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这是 help xtabond2后的描述
Description
xtabond2 can fit two closely related dynamic panel data models. The first is the Arellano-Bond (1991) estimator, which
is also available with xtabond, though without the two-step standard error correction described below. It is sometimes
called "difference GMM." The second is an augmented version outlined by Arellano and Bover (1995) and fully developed
by Blundell and Bond (1998). It is known as "system GMM." Roodman (2006) provides a pedagogic introduction to linear
GMM, these estimators, and xtabond2. The estimators are designed for dynamic "small-T, large-N" panels that may
contain fixed effects and--separate from those fixed effects--idiosyncratic errors that are heteroskedastic and
correlated within but not across individuals. Consider the model:
y_it = x_it * b_1 + w_it * b_2 + u_it i=1,...,N; t=1,...,T
u_it = v_i + e_it,
where
v_i are unobserved individual-level effects;
e_it are the observation-specific errors;
x_it is a vector of strictly exogenous covariates (ones dependent on neither current nor past e_it);
w_it is a vector of predetermined covariates (which may include the lag of y) and endogenous covariates, all of which
may be correlated with the v_i (Predetermined variables are potentially correlated with past errors.
Endogenous ones are potentially correlated with past and present errors.);
b_1 and b_2 are vectors of parameters to be estimated;
and E[v_i]=E[e_it]=E[v_i*e_it]=0, and E[e_it*e_js]=0 for each i, j, t, s, i<>j.
First-differencing the equation removes the v_i, thus eliminating a potential source of omitted variable bias in
estimation. However, differencing variables that are predetermined but not strictly exogenous makes them endogenous
since the w_it in some D.w_it = w_it � w_i,t-1 is correlated with the e_i,t-1 in D.e_it. Following Holt-Eakin, Newey,
and Rosen (1988), Arellano and Bond (1991) develop a Generalized Method of Moments estimator that instruments the
differenced variables that are not strictly exogenous with all their available lags in levels. (Strictly exogenous
variables are uncorrelated with current and past errors.) Arellano and Bond also develop an appropriate test for
autocorrelation, which, if present, can render some lags invalid as instruments.
A problem with the original Arellano-Bond estimator is that lagged levels are poor instruments for first differences if
the variables are close to a random walk. Arellano and Bover (1995) describe how, if the original equation in levels
is added to the system, additional instruments can be brought to bear to increase efficiency. In this equation,
variables in levels are instrumented with suitable lags of their own first differences. The assumption needed is that
these differences are uncorrelated with the unobserved country effects. Blundell and Bond show that this assumption in
turn depends on a more precise one about initial conditions.
xtabond2 implements both estimators--twice. The version in Stata�s ado programming language is slow but compatible
with Stata 7 and 8. The Mata version is usually faster, and runs in Stata 9.1 or later. (Upgrading from 9.0 to 9.1 is
free.) The xtabond2 option nomata prevents the use of Mata even when it is available.
The Mata version also includes the option to use the forward orthogonal deviations transform instead of first
differencing. Proposed by Arellano and Bover (1995) the orthogonal deviations transform, rather than subtracting the
previous observation, subtracts the average of all available future observations. The result is then multiplied by a
scale factor chosen to yield the nice but relatively unimportant property that if the original e_it are i.i.d., then so
are the transformed ones (see Arellano and Bover (1995) and Roodman (2006)). Like differencing, taking orthogonal
deviations removes fixed effects. Because lagged observations of a variable do not enter the formula for the
transformation, they remain orthogonal to the transformed errors (assuming no serial correlation), and available as
instruments. In fact, for consistency, the software stores the orthogonal deviation of an observation one period late,
so that, as with differencing, observations for period 1 are missing and, for an instrumenting variable w, w_i,t-1
enters the formula for the transformed observation stored at i,t. With this move, exactly the same lags of variables
are valid as instruments under the two transformations.
On balanced panels, GMM estimators based on the two transforms return numerically identical coefficient estimates,
holding the instrument set fixed (Arellano and Bover 1995). But orthogonal deviations has the virtue of preserving
sample size in panels with gaps. If some e_it is missing, for example, neither D.e_it nor D.e_i,t+1 can be computed.
But the orthogonal deviation can be computed for every complete observation except the last for each individual.
(First differencing can do no better since it must drop the first observation for each individual.) Note that
"difference GMM" is still called that even when orthogonal deviations are used. We will refer to the equation in
differences or orthogonal deviations as the transformed equation. In system GMM with orthogonal deviations, the levels
or untransformed equation is still instrumented with differences as described above.
xtabond2 reports the Arellano-Bond test for autocorrelation, which is applied to the differenced residuals in order to
purge the unobserved and perfectly autocorrelated v_i. AR(1) is expected in first differences, because D.e_i,t = e_i,t
- e_i,t-1 should correlate with D.e_i,t-1 = e_i,t-1 - e_i,t-2 since they share the e_i,t-1 term. So to check for AR(1)
in levels, look for AR(2) in differences, on the idea that this will detect the relationship between the e_i,t-1 in
D.e_i,t and the e_i,t-2 in D.e_i,t-2. This reasoning does not work for orthogonal deviations, in which the residuals
for an individual are all mathematically interrelated, thus contaminated from the point of view of detecting AR in the
e_it. So the test is run on differenced residuals even after estimation in deviations. Autocorrelation indicates that
lags of the dependent variable (and any other variables used as instruments that are not strictly exogenous), are in
fact endogenous, thus bad instruments. For example, if there is AR(s), then y_i,t-s would be correlated with e_i,t-s,
which would be correlated with D.e_i,t-s, which would be correlated with D.e_i,t.
xtabond2 also reports tests of over-identifying restrictions--of whether the instruments, as a group, appear exogenous.
For one-step, non-robust estimation, it reports the Sargan statistic, which is the minimized value of the one-step GMM
criterion function. The Sargan statistic is not robust to heteroskedasticity or autocorellation. So for one-step,
robust estimation (and for all two-step estimation), xtabond2 also reports the Hansen J statistic, which is the
minimized value of the two-step GMM criterion function, and is robust. xtabond2 still reports the Sargan statistic in
these cases because the J test has its own problem: it can be greatly weakened by instrument proliferation. The Mata
version goes further, reporting difference-in-Sargan statistics (really, difference-in-Hansen statistics, except in
one-step robust estimation), which test for whether subsets of instruments are valid. To be precise, it reports one
test for each group of instruments defined by an ivstyle() or gmmstyle() option (explained below). So replacing
gmmstyle(x y) in a command line with gmmstyle(x) gmmstyle(y) will yield the same estimate but distinct
difference-in-Sargan/Hansen tests. In addition, including the split suboption in a gmmstyle() option in system GMM
splits an instrument group in two for difference-in-Sargan/Hansen purposes, one each for the transformed equation and
levels equations. This is especially useful for testing the instruments for the levels equation based on lagged
differences of the dependent variable, which are the most suspect in system GMM and the subject of the "initial
conditions" in the title of Blundell and Bond (1998). In the same vein, in system GMM, xtabond2 also tests all the
GMM-type instruments for the levels equation as a group. All of these tests, however, are weak when the instrument
count is high. Difference-in-Sargan/Hansen tests are are computationally intensive since they involve re-estimating
the model for each test; the nodiffsargan option is available to prevent them.
As linear GMM estimators, the Arellano-Bond and Blundell-Bond estimators have one- and two-step variants. But though
two-step is asymptotically more efficient, the reported two-step standard errors tend to be severely downward biased
(Arellano and Bond 1991; Blundell and Bond 1998). To compensate, xtabond2 makes available a finite-sample correction
to the two-step covariance matrix derived by Windmeijer (2005). This can make two-step robust estimations more
efficient than one-step robust, especially for system GMM.
Standard errors can also be "bootstrapped"--but not with the bootstrap command. That command builds temporary data sets
by sampling the real one with replacement. And having multiple observations for a given observational unit and time
period violates panel structure. Instead, use jacknife, perhaps with the cluster() option, clustering on the panel
identifier variable, in order to drop each observational unit in turn.
The syntax of xtabond2 differs substantially from that of xtabond. xtabond2 almost completely decouples specification
of regressors from specification of instruments. As a result, most variables used will appear twice in an xtabond2
command line. xtabond2 requires the initial varlist of the command line to include all regressors except for the
optional constant term, be they strictly exogenous, predetermined, or endogenous. Variables used to form instruments
then appear in gmmstyle() or ivstyle() options after the comma. The result is a loss of parsimony, but fuller control
over the instrument matrix. Variables can be used as the basis for "GMM-style" instrument sets without being included
as regressors, or vice versa.
The gmmstyle() and ivstyle() options also have suboptions that allow further customization of the instrument matrix.
But a warning: these suboptions, along with the h() and arlevels options, can be confusing to new users and are usually
not needed in standard applications. They are best ignored when first learning xtabond2.
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