Category Advanced Econometrics Takeshi Amemiya

Log-Linear Model

A log-linear model refers to a particular parameterization of a multivariate model. We shall discuss it in the context of the 2 X 2 model given in Table 9.1. For the moment we shall assume that there are no independent variables and that there is no constraint among the probabilities; therefore the model is completely characterized by specifying any three of the four probabilities appearing in Table 9.1. We shall call Table 9.1 the basic parameterization and shall consider two alternative parameterizations.

The first alternative parameterization is a logit model and is given in Table

9.4, where d is the normalization chosen to make the sum of probabilities equal to unity. The second alternative parameterization is called a log-linear model and is given in Table 9...

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A Special Case of Independence

Dudley and Montmarquette (1976) analyzed whether or not the United States gives foreign aid to a particular country and, if it does, how much foreign aid it gives using a special case of the model (10.7.1), where the independence of uu and u2t is assumed. In their model the sign of y* determines whether aid is given to the rth country, and yf,- determines the actual amount of aid. They used the probit MLE to estimate (assuming ax = 1) and the least squares regression of y2i on x2i to estimate f)2. The LS estimator of ft2 is consistent in their model because of the assumed independence between uu and u2i. This makes their model computationally advantageous...

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Generalized Least Squares Theory

1. If rank (ЩХ) = K, fiis uniquely determined by (6.1.2).

2. Farebrother (1980) presented the relevant tables for the case in which there is no intercept.

3. Breusch (1978) and Godfrey (1978) showed that Durbin’s test is identical to Rao’s score test (see Section 4.3.1). See also Breusch and Pagan (1980) for the Lagrange multiplier test closely related to Durbin’s test.

4. In the special case in which N=2 and fi changes with і (so that we must estimate both /?, and fi2), statistic (6.3.9) is a simple transformation of the F statistic (1.5.44). In this case (1.5.44) is preferred because the distribution given there is exact.

5. It is not essential to use the unbiased estimators a) here. If 6} are used, the distribution of у is only trivially modified.


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Bianchi and Calzolari (1980) proposed a method by which we can calculate the mean squared prediction error matrix of a vector predictor based on any estimator of the nonlinear simultaneous equations model. Suppose the struc­tural equations can be written as f(yp, xp, a) = up at the prediction period p and we can solve for yp as yp = g(xp, a, up). Define the predictor $p based on the estimator a by % = g(xp, a, 0). (Note that yp is an jV-vector.) We call this
the deterministic predictor. Then we have

Подпись:E(Ур ~ fP)(YP ~ fpY

= E[g(xp, a, up)~ g(xp, a, 0)] [g(xp, a, up)~ g(xp, a, 0)]’ + o, 0) – g(xp, a, 0)] [g(xp, a, 0) – g{xp, a, 0)]’

= Aj + A2.

Bianchi and Calzolari suggested that A, be evaluated by simulation...

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Panel Data QR Models

Panel data consist of observations made on individuals over time. We shall consider models in which these observations are discrete. If observations are independent both over individuals and over time, a panel data QR model does not pose any new problem. It is just another QR model, possibly with more observations than usual. Special problems arise only when observations are assumed to be correlated over time. Serial correlation, or temporal correla­tion, is certainly more common than correlation among observations of dif­ferent individuals or cross-section units. In this section we shall consider various panel data QR models with various ways of specifying serial correla­tion and shall study the problem of estimation in these models.

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Type 5 Tobit Model: P(yі < 0, ya) • P(y, > 0, ya)

10.10.1 Definition and Estimation

The Type 5 Tobit model is obtained from the Type 4 model (10.9.1) by omitting the equation for yu. We merely observe the sign of yf,. Thus the model is defined by

Уи = *иРі + Щі (10.10.1)

У*і = XliPl + m2i У *t = x’uPi + m3(

Угі = У *1 if У и > 0

= 0 if yf, ё 0

Ун = У it if Уи = о

= 0 if y*>0, і = 1, 2, . . . , п,


where (uu, u2i, u3i} are i. i.d. drawings from a trivariate normal distribution. The likelihood function of the model is

where f3 and are as defined in (10.9.2). Because this model is somewhat

simpler than Type 4, the estimation methods discussed in the preceding section apply to this model a fortiori. Hence, we shall go immediately into the discussion of applications.

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