Category Advanced Econometrics Takeshi Amemiya

Exact Distributions of the Limited Information Maximum Likelihood Estimator and the Two-Stage Least Squares Estimator

The exact finite sample distributions of the two estimators differ. We shall discuss briefly the main results about these distributions as well as their ap­proximations. The discussion is very brief because there are several excellent survey articles on this topic by Mariano (1982), Anderson (1982), Taylor (1982), and Phillips (1983).

In the early years (say, until the 1960s) most of the results were obtained by Monte Carlo studies; a summary of these results has been given by Johnston (1972). The conclusions of these Monte Carlo studies concerning the choice between LIML and 2SLS were inconclusive, although they gave a slight edge to 2SLS in terms of the mean squared error or similar moment criteria...

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Multivariate Nested Logit Model

The model to be discussed in this subsection is identical to the nested logit model discussed in Section 9.3.5. We shall merely give an example of its use in a multivariate situation. We noted earlier that the nested logit model is useful whenever a set of alternatives can be classified into classes each of which contains similar alternatives. It is useful in a multivariate situation because the alternatives can be naturally classified according to the outcome of one of the variables. For example, in a 2 X 2 case such as in Table 9.1, the four alterna­tives can be classified according to whether у, = 1 or y, = 0. Using a parame­terization similar to Example 9.3.5, we can specialize the nested logit model to a 2 X 2 multivariate model as follows:

Р(Уі = 1) – ві exp (z’,y)[exp (pj’x’nP) +...

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Type 2 Tobit Model: P(y1 < 0) • P(y, > 0, y2)

10.7.1 Definition and Estimation

The Type 2 Tobit model is defined as follows:

у*и = хА + Щі (Ю.7.1)

У 21 = *2ifi2 + U2i

У2і = У*і if ^>0

= 0 if /= 1, 2,. . . , n,

where {uu, u2i} are i. i.d. drawings from a bivariate normal distribution with zero mean, variances a and a, and covariance a12. It is assumed that only the sign of yf,- is observed and that y* is observed only when y* > 0. It is assumed that x1( are observed for all і but that x2i need not be observed for і such that yf, S 0. We may also define, as in (10.4.3),

w„-l if yf(> 0 (10.7.2)

= 0 if yfi S 0.

Then {wu, y2i] constitute the observed sample of the model. It should be noted that, unlike the Type 1 Tobit, y2i may take negative values.11 As in (10.2.4), y2i = 0 merely signifies the event yf, S 0.

The likelihoo...

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Time Series Analysis

1. We are using the approximation sign s* to mean that most elements of the matri­ces of both sides are equal.

2. The subscript p denotes the order of the autoregression. The size of the matrix should be inferred from the context.

3. Almon (1965) suggested an alternative method of computing f} based on^jgran – gian interpolation polynomials. Cooper (1972) stated that although the metlfigwlh, cussed in the text is easier to understand, Almon’s method is computationally»^? perior.

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Tests of Hypotheses, Prediction, and Computation

8.3.1 Tests of Hypotheses

Suppose we want to test a hypothesis of the form h(a) = 0 in model (8.1.1), where h is a ^-vector of nonlinear functions. Because we have not specified the distribution ofY„ we could not use the three tests defined in (4.5.3), (4.5.4), and (4.5.5) even if we assumed the normality of u. But we can use two test statistics: (1) the generalized Wald test statistic analogous to (4.5.21), and (2) the difference between the constrained and the unconstrained sums of squared residuals (denoted SSRD). Let a and St be the solutions of the uncon­strained and the constrained minimization of (8.1.2), respectively. Then the generalized Wald statistic is given by

Wald = j~ h(a)'[H(G’Pw£)-1H/]-1h(a), (8.3.1)

where ft = [dh/da’k and 6 = [df/Sa’]*, and the SSRD statistic is given...

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Generalized Maximum Likelihood Estimator

Cosslett (1983) proposed maximizing the log likelihood function (9.2.7) of a binary QR model with respct to P and F, subject to the condition that F is a distribution function. The log likelihood function, denoted here as у/, is

¥(fi, Л = І {у, log F(x’fl) + (1 – y,) log [1 – F(x’M). (9.6.33)

f-i

The consistency proof of Kiefer and Wolfowite (1956) applies to this kind of model. Cosslett showed how to compute MLE fi and F and derived conditions for the consistency of MLE, translating the general conditions of Kiefer and Wolfowitz into this particular model. The conditions Cosslett found, which are not reproduced here, are quite reasonable and likely to hold in most practical applications.

Clearly some kind of normalization is needed on fi and /’before we maxi­mize (9.6.33)...

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