jx a;v j = f W u jx ;V j so that we may ientify f W u jx ;V j through f W u jx ;V j (ujx ) = f Wa u;x a jv (u; j x ) R fw u a ;XajV j (u; x )u for all x 2 X. By equation (2.3) an the injectivity of the ienti e operator L W u jx ;V j, we have L XjX L X jv j = L X;W u jv j L W u jx ;V j : (2.20) The left-han sie of equation (2.20) equals an operator with the kernel function f X;X jv j f XjX f X jv j. Since the right-han sie of equation (2.20) has been ienti e, the kernel f X;X jv j on the left-han sie is also ienti e. We may then ientify f XjX through f XjX (xjx ) = f X;X jv j (x; x ) R fx;x jv j (x; x )x for all x 2 X. 8

20 Finally, assumption 2. an the injectivity of L XjX imply that the ensity of interest f X ;W;Y is ienti e through f X ;W;Y = L XjX f X;W;Y : We summarize the ienti cation result in the following theorem: Theorem 2.4 Suppose assumptions hol. Then, the ensities f X;W;Y uniquely etermine f X ;W;Y, f XjX, an f XajX a. an f Xa;W a Remark 2.5 () When there exist extra common covariates in the two samples, we may consier more generally e ne W u an W u a, or relax assumptions on the error istributions in the auxiliary sample. On the one han, this ienti cation theorem still hols when we replace W u an W u a with a scalar measurable function of W an W a, respectively. For example, let g be a known scalar measurable function. Then the ienti cation theorem is still vali when assumptions 2.2, 2.3, an 2.5(ii iii) hol with W u = g(w ) an W u a = g(w a ). On the other han, we may relax assumptions 2. an 2.2(ii) to allow the error istributions to be conitional on the true values an the extra common covariates. (2) The ienti cation theorem oes not require that the two samples be inepenent of each other. 3 Sieve Quasi Likelihoo Estimation an Inference Our ienti cation result is very general an oes not require the two samples to be inepenent. However, for many applications, it is reasonable to assume that there are two ranom samples X i ; Wi T ; Yi T n an X i= aj ; Waj T n a that are mutually inepenent. j= As shown in Section 2, the ensities f Y jx ;W, f XjX ; f W u jx ;W v; f X jw v, f X ajxa, an f X a jwa v are nonparametrically ienti e uner assumptions Nevertheless, in empirical stuies, we typically have either a semiparametric or a parametric speci cation of the conitional ensity f Y jx ;W as the moel of interest. In this section, we treat the other ensities f XjX ; f W u jx ;W v; f X jw v, f X ajxa, an f Xa jw a v as unknown nuisance functions, but consier a parametrically speci e conitional ensity of Y given (X ; W T ): fg(yjx ; w; ) : 2 g, a compact subset of R, <. De ne Z 0 arg max 2 [log g(yjx ; w; )]f Y jx ;W (yjx ; w)y. The latent parametric moel is correctly speci e if g(yjx ; w; 0 ) = f Y jx ;W (yjx ; w) for almost all y; x ; w T (an 0 is calle true parameter value); otherwise it is misspeci e (an 0 is calle pseuo-true parameter value); see, e.g., White (982). 9

21 In this section we provie a root-n consistent an asymptotically normally istribute sieve MLE of 0, regarless of whether the latent parametric moel g(yjx ; w; ) is correctly speci e or not. When g(yjx ; w; ) is misspeci e, the estimator is better calle the sieve quasi MLE than the sieve MLE. (In this paper we have use both terminologies since we allow the latent moel g(yjx ; w; ) to either correctly or incorrectly specify the true latent conitional ensity f Y jx ;W.) Uner the correct speci cation of the latent moel, we show that the sieve MLE of 0 is automatically semiparametrically e cient, an provie a simple consistent estimator of its asymptotic variance. In aition, we provie a sieve likelihoo ratio moel selection test of two non-neste parametric speci cations of f Y jx ;W when both coul be misspeci e. To simplify notation without loss of generality, in this section we assume W T = (W u ; W v ), Wa T = (Wa u ; Wa v ) with W v ; Wa v 2 fv ; v 2 ; :::; v J g. We e ne V j as the subset of W with W v or Wa v equal to v j. Also, we assume that all the variables Y, W u, X, Wa u, an X a are scalars, an each has possibly unboune support (i.e., each coul have the whole real line as its support). 3. Sieve likelihoo estimation uner possible misspeci cation Let 0 ( T 0 ; f 0 ; f 0a ; f 02 ; f 02a ; f 03 ) T ( T 0 ; f XjX ; f XajX a ; f X jw v; f X a jw v a ; f W u jx ;W v)t enote the true parameter value, in which 0 is really pseuo-true when the parametric moel g(yjx ; w; ) is incorrectly speci e for the unknown true ensity f Y jx ;W. We introuce a ummy ranom variable S, with S = inicating the primary sample an S = 0 inicating the auxiliary sample. Then we have a big combine sample Z T t S t X t ; S t Wt T ; S t Yt T ; S t ; ( S t )X t ; ( S t )Wt T n+na t= such that fx t ; Wt T ; Yt T ; S t = g n t= is the primary sample an fx t ; Wt T ; S t = 0g n+na t=n+ is the auxiliary sample. Denote p Pr(S t = ) 2 (0; ). Then the observe joint likelihoo for 0 is in which n+n Y a t= [p f(x t ; W t ; Y t js t = ; 0 )] St [( p) f(x t ; W t js t = 0; 0 )] St ; f(x t ; W t ; Y t js t = ; 0 ) f X;W;Y (X t ; W t ; Y t ; 0 ; f 0 ; f 02 ; f 03 ) Z = f W v(wt v ) f 0 (X t jx )g(y t jx ; W t ; 0 )f 03 (Wt u jx ; Wt v )f 02 (x jwt v )x ; 20