Erik Brouwer <[email protected]> estimated a stochastic frontier model
in Stata and obtained large z-statistics.
Specifically, the estimation command
. frontier lnTK lnZT, d(e) cost;
produced
Stoc. frontier normal/exponential model Number of obs = 15
Wald chi2(1) = 1.115e+12
Log likelihood = 8.5130782 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
lnTK | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lnZT | 1.244566 1.18e-06 . 0.000 1.244563 1.244568
_cons | -4.327623 .0000181 . 0.000 -4.327659 -4.327588
-------------+----------------------------------------------------------------
/lnsig2v | -39.73655 977.6138 -0.04 0.968 -1955.824 1876.351
/lnsig2u | -3.135077 .5163978 -6.07 0.000 -4.147198 -2.122956
-------------+----------------------------------------------------------------
sigma_v | 2.35e-09 1.15e-06 0 .
sigma_u | .2085579 .0538494 .1257324 .3459441
sigma2 | .0434964 .0224614 -.0005272 .08752
lambda | 8.87e+07 .0538494 8.87e+07 8.87e+07
------------------------------------------------------------------------------
Likelihood-ratio test of sigma_u=0: chibar2(01) = 7.84 Prob>=chibar2 = 0.003
Some of the z-statistics are missing because their corresponding standard
errors are so small.
The pattern of extremely small and very large standard errors indicates that
the Hessian is not well-conditioned at the point which the algorithm has
converged. An ill-conditioned Hessian implies that the parameters are not
well-identified by the data for this model.
As discussed by Drukker and Wiggins (2004), Erik might want to begin dealing
with this problem by checking that all of the variables are on about the
same scale. Simply rescaling the variables could produce a
better-conditioned Hessian and eliminate the problem of the missing
z-statistics.
However, the coefficients do not provide any clear indication of a scaling
problem. Instead, if we accept the given solution point, the output
indicates that there is very strong evidence against the presence of an
inefficiency term in the model. This raises the possibility that the
problems with numerically identifying the parameters of interest may be due
to model misspecification.
Finally, Erik asked how is it possible that two packages could produce very
different Z-statistics when the parameter estimates are very similar. It
might be that the different packages are using different estimators of the
variance-covariance matrix. By default, -frontier- in Stata uses the
inverse of the average of the Hessian at the point of convergence. It could
be the other package is using another estimator, such as the inverse of the
average outer product of the gradient (OPG) estimator. (See Wooldridge
(2002) chapter 13 for a discussion of the different estimators.)
--David
[email protected]
References
----------
David M. Drukker and Vince Wiggins. 2004. "Verifying the solution from a
Nonlinear Solver: A Case Study: Comment". American Economic Review,
Forthcoming.
Jeffry M. Wooldridge. 2002. Econometric Analysis of Cross Section and Panel
Data. Cambridge, Mass: MIT Press.
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