A Bounds Test of Equality Between Sets of Coefficients in Two Linear Regressions WhenDisturbance Variances are UnequalAuthor(s): Masahito KobayashiSource: Journal of the American Statistical Association, Vol. 81, No. 394 (Jun., 1986), pp. 510-513Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2289242 .

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A Bounds Test of Equality Between Sets of Coefficients in Two Linear Regressions When

Disturbance Variances Are Unequal MASAHITO KOBAYASHI*

This article considers the Wald test statistic for testing equality between sets of coefficients in two linear regressions when the disturbance variances are unequal. It is shown that the distri- bution of the test statistic under the null hypothesis is bounded, asymptotically up to the second order, by the distributions of two F variates multiplied by the number of the regressors.

KEY WORDS: Behrens-Fisher problem; F distribution; Wald test statistic.

1. INTRODUCTION

I consider tests of the hypothesis PI = ,62 in the two linear regressions

yi=X iA +ei, i= 1,2, (1.1)

where Xi is a full rank ni X p matrix and ei follows a multivariate normal distribution with mean zero and covariance matrix urI. Assume that e1 and e2 are distributed independently.

When we can assume that U2 = o2, this hypothesis can be checked by several small sample tests (e.g., see Chow 1960). These tests are erroneous (as is shown in Toyoda 1974) in the case in which U2 = o2, which is this article's main concern. A small sample test that can be employed in this case was suggested by Jayatissa (1977). Watt (1979) showed numeri- cally, however, that in most situations the asymptotic Wald test is preferable to the Jayatissa test from the viewpoint of power, although the actual level of the Wald test is larger than the nominal level, which is given from the chi-squared table; the exact significance level is practically unobtainable, since the distribution of the test statistic depends complicatedly on the regressor variables and the disturbance variances.

Rothenberg (1984) proposed an adjusting procedure of crit- ical values of the Wald test so that the adjusted test may have the assigned significance level asymptotically up to the second order; his work is a generalization of the well-known result by Welch (1947) for the Behrens-Fisher problem. But, as he ad- mits, the formula for the adjustment is somewhat complicated; a simpler test might be useful in some cases.

I propose a bounds test using the result that the distribution of the Wald test statistic under the null hypothesis is bounded by the distribution functions of F variates, multiplied by p, with degrees of freedom (p, min(n1 - p, n2 - p)) and (p, n, + n2 - 2p) asymptotically up to the second order-namely, up to order 1Ini. This conclusion is, in a certain sense, a gener- alization of the result by Mickey and Brown (1966), in which they showed that the small sample distribution of the Behrens- Fisher statistic is bounded by two Student-t distributions.

* Masahito Kobayashi is with the Institute of Economic Research, Kyoto University, Sakyo-ku, Kyoto 606, Japan. The author thanks K. Takeuchi, T. Sawa, K. Ohtani, and an anonymous referee for their helpful comments.

2. MAIN RESULT Consider the Wald test statistic-namely,

T-= (b,~ - b2)'(sy1 + sE2) '(b, - b2), (2.1)

where bi and s2 are the least squares estimators for /, and ci, respectively, and 1i denotes (X'Xi) '.

The main purpose of this section is to prove that

inf Pr(Tlp < t) = F(t; p, min(n1 - p, n2 - p)) + O(1/n?)

sup Pr(Tlp < t) = F(t; p, n, + n2 - 2p) + O(1/n?), (2.2)

assuming that t > p + 2, where F(t; p, n) denotes the distri- bution function of an F variate with degrees of freedom (p, n). By x = O(1/n?) I mean that n?x is bounded when ni increases; the supremum and the infimum are evaluated by varying the values of the regressor variables and the disturbance variances. I henceforth refer to ni - p by fi. The supremum is actually taken when 1/(1 + 'iU2/a2) - fl/(fl + f2), i = 1, . . . p, where u,u, . . ., ,up are the eigenvalues of (X2X2)-'X,X1, and Pr(T < t) approaches the infimum as 1/(1 + 21 approaches unity if f I < f2 and zero if f1 2 f2

Their derivation is summarized as follows: First, we obtain an asymptotic expansion of the distribution function of the Wald test statistic, or Pr(T < t), up to the second order-namely, up to order 1/ni. Next we obtain a lower limit and an upper limit to this asymptotic probability. We then show that these limits agree with the distribution functions of the F variates up to the required order. Table 1 shows upper critical values (UCv'.s) and lower critical values (LCV's). A full table can be easity given using the F table.

Table 2 gives Pr(Tlp > LCV) and Pr(Tlp > UCV) for p 2 and f, = f2 = 12 by means of numerical integration. Their estimated errors are of order 10-5, at most. As far as Table 2 shows, the assigned level of significance, .05, is actually bounded by Pr(Tlp > UCV) and Pr(Tlp > LCV).

A small sample bounds test has been given by Ohtani and Kobayashi (1984). They obtained a lower bound and an upper bound on the Wald test statistic-namely,

p p Ez?/max(2a,51)CTCE Zi /minl( I/al S2/ 2)

i=I i=l

where z's are independent standard normal variates; they sug- gested a small sample bounds test. The table of upper and lower critical values for this test is given in their paper. For example, their test in terms of Tlp has the lower critical value 2.84 and

? 1986 American Statistical Association Journal of the American Statistical Association

June 1986, Vol. 81, No. 394, Theory and Methods

510

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Kobayashi: Testing Equality Between Two Linear Regressions 511

Table 1. Lower Critical Values (LCV's) and Upper Critical Values (UCV's) of Tlp for Significance Level .05

p =2 p=3 p=4

f, f2 LCV UCV LCV UCV LCV UCV

12 12 3.40 3.89 3.01 3.49 2.78 3.26 20 20 3.23 3.49 2.84 3.10 2.61 2.87 40 20 3.15 3.49 2.76 3.10 2.53 2.87 40 40 3.11 3.23 2.72 2.84 2.49 2.61 60 20 3.11 3.49 2.72 3.10 2.49 2.87 60 60 3.07 3.15 2.68 2.76 2.45 2.53 00 X0 3.00 3.00 2.61 2.61 2.37 2.37

the upper critical value 4.78, for f1 = f2 = 12, p = 2, and the significance level .05, whereas the nominal critical value given in the chi-squared table is 3.00. On the other hand, my test has the lower critical value 3.40 and the upper critical value 3.88 in this case. For fI = f2 = 60 and p = 2, the lower value and the upper value of their test are 2.79 and 3.47, whereas my test has 3.07 and 3.15. It seems that my test has much narrower inconclusive regions than theirs, even when the sample sizes are large.

3. ALGEBRAIC DETAILS

First, I show that the Wald test statistic defined in (2.1) can be expressed as

p

T = > Z[60iX1/f1 + (1 - i)x2/f2]', (3.1) i=1

where (z,, . . , zp) has a normal distribution with a mean of zero and a unit covariance matrix. Then 0 < 61, . . ., Op < 1, and Xi is a chi-squared variate with degrees of freedom fi = ni - p. I also show that these variates are stochastically independent.

Since b, - b2 is, under the null hypothesis, normally dis- tributed with mean zero and covariance matrix U21, + U222,

which is denoted by X, we have

T= v't2 ,-1 1/2V,

by defining v' as (b, - b2)'YI 2 and i as s,1 + s22. Note

Table 2. Pr(Tlp > LCV) and Pr(Tlp > UCV) for Significance Level .05, p = 2, f1 = 12, f2 = 12 (LCV = 3.40, UCV = 3.89)

62

61 .01 .20 .40 .60 .80 .99

.01 .0493 .0443 .0419 .0415 .0430 .0473 .0666 .0611 .0584 .0580 .0599 .0650

.20 .0394 .0371 .0368 .0385 .0430 .0555 .0529 .0527 .0548 .0599

.40 - .0350 .0349 .0368 .0415 - - .0505 .0504 .0527 .0580

.60 - - .0350 .0371 .0419 - - - .0505 .0529 .0584

.80 - - .0394 .0443 - - - - .0555 .0611

.99 .0493 - - - - - .0666

NOTE: The upper entries show Pr(T/p > UCV) and the lower entries show Pr(T/p> LCV).

that v is normally distributed with mean zero and unit covari- ance matrix. Then, by an orthogonal transformation, we have

p

T = >Zi

where A . . .A, P are the eigenvalues of 1/22-1E1/2 and (z1, . . , zp) is the standardized normal variate employed in (3.1). Now I show that these eigenvalues are

i= ( + e24u)/(s1 + s2ui) i = 1, . . . ,

where ,u,, . . , ,up are the eigenvalues of jI 112* Note that the characteristic equation 1/2i -112 - 1 I| =

O is equivalent to the equation |1 - Al= 0; I solve the latter. We can diagonalize 1, as well as i, by a nonsingular matrix U such that

U'l,U = I, U'12U = diag(,u, . ,up).

We have U'YU = U2,I + U2 diag(,u ,. . . , up), where diag(,u, . ., up) denotes a diagonal matrix that has ,ui as the (i, i)

element. Analogously, we can have U'IU = s2, + s2diag(,u, . . ., ,up). Since

IU'I | 1-A | UI = |U2,I + U2diag(fL, . . .

- 2dsia + 52 diag(u, . . ' ,u

we can obtain the desired roots Al, . . ., AP for A-il = 0, using IUI $ 0. Referring to 1/(1 + UiU2/U2) by 6i and s?/ vI? by Xi/fi we have Ai = [6iX I/fI + (1 - OA)x2 f2V Thus we have finally obtained (3.1).

Write the statistic T as

T = ,2 E rii, j=l i=l

where ri = z?/1P I zj2. Then we have

Pr(T < t) = E[G (t ri> (3.2)

where G(t) is the distribution function of a chi-squared variate with degrees of freedom p, since Xi (and accordingly, Ai) and ri are independent of Ep=. 4Z2; the independency of ri and

4= z2was verified by Rao (1973, p. 164) in a slightly different form.

Let wi denote Xi/fi. The term G(t/(EP= riAi)) inside the expectation on the right-hand side of (3.2) can be expanded into the Taylor series in powers of wi - 1, which is stochas-

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512 Journal of the American Statistical Association, June 1986

tically of order ni 1/2. Then we have -2

E[G(tlE riAi)] = E G(t) + E (wi - 1)aGlawi

2 2 ] + E E( - - 1)2G/l3wiawj +

i=1 j=1

where a2G/awiawj, for example, denotes a2G(t/l P= lrk4k)/awiawj evaluated at w, = 1 and w2 = 1-namely, at Ai = 1 and 'riAi= 1. I first take the expected value only with respect to w, and

w2, noting that wi is independent of r's. I then substitute E(wi - 1) = 0, E(w1 - 1)(W2 - 1) = 0, and E(wi - 1)2 = 2/ fi; I neglect higher-order moments of w, - 1 and w2- 1, since they are of smaller order than n7_' . Then, evaluating 02GI dw1, and so forth, we have

Pr(T < t)

- G(t) + 2E(G"(t)t2[E riOi]2

+ 2tG'(t){[ riOi]2 - E riO)I})IfI

+ 2E(GPP(t)t2[E ri(I - Q)]2

+ 2tG'(t){[E ri(1 - Qi)2 _ E ri(1 - i)2]})/f2

+ O(1/In?). (3.3)

Note that the first term, which is the principal term, is the distribution function of a chi-squared variate, which is em- ployed in the traditional asymptotic test.

Now take the expected value of (3.3) with respect to r's. Let Q(01, . . . , Op) denote the terms of order of 1/f and 1/f2 in (3.3). Then, after some algebra, we have

Q(01, ,Op) = 2c{[O i + p > ojoj fl

+ [ (1 - Oi)2 + p (1 - Oj)(1 - Oj)J f2it (3.4)

where

c = [3G"(t)t2 + 2tG'(t)(I - p)]/(p2 + 2p)

p = t2(G"(t) + 2G'(t)lt)l[(p2 + 2p)c],

since we have

E(ri) = Ip,

E(rirj) = l/(p2 + 2p), i#j

= 3/(p2 + 2p), i =j.

These equations can be shown using

[- ] -I[ / D

E [L ( zjE[z1E Z k i zj = E[ZJ Z],

and E[zjz'] = 3, j = k

= 1, j=k, which can be easily shown from the independence of ri and

__ 1 __2

Note that G"(t) and G"(t)/G'(t) + 2/t = (p!2 + 1)/t -2 are negative, since t > p + 2. This condition is normally satisfied for a critical value t corresponding to a small signif- icance level, since t is supposed to be much larger than p, which is, roughly speaking, the mean of the test statistic. Then we can safely say that p > 0, c < 0 in (3.4).

Now I show that Q has its minimum at (01, . . , Op) = (0, ., O) if f1 I f2, and at (1, . . ., 1) if f, < f2in the region

S = {(0l,. . . , Op); 0 < 0i < 1}, and that Q has its maximum at (f11(f, + f2), . . . , fI/(fI + f2)) in S; note that the region S is the closure of the parameter space of 0's. In the following I mainly consider the case in which f, ' f2; the other case can be dealt with analogously.

First, I verify that the contour of Q(01, . . , Op) represents an elliptic quadric hypersurface in a p-dimensional space. By moving the origin, we can see that this contour coincides with a quadric hypersurface that is defined by 0'AO = constant, where 0 = (01, . .. ,Op)' and

1pP

A=[P p1j. 1

This matrix is positive definite for my p, since it can be shown that Qlc is always positive, noting that G"(t) < 0, G'(t) > 0,

ri = 1, and Cauchy's inequality (X ri0i)2 X< ; rjO2 X ri in (3.3). Thus we have seen that this contour is an elliptic quadric.

Next, let us examine the shape of this elliptic. It can be easily shown that the preceding matrix has a distinct eigenvalue 1 + (p - l)p, which corresponds to the eigenvector (1, . . ., 1), and an eigenvalue 1 - p with multiplicity p - 1; the former is the largest, since p > 0. Besides, it can be seen that the original elliptic has the center at (fil(fi + f2), . . , f,l(f1 + f2)). Then the elliptic has the shortest principal axis along the line from the center to (1, . . , 1) and the other longer principal axes of equal length.

Then I have only to show that the aforementioned elliptic that has (1, . . . , 1) upon its surface contains the region S- in order to show that, in the region S, the function Q takes its minimum at (1, . . , 1), since this elliptic is a contour of the function Q.

First, I show that the farthest point in S from the center of the elliptic is (1, . . . , 1). Consider a triangle that consists of a vertex of the cubic region S, the center of the elliptic, and (2, . . * 2). Note that the center of the elliptic is located between the origin and (2, . . , 2). In this triangle, the segment between the center of the elliptic and the vertex is shorter than the sum of the other sides, which is equal to the segment between (1, . . , 1) and the center of the elliptic in length. Then we can see that (1, . . . , 1) is the farthest point from the center of the elliptic. In other words, it is verified that .a hy- persphere that has (1, . . . , 1) upon its surface, with the same center as the elliptic, contains the region S.

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Kobayashi: Testing Equality Between Two Linear Regressions 513

Besides, the elliptic that has (1, . , 1) upon its surface contains this hypersphere entirely, since the elliptic has the shortest principal axis along the line from the center to (1, . . ., 1). Then it follows that the elliptic contains the cubic

region S. Thus I have proved that Q has its minimum at (1, . 1) if f, < f2. On the other hand, by the symmetry of the elliptic, we can see that Q has its minimum at (0, . . . , 0) if fI > f2. It is also obvious that Q takes its maximum value at the center of the elliptic.

Now I can verify the desired result. Note that the statistic TI p, or ;=X lz[61 x,/f1 + (1 - OA)x2f2]-'/P, is distributed as an F variate with degree of freedom (p, f ) at (Or, . . , OP) = (1, . . , 1), with degree of freedom (p, f2) at (0, . .

0), and with degrees of freedom (p, f, + f2) at (f,I(f, + f2),

* * *, flI(f, + f2)). Then we see that the distribution of Tlp (namely, the Wald statistic divided by the number of regressors) is bounded, asymptotically up to order 1/ni, by the distribution functions of the two F variates F(p, f + f2) and F(p, min(f, f2)). Thus we have the main result (2.2).

[Received February 1985. Revised August 1985.]

REFERENCES

Chow, G. C. (1960), "Tests of Equality Between Sets of Coefficients in Two Linear Regressions," Econometrica, 28, 591-605.

Jayatissa, W. A. (1977), "Tests of Equality Between Sets of Coefficients in Two Linear Regressions When Disturbance Variances Are Unequal," Econ- ometrica, 45, 1291-1298.

Mickey, M. R., and Brown, M. B. (1966), "Bounds on the Distribution Functions of the Behrens-Fisher Statistic," Annals of Mathematical Statis- tics, 37, 639-642.

Ohtani, K., and Kobayashi, M. (1984), "The Wald Bounds Test for Equality Between Sets of Coefficients in Two Linear Regressions Under Het- eroscedasticity," unpublished mimeograph.

Rao, C. R. (1973), Linear Statistical Inference and Its Applications (2nd ed.), New York: John Wiley.

Rothenberg, T. J. (1984), "Hypothesis Testing in Linear Models When the Error Covariance Matrix Is Nonscalar," Econometrica, 52, 827-842.

Toyoda, T. (1974), "Use of the Chow Test Under Heteroscedasticity," Econ- ometrica, 42, 601-608.

Watt, P. A. (1979), "Tests of Equality Between Sets of Coefficients in Two Linear Regressions When Disturbance Variances Are Unequal: Some Small Sample Properties," Manchester School of Economic and Social Studies, 47, 391-396.

Welch, B. L. (1947), "The Generalization of 'the Student's' Problem When Several Different Population Variances Are Involved," Biometrika, 34, 28- 35.

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