Thomas Blake Pepinsky <[email protected]> asked about how the
constant term changed in -xtabond- between Stata 9 and Stata 10
Here is the short answer. In Stata 9, -xtabond- reported the estimates of
the parameters of the first differenced model, so the reported constant
estimate was an estimate of the coefficient on a time trend in the model.
In Stata 10, -xtabond- reports estimates of the parameters of the level
model, so the reported constant estimate is an estimate of the constant in
the level model. In Stata 10, -xtabond- can estimate both a time-trend and
a constant.
Here is a more detailed answer.
I begin with a simple example using the version 9 -xtabond- applied to the
familiar Arellano-Bond data. What the variables measure is not important
except that this is yearly data so year is a time-trend.
. webuse abdata
. version 9: xtabond n w k year, noconstant
Arellano-Bond dynamic panel-data estimation Number of obs = 751
Group variable: id Number of groups = 140
Wald chi2(.) = .
Time variable: year Obs per group: min = 5
avg = 5.364286
max = 7
One-step results
------------------------------------------------------------------------------
D.n | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
n |
LD. | .2206198 .0621776 3.55 0.000 .098754 .3424856
w |
D1. | -.4588164 .0493884 -9.29 0.000 -.5556158 -.3620169
k |
D1. | .3516235 .0277573 12.67 0.000 .2972202 .4060267
year |
D1. | -.0239647 .0025211 -9.51 0.000 -.0289059 -.0190234
------------------------------------------------------------------------------
Sargan test of over-identifying restrictions:
chi2(27) = 119.42 Prob > chi2 = 0.0000
Arellano-Bond test that average autocovariance in residuals of order 1 is 0:
H0: no autocorrelation z = -1.82 Pr > z = 0.0688
Arellano-Bond test that average autocovariance in residuals of order 2 is 0:
H0: no autocorrelation z = -0.86 Pr > z = 0.3922
The variables in the estimates table are in first differences because the
estimated parameters are from the first-differenced model.
First-differencing causes the year variable to be collinear with the
constant, I specified the -noconstant- option.
Now, I will issue a similar command in Stata 10, allowing for a constant in
the model.
. xtabond n w k year
Arellano-Bond dynamic panel-data estimation Number of obs = 751
Group variable: id Number of groups = 140
Time variable: year
Obs per group: min = 5
avg = 5.364286
max = 7
Number of instruments = 32 Wald chi2(4) = 1974.00
Prob > chi2 = 0.0000
One-step results
------------------------------------------------------------------------------
n | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
n |
L1. | .2206198 .0622192 3.55 0.000 .0986724 .3425673
w | -.4588164 .0494215 -9.28 0.000 -.5556807 -.3619521
k | .3516235 .0277759 12.66 0.000 .2971838 .4060632
year | -.0239647 .0025228 -9.50 0.000 -.0289092 -.0190201
_cons | 49.85197 4.982592 10.01 0.000 40.08627 59.61767
------------------------------------------------------------------------------
Instruments for differenced equation
GMM-type: L(2/.).n
Standard: D.w D.k D.year
Instruments for level equation
Standard: _cons
The estimates table is in levels. The estimated coefficient on year is an
estimate of the coefficient on the time-trend year and the estimated _cons
is an estimate of the constant term in the level model.
When comparing the two sets of results, I see that the estimated
coefficients on D.year and year are the same, as expected. Similarly, we
also have equality of the estimates for the coefficients on LD.n and L.n,
D.w and w, and D.k and k. There is no constant in the differenced model
because it would be collinear with D.year. The estimated constant in the
Stata 10 output is for the constant in the level model. The difference in
the standard errors is caused the additional parameter that is estimated in
the Stata 10 -xtabond.
The constant term for the level model is estimated by including a level
moment condition. Technically, this moment condition is from an
Arellano-Bover/Blundell-Bond estimator, however including only the constant
in an Arellano-Bond estimator leaves the other parameters unchanged.
The estimated constant for the level model is useful when predicting levels
of the dependent variable.
The following example illustrates that dropping the constant from the model
does not change the remaining estimated parameters. I also point out that
the standard errors now match those from the Stata 9 output, because the
number of parameters is the same.
. xtabond n w k year, noconstant
Arellano-Bond dynamic panel-data estimation Number of obs = 751
Group variable: id Number of groups = 140
Time variable: year
Obs per group: min = 5
avg = 5.364286
max = 7
Number of instruments = 31 Wald chi2(4) = 1976.65
Prob > chi2 = 0.0000
One-step results
------------------------------------------------------------------------------
n | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
n |
L1. | .2206198 .0621776 3.55 0.000 .098754 .3424856
w | -.4588164 .0493884 -9.29 0.000 -.5556158 -.3620169
k | .3516235 .0277573 12.67 0.000 .2972202 .4060267
year | -.0239647 .0025211 -9.51 0.000 -.0289059 -.0190234
------------------------------------------------------------------------------
Instruments for differenced equation
GMM-type: L(2/.).n
Standard: D.w D.k D.year
Reiterating the main points,
1) in the Stata 9 -xtabond-, the output table was for the parameters
in the differenced model;
2) in the Stata 9 -xtabond-, the estimated constant is an estimate of the
coefficient on a time trend;
3) in the Stata 10 -xtabond-, the output table is for the parameters
of the level model;
4) in the Stata 10 -xtabond-, the estimated constant is an estimate
of the constant in the level model; and
5) including a constant in the Stata 10 -xtabond- does not affect any of the
other parameter estimates.
I hope that this helps.
David
<[email protected]>
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