A COMPANION TO Theoretical Econometrics

The General Multivariate Parametric Problem

Consider the setting in (25.3) when { = p + v, and v ~ N(0, Q), is an available unrestricted estimator. Consider the restricted estimator p as the solution to the following quadratic programming (QP) problem:

min({ - p)'Q-1(p - p), subject to p > 0. (25.5)

Then the likelihood ratio (LR) test of the hypothesis in (25.3) is:

LR = p'Q-1p.

Several researchers, for instance Kudo (1963) and Perlman (1969), have estab­lished the distribution of the LR statistic under the null as:

Sup^. pr^LR > ca) = pr0#(LR > ca)

P

= X w(1, P, Q) X Pr(X(0 > ca) (25.7)

i=0

a weighted sum of chi-squared variates for an exact test of size a. The weights w(i, ) sum to unity and each is the probability of " having i positive elements.

If the null hypothesis is one of inequality restrictions, a similar distribution theory applies. To see this, consider:

H0 : ц > 0 vs. H1 : ц Є Rp, (25.8)

where { = ц + v, and v ~ N(0, Q). Let " be the restricted estimator from the following QP problem:

D = min ({ - q)Q-1({ - ц) subject to ц > 0. (25.9)

D is the LR statistic for (25.8). Perlman (1969) showed that the power function is monotonic in this case. In view of this result, taking Ca as the critical level of a test of size a, we may use the same distribution theory as in (25.7) above except that the weight w (i.) will be the probability of " having exactly p - i positive elements.

There is a relatively extensive literature dealing with the computation of the weights w(). Their computation requires evaluation of multivariate integrals which become tedious for p > 8. For example, Kudo (1963) provides exact expressions for p < 4, and Bohrer and Chow (1978) provide computational algorithms for p < 10. But these can be slow for large p. Kodde and Palm (1986) suggest an attractive bounds test solution which requires obtaining lower and upper bounds, cl and cu, to the critical value, as follows:

ai = ipr(x 21) > Ci), and

au = ipr(X2p-1) > Cu) + ipr(X 2p) > Cu) (25.10)

The null in (25.8) is rejected if D > cu, but is inconclusive when cl < D < cu. Advances in Monte Carlo integration suggest resampling techniques may be used for large p, especially if the bounds test is inconclusive.

In the case of a single hypothesis (ц1 = 0), the above test is the one-sided UMP test. In this situation:

pr(LR > ca) = pr( і X(0) + і X(1) > ca) = a

The standard two-sided test would be based on the critical values c' from a хЦ distribution. But pr(x2i) > c'a) = a makes clear that c'a > ca, indicating the substantial power loss which was demonstrated by Bartholomew (1959a, 1959b) and others.

In the two dimension (p = 2) case, under the null we have:

pr(LR > ca) = w(2, 0)x2q) + w(2, 1)%^ + w(2, 2)Xp) (25.12)

where w(2, 0) = pr[LR = 0] = pr[{ < 0, {2 < 0], w(2, 1) = -1, w(2, 2) = -1 - w(2, 0). While difficult to establish analytically, the power gains over the standard case can be substantial in higher dimensions where UMP tests do not generally exist. See Kudo (1963) and GHM (1982).