bollen (1995 sm)
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a n
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)
pro-
posed
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
pro-
vides 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
asymp-
totically normal with a known asymptotic covariance matrix.
The observed
random
variables can
come
from
nonnormal
distributions.
Several
hypothetical
cases and
an
empirical
exam-
ple
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
helpfuldiscussions 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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KENNETH A.
BOLLEN
1.
INTRODUCTION
Structural
equation
models
(SEMs),
sometimes called LISREL mod-
els,
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
analy-
sis
techniques
as
special
cases
(Joreskog
and
Sorbom
1993).
Though
it is
straightforward
o 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"vari-
ables
(Marsden 1983),
the treatment of
models
with
equations
that
are nonlinear in latent
or
unobserved variables
is not
fully
devel-
oped. 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
interac-
tions 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
esti-
mator 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,andSection4 discussesthe selection of instrumen-
tal 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
n
Section
6 in
the conclusion.
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MODELS
THAT ARE NONLINEAR IN LATENT VARIABLES
2.
LITERATURE REVIEW
2.1. Literatureon
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 randomvariable,L1and
L2
are latentrandom
variables and
,
is
a random disturbanceterm 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
ndependently
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 normallydistributedvariables, allows one to consis-
tently
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
non-
product
observed
variables must be
known;
tests of
statistical
signifi-
cance
of
parameter
estimates
are
not
provided;
it offers no
methods
for estimating equation intercepts; and the robustness of the esti-
mates
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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KENNETH A. BOLLEN
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
dis-
tributed
explanatory
variables,
and
provides
estimates
of
the stan-
dard
errors
of the
resulting
coefficient estimates.
Single
indicators
and known
error
variances
(and
error covariances
f
present)
are still
required,
however.
Heise's
(1986)
and Feucht's
(1989)
Monte Carlo
simulation results
provide
mixed evidence
on
the value
of
these sin-
gle
indicator
approaches
to
including
interactionsof latent variables.
Kenny
and Judd
(1984) give
an alternativemethod
of
incorpo-
rating
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
nto 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) treatmultipleindicators,add one more indi-
cator 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 nonautocorre-
lated,
and are
independent
of
L1,
L2,
62,
63,
,
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
226
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MODELS THAT ARE NONLINEAR
IN
LATENT
VARIABLES
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 s
L1,
L2,
62
to
E5,
1,
L1L2,
L
13,
L1E,
L2E2,
L2E4,
E2E3,
E2E5,
E364,
and
E465.
Estimating
the measurement
equations
(6)
to
(9)
in-
volves linear and nonlinear
constraints on the
parameters.
For in-
stance,
in
equation
(7)
the factor
loadings
for
L1L2
and for
L2E2
are
both
equal
to
A52
n
equation
(5).
Equation
(9)
for
y4y5
has
a nonlinear
constraint on the coefficient
for the
L1L2
variable. Additional
restric-
tions occur
for the variances of the
product
latent variables
in
equa-
tions
(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 restric-
tions are in
Kenny
and Judd
(1984).
The introduction
of the nonlinear
constraints
implied
by
the
model and
assumptions
allows consistent
estimation of the coeffi-
cients 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 Higginsand Judd(1990)for anotherempiricalapplication.
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
assump-
tions. Another is
the
proliferation
of
product
latent
variables,
distur-
bances,
and
observed variables that
occurs with
this method.
Even
a
relatively
simple
model
requires
many
terms
when
multiple
indica-
tors 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
prop-
erties of
the model with raw
rather than
deviation
scores are not
known.
2.2.
Literature
on Instrumental
Variables
and
2SLS
Other literaturehas been less concerned with nonlinearfunctions 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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KENNETH
A.
BOLLEN
IV and 2SLS
techniques
are
helpful
when
an
explanatory
vari-
able
in a
regression equation
is correlated
with
the
disturbance term
of
the
equation.
An IV is a variable
that is
correlated with
an
"endoge-
nous"
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
vari-
able
in
the
original equation.
The "second
stage"
of
2SLS
is the OLS
regression
of the
original
dependent
variable
on this
predicted
endoge-
nous
explanatory
variable
and
the other
explanatory
variables.
It
pro-
vides 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) pro-
posed
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
func-
tions 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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MODELS
THAT ARE NONLINEAR
IN LATENT VARIABLES
and
applies
to the
measurement model as well as to
the latent vari-
able model.
Suppose
that
the
model for the latent
variablesis
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
constantcoeffi-
cients 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
non-
autocorrelated 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 indica-
tors.
A
second
equation
provides
the
measurementmodel
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
s
an
i.i.d. random
error
of
measurement that has a
mean of
zero and that
is
independent
of L
andf(L). If a "latentvariable"is perfectlymeasured, then the corre-
sponding
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) correspondsto the latent variablemodel, and
equation (11)
to
the
measurement
(or
confirmatory
actor
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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KENNETH A. BOLLEN
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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MODELS THAT ARE NONLINEAR
IN
LATENT VARIABLES
Yi
=
aLi
+
B1Y1
+
B2ifYl
-
E1)
-
Bli
1
+
Ei
+
i,
(18)
where Yiis one of the indicators that scales a latent variable. The i
subscript signifies
the
ith
row of
the
matrix or
vector-so,
for
in-
stance,
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
n
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
in-
stance,
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
s
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
disturbanceterm
Ui
=
B2ig2(l,E1)
-
Bli
1
+
Ei
+
i'-
(23)
In
general
ui
will
be
correlated with
the
right-hand
side vari-
ables 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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KENNETH
A.
BOLLEN
hand
side
variables,
a
two-stage
least
squares
(2SLS)
estimator
pro-
vides 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 mea-
surement 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 variablesin
y,
that have nonzero coefficientsin the
yi
equation,
and
gl(yl)(i)
to be
the
N row
matrix
of values for
the vari-
ables in
gl(yl)
that
have nonzero coefficients in
the
Yiequation.
The N
x
1
vector
Yi
contains the N
values of
Yi
n
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 vari-
ables,
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-handside matricesof equations (25)
to
(27)
are
finite,
that
Xv,iv
s
nonsingular,
and that
XviZi
s
nonzero.
These
assumptions
require
that the
instrumental
variables
(IVs)
cor-
relate 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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MODELS THAT ARE NONLINEAR
IN LATENT VARIABLES
observed variables
(y's)
that are
part
of the model or
nonlinear func-
tions
of
such
observed variables.
Assume
that
E[uiui]
=
o2I
so that the
composite
disturbance s
homoscedastic
and nonautocorrelated. Whether
E[ui]
=
0
will
de-
pend
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
s
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^' ,)-1^. (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
2),
(30)
where AN
refers to an
asymptotically
normal
distribution.
The
previ-
ous
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
A
N,),
(32)
and an
estimate
of the
asymptotic
covariance matrix
of
Ai
is
acov
(Ai)
=
a
ui(Z'iZi)-.
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KENNETH
A. BOLLEN
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
statisti-
cal
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
equa-
tion
(11). Substituting
equation
(15),
y,
-
e1,
for
L in
equation
(11)
leads
to
equation
(17).
Analogous
to
equation
(18)
from
the
latent
variablemodel, 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
+
+
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 proceduresdeveloped in the preced-
ing
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
or 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,
distur-
bances,
and latent
variables
of the
model are
important
aids
to
select-
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MODELS THAT ARE NONLINEAR
IN LATENT
VARIABLES
ing
appropriate
IVs.
A
general
procedure
for
selecting
IVs
has
sev-
eral
steps.
Assume
that
vi
is
a variable that
might
be a suitable
instrumentalvariable. The
following steps help
to
evaluate
its
eligibil-
ity: (1)
Form
COV(vi,
ui);
(2)
if
vi
is an
endogenous
variable,
substi-
tute
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,
2Y6,
3Ys,
nd
Y3Y6
ften
will
qualify
as IVs. Determination
of
their
eligibility
follows the
same
steps
of
writing
a reduced-form
expression
for
each variable
in
the
product, obtainingthe productof the reducedforms, 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 vari-
ables
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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KENNETH A.
BOLLEN
where
u1
=
-311E2
-
f12E3
-
813y2E3
-
f813y3E2
+
813E2E3
+
1.
Allther.h.s.
variables
of
equation
(38)
are correlated with the
composite
distur-
bance,
u1.
The
y4
and
y5
variables are
indicatorsof L1
and
L2,
respec-
tively (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
eads
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 influ-
enced
by Li
are
unacceptable
since
they
too will correlate
with
Si.
Also,
if icorrelateswith rj, hen the indicatorsof Ljarenot suitable asIVs in
the latent
variable
model.
The
B1iEl
term
means
that
any
of the
scaling
indicators
for the
latent
variablesthat
appear
on
the
right-hand
ide
of
the
yi equation
cannot be
IVs. Nor can
y's
whose
errors
of
measure-
ment
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
correlatewith 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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MODELS
THAT ARE NONLINEAR
IN LATENT
VARIABLES
these
components
of
ui.
An indicator whose
error
correlates
with
Ei
s
ineligible.
Scaling
indicators for latent variables that affect
Yi
or indi-
cators whose errors of measurementcorrelate with the errorsof 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
n
the
latent variable model
(see
equation
[23]),
i
does
not
appear
in the
composite
disturbance
for
the mea-
surement
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.,