st: Paul Allison's SEM model with fixed effects, reciprocal effects, and lagged predictors

[Jeremy Reynolds <jeremyr@uga.edu>]

Dear Statalist,

On June 29th, 2013, I asked for help estimating an SEM model that
contains fixed effects, reciprocal effects, and lagged predictors.
The model was suggested by Paul Allison (see references at the end of
this post).  I thought others might be interested in the solution, so
I have posted it below.

Jeremy


The models below estimate the relationship between the proportion of
women in an occupation (pf*) and the occupation's median wage
(mdwgf*).
They contains fixed effects (FE) and lagged measures of the dependent variable.
They also contain a correlation between the error term in the first
equation  and all future values of the cross-lagged IV (in this case
there is only one future value).
To add this special correlation, we have to play a little trick on
stata that was suggested by Richard Williams.
Specifically, Stata does does not permit correlations between
exogenous variables and error variables (SEM manual pg 240).
Howevever, mathematically, error terms are just special cases of
latent variables.
Therefore, we set the covariance for the error term in the first
equation (e.g., e.pf2) to zero using the cov() option.
This prevents it from covarying with anyting and effectively removes
it from the model.
The latent variable ERR1 takes the place of the removed error term and
is only allowed to covary with future values of the cross-lagged
variable (e.g., mdwgf3).
This special new variable should not be correlated with any other
exogenous variables including the fixed effects (FE)
These additional constraints on how ERR1 correlates with other
variables are specified in the cov() option.
Note: The results below match Allison's results with MPlus, but to
match his SAS results, the method(mlmv) option must be removed.

************************************
*Median Wage, Proportion Female Example
************************************
use "http://www.statisticalhorizons.com/wp-content/uploads/occ.dta";, clear

#delimit ;
sem (pf2 <- mdwgf1@c1 pf1@c2 FE@1 ERR1@1) // fixed effects and latent
var (ERR1) to replace error term
    (pf3 <- mdwgf2@c1 pf2@c2 FE@1)
    (pf4 <- mdwgf3@c1 pf3@c2 FE@1),
    cov(e.pf2@0) // covariance of error term set to zero to remove it
from the model
    cov(ERR1*_oexogenous@0 ERR1*FE@0 ERR1*mdwgf3) // ERR1 should only
correlate with future values of cross-lagged IV
    nocapslatent latent(FE ERR1)
    method(mlmv);
#delimit cr

*Model 2 (2nd half of model)
#delimit ;
sem (mdwgf2 <- pf1@c1 mdwgf1@c2 FE@1 ERR1@1) // fixed effects and
latent var (ERR1) to replace error term
    (mdwgf3 <- pf2@c1 mdwgf2@c2 FE@1)
    (mdwgf4 <- pf3@c1 mdwgf3@c2 FE@1),
    cov(e.mdwgf2@0) // covariance of error term set to zero to remove
it from the model
    cov(ERR1*_oexogenous@0 ERR1*FE@0 ERR1*pf3) // ERR1 should only
correlate with future values of cross-lagged IV
    nocapslatent latent(FE ERR1)
    method(mlmv);
#delimit cr

The model estimated by the code above is described in the works below.
http://www.statisticalhorizons.com/wp-content/uploads/2012/01/Causal-Inference.pdf
England et al. Social Science Research 36: 2007
Fixed Effects Regression Models Using SAS, 2005 by Paul Allison
Fixed Effects Regression Models, Sage green book # 160, 2009 by Paul Allison

-- 
********************
Dr. Jeremy Reynolds
Associate Professor
Undergraduate Coordinator
Department of Sociology
116 Baldwin Hall
University of Georgia
Athens, GA 30602-1611
Phone: (706) 583-8072
Web: http://uga.edu/soc/people/faculty/reynolds_jeremy.php
Fax: (706) 542-4320
*
*   For searches and help try:
*   http://www.stata.com/help.cgi?search
*   http://www.stata.com/support/faqs/resources/statalist-faq/
*   http://www.ats.ucla.edu/stat/stata/