# Uniform linear hypothesis in multivariate regression models

Multivariate linear regression (MLR) models involve a set of p regression equa­tions with cross-correlated errors. When regressors may differ across equations, the model is known as the seemingly unrelated regression model (SUR or SURE; Zellner, 1962). The MLR model can be expressed as follows: Y = XB + U,

where Y = (Y1,… ,Yp) is an n x p matrix of observation on p dependent variables, X is an n x k full-column rank matrix of fixed regressors, B = [pv…, Pp] is a k x p matrix of unknown coefficients and U = [Uv…, Up] = [Q1,…, Un]’ is an n x p matrix of random disturbances with covariance matrix X where det (X) Ф 0. To derive the distribution of the relevant test statistics, we also assume the following:

A = W, i = 1,…, n, (23.19)

where the vector w = vec(W1,…, Wn) has a known distribution and J is an unknown, nonsingular matrix; for further reference, let W = [W1, …, Wn]’ = UG, where G = J-1. In particular, this condition will be satisfied when the normality assumption is imposed. An alternative representation of the model is

p

Yj = a j + X VjkXik, i = 1, …, n, j = 1, …, p. (23.20)

k =1

Uniform linear (UL) constraints take the special form

H0 : RBC = D, (23.21)

where R is a known r x k matrix of rank r < k, C is a known p x c matrix of rank c < p, and D is a known r x c matrix. An example is the case where the same hypothesis is tested for all equations

H01 : Rp,. = 5i, i = 1,…, p, (23.22)

which corresponds to C = Ip. Here we shall focus on hypotheses of the form

(23.22) for ease of exposition; see Dufour and Khalaf (1998c) for the general case.

Stewart (1997) discusses several econometric applications where the problem can be stated in terms of UL hypotheses. A prominent example includes the multivariate test of the capital asset pricing model (CAPM). Let rjt, j = 1,…, p, be security returns for period t, t = 1, …, T. If it is assumed that a riskless asset rF exists, then efficiency can be tested based on the following MLR-based CAPM model:

rjt – rFt = aj + Pj (rMt – rFt) + j = 1 . . . , P, t = 1 . . . , ^

where rMt are the returns on the market benchmark. The hypothesis of efficiency implies that the intercepts a; are jointly equal to zero. The latter hypothesis is a special case of (23.22) where R is the 1 x p vector (1, 0,…, 0). Another example concerns demand analysis. It can be shown (see, e. g., Berndt, 1991, ch. 9) that the translog demand specification yields a model of the form (23.20) where the hypothesis of linear homogeneity corresponds to

p

H0 : X Pjk = 0, j = 1, …, p. (23.23)

k=1

Table 23.3 Empirical type I errors of multivariate tests: uniform linear hypotheses

Sample size 5 equations 7 equations 8 equations

 LR LRc LRMC LR LRc LRMC LR LRc LRMC

 20 0.295 0.1 0.051 0.599 0.25 0.047 0.76 0.404 0.046 25 0.174 0.075 0.049 0.384 0.145 0.036 0.492 0.19 0.042 40 0.13 0.066 0.056 0.191 0.068 0.051 0.23 0.087 0.051 50 0.097 0.058 0.055 0.138 0.066 0.05 0.191 0.073 0.053 100 0.07 0.052 0.042 0.078 0.051 0.041 0.096 0.052 0.049
 LR, LRc, LRMC denote (respectively) the standard LR test, the Bartlett corrected test and the (corresponding) MC test.

In this context, the likelihood ratio (LR) criterion is:

LR = n 1п(Л), Л = |й’0й0|/|й’й|, (23.24)

where й’0й0 and U’U are respectively the constrained and unconstrained SSE (sum of square error) matrices. On observing that, under the null hypothesis,

й’й = G -1W’MW{G -1)’, (23.25)

й0й0 = G -1W M0W{G -1)’, (23.26)

where M0 = I – X(X’X)-1(X’X – R'(R(X’X)-1R’)-1R)(X’X)-1X’ and M = I – X(X’X)-1X’, we can then rewrite Л in the form

Л = | WM0W |/| WMW |, (23.27)

where the matrix W = UG’ has a distribution which does not involve nuisance parameters. As shown in Dufour and Khalaf (1998c), decomposition (23.27) obtains only in the case of UL constraints. In Section 4 we will exploit the latter result to obtain exact MC tests based on the LR statistic.

To illustrate the performance of the various relevant tests, we consider a simula­tion experiment modeled after demand homogeneity tests, i. e. (23.20) and (23.23) with p = 5, 7, 8, n = 20, 25, 40, 50, 100. The regressors are independently drawn from the normal distribution; the errors are independently generated as iid N (0, X) with X = GG’ and the elements of G drawn (once) from a normal distribution. The coefficients for all experiments are available from Dufour and Khalaf (1998c). The statistics examined are the relevant LR criteria defined by (23.24) and the Bartlett-corrected LR test (Attfield, 1995, section 3.3). The results are summarized in Table 23.3. We report the tests’ empirical size, based on a nominal size of 5 percent and 1,000 replications. It is evident that the asymptotic LR test overrejects substantially. Second, the Bartlett correction, though providing some improve­ment, fails in larger systems. In this regard, it is worth noting that Attfield (1995, section 3.3) had conducted a similar Monte Carlo study to demonstrate the effec­tiveness of Bartlett adjustments in this framework, however the example analyzed was restricted to a two-equations model. We will discuss the MC test results in Section 4.

To conclude this section, it is worth noting that an exact test is available for hypotheses of the form H0 : RBC = D, where min(r, c) < 2. Indeed, Laitinen (1978) in the context of the tests of demand homogeneity and Gibbons, Ross, and Shanken (1989), for the problem of testing the CAPM efficiency hypothesis, independently show that a transformation of the relevant LR criterion has an exact F-distribution given normality of asset returns.2