Generalized method of moments (GMM)
Stata’s gmm
command makes generalized method of moments
estimation as simple as nonlinear least-squares estimation and nonlinear seemingly
unrelated regression. Just specify your residual equations by using
substitutable expressions, list your instruments, select a weight
matrix, and obtain your results.
Here we fit a Poisson model of the number of doctor visits as a function
of gender, income, and whether a person has a chronic disease or private
health insurance. We have reason to believe that income is endogenous,
so we use age and race as instruments.
By default, gmm uses the two-step estimator and a weight matrix
that assumes the errors are independent but not identically distributed. By using the wmatrix() and vce()
options, you can request weight and variance–covariance matrices
appropriate for errors that are independent and identically
distributed, are independent but not identically distributed, exhibit
group-level clustering, or that exhibit heteroskedasticity and
autocorrelation. In addition, you can obtain standard errors via
the bootstrap or the jackknife.
When fitting a model by GMM, you should check to see whether the
instruments you use really satisfy the orthogonality condition—i.e.,
whether they are uncorrelated with the errors. In GMM estimation,
Hansen’s J statistic is the most common test statistic. In our
example, whether our instruments are valid is certainly open for
debate—age likely influences the number of doctor visits—and we can
test their validity by using estat overid to obtain Hansen’s J
statistic:
The test statistic has a χ2 distribution under the null hypothesis
that the instruments are valid. The significant statistic indicates
that one or more of our instruments are not valid (assuming that the
model is otherwise correctly specified).
gmm’s numerical derivative routines are very accurate, so most of
the time you do not need to spend time taking analytic derivatives.
However, if speed is of the essence or if you plan to fit the same model
repeatedly, analytic derivatives can be a boon. gmm provides a
simple way to specify derivatives. We could fit our previous model
with analytic derivatives by specifying
The notation {xb:private chronic female income} is a shorthand to
create a linear combination of variables; we could have specified
{xb_private}*private + {xb_chronic}*chronic + {xb_female}*female +
{xb_income}*income and declared each parameter explicitly. By using
the shorter notation, we need only specify the derivative with respect
to the entire linear combination instead of each parameter individually.
Moreover, once we declare our linear combination, we can subsequently refer to
it as xb: without having to specify the variables associated with it.
In addition to standard instruments, gmm allows you to create
panel-style instruments used in dynamic and other panel models with
endogenous regressors. In these models, you typically use lagged
values of regressors as instruments; the number of lagged values
available increases as the time dimension increases. gmm’s
xtinstruments() option makes creating these instruments a snap.
For more complicated analyses, gmm allows you to write a program
to evaluate your residual equations instead of using substitutable
expressions. These programs are structured like those that
ml,
nl, and
nlsur use.
Here is our Poisson model fit using
the moment-evaluator program version of gmm:
The programmable version of gmm is eminently flexible. You just
need to tell it how many moment equations you have and the number of
parameters in your model. You can optionally specify instruments and
select the type of weight matrix to use and standard errors to report.
Explore more about GMM features
in Stata.
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