Category Advanced Econometrics Takeshi Amemiya

Hsiao’s Model

Hsiao’s model (1974, 1975) is obtained as a special case of the model of the preceding subsection by assuming 2,, and 2* are diagonal and putting 2€ =

Hsiao (1975) proposed the following method of estimating 2,,, 2Л, and a1: For simplicity assume that X does not contain a constant term. A simple modification of the subsequent discussion necessary for the case in which X contains the constant term is given in the appendix of Hsiao (1975). Consider the time series equation for the /th individual:

Уі = ВД + Л) + X? A + €/. (6.7.8)

If we treat Ці as if it were a vector of unknown constants (which is permissible so far as the estimation of 2д and a2 is concerned), model (6.7.8) is the heteroscedastic regression model considered in Section 6.5.4...

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Multinomial Discriminant Analysis

The DA model of Section 9.2.8 can be generalized to yield a multinomial DA model defined by

X? l(y< =j) ~ Mjij, 2;)



Р(Уі =j) = Qj


for / = 1, 2,. . ., n and j = 0, 1,. .

., m. By Bayes’s rule we obtain


Р(Уі=М?)= m8j(xT)qj, X g^f)Qk

where gj is the density function of 2,). Just as we obtained (9.2.48) from

(9.2.46) , we can obtain from (9.3.48)

^р^-^Дц+^ + хГАх,*), (9.3.49)

where РЛ1), fiA2), and A are similar to (9.2.49), (9.2.50), and (9.2.51) except that the subscripts 1 and 0 should be changed to j and 0, respectively.

As before, the term xf’Axf drops out if all the 2’s are identical. If we write Дко fi’x2)xT = the DA model with identical variances can be written exactly in the form of (9.3...

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Tobit Maximum Likelihood Estimator

The Tobit MLE maximizes the likelihood function (10.2.5). Under the as­sumptions given after (10.2.4), Amemiya (1973c) proved its consistency and asymptotic normality. If we define 0 = (/?’, a2)’, the asymptotic variance-co­variance matrix of the Tobit MLE 0 is given by


Подпись: ve =X X **

Подпись: (10.4.36);-i t-i

X W І c‘

1-1 i-1 J where

а і: = – а 2{хаф, – [ф?/(1 ~ Ф,)] – Ф,},

b, = (/2)а-(х’а)2фі + ф,- Ш/(1 – Ф,)]}, and

q = -(1/4)ст-4{(х’а)3ф, + (х’а)фі – [{х’а)фУ( – Ф,)] – 2Ф,);

and фі and Ф, stand for ф(х,-0£) and Ф(х<а), respectively.

The Tobit MLE must be computed iteratively. Olsen (1978) proved the global concavity of log L in the Tobit model in terms of the transformed parameters a = fit a and h = er’...

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Nonstationary Models

So far we have assumed X)k(t) = Ajfc for all t (constant hazard rate). Now we shall remove this assumption. Such models are called nonstationary or semi – Markov.

Suppose a typical individual stayed in state yin period (0, t) and then moved to state к in period (t, t + AO – We call this event A and derive its probability P(A), generalizing (11.2.2) and (11.2.3). Defining m = t/At and using log (1 — є) = —є for small e, we obtain for sufficiently large m


image867The likelihood function of this individual is the last expression in (11.2.50) except for t/m.

Let us obtain the likelihood function of the same event history we consid­ered in the discussion preceding (11.2.4). The likelihood function now be­comes




As in (11.2...

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Three-Stage Least Squares Estimator

In this section we shall again consider the full information model defined by

(7.1.1) . The 3SLS estimator of a in (7.1.5) can be defined as a special case of G2SLS applied to the same equation. The reduced form equation comparable to (7.3.22) is provided by

Z = xn + V, (7.4.1)

where X = I ® X, П = diagtdl,, J, ),(П2, J2), . . . , (Пдг, J*)], V = diagKVHOMV^O), . . . , (VN, 0)], and J, = (X’X)-‘X’X,.

To define 3SLS (proposed by Zellner and Theil, 1962), we need a consistent estimator of 2, which can be obtained as follows:

Step 1. Obtain the 2SLS estimator of a,-, /=1,2,. . . , N.

Step 2. Calculate fl, = у,- — Z,/*,^, /=1,2,. . . , N.

Step 3. Estimate аи by = T~lfiju,.

Next, inserting Z = Z, X = X, and = X ® I into (7.3...

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Choice-Based Sampling

9.5.1 Introduction

Consider the multinominal QR model (9.3.1) or its special case (9.3.4). Up until now we have specified only the conditional probabilities of alternatives j = 0, 1,. . . , m given a vector of exogenous or independent variables x and have based our statistical inference on the conditional probabilities. Thus we have been justified in treating x as a vector of known constants just as in the classical linear regression model of Chapter 1. We shall now treat both j and x as random variables and consider different sampling schemes that specify how j and x are sampled.

First, we shall list a few basic symbols frequently used in the subsequent discussion:

F(y|x, P) or P(j) Conditional probability the y’th

alternative is chosen, given the exogenous variables x

Подпись: The above evaluated at the true value of fi True density of x10 Density according to which a researcher draws x Probability according to which a researcher draws j
Подпись: P(j|x, fio) or P0U) /(x) g(x)
Подпись: HU) QU) = QUfi) = SPU*, dx QoU) = QUPo) = SPUx, fio)f(x) dx


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