Palgrave Handbook of Econometrics: Applied Econometrics

William Greene 477

most of what follows, we will not need this level of generality, but the models to
be developed will accommodate it.
We will formulate a model that describes the consumer choice in probabilistic
terms. (A bit more of the underlying behavioral theory is presented in section 11.8.)
The “model” will consist of a probability distribution defined over the set of
choices:

Pit,j=Prob(consumerimakes choicejat timet|choice set), j=1,...,Jit.

The manner in which the probabilities arise is an essential feature of the model. As
noted earlier, choices are dependent on the environment in which they are made,
which we characterize in terms of income,y, and prices,x. Individual heterogene-
ity may be measured by such indicators as family size, gender, location, etc., which
we collect in a set of variables,z, and unmeasured, and therefore random from the
point of view of the analyst, indicators, which we denote asu. Common elements
of the choice mechanism that constitute the interesting quantities that the analyst
seeks to draw statistical inference about will be parameters,β,γ, etc.^3 For purposes
of translating the underlying choice process into an estimable econometric model,
we define the choice indicators:

dit,j=1 if individualimakes choicejat timet, and 0 otherwise.

With all this in place, our discrete probability distribution will be defined by:

Pit,j=Prob

(

dit,j= 1

∣

∣Xit,zit,uit,β,γ,...

)

, j=1,...,Jit

whereXitis the set of attributes of allJitchoices in the choice set for individual
iat timet. Note that being characteristics of the individual, and not the choices,
zitanduitdo not vary across the choices. Whether the preference parameters,
β,γ,..., should be allowed to vary (i.e., whether they do vary) across individuals –
i.e., whether the parameters of the utility functions are heterogeneous – is a ques-
tion that we will pursue at several points below. We will assume (not completely
innocently) that in any choice situation, the individual actually makes a choice. It
follows that:
∑Jit

j= 1

dit,j=1 and

∑Jit

j= 1

Pit,j=1.

The “model” consists of the interesting or useful features ofPit,j. The preceding
discussion assumes that, at timet, the consumer makes a single decision. It will be
necessary in section 11.4 to extend the model to cases of two or more decisions.
This is straightforward, but requires a small change in notation and interpretation.
We will defer that extension until we encounter it in the discussion in section 11.4.
We close this section with some definitions of terms that will be used throughout
the text. The individualcharacteristics, such as gender or education, are denotedzit.
Attributes of the choices, such as prices, are denotedxit,j. We denote bybinomial
ormultinomial choice, the single choice made between either two or more than two