Giovanni Bruno <[email protected]> pointed out that there is an
issue with
> [the formula for] S2 (last formula at page 32 in the xt manual), which
> is based on A2 that necessarily uses 1-step residuals. This is not
> the 2-step Sargan in Arellano and Bond (1991) paper, formula (10) p.
> 282 that employs 2-step residuals.
Giovani is correct that equation (10) in Arellano and Bond (1991) uses the
two-step residuals to define the Sargan statistic for the two-step
estimator. However, in this equation Arellano and Bond (1991) deviated from
the common definition of the two-step Sargan statistic as the value of the
GMM objective function at its two-step optimum. The definition implemented
in -xtabond- is the more common definition of the Sargan-Hansen statistic.
To clarify define,
A_2 = (\sum_i Z_i'e1_i'e1_i Z_i)^{-1}
S_2 = (\sum_i e2_i'Z_i)A_2(\sum_i Z_i' e2)
where e1_i are one-step residuals
e2_i are two-step residuals
Z_i is the matrix of instruments for each panel i.
The coefficients beta that minimize S_2 with residuals e2_i are the GMM
estimates. This is the Hansen (1982) formulation in which the weight matrix
A_2 is a function of the residuals from the previous step, but not the
current step.
As Mark Schaffer <[email protected]> pointed out, the Sargan statistic
with A_2 as function of e2_i still converges to a chi-squared distribution,
so there is nothing wrong with equation in Arellano and Bond (1991).
However, as Mark also noted, that definition differs from the standard
textbook version in which A_2 is a function of e1_i which is discussed in
Hayashi(2000) and Davidson and MacKinnon(2004, 1993). Thus, the A_2 as a
function of e1_i is well documented.
--David
[email protected]
References
Arellano, M. and Bond, S. (1991) "Some tests of specification for panel data:
Monte Carlo evidence and an application to Employment Equations", The Review
of Economic Studies, 58(2) 277-297.
Davidson, R and MacKinnon J. (2004) Econometric Theory and Methods.
New York:Oxford University Press.
Davidson, R and MacKinnon J. (1993) Estimation and Inference in Econometrics.
New York:Oxford University Press.
Hansen, L.P. (1982) "Large Sample properties of GMM estimators" Econometrica
50(4) 1029-1054.
Hayashi, F. (2000) Econometrics. New York:Princeton University Press.
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