Category Advanced Econometrics Takeshi Amemiya

Global Concavity of the Likelihood Function in the Logit and Probit Models

Подпись: log L(P) = log L(j}) image552 Подпись: (9.2.18)

Global concavity means that d2 log L/dfldf}’ is a negative definite matrix for fi Є В. Because we have by a Taylor expansion


where P* lies between P and p, global concavity implies that log L(P) > log L(P) for P Ф P’iiP is a solution of (9.2.8). We shall prove global concavity for logit and probit models.

Подпись: ал dx Подпись: ... , а*л .. _.. ал Л(1-Л) and ^ = <1 -2Л)—. Подпись: (9.2.19)

For the logit model we have

Inserting (9.2.19) into (9.2.12) with F— Л yields

Д2 T n

-ЩІ’———- 2А,(1-Л,)х,.х?. (9.2.20)

where = Л(х’іР). Thus the global concavity follows from Assumption 9.2.3.

Подпись: a2 log L apap' Подпись: - 2 <M>r2( 1 - Ф/Г2[(У, - ^УІФ, + Ф 1)ф( і-1 Подпись: (9.2.21)

A proof of global concavity for the probit model is a little more complicated. Putting F, = Фf, fi = фі, and/• = — х’іРФі, where ф is the density function of ЩО, 1), into (9.2.12) yields

+ {уі – Ф/)Ф,(і – ф,)х<0]х, х;...

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One-Factor Models

The individual likelihood function (9.7.12) involves a Г-tuple normal inte­gral, and therefore its estimation is computationally infeasible for large Г (say, greater than 5). For this reason we shah consider in this subsection one-factor models that lead to a simplification of the likelihood function.

We assume

»й = а»и* + ей, (9.7.13)

where (a,), t = 1, 2,. . . , Г, are unknown parameters, u, and {ей} are nor­mally distributed independent of each other, and (eft) are serially indepen­dent. We suppress subscript і as before and express (9.7.13) in obvious vector notation as

V = atM + 6, (9.7.14)

where v, a, and e are Г-vectors and и is a scalar. Then the joint probability of у can be written as

P(y) = EuF[y/ * (2y -1); D * (2y – l)(2y -1)’], (9.7.15)

where у/ now include...

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Multinomial Generalizations

In all the models we have considered so far in Section 10.10, the sign of у ft determined two basic categories of observations, such as union members versus nonunion members, states with an antidiscrimination law versus those without, or college-goers versus non-college-goers. By a multinomial general­ization of Type 5, we mean a model in which observations are classified into more than two categories. We shall devote most of this subsection to a discus­sion of the article by Duncan (1980).

Duncan presented a model of joint determination of the location of a firm and its input – output vectors. A firm chooses the location for which profits are maximized, and only the input-output vector for the chosen location is observed...

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Other Models

In the preceding subsections we have discussed models in which cross-sec­tion-specific components are independent across individuals and time-spe­cific components are independent over time periods. We shall cite a few references for each of the other types of random coefficients models. They are classified into three types on the basis of the type of regression coefficients: (1) Regression coefficients are nonstochastic, and they either continuously or discretely change over cross-section units or time periods. (2) Regression coefficients follow a stochastic, dynamic process over time. (3) Regression coefficients are themselves dependent variables of separate regression models. Note that type 1 is strictly not a RCM, but we have mentioned it here because of its similarity to RCM...

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Higher-Level Nested Logit Model

The nested logit model defined in the preceding section can be regarded as implying two levels of nesting because the responses are classified into S groups and each group is further classified into the individual elements. In this section we shall consider a three-level nested logit. A generalization to a higher level can be easily deduced from the three-level case.

Figure 9.1 shows examples of two-level and three-level nested logit models for the case of eight responses.

Following McFadden (1981), we can generalize (9.3.58) to the three-level case by defining

Подпись:Подпись:Подпись:Део. е,,. . . ,ej

: exp (-? L?. Ls. яр ^чГТ)’

Then (9.3.59) and (9.3.60) can be generalized to

pry-,-Id – exp (ft/A)_.

M 2exPWA)’


Г "I

as 2 exP (A//A)

2 ^ – M – —T——————- V-.

je * 2 a* 2exp &*//a)

тєс, L/eBt &#...

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Properties of the Tobit Maximum Likelihood Estimator under Nonstandard Assumptions

In this section we shall discuss the properties of the Tobit MLE—the estima­tor that maximizes (10.2.5)—under various types of nonstandard assump­tions: heteroscedasticity, serial correlation, and nonnormality. It will be shown that the Tobit MLE remains consistent under serial correlation but not under heteroscedasticity or nonnormality. The same is true of the other esti­mators considered earlier. This result contrasts with the classical regression model in which the least squares estimator (the MLE under the normality assumption) is generally consistent under all of the three types of nonstandard assumptions mentioned earlier.

Before proceeding with a rigorous argument, we shall give an intuitive explanation of the aforementioned result. By considering (10.4...

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