Category Advanced Econometrics Takeshi Amemiya

Nonlinear Limited Information Maximum Likelihood Estimator

In the preceding section we assumed the model (8.1.1) without specifying the model for Y( or assuming the normality of u, and derived the asymptotic distribution of the class of NL2S estimators and the optimal member of the class—BNL2S. In this section we shall specify the model for Y, and shall assume that all the error terms are normally distributed; under these assump­tions we shall derive the nonlinear limited information maximum likelihood (NLLI) estimator, which is asymptotically more efficient than BNL2S. The NLLI estimator takes advantage of the added assumptions, and consequently its asymptotic properties depend crucially on the validity of the assumptions. Thus we are aiming at a higher efficiency at the possible sacrifice of robustness.

Assume, in addition to (8.1.1),

image527(8.1...

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Results of Cosslett: Part II

Cosslett (1981b) summarized results obtained elsewhere, especially from his earlier papers (Cosslett, 1978, 1981a). He also included a numerical evalua­tion of the asymptotic bias and variance of various estimators. We shall first discuss CBMLE of the generalized choice-based sample model with unknown /and known Q. Cosslett (1981b) merely stated the consistency, asymptotic normality, and asymptotic efficiency of the estimator, which are proved in Cosslett (1978). The discussion here will be brief because the results are analo­gous to those given in the previous subsection.

The log likelihood function we shall consider in this subsection is similar to

(9.5.47) except that the last term is simplified because Q is now known. Thus
we have

log L2 = 2 log Р(ЛІХ» Л + 2 log/(*<) (9.5.63)

i-...

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Other Examples of Type 3 Tobit Models

Roberts, Maddala, and Enholm (1978) estimated two types of simultaneous equations Tobit models to explain how utility rates are determined. One of their models has a reduced form that is essentially Type 3 Tobit; the other is a simple extension of Type 3.

The structural equations of their first model are

У*і = x’ufiz + “2; (10.8.14)

and

yft = УУ*і + + Щ i> (10.8.15)

where y*t is the rate requested by the fth utility firm, y*t is the rate granted for the fth firm, x2i includes the embedded cost of capital and the last rate granted minus the current rate being earned, and x3/ includes only the last variable mentioned. It is assumed that y* and y% are observed only if

у*/ = г(х. + Vj > 0, (10.8.16)

where z, include the earnings characteristics of the fth firm...

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Model of Lee

In the model of Lee (1978), y2i represents the logarithm of the wage rate of the rth worker in case he or she joins the union and y3i represents the same in case he or she does not join the union. Whether or not the worker joins the union is determined by the sign of the variable

Подпись: (10.10.3)Уи = УІі-УІ + *іа + і>і-

Because we observe only y2i if the worker joins the union and y3i if the worker does not, the logarithm of the observed wage, denoted yt, is defined by

Подпись:Уі = Уи if Тн>0

= y’ti if j/J) s 0.

Lee assumed that x2 and x3 (the independent variables in the y* and y3 equations) include the individual characteristics of firms and workers such as regional location, city size, education, experience, race, sex, and health, whereas z includes certain other individual characteristics and...

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Two Error Components Model

In 2ECM there is no time-specific error component. Thus the model is a special case of 3ECM obtained by putting o = 0. This model was first used in econometric applications by Balestra and Nerlove (1966). However, because in the Balestra-Nerlove model a lagged endogenous variable is included among regressors, which causes certain additional statistical problems, we shall discuss it separately in Section 6.6.3.

In matrix notation, 2ECM is defined by

y = X0 + u

(6.6.18)

and

u = 1^ + e.

(6.6.19)

The covariance matrix ft = £iiu’ is given by

ft = a* A + oJIjvt,

(6.6.20)

and its inverse by

n~l=±(lNT-yl A),

(6.6.21)

where у, = а*/{а + Та*) as before.

We can define GLS and FGLS as in Section 6.6.1...

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Comparison of the Maximum Likelihood Estimator and the Minimum Chi-Square Estimator

In a simple model where the vector x( consists of 1 and a single independent variable and where T is small, the exact mean and variance of MLE and the MIN x2 estimator can be computed by a direct method. Berkson (1955,1957) did so for the logit and the probit model, respectively, and found the exact mean squared error of the MIN x2 estimator to be smaller in all the examples considered.

Amemiya (1980b) obtained the formulae for the bias to the order of л~1 and the mean squared error to the order of л-2 of MLE and MIN x2 in a general logit model.3 The method employed in this study is as follows: Using (9.2.19) and the sampling scheme described in Section 9.2.5, the normal equation

(9.2.8) is reduced to

We can regard (9.2.40) as defining the MLE fi implicitly as a function of Pi, P2,...

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