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Some aspects of testing non-nested hypotheses
Journal
of Econometrics
SOME
21 (1983) 213-228.
ASPECTS
OF TESTING HYPOTHESES
Naorayex University
Received
North-Holland
Publishing
NON-NESTED
K. DASTOOR*
of Liverpool, Liverpool L69 3BX,
March
Company
1979, final version received
UK
May 1982
This paper examines some aspects of testing non-nested hypotheses. An inequality between the Cox and Atkinson statistics is noted, and the necessary and sufficient condition for the Atkinson test to be consistent is derived. A new test procedure is also outlined. The rest of the paper illustrates the various test statistics, their properties, and relationships for competing linear regression models.
1. Introduction The last decade or so has witnessed numerous papers on tests of separate families of hypotheses, see e.g. the bibliography by Pereira (1977a). The same topic has been studied by econometricians under the heading of ‘non-nested hypotheses’. The problem can be viewed as an examination of the probability density function (p.d.f.) of an (n x 1) vector random variable (T.v.) Y with respect to the hypotheses H,: Y has p.d.f. f(y, a,),
MOE Qo,
H,: Y has p.d.f. g(y,a,),
a1 EQl,
and
where the parameter spaces Sz, and Q, are such that Q, n Q, #L’, and 52, n 52,#ill; and cl0 and a, are the parameters of the p.d.f.‘s f and g, respectively. If 52, n 52,=52, (or 52, n 8, =5),), then the hypotheses are said to be nested. A researcher may wish to choose a particular model from amongst the set available (discrimination), in which case the procedures discussed by Amemiya (1980) could be used. Alternatively, he may wish to test the ‘truth’ *An earlier version of this paper was presented at the European Meeting of the Econometric Society, Amsterdam, September 1981. The author is grateful to M. Chatterji, K. Holden, M. McAleer, A.R. Nobay, J. Richmond, N.E. Savin, M.R. Wickens and anonymous referees for helpful comments on earlier versions of the paper, and to N.R. Ericsson for pointing out an error.
0304-4076/83/0000-0000/$03.00
0
1983 North-Holland
214
N.K. Dastoor,
Testing non-nested hypotheses
of a particular model and this can be accomplished by using an appropriate significance test. This paper is concerned with significance testing as, presumably, an economic researcher would be more interested in the ‘truth’ of a particular model than in choosing from among a given set of models. Since to each hypothesis there corresponds a unique model, the terms ‘hypothesis’ and ‘model’ will be used synonymously. To test the ‘truth’ of H,, against the non-nested alternative Hi, the relevant statistic is calculated assuming H, is true. Then, on the basis of the observed data, H, is either rejected or not rejected. The roles of H, and H, are then reversed and the procedure repeated. So the two tests yield four possible outcomes. Two points should be noted. Firstly, for each hypothesis assumed in turn, the level of significance is a subjective decision. Secondly, the optimal level of significance for such a test procedure requires further research. The next section outlines the Cox and Atkinson procedures and notes an inequality between the two statistics. In section 3, the necessary and sufficient condition for the Atkinson test to be consistent is derived. The R procedure is proposed in the fourth section, and in section 5 some R statistics are derived for competing linear regression models. For the same models, the relationship between an R test and other non-nested tests is examined in section 6. Finally, section 7 contains the conclusion.
2. Some proposed statistics Suppose a researcher wishes to test H, and H, as stated in the introduction. Cox (1961, 1962) first considered this problem and, to test Ho, suggested the statistic
where L,(a,) {L,(a,)} is the log-likelihood function under H, {Hi}; I&, {gl} is the maximum likelihood estimate of CQ {ai} under H, {Hi}; &lo= &r; and both the expectation and probability limit are carried out plim+ m under H,. Cox (1961, 1962) showed that asymptotically, under H,, TC,-N(0, V,) distributed as a N(0, 1) and therefore NC, = TC,/{ I$} * is asymptotically variate,’ where v0 is a consistent estimator of V, under H,. If NC, is significantly different from zero, then H, is rejected; otherwise H, is not rejected. To test H,, the roles of Ho and H, are reversed and NC, is similarly constructed. Hence, the two tests yield four possible conclusions.’ ‘See Fisher and McAleer (1979) and Dastoor (1981) for the correct interpretation of the Cox procedure; and Pesaran (1974) and Pesaran and Deaton (1978) for an application of the Cox test to linear and nonlinear non-nested regression models, respectively.
N.K. Dastoor,
An alternative the p.d.f.,
method,
Testing non-nested hypotheses
suggested
215
by Cox (1961, p. 110) is to test for 1 in
h(Y,2,go> cd = {f(y, a,)>’-“{dy, ~,))A/~{f(y, %,,}’-“(g(y, cc,)~” dy, where i=O (1) corresponds this suggestion and obtained,
to Ho {Hi} above. Atkinson under Ho, the statistic
(1970) investigated
which is asymptotically distributed as a N(0, I’,) variate. Therefore, under Ho, NA, = TAO/{ vo>+ IS . asymptotically N(0, 1). If N& is significantly different from zero, then Ho is rejected; otherwise Ho is not rejected. To test H,, NA, is similarly constructed and, as in the Cox procedure, four conclusions are possible. An inequality between the Cox and Atkinson statistics can be easily noted. Since both TC, and TAO have the same asymptotic variance under Ho, only the numerators of NC, and NA, need to be considered. Now, _I!,,(&,) and L,(&,,) are evaluations of the same function, L,(cc,), but at different points in the domain, i.e., at points s1 and &io, respectively. As L,($,) is the maximized value of L,(a,), it follows that L,(&,)~L,(&,,). Hence, ‘from (1) and (2), NC0 5 NA, irrespective of the particular models considered by Ho and H,.2 In practice, E, is replaced by plim, in both statistics and this does not alter the inequality. Similarly, it is seen that NC, 5 NA,. These inequalities imply that with identical data sets the two approaches could yield conflicting results for a given level of significance. Atkinson (1970, p. 335) justifies the use of Glo by claiming that both statistics will be biased, but the bias of TAO will be less. Breusch and Pagan (1980) have argued that both the Cox and Atkinson statistics can be viewed as Lagrange Multiplier (LM) tests on h(y, A, uo, a,), and it therefore, seems to them logical to adopt L,(Fx,,), instead of L,(&,), ‘from an LM viewpoint as the score is then unbiased’.3 Subsequently, Pesaran (1981) has shown that the analysis of Breusch and Pagan (1980) is valid only if c~i is known - an unacceptable assumption. Nonetheless, the Atkinson statistic is still valid, but cannot be interpreted as an LM test arising from the comprehensive model h(y,&cc,,a,). Therefore, the choice between the Cox and Atkinson tests must be made on other grounds. Atkinson (1970, p. 324) also suggested the linear combination (1 - i.)f(y, ao) + 1g(y, a,), which was studied in greater detail by Quandt (1974); and Pesaran (1982b) has mentioned a CES type mixture which contains both the ‘Dastoor (1978) and Fisher and McAleer (1981) obtained linear and nonlinear regression models, respectively. ‘Breusch and Pagan (1980, p. 249).
these
inequalities
for competing
N.K. Dastoor,
216
Testing non-nested hypotheses
exponential and linear combinations as special cases. Of course, the formulation of general models need not be restricted to the p.d.f.‘s of the random variable I: Suppose a researcher wishes to test the non-nested nonlinear regression models ~o:Y=f(XP)+~o,
~o-w, dZ”)>
N,:.?J=g(Z,y)+s,,
%-NO,~:~"),
and
where y is an (n x 1) vector of observations on the dependent variable;f{g} is either a linear or nonlinear function of X {Z}, a matrix of predetermined variables, and /I {y}, the (k, x 1) {(k, x l)} vector of parameters; and ~~ {Ed} is an (n x 1) vector of disturbance terms. To test Ho, Davidson and MacKinnon (1981) proposed testing for p =0 in the regression Y=
Cl- !Mx, B)+ Kdz, i) + -%
(3)
where g(Z, j) is the ‘(n x 1) vector of predictions from Hr. They showed that, under Ho, the t statistic for p=O from regression (3) will be asymptotically distributed as a N(0, 1) variate; and also outlined how their procedure could be generalized to cater for more than two non-nested regression models. Since plim,,,,, Jo= yo, Fisher and McAleer (1981) suggested testing for p = 0 in the regression Y=
(I- P).f(x,b) + MLg(z, 30)+ E.
(4)
Under Ho, the tests of Davidson and MacKinnon (1981) and Fisher and McAleer (1981) are asymptotically equivalent. But McAleer (1981a) has shown that if Ho is a linear regression, then the t statistic from regression (4) will be distributed as a t(n - k, - 1) variate under Ho and is therefore a small sample test. McAleer (1981b) has extended regression (4) to test for Ho against rnz 1 non-nested regression models. If Ho is a linear regression then, under Ho, the conventional F statistic is distributed as an F(m, II- k, - m) variate, therefore resulting in a small sample test. As above, an exponential (and CES type) combination of the deterministic components of the regression models could also be considered. For competing linear regression models, Davidson and MacKinnon (1980) conjecture that the various combinations yield similar results, but recommend the linear combination for its simplicity. Therefore, a choice between the numerous statistics is a topic for further research.
N.K. Dastoor, Testing non-nested hypotheses
3. Consistency
217
of the Atkinson statistic
Pereira (1977b) showed that in some cases it is possible for the Atkinson test to be inconsistent. His arguments are based on Cox’s own interpretation of the statistic. Since, under H,, plim,,,,, (X,/n} is strictly negative, Cox (1962, p. 407) suggested that if TC, is significantly negative then there is ‘evidence of departure in the direction of H,'.Pereira (1977b, p. 110) also used the definition that ‘a test of a hypothesis H, against a class of alternatives H, is said to be consistent if, when any member of HI holds, the probability of rejecting H, tends to one as the sample size tends to infinity’. Under H,, Pereira (1977b, p. 110) showed that for some values of {T&/n} could be non-negative and therefore (using both the a1 plim,,,+, definition above and Cox’s interpretation of the statistic) concluded that the Atkinson test is not consistent for these values of a,. So according to Pereira, the Atkinson test is consistent if it can be shown that { T’,/n} is strictly negative for all values of CI~. This is in fact a plim+, sufficient (but not necessary) condition for the Atkinson test to be consistent. In order to obtain the necessary and sufficient condition, three points have to be noted. Firstly, Pereira (1977b, p. 111) noted that ‘although under the null hypothesis two statistics may be asymptotically equivalent, under the alternative hypothesis they may not be’. Therefore, under H,, there is no reason to expect the Atkinson statistic to behave in the same way as the Cox statistic. Secondly, as argued by Fisher and McAleer (1979), if a researcher is interested in the ‘truth’ of a model (significance testing), then a two-tailed test should be utilized. So if plim,,.,, { TA,/n} is strictly positive, for some values then this also implies rejection of H, with probability one (as of a,, NA, = + co) and so the Atkinson test will still be consistent for plim,,,+, these values of CQ. Thirdly, Dastoor (1981) has shown that a significantly negative value for NC, does not imply that H, is the ‘true’ model. So the null hypothesis can be either not rejected, or rejected with no implication for the alternative. Hence, a two-tailed test must be utilized. Therefore, a necessary and sufficient condition for the Atkinson test to be consistent is that plim,,,,, {T&/n) # 0 for all valuesof c(~. The consistency of the Atkinson test was also examined by Dastoor (1978), for competing linear regression models, and by Fisher and McAleer (1981), for nonlinear regression models. As both papers showed that is strictly negative for all values of c(~, the sufficient plim,,,+, VWn) condition of Pereira is satisfied and so the Atkinson test is consistent for these cases. 4. The R procedure4 In this section, 4The R procedure
a test based is named
in honour
on a comparison
between
of Rusty, an exquisitefelis
two estimators
domesticus.
is
218
proposed for
and its methodology H,:
against
Testing non-nested hypotheses
N.K. Dastoor,
Y is a r.v. with unknown
the non-nested H,:
outlined.
Suppose
a researcher
parameters
c(~E Q,
parameters
CI~ls2,
wishes to test
alternative,
Y is a r.v. with unknown
These are the same hypotheses stated in the introduction, but with the distributional assumptions relaxed. In many cases, a; can be partitioned as the vector (8;,y;), where 8, will be the parameter of interest and y1 treated as a nuisance parameter. To test H,, the R procedure is based on testing the to the zero vector, where 8, is any consistent proximity of Go = P, -a,, estimator of 8, under H,, and gl,, = {plim,,,,, Pl}aO=B,. If all the parameters of H, are of interest then, clearly, &, EL?, --klO. Under H,, ij, would on average be expected to be close to the zero vector, while if H, does not hold then on average do will be different from zero. The proximity of $,, to the zero vector is judged by means of an appropriate significance test. The an R, asymptotic distribution of tiO, under H,, is utilized in constructing statistic. On the basis of the significance test, H, is either rejected or not rejected. Clearly, numerous R, statistics can be constructed depending on the choice of 8,. The R procedure shares the same motivation as the Cox procedure. ‘We test whether H, {H,} can be rejected by considering the actual performance of H, {H,) relative to the expected performance of H, {H0}‘.5 As a criterion of performance, Cox used the likelihood functions whereas the R procedure directly considers the parameters of interest. Tests based on the difference between estimators have been considered, in different contexts, by Hausman (1978) and White (1980a, b).6 Cox (1961, section 10) investigated conditions under which TC, was equivalent to testing whether a certain asymptotically linear combination of i1 -k10 differed significantly from zero, but did not suggest the test outlined above. To test H,, the roles of H, and H, are reversed and an R, statistic constructed from the asymptotic distribution of Q1 =e,-g,,. If all the parameters of H, are of interest then e1 E&~-&,~. On the basis of RI, HI is either rejected or not rejected. Combining the results from R, and R,, gives the four possihle conclusions demonstrated in table 1.
5. Some R statistics for competing linear regression models Some R statistics are now derived, for competing linear regression models, and their computation and asymptotic power characteristics examined. ‘Fisher and McAleer (1979, p. 147). ‘On completion of this paper, the author proposed the R procedure.
learnt
of a paper
by Deaton
(1980) which has also
N.K. Dastoor,
219
Testing non-nested hypotheses Table 1
Outcomes
and conclusions
of the R procedure.
R, test outcomes
Suppose
against
R, test outcomes
Do not reject H$:E,[FjJ=O
Reject H$‘: Eo[fo]
Do not reject H$:E,[tj,]=O
Reject neither H, nor H,
Do not reject H,
Reject H;l:E,[ij,]=O
Do not reject H,
Reject both H, and H,
a researcher
=0
wishes to test for
the alternative
where y is an (n x 1) vector of observations on the dependent variable; X {Z} is an (n x k,) {(n x k,)} matrix of non-stochastic explanatory variables; b {y} is a (k, x 1) {(k, x l)} vector of unknown regression coefficients; and q, {si} is an (n x 1) vector of disturbance terms. It is assumed that: all the columns of X cannot be obtained as a linear combination of the columns of 2, and vice versa; the limits of XIX/n, Z’Z/n and Z’X/n exist with the first two positive definite. Hence, plim,,,,, K~X’.Q =0 and plim,,,,, nPIZ’.sO=O. Consider the derivation of an R, statistic. Since 6: is a function of 3 and &fO is a function of iO, an R, statistic can be based on
to-?-io,
(5)
9 = (ZZ) - ‘z’y
(6)
where
is the OLS estimator
of y under
H,,
= (2’2) - ‘Z’X(XX)
- ix’y,
and $0 =(j?, &-@=(y’X(XX)-‘, is an estimate
of crb=(/I’, 0;) under
nP(y-X&(y-X4) H,.
(7)
N.K. Dastoor, Testing non-nested hypotheses
22G
Substituting
(6) and (7) into (5) gives
‘8)
rio = QY, where Q =(Z’Z)-‘Z’M,
and
M, =Z,-X(X’X)-lX’.
Assuming the elements of Z’M, are uniformly bounded and the elements of E,, independently and identically distributed, the multivariate central limit theorem can be applied to (8) to show that asymptotically under HCI J n 90 - N(O, @), where CE lim nQQ’. n+cc
(9)
Note that if E,,- N(O,oil,), then central limit theorem. Typically, ,FYwill be singular dependent; e.g., X and Z could to both hypotheses. In this case,
the above result follows without
applying
as the columns of X and Z will be linearly contain some explanatory variables common define
&,=GZ(&W-(,,‘&,), where C- is a g-inverse Substituting
(10) of C.
both (8) and the sample equivalent
of (9), into (lo), gives (11)
K, = c, “y’Q’(QQ’) -QY. It can be shown, Ro-x2(k,
Pollock
the
(1979, p. 320), that asymptotically
under
H,
-P)>
where p E k, + k, - rank(W),
W = [X i Z] and rank (Q) = k, - p.
Since the matrix Q’(QQ’))Q is invariant for any choice of g-inverse, Rao (1973, p. 26), R, given by (11) is also invariant for any choice of g-inverse.’ If the columns of X and Z are linearly independent then C- in (10) is replaced by C- ’ in which~ case, asymptotically under Ho, R, - X2(k,) as p = 0. In practice, C$ in (11) is replaced by a consistent estimator under H, and the sample statistic evaluated. If the sample statistic is greater than an appropriate critical value, then H, is rejected; otherwise H, is not rejected. ‘The author
is grateful
to an anonymous
referee for suggesting
the above exposition.
N.K. Dastoor,
Testing non-nested
hypotheses
221
Consider the computation of R,. Let ZE [XA i Z] where A is a known (k, x p) matrix; XA contains the p columns of Z that are linearly dependent on the columns of X; and Z contains the (k, -p) columns of Z that are linearly independent of the columns of X. Then, noting that MOZ = [O j M,.2?J, a g-inverse of QQ’ is given by
(QQ') - =
(12)
V-3
where the centre matrix is conformably Substituting (12) into (11) yields
--
R, = 0; *y'M()z(zM,z)Consider two commonly by c?;=n-l(y-X&‘(y-Xfi, statistic is
partitioned.
l.zMoy.
(13)
employed estimates of ci. Firstly, if 0: is estimated th e restricted estimate, then one form of the
Rk”=&;*y’MOZ(Z - -, M,z))-‘Z’M,y, which is also the LM statistic regression
for testing
the null hypothesis
y=xj?+Zd+&. Secondly,
Hi:6 = 0 in the
(14)
let r~i be estimated
by
an unrestricted estimate, where p and 8 are the OLS estimates of /? and 6, respectively, from regression (14). Then from (13), another form of the statistic is
n(k, - PP’, R’=(n-k,-k,
+p)’
where F, is the usual numerical F statistic to test for Hi:6 =0 in (14).8 Hence Ry is the Wald statistic for testing Hd,:6 =0 in regression (14). So far the R, statistics were derived without distributional assumptions on q,. But, if .q, -N(O, oil,,) then from (15) it follows that, under H,, Rr is ‘The asymptotically
equivalent
form R,=(k,
- p)F, could also be used.
222
N.K. Dastoor,
Testing non-nested hypotheses
to the usual F test of Hd,:6 =0 in (14). In this case,
equivalent F
0
Jn-ko-k,+d n(k,-p)
Rw
’
is distributed under Ho as an F(k, -p, n- k, - k, +p) variate, and so the R procedure results in a small sample test. The asymptotic power characteristics of RbM and RT, under H,, are now briefly examined. Let ylol~pliml,~,,~o#O, then it can be shown that asymptotically under H,
Using (lo), the sample value of RbM can be rewritten RfjM= 8; ‘(&
ijo)‘(nQQ’)-(6
as
ij,).
Since lim (nQQ’)- = Cn-rcc asymptotically
under
and
plim, 6; = c:, 2 al, n+m
H,,
RbM - a:cr,lz~2(k1 -p, v’), i.e., under x2 variate
(16)
distributed as CJ~C,; times a non-central H,, RbM IS . asymptotically with k, -p degrees of freedom and non-centrality parameter
v2=(T;2(~rol)lc-(~ro1). Since, under Similarly,
(17)
H,, v2 = O(n) the test is consistent.’ value of Rr can be rewritten
using (lo), the sample
Rr = E-‘(&
fio)‘(nQQ’)-(&
as
ijo).
Since lim (nQQ’)- = C II’30 asymptotically
under
and
plim, C2 = o:, n-tm
H, ,
R?‘-x2(k,-P,v’),
(18)
where v2 is as defined in (17). 9Cf. Kendall
and Stuart
(1979,
pp. 247-249).
N.K.
223
Dastoor, Testing non-nested hypotheses
Let c be the critical value (at the c1 level of significance) of the ~‘(k, -p) distribution and let F(x) be the distribution function of x under H,. Then under H, the asymptotic power of RbM is given by
PLM=
7c dF{ Rb”},
and using (16) PLM= 7 dF{X’(k, -p, v”)}, a where a=co;*ai,. Similarly,
using (18) the asymptotic
power of R$’ under
H, is given by
Pw = 7 dF{X2(k, --p, v”)}. Since
this implies
azc,
and therefore
PLM-I PW with Hence, asymptotically power of Rb”.
under
H,,
lim PLM= lim Pw= l.‘O II+30 n-rm
under
H,, the power
of Rr is at least as great as the
On the other hand, since RbM and RT are equivalent to the usual LM and Wald statistics, respectively, the inequalities first obtained by Savin (1976) [later extended by Berndt and Savin (1977) and Breusch (1979)] can be used to note that under Ho RLM 0 -I RW 0, which implies the tests have different sizes. Therefore, it appears that asymptotically RbM and RT exhibit the usual trade-off between the type I and type II errors. The R, test above considered y as the parameter of interest. If CJ: is considered as the parameter of interest, then an R, test could be based on “‘For
similar power comparisons,
but in a different
context,
see Wu (1973, pp. 741-742)
N.K. Dastoor, Testing non-nested hypotheses
224
the asymptotic distribution of 6:--6:,, under H,, provided lim,,, n-‘X’Z exists and is non-zero.” Then the resulting R, statistic will be Rb =(NL#, which under H, is asymptotically distributed as a x’(l) variate, where NL, is the statistic proposed by Fisher and McAleer (1981). Therefore, the NL, statistic can be viewed as an R test. Recently, Mizon and Richard (1982) have shown that if the R test is based A . statistic is equivalent to the on c~~-c(r~ then, as expected, the resulting conventional F test for H$6=0 in regression (14). They also argue that this F, statistic actually tests the null hypothesis H,, while the other non-nested procedures test for a wider class of models than H,. This would tend to favour the use of Rb”, Rr and F,, but Pesaran (1982a) has shown that the Cox (1961, 1962) and Davidson and MacKinnon (1981) procedures could be more powerful than F, under certain local alternatives. Therefore, a choice between the numerous statistics demands further research. Finally, it should be noted that to test for H, the roles of H, and H, are reversed and a similar analysis carried out.
6. Relationship
of r&,with other tests for competing linear regression models
For the hypotheses considered in section 5, this section demonstrates that other tests of 23, can be viewed as particular comparisons of i and jjO. For the purposes of comparison, it is also assumed that, under H,, .sON N(0, d;Z,), and lim, _ a, n - ‘X’Z exists and is non-zero. To test H,, the Cox statistic is based on [see Pesaran (1974, p. 157)] the asymptotic distribution of n-+X,
=(Jn/2)log{r?~/6-fo).
(19)
Since n8; = y’y - i3,X’X?, n&f, = nr3; + FX!{ I” - Z(ZZ)
- ‘Z}XB,
(20)
and
‘Z’X?,
Z& = Z(zlZ) eq. (20) can be rewritten
as
nc?f, = y’y - f&ZZfo.
Substituting “If lim,,, cq,] = 0.
(21)
(21) into (19), and noting n- ‘X’Z=O
then
the
test
that 6: = y’y -$YZ’Zf, gives
is ‘undefined
as, asymptotically
under
H,,
I/b[BT-
N.K. Dastoor, Testing non-nested hypotheses
225
n-WC‘, = h(F) - h(j,), where h(y) = (Jn/2)
log(Y’Y - r’ZZy>.
Therefore, under H,, the Cox test can be viewed as a particular type of comparison between 9 and $jO. The Atkinson statistic, to test Ho, is based on the asymptotic distribution of n-*T’& which can be written as [cf. Fisher and McAleer (1981, p. 112)] n ~~TA,=(2~~-:,)-‘I(y-Z~,)‘(y-Z~,)-n~:,}.
Substituting gives
(21) into the numerator
of (22) and noting
that j&Z’Y=FbZ’Zf,
where the (k, x 1) vector S,= -8.;t(n-‘Z’Z)?, and Go is as defined in (5). procedure, for H,, tests whether a Since plim,, n-r m S, exists, the Atkinson certain asymptotically linear combination of ,,&fie is significantly different from zero. The J test for H,, proposed by Davidson and MacKinnon (1981) tests for p=O in regression (3) which, for competing linear regression models, reduces to y=(l-p)xp+pzf+E.
(23)
The J test is then based on the asymptotic estimator of p from (23). It is easily seen that
distribution
of bJ, the OLS
~~,=~{~‘Z’MoZ~t)-‘~‘Z’M,y.
(24)
Z’M,y=Z’Z&,
(25)
Since
eq. (24) can be rewritten
as
where the (k, x 1) vector S,=Z’Z~{f’Z’M,Zjj} As plim,,,,, JE-C
S, exists, the J procedure
_ ‘. for H, also tests whether
a certain
226
N.K. Dastoor,
Testing non-nested hypotheses
asymptotically linear combination of &&, is significantly different from zero. The JA test for H,, proposed by Fisher and McAleer (1981) and McAleer (198 1a), tests for P = 0 in regression (4) which, for competing linear regression models, reduces to
y=(l -p)X/l+pZj$+E.
(26)
The J.4 test is then based on the asymptotic estimator of p from (26). It is easily seen that
and substituting
distribution
of bJA, the OLS
(25) into (27) yields
where the (k, x 1) vector S,,=Z’Zy^,ly^bZ’MoZy^o}-‘. Since plim,,,,,
S,,
(=plim,,.,,
S,) exists, the JA procedure
tests whether a certain asymptotically significantly different from zero.
linear
combination
for H, also of
$4,
is
7. Conclusion This paper first discussed the Cox and Atkinson procedures for testing two non-nested hypotheses, and later derived the necessary and sufficient condition for the Atkinson test to be consistent. A different approach, the R procedure, was also suggested. For competing linear regression models, two R statistics were derived in the absence of distributional assumptions. If normality is also assumed, then an R test is equivalent to the conventional F test therefore resulting in a small sample test. The rest of the paper illustrated the various test statistics, their properties, and relationships for competing linear regression models. It is interesting to note that all the tests proposed for two non-nested hypotheses utilize two tests of nested hypotheses. For example, to test H, against H, of section 5: the Cox procedure tests the null hypotheses Hg: E,[TC,] = 0 against the alternative Hy: E[TC,] #O; the Atkinson procedure tests H$: A=0 against Hf: A#0 in the comprehensive model h(y, 2, Q, a,); the Davidson and MacKinnon (1981) and Fisher and McAleer (1981) procedures test H$:p=O against H'f :p#O in regressions (23) and (26), respectively; and the R procedure tests H~:E,[ijo] =0 against HF: E[i),J #O.
N.K. Dastoor, Testing non-nested
hypotheses
227
To test H, against H,, the roles of Ho and H, are reversed and the second set of nested tests carried out. Recently, Pesaran (1982a) and Ericsson (1981) have investigated the behaviour of non-nested tests under certain local alternatives, and McAleer et al. (1982) have examined the consistency of these tests when the stated alternative is subject to misspecifkation. But, clearly, further research is still required to enable a choice to be made from amongst the plethora of available statistics. References Amemiya, T., 1980, Selection of regressors, International Economic Review 21, 331-354. Atkinson. A.C.. 1970. A method for discriminating between models, Journal of the Royal Statistical Society B 32, 3233345. Berndt, E.R. and N.E. Savin, 1977, Conflict among criteria for testing hypotheses in the multivariate linear regression model, Econometrica 45, 1263-1277. Breusch, T.S., 1979, Conflict among criteria for testing hypotheses: Extensions and comments, Econometrica 47, 2033207. Breusch, T.S. and A.R. Pagan, 1980, The Lagrange multiplier test and its applications to model specification in econometrics, Review of Economic Studies 47, 239-253. Cox, D.R., 1961, Tests of separate families of hypotheses, in: Proceedings of fourth Berkeley symposium on mathematical statistics and probability, Vol. 1 (University of California Press, Berkeley, CA) 1055123. Cox, D.R., 1962, Further results on tests of separate families of hypotheses, Journal of the Royal Statistical Society B 24, 406424. Dastoor, N.K., 1978, Non-nested hypothesis testing: The A.C. Atkinson approach to linear regression models, Department of Economics discussion paper no. 117 (University of Essex, Colchester). Dastoor, N.K., 1981, A note on the interpretation of the Cox procedure for non-nested hypotheses, Economics Letters 8, 113-l 19. Davidson, R. and J.G. MacKinnon, 1980, On a simple procedure for testing non-nested regression models, Economics Letters 5, 4548. Davidson, R. and J.G. MacKinnon, 1981, Several tests for model specification in the presence of alternative hypotheses, Econometrica 49, 781-793. or does the consumption function exist?, Deaton, A., 1980, Model selection procedures, Department of Economics discussion paper no. 81/92 (University of Bristol, Bristol). Ericsson, N.R., 1981, Asymptotic properties of statistics for testing non-nested hypotheses, Review of Economic Studies, forthcoming. Fisher, G.R. and M. McAleer, 1979, On the interpretation of the Cox test in econometrics, Economics Letters 4, 1455150. Fisher, G.R. and M. McAleer, 1981, Alternative procedures and associated tests of significance for non-nested hypotheses, Journal of Econometrics 16, 103-l 19. Hausman, J.A., 1978, Specification tests in econometrics, Econometrica 46, 1251-1271. Kendall, M. and A. Stuart, 1979, The advanced theory of statistics, Vol. 2: Inference and relationship, 4th ed. (Charles Grifftn, London). McAleer, M., 198la, A small sample test for non-nested regression models, Economics Letters 7, 335-338. McAleer, M., 198lb, Exact tests of a model against non-nested alternatives, Biometrika, forthcoming. 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. Mizon, G.E. and J.-F. Richard, 1982, The encompassing principle and its application to nonnested hypotheses, Paper presented at the S.S.R.C. Econometric Study Group, Model specification and testing conference, University of Warwick, March.
228
N.K. Dastoor,
Testing non-nested hypotheses
Pereira, B. de B., 1977a, Discriminating among separate models: A bibliography, International Statistical Review 45, 1633172. Pereira, B. de B., 1977b, A note on the consistency and on the finite sample comparisons of some tests of separate families of hypotheses, Biometrika 64, 109-l 13. Pesaran, M.H., 1974, On the general problem of model selection, Review of Economic Studies 41, 153-171. Pesaran, M.H., 1981, Pitfalls of testing non-nested hypotheses by the Lagrange multiplier method, Journal of Econometrics 17, 3233331. Pesaran, M.H., 1982a, Comparison of local power of alternative tests of non-nested regression models, Econometrica 50, 1287-1305. Pesaran, M.H., 1982b, On the comprehensive method of testing non-nested regression models, Journal of Econometrics 18, 2633274. Pesaran, M.H. and AS. Deaton,. 1978, Testing non-nested nonlinear regression models, Econometrica 46, 6777694. Pollock, D.S.G., 1979, The algebra of econometrics (Wiley, New York). Quandt, R.E., 1974, A comparison of methods for testing nonnested hypotheses, Review of Economics and Statistics 56, 92-99. Rao, CR., 1973, Linear statistical inference and its applications, 2nd ed. (Wiley, New York). Savin, N.E., 1976, Conflict among testing procedures in linear regression model with autoregressive disturbances, Econometrica 44, 1303-1315. White, H., 198Oa, Using least squares to approximate unknown regression functions, International Economic Review 21, 1499170. White, H., 1980b, Nonlinear regression on cross-section data, Econometrica 48, 721-746. Wu, De-Min, 1973, Alternative tests of independence between stochastic regressors and disturbances, Econometrica 41, 733-750.
of Econometrics
SOME
21 (1983) 213-228.
ASPECTS
OF TESTING HYPOTHESES
Naorayex University
Received
North-Holland
Publishing
NON-NESTED
K. DASTOOR*
of Liverpool, Liverpool L69 3BX,
March
Company
1979, final version received
UK
May 1982
This paper examines some aspects of testing non-nested hypotheses. An inequality between the Cox and Atkinson statistics is noted, and the necessary and sufficient condition for the Atkinson test to be consistent is derived. A new test procedure is also outlined. The rest of the paper illustrates the various test statistics, their properties, and relationships for competing linear regression models.
1. Introduction The last decade or so has witnessed numerous papers on tests of separate families of hypotheses, see e.g. the bibliography by Pereira (1977a). The same topic has been studied by econometricians under the heading of ‘non-nested hypotheses’. The problem can be viewed as an examination of the probability density function (p.d.f.) of an (n x 1) vector random variable (T.v.) Y with respect to the hypotheses H,: Y has p.d.f. f(y, a,),
MOE Qo,
H,: Y has p.d.f. g(y,a,),
a1 EQl,
and
where the parameter spaces Sz, and Q, are such that Q, n Q, #L’, and 52, n 52,#ill; and cl0 and a, are the parameters of the p.d.f.‘s f and g, respectively. If 52, n 52,=52, (or 52, n 8, =5),), then the hypotheses are said to be nested. A researcher may wish to choose a particular model from amongst the set available (discrimination), in which case the procedures discussed by Amemiya (1980) could be used. Alternatively, he may wish to test the ‘truth’ *An earlier version of this paper was presented at the European Meeting of the Econometric Society, Amsterdam, September 1981. The author is grateful to M. Chatterji, K. Holden, M. McAleer, A.R. Nobay, J. Richmond, N.E. Savin, M.R. Wickens and anonymous referees for helpful comments on earlier versions of the paper, and to N.R. Ericsson for pointing out an error.
0304-4076/83/0000-0000/$03.00
0
1983 North-Holland
214
N.K. Dastoor,
Testing non-nested hypotheses
of a particular model and this can be accomplished by using an appropriate significance test. This paper is concerned with significance testing as, presumably, an economic researcher would be more interested in the ‘truth’ of a particular model than in choosing from among a given set of models. Since to each hypothesis there corresponds a unique model, the terms ‘hypothesis’ and ‘model’ will be used synonymously. To test the ‘truth’ of H,, against the non-nested alternative Hi, the relevant statistic is calculated assuming H, is true. Then, on the basis of the observed data, H, is either rejected or not rejected. The roles of H, and H, are then reversed and the procedure repeated. So the two tests yield four possible outcomes. Two points should be noted. Firstly, for each hypothesis assumed in turn, the level of significance is a subjective decision. Secondly, the optimal level of significance for such a test procedure requires further research. The next section outlines the Cox and Atkinson procedures and notes an inequality between the two statistics. In section 3, the necessary and sufficient condition for the Atkinson test to be consistent is derived. The R procedure is proposed in the fourth section, and in section 5 some R statistics are derived for competing linear regression models. For the same models, the relationship between an R test and other non-nested tests is examined in section 6. Finally, section 7 contains the conclusion.
2. Some proposed statistics Suppose a researcher wishes to test H, and H, as stated in the introduction. Cox (1961, 1962) first considered this problem and, to test Ho, suggested the statistic
where L,(a,) {L,(a,)} is the log-likelihood function under H, {Hi}; I&, {gl} is the maximum likelihood estimate of CQ {ai} under H, {Hi}; &lo= &r; and both the expectation and probability limit are carried out plim+ m under H,. Cox (1961, 1962) showed that asymptotically, under H,, TC,-N(0, V,) distributed as a N(0, 1) and therefore NC, = TC,/{ I$} * is asymptotically variate,’ where v0 is a consistent estimator of V, under H,. If NC, is significantly different from zero, then H, is rejected; otherwise H, is not rejected. To test H,, the roles of Ho and H, are reversed and NC, is similarly constructed. Hence, the two tests yield four possible conclusions.’ ‘See Fisher and McAleer (1979) and Dastoor (1981) for the correct interpretation of the Cox procedure; and Pesaran (1974) and Pesaran and Deaton (1978) for an application of the Cox test to linear and nonlinear non-nested regression models, respectively.
N.K. Dastoor,
An alternative the p.d.f.,
method,
Testing non-nested hypotheses
suggested
215
by Cox (1961, p. 110) is to test for 1 in
h(Y,2,go> cd = {f(y, a,)>’-“{dy, ~,))A/~{f(y, %,,}’-“(g(y, cc,)~” dy, where i=O (1) corresponds this suggestion and obtained,
to Ho {Hi} above. Atkinson under Ho, the statistic
(1970) investigated
which is asymptotically distributed as a N(0, I’,) variate. Therefore, under Ho, NA, = TAO/{ vo>+ IS . asymptotically N(0, 1). If N& is significantly different from zero, then Ho is rejected; otherwise Ho is not rejected. To test H,, NA, is similarly constructed and, as in the Cox procedure, four conclusions are possible. An inequality between the Cox and Atkinson statistics can be easily noted. Since both TC, and TAO have the same asymptotic variance under Ho, only the numerators of NC, and NA, need to be considered. Now, _I!,,(&,) and L,(&,,) are evaluations of the same function, L,(cc,), but at different points in the domain, i.e., at points s1 and &io, respectively. As L,($,) is the maximized value of L,(a,), it follows that L,(&,)~L,(&,,). Hence, ‘from (1) and (2), NC0 5 NA, irrespective of the particular models considered by Ho and H,.2 In practice, E, is replaced by plim, in both statistics and this does not alter the inequality. Similarly, it is seen that NC, 5 NA,. These inequalities imply that with identical data sets the two approaches could yield conflicting results for a given level of significance. Atkinson (1970, p. 335) justifies the use of Glo by claiming that both statistics will be biased, but the bias of TAO will be less. Breusch and Pagan (1980) have argued that both the Cox and Atkinson statistics can be viewed as Lagrange Multiplier (LM) tests on h(y, A, uo, a,), and it therefore, seems to them logical to adopt L,(Fx,,), instead of L,(&,), ‘from an LM viewpoint as the score is then unbiased’.3 Subsequently, Pesaran (1981) has shown that the analysis of Breusch and Pagan (1980) is valid only if c~i is known - an unacceptable assumption. Nonetheless, the Atkinson statistic is still valid, but cannot be interpreted as an LM test arising from the comprehensive model h(y,&cc,,a,). Therefore, the choice between the Cox and Atkinson tests must be made on other grounds. Atkinson (1970, p. 324) also suggested the linear combination (1 - i.)f(y, ao) + 1g(y, a,), which was studied in greater detail by Quandt (1974); and Pesaran (1982b) has mentioned a CES type mixture which contains both the ‘Dastoor (1978) and Fisher and McAleer (1981) obtained linear and nonlinear regression models, respectively. ‘Breusch and Pagan (1980, p. 249).
these
inequalities
for competing
N.K. Dastoor,
216
Testing non-nested hypotheses
exponential and linear combinations as special cases. Of course, the formulation of general models need not be restricted to the p.d.f.‘s of the random variable I: Suppose a researcher wishes to test the non-nested nonlinear regression models ~o:Y=f(XP)+~o,
~o-w, dZ”)>
N,:.?J=g(Z,y)+s,,
%-NO,~:~"),
and
where y is an (n x 1) vector of observations on the dependent variable;f{g} is either a linear or nonlinear function of X {Z}, a matrix of predetermined variables, and /I {y}, the (k, x 1) {(k, x l)} vector of parameters; and ~~ {Ed} is an (n x 1) vector of disturbance terms. To test Ho, Davidson and MacKinnon (1981) proposed testing for p =0 in the regression Y=
Cl- !Mx, B)+ Kdz, i) + -%
(3)
where g(Z, j) is the ‘(n x 1) vector of predictions from Hr. They showed that, under Ho, the t statistic for p=O from regression (3) will be asymptotically distributed as a N(0, 1) variate; and also outlined how their procedure could be generalized to cater for more than two non-nested regression models. Since plim,,,,, Jo= yo, Fisher and McAleer (1981) suggested testing for p = 0 in the regression Y=
(I- P).f(x,b) + MLg(z, 30)+ E.
(4)
Under Ho, the tests of Davidson and MacKinnon (1981) and Fisher and McAleer (1981) are asymptotically equivalent. But McAleer (1981a) has shown that if Ho is a linear regression, then the t statistic from regression (4) will be distributed as a t(n - k, - 1) variate under Ho and is therefore a small sample test. McAleer (1981b) has extended regression (4) to test for Ho against rnz 1 non-nested regression models. If Ho is a linear regression then, under Ho, the conventional F statistic is distributed as an F(m, II- k, - m) variate, therefore resulting in a small sample test. As above, an exponential (and CES type) combination of the deterministic components of the regression models could also be considered. For competing linear regression models, Davidson and MacKinnon (1980) conjecture that the various combinations yield similar results, but recommend the linear combination for its simplicity. Therefore, a choice between the numerous statistics is a topic for further research.
N.K. Dastoor, Testing non-nested hypotheses
3. Consistency
217
of the Atkinson statistic
Pereira (1977b) showed that in some cases it is possible for the Atkinson test to be inconsistent. His arguments are based on Cox’s own interpretation of the statistic. Since, under H,, plim,,,,, (X,/n} is strictly negative, Cox (1962, p. 407) suggested that if TC, is significantly negative then there is ‘evidence of departure in the direction of H,'.Pereira (1977b, p. 110) also used the definition that ‘a test of a hypothesis H, against a class of alternatives H, is said to be consistent if, when any member of HI holds, the probability of rejecting H, tends to one as the sample size tends to infinity’. Under H,, Pereira (1977b, p. 110) showed that for some values of {T&/n} could be non-negative and therefore (using both the a1 plim,,,+, definition above and Cox’s interpretation of the statistic) concluded that the Atkinson test is not consistent for these values of a,. So according to Pereira, the Atkinson test is consistent if it can be shown that { T’,/n} is strictly negative for all values of CI~. This is in fact a plim+, sufficient (but not necessary) condition for the Atkinson test to be consistent. In order to obtain the necessary and sufficient condition, three points have to be noted. Firstly, Pereira (1977b, p. 111) noted that ‘although under the null hypothesis two statistics may be asymptotically equivalent, under the alternative hypothesis they may not be’. Therefore, under H,, there is no reason to expect the Atkinson statistic to behave in the same way as the Cox statistic. Secondly, as argued by Fisher and McAleer (1979), if a researcher is interested in the ‘truth’ of a model (significance testing), then a two-tailed test should be utilized. So if plim,,.,, { TA,/n} is strictly positive, for some values then this also implies rejection of H, with probability one (as of a,, NA, = + co) and so the Atkinson test will still be consistent for plim,,,+, these values of CQ. Thirdly, Dastoor (1981) has shown that a significantly negative value for NC, does not imply that H, is the ‘true’ model. So the null hypothesis can be either not rejected, or rejected with no implication for the alternative. Hence, a two-tailed test must be utilized. Therefore, a necessary and sufficient condition for the Atkinson test to be consistent is that plim,,,,, {T&/n) # 0 for all valuesof c(~. The consistency of the Atkinson test was also examined by Dastoor (1978), for competing linear regression models, and by Fisher and McAleer (1981), for nonlinear regression models. As both papers showed that is strictly negative for all values of c(~, the sufficient plim,,,+, VWn) condition of Pereira is satisfied and so the Atkinson test is consistent for these cases. 4. The R procedure4 In this section, 4The R procedure
a test based is named
in honour
on a comparison
between
of Rusty, an exquisitefelis
two estimators
domesticus.
is
218
proposed for
and its methodology H,:
against
Testing non-nested hypotheses
N.K. Dastoor,
Y is a r.v. with unknown
the non-nested H,:
outlined.
Suppose
a researcher
parameters
c(~E Q,
parameters
CI~ls2,
wishes to test
alternative,
Y is a r.v. with unknown
These are the same hypotheses stated in the introduction, but with the distributional assumptions relaxed. In many cases, a; can be partitioned as the vector (8;,y;), where 8, will be the parameter of interest and y1 treated as a nuisance parameter. To test H,, the R procedure is based on testing the to the zero vector, where 8, is any consistent proximity of Go = P, -a,, estimator of 8, under H,, and gl,, = {plim,,,,, Pl}aO=B,. If all the parameters of H, are of interest then, clearly, &, EL?, --klO. Under H,, ij, would on average be expected to be close to the zero vector, while if H, does not hold then on average do will be different from zero. The proximity of $,, to the zero vector is judged by means of an appropriate significance test. The an R, asymptotic distribution of tiO, under H,, is utilized in constructing statistic. On the basis of the significance test, H, is either rejected or not rejected. Clearly, numerous R, statistics can be constructed depending on the choice of 8,. The R procedure shares the same motivation as the Cox procedure. ‘We test whether H, {H,} can be rejected by considering the actual performance of H, {H,) relative to the expected performance of H, {H0}‘.5 As a criterion of performance, Cox used the likelihood functions whereas the R procedure directly considers the parameters of interest. Tests based on the difference between estimators have been considered, in different contexts, by Hausman (1978) and White (1980a, b).6 Cox (1961, section 10) investigated conditions under which TC, was equivalent to testing whether a certain asymptotically linear combination of i1 -k10 differed significantly from zero, but did not suggest the test outlined above. To test H,, the roles of H, and H, are reversed and an R, statistic constructed from the asymptotic distribution of Q1 =e,-g,,. If all the parameters of H, are of interest then e1 E&~-&,~. On the basis of RI, HI is either rejected or not rejected. Combining the results from R, and R,, gives the four possihle conclusions demonstrated in table 1.
5. Some R statistics for competing linear regression models Some R statistics are now derived, for competing linear regression models, and their computation and asymptotic power characteristics examined. ‘Fisher and McAleer (1979, p. 147). ‘On completion of this paper, the author proposed the R procedure.
learnt
of a paper
by Deaton
(1980) which has also
N.K. Dastoor,
219
Testing non-nested hypotheses Table 1
Outcomes
and conclusions
of the R procedure.
R, test outcomes
Suppose
against
R, test outcomes
Do not reject H$:E,[FjJ=O
Reject H$‘: Eo[fo]
Do not reject H$:E,[tj,]=O
Reject neither H, nor H,
Do not reject H,
Reject H;l:E,[ij,]=O
Do not reject H,
Reject both H, and H,
a researcher
=0
wishes to test for
the alternative
where y is an (n x 1) vector of observations on the dependent variable; X {Z} is an (n x k,) {(n x k,)} matrix of non-stochastic explanatory variables; b {y} is a (k, x 1) {(k, x l)} vector of unknown regression coefficients; and q, {si} is an (n x 1) vector of disturbance terms. It is assumed that: all the columns of X cannot be obtained as a linear combination of the columns of 2, and vice versa; the limits of XIX/n, Z’Z/n and Z’X/n exist with the first two positive definite. Hence, plim,,,,, K~X’.Q =0 and plim,,,,, nPIZ’.sO=O. Consider the derivation of an R, statistic. Since 6: is a function of 3 and &fO is a function of iO, an R, statistic can be based on
to-?-io,
(5)
9 = (ZZ) - ‘z’y
(6)
where
is the OLS estimator
of y under
H,,
= (2’2) - ‘Z’X(XX)
- ix’y,
and $0 =(j?, &-@=(y’X(XX)-‘, is an estimate
of crb=(/I’, 0;) under
nP(y-X&(y-X4) H,.
(7)
N.K. Dastoor, Testing non-nested hypotheses
22G
Substituting
(6) and (7) into (5) gives
‘8)
rio = QY, where Q =(Z’Z)-‘Z’M,
and
M, =Z,-X(X’X)-lX’.
Assuming the elements of Z’M, are uniformly bounded and the elements of E,, independently and identically distributed, the multivariate central limit theorem can be applied to (8) to show that asymptotically under HCI J n 90 - N(O, @), where CE lim nQQ’. n+cc
(9)
Note that if E,,- N(O,oil,), then central limit theorem. Typically, ,FYwill be singular dependent; e.g., X and Z could to both hypotheses. In this case,
the above result follows without
applying
as the columns of X and Z will be linearly contain some explanatory variables common define
&,=GZ(&W-(,,‘&,), where C- is a g-inverse Substituting
(10) of C.
both (8) and the sample equivalent
of (9), into (lo), gives (11)
K, = c, “y’Q’(QQ’) -QY. It can be shown, Ro-x2(k,
Pollock
the
(1979, p. 320), that asymptotically
under
H,
-P)>
where p E k, + k, - rank(W),
W = [X i Z] and rank (Q) = k, - p.
Since the matrix Q’(QQ’))Q is invariant for any choice of g-inverse, Rao (1973, p. 26), R, given by (11) is also invariant for any choice of g-inverse.’ If the columns of X and Z are linearly independent then C- in (10) is replaced by C- ’ in which~ case, asymptotically under Ho, R, - X2(k,) as p = 0. In practice, C$ in (11) is replaced by a consistent estimator under H, and the sample statistic evaluated. If the sample statistic is greater than an appropriate critical value, then H, is rejected; otherwise H, is not rejected. ‘The author
is grateful
to an anonymous
referee for suggesting
the above exposition.
N.K. Dastoor,
Testing non-nested
hypotheses
221
Consider the computation of R,. Let ZE [XA i Z] where A is a known (k, x p) matrix; XA contains the p columns of Z that are linearly dependent on the columns of X; and Z contains the (k, -p) columns of Z that are linearly independent of the columns of X. Then, noting that MOZ = [O j M,.2?J, a g-inverse of QQ’ is given by
(QQ') - =
(12)
V-3
where the centre matrix is conformably Substituting (12) into (11) yields
--
R, = 0; *y'M()z(zM,z)Consider two commonly by c?;=n-l(y-X&‘(y-Xfi, statistic is
partitioned.
l.zMoy.
(13)
employed estimates of ci. Firstly, if 0: is estimated th e restricted estimate, then one form of the
Rk”=&;*y’MOZ(Z - -, M,z))-‘Z’M,y, which is also the LM statistic regression
for testing
the null hypothesis
y=xj?+Zd+&. Secondly,
Hi:6 = 0 in the
(14)
let r~i be estimated
by
an unrestricted estimate, where p and 8 are the OLS estimates of /? and 6, respectively, from regression (14). Then from (13), another form of the statistic is
n(k, - PP’, R’=(n-k,-k,
+p)’
where F, is the usual numerical F statistic to test for Hi:6 =0 in (14).8 Hence Ry is the Wald statistic for testing Hd,:6 =0 in regression (14). So far the R, statistics were derived without distributional assumptions on q,. But, if .q, -N(O, oil,,) then from (15) it follows that, under H,, Rr is ‘The asymptotically
equivalent
form R,=(k,
- p)F, could also be used.
222
N.K. Dastoor,
Testing non-nested hypotheses
to the usual F test of Hd,:6 =0 in (14). In this case,
equivalent F
0
Jn-ko-k,+d n(k,-p)
Rw
’
is distributed under Ho as an F(k, -p, n- k, - k, +p) variate, and so the R procedure results in a small sample test. The asymptotic power characteristics of RbM and RT, under H,, are now briefly examined. Let ylol~pliml,~,,~o#O, then it can be shown that asymptotically under H,
Using (lo), the sample value of RbM can be rewritten RfjM= 8; ‘(&
ijo)‘(nQQ’)-(6
as
ij,).
Since lim (nQQ’)- = Cn-rcc asymptotically
under
and
plim, 6; = c:, 2 al, n+m
H,,
RbM - a:cr,lz~2(k1 -p, v’), i.e., under x2 variate
(16)
distributed as CJ~C,; times a non-central H,, RbM IS . asymptotically with k, -p degrees of freedom and non-centrality parameter
v2=(T;2(~rol)lc-(~ro1). Since, under Similarly,
(17)
H,, v2 = O(n) the test is consistent.’ value of Rr can be rewritten
using (lo), the sample
Rr = E-‘(&
fio)‘(nQQ’)-(&
as
ijo).
Since lim (nQQ’)- = C II’30 asymptotically
under
and
plim, C2 = o:, n-tm
H, ,
R?‘-x2(k,-P,v’),
(18)
where v2 is as defined in (17). 9Cf. Kendall
and Stuart
(1979,
pp. 247-249).
N.K.
223
Dastoor, Testing non-nested hypotheses
Let c be the critical value (at the c1 level of significance) of the ~‘(k, -p) distribution and let F(x) be the distribution function of x under H,. Then under H, the asymptotic power of RbM is given by
PLM=
7c dF{ Rb”},
and using (16) PLM= 7 dF{X’(k, -p, v”)}, a where a=co;*ai,. Similarly,
using (18) the asymptotic
power of R$’ under
H, is given by
Pw = 7 dF{X2(k, --p, v”)}. Since
this implies
azc,
and therefore
PLM-I PW with Hence, asymptotically power of Rb”.
under
H,,
lim PLM= lim Pw= l.‘O II+30 n-rm
under
H,, the power
of Rr is at least as great as the
On the other hand, since RbM and RT are equivalent to the usual LM and Wald statistics, respectively, the inequalities first obtained by Savin (1976) [later extended by Berndt and Savin (1977) and Breusch (1979)] can be used to note that under Ho RLM 0 -I RW 0, which implies the tests have different sizes. Therefore, it appears that asymptotically RbM and RT exhibit the usual trade-off between the type I and type II errors. The R, test above considered y as the parameter of interest. If CJ: is considered as the parameter of interest, then an R, test could be based on “‘For
similar power comparisons,
but in a different
context,
see Wu (1973, pp. 741-742)
N.K. Dastoor, Testing non-nested hypotheses
224
the asymptotic distribution of 6:--6:,, under H,, provided lim,,, n-‘X’Z exists and is non-zero.” Then the resulting R, statistic will be Rb =(NL#, which under H, is asymptotically distributed as a x’(l) variate, where NL, is the statistic proposed by Fisher and McAleer (1981). Therefore, the NL, statistic can be viewed as an R test. Recently, Mizon and Richard (1982) have shown that if the R test is based A . statistic is equivalent to the on c~~-c(r~ then, as expected, the resulting conventional F test for H$6=0 in regression (14). They also argue that this F, statistic actually tests the null hypothesis H,, while the other non-nested procedures test for a wider class of models than H,. This would tend to favour the use of Rb”, Rr and F,, but Pesaran (1982a) has shown that the Cox (1961, 1962) and Davidson and MacKinnon (1981) procedures could be more powerful than F, under certain local alternatives. Therefore, a choice between the numerous statistics demands further research. Finally, it should be noted that to test for H, the roles of H, and H, are reversed and a similar analysis carried out.
6. Relationship
of r&,with other tests for competing linear regression models
For the hypotheses considered in section 5, this section demonstrates that other tests of 23, can be viewed as particular comparisons of i and jjO. For the purposes of comparison, it is also assumed that, under H,, .sON N(0, d;Z,), and lim, _ a, n - ‘X’Z exists and is non-zero. To test H,, the Cox statistic is based on [see Pesaran (1974, p. 157)] the asymptotic distribution of n-+X,
=(Jn/2)log{r?~/6-fo).
(19)
Since n8; = y’y - i3,X’X?, n&f, = nr3; + FX!{ I” - Z(ZZ)
- ‘Z}XB,
(20)
and
‘Z’X?,
Z& = Z(zlZ) eq. (20) can be rewritten
as
nc?f, = y’y - f&ZZfo.
Substituting “If lim,,, cq,] = 0.
(21)
(21) into (19), and noting n- ‘X’Z=O
then
the
test
that 6: = y’y -$YZ’Zf, gives
is ‘undefined
as, asymptotically
under
H,,
I/b[BT-
N.K. Dastoor, Testing non-nested hypotheses
225
n-WC‘, = h(F) - h(j,), where h(y) = (Jn/2)
log(Y’Y - r’ZZy>.
Therefore, under H,, the Cox test can be viewed as a particular type of comparison between 9 and $jO. The Atkinson statistic, to test Ho, is based on the asymptotic distribution of n-*T’& which can be written as [cf. Fisher and McAleer (1981, p. 112)] n ~~TA,=(2~~-:,)-‘I(y-Z~,)‘(y-Z~,)-n~:,}.
Substituting gives
(21) into the numerator
of (22) and noting
that j&Z’Y=FbZ’Zf,
where the (k, x 1) vector S,= -8.;t(n-‘Z’Z)?, and Go is as defined in (5). procedure, for H,, tests whether a Since plim,, n-r m S, exists, the Atkinson certain asymptotically linear combination of ,,&fie is significantly different from zero. The J test for H,, proposed by Davidson and MacKinnon (1981) tests for p=O in regression (3) which, for competing linear regression models, reduces to y=(l-p)xp+pzf+E.
(23)
The J test is then based on the asymptotic estimator of p from (23). It is easily seen that
distribution
of bJ, the OLS
~~,=~{~‘Z’MoZ~t)-‘~‘Z’M,y.
(24)
Z’M,y=Z’Z&,
(25)
Since
eq. (24) can be rewritten
as
where the (k, x 1) vector S,=Z’Z~{f’Z’M,Zjj} As plim,,,,, JE-C
S, exists, the J procedure
_ ‘. for H, also tests whether
a certain
226
N.K. Dastoor,
Testing non-nested hypotheses
asymptotically linear combination of &&, is significantly different from zero. The JA test for H,, proposed by Fisher and McAleer (1981) and McAleer (198 1a), tests for P = 0 in regression (4) which, for competing linear regression models, reduces to
y=(l -p)X/l+pZj$+E.
(26)
The J.4 test is then based on the asymptotic estimator of p from (26). It is easily seen that
and substituting
distribution
of bJA, the OLS
(25) into (27) yields
where the (k, x 1) vector S,,=Z’Zy^,ly^bZ’MoZy^o}-‘. Since plim,,,,,
S,,
(=plim,,.,,
S,) exists, the JA procedure
tests whether a certain asymptotically significantly different from zero.
linear
combination
for H, also of
$4,
is
7. Conclusion This paper first discussed the Cox and Atkinson procedures for testing two non-nested hypotheses, and later derived the necessary and sufficient condition for the Atkinson test to be consistent. A different approach, the R procedure, was also suggested. For competing linear regression models, two R statistics were derived in the absence of distributional assumptions. If normality is also assumed, then an R test is equivalent to the conventional F test therefore resulting in a small sample test. The rest of the paper illustrated the various test statistics, their properties, and relationships for competing linear regression models. It is interesting to note that all the tests proposed for two non-nested hypotheses utilize two tests of nested hypotheses. For example, to test H, against H, of section 5: the Cox procedure tests the null hypotheses Hg: E,[TC,] = 0 against the alternative Hy: E[TC,] #O; the Atkinson procedure tests H$: A=0 against Hf: A#0 in the comprehensive model h(y, 2, Q, a,); the Davidson and MacKinnon (1981) and Fisher and McAleer (1981) procedures test H$:p=O against H'f :p#O in regressions (23) and (26), respectively; and the R procedure tests H~:E,[ijo] =0 against HF: E[i),J #O.
N.K. Dastoor, Testing non-nested
hypotheses
227
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