William Greene 517Harris and Zhao (2007) suggested alternative parameterizations that circumvent
these problems – a restricted version:
μi,j=exp(μj+v′iπj),and a counterpart to Pudney and Shields’ (2000) formulation:
μi,j=exp(v′iπj).^2311.5.1 Specification analysis
As in the binary choice case, the analysis of micro-level data is likely to encounter
individual heterogeneity, not only in the means of utilities (xi,zi), but also in the
scaling ofUi∗, i.e., in the variance ofεi. Building heteroskedasticity into the model,
as in the binary choice model shown earlier, is straightforward. If:
E[εi^2 |vi]=[exp(v′iτ)]^2 ,then the log-likelihood would become:
lnL=∑j
i= 1
ln[
F(
μyi− 1 −x′iβ−z′iγ
exp(v′iτ))
−F(
μyi−x′iβ−z′iγ
exp(vi′τ))]
.As before, this complicates (even further) the interpretation of the model compo-
nents and the partial effects.
There is no direct test for the distribution, since the alternatives are not nested.
The Vuong test is a possibility, although the power of this test and its characteristics
remain to be examined both analytically and empirically.
11.5.2 Bivariate ordered probit models
There are several extensions of the ordered probit model that follow the logic of
the bivariate probit model we examined in Section 11.4. A direct analog to the
base case two-equation model was used by Butler, Finegan and Siegfried (1998),
who analyzed the relationship between the level of calculus attained and grades in
intermediate economics courses for a sample of Vanderbilt students. The two-step
estimation approach involved the following strategy. (We are stylizing the precise
formulation a bit in order to compress the description.) Step 1 involved a direct
application of the ordered probit model to the level of calculus achievement, which
is coded 0, 1,...,6:
m∗i=x′iβ+εi,εi|xi∼N[0, 1],
mi=0if−∞<m∗i≤0,
1if0<m∗i≤μ 1 ,
...
6ifμ 5 <m∗i<+∞.