Category Advanced Econometrics Takeshi Amemiya

Generalized Maximum Likelihood Estimator

Cosslett (1983) proposed maximizing the log likelihood function (9.2.7) of a binary QR model with respct to P and F, subject to the condition that F is a distribution function. The log likelihood function, denoted here as у/, is

¥(fi, Л = І {у, log F(x’fl) + (1 – y,) log [1 – F(x’M). (9.6.33)


The consistency proof of Kiefer and Wolfowite (1956) applies to this kind of model. Cosslett showed how to compute MLE fi and F and derived conditions for the consistency of MLE, translating the general conditions of Kiefer and Wolfowitz into this particular model. The conditions Cosslett found, which are not reproduced here, are quite reasonable and likely to hold in most practical applications.

Clearly some kind of normalization is needed on fi and /’before we maxi­mize (9.6.33)...

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Model of Tomes

Tomes (1981) studied a simultaneous relationship between inheritance and the recipient’s income. Although it is not stated explicitly, Tomes’ model can be defined by

Подпись: (10.9.7) (10.9.8) .У?/ = УіУ2і + x’ufii + “н> У* = У2Уи + + м2,’


Уи = Уи if J>u>0 (10.9.9)

= 0 if yftSO,

where у fr is the potential inheritance, yu is the actual inheritance, and y2t is the recipient’s income. Note that this model differs from Nelson’s model defined by (10.9.5) and (10.9.6) only in that yu, not yf,, appears in the right-hand side of (10.9.8). Assuming yxy2< 1 for the logical consistency of the model (as in Amemiya, 1974b, mentioned in Section 10.6), we can rewrite (10.9.7) as

У*=(1 ~Уі УаГЧУі (*2 tPi + «2.) + х’і A + «и! (10.9.10)

and (10.9.8) as

У2І = У2,} = (1 – У УіҐ’Ш^иРі + Ml,...

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Random Coefficients Models

Random coefficients models (RCM) are models in which the regression coef­ficients are random variables, the means, variances, and covariances of which are unknown parameters to estimate. The Hildreth and Houck model, which we discussed in Section 6.5.4, is a special case of RCM. The error components models, which we discussed in Section 6.6, are also special cases of RCM. We shall discuss models for panel data in which the regression coefficients contain individual-specific and time-specific components that are independent across individuals and over time periods. We shall discuss in succession models proposed by Kelejian and Stephan (1983), Hsiao (1974,1975), Swamy (1970), and Swamy and Mehta (1977)...

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Multinomial Models

9.3.1 Statistical Inference

In this section we shall define a general multinomial QR model and shall discuss maximum likelihood and minimum chi-square estimation of this model, along with the associated test statistics. In the subsequent sections we shall discuss various types of the multinomial QR model and the specific problems that arise with them.

Assuming that the dependent variable y, takes m, + 1 values 0, 1,2,. . . , mh we define a general multinomial QR model as

Р(Уі=Л = Р0(х*,в), i’=l,2,. . . ,n and (9.3.1)

7 = 1,2,.. . ,m„

where x* and в are vectors of independent variables and parameters, respec­tively. (Strictly speaking, we should write j as jt, but we shall suppress the subscript /’.) We sometimes write (9.3.1) simply as Pi} = Ftj...

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Heckman’s Two-Step Estimator

Heckman (1976a) proposed a two-step estimator in a two-equation general­ization of the Tobit model, which we shall call the Type 3 Tobit model. But his estimator can also be used in the standard Tobit model, as well as in more complex Tobit models, with only a minor adjustment. We shall discuss the estimator in the context of the standard Tobit model because all the basic features of the method can be revealed in this model. However, we should keep in mind that since the method requires the computation of the probit MLE, which itself requires an iterative method, the computational advantage of the method over the Tobit MLE (which is more efficient) is not as great in the standard Tobit model as it is in more complex Tobit models.

To explain this estimator, it is useful to rewrite (10.4...

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Durations as Dependent Variables of a Regression Equation

Suppose that each individual experiences one complete spell. Then the likeli­hood function is


L = J"JA, exp (—A^j). (11.2.25)


The case of a person having more than one complete spell can be handled by behaving as if these spells belonged to different individuals. Assume as before

Аі = exp (0′ xt). (11.2.26)

But, here, we have absorbed the constant term a into fi as there is no need to separate it out.

We shall derive the asymptotic covariance matrix of the MLE fi. We have

which is the continuous-time analog of (11.1.65). Therefore we have r^log L

Подпись: (11.2.29)

image854 Подпись: (11.2.30)
Подпись: exp (—A,-z) dz = y, л,-
Подпись: (11.2.28)

Therefore, by Theorem 4.2.4, fi is asymptotically normal with asymptotic covariance matrix

Now, suppose we use log /,• as the dependent variable of a linear regression equation...

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