INTRODUCTION TO STATISTICS AND ECONOMETRICS

# Student’s t Test

The t test is ideal when we have a single constraint, that is, q = 1. The F test, discussed in the next section, must be used if q > 1.

Since (J is normal as shown above, we have

(12.4.7) Q'0 ~ N[c, a2Q' (X'X^Q] under the null hypothesis (that is, if Q'3 = c). Note that here Q' is a row vector and c is a scalar. Therefore,

Q'0-r

(12.4.8) т z ~ N(0, 1).

<Wq'(x'x)-1q

The random variables defined in (12.4.2) and (12.4.8) are independent because of Theorem 12.4.2. Hence, by Definition 2 of the Appendix, we have

QP ~ c

ctVQ'(X'X)_1Q

Student’s t with T — К degrees of freedom, where a is the square root of

л

the unbiased estimator of a defined in equation (12.2.29). Note that the denominator in (12.4.9) is an estimate of the standard deviation of the numerator. The null hypothesis Q'P = c can be tested by the statistic

(12.4.9) . We use a one-tail or two-tail test, depending on the alternative hypothesis.

The following are some of the values of Q and c that frequently occur in practice:

The ith element of Q is unity and all other elements are zero. Then the null hypothesis is simply (3, = c.

The zth and 7th elements of Q are 1 and — 1, respectively, and c = 0. Then the null hypothesis becomes (3, = (37.

Q is a А-vector of ones. Then the null hypothesis becomes Х,:=1рг = c.

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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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