Category Advanced Econometrics Takeshi Amemiya

Multinomial Probit Model

Let Uj, j = 0, 1, 2,. . ., m, be the stochastic utility associated with the yth alternative for a particular individual. By the multinomial probit model, we mean the model in which {Uj) are jointly normally distributed. Such a model was first proposed by Aitchison and Bennett (1970). This model has rarely been used in practice until recently because of its computational difficulty (except, of course, when m = 1, in which case the model is reduced to a binary
probit model, which is discussed in Section 9.2). To illustrate the complexity of the problem, consider the case of m = 2. Then, to evaluate P(y = 2), for example, we must calculate the multiple integral

Подпись: (9.3.72)


P(y~L) = P(U2 > Ul, U2> U0)

where / is a trivariate normal density...

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Tests for Normality

The fact that the Tobit MLE is generally inconsistent when the true distribu­tion is nonnormal makes it important for a researcher to test whether the data are generated by a normal distribution. Nelson (1981) devised tests for nor­mality in the i. i.d. censored sample model and the Tobit model. His tests are applications of the specification test of Hausman (1978) (see Section 4.5.1).

Nelson’s i. i.d. censored model is defined by

Уі = уТ if yf>0 = 0 if yf = 0, /’= 1,2,. . . ,n,

where yf — Щц, a2) under the null hypothesis. Nelson considered the esti­mation of P(yf > 0). Its MLE is Ф(р/а), where fi and a are the MLE of the respective parameters. A consistent estimator is provided by щ/п, where, as before, n{ is the number of positive observations of yt...

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Recent Developments in Regression Analysis

1. A much more detailed account of this topic can be found in Amemiya (1980a).

2. We use the term estimator here, but all the definitions and the results of this subsection remain the same if we interpret d as any decision function mapping Y into 9.

3. We assume that the losses do not depend on the parameters of the models. Otherwise, the choice of models and the estimation of the parameters cannot be sequentially analyzed, which would immensely complicate the problem. However, we do not claim that our assumption is realistic. We adopt this simplification for the purpose of illustrating certain basic ideas.

4. For ways to get around this problem, see, for example, Akaike (1976) and Schwartz (1978).


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Estimation in a System of Equations

8.2.1 Introduction

Define a system of N nonlinear simultaneous equations by

fu(y,, <*i) = ua, і = 1, 2,. . . , N, t = 1, 2,. . . , T,


where y, is an N-vector of endogenous variables, x, is a vector of exogenous variables, and at is a vector of unknown parameters. We assume that the

N-vector u, = (uu, u2t,. . . , uNt)’ is an i. i.d. vector random variable with zero mean and variance-covariance matrix 2. Not all of the elements of vectors yf and x( may actually appear in the arguments of each f,. We assume that each equation has its own vector of parameters a, and that there are no constraints among a, but the subsequent results can easily be modified if each at can be parametrically expressed as where the number of ele­ments in в is less than 2fLt K,.

Strictly speaking, (8.2...

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Distribution-Free Methods

In this section we shall discuss two important articles by Manski (1975) and Cosslett (1983). Both articles are concerned with the distribution-free estima­tion of parameters in QR models—Manski for multinomial models and Cosslett for binary models. However, their approaches are different: Manski proposed a nonparametric method called maximum score estimation whereas Cosslett proposed generalized maximum likelihood estimation in which the likelihood function is maximized with respect to both F and fi in the model

(9.2.1) . Both authors proved only consistency of their estimator but not asymptotic normality.

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Type 4 Tobit Model: P(y, < 0, y3) • P(y,, y2)

10.9.1 Definition and Estimation

The Type 4 Tobit model is defined as follows:

У и = x’uA + Uu

Подпись: (10.9.1)У*і = *2.7*2 + “2,

Уіі — хЗіРз + U3i Уи ~ У*і if Уи>0

= 0 if yf, ^ 0

Уи = Уіі if У и > 0

= 0 if yf, S0

Узі = У*і if Уи = 0

= 0 if yf,>0, /= 1,2,. . . ,/1,

where {uu, и2/. «зЛ are i. i.d. drawings from a trivariate normal distribution.

This model differs from Type 3 defined by (10.8.1) only by the addition of yf,, which is observed only if yf, £ 0. The estimation of this model is not significantly different from that of Type 3. The likelihood function can be written as


where f3 ( • , •) is the joint density of yf, and Тз< and^( •, •) is the joint density of у * and y*...

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