Model-implied instrumental variable (MIIV) search
A key step in the MIIV-2SLS approach is to transform the SEM by replacing the latent variables with their scaling indicators minus their errors. Upon substitution the SEM is transformed from a model with latent variables to one containing observed variables with composite errors. The miivs function automatically makes this transformation. The miivs function will also identify equation-specific model-implied instrumental variables in simultaneous equation models without latent variables.
miivs(model)
model: A model specified using lavaan model syntax. See the model argument within the lavaanify
function for more information. See the documentation below for a description of how to specify the scaling indicator in latent variable models and impose equality constraints on the parameter estimates.
A list of model equations.
model A model specified using the model syntax employed by lavaan. The following model syntax operators are currently supported: =~, ~, ~~ and *. See below for details on default behaviors, how to specify the scaling indicator in latent variable models, and how to impose equality constraints on the parameter estimates.
Example using Syntax Operators
In the model below, 'L1 =~ Z1 + Z2 + Z3' indicates the latent variable L1 is measured by 3 indicators, Z1, Z2, and Z3. Likewise, L2 is measured by 3 indicators, Z4, Z5, and Z6. The statement 'L1 ~ L2' specifies latent variable L1 is regressed on latent variable L2. 'Z1 ~~ Z2' indicates the error of Z2 is allowed to covary with the error of Z3. The label LA3 appended to Z3 and Z6 in the measurement model equations constrains the factor loadings for Z3 and Z6 to equality. For additional details on constraints see Equality Constraints and Parameter Restrictions.
model \<- '
L1 =~ Z1 + Z2 + LA3*Z3
L2 =~ Z4 + Z5 + LA3*Z6
L1 ~ L2
Z2 ~~ Z3
'
Scaling Indicators
Following the lavaan model syntax, latent variables are defined using the =~ operator. For first order factors, the scaling indicator chosen is the first observed variable on the RHS of an equation. For the model below Z1 would be chosen as the scaling indicator for L1 and Z4 would be chosen as the scaling indicator for L2.
model \<- '
L1 =~ Z1 + Z2 + Z3
L2 =~ Z4 + Z5 + Z6
'
Equality Constraints and Parameter Restrictions
Within- and across-equation equality constraints on the factor loading and regression coefficients can be imposed directly in the model syntax. To specify equality constraints between different parameters equivalent labels should be prepended to the variable name using the
operator. For example, we could constrain the factor
loadings for two non-scaling indicators of latent factor L1 to
equality using the following model syntax.
model \<- '
L1 =~ Z1 + LA2*Z2 + LA2*Z3
L2 =~ Z4 + Z5 + Z6
'
Researchers can also constrain the factor loadings and regression
coefficients to specific numeric values in a similar fashion. Below
we constrain the regression coefficient of L1 on L2
to 1.
model \<- '
L1 =~ Z1 + Z2 + Z3
L2 =~ Z4 + Z5 + Z6
L3 =~ Z7 + Z8 + Z9
L1 ~ 1*L2 + L3
'
Higher-order Factor Models
For example, in the model below, the scaling indicator for the
higher-order factor H1 is taken to be Z1, the scaling
indicator that would have been assigned to the first lower-order
factor L1. The intercepts for lower-order latent variables
are set to zero, by default
model \<- '
H1 =~ L1 + L2 + L3
L1 =~ Z1 + Z2 + Z3
L2 =~ Z4 + Z5 + Z6
L3 =~ Z7 + Z8 + Z9
'
Model Defaults
In addition to those relationships specified in the model syntax
MIIVsem will automatically include the intercepts of any
observed or latent endogenous variable. The intercepts
for any scaling indicators and lower-order latent variables are
set to zero. Covariances among exogenous latent
and observed variables are included by default when var.cov = TRUE. Where appropriate the covariances of the errors
of latent and observed dependent variables are also automatically
included in the model specification. These defaults correspond
to those used by lavaan and auto = TRUE, except that
endogenous latent variable intercepts are estimated by default,
and the intercepts of scaling indicators are fixed to zero.
Invalid Specifications
Certain model specifications are not currently supported. For example,
the scaling indicator of a latent variable is not permitted to
cross-load on another latent variable. In the model below
Z1, the scaling indicator for L1, cross-loads on the latent
variable L2. Executing a search on the model below will
result in the warning: miivs: scaling indicators with a factor complexity greater than 1 are not currently supported.
model \<- '
L1 =~ Z1 + Z2 + Z3
L2 =~ Z4 + Z5 + Z6 + Z1
'
In addition, MIIVsem does not currently support relations
where the scaling indicator of a latent variable is also the
dependent variable in a regression equation. For example, the
model below would not be valid under the current algorithm.
model \<- '
L1 =~ Z1 + Z2 + Z3
Z1 ~ Z4
Z4 ~ Z5 + Z6
'
The miivs function displays a table containing the following information for each equation in the system:
LHS The "dependent" variable.RHS The right hand side variables of the transformed equation.MIIVs The model-implied instrumental variables for each equation.Bollen, K. A. (1996). An Alternative 2SLS Estimator for Latent Variable Models. Psychometrika, 61, 109-121.
Bentler, P. M., and Weeks, D. G. (1980). Linear Structural Equations with Latent Variables. Psychometrika, 45, 289–308.
miive