The dynamic model is given by
Finally we may have a panel with . In this case these three
approaches (methods 2-4) are not applicable. Therefore another method is
implemented using only
and
as instruments for the
lagged differences:
The computation of the GMM estimator may be computationally burdensome when the number of instruments gets large. Therefore, it is highly recommended to start with the simplest GMM estimator (i.e. method 1) and then to try the more computer intensive methods 2-4. For a more efficient estimator, the standard errors of the coefficients should be smaller than for less efficient estimators. Therefore, it is expected that the standard errors tend to decrease with a more efficient GMM method. However in small samples the difference may be small or one may even encounter situations where the standard errors increase with an (asymptotically) more efficient estimator. This may occur by chance in a limited sample size or it may indicate a serious misspecification of the model.
To estimate the optimal weight matrix of the GMM estimate, two different approaches can be used. First, under the standard assumptions of the errors, the weight matrix may be estimated as
To assess the validity of the model specification, Hansen's
misspecification statistic is used. This statistic tests the validity of the
overidentifying restrictions resulting from the difference of the
number of conditional moments and the number of estimated coefficients.
If the model is correctly specified, the statistic is distributed
and the
-value of the statistic is given in the output string of the
pandyn
quantlet. Furthermore, a Hausman test (computed as a
conditional moments test) can be used to test the hypothesis that the
individual effects are correlated with the explanatory variables.
The data set z is similarly arranged as in the case of a static panel
data estimation. If the data set is unbalanced, the identification
number of the cross-section unit and the time period are given in the first two
columns. However, the quantlet
pandyn
only uses the time index
so that the first column may have arbitrary values. The following
columns are the dependent variable
and the explanatory variables
, where all explanatory variables must vary
in time. This is necessary because if the variables are constant,
they become zero after performing differences.
If the data set is in a balanced form, the first two columns can be
dropped and the common number of time periods is given. Furthermore, the
number of lagged dependent variables must be indicated. Accordingly,
the quantlet is called by
{output,beta} = pandyn(z,p,IVmeth {,T})The output table is returned in the string output and the coefficient estimates are stored in the vector
For the two-stage GMM estimator the weight matrix is computed using a consistent first-step estimate of the model. This can be done using the following estimation stages:
{out1,beta} = pandyn(z,p,IVmeth {,T}) out2 = pandyn(z,p,IVmeth,beta {,T}) out2The output for the second estimation stage is presented in the string out2. In small samples, the two-stage GMM estimator may have poor small sample properties. Thus, if the results of the two stages differ substantially, it is recommended to use the (more stable) first-stage estimator.