Those who are not familiar with matrix analysis should study Chapter 11 before reading this section. The results of this chapter will not be needed to understand Chapter 10. Insofar as possible, we shall illustrate our results in the two-dimensional case.

We consider the problem of testing H0: 0 = 00 against Ну 0 Ф 0O, where 0 is a A-dimensional vector of parameters. We are to use the test statistic 0 ~ N(Q, X), where X is а К X К variance-covariance matrix: that is, X = £(0 — 0)(0 — 0)’. (Throughout this section a matrix is denoted by a boldface capital letter and a vector by a boldface lower-case letter.) In


FIGURE 9.9 Critical region for testing about two parameters

Section 9.7.1 we consider the case where X is completely known, and in Section 9.7...

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Tobin (1958) proposed

the following important model:


yf = Хг’р + Ui



Уі = x’p + и{

if yf > о

= 0

if yf >0, і = 1, 2…………………. n,

where (wj are assumed to be i. i.d. N(0, cr2) and хг is a known nonstochastic vector. It is assumed that {yj and (x,) are observed for all i, but {y*} are unobserved if y* < 0. This model is called the censored regression model or the Tobit model (after Tobin, in analogy to probit). If the observations corresponding to y* < 0 are totally lost, that is, if {x,} are not observed whenever y* < 0, and if the researcher does not know how many obser­vations exist for which y* < 0, the model is called the truncated regression model.

Tobin used this model to explain a household’s expenditure (y) on a ...

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Подпись: 11In Chapter 10 we discussed the bivariate regression model using summa­tion notation. In this chapter we present basic results in matrix analysis. The multiple regression model with many independent variables can be much more effectively analyzed by using vector and matrix notation. Since our goal is to familiarize the reader with basic results, we prove only those theorems which are so fundamental that the reader can learn important facts from the process of proof itself. For the other proofs we refer the reader to Bellman (1970).

Symmetric matrices play a major role in statistics, and Bellman’s discus­sion of them is especially good. Additional useful results, especially with respect to nonsymmetric matrices, may be found in a compact paperback volume, Marcus and Mine (1964)...

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In Section 7.4.1 we show that the maximum likelihood estimator is the best unbiased estimator under certain conditions. We show this by means of the Cramer-Rao lower bound. In Sections 7.4.2 and 7.4.3 we show the consistency and the asymptotic normality of the maximum likelihood estimator under general conditions. In Section 7.4.3 we define the con­cept of asymptotic efficiency, which is closely related to the Cramer-Rao lower bound. In Section 7.4.4 examples are given. To avoid mathematical complexity, some results are given without full mathematical rigor. For a rigorous discussion, see Amemiya (1985).

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Known Variance-Covariance Matrix

In this subsection we develop the theory of generalized least squares under the assumption that X is known (known up to a scalar multiple, to be precise); in the remaining subsections we discuss various ways the ele­ments of X are specified as a function of a finite number of parameters so that they can be consistently estimated.

Since X is symmetric, by Theorem 11.5.1 we can find an orthogonal matrix H which diagonalizes X as H’XH = A, where A is the diagonal matrix consisting of the characteristic roots of X. Moreover, since X is positive definite, the diagonal elements of A are positive by Theorem 11.5.10. Using (11.5.4), we define X_1/2 = HA’1/2H’, where A“1/2 =

_ і /о

Z>{, }, where X, is the ith diagonal element of A. Premultiplying

(13.1.1) by X_1/2, we obtain (13.1...

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Variance-Covariance Matrix Assumed Known

Consider the case of К = 2. We can write 0 = (0b 02)’ and 0O = (0ю, 02o)’ • It is intuitively reasonable that an optimal critical region should be outside some enclosure containing 0O, as depicted in Figure 9.9. What should be the specific shape of the enclosure?

An obvious first choice would be a circle with 0O at its center. That would amount to the test:

Подпись: (9.7.1)


Reject HQ if (01 — 0ioC + (02 — 02oT > c

for some c, where c is chosen so as to make the probability of Type I error equal to a given value a. An undesirable feature of this choice can be demonstrated as follows: Suppose F0i is much larger than V02...

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