Test of Linear Hypotheses

In this section we shall regard the linear constraints (1.4.1) as a testable hy­pothesis, calling it the null hypothesis. Throughout the section we shall as­sume Model 1 with normality because the distributions of the commonly used test statistics are derived under the assumption of normality. We shall discuss the t test, the Ftest, and a test of structural change (a special case of the Ftest).

1.5.1 The t Test

The t test is an ideal test to use when we have a single constraint, that is, q = 1. The Ftest, which will be discussed in the next section, will be used if q>. Because 0 is normal, as shown in Eq. (1.3.4), we have

Q’0 ~ N[c, <72Q'(X’X)_,Q] (1.5.1)

Подпись: Q'0-с [<r2Q' (X' X)- *Q]1/2 Подпись: що, і). Подпись: (1.5.2)

under the null-hypothesis (that is, if Q’0 = c). With q = 1, Q’ is a row vector and c is a scalar. Therefore

image060 Подпись: XT-K‘ Подпись: (1.5.3)

This is the test statistic one would use if a were known. As we have shown in Eq. (1.3.7), we have

The random variables (1.5.2) and (1.5.3) easily can be shown to be indepen­dent by using Theorem 6 of Appendix 2 or by noting Е6/Г = 0, which implies that б and 0 are independent because they are normal, which in turn implies that u’ б and 0 are independent. Hence, by Theorem 3 of Appendix 2 we have

Q’0— c 0

[<T2Q'(X’Xr1Q]1’2 ~ St~k’ (L5’4)

which is Student’s t with Г— К degrees of freedom, where a is the square root of the unbiased estimator of a2 defined in Eq. (1.2.18). Note that the denomi­nator in (1.5.4) is an estimate of the standard deviation of the numerator. Thus the null hypothesis Q’0= c can be tested by the statistic (1.5.4). We can use a one-tail or two-tail test, depending on the alternative hypothesis.

In Chapter 3 we shall show that even if u is not normal, (1.5.2) holds asymptotically (that is, approximately when T is large) under general condi­tions. We shall also show that er2 converges to a2 in probability as T goes to
infinity (the exact definition will be given there). Therefore under general distributions of u the statistic defined in (1.5.4) is asymptotically distributed as N(0,1) and can be used to test the hypothesis using the standard normal table.6 In this case a2 may be used in place of a2 because a2 also converges to a2 in probability.

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