On exact and asymptotic tests of non-nested models

Statistics & Probability Letters 5 (1987) 19-22 North-Holland

ON EXACT AND ASYMPTOTIC

January 1987

TESTS OF NON-NESTED

MODELS

Anil K. B E R A Department of Economics, Unioersity of Illinois, Champaign, IL 61820, USA

Michael M c A L E E R Department of Statistics, Australian National Unioersity, Canberra, Australia Received December 1982 Revised August 1986

Abstract: In this note we establish finite sample relations among some exact and asymptotic tests of non-nested linear regression models.

Kevwords: Atkinson test, Cox test, finite sample distribution, non-nested regression models.

In this note we consider the relations among alternative tests of two non-nested linear regression models, H 0 a n d H 1, where the models are given by

y = X f l + u o, uo-N(O, I.o2),

(1)

1-11" y = Z'y + Ux, u x - N(O, 1,,o2).

(2)

H0:

In equations (1) a n d (2), y is an n × 1 vactor of observations on the dependent variable, X and Z are n x k and n × g matrices of n observations on k and g linearly independent non-stochastic regressors, 13 and y are k × 1 and g × 1 vectors of u n k n o w n parameters, and u 0 and u x are n × 1 vectors of disturbances. It is a s s u m e d that the matrices n - I x ' x and n - I z ' z converge to welldefined finite positive definite limits, and n - I x ' z is non-zero and also converges to a non-zero finite matrix. The two models are non-nested in the sense that at least one column of each regressor matrix cannot be expressed as a linear combination of the columns of the other. Denoting 0~ = (fl', 02) and 0~ = ( y ' , o2), the Cox (1961, 1962) test of H 0, as developed by Pesaran (1974), is based upon the statistic

T o = ( l o - l l ) - n [ p l i m o n-l(fo-ll)]eo_6o,

(3)

in which /,=-½n

log2cr-½nlogS/2-½n

(i=0,1)

is the maximized log-likelihood function under H i and plim o denotes probability limit under H o. The circumflex denotes maximum likelihood estimate, in which case

~2=n-ly'(I-P)y,

e = x ( x , x ) - l x ,,

82=n-ly'(1-Q)y

and Q = Z ( Z ' Z ) - x z ".

The Atkinson (1970) variation of the Cox test of H o is based u p o n (see Fisher and McAleer (1981))

TAo= (io - 4 0 ) - n [ p l i m 0 n - l ( 1 0 - fl0)] 0o=0O,

(4) in which

l]0=

log

log a?0

-½n(n-ly'( I - QP)'(1- QP)y)/620, and 620=62 +

n-ly'P(1-a)Py=n-ly'(1-PQP)y.

N o t e that plimo(62 - 0^2 1 0 ) = 0 and the expectation of QPy equals the expectation of Qy under H o.

0167-7152/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)

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STATISTICS & PROBABILITY LETTERS

The Atkinson version differs from the Cox test in that the former evaluates the entire test statistic under the null hypothesis. The second terms in (3) and (4) are asymptotically equivalent under H 0. Since, in addition, plim 0 812 = plim o olO ^2 =plim 0 n-Xy'(I-QP)'(I-QP)y = 02 ,

it follows that TO and TA o are asymptotically equivalent under H o. The important point to note in the preceding discussion is that both the Cox and Atkinson procedures modify a likelihood ratio by subtracting its asymptotic expectation evaluated at consistent estimates of the parameters under the null model. Denoting a consistent estimate of the asymptotic variance of TA o under H o as l~0, the Atkinson test, namely NA o = TAo/(Vo) 1/2, is approximately distributed as N(0, 1) under H 0 in large samples (see Atkinson (1970, p. 335) and Fisher and McAleer (1981, p. 111)). Fischer and McAleer (1981) show that NA o may be written as

NA o =

8o( y ' P Q ( I - P )QPy ) 1/z

NA o = - y ' P O ( 1 - P) y (5)

Apart from 8o, NA o may be calculated from the artificial regression (6)

The t-ratio for the ordinary least squares (OLS) estimate of a in (6), namely

t(a)=y'PQ(1-P)y / { 8 ( y ' P Q ( I - P )QPy)X/2}, in which

82=(n-k-1)-l{y'(I-P)y - ( y'PQ( I - P) y ) 2 / ( y , p Q ( I - P)QPy ) } is the estimated error variance from (6), is asymp20

y = (1 - a)Xfl + aQy + u.

(7)

A two-tailed test of a = 0 in (7) is a test of H 0, whereas a test of H x is performed by testing )~ = 0 in the artificial regression (8)

If Py is substituted for Xfl in (7), we have the C test of H o, namely the test of a = 0 in

which is equivalent to

y = Xfl + aQPy + u.

totically equivalent to NA o under H 0. This result follows from the fact that plim 0 82 = plim 0 82 = 02 . In the remainder of this note, we will concentrate u p o n asymptotic approximations to the Cox test based u p o n artificial regression such as the one given in (6). Three simple asymptotic procedures, namely the J, P, and C tests have been developed by Davidson and M a c K i n n o n (1981) for testing nonnested models. These tests are approximations to the Cox test and are also very straightforward to calculate. In the linear case, the J and P tests are algebraically equivalent, and both are preferred to the C test. We shall provide below the finite sample relations between the J and C tests. The sample means of y under H 0 and H 1 are given by Py = Xfl and Qy = Z'~, respectively, and the J test is the t-ratio for the OLS estimate of a in the artifical regression

y = ( 1 - X ) Z y + XPy + u.

½ n ( 8 7 - 8 2 o ) + ½y'( I - P )Q( I - P ) y

/{8o(y'PQ(I-P)QPy)I/2}.

January 1987

y = ( 1 - a)Py + aQy + u, or ( I - P ) y = a ( Q - P ) y + u .

(9)

Davidson and MacKinnon (1981, p. 783) recomm e n d the C test as a preliminary test of H 0 because asymptotically it " . . . h a s variance less that unity under H0." A heuristic argument that the C test is undersized (i.e., the actual size of the test is less than its nominal size) when both a and the error variance are estimated by maximum likelihood methods is given below. Denoting the large sample tests of a = 0 (i.e., the t-ratios based u p o n m a x i m u m likelihood estimation) from (7) and (9) as i(&s) and i(gtc), respectively, it follows that /2(ks ) = (y,Q( I-

P ) y ) 2 / ( 82(y,Q ( I - P)Q_y)),

iZ( kc) = ( y ' Q ( I - p ) y ) 2 / ( ~. ( y , ( Q _ p ) Z y ) ) ,

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STATISTICS& PROBABILITYLETTERS

( I - - P ) y , the vector of residuals obtained from

where

8f =n

-1

{ y'(I- P)y- (y'Q(I- p)y)2 /(y'Q(I- P)Qy}

and

8~= n-l{ y ' ( 1 - P ) y -

(y'Q(I- p)y)2 /(y'(e-p)2y))

are the m a x i m u m likelihood estimates of the error variances from (7) and (9), respectively. Since

y'( Q - p )2y - y ' Q ( 1 - P )Qy

is strictly positive (even asymptotically) when H 1 is not fitting perfectly, it follows that [ 2 ( S j ) > [2(&c).

(10)

Since the J test has correct size asymptotically, the C test is undersized. It should be noted that there are two inherent problems with the C test. First, the formula for the variance of &c is incorrect. Second, the estimated error variance used in calculating the t-ratio for &c is inappropriate for finite samples. When the two tests are calculated from a least squares package, the denominators of 82 and 82 become (n - k - 1) and (n - 1), respectively, so that the J and C tests are given by t ( d j ) and t(&c). The relationships between the tests based u p o n maxim u m likelihood and least squares procedures are nt2(~tj) = (n -- k - 1)i2(~:) and nt2(Stc) = (n - 1)/2(&c), so that the inequality between the J and C tests may be written as

t2(Sj)>(n-k-1)t2(&c).

least squares estimation of (1). It is noteworthy that Q_y is not distributed independently of ( 1 - P)y. Therefore, the J test is not an exact test for H 0 when there is more than one column in Z that is linearly independent of the columns of X. However, when Z contains only one column that is linearly independent of those of X (e.g. when there is one regressor in H 1 that is not in H 0), the J test of H 0 does have the exact t distribution in small samples. Following Atkinson (1970), Fisher and McAleer (1981, p. 109) evaluate the entire statistic by replacing Qy in (7) with QPy, as in (6), leading to y = (1 - a) Xfl + aQPy + u,

=y'(I-Q)P(I-Q)y

(n-1)

January 1987

(11)

Strictly speaking, there should be (n - k - 1) degrees of freedom for t ( ~ c ) , as well as for t(~j), since k degrees of freedom are lost in estimating fl to obtain (9) from (7). The inequality given by (11) refers solely to the t-ratios obtained from OLS estimation of (7) and (9). In spite of inequality (10), when k is large relative to n and OLS estimation is used, it is clear from (11) that t(&j) may be exceeded by t(&c). Under H 0, Py is distributed independently of

(12)

in which a two-tailed test of a = 0 is a test of H 0. Milliken and Graybill (1970) have shown that if one or more functions of the linear predictions of the linear null model, namely Py, are added to that model, the test statistic of the significance of the additional regressors will have the exact F distribution under the null hypothesis when the disturbances are normal. Thus, the JA test, which is the t-ratio for the OLS estimate of a, has the exact t distribution with ( n - k - 1) degrees of freedom under H 0. If the true spirit of the Cox principle is to be observed, however, the roles of the null and alternative models should be reversed. The JA test of H 1, which is the t-ratio for the OLS estimate of ~ in y = (1 - X)Z'y + XPQy + u,

(13)

has the t-distribution with ( n - g 1) degrees of freedom under H 1. There is presently very little known about the power of the Milliken and Graybill procedure, and indeed the test may not even be unbiased. Pesaran (1982) has shown that the JA and J tests have equal power in large samples when H 0 is tested against local alternatives and k >/g. However, httle is k n o w n of power considerations in the interesting case of fixed alternatives, or when k < g. Of course, it is possible for k to exceed g through underspecifying the alternative, in which case the tests of the null may not even be consistent. For more on this, see McAleer et al. (1982), where it is also shown that overspecification does n o t affect the consistency of the tests. 21

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STATISTICS & PROBABILITY LETTERS

Finally, we derive the exact versions of the P and C tests. Transforming b o t h sides of (12) b y ( I - P ) annihilates X and yields the P A test of H 0, n a m e l y the test of a = 0 in (I-

P)y=a(I-

P ) Q P y + u.

(14)

It should b e n o t e d that there are only ( n - k ) i n d e p e n d e n t observations in (14) since the k elements of fl have already been estimated. Therefore, the t-ratio for alva from (14) m u s t be multiplied b y (n - k - 1 ) 1 / 2 / ( n - 1) 1/2 to o b t a i n a test statistic which has the exact t distribution with (n - k - 1) degrees of freedom. If Xfl is replaced b y P y in (12), rearrangement yields (I-

P ) y = -a(1-

Q ) P y + u,

(15)

in which the test of a = 0 is the C A test of H 0. N o t e that transforming b o t h sides of (15) b y ( 1 - P ) yields the P A test in (14).

Acknowledgement The authors w o u l d like to t h a n k a referee for helpful comments. The financial s u p p o r t of the Research Board a n d the Bureau o f E c o n o m i c and Business Research of the University of Illinois is gratefully acknowledged.

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January 1987

References Atkinson, A.C. (1970), A method for discriminating between models, Journal of the Royal Statistical Society B 32, 323-344. Cox, D.R. (1961), Tests of separate families of hypotheses, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability 1 (Berkeley, University of California Press) pp. 105-123. Cox, D.R. (1962), Further results on tests of separate families of hypotheses, Journal of the Royal Statistical Socie(v B 24, 406-424. Davidson, R.. and J.G. MacKinnon (1981), Several tests for model specification in the presence of alternative hypotheses, Econometrica 49, 781-793. Fisher, G.R. and M. McAleer (1981), Alternative procedures and associated tests of significance for non-nested hypotheses, Journal of Econometrics 16, 103-119. McAleer, M., G.R.. Fisher and P. Volker (1982), Separate misspecified regressions and the U.S. long run demand for money function, Review of Economics and Statistics 64, 572-583. Milliken, G.A. and F.A. Graybill (1970), Extensions of the general linear hypothesis model, Journal of the American Statistical Association 65, 797-807. Pesaran, M.H. (1974), On the general problem of model selection, Review of Economic Studies 41, 153-171. Pesaran, M.H. (1982), Comparison of local power of alternative tests of non-nested regression models, Econometrica 50, 1287-1305.