Category Advanced Econometrics Takeshi Amemiya

Nonlinear Full Information Maximum Likelihood Estimator

In this subsection we shall consider the maximum likelihood estimator of model (8.2.1) under the normality assumption of uu. To do so we must assume that (8.2.1) defines a one-to-one correspondence between y, and uf. This assumption enables us to write down the likelihood function in the usual way as the product of the density of u, and the Jacobian. Unfortunately, this is a rather stringent assumption, which considerably limits the usefulness of the nonlinear full information maximum likelihood (NLFI) estimator in prac­tice. There are two types of problems: (1) There may be no solution for у for some values of u. (2) There may be more than one solution for у for some values of u...

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Maximum Score Estimator—A Multinomial Case

The multinomial QR model considered by Manski has the following struc­ture. The utility of the і th person when he or she chooses the jth alternative is given by

Uij^XvPo + bj’ /= 1, 2,. . . , я, 7 = 0, 1,… , m,

(9.6.23)

where we assume

Assumption 9.6.1. {e^} are i. i.d. for both і and j.

Assumption 9.6.2. (xj,, x<,,. . . , x-m)’ = x, is a sequence of {m + 1 ^-di­mensional i. i.d. random vectors, distributed independently of {e,-,}, with a joint density #(x) such that g(x) > 0 for all x.

Assumption 9.6.3. The parameter space В is defined by В = {filfi’fi = 1).

Each person chooses the alternative for which the utility is maximized. Therefore, if we represent the event of the ith person choosing the 7 th alterna­tive by a binary random variable yv, we have

ytj =1 if Uff > £/л for a...

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Model of Nelson and Olson

The empirical model actually estimated by Nelson and Olson (1978) is more general than Type 4 and is a general simultaneous equations Tobit model (10.9.4). The Nelson-Olson empirical model involves four elements of the vector y*:

yf Time spent on vocational school training, completely observed if yf > 0, and otherwise observed to lie in the interval (—°°, 0] yf Time spent on college education, observed to lie in one of the three intervals (—<», 0], (0, 1], and (1, °°) yf Wage, always completely observed yf Hours worked, always completely observed

These variables are related to each other by simultaneous equations...

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Aggregate Prediction

We shall consider the problem of predicting the aggregate proportion r = і yt in the QR model (9.2.1). This is often an important practical prob­lem for a policymaker. For example, in the transport choice model of Example

9.2.2, a policymaker would like to know the proportion of people in a commu­nity who use the transit when a new fare and the other values of the indepen­dent variables x prevail. It is assumed that fi (suppressing the subscript 0) has been estimated from the past sample. Moreover, to simplify the analysis, we shall assume for the time being that the estimated value of/? is equal to the true value.

The prediction of r should be done on the basis of the conditional distribu­tion of r given (x,). When n is large, the following asymptotic distribution is accurate:

image587(9.2...

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Least Squares Estimator

From Figure 10.1 it is clear that the least squares regression of expenditure on income using all the observations including zero expenditures yields biased estimates. Although it is not so clear from the figure, the least squares regres­sion using only the positive expenditures also yields biased estimates. These facts can be mathematically demonstrated as follows.

First, consider the regression using only positive observations of yt. We obtain from (10.2.3) and (10.2.4)

Е(Уіу, > 0) = x’t0 + E(UjUj > – xtfi). (10.4.5)

The last term of the right-hand side of (10.4.5) is generally nonzero (even without assuming щ is normal). This implies the biasedness of the LS estima­tor using positive observation on yt under more general models than the standard Tobit model...

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Number of Completed Spells

The likelihood function (11.2.5) depends on the observed durations ti, t2, ■ ■ ■ , tr only through r and T. In other words, r and Tconstitute the sufficient statistics. This is a property of a stationary model. We shall show an alternative way of deriving the equivalent likelihood function.

We shall first derive the probability of observing two completed spells in total unemployment time T, denoted P(2, T). The assumption that there are two completed unemployment spells implies that the third spell is incomplete (its duration may be exactly 0). Denoting the duration of the three spells by t,, t2, and t3, we have

P(2, T) = P(0 ё t{ < T, 0 < t2 § T—tt, t3 Ш T-tx ~ h) (11.2.12)

к exp (—Az2){exp [-А(Г – z, – z2)]} dz2^j dzx

{kTfe~2T

2

It is easy to deduce from the derivation in (11...

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