INTRODUCTION TO STATISTICS AND ECONOMETRICS

# ELEMENTS OF MATRIX ANALYSIS

In 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). Graybill (1969) described specific ap­plications in statistics. For concise introductions to matrix analysis see, for example, Johnston (1984, chapter 4), Anderson (1984, appendix), or Amemiya (1985, appendix).

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## INTRODUCTION TO STATISTICS AND ECONOMETRICS

### EXAMPLES OF HYPOTHESIS TESTS

In the preceding sections we have studied the theory of hypothesis testing. In this section we shall apply it to various practical problems. EXAMPLE 9.6.1 (mean of binomial) It is …

### Multinomial Model

We illustrate the multinomial model by considering the case of three alternatives, which for convenience we associate with three integers 1, 2, and 3. One example of the three-response model …

### Tests for Structural Change

Suppose we have two regression regimes (10.3.7) and Уь = a + 3i*T + ut, t= 1, 2, . . ■ • ,Ti (10.3.8) Tit = a + 32*21 + …

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