Handbook of Econometrics Vols1-5 _ Chapter 23 pot

Chaprer 23 LATENT VARIABLE MODELS IN ECONOMETRICS DENNIS J AIGNER L’niversio, of Southern California CHENG HSIAO Unil*ersi{v of Toronto ARIE KAPTEYN Tilburg TOM Uniuersiy WANSBEEK* Nerherlands Central Bureau of Statistics Contents Introduction 1.1 Background 1.2 Our single-equation heritage 1.3 Multiple equations 1.4 Simultaneous equations 1S The power of a dynamic specification 1.6 Prologue Contrasts and similarities between structural and functional models 2.1 ML estimation in structural and functional models 2.2 Identification 2.3 Efficiency 2.4 The ultrastructural relations Single-equation models 3.1 Non-normality and identification: An example 3.2 Estimation in non-normal structural models 1323 1323 1324 1326 1327 1328 1329 1329 1330 1332 1335 1336 1337 1337 1338 *The authors would like to express their thanks to Zvi Griliches, Hans Schneeweiss, Edward Learner, Peter Bentler, Jerry Hausman, Jim Heckman, Wouter Keller, Franz Palm, and Wynand van de Ven for helpful comments on an early draft of this chapter and to Denzil Fiebig for considerable editorial assistance in its preparation Sharon Koga has our special thanks for typing the manuscript C Hsiao also wishes to thank the Social Sciences and Humanities Research Council of Canada and the National Science Foundation, and Tom Wansbeek the Netherlands Organization for the Advancement of Pure Research (Z.W.O.) for research support Handbook Q Elsmier of Econometrics, Vohlme II, Edited 61, Z Griliches Science Publishers BV, 1984 and M.D Intriligaror D J Aiper 1322 3.3 A non-normal model with extraneous information 3.4 Identifying restrictions in normal structural and functional models 3.5 Non-linear models 3.6 Should we include poor proxies? 3.7 Prediction and aggregation 3.8 Bounds on parameters in underidentified models 3.9 Tests for measurement error 3.10 Repeated observations 3.11 Bayesian analysis Multiple equations 4.1 4.2 4.3 Instrumental variables Factor analysis The MIMIC model and extensions Simultaneous equations 5.1 The case of D known 5.2 Identification and estimation 5.3 The analysis of covariance structures Dynamic models 6.1 6.2 Identification of single-equation models Identification of dynamic simultaneous equation models 6.3 Estimation of dynamic error-shock models References et al 1340 1341 1344 1345 1346 1347 1349 1350 1352 1353 1354 1357 1359 1362 1363 1363 1369 1372 1372 1377 1380 1386 Ch 23: L.utent 1.1 Vuriuble Models in Econometrics 1323 Introduction Background Although it may be intuitively clear what a “latent variable” is, it is appropriate at the very outset of this discussion to make sure we all agree on a definition Indeed, judging by a recent paper by a noted psychometrician [Bentler (1982)], the definition may not be so obvious The essential characteristic of a latent variable, according to Bentler, is revealed by the fact that the system of linear structural equations in which it appears cannot be manipulated so as to express the variable as a function of measured variables only This definition has no particular implication for the ultimate identifiability of the parameters of the structural model itself However, it does imply that for a linear structural equation system to be called a “latent variable model” there must be at least one more independent variable than the number of measured variables Usage of the term “independent” variable as contrasted with “exogenous” variable, the more common phrase in econometrics, includes measurement errors and the equation residuals themselves Bentler’s more general definition covers the case where the covariance matrices of the independent and measured variables are singular From this definition, while the residual in an otherwise classical single-equation linear regression model is not a measured variable it is also not a latent variable because it can be expressed (in the population) as a linear combination of measured variables There are, therefore, three sorts of variables extant: measured, unmeasured and latent The distinction between an unmeasured variable and a latent one seems not to be very important except in the case of the so-called functional errors-in-variables model For otherwise, in the structural model, the equation disturbance, observation errors, and truly exogenous but unmeasured variables share a similar interpretation and treatment in the identification and estimation of such models In the functional model, the “true” values of exogenous variables are fixed variates and therefore are best thought of as nuisance parameters that may have to be estimated en route to getting consistent estimates of the primary structural parameters of interest Since 1970 there has been a resurgence of interest in econometrics in the topic of errors-in-variables models or, as we shall hereinafter refer to them, models involving latent variables That interest in such models had to be restimulated at all may seem surprising, since there can be no doubt that economic quantities frequently are measured with error and, moreover, that many applications depend on the use of observable proxies for otherwise unobservable conceptual variables D A igner et al 1324 Yet even a cursory reading of recent econometrics texts will show that the historical emphasis in our discipline is placed on models without measurement error in the variables and instead with stochastic “shocks” in the equations TO the extent that the topic is treated, one normally will find a sentence alluding to the result that for a classical single-equation regression model, measurement error in the dependent variable, y, causes no particular problem because it can be subsumed within the equation’s disturbance term.’ And, when it comes to the matter of measurement errors in independent variables, the reader will usually be convinced of the futility of consistent parameter estimation in such instances unless repeated observations on y are available at each data point or strong a priori information can be employed And the presentation usually ends just about there We are left with the impression that the errors-in-variables “problem” is bad enough in the classical regression model; surely it must be worse in more complicated models But in fact this is not the case For example, in a simultaneous equations setting one may employ overidentifying restrictions that appear in the system in order to identify observation error variances and hence to obtain consistent parameter estimates (Not always, to be sure, but at least sometimes.) This was recognized as long ago as 1947 in an unpublished paper by Anderson and Hurwicz, referenced (with an example) by Chemoff and Rubin (1953) in one of the early Cowles Commission volumes Moreover, dynamics in an equation can also be helpful in parameter identification, ceteris paribus Finally, restrictions on a model’s covariante structure, which are commonplace in sociometric and psychometric modelling, may also serve to aid identification [See, for example, Bentler and Weeks (1980).] These are the three main themes of research with which we will be concerned throughout this essay After brief expositions in this Introduction, each topic is treated in depth in a subsequent section 1.2 Our single-equation heritage (Sections and 3) There is no reason to spend time and space at this point recreating the discussion of econometrics texts on the subject of errors of measurement in the independent variables of an otherwise conventional single-equation regression model But the setting does provide a useful jumping-off-place for much of what follows Let each observation ( y,, xi) in a random sample be generated by the stochastic relationships: _Y, Vi + u 1) = xi = E, + u, 17,=cr+P&+q, 0.1) i=l , n 0.3) ‘That is to say, the presence of measurement error iny does not alter the properties of least squares estimates of regression coefficients But the variance of the measurement error remains hopelessly entangled with that of the disturbance term Ch 23: Latent Variable Models in Econometrics 1325 Equation (1.3) is the heart of the model, and we shall assume E( VilEi) = (Y &, + so that I$&,) = and E(&e,) = Also, we denote I!($) = ueE.Equations (1.1) and (1.2) involve the measurement errors, and their properties are taken to be E(u,) = E(ui) = 0, E(u;) = a,,, E(u;) = a”” and E(u,ui) = Furthermore, we will assume that the measurement errors are each uncorrelated with E, and with the latent variables vi and 5, Inserting the expressions ti = xi - ui and T),= y, - ui into (1.3), we get: y,=a+px,+w;, 0.4) where w, = ei + ui - /3uj Now since E(uilxi) # 0, we readily conclude that least squares methods will yield biased estimates of (Yand /3 By assuming all random variables are normally distributed we eliminate any concern over estimation of the 5;‘s as “nuisance” parameters This is the so-called structural latent variables model, as contrasted to the functional model, wherein the &‘s are assumed to be fixed variates (Section 2) Even so, under the normality assumption no consistent estimators of the primary.parameters of interest exist This can easily be seen by writing out the so-called “covariance” equations that relate consistently estimable variances and covariances of the observables ( y, and x,) to the underlying parameters of the model Under the assumption of joint normality, these equations exhaust the available information and so provide necessary and sufficient conditions for identification They are obtained by “covarying” (1.4) with y, and x,, respectively Doing so, we obtain: Uyx = PU,, - I%>,, (1.5) Uxx = a[6 + UCL, Obviously, there are but three equations (involving three consistently estimable quantities, uYY, and a,,) and five parameters to be estimated Even if we agree a,, to give up any hope of disentangling the influences of si and ui (by defining, say, u2 = a_ + a,,) and recognize that the equation uXX uEE uoC, = + will always be used to identify art alone, we are still left with two equations in three unknowns (p, u2, and Q) The initial theme in the literature develops from this point One suggestion to achieve identification in (1.5) is to assume we know something about a,,, re&ue to u2 or uuu relative to uXX.Suppose this a priori information is in the form h = ~,,/a* Then we have a,, = Au2 and (1.5a) D J Aigner 1326 et (11 From this it follows that p is a solution to: p2xu,, - P( huv, - e,J- y,,X= 0, (1.6) and that (1.7) lJ2=u YY Pa,, Clearly this is but one of several possible forms that the prior information may take In Section 3.2 we discuss various alternatives A Bayesian treatment suggests itself as well (Section 3.11) In the absence of such information, a very practical question arises It is whether, in the context of a classical regression model where one of the independent variables is measured with error, that variable should be discarded or not, a case of choosing between two second-best states of the world, where inconsistent parameter estimates are forthcoming either from the errors-in-variables problem or through specification bias As is well known, in the absence of an errors-ofobservation problem in any of the independent variables, discarding one or more of them from the model may, in the face of severe multicollinearity, be an appropriate strategy under a mean-square-error (MSE) criterion False restrictions imposed cause bias but reduce the variances on estimated coefficients (Section 3.6) 1.3 Multiple equations (Section 4) Suppose that instead of having the type of information described previously to help identify the parameters of the simple model given by (l.l)-(1.3), there exists a z,, observable, with the properties that zi is correlated with xi but uncorrelated with w, This is tantamount to saying there exists another equation relating z, to x,, for example, x,=yz,+8i, (1.8) with E(z,&)= 0, E(6,)= and E(62)= uss Treating (1.4) and (1.8) as our structure (multinormality is again assumed) and forming the covariance equations, we get, in addition to (1.5): apz = P&Z Ux.x= YUZ, + *,,, (1.9) uZX Yo;z = It is apparent that the parameters of (1.8) are identified through the last two of these equations If, as before, we treat a,, + uuU as a single parameter, u2, then (1.5) and the first equation of (1.9) will suffice to identify p, u*, uL,“,and act This simple example serves to illustrate how additional equations containing the same latent variable may serve to achieve identification This “multiple 1321 Ch 23: L.utent Variable Models in Econometrics equations” approach, explored by Zellner (1970) and Goldberger (1972b), spawned the revival of latent variable models in the seventies 1.4 Simultaneous equations (Section 5) From our consideration of (1.4) and (1.8) together, we saw how the existence of an instrumental variable (equation) for an independent variable subject to measurement error could resolve the identification problem posed This is equivalent to suggesting that an overidentifying restriction exists somewhere in the system of equations from which (1.4) is extracted that can be utilized to provide an instrument for a variable like xi But it is not the case that overidentifying restrictions can be traded-off against measurement error variances without qualification Indeed, the locations of exogenous variables measured with error and overidentifying restrictions appearing elsewhere in the equation system are crucial To elaborate, consider the following equation system, which is dealt with in detail in Section 5.2: + Err I$+ P12Yz= Y1151 P21YI+ (1.10) Y2252 + Y23& + -529 Y2 = where [, ( j = 1,2,3) denote the latent exogenous variables in the system Were the latent exogenous variables regarded as obseruable, the first equation is-conditioned on this supposition-overidentified (one overidentifying restriction) while the second equation is conditionally just-identified Therefore, at most one measurement error variance can be identified Consider first the specifications x1 = [r + ut, x2 = t2, x3 = t3, and let ull denote the variance of ut The corresponding system of covariance equations turns out to be: Yll(%,x, - 000 =&&& I I -[ ( Y22%2x1+ 0 Y23%3x, > ( Yll%x,x* Y2Pxx,x, + YdJx,x* > Yll~x,x, (Y*2%*x, + Y*3%x,x, ) (1.11) which, under the assumption of multinormality we have been using throughout the development, is sufficient to examine the state of identification of all parameters In this instance, there are six equations available to determine the six D J Aigner et al 1328 unknowns, &, &,, Yllt Y221 Y23, and qI It is clear that equations @ and @ in (1.11) can be used to solve for /Ii2 and yli, leaving @ to solve for ull The remaining three equations can be solved for p2i, y22, ~23, SO in this case all parameters are identified Were the observation error instead to have been associated with t2, we would find a different conclusion Under that specification, pi2 and yll are overdetermined, whereas there are only three covariance equations available to solve for fi2t, y22, ~23, and u22 Hence, these latter four parameters [all of them associated with the second equation in (MO)] are not identified 1.5 The power of a dynamic specification (Section 6) Up to this point in our introduction we have said nothing about the existence of dynamics in any of the equations or equation systems of interest Indeed, the results presented and discussed so far apply only to models depicting contemporaneous behavior When dynamics are introduced into either the dependent or the independent variables in a linear model with measurement error, the results are usually beneficial To illustrate, we will once again revert to a single-equation setting, one that parallels the development of (1.4) In particular, suppose that the sample at hand is a set of time-series observations and that (1.4) is instead: 7, = P-$-l + E,, Yt = q, + uf, (1.12) t =l, ,T, with all the appropriate previous assumptions imposed, except that now we will also use IpI