William Greene 487An empirical conundrum can arise when doing inference about partial effects
rather than coefficients. For any particular variable,wk, the preceding theory does
not guarantee that both the estimated coefficient,θk, and the associated partial
effect,δk, will be “statistically significant,” or statistically insignificant. In the event
of a conflict, one is left with the uncomfortable problem of simultaneously reject-
ing and not rejecting the hypothesis that a variable should appear in the model.
Opinions differ on how to proceed. Arguably, the inference should be aboutθk,
notδk, since in the latter case, one is testing a hypothesis about a function of all
the coefficients, not just the one of interest.
11.3.2.4 Hypothesis tests
Conventional hypothesis tests about restrictions on the model coefficients,θ, can
be carried out using any of the three familiar procedures. Given the simplicity of
the computations for the MLE, the likelihood ratio test is a natural candidate. The
likelihood ratio statistic is:
λLR= 2 [lnL 1 −lnL 0 ]where “1” and “0” indicate the values of the log-likelihood computed at the un-
restricted (alternative) estimator and the restricted (null) estimator, respectively.
A hypothesis that is usually of interest in this setting is the null hypothesis that
all coefficients save for the constant term are equal to zero. In this instance, it is
simple to show that, regardless of the assumed distribution:
lnL 0 =n[P 1 lnP 1 +P 0 lnP 0 ],whereP 1 is the proportion of observations for whichdiequals one, which is also
d=^1 nni= 1 di, andP 0 = 1 −P 1. Wald statistics use the familiar results, all based on
the unrestricted model. The general procedure assesses departures from the null
hypothesis
H 0 :r(θ,c)= 0 ,wherer(θ,c) is a vector ofJfunctionally independent restrictions onθandcis a
vector of constants. The typical case is the set of linear restrictions, H 0 :rθ−c=0,
whereris a matrix of constants. The Wald statistic for testing the null hypothesis
is constructed using the delta method to obtain an asymptotic covariance matrix
forr(θ,c). The statistic is:
λWALD=[r(θ,c)]′[R(θ,c)VR(θ,c)′]−^1 [r(θ,c)],whereR(θ,c)=∂r(θ,c)/∂θ′and all computations are carried out using the unre-
stricted maximum likelihood estimator. The standard “t-test” of the significance of
a coefficient is the most familiar example. The Lagrange multiplier (LM) statistic is:
λLM=g^0 ′V^0 g^0 ,where “0” indicates that the computations are done using the restricted estimator
andVis any of the estimators of the asymptotic covariance matrix of the MLE