A. Colin Cameron 747Specifically, Monte Carlo studies find considerable over-rejection due to the finite
sample test size being considerably larger than the asymptotic size. A bootstrap
with asymptotic refinement, however, can correct this problem. The LM test can
be extended to non-likelihood settings, and can be robustified.
The likelihood ratio test generally does not extend to non-likelihood settings,
though it does for optimal GMM estimation. Newey and West (1987) generalize
the three classical tests from the likelihood framework to the GMM framework.
14.4.3 Model specification tests
Various model specification tests have been developed that do not rely on
hypothesis tests of the formc(θ)= 0.
The Hausman (1978) test contrasts two estimators that may be the same under
a null hypothesis and differ under an alternative hypothesis. For example, one
can compare OLS to the 2SLS estimator and conclude that there is endogeneity
if the two estimators differ. Denote the two estimators bŷθand ̃θ, and testH 0 :
plim(̂θ− ̃θ)= 0 using the statistic:
H=(̂θ− ̃θ)′(̂V[̂θ− ̃θ])−^1 [̂θ− ̃θ],which is chi-squared distributed underH 0. Implementation requires estimating the
variance matrix of the difference in the estimators. The original approach was to
assume that one estimator, saŷθ, is efficient under the null, in which case V[̂θ− ̃θ]=
V[ ̃θ]−V[̂θ]. This is the standard method used today, even though it is generally
incorrect since, from section 14.4.1, most applied studies use heteroskedastic-
robust or cluster-robust standard errors that presume the estimator is, in fact,
inefficient. One should instead use alternative methods to estimate V[̂θ− ̃θ], such
as the bootstrap.
Moment tests are tests of whether or not a population moment condition is
supported by the data. Specifically, they test:
H 0 :E[m(wi,θ)]= 0
Ha:E[m(wi,θ)] = 0.An obvious test is based on whether the corresponding sample momentm̂ =
N−^1
∑
im(wi,̂θ)is close to zero. The test statistic is:M=m̂′(̂V[m̂])−^1 m̂,whereMis chi-squared distributed underH 0 and the challenge is to estimatêV[m̂].
One leading example is an overidentifying restrictions (OIR) test. Then GMM
estimation based on E[m(wi,θ)]= 0 cannot exactly imposem̂= 0 if the model
is overidentified. If GMM with an optimal weighting matrix is used then Hansen
(1982) showed thatMis chi-squared distributed underH 0 with degrees of freedom
equal to the number of overidentifying restrictions.
A second class of examples are conditional moment tests, where some model
restrictions are used in estimation while other restrictions, not imposed in