Multivariate Behavioral Research, 1988, 23, 69-86
The Impact of Specification Error on the Estimation,
Testing, and Improvement of Structural Equation Models
David Kaplan
Graduate School of Education, University of California, Los Angeles
The purpose of this paper is to assess the impact of misspecification on the estimation,
testing, and improvement of structural equation models. A population study is conducted whereby a prototypical latent variable model is misspecified in various ways.
Measurement model and structural model misspecifications are considered separately
and together. The maximum likelihood estimator (ML) is compared to a limited
information two-stage least squares (2SLS) estimator implemented in LISREL. The
ratio of chi-square to its degrees of freedom and power of the likelihood ratio test is
assessed for each misspecification. The modification index provided by LISREL is also
studied. Results indicate that ML and 2SLS estimates of measurement and structural
parameters are both affected by measurement model misspecification. For misspecification of the structural part, ML is shown to propagate errors throughout the structural
parameters whereas 2SLS isolates errors only in the parameters of the misspecified
equation. Results also show that relying on the ratio of chi-square to degrees of freedom
a s a n index of fit may lead to accepting models with severe parameter bias. Finally, the
modification index is shown to be an unreliable indicator of the location of a specification
error.
The purpose of this paper is to study the impact of specification error on
the estimation, testing, and improvement of structural equation models. The behavior of the maximum likelihood (ML) estimator will be
compared to a recently proposed two-stage least squares (2SLS) estimator for a prototypical latent variable model. The focus of attention
will be on misspecification of the measurement as well as structural
components of the model. Errors due to violations of distributional
assumptions will not be considered (see Boomsma, 1983; Muthen &
Kaplan, 1985). Performance will be judged in terms of large sample
bias. Furthermore, the standard output of LISREL will be examined in
order to assess the total impact of specification error on testing and
subsequent improvement of structural models. In particular we will
study the chi-squareldegrees of freedom ratio and the maximum
modification index. Section 2 will discuss the problem of specification
error with respect to estimation, testing, and model improvement. In
Section 3 the design and results of a population study of specification
error will be presented. Section 4 will conclude.
The author would like to thank Bengt Muthen, Albert Satorra and a n anonymous
referee for their valuable comments, and Katherine Fry for drawing the path diagram.
Request for reprints should be sent to the author, Department of Educational Studies,
University of Delaware, Newark, DE 1.9716.
JANUARY 1988
69
David Kaplan
Specification Error in Structural Equation Models
Specification Error and Model Estimation
The problem of model specification error is present in the simplest
of regression models as well as complex simultaneous equation models
considered in econometrics (e.g. Amemiya, 1985) and sociology (e.g.
Duncan, 1975). In the case of the single equation linear regression
model, the problem is often formulated in terms of restricted least
squares estimation. Under the null hypothesis of correct specification,
the restrictions are true and estimation of the remaining parameters
yield best linear unbiased estimates. If the restrictions are false,
however, this implies that variables are incorrectly omitted from the
equation. Thus estimates of the remaining regression coefficients are
biased and the bias is proportional to the values of the parameters
associated with the omitted variables (see Judge, Griffiths, Hill,
Lutkepohl, & Lee, 1985, pp. 857-859).
In the context of simultaneous equation estimation, research on
specification error has primarily come out of the econometric tradition
(see White, 1982). Results indicate that full information estimators
such as ML and three-stage least squares (3SLS) tend to propagate
specification errors throughout the system of equations resulting in
inconsistent estimates of the model parameters (Intriligator, 1978). In
contrast, limited information (single equation) estimators such as
2SLS have been found to isolate specification error in the misspecified
equation (Cragg, 1968). Thus the parameters of the particular misspecified equation are not consistently estimated.
In the psychometric literature results on the impact of specification error in the estimation of covariance structure models are limited.
In the context of recursive path analysis, Gallini (1983)has studied the
impact of a variety of misspecifications on a model of aptitude and
achievement. Among other things, her results show that parameter
bias due to omitted variables can be severe if the omitted variables are
strongly related to the exogenous variables. However, since recursive
path models are a special case of simultaneous equation models, her
results only confirm previous econometric findings. Furthermore, her
study is based on a real data set where the true structure (and hence
the severity of the misspecification) is unknown. This study expands on
Gallini by examining misspecification in a population framework
thereby allowing for a precise study of how specification error can lead
to biased estimates. In addition we compare full and limited information estimators.
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MULTIVARIATE BEHAVIORAL RESEARCH
David Kaplan
SpecificationError and Model Testing
The issue of model testing under misspecification has recently
received attention in the psychometric and sociometric literature. For
example, the work of Saris and his colleagues (Saris, den Ronden, &
Satorra, 1985; Satorra & Saris, 1985) has focused primarily on the
issue of testing under misspecification. Their position is that the use of
ad hoc indices such as the chi-squareldegrees of freedom ratio, hereafter X2/df,detract from the importance of hypothesis testing which they
feel is essential for assessing the meaningfulness of any set of results.
Although x2/df was originally designed to be used for large sample
sizes, researchers tend to use this ratio with almost any sample size
(the lowest sample size reported by Saris, et al., 1985, was 65).
Furthermore there is virtually no agreement as to what constitutes an
acceptable value of the index (Saris, et al., 1985). Given the lack of an
agreed upon value and the use of the ratio for almost any sample size,
Saris and his colleagues are concerned that ". . . researchers only use it
in order to have a rationale for accepting the model. Testing the model
this way is, it seems, not necessary at all." (Saris, et al., 1985, p. 5).
In response to the inadequacy of this index, Saris, den Rondon, and
Satorra (1985) argue that it is important to assess the power of the test
statistic for rejecting null hypotheses. A procedure for determining the
power of the test statistic using such standard software as LISREL has
been given in Satorra and Saris (1985). The application of this
procedure for real data is reported in Saris, et al., (1985).In addition to
parameter estimate bias this paper also addresses issues of power
when models are misspecified. Furthermore, we will study the extent
to which ad hoc indices such as X21dfcan give misleading results.
Specification Error and Model Improvement
Research on model improvement can be roughly divided into two
categories: (a) methods based on residuals (Costner & Schoenberg,
1973), (b) methods based on Lagrange multipliers (Byron, 1972;
MacCallum 1986; Saris, de Pijper, & Zegwaart, 1979; Sorbom, 1975).
Of relevance to this paper is the use of modification indices (MI'S)
in specification error analysis. The modification index is a function of
the Lagrange multiplier diagnostic and is calculated as the (Nl2)times
the ratio first-order derivative and the second-order derivative of the
fitting function evaluated at the fixed parameter, scaled in a chisquare metric (Joreskog & Sorbom, 1984).
MacCallum (1986) has recently addressed the issue of utilizing
JANUARY 1988
71
David Kaplan
MI'S for specification searches in the context of covariance structure
models. Considering certain true population models, MacCallum generated twenty samples of sizes N=300 and N=100 from population
covariance matrices based on these models and introduced various
errors of omissions and inclusions. Among other things, MacCallum
concludes that the likelihood of success in recovering correct models
increases when models are close to the true model, sample sizes are
large, and restrictive search strategies are used.
A major problem encountered in MacCallum's study is the occurrence of Type I1 errors, which he attributes in part to the occurrence of
low power. The frequency of Type I1 errors could have been predicted,
however, if the power of the test statistics associated with the various
misspecifications had been calculated using Satorra-Saris procedures.
In addition, MacCallum found that only about one-half of the restrictive searches led to uncovering the true model. Thus the routine use of
the MI for specification searches seems unwarranted and even MacCallum goes on to say that the results are generally discouraging for a
variety of realistic situations. In addition to studying the modification
index this study adds to MacCallum's paper by considering misspecification of both the measurement and structural parts of the
model as well as issues of parameter estimate bias, power, and the
behavior of the X2/dfratio.
Design and Results of a Population Study
A population study was undertaken to assess the effects of misspecification on the estimation, testing, and subsequent improvement
of a latent variable structural equation model. A prototype nonrecursive latent variable model with realistic parameter values was chosen
to allow for a wide variety of possible misspecifications. We will
consider the impact of misspecification on ML estimation. For comparative purposes we also include the 2SLS estimator of Hagglund (1982,
Joreskog, 1983). Since the 2SLS estimator is typically used to obtain
starting values and is not used as an estimator in its own right, a brief
description of how it is implemented in LISREL may be useful.
The 2SLS estimator as implemented in LISREL is a multi-stage
estimator in the sense that measurement parameters are estimated
first via a 2SLS estimator and then the structural parameters are
estimated via 2SLS conditional on the values obtained for measurement parameters. The 2SLS estimator of the measurement part is an
72
MULTIVARIATE BEHAVIORAL RESEARCH
David Kaplan
Figure 1.
A hypothetical latent variable structural model
extension of the instrumental variable estimator of Hagglund (1982).
Once the measurement parameters have been estimated using Hagglund's 2SLS estimator, the parameters of each structural equation
are estimated separately via 2SLS using all exogenous constructs as
instruments (see Joreskog and Sorbom, 1984, p. 1.35).
Inclusion of the 2SLS estimator in this study is motivated by the
fact that it is a limited information estimator and may, in some cases,
be less sensitive than ML to misspecification. It should be noted
however that, because of the multi-stage nature of the 2SLS estimator,
its superiority over ML will be limited to very special cases-namely
nonrecursive path analysis models or nonrecursive latent variable
models with well specified measurement parts. For recursive models
ML and 2SLS give identical results. Nevertheless it is of interest to
compare ML and 2SLS for more general models.
Below we will consider specification error of the measurement part
of the model, then specification error of the structural part of the
model, and end with specification error of both measurement and
structural parts. The model to be considered is described in the
LISREL manual (p. III.94), and is presented in Figure 1. Parameter
values are given for the model in Table 2 and repeated for all
remaining tables. The model in Figure 1 follows the general form of a
system of linear equations among a set of latent variables
JANUARY 1988
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David Kaplan
where q is a vector of endogenous constructs, 6 is a vector of exogenous
constructs, p is a matrix of coefficients relating endogenous constructs
to each other, r is a matrix relating endogenous constructs to
exogenous constructs, and 5 is a vector of disturbances where var(lJ =
?v.
The unobservable constructs q and 6 are related to observable
variables y and x, respectively via the following measurement relations:
where A, and A, are matrices of factor loadings, and E and 6 are vectors
) 0, and var(6) = Og.
of measurement errors, with v a r ( ~ =
For future reference it will be useful to have the specific structural
equations presented. The first and second structural equations can be
written respectively as
The population covariance matrix 2 was provided in the LISREL
manual and was used as the input covariance matrix. For the two
estimators, both gave estimates that agreed with the true values, as
would be expected under the null hypothesis of correct specification.
Given the model in Figure 1, a variety of misspecifications were
imposed. Due to the large number of possible misspecifications only a
subset will be presented and these are shown in the first column of
Table 1.
It is known that when the null (model) hypothesis is true, the
distribution of the likelihood ratio test statistic follows asymptotically
a central chi-square distribution characterized by the degrees of
freedom of the model. If the null hypothesis is false however, the
distribution of the test statistic no longer follows a central chi-square
distribution but instead approaches a noncentral chi-square distribution provided the misspecification is not too severe. The noncentral
chi-square distribution is characterized by the degrees of freedom and
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MULTIVARIATE BEHAVIORAL RESEARCH
David Kaplan
Table 1
Summaru of Po~ulationStudy: T U Dof
~ Miss~ecification.Power, Leraest
MMI Parameter, and Expected Chi-Spuare/df ratio
...................................................................
Mi sspecif ied
Parameter
Noncentrali t y
Parameter
d.f.
Power
Largest MMI
Parameter g2'/dfa
...................................................................
Measurement Misspecification
'~31 = O
179.28
34
1.000
Xx31
6.273
'x52 = O
XXs3 = free
644.95
33
1.000
XXs2
20.549
removed
122.88
21
1.000
XXl2
6.58 1
Structural Nisspecification
9.77
34
0.300
$21
1.287
$21 = o
812 = 0
1 20.73
34
1.000
2(i3
4.55 1
=O
50.03
34
0.993
'd13
2.47 1
Measurement and Structural Misspecification
35
1.000
XX3,
X X 3 i = 0 $ i 2 = 0 291.05
hXs2= 0,
764.79
34
1.000
9.3 16
23.494
I x 5 3 = free,
812 = 0
...................................................................
a
x2' i s the expected value of g2 and i s defined as the sum of the
degrees of freedom and the noncentrality parameter. See text.
the noncentrality parameter. The noncentrality parameter is an index
of how far the central chi-square distribution shifts to the right when
the null hypothesis is false.
When analyzing a misspecified population model such as we do
below, the "CHI-SQUARE" value given by LISREL is in fact the
noncentrality parameter of the noncentral chi-square distribution
corresponding to the asymptotic distribution of the test statistic under
the alternative hypothesis (Satorra & Saris, 1985). This noncentrality
JANUARY 1988
75
David Kaplan
parameter, in addition to the degrees of freedom of the model, can be
used to determine the power of the test statistic for rejection of a false
null hypothesis using tables of power such as those given in Saris and
Stronkhorst (1984). Thus columns two, three, and four of Table 1give,
respectively, the noncentrality parameter, degrees of freedom, and
power of the test for each misspecificaton considered. A sample size of
500 was chosen to obtain this information. Here an examination of
power provides a rough indication of the extent to which one would
detect the misspecification in a finite sample.
The fifth column of Table 1 gives the parameter associated with
the LISREL "Maximum Modification Index" (MMI). Regarding the
MMI one can see immediately that it does not always point to the
correct parameter associated with the misspecification. This is because
the MMI uses the expected drop in chi-square as a criterion for flagging
a certain parameter to be freed and this may not point to the
misspecified parameter. Although in some cases the MMI coincides
with the true misspecified parameter, users of LISREL (or other
programs that provide this index) need to exercise caution when using
the MMI for improvement of model specification. This is perhaps
especially true in those cases where the power is high for non-trivial
misspecifications since it is these cases that are of the most concern.
In addition, the last column in Table 1 also provides the X2':df'
ratio. Here x2' is the expected value of the noncentral chi-square
distribution associated with the LISREL goodness-of-fit statistic and is
calculated as the sum of the degrees of freedom and the noncentrality
parameter (Kendall & Stuart, 1979). In applied research, using finite
samples, this ratio is used as an ad hoc index of fit when the sample
size is deemed too large. For our purposes we may consider this index
as the value that would be expected in finite samples. It can be seen
that different types of misspecification can give rise to various values
of the X2'/dfratio. For some values one may argue that the model "fits,"
while for other values the model clearly does not "fit." Nevertheless, as
we will see below, very severe parameter bias could be obtained
regardless of whether the model fits as judged from the x2'idf ratio.
Misspecification of the measurement part
In what follows we will consider the impact of misspecification on
parameter estimate bias. Cases 1thru 10 correspond directly to the ten
misspecifications displayed in Table 1. The impact of measurement
misspecification on ML and 2SLS parameter estimates of the model are
given in Cases 1thru 3 of Table 2. Bias is defined as the percentage of
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MULTIVARIATE BEHAVIORAL RESEARCH
David Kaplan
Table 2
ML and 2SLS Parameter Estimate Bias For Measurement M i ~ ~ ~ e C i f i C a t i ~ n
Parameter True Value
Case 1
Case 2
Case 3
...................................................................
Measurement Parameters
0, oa
0, 0
0, 0
0, 0
0, 0
0, 0
---0, 0
78, 5 2
0, 0
---0, 0
-77, -77
0, 0
0, 0
a Values i n t h i s table and remaining tables represent percent bias
calculated as (estimate - true value)/true value * 100. The f i r s t
value i s f o r HL, the second value i s f o r 2SLS.
The symbol * means the occurance of a Heywood case.
under- or overestimation of the observed parameter estimate relative
to the true value. From here on we consider a bias over 10% as serious.
Case 1 shows the effect of misspecification when setting a loading
incorrectly to zero. The results show that ML gives rise to a Heywood
case. Here the researcher would presumably stop and examine the
model specification carefully. Inspection of 2SLS estimates reveals no
JANUARY 1988
77
David Kaplan
Table 2 (cont'd)
ML and 2SLS Structural Parameter Estimate Bias For Measurement
Hisspecification
...................................................................
Parameter True Value
Case 1
Case 2
Case 3
Structural Parameters
812
#2
0.493
0.555
-2, o
0, 0
0, 0
-4, -39
5, 8
-245, -2 19
such Heywood case. Nevertheless for both ML and 2SLS we see that
certain measurement and structural parameters are affected by the
misspecification of the measurement part. The incorrect measurement
parameter estimates are associated with the indicator x3 and its
relation to the measurement of 52. The incorrect structural parameter
estimates are due to the conditional estimation of structural parameters given the measurement estimates as discussed earlier.
Case 2 shows the results of misspecifying a measurement relation
by assuming that a variable (xg)is an indicator of one factor (b)when
it is truly the indicator of another factor (t2). In this case ML does not
78
MULTIVARIATE BEHAVIORAL RESEARCH
David Kaplan
produce a Heywood case. ML and 2SLS estimates associated with the
inappropriate measurement relation are biased to a considerable
degree. ML estimates of the structural parameters do not appear to be
as adversely affected as 2SLS, with the exception of $22. Here the
conditional estimation feature of the 2SLS estimator seems to give rise
to more bias than that due to the full information aspect of ML.
A final measurement misspecification is considered in Case 3.
Here we assume that
and its indicators are not included in the
model. This also implies that the structural relationships associated
with t1are not included. Thus, a total of nine parameters are removed.
This is perhaps a common form of misspecification arising from
incomplete knowledge of the constructs involved in fully specifying a
theoretical model. The impact of this misspecification is severe. Many
parameters are affected to a great degree with ML and 2SLS performing about the same. Thus here we find the multi-stage 2SLS estimator
to be as adversely affected as ML.
For the type of measurement misspecification considered here we
find that in two instances the X2'ldf is less than 10.0. In applied
literature the value 10.0 is sometimes used as a criterion for acceptable
fit when using this index. We can see that if models were not rejected
on this basis, the parameter estimates reported would be severely
biased.
Misspecification of the structural part
Next, consider the case where the measurement model is assumed
to be correctly specified but specification errors may occur in the
structural part of the model. Very little measurement parameter bias
was found for these cases therefore only results for the structural
parameters are presented. These results are displayed in Table 3. The
results demonstrate rather clearly the extent to which ML is affected
by misspecification of the structural part of the model. In particular,
the number of biased ML parameters exceed those of 2SLS. This is due
to the fact that, as stated earlier, ML propagates errors throughout the
system of equations whereas 2SLS isolates misspecifications in the
equations that are misspecified.
The size of the bias appears to be strongly related to the size of the
parameter that is incorrectly fixed to zero. This can be seen by
comparing the results from Case 6 (yll = 0.400) with Case 7 (yzl =
-1.000). In Case 7 we see very large parameter estimate bias. By
contrast note the relative robustness of ML and 2SLS for Case 6.
Though both sets of estimates are biased to a great degree careful
JANUARY 1988
79
David Kaplan
Table 3
c
MissDeciflcatlon
...................................................................
Parameter
True Value
Case 4
case 5
Case 6
Case 7
...................................................................
8 12
82 1
0.493
0.595
Structural Parameters
----2, 0
----
-56, 0
-2, 2
-6, 0
17, 0
-346, - 130
inspection shows that only the 2SLS parameter estimates associated
with the misspecified equation are affected by the misspecification. For
example, consider the equation for qz given in Equation 5 and shown
in Figure 1. Now consider the results of misspecification yzl = 0 in
Case 7. The affected 2SLS estimates are PZ1,yz3,GZ1,and I J J ~ which
~,
are
precisely the remaining parameters of Equation 5 when yzl = 0.
Similar results for 2SLS can be seen for Cases 4 thru 6.
With regard to testing, the inadequacy of the X2'ldfratio is clear
when examining the structural misspecifications presented here. All
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MULTIVARIATE BEHAVIORAL RESEARCH
David Kaplan
misspecified models presented in this section have an index value less
than 10.0 and four out of five have index values less than 5.0. It should
be noted that different sample sizes would perhaps give rise to varying
values of this index. However, in applied work it would be tempting to
conclude that these models are consistent with the data. Nevertheless,
from an inspection of Cases 4 thru 7 of Table 3 it can be seen that the
parameter estimates are virtually meaningless.
As an example of how misspecification can affect conclusions regarding model improvement Cases 4 and 5 offer an interesting contrast. From
Table 1 note that the misspecification pzl = 0 (Case 4) gives rise to low
power but very high parameter bias. For a given sample we would
perhaps not reject the null hypothesis, but the parameter estimates could
be biased by a considerable degree. Furthermore, though the power is
low, the MMI in this case does point to the correct parameter to be freed.
For Case 5 (plz = 01, however, there is certainly a good deal of power for
rejection but the use of the MMI for model improvement would be
misleading because it flags y13to be freed.
Misspecification of the measurement and structural parts
Perhaps the most realistic case of misspecification occurs when
more than one structural parameter is misspecified or when both the
measurement model and structural model are misspecified. The results
of a selected set of cases are shown in Table 4, Cases 8 through 10.
Case 8 gives the results of more than one structural misspecification. Here we specify a recursive model with yll, Plz, and 4~~~= 0. First,
it should be noted that the MMI does not flag the correct parameter to
be freed. In another analysis, however, we set yzl = 0 and left yll free.
In that analysis the MMI did point to yzl. Thus once again we can see
the unreliability of the MMI.
Case 9 shows the affect of fixing kx31 = 0 and plz = 0. ML estimation
yields a Heywood case as we saw in Table 2. The 2SLS estimates are
affected in both the measurement part and structural part as expected.
Because the measurement model is misspecified, 2SLS estimates of
structural parameters are more adversely affected in the sense that the
misspecification is not isolated to only the misspecified equation. A
similar result emerges when inspecting Case 10.
With respect to testing, a decision to not reject these models based on
the X2'ldfratio would depend on what is considered an acceptable value.
For example, Cases 8 and 9 have X2'ldfvalues that could be considered
low, whereas the value in Case 10 is very high. Regardless of the value,
it can be seen that parameter estimates are severely biased.
JANUARY 1988
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David Kaplan
Table 4
ML and 2SLS Measurement Parameter Estimate Bias For Measurement and
Structural Part Miss~ecif
ication
Parameter True Value
Case 8
Case 9
Case 10
...................................................................
Measurement Parameters
0, 0
0, 0
-1, 0
3, 0
-4, 0
0, 0
0, 0
0, 0
0, 0
---79, 52
0, 0
Summary and Concluding Remarks
The purpose of this paper was to assess the impact of misspecification on parameter estimation, model testing, and model improvement.
The above findings were the result of a population study wherein the
true parameter values as well as the precise location of the specification error were known. This information is usually not available to the
substantive researcher. Nevertheless the results presented here can be
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MULTIVARIATE BEHAVIORAL RESEARCH
David Kaplan
Table 4 (Cont'd)
7Sl S Structural Parameter Estimate Bias For Measurement and
Structural H i s s ~ e c i f i c a t i o n
Parameter True Value
Case 8
Case 9
Case 10
...................................................................
$12
0.493
82 1
0.595
Structural Parameters
------77, 0
-71, 0
----68, -39
taken as an indication of what might happen in practice.
With regard to estimation under measurement model misspecification, ML and 2SLS were both affected with ML performing
slightly worse in terms of large sample bias. Also, it appears that the
exclusion of important constructs can be the most serious form of
measurement misspecification. For misspecification of the structural
part, ML propagated parameter estimate bias throughout the structural parameters as expected and was seen to affect some measurement parameters as well. 2SLS clearly out-performed ML for structural misspecification and behaved according to standard econometric
JANUARY 1988
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David Kaplan
theory. Combinations of measurement and structural rnisspecification
yielded the worse case scenario in terms of the actual number of ML
estimates exceeding 10% bias compared to 2SLS estimates. Thus, due
to the bias that can result from model rnisspecification, caution needs
to be exercised when routinely presenting parameter estimates from
misfitting models. Presentation of parameter estimates from misfitting
models might be justified if a power analysis reveals only trivial
specification errors are being detected. In this case, it may be that,
estimates are only biased to a small degree.
An interesting result that emerged was the fact that under no case
of misspecification were the measurement parameters for the y variables affected. This result appears to be due to the correlation among the
parameter estimates. An inspection of the parameter correlation
matrix for the population model under no rnisspecification shows that
the measurement parameters for the y variables are virtually
uncorrelated with other parameters in the model. This finding explains, for example, Case 3 where the removal of the construct did
not affect the measurement parameters for t h e y variables. It should be
noted however, that this lack of correlation among parameters is model
dependent and may not hold for other models or other parameter values.
Comparisons of ML and 2SLS rested on the econometric results of
rnisspecification in simultaneous equation systems. A discussion on
formally comparing full information and limited information estimators in the context of specification error in econometric simultaneous
equation models is given in Hausman (1978). Hausman noted that
under the null hypothesis of correct specification full information
estimators such as ML will be consistent as well as asymptotically
efficient. Under the alternative hypothesis the incorrect parameters
will only be efficient. By contrast, single equation estimators such as
2SLS are consistent for all but the misspecified equations. From these
results Hausman derived a chi-square test of rnisspecification.
Because of the multi-stage nature of the 2SLS estimator, as
implemented in LISREL, a Hausman-type specification error test
would be limited to nonrecursive path analysis models, or
nonrecursive latent variable models with well specified measurement
parts. Furthermore, the Hausman test requires expressions for the
asymptotic covariance matrix of the 2SLS estimator in order to obtain
standard errors. Although these expressions exist (see e.g. Hagglund,
1982; Joreskog, 1983) they are not implemented in LISREL. If the
2SLS standard errors can be computed, a Hausman test might be
useful for these limited cases.
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MULTIVARIATE BEHAVIORAL RESEARCH
David Kaplan
With regard to testing, we find that in the population the ad hoc
X2'ldfratio can lead to the acceptance of certain models that have
severely biased parameter values. It may be the case that in the
sample certain models with certain parameter values would give rise
to a low value of this index and low parameter bias; however, it would
be impossible to predict ahead of time which models would lead to this
situation. Thus, in agreement with Saris, et al., (1985),chi-square
should be taken seriously as a means of formally testing model
specification. The results lead us to disagree with, for example,
MacCallum (1986, p. 118) who strongly emphasizes that other measures of model fit (i.e. X2/df)be used to evaluate models.
Finally, regarding model improvement, we have seen how the MMI
can give misleading results and argue that caution needs to be exercised
when using the MMI for the detection of a specification error. Thus, the
routine use of the MMI cannot generally be recommended for improvement of model specifications. Instead, future research should focus on the
development of alternative methods for model improvement.
References
Amemiya, T. (1985). Advanced Econometrics. Cambridge: Harvard University Press.
Boomsma, A. (1983). On the robustness of LISREL (maximum likelihood estimation)
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MULTIVARIATE BEHAVIORAL RESEARCH