14 and is included here for the sake of com pleteness.

C o r o lla r y 6,10,- Under the assum ptions of Theorem 6.1 and *

(i) xt is w eakly exogenous for 9 , and *

(ii) xt I 9 t - N [ pxt» Hxt 1 * the functions Pxt and Hxt obey the same regularity conditions as jit and h t , and have the form in (35),

* 1 2

the LM test for strong exogeneity of xt for 9 is given by Sg© = %nx ( nx + 3) T Ro

from the regression of ^ N

f d\Llxt 9 l J « w \ f vech [ vt vt - Hxt ] on 9fd2xt 9 ß x i' dßx2'

9 vech Hixt 9 vech H ixt 9 vech H ^ 9 vech H2xt

V 9 ßxi' 9axi' 9ßx2/ 9ax2'

J

/N _1 /N _1 A 1

in the metric of diag { H x t» P ( Hit ® H it ) P ' } , where P is the m atrix such that

A A

vec Hxt = P' vech Hxt , and all functions are evaluated at 9 , 9X , ßx2 = 0 > and aX2

0 . Under Ho , s_{se Xdim (ßx2 ) + dim (a ^ )}

Proof: Given weak exogeneity, we need only test that the conditional moments of Xt do not depend on the past of y t , and this is accomplished w hen ßx2 = 0 and

0Cx2 = 0 in view of (35) . The test is a variable addition test and is a particular

case of Theorem 9.14 . The additions to the auxiliary regression follow obviously.

To construct a full test for strong exogeneity observe that sic and Sse are asym ptotically dependent in general, but the auxiliary regression that incorporates the additional variables in Corollaries 6.9 and 6.10 (avoiding redundancies if necessary) provides a statistic Sic.se that is asym ptotically independent of spf_is , and thus Sfse = spf.iS + Sic-se gives the appropriate statistic. An alternative approach to m easure the feedback betw een tim e series rather th an constructing G -causality tests has been put forward by Geweke

[1982,1986b], and this may be adapted to the heteroskedastic case given fourth order stationarity.

§ 6.5 Testing normality

W hite and MacDonald [1980] proposed constructing well known tests for norm ality using LS residuals and compare different tests in some Monte Carlo experim ents. An alternative approach is adopted by Jarque and Bera [1980], by embedding norm ality into a more general class of distributions - the Pearson fam ily - and using the LM principle. The approach has been

extended to other situations by Lee [1 9 8 2 ,1984a, 1984b] and Bera et al [1984]. We follow the approach of Jarque and Bera, and thus the LM test derived below is a sim ple generalization of theirs that allows for heteroskedasticity under the null and for additional evolution of the conditional third and fourth m om ents. The Pearson fam ily pdf for yt conditional on 7 1 is given by

ft y t l ^ t ) = exp { ( p t ( u t) } / b t , w h ere

and

<pt (u t) = *________ bit (bi,9) - ut________ ht(0) - bit (bi,0)ut + b2t (b2,9)ut

dut ,

& J exp { (pt (ut) } dut ,

and the bjt are measurable functions of & t such that bjt = 0 when bj = 0 , j = 1, 2 . This parameterization is similar to the linear-in-a structure for the

variance, where a = 0 implies homoskedasticity. The distributional

parameter bit is closely related to symmetry because bit = ht(9)1/2 Sk(yt) > Sk being Pearson's measure of skewness (Kendall and Stuart [1968], pp.85 and 149). Given symmetry, b2t is a monotonically increasing function of the

kurtosis measure y2(yt) (fourth cumulant), and has the same sign. Therefore, symmetry implies bit = 0 , and adding mesokurtosis implies b2t = 0 . It is easily seen that bit = b2t = 0 results in the conditional normal distribution. These relations between distributional parameters and the third and fourth moments suggests some plausible parameterizations for the bjt in coherency with the specification of the first two moments. For example, in the ARCH model we might propose bit = bit (uj?.j; j > 0) and b2t = b2t (ujf j ; j > 0).

If we define b = ( bT , b2' )' and £ = ( 0', b ')', the log-likelihood function is

T T T

m

*

t=l t=l t=l

The null hypothesis of normality in (1) can then be expressed as Ho : bi = 0 and b2 = 0 ,

and if we denote by nj the dimension of bj and rjt = rjt (9 = 3bjt /9bj , j = 1, 2 , and follow the appendix to Bera and Jarque [1982] we find that under Ho

dit = — and

d2t = 77“ = l r2t ( h[2 ut - 3 ), ob2

The LM statistic is based on the subvector of the score d = (d i', d2' )' T

= T4 X d t , where dt = (d u ', d2t' Y , which shows that this test is an efficiency t=l

test as those considered in § 5.3 . We then have

an d

sk = T d2' [ 6 T W R2 - f T W a-1 § V(0) T 1 S' Q-1 R21 1 d2 - i»

under the null hypothesis of conditional normahty, where Rj = (rji ,...,rjT ) ' ,

A A A

j = 1,2 , V(0) is a consistent estim ator of V(9) , and all evaluations are made at 9 under Ho . Moreover sg and Sk are asym ptotically independent and the LM test for norm ality is given by

under Ho .

Proof: In the notation of § 5.3 , let ma = m(Q-1 Ri , 1 , 0 ; 9 ) , mb = m(Q-2 Ri , 3 , 0 ; 9 ) , and m^ = m(iV2 R2 , 4 , 0 ; 9 ) . Let Vj =

lim var [ T172 > = lim cov [ T1/2 mi t T172 m.. ] t for i , j = a , b , c .

T —^°° T—)°°

From Theorem 5.10 and using (5.35) , (5.37) and (5.38) we obtain

Va = e {T*1 R J Q-1 RJ*1 - Ö {T-1 R J a -1 X] V( ß) Ö {T-1 X' a -1 Ri) , - (36a) Vb = 15Ö {T-i Ri' Q-1 RJ-1 - 9Ö {T-1 RJ Q-1 X} V( ß) ÖfT1 X' Q*1 RJ , - (36b)

Vc = 96Ö{T-! RJ R2}-1 - 360{T-i r2' q-i S) V(9) SfT-1 S' Q-1 R2] , - (37) and from Theorem 5.12 we get

Vab = 3Ö{T-i RJ Q-1 RJ-1 - 3e{T-! RJ Q-1 X) V(ß) SfT-1 X' Or1RJ , - (38) Sn — Ss + Sk

and

Vac = - 6 0{T-1 RJ a-1 X] V(9) C{T-! S' ß-1 R2} , - (39a) and

Vbc = - i s err-1 r j a -1 x i v (9 ) sr r -1 s' a -1 r2} . - (39b) Therefore, from (36) and (38),

1

while V( d2 ) follows from d2 = ^ me . Independence is established using (39) in

and the asymptotic distribution of the statistics under Hq follows from

The theorem provides three test-statistics to assess normality: s3

emphasizes departures from symmetry, Sk emphasizes on departures from mesokurtosis, and sn is an omnibus test in the third and fourth moments. The theorem is sufficiently general to allow for different parameterizations of bit and b2t reflecting alternative propositions about conditional skewness and kurtosis. However, if the objective of the researcher is simply that of

producing a diagnostic to evaluate normality, it may appear that the specification of such functions consumes too much time and this is a

drawback to the use of the test. For this reason, it may be of interest to have a standard statistic which gives a good indication of whether more careful thought should be given to non-normality before proceeding with further inference. To produce such a standard statistic we make bit and b2t constants, so that ru = r2t = 1 and ni = n2 = 1 , and we have

C orollary 6.12.~ Under the assumptions of Theorem 6.11 with bit = bi and

b2t = b2 , Corollary 5.11 .

□

T