Category Advanced Econometrics Takeshi Amemiya

Generalized Extreme-Value Model

McFadden (1978) introduced the generalized extreme-value (GEV) distribu­tion defined by

F(€ue2,. . . ,ej (9.3.67)

= exp {—(/[exp (-eO, exp (-e2), . . . , exp ( Cm)]), where G satisfies the conditions,

(i)

G(ux, u2,. . .

., мт) ё 0,

«і, «2. • • • > мтё0.

(ІІ)

G(olu1,olu2,

. . ., aum)

= otG(ux, u2,. . ., uj.

(iii)

> A

if к is odd

if к is even, k= 1,

duhduh. . .

_ != U

dulk

SO

If Uj — fij + €j and the alternative with the highest utility is chosen as before, (9.3.67) implies the GEV model

„ exp {/ii)G,[ep (fij), exp (fi2. . ., exp (jMm)]

/-УТ / / / | > (У. З.Оо)

3 (/[exp (fj. il exp (fj2),. . ., exp (yUm)]

where Gj is the derivative of G with respect to its yth argument.

Both the nested logit model and the higher...

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Heteroscedasticity

Hurd (1979) evaluated the probability limit of the truncated T obit MLE when a certain type of heteroscedasticity is present in two simple truncated Tobit models: (1) the i. i.d. case (that is, the case of the regressor consisting only of a constant term) and (2) the case of a constant term plus one independent variable. Recall that the truncated Tobit model is the one in which no infor­mation is available for those observations for which yf < 0 and therefore the MLE maximizes (10.2.6) rather than (10.2.5).

In the i. i.d. case Hurd created heteroscedasticity by generating rn observa­tions from N(n, (72i) and (1 — r)n observations from N(p, a). In each case he recorded only positive observations. Let >>(,/= 1, 2,. . . , и,, be the recorded observations. (Note nx = rt)...

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Distribution Theory

The theorems listed in this appendix, as well as many other results concerning the distribution of a univariate continuous random variable, can be found in Johnson and Kotz (1970a, b). “Rao” stands for Rao (1973) and “Plackett” for Plackett (1960).

1. Chi-Square Distribution. (Rao, p. 166.) Let an «-component random vector z be distributed as N(0,1). Then the distribution of z’z is called the chi-square distribution with n degrees of freedom. Symbolically we write z’z ~ Xn – Its density is given by

f(x) = 2~n/2T{n/2)e~x/2x~n/2~

where Г(р) = f qXp~ 1 e~x dk is called a gamma function. Its mean is n and its variance 2 n.

2. (Rao, p. 186). Let z ~ ЛГ(0,1) and let A be a symmetric and idempotent matrix with rank n. Then z’ Az ~ xl-

3.

image918

Student’s t Distribution. (Rao, p...

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Box-Cox Transformation

Подпись:Подпись:Box and Cox (1964) proposed the model3
z,(A) = x# + u„

where, forj>, > 0,

— 1

Z,(A)== )T if ХФ0

= log y, if A = 0.

Note that because lim^oCy? — 1 )/A = log y,, z,(A) is continuous at A = 0. It is assumed that (и,) are i. i.d. with Eu, = 0 and Vu, = a2.

The transformation z,(A) is attractive because it contains y, and log y, as special cases, and therefore the choice between y, and log y, can be made within the framework of classical statistical inference.

Box and Cox proposed estimating A, fi, and a1 by the method of maximum likelihood assuming the normality of ut. However, u, cannot be normally distributed unless A = 0 because z,(A) is subject to the following bounds:

z,(A)S-I if A>0 (8.1.14)

if A <0.

A

Later we shall discuss their implications on the properties of the Box – Cox M...

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Results of Cosslett: Part I

Cosslett (1981a) proved the consistency and the asymptotic normality of CBMLE in the model where both /and Q are unknown and also proved that CBMLE asymptotically attains the Cramer-Rao lower bound. These results require much ingenuity and deep analysis because maximizing a likelihood function with respect to a density function /as well as parameters fi creates a new and difficult problem that cannot be handled by the standard asymptotic theory of MLE. As Cosslett noted, his model does not even satisfy the condi­tions of Kiefer and Wolfowitz (1956) for consistency of MLE in the presence of infinitely many nuisance parameters.

Cosslett’s sampling scheme is a generalization of the choice-based sampling we have hitherto considered...

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Amemiya’s Least Squares and Generalized Least Squares Estimators

Amemiya (1978c, 1979) proposed a general method of obtaining the esti­mates of the structural parameters from given reduced-form parameter esti­mates in general Tobit-type models and derived the asymptotic distribution. Suppose that a structural equation and the corresponding reduced-form equations are given by

y = Yy + X,)? + u – (10.8.11)

[y> Y] = X[jr, П] + V,

where X, is a subset of X. Then the structural parameters у and fi are related to the reduced-form parameters n and П in the following way:

п = Пу + ЗР, (10.8.12)

where J is a known matrix consisting of only ones and zeros. If, for example, Xj constitutes the first Kx columns of X (K = Kx+ K2), then we have J = (I, 0)’, where I is the identity matrix of size Kx and 0 is the K2 X Kx matrix of zeros...

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