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a6n STRUCTURAL EQUATION MODELS THAT ARE NONLINEAR IN LATENT VARIABLES: A LEAST- SQUARES ESTIMATOR

Kenneth A. Bollen*

Busemeyer and Jones (1983) and Kenny and Judd (1984) proposed methods to include interactions of latent variables in structural equation models (SEMs). Despite the value of these works, their methods are limited by the required distributional assumptions, by their complexity in implementation, and by the unknown distributions of the estimators. This paper provides a framework for analyzing SEMs ("LISREL" models) that include nonlinear functions of latent or a mix of latent and observed variables in their equations. It permits such nonlinear functions in equations that are part of latent variable models or measurement models. I estimate the coefficient parameters with a two-stage least squares estimator that is consistent and asymptotically normal with a known asymptotic covariance matrix. The observed random variables can come from nonnormal distributions. Several hypothetical cases and an empirical example illustrate the method.

My thanks to Scott Long, the referees, and Peter Marsden for their comments on this paper and to Laura Stoker and John Zaller for their helpful discussions on the empirical example. I gratefully acknowledge the support from the Center for Advanced Study in the Behavioral Sciences and the Sociology Program of the National Science Foundation (SES-9121564).

*University of North Carolina at Chapel Hill

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1. INTRODUCTION

Structural equation models (SEMs), sometimes called LISREL models, are widely used in the social sciences. These general models include multiple regression, confirmatory factor analysis, classical simultaneous equation models, and a variety of other common analysis techniques as special cases (Joreskog and Sorbom 1993). Though it is straightforward to include nonlinear functions of exogenous or predetermined observed variables into these models (Bollen 1989, pp. 128-29) or to incorporate cross-product terms of "block" variables (Marsden 1983), the treatment of models with equations that are nonlinear in latent or unobserved variables is not fully developed. Typical examples are equations that include the product of two latent variables or the square of a latent variable as explanatory variables.

Researchers using SEMs have proposed two major solutions to this problem. One is based on the work of Busemeyer and Jones (1983), Bohrnstedt and Marwell (1978), Feucht (1989), and Heise (1986). The other derives from the work of Kenny and Judd (1984). These papers take important steps toward allowing product interactions and squared terms of latent variables into SEMs, but they have several limitations.

This paper provides a more general framework for analyzing SEMs that include nonlinear functions of latent or a mix of latent and observed variables. In addition, I propose a limited information estimator for such models that is based on a two-stage least squares (2SLS) procedure described in Bollen (forthcoming). Unlike the other methods, this estimator is simple, easy to implement, and has known asymptotic properties that do not depend on the normality of the observed random variables.

The next section reviews the literature on product interactions and squares of latent variables in SEMs and instrumental variable/ 2SLS methods. Section 3 presents the notation, model assumptions, and the estimator, and Section 4 discusses the selection of instrumental variables (IVs) that are needed to implement the procedure. Section 5 includes three hypothetical examples and one empirical example to illustrate the methodology. The results are summarized in Section 6 in the conclusion.

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2. LITERATURE REVIEW

2.1. Literature on Products of Latent Variables

An early study in the SEMs literature on incorporating products of latent variables in models was by Busemeyer and Jones (1983). Busemeyer and Jones focus on a single equation:

y, = /311L + f312L2 + /13L1L2 + 1, (1)

where y, is an observed random variable, L1 and L2 are latent random variables and , is a random disturbance term with a mean of 0. The latent variables L1 and L2 are each measured with a single indicator such that

Y2 = L1 + e2 (2)

Y3 L2 + 63, (3)

where E(ei) is zero, and E2, E3, and 5, are distributed independently of L1 and L2 and of each other. The terms L1, L2, E2, E3, and , are random variables from normal distributions; e2, 63, and s are each homoscedastic and nonautocorrelated; and y1, L1, L2, Y2, L1L2, and y3 are deviated from their means.

Busemeyer and Jones (1983) show that knowledge of the error variances (or reliability) of Y2 and Y3, together with the results from Bohrnstedt and Marwell (1978) on estimating the reliability of the product of two normally distributed variables, allows one to consistently estimate the covariance matrix of yi, L1, L2, and L1L2. This in turn yields a consistent estimator of the parameters /3,, 312, and /13 in equation (1).

The major limitations of this method are: it allows only a single indicator per latent variable; the error variances of the nonproduct observed variables must be known; tests of statistical significance of parameter estimates are not provided; it offers no methods for estimating equation intercepts; and the robustness of the estimates to violations of the normality and independence assumptions for the nonproduct latent variables and nonproduct disturbances is not given (Bollen 1989, pp. 407-8).

Feucht (1989) draws on Fuller's (1980) work and suggests

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modifications that overcome some of these limitations. The Feucht- Fuller method ensures that the moment matrix that is corrected for measurement error is positive-definite, allows for nonnormally distributed explanatory variables, and provides estimates of the standard errors of the resulting coefficient estimates. Single indicators and known error variances (and error covariances if present) are still required, however. Heise's (1986) and Feucht's (1989) Monte Carlo simulation results provide mixed evidence on the value of these single indicator approaches to including interactions of latent variables.

Kenny and Judd (1984) give an alternative method of incorporating squares of or product interactions of latent variables into SEMs (see also Wong and Long 1987; Hayduk 1987; Bollen 1989). Their method allows multiple measures of each latent variable. Prod- ucts of these indicators are incorporated into the model as indicators of the products of the latent variables.

To illustrate the Kenny-Judd method, consider the example including an interaction of latent variables in equation (1) and the indicators of Y2 for L1 and Y3 for L2 in equations (2) and (3). Since Kenny and Judd (1984) treat multiple indicators, add one more indicator each for L1 and L2, as in equations (4) and (5):

y4 = A41 + 64 (4)

y5 = 52L2 + 65 (5)

In addition to the assumptions already made for equations (1) to (3), the assumptions are that 64 and E5 have means of zero, come from normal distributions, are each homoscedastic and nonautocorrelated, and are independent of L1, L2, 62, 63, 6 , and of each other. All y variables are deviated from their means.

Kenny and Judd (1984) suggest that analysts form indicators of the interaction term, L1L2, by taking two-way products of the indicators of L1 with the indicators of L2. This results in four new measurement equations for the indicators of L1L2:

Y2Y3 = L1L2 + L1E3 + L2E2 + E2E3 (6)

Y2Y = A52L1L2 + LE5 5L2 + E5LE + (7)

43 = A41L1L2 + L2E4 + A41L1E3 + E3E4 (8)

Y4Y5 = A41A52L1L2 + A41L165 + A52L264 + 6465

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(9)


Equations (1) to (9) give the full model to estimate under the Kenny- Judd approach. This involves the introduction of a number of latent variables and combinations of latent and error variables. The list of such variables is L1, L2, 62 to E5, 1, L1L2, L 13, L1E, L2E2, L2E4, E2E3, E2E5,

E364, and E465. Estimating the measurement equations (6) to (9) involves linear and nonlinear constraints on the parameters. For instance, in equation (7) the factor loadings for L1L2 and for L2E2 are both equal to A52 in equation (5). Equation (9) for y4y5 has a nonlinear constraint on the coefficient for the L1L2 variable. Additional restrictions occur for the variances of the product latent variables in equations (6) to (9). Under the assumption that L1 and L2 come from normal distributions, the variance of L1L2 must be kept equal to VAR(L1)VAR(L2) + [COV(L1,L2)]2. Other examples of the restrictions are in Kenny and Judd (1984).

The introduction of the nonlinear constraints implied by the model and assumptions allows consistent estimation of the coefficients of the terms that are nonlinear in the latent variables. Kenny and Judd use a GLS fitting function (Browne 1984) to estimate their model. See Higgins and Judd (1990) for another empirical application.

The Kenny-Judd method represents an advance in the ability to handle interactions and squares of latent variables, but it still has limitations. One is the lack of knowledge about the robustness of the method to the failure of the normality and independence assumptions. Another is the proliferation of product latent variables, disturbances, and observed variables that occurs with this method. Even a relatively simple model requires many terms when multiple indicators are available for each latent variable involved in the product interaction. Each of the new terms and the accompanying nonlinear constraints must be entered explicitly into the model. Also, the properties of the model with raw rather than deviation scores are not known.

2.2. Literature on Instrumental Variables and 2SLS

Other literature has been less concerned with nonlinear functions of latent variables but is relevant to this paper. This is the econometric literature on instrumental variables (IV) and two-stage least squares (2SLS) estimation. Most econometric texts (e.g., Johnston 1984; Judge et al. 1985) provide overviews of these methods.

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IV and 2SLS techniques are helpful when an explanatory variable in a regression equation is correlated with the disturbance term of the equation. An IV is a variable that is correlated with an "endogenous" explanatory variable, but it is uncorrelated with the disturbance term. In 2SLS the predicted value of the endogenous explanatory variable, from a "first-stage" ordinary least squares (OLS) regression of the explanatory variable on the IV, replaces the explanatory variable in the original equation. The "second stage" of 2SLS is the OLS

regression of the original dependent variable on this predicted endogenous explanatory variable and the other explanatory variables. It provides a consistent estimator of the coefficient in the original equation. When more than one IV is available, the 2SLS estimator is an IV estimator that uses an optimal combination of instruments.

Random measurement error in an explanatory variable cre- ates a correlation between it and the disturbance. The bulk of econometric research on IV and measurement error is restricted to bivariate or multiple regression models with a single explanatory variable measured with error. Reiers0l (1941) was one of the first to

suggest the use of IV methods as a correction for an explanatory variable measured with error. Extensions of these methods allow an

explanatory variable to have more than one measure or expand to a two- to three-equation model (e.g., Bowden and Turkington 1984,

pp. 3-7, 58-62; Aigner et al. 1984). IV methods for models that have nonlinear functions of observed variables also are available (see Bowden and Turkington, 1984).

Madansky (1964), Hagglund (1982), and Joreskog (1983) proposed IV/2SLS methods to estimate factor analysis models. Bollen

(forthcoming) developed a 2SLS estimator for the latent variable models as well. But none of these authors dealt with nonlinear functions of latent variables. The next section develops a general model and method that makes use of the 2SLS estimator for such models.

3. MODEL AND ESTIMATOR

Busemeyer and Jones (1983), Kenny and Judd (1984), and Feucht

(1989) concentrated either on the product of two latent variables or the square of a latent variable in a single equation latent variable model. A more general approach permits any number of equations, allows other nonlinear functions of the latent or observed variables,

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and applies to the measurement model as well as to the latent variable model. Suppose that the model for the latent variables is

L = acL + BlL + B2fL) + , (10)

where L is an m x 1 vector of latent variables, aL is an m x 1 vector of intercept terms, B1 is an m x m matrix containing constant coefficients for the effects of L on other L's, f(L) is an n x 1 vector of functions that are nonlinear in L, B2 is an m x n matrix containing constant coefficients for the effects of f(L) on L, and ; is an m x 1 vector of disturbances with E(S) equal to zero and each , is i.i.d. That is, the disturbance for each equation is homoscedastic and nonautocorrelated across observations, though the variance and other distributional traits of i can differ from j for i = j. Typically some elements of L orf(L) are "predetermined" or exogenous in the sense that they are uncorrelated with, or even independently distributed of '.

The latent variables in L are observable through their indicators. A second equation provides the measurement model linking the latent to the observed variables

y = ay + AL + A2f(L) + e, (11)

where y is a p x 1 vector of random variables that are observed, a is a p x 1 vector of intercept constants for the measurement equations, A1 and A2 are p x m and p x n constant coefficient matrices for L and f(L), and e is ap x 1 vector, where each Ei is an i.i.d. random error of measurement that has a mean of zero and that is independent of L andf(L). If a "latent variable" is perfectly measured, then the corresponding element of ay is zero, the corresponding row of A1 has a 1 in the column that matches the latent variable and zeros in the rest of the row, and the corresponding row of A2 is zero, as is the matching element in E.

If B2 and A2 are zero, then the model in equations (10) and (11) matches general SEMs with intercept terms such as Joreskog and Sorbom's (1993) LISREL model. In the case of the LISREL model, equation (10) corresponds to the latent variable model, and equation (11) to the measurement (or confirmatory factor analysis) model. What is distinctive about the model of equations (10) and (11) is its inclusion of f(L). This permits effects that are nonlinear in the latent variables. The nonlinear terms can enter the latent vari-

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able or the measurement model. Thus the model is a generalization of the usual SEM.1

To help identify the model, assume that each latent variable has an indicator that "scales" the latent variable such that

Yi = Li + E (12)

This assumption does not rule out multiple indicators for a latent variable, nor does it require that indicators be influenced by no more than one latent variable. It requires only that there be at least one indicator per latent variable that "loads" exclusively on that latent variable and that scales it by virtue of having a loading of unity. Other scaling choices are possible, but the failure to assign a scale leads to an underidentified model (see e.g., Bollen 1989, pp. 152-54, 307-9).

Partition y such that the m y's that scale the latent variables occur first (as vector yl) and the other (p - m) y's second (as vector

Y2). This leads to

Y Y2 ](13)

where

Y = L + eI (14)

and

L = Y - E1. (15)

Substituting equation (15) into (10) transforms (10) into an equation for the observed scaling variables rather than one for the latent variables:

y, = aL + Bly, + B2f(, - E,) + El - Bll + (16)

Similarly use equation (15) to rewrite the measurement model in

equation (11) to exclude L:

y = ay + Al,y + A2f(y1 - E1)- Ale1 + E. (17)

Consider a single equation from the latent variable model in

equation (16):

'One can also view this as the "all Y model" (Bollen 1989, ch. 9) with the addition of nonlinear functions of the latent variables.

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Yi = aLi + B1Y1 + B2ifYl - E1) - Bli 1 + Ei + i, (18)

where Yi is one of the indicators that scales a latent variable. The i subscript signifies the ith row of the matrix or vector-so, for instance, B,l is the ith row of B1 and Ei is the ith element in the e1 vector. In one broad and useful class of models, the nonlinear function of the latent variables is expressible as

f(Yl - E1) = gl(yl) + g2(y1,E1), (19)

where gl(.) and g2(.) are functions of the respective variables in paren- theses. This class of models includes the common cases of product interactions and quadratic terms of latent variables that Busemeyer and Jones (1983) and Kenny and Judd (1984) examined. For instance, supposef(L) is a scalar that consists of the product L1L2. Then f(Yl - e1) equals the scalar

f(Y - El) = (Yl - E1)(2 - E2)

= YlY2 - Y - Y2E1 + E1l2, (20)

where YlY2 is gl(y1) and the last three terms are g2(yi,el). Or if f(L) equals L2, then f(y - E,) is the scalar

f(Yl - E1) = Y2 - 2yle1 + El, (21)

where y2 is gl(yl) and the remaining terms are g2(y1,el). The decomposition in equation (19) is useful because it allows

one to place the g2(Yl, el) component in the residual while keeping gl(yl,) in the main part of the model. For these and other functions that are expressible as in equation (19), I can write equation (18) as

yi = aLi + Bliy1 + B2igl(Yl) + Ui, (22)

where ui is the composite disturbance term

Ui = B2ig2(l,E1) - Bli 1 + Ei + i'- (23)

In general ui will be correlated with the right-hand side variables in equation (22), and that makes ordinary least squares an inconsistent estimator of aLi, Bli, and B2,. An exception would occur if all the right-hand side variables in the equation are measured without error and are uncorrelated with the equation disturbance, Vi. In the more general case where the disturbance correlates with the right-

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hand side variables, a two-stage least squares (2SLS) estimator provides a consistent estimator of these parameters.

The literature review described special cases where the 2SLS estimator has been successful. Here I develop a 2SLS estimator that applies to general SEMs, including the latent variable and the measurement model. And the 2SLS estimator allows for equations that are nonlinear in the latent or observed variables, requiring only that they be linear in the parameters.

To develop this procedure, I modify the notation somewhat. Define N to be the number of cases, y1(l to be the N row matrix of values for the variables in y, that have nonzero coefficients in the yi equation, and gl(yl)(i) to be the N row matrix of values for the variables in gl(yl) that have nonzero coefficients in the Yi equation. The N x 1 vector Yi contains the N values of Yi in the sample, and ui is an N x 1 vector of the values of ui. Let B1(i) be a column vector of the coefficients that correspond to yl(i) and B2(' be the coefficient column vector for gl(y1)(') with all coefficients being identified parameters. Define Zi = [1 : yl( ' gl(yl)(i) ] and A' = [a'i:P B (i)]. Then rewrite

equation (22) as

Yi = ZiAi + ui. (24)

The 2SLS estimator requires a matrix of instrumental variables, say Vi, that satisfy the assumptions

1 plim ( V i Zi) = izi (25)

1 plim ( - V;Vi) = I?ivi (26)

1 plim ( - V'iui ) = 0, (27)

where plim stands for the probability limit as N goes to infinity. Other assumptions are that the variables in Zi have finite variances and covariances, that the right-hand side matrices of equations (25) to (27) are finite, that Xv,iv is nonsingular, and that XviZi is nonzero. These assumptions require that the instrumental variables (IVs) correlate with Zi and that the IVs not correlate with the composite disturbance ui. As I explain in the next section, the IVs will be

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observed variables (y's) that are part of the model or nonlinear functions of such observed variables.

Assume that E[uiui] = o2I so that the composite disturbance is homoscedastic and nonautocorrelated. Whether E[ui] = 0 will depend on the nonlinear function of the latent variables that occurs in the original model. For now assume that the model is such that the mean of the composite disturbance is zero; later two of the examples will illustrate the consequences that follow when this assumption is false.

In general the ui of equation (24) will correlate with one or more of the variables in Zi. This rules out the use of single-stage OLS to estimate Ai. The first stage of the 2SLS estimator is to perform an OLS regression of Zi on Vi, with coefficients

(ViVi) -1V'zi. (28)

The Vi matrix is then postmultiplied by this coefficient to form Zi ( =

Vi(V'iV)-~ V'Zi), the predicted Zi matrix. The second stage in the 2SLS estimation of A, is the OLS

regression of y, on Zi which gives coefficients

A, = (Z^' Z,)-1^. Z (29)

As is well known, the 2SLS estimator is a consistent estimator of Ai (e.g., see Johnston 1984, pp. 478-79). Assume that

1 Z'i u- AN(0, a r2), (30)

where AN refers to an asymptotically normal distribution. The previous assumptions in equations (25) to (27) imply that

1 plim ( ZiZi)-' ,Z - (31)

The asymptotic distribution of Ai is then D D - 2 -1

N(Ai - A) D- N(0, a uA N,), (32)

and an estimate of the asymptotic covariance matrix of Ai is

acov (Ai) = a ui(Z'iZi)-.

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where 6u = (Yi - ZiAi)' (y - ZiAi)/N. Thus the preceding procedure provides a consistent estimator of the coefficients for the linear and nonlinear terms in equation (22) as well as a measure of their statistical variability.

I have limited the discussion to the latent variable model in equation (10) that allows effects that are nonlinear in the latent variables for the class of models described in equations (22) and (23). A similar series of steps applies to the measurement model in equation (11). Substituting equation (15), y, - e1, for L in equation (11) leads to equation (17). Analogous to equation (18) from the latent variable model, a single equation for the measurement model is

Yi = ai + AliY 1 + A2iAf(y - el) - Aliel + Ei. (34)

Considering the gl(.) and g2(.) functions as before leads to

Yi = ayi + + A 2ig1) + i) + , (35)

where

Ui = A2ig2(YlEl) - Alil + Ei. (36)

An appropriate redefinition of Zi, Ai, and ui leads back to equation (24), Yi = ZiAi + ui. Under the assumptions detailed for the latent variable model, one can obtain a consistent 2SLS estimator of A, with a known asymptotic distribution.

4. INSTRUMENTAL VARIABLE SELECTION

Key to the success of using the procedures developed in the preceding section is finding appropriate instrumental variables (IVs) that satisfy the conditions for IVs and that lead to an identified model. When treating the selection of IVs, many econometric texts do not explain methods for finding the IVs. In contrast, the 2SLS procedure here depends on the model structure for the creation and selection of IVs. Indeed, the structure of the full model is essential in finding IVs, as is the idea that nonlinear functions of some of the observed variables can serve as IVs.

In practice the most challenging task is to find IVs that are uncorrelated with the composite disturbance ui. Equations (25) to (27) along with the pattern of correlations among the errors, disturbances, and latent variables of the model are important aids to select-

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ing appropriate IVs. A general procedure for selecting IVs has several steps. Assume that vi is a variable that might be a suitable instrumental variable. The following steps help to evaluate its eligibility: (1) Form COV(vi, ui); (2) if vi is an endogenous variable, substitute its reduced-form equation for it; (3) substitute the right-hand side of equation (23) or (36) for ui; and (4) take the covariance of the resulting terms and see if it is zero. If so, then vi passes this condition for an IV.

A similar series of steps applies in the search for IVs that are nonlinear functions of the observed variables. For instance, when modeling the product of two latent variables, products of indicators that do not "scale" the respective latent variables are often suitable for use as IVs. Suppose that Yi scales the first latent variable and Y2 and Y3 are additional measures of the same latent variable. Similarly, suppose that the y4 variable scales the second latent variable and y5 and Y6 are two other indicators. Then Y2Y5, Y2Y6, Y3Ys, and Y3Y6 often will

qualify as IVs. Determination of their eligibility follows the same steps of writing a reduced-form expression for each variable in the product, obtaining the product of the reduced forms, and calculating its covariance with ui to see if it is zero. If so, this product of the observed variables can serve as an IV.

Researchers can sometimes form another IV by regressing each observed variable in the product term on all of the individual and product IVs of observed variables and calculating the predicted values from the linear regressions for each component (e.g., 9Y and Y2). Then one forms 9192 as an additional IV for the model. This latter IV follows a suggestion of Bowden and Turkington (1981) about the creation of IV for nonlinear functions of endogenous observed variables in econometric models.

The Kenny and Judd (1984) example discussed above provides an illustration of the selection of IVs for the 2SLS method. Recall that the latent variable equation was

y, = 311L, + 312L2 + 3L,L2 + 1, (37)

with Y2 and Y3 the indicators that scale L1 and L2, respectively (see equations [2] and [3]). Substituting (Y2 - E2) for L1 and (Y3 - 63) for L2 leads to

Yl = P31Y2 + P/12Y3 + 133Y2Y3 + ul,

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where u1 = -311E2 - f12E3 - 813y2E3 - f813y3E2 + 813E2E3 + 1. Allther.h.s.

variables of equation (38) are correlated with the composite disturbance, u1. The y4 and y5 variables are indicators of L1 and L2, respectively (see equations [4] and [5]). The Y4, Y5, and Y4Ys variables satisfy the conditions for IVs, as the reader can confirm. Regressing y, and Y2 on these IVs and forming 91 and 92 and then calculating YP12 leads to another IV. The 2SLS estimator using all four IVs (Y4, y5, y4y5, and 9192) is a consistent estimator of the coefficients in equation (37).

Though the specific steps outlined above apply to any model, some general guidelines for ruling out IVs emerge from closer exami- nation of the composite disturbance ui. For the latent variable model in equation (22), equation (23) defines ui; it is repeated here for easy reference:

Ui = B2ig2(yl,E1) - Bli + Ei + vi (39)

Note that the latent variable model only has equations for the latent endogenous variables, so we do not have any equations to estimate for the latent exogenous variables in the latent variable model. Any variables correlated with i are ineligible as IVs (except in the im- probable situation in which a variable has an exactly equal but oppo- site in sign covariance with the remaining components of ui). In the typical situation, this means that other y's that are indicators of an endogenous Li are ineligible as IVs in the latent variable model since these other indicators correlate with V.2

Less obvious is that indicators of latent variables that are influenced by Li are unacceptable since they too will correlate with Si. Also, if i correlates with rj, then the indicators of Lj are not suitable as IVs in the latent variable model. The B1i El term means that any of the scaling indicators for the latent variables that appear on the right-hand side of the yi equation cannot be IVs. Nor can y's whose errors of measurement correlate with the errors of such scaling indicators serve as IVs. Furthermore, any y's that have errors that correlate with Ei are ruled out as IVs. Finally, IVs must be uncorrelated with B2ig(yl,E1). In many cases variables that do not correlate with the other terms in ui will not correlate with this one, but there are exceptions.

In the measurement model the composite disturbance ui equals A2ig2(Y,1) -

AliE1 + Ei. The IVs must be uncorrelated with

2Remember that I am referring to the latent variable model here. In measurement models some of the other indicators of the same latent variable can serve as IVs.

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these components of ui. An indicator whose error correlates with Ei is ineligible. Scaling indicators for latent variables that affect Yi or indicators whose errors of measurement correlate with the errors of such scaling indicators cannot qualify as IVs either. Last, the IVs must be uncorrelated with the nonlinear term, A2ig2(yl,El). Note that unlike the composite disturbance in the latent variable model (see equation [23]), i does not appear in the composite disturbance for the measurement equation. This means that some of the observed variables that correlate with i and are hence ineligible as IVs for the latent variable equation might still be suitable IVs for equations from the measurement model.

Another consideration in selecting IVs is that some variables might technically meet the conditions to be an IV, but they may not work well in practice. For instance, if the IVs collectively are poorly correlated with the variables that they are to replace, the resulting 2SLS estimates may be unstable and far from the true parameters. Analysts can check this by examining the R2's from the first stage of the 2SLS procedure. Low values (e.g., < 0.1) suggest that the IV method may not be suitable and that better IVs must be sought.

A final requirement for IVs is tied to the identification of an equation. In general there should be at least as many IVs as there are coefficients to estimate. Too few IVs leads to underidentification. More IVs than coefficients implies overidentification.

5. EXAMPLES

Several examples in this section illustrate the selection of IVs and the 2SLS approach in models with product interactions or powers of latent variables. These are the nonlinear functions of most interest in SEMs and are the functions discussed by Kenny and Judd (1984). Unlike Kenny and Judd, I also give a case of a factor analysis model (i.e., a "measurement model") with a product interaction.

Example 5.1: Product Interaction in a Measurement Model In the first example the measurement model is

Yi 0 1 0 0 Y2 aY2 A21 0 L1 2 Y3 = 3 + A31 A32 L + A33 [LL2 ] + e3 .(40) Y4 ay4 02

L 2 0 E4

Ys 0 0 1 0 65

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1 1

1 2 3 ?54 5

FIGURE 1. Product interaction in a measurement model.

The variables y, and y5 are the scaling indicators for L1 and L2, respectively. Assume that the errors of measurement, the E's, are

mutually independent, each Ei is i.i.d., has a mean of zero, and is

independent of L1 and L2.3 The path diagram for the example is in

Figure 1. Most of the symbols are conventional: Variables enclosed in

ellipses are latent variables, those in boxes are observed variables, and errors are unenclosed. Single-headed straight arrows stand for the linear influence of the variable at the base of the arrow on the variable at the head of the arrow. Curved two-headed arrows signify a linear association between two variables that is unexplained by the model. The less conventional symbols are the single-headed "sawtooth" arrows and the ellipse with L1L2 inside. The sawtooth arrows stand for nonlinear relations between the variables at the base and the head of

3Here as elsewhere in the paper the i.i.d. assumption applies to a single random variable at a time. That is, the distribution of Ei might well differ from that of Ej for i not equal to j. But I assume that Ei is independently and identically distributed across observations.

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the arrow. Within the ellipse, I define the nonlinear function of the latent variables, in this case L1L2. No disturbance leads to the product interaction since it is an exact nonlinear function of L1 and L2. No disturbances point to L1 and L2 since they are exogenous variables.

An example of where such a model might arise is in the testing area. A job task might require the combination of high intelligence (L1) and strong concentration (L2). Suppose that two scales measure intelligence (y, and Y2) and two scales measure the ability to maintain concentration (y4 and Y5). Another test is devised to measure the combination of intelligence and concentration. If Y3 is the test hy- pothesized to measure intelligence*concentration, one could estimate a model like that in Figure 1 to evaluate how well it performs. Or in a macrocomparative research project a researcher might have two scales that measure political democracy in a nation and two scales that measure its political stability. Another scale is suspected of measuring "democratic stability," the product of the two latent variables. The model in Figure 1 could address this question.

There are three equations to estimate to obtain the factor loadings in the measurement model. Using the equation for Yl, which scales L1, apply the general expression in equation (34) to write the Y2 equation as

Y2 = Cy2 + A21Yl - A211 + E2. (41)

The expected value of the composite disturbance term (-A21e + e2) is zero. Variables y, and Y2 are ineligible as IVs because of their correlation with the composite disturbance term. However, y3, y4, and y5 can serve as IVs since they correlate with y, and are uncorrelated with the composite disturbance. An analogous procedure applies to estimating the Y4 equation.

The Y3 equation contains the nonlinear product interaction term. Use equation (34) and the equations for the scaling indicators y, and Y3 to write the Y3 equation as

Y3 = ay3 + A31Y1 + 32Y5 + A33YlY5 + U3, (42)

where u3 is

U3 = -A31E1- A32E5 - 33 (Y1E5 + Y5E1 - E1E5) + E3. (43)

The expected value of u3 is zero. The eligible IVs among y, to y5 are Y2 and Y4. The reader can verify this by following the steps on select-

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KENNETH A. BOLLEN

?

- ^

Y3 3- t

2e Y

FIGURE 2. Latent variable model with a squared latent variable.

ing IVs described earlier: form COV(y2, u3) and COV(y4, u3); substitute in the reduced forms for Y2 and y4;4 and confirm that these covariances are zero. Since this equation has a nonlinear function, Y2Y4 can also serve as an IV since COV(y2y4, u3) is zero. Following an earlier suggestion, 9195 is another possible IV. The 9^ and 95 that make up the Y9Y9 term are from the regressions of y, and Y5 on Y2, y4, and Y2Y4.

Once the researcher determines the IVs for each equation, she or he can use the 2SLS estimator in equation (29) and the estimated asymptotic covariance matrix in equation (33) to obtain parameter estimates and asymptotic standard errors.

Example 5.2: Latent Variable Model with a Squared Latent Variable Figure 2 contains the path diagram for the second example in which a latent variable, L3, is a function of another latent variable, L1, and its square. Assume that the expected values of e1 to 63 are zero; they may come from different distributions but each error variable is i.i.d., and E1 to E3 are mutually independent of each other and of L1 and L3. The

y, variable scales L1. The disturbance '3 has a mean of zero, is i.i.d., and is independent of E1 to E3 and L1. The equation for the latent variable model is

L3 = CYL3 + / 331L1 +332L + 3. (44)

Using equation (16) and the scaling equations y, = L1 + el and y3 =

L3 + E3, write this equation as

4The structural and reduced form equations for Y2 are the same since L1 is the only right-hand side explanatory variable and it is exogenous. Analogous arguments hold for y4, where L2 is exogenous. See equation (40).

240


Y3 = L3 + 31Y1 + 32Y + 3, (45)

where

U3 = -31E1 - 2P32Yl11 + 332E1 + E3 + ;3- (46)

The expected value of the composite disturbance, U3, is /32VAR(E1). For a homoscedastic error, the mean of the disturbance deviates from zero by a constant. Thus, instead of consistently estimating aL3, the 2SLS estimator will estimate aL3 + /32VAR(E1). The other coefficient estimators remain consistent. If a consistent estimator of the intercept is required, then a researcher should subtract a consistent estimator of ,332VAR(E1) from the intercept given by the 2SLS results.

Selection of IVs follows the same procedure as before. Each IV needs to be uncorrelated with the composite disturbance, U3. The y, and y3 variables are eliminated because of their correlations with the disturbance. The covariance of Y2 with u3 is

COV(y2,u3) = COV(A21L1 + E2, -331E1 -

2P32Yle1

+ 0321 + E3 + 3. (47)

Given the assumption that L1 and 62 are independent of e1, 63, and s3, this covariance is zero and Y2 is eligible as an IV. In addition, its square also can serve as an IV since the square of Y2 also has zero covariance with u3.

Example 5.3: Latent Variable Model with a Cubed Latent Variable The third example extends Example 5.2 by adding a cube of L1 to the latent variable model. This leads to the equation

L3 = ?L3 -+ f331Ll + ]32L~ + -3L33L3 + '3. (48)

Substituting (y1 - El) for L1 and (y3 - E3) for L3 leads to

Y3 = L3 + P31Y1 + 3Y + 333Y + 3, (49)

where

U3 = -3311 - 232y1E 1 + P32E3 - 333321 1 + 3 2333Y1E - 333E + E3 + 3- (50)

The expected value of the composite disturbance term is

E(u3) = [(332 + 3P331,yl)VAR(E1) - 333E(E )],

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(51)

KENNETH A. BOLLEN

where /Yl is the mean of Yl. As in the previous model with a squared latent variable, for homoscedastic and equal skew measurement errors, the net effect of this alone is to lead to an inconsistent estimator of the intercept term. However, the cubed term complicates the situation because inclusion of the cubed term eliminates some of the

potential IVs. For instance, Y2 was a suitable IV in the previous example with a squared term. However, in this example the covariance of Y2 with u3 is

COV(y2,u3) = COV(A21L, + E2, - 31E - 2332YlE1 + 332E - 3/333YE1 + 31333Y1 El - 333E + E3 + 3)

=-3A21, 33VAR(Ll)VAR(E1). (52)

This follows after substituting (L1 + El) in for yl, multiplying out, taking expectations, and simplifying. The nonzero value means that

Y2 is no longer an appropriate IV. One also can demonstrate that the

square of Y2 correlates with u3 and thus cannot serve as an IV. Having additional variables in the model might create suitable IVs, but as it

stands, the IV method does not work for this model. This example provides a warning against generalizing from the example with only a

square of a latent variable to ones that include higher powers of latent variables. Examining the covariance of a variable with the

composite disturbance of a model helps to determine the eligibility of a potential IV.

Example 5.4: Empirical Example: Feelings Toward President Bush The previous three hypothetical examples illustrate the basic opera- tions of the procedure. The last illustration uses empirical data in a more complicated and realistic situation. Figure 3 is a path diagram that relates four latent variables to 16 observed variables. Unlike the

previous examples where the interaction terms were functions of two latent endogenous variables, this case has an interaction between an observed exogenous variable and a latent endogenous one. The y, to

Yll variables are exogenous. Other assumptions are that E12 to E16 and

1l to '4 each have expected values of zero and each is i.i.d., and they are mutually independent and are independent of Yl to Yl. The error variables E12 to E16 are also independent of LI to L4.

Perceived financial status (Lj) and identification with the Democratic Party (L2) each have two effect indicators, Y14 and Y15 for

242


'i[- 15

FIGURE 3. Model of feelings toward President Bush with product interaction.

L1 and Y12 and Y13 for L2. The L3 variable is a product interaction of

years of education (y1) times Democratic Party identification (L2). The last latent variable is feelings toward President Bush (L4) and it has a single indicator, Y16. The data are from the 1992 National Elec- tion Study, available through the Interuniversity Consortium for Po- litical and Social Research in Ann Arbor, Michigan. Table 1 lists the variables and their meanings (n = 1944).

Consider the latent variable equation for L4 (Feelings toward President Bush), the only equation with an interaction term. The L4

equation is

L4 = L4 + /341L1 + /342L2 + P43L2Y1 + ?44Y1 + 345Y3 + 4. (53)

Substituting in equations for the scaling indicators, Y12, Y14, and Y16, rewrite this in terms of observed variables:

Y16 = L4 + P41Y14 + P42Y12 + 43Y12Y1 + 344Y1 + 345Y3 + U4, (54)

where

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!4

KENNETH A. BOLLEN

TABLE 1 Variables Used in Model of Feelings Toward President Bush (N = 1944)

Latent Variables

L, = Respondent's self-perception of financial situation L2 = Identification with Democratic Party L3 = L2*Yl L4 = Feelings toward President Bush

Observed Variables

Yi = Years of education Y2 = 1 if female, 0 if male

y3 = 1 if Black, 0 if otherwise y4 = Natural logarithm of 1991 family income before taxes y5 = 1 if put off buying things, 0 otherwise Y6 = 1 if put off medical treatment, 0 otherwise y7 = 1 if borrowed money, 0 otherwise Y8 = 1 if dipped into savings, 0 otherwise

yg = 1 if tried to work more hours, 0 otherwise Yo = 1 if not able to save any money, 0 otherwise

yv = 1 if fallen behind in rent or house payments, 0 otherwise

Y12 = Degree of self-identification with Democratic Party Yl3 = Feeling thermometer rating of Democratic Party divided by 10

Y14 = Self-rating of current financial situation relative to one year ago y15 = Self-rating of relation between change in cost of living and income

Y16 = Feeling thermometer rating of President Bush divided by 10

Source: All observed variables come from the 1992 National Election Study that is available from the Interuniversity Consortium for Political and Social Research at the University of Michigan, Ann Arbor.

U4 = -f41E14 -

42E12 - 343E12Yl + E16 + 4- (55)

The E(u4) is zero. The Y14, Y12, and Y1i2Y variables are correlated with

u4, and hence ineligible as IVs. The following variables qualify as IVs: yl to Yl1, Y13, Y15, and Yl3Y. I form another IV by regressing Y12 on the preceding list of IVs and then creating Y12y. All of these are the IVs for the analysis. The R2's for the first stage regressions of the 2SLS estimator for Y14, Y12, and Y12Yi are 0.39, 0.44, and 0.45, respectively. These R2's are sufficiently high to use these IVs.

The 2SLS estimates and asymptotic critical ratio estimates for equation (53) are

244


4 = 9.206 - 0.599 L1 - 0.166 L2 - 0.034 (56) (12.04) (-7.37) (-0.88) (-2.54)

L2y1- 0.009 y, -0.362 y3 (-0.16) (-2.11)

(estimate/asymp. s.e.)

The 2SLS coefficient estimates are from the PROC SYSLIN procedure in SAS (SAS Institute, 1988). The SAS code for this example is in the appendix to this chapter. I multiplied the estimated asymptotic standard errors from the SAS output by V/(N-6)/N so that they match the estimated asymptotic standard errors formula in equation (33), since a degrees of freedom "correction" is not part of the asymptotic theory. The critical ratios above are the coefficient estimates divided by their respective standard errors.

Perceived financial status (L1) is scaled so that high scores signify a poorer financial situation. Democratic Party identification (L2) has stronger identification receiving higher scores. Thus the results indicate that on average respondents with the poorest perceived financial situation have more negative feelings toward Bush than those with good financial situations, net of the other variables. Similarly being African-American (y3) has an expected negative effect on reactions to Bush, statistically controlling for the other explanatory variables. Since the coefficient estimate of the interaction term (L2y,) is statistically significant, I cannot interpret the effects of party identification apart from education and vice versa. However, the coefficients indicate that the slope for party identification (L2) becomes more negative for high levels of education than for low. The slope for education (yl) also becomes more negative as identification with the Democratic party grows stronger.

6. CONCLUSIONS

An obstacle in the application of structural equation models has been the difficulty of including nonlinear functions of latent variables in the model. Researchers estimating regression models and ANOVA models routinely test for interactions in observed variables, yet few researchers test for the effects of interactions or squares of latent variables in general SEMs. Those who do have used Kenny and Judd's (1984) technique in models with multiple indicators of the latent variables. Here I propose an alternative method.

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KENNETH A. BOLLEN

TABLE 2 Comparison of Kenny and Judd (1984) and 2SLS Methods for Nonlinear

Functions of Latent Variables

Kenny and Judd (1984) 2SLS

1. Only for nonlinear functions that are squares of or products of latent variables?

2. Type of estimator? 3. Introduce products of

latent variables and disturbances into the model?

4. Need to form products of observed variables?

5. Add measurement equations to the model for product indicators?

6. Impose nonlinear constraints on some coefficients and variances?

7. Are equation intercepts estimated with

significance tests? 8. Assume normality for

nonproduct observed variables?

9. Are small to moderate sample size properties known?

Yes Full-information

Yes Yes

(to use as indicators)

Yes

Yes

No

Yes

No

No Limited-information

No Yes

(to use as IVs)

No

No

Yes

No

No

Table 2 gives a summary comparison of the characteristics of the Kenny-Judd technique and the 2SLS technique developed in this

paper. The first comparison in Table 2 shows that the Kenny and Judd (1984) method was devised for the common case of product interactions and squares of latent variables. The 2SLS method han- dles these as well as other models that are nonlinear in the latent

variables, though this paper concentrated on the interaction and

quadratic cases.5

5Both methods focus on models that are linear in the parameters.

246


The second point of contrast is that the Kenny and Judd method uses a full information (FI) estimator whereas the 2SLS is a limited information (LI) estimator. An advantage of a FI estimator is that it incorporates information on all equations and restrictions in the model at once. This can lead to greater efficiency of the estimator relative to LI estimators (e.g., see Johnston 1984, pp. 490-92). A drawback is that specification error in one part of the system may spread to other parts of the system with an FI estimator, whereas an LI estimator may better isolate the error. Given the approximate nature of virtually all structural equation models, any efficiency gains may be more than offset by the biases spread by specification errors.

I have been careful not to characterize the Kenny and Judd estimator as a full information maximum likelihood (ML) because although the ML fitting function might be used with their procedure, it is not a true ML estimator. As Kenny and Judd recognized (1984, p. 208), even if the nonproduct indicators come from normal distributions, the product variables in the model cannot originate from a normal distribution. Work on the robustness of the ML estimator to nonnormality looks promising (e.g., Satorra 1990), but it is not clear whether models with the product indicators and nonlinear constraints will satisfy the robustness conditions. The weighted least squares estimator (Browne 1984; Joreskog and Sorbom 1993) for arbitrary distributions is a possible alternative, but recent Monte Carlo simulation work suggests that it may require unrealistically large sample sizes for the asymptotic properties to hold (Muthen and Kaplan 1992).

The properties of the 2SLS estimator are large sample properties. Monte Carlo simulations and some analytical evidence are available for its finite sample behavior in observed variable econometric models (Judge et al. 1985). Similarly, some Monte Carlo evidence is available for using a 2SLS for factor analysis models (Hagglund 1983; Lukashov 1994). Overall, the evidence suggests that it compares favorably with FI estimators such as maximum likelihood in terms of unbiasedness and variance, but no Monte Carlo evidence is available for the linear latent variable model (Bollen, forthcoming) or for models with nonlinear latent variables.

The third point of comparison is whether the procedure requires that products of latent variables and disturbances be intro- duced into the model. This is true for the Kenny and Judd (1984)

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KENNETH A. BOLLEN

method but not for the 2SLS one. An example of this was given in the literature review section of this paper. Both methods form products of the observed variables, but there are differences. In the Kenny and Judd method these are to be indicators of the nonlinear terms in the model. In the 2SLS method they are to be IVs for estimating the model. Also, the Kenny and Judd method introduces more of these product terms than does the 2SLS method. For instance, in the example in the literature review section the Kenny and Judd method introduces four product indicators (see equations [6] to [9]). Using the 2SLS method for this same example would require only the y4y5 product variable to add to the IVs.

The fifth comparison in Table 2 shows that the Kenny and Judd procedure adds one measurement equation for each of the product indicators added to the model. In the literature review example, this led to four additional measurement equations, (6) to (9). In addition, the Kenny and Judd method imposes nonlinear restrictions on some coefficients and variances.6 The 2SLS method neither requires additional measurement equations nor does it require nonlinear constraints.

Kenny and Judd developed their technique assuming that variables were deviated from their means. They do not estimate intercepts or test their statistical significance. The 2SLS method is not so restricted, and it straightforwardly yields the estimates of the intercepts and their asymptotic standard errors.

Another contrast is that the Kenny and Judd method assumes that all observed variables come from normal distributions whereas the 2SLS estimator does not. The robustness of the Kenny and Judd technique to nonnormality is unknown. The final point of comparison is that the small to moderate sample size properties of both estimators are not known. This is a disadvantage of both estimators and reason to be cautious in applying either.

Though the 2SLS method compares favorably with the other methods of handling nonlinear functions of latent variables, it has at least two other properties worth noting. As with all SEMs, the variable that scales the latent variable should be chosen with care. Gener- ally, the scaling variable should be the observed variable with the

6Joreskog and Sorbom's (1993) LISREL 8 allows nonlinear restrictions in estimation.

248


greatest association with the latent variable. Cudeck (1991) and Jennrich (1987) have examined methods of choosing scaling variables for exploratory factor analyses estimated with IV methods. Their methods also could be useful in the context of the 2SLS procedures presented here. The sensitivity of the estimator to the choice of the scaling indicator for each latent variable requires further research.

Another limitation is that I have treated only models where each latent variable has at least one scaling indicator that is not influenced by other latent variables in the model. The other indicators can load on additional factors provided that the model is identified. Most models have scaling indicators that satisfy this condition. In some models, such as multitrait-multimethod ones, this may not be true. Modifications to the method would be required to handle these situations.

APPENDIX: SAS CODE FOR EMPIRICAL EXAMPLE7

data two; set one; proc reg; model yl2=yl--yll y13 y15 yl3yl; output out=three p=yl2hat; data four; set three: yl2hatyl =yl2hat*yl; proc syslin 2sls; endogenous y16 y14 y12 yl2yl; instruments yl--yll y13 y15 yl3yl yl2hatyl; model y16=y14 y12 yl2yl yl y3;

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Bohrnstedt, G. W., and Marwell, G. 1978. "The Reliability of Products of Two

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