Category Advanced Econometrics Takeshi Amemiya

Disequilibrium Models

Disequilibrium models constitute an extensive area of research, about which numerous papers have been written. Some of the early econometric models have been surveyed by Maddala and Nelson (1974). A more extensive and


up-to-date survey has been given by Quandt (1982). See, also, the article by Hartley (1976a) for a connection between a disequilibrium model and the standard Tobit model. Here we shall mention two basic models first discussed in the pioneering work of Fair and JafFee (1972).

The simplest disequilibrium model of Fair and JafFee is a special case of the Type 5 model (10.10.1), in which y*,- is the quantity demanded in the ith period, y*i is the quantity supplied in the ith period, and y* = y* — y*...

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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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