Tests for Gaussianity and linearity of multivariate stationary time series

:•

joumalof statistical planning

ELSEVIER

Journal of Statistical Planning and Inference 68 (1998) 373-386

and inference

Tests for Gaussianity and linearity of multivariate stationary time series T. Subba Rao*, W.K. Wong 1 Institute of Science and Technology, University of Manchester, Manchester M60 I QD, UK Received 11 April 1996; received in revised form 22 January 1997; accepted 10 February 1997

Abstract We propose tests for Gaussianity of a vector stationary time series based on multivariate measures of skewness and kurtosis. The tests are illustrated by two real sets of data. We discuss briefly some properties of linear transforms of vector time series, and stress the need for separate tests for linearity. © 1998 Elsevier Science B.V. All rights reserved.

1. Introduction The usual assumption often made in studying vector time series is that the series is stationary and Gaussian. It is important that we should have statistical tests for testing these two hypotheses. In this paper, assuming the series is stationary, we propose statistical tests for testing for Gaussianity. W e say the vector time series is Gaussian, if all subsets have joint normal distributions. In recent years, in the case o f univariate time series, several tests have been proposed for testing Gaussianity and linearity. The first test for testing linearity, using a frequency domain approach, was due to Subba Rao and Gabr (1980). Lomnicki (1961) proposed a test for Gaussianity, based on measures o f skewness and kurtosis. This is similar to the classical test for Gaussianity in the case o f independent observations. In 1970, Mardia defined multivariate measures o f skewness and kurtosis and based on them he proposed a test for multivariate normality. Our objective here is to extend these to the time-series context. These tests are scale invariant. Let X be a d-dimensional random column vector with mean p and variance covariance matrix E. The population measure o f skewness is defined as (Mardia, 1970)

flhd = [E( Y ® Y ® Y)]'[~ ® E ® ~ ] - I [ E ( Y ® Y ® Y)], * Corresponding author. E-mail: [email protected]. I Now at the University of Stirling, Scotland. 0378-3758/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PH S0378-3758(97)00150-X

(1.1)

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T Subba Rao, W.K. Won.q/Journal o f Statistical Plannin9 and Inference 68 (1998) 373-386

where Y = X - p, and ® stands for the Kronecker product. If X is Gaussian, then ill,a = 0. Therefore, the rejection of ill,d = 0 indicates skewness in the distribution of X. The measure of kurtosis is defined as

il2,d = E[ Y' ~, -l y]2 = E[ S SffiJ yi Yj ]2,

(1.2)

where Lr,-l=(a ij) and Y ~ = ( Y b Y 2 . . . . . Ya). If X is Gaussian, it is easy to show

il2,d = d(d + 2). We can write (1.1) alternatively as d

ill,d =

d

E E rr' ss' tt' rst r's' t' O" O" O" # 1 1 1 1 2 1 1 1 , r,s,t=l r~,s~,F--1

(1.3)

where #lllm = E ( Y r Y Y t ) . We use the form (1.3).

2. Sample measures of skewness and kurtosis

Let {Xt;t ¢ Z} be a d-dimensional multivariate time series and let X I , X 2 , . . . , X n be a sample. Let X[=(XI,t,Xz, t . . . . . Xd, t). We say Xt is Gaussian if for any finite r~>2, the random vector (Xil,t,Xi2,t . . . . . X/,,t)', where 1 <~ij<~d; j = 1,2 . . . . . r and tj is an integer, is multivariate normal. We assume {Xt} is stationary. We further assume

q=-~

17iAq)l < ~ ,

7ij(q) = E ( X i , t + q - P i ) ( X j , t - # j ) , # i = E ( g / , t ) , We define the sample moments as follows:

where

1 nX,

mi ------ ~

(i = 1,2 .....

d).

We have 7ij(0)

= 7ij-

i,t,

n t=l

mili2...ir = -

(2.1)

Y i , , t Y i 2 , t . . . Nit,t,

n t=l

1 n Mi, i2...ir = - E nt-i

(Si,,, -

mi, )(Xl2,t -- mi2)... (X#,t

-

mir).

The sample variance covariance matrix is defined as S = 1 ~ (X, - X)(Xt - X)',

(2.2)

nt- 1

where A" = (ml, m 2 . . . . . md). Following Mardia (1970), we define the sample measures of skewness and kurtosis as follows:

bl,a

~

~

il,jl,kl i2,jz,k2

ci,i2cj,y2~k, ~ o o o k2H lvliljlkllVii2j2k2

(2.3)

T. Subba Rao, 14(K Wong/Journal of Statistical Planning and Inference 68 (1998) 373-386

375

Table 1 Two types of non-zero indecomposable partitions

Type 1 ili2 ]jlj2 ]klk2 ilj21jlizlklk2 ilk2[jli2lklJ2

ili2 Ijlk2 I klj2 ilj2lJlk2lkli2 ilk2]jlj2[kli2

ili2 ]jlkl Ij2k2 jli2lilkllj2k2 kli2[iljl[j2k2

Type 2 ilj2 ]jlkl 1i2k2 jlj2[ilkllizk2 klJ2liljl[izk2

ilk2 Ijlkl li2j2 jlk2lilklli2j2 klk2[ilJl]izj2

and d

b2,d=

~

siJSkfMijk~,

(2.4)

i,j,k,{-1

where S -~ = ( S gj) and bl,a is invariant under linear transformation, we can assume E(Xt) = 0 , E(XtXt' ) = ~, = I. Since S converges to X = I in probability, bl,d is equivalent to ~r,s,t[Mrst] z. Hence, we can study the asymptotic distribution of ~r,s,t[M~.t] 2. Under the assumption that {X,} is Gaussian and E(Xt) = 0, we can show (see Wong (1993) for details)

E(mi) = O, cum(mi, m j ) = n1 ~, 7ij(q) + o( n-1 ),

E(mij ) ----7ij, O<3 c u m ( m i u 1 , I'ni2j2 ) = n1 _~ [7i, i:(q)7),/, (q) + 7i,j2(q)Tj, i2(q)] + o( n-1 ),

(2.5)

E ( mijk ) = O, cum(mi,j,k,,mi2j:k~)= !

~

q=--oo

[Ti~i~(q)Tj,j2(q)Tk~k2(q)[6] + 7i, i~(q)Tj, k, Tj2k~[9]]

+ o(n -1 ).

(2.6)

The expression (2.6) contains fifteen indecomposable partitions of [iljlkl[izj2k2]. In Table 1, we give all the 15 non-zero indecomposable partitions in (2.6). Note that the notation used in McCullagh (1987) is adopted here: only a typical example of the distinct types of partitions, together with the number of partitions in each type, needs to be written. Thus the first summand of the RHS of (2.6) corresponds to the 6 partitions of Type 1, in which all autocovariances are of lag q. Type 2 partitions referred by second summand in (2.6) are products of autocovariances with lags q and 0. Here we have used 7ij for 7gj(0). Also, we have E ( mijkl ) = 7ij Ykl [3 ], oo

~ [Yiti2(q)Tjd2(q)yk, k2(q)yl,12(q)[24 ] cum(mi~j, k~Ii, mi2j2k212 ) = n1 q=_~ + Yi~is(q)';/,j2 (q)7,~ 1,7k~t~[72]] + o(n- 1).

(2.7)

376 T. Subba Rao. W.K. Wono/Journal of Statistical Plannin9 and Inference 68 (1998) 373-386 Table 2 Two types of non-zero indecomposable partitions Type 1

Type 2

lit2 Jjlj2 Iklk2 [ lll2 lll2 Jjlj2 Jkll2 [ llk2 tit2 Jjlk2 Iklj2 l lll2 tit2 ]jlk2 lkll2 l llJ2 q~2 [jil2 lklJ2 l ljk2 ilZ2 [jll2 [klk2 l llJ2 Zlj2 [jli2 [klk2 1112 q J2 IJli2 Ikll2 llk2 llJ2lJlk2[kli2 1112 tlJ2[jlk2lkll2 lli2 tlJ2Jjll2lkli2 llk2 tlJ2[jll2lklk2 lli2 ilk21jli2[klj2 1112 ilk2[jli2[kll2 llJ2 ilk2ljjj21kli2 lll2 ilk2 IJlj2 [kll2 J lli2 ilk2 IJll2 Ikliz l llj2 ilk2 [jtl2 IklJ21lli2 i112 jjli2 I klJ2 [llk2 i112 [jli2 Iklk2 [ llj2 ill2 [jlj2 Ikli2 [ llk2 i112 Jjlj2 j klk2 J lli2 ill2 Ijlk2 Ikliz J llj2 ill2 Jjlkz Iklj2 [ lli2

ili2 IJlj2 Iklll ilJ2 Ijli2 [kill ilk2 IJli2 Ikllj ill2 [jli2 Jklll ill2 Iklj2 [jill ilj2 Ikli2 t jill ilk2 Ikli2 ]jill ill2 I kli2 [jill ili2 [ llj2 ]jlkl ilj2111i2 [jlkl ilk2 J lti2 Jjlkl ill2llli2 [jlkl jli2 IklJ2 Jilll JlJ2 [kli2 Jilll ilk2 I kli2 Jilll jll2 Ikli2 Iilll jli2 J llj2 [ilkl jlj2 [ lli2 J ilkl jlk2Jlli2lilkl jll2llli2 Jilkl kli2 Ill j2 [iljl klj2 l lli2 liljl klk2 I lji2 [ijjj kll2 I lli2 I ilJl

Jk212 1k212 1J212 IJ2k2 [k212 ]k212 ]j212 Ij2k2 [k212 [k212 IJ212 ]j2k2 1k212 Jk212 Jj212 [j2k2 lk2/2 [k212 lj2/2 IJ2k2 Jk212 [k212 [j212 Ijzk2

ili2 Ijlk2 [kill 1J212 ilj2 [jlk2 Ikjlj li212 ilk2 IJlJ2 Jklll 1i212 i112 IJlJ2 [kill 1i2k2 ilizlklk2 ljill Jj212 ilj2 Jklk2 [J1111i212 ilk2 JklJ2 ljill Ji212 ijl2 IklJ2 ljill [i2k2 lli2 I llk2 }jlkl 1J212 tlj2 I ljk2 [jlkl 1i212 ilk2 [ llj2 ]jlkl 1i212 ill2 [ llj2 Ijlkl li2k2 Jliz Jklkz l illl 1J212 )l j2 [ klk2 [ ij II i212 ilk2 Ikjj2 ]Jill i212 jIl2 [klj2 [i111 i2k2 )li2 lltk2 lilkl j212 .llj2 l llk2 Iilkl i212 jlk2 l llj2 ]ilkl i212 jl l2 [ llj2 l ilkl i2k2 kli2 [Ilk2 IfiJl ]212 klJ2 J llk2 [ilJl i212 klk2 J llj2 I iljl i212 k112111J21iljl i2k2

lli2 lJ112 Jkl Ii IJ2k2 llj2 Jjl l2 ]kill [i2k2 ilk2 JJ112 I kl Ii I i2j2 i ll2lJlk21klllli2J2 tli2 [kl12 ]jill Ij2k2 tlj2 Ikll2 ]jill li2k2 ilk2lkll2JjlllJi2J2 ill2 Iklk2 [jill li2J2 lliz l lll2 [jlkl Jj2k2 ilj2 J lll21jlkl li2k2 ilk2 11112[jlkl 1i2j2 i112 [Ilk2 [jlkl l i2j2 jliz ]kllz lilll [j2k2 JlJ2 Ikll2 lilll 1i2k2 jlk2 Ik112 [illl Ji2j2 Jll2 [klk2 [illl 1i2j2 Jli2 [ 1112 l ilk1 IJ2k2 JIJ2 [ ll lz l ilkl 1i2k2 jlk2 [ l112 I ilkl ] i2j2 j112 I llk2 l ilkl [ i2h kli2 [Ill2 [iljl IJ2k2 klj2 [ Ill2 Jiljl li2k2 klk2 11112[ilJl [i2J2 kl12 [Ilk2 I iljl li2J2

The 96 indecomposable partitions mentioned in (2.7) are given in Table 2. From the above results, it is obvious that the sample moments are unbiased and consistent. For a zero mean Gaussian distribution, odd moments are zero, and cross covariances such as c u m ( m i , , m i , j , ) , cum(mi,j,,mi,j,~,), etc. are also zero. Therefore, we have to consider only the even moments. We can show cum(mit, mi2j2k2 ) = n1 ~ 7i,i:(q)yj2k2[3] q'- o(n_ t ), (2.8) cum(miljl, mi2j2k212 ) = n1 ~ 7i, i2(q)Tj~j2(q)Tk212 [12] + o ( n - l ) .

We can establish the asymptotic joint normality of the sample covariances; and we state the results in the following theorem (for a proof see Wong, 1993). Theorem 1. L e t {X,} be a zero m e a n d - d i m e n s i o n a l stationary Gaussian p r o c e s s with absolutely s u m m a b l e covariances. Then as n ~ oe, the distribution o f [v/nmi, x/n(mij - 7ij), x/nmijk, x/n(mijkt - 7ijTkt [3] )] converoes to the multivariate n o r m a l distribution with m e a n zero a n d variance covariance m a t r i x with elements 9iven by (2.5)-(2.8).

T. Subba Rao, V~I~ WonglJournal of Statistical Planning and Inference 68 (1998) 373-386

377

The test statistics bl,a, and b2,a we propose are functions of the sample central moments Mijk,Mijkl. Hence, we need the cumulants o f Mijk,Mijk! and the joint distribution of (Mi,Mij,Mijk,Mijkl). We observe (for details see Wong, 1993), v / n { ( M i j -- 7ij) -- (mij - ~ij)} P.~O,

v/n{(Mijk - (mqk -- miYjk)[3]} p O,

(2.10)

v/-n { ( Mijkt -- ?ij ?kt[ 3 ] ) - ( mijkt - ?ij Ykl[3 ] ) } P-~O,

(2.11)

where Yn p 0 means that Yn tends to zero in probability as n ~ oo. Thus the asymptotic covariances of v~[M/j - ?ij] and v/n[Mijkl -- ?ij?kt[3]] and the cross covariances between them are the same as the expressions given for mtj, mijkt in (2.5), (2.7) and (2.8). From (2.11 ), we have lim n cum[Mi,j,k,

,Mi2j2k2]

n ----~o o

---- l i m 11---+0 0

n cum[mfijlk I -- mil ]:jlkl [3], mi2j2kz -- mi2"~j2kz[3]]

O~

=

(2.12)

~ [Ti, i2(q)Tjtj2(q)Tk, kz(q)[6]] • q=-oc

In view of (2.9)-(2.11), we state the following theorem without proof.

Theorem 2. Let {Xt} be a zero mean d-dimensional stationary Gaussian process with absolutely summable covariances. Then as n --~ ~ , the distribution o f v~(Mij - ?iy), ~.Mijk and [v~[Mijkl - ?ij?kt[3]] converges to the multivariate normal distribution. The means are zero and variances and covariances o f the second- and fourth-order central moments are the same as the corresponding non-central moments as given by (2.5)-(2.8). However, for the third-order central moment the asymptotic covariance is given by (2.12). The third-order central moment is asymptotically independent of the other two central moments.

3. Asymptotic distributions of skewness and kurtosis We have shown earlier that the sample skewness m e a s u r e b|, d is equivalent to ~'-~r,s,t[Mrst] 2 in probability. Therefore, we consider the asymptotic distribution of d

n ~ [Mrst] 2

(3.1)

r,s,t

which has asymptotically the same distribution

as

nbld.

378

T. Subba Rao, W.K. Wony/Journal of Statistical Plannin9 and Inference 68 (1998) 373~86

Though the number of elements in the summation (3.1) are d 3, the distinct number of elements are only q = [d(d + 1 )(d + 2)]/6. We define a random column vector ~ of order q × 1 containing these distinct elements, ~t =- (v~M111, x/-nM112,. •., v/nMclacl) •

We can now write (3.1) as n ~ [Mrst] 2 = ~'Dff, rst

(3.2)

where D is a diagonal matrix of order q × q, and elements distinct elements of ~, and are given by

dgjk corresponds to the

if i = j = k , 6 if i,j,k are all different, 1

dgjk =

3

otherwise.

nbl,a = n(M~l 1 + 3M212 + 3M221 + M222) = ~'D~, where ~'=(v/-nMl11,v/-dM112, x/~M221,v/-dM222), and diagonal ( D ) = ( 1 , 3 , 3 , 1 ) . Similarly if d = 3, q = 10, nbl,a = n[M(11 + 3M212 + 3M~13 + 3M222 + 3M233 +6M(23 + 3M233 + 3)142223+ For example, if d = 2, q = 4,

M222 +M3233 ) ----_~ t / ) if, where ~' = (v'~Ml 11, X/riM112, x/~M113, V/-nM122, v~M133, x/nM123, v/-nM233,x/~M223,x/riM222,v~M333) and the diagonal D = (1,3, 3, 3, 3, 6, 3, 3, 1, 1 ). Since ~ is asymptotically normal with mean zero and variance-covariance matrix ,~, y,1/21-ly.l/2 y.-1/2 I..=~'~,-fl/2Qy.U1/2~, where Q is a symmetwe can write ~ ' D ( = . _?.ty.-1/2 ~ _~ v _ ( _ff ric matrix and, hence, there exists an orthogonal matrix F such that Q = FtAF where the columns of F are the eigenvectors of Q and A is a diagonal matrix with elements corresponding to the eigenvalues of Q. Hence, we have q

~'D~=~ t T.~--1/2 F t AF ! ~,~--1/2 ~ = Z ' A Z = ~ 2iZi2,

(3.3)

i=1

where Z = l " ~ l / 2 ~ and Z is a multivariate normal with mean zero and variance covariance matrix I, and (21,22 . . . . . •q) are the eigenvalues of Q. Hence for large n, nbl,a is a weighted Z2 with q degrees of freedom. The results of Mardia (1970) can be obtained by noting that Xl, X2,..., Xn is a random sample with D,~¢ = 61, and Lomnicki's (1961) result is obtained by setting d = 1 in the above. We now consider the asymptotic distribution of b2,d. We have b2,d-~- ~ siJsklMijkl • i,j,k,l

Since

Mij = Sij, Z,SSiJSij =d b2,a -

d(d + 2) = ~ i,j,k,I

and ~i,j,k,l siJSkISiISj k = d, we can consider the statistic siJskl[Mijkl

--

MijMkt[3]].

(3.4)

T. Subba Rao, W.K. Wono/Journal o f Statistical Plannino and Inference 68 (1998) 373 386

379

As S ~ E in probability and in view of our assumption that _r = I, we have d

[b2,d - d(d + 2)] = ~ 7iJykt{Mijkl -- MijMkl[3]} P-+O. ijkl

But

--

{(Mij

7ij)(Mkl - 7kt)}[3].

-

(3.5)

Since the last term of the right-hand-side expression of (3.5) is of smaller order of magnitude than the first two terms, to this order of approximation we can write Mijkl - - M i j M k l [ 3 ]

= (Mijkl - 7ijVkl[3]) -- ( M i j - 7ij)Tkl[6].

(3.6)

The asymptotic distribution of (3.6) is the same as that of (mijkl- 7ij~'kt[3])--{(mij7ij)7k/[6]} which is asymptotically normal with mean zero and variance given by the following expression. We have nlirn n cum[ mitjl k~It -- mi,jl "YklIi [6], mi2j2k212 -- mi2j2 "~k212[6]] O<3

=

~

{7i,i2(q)Ty,j2(q)Vk,k:(q)~tt12(q)[24]},

(3.7)

where v ( q ) = (~ij(q)). If we now define the standardised variance-covariance matrix of lag q, as ~(q)=F_,-l/2v(q)Z-l/2, E = y ( O ) = E ( X t - l~)(Xt- p)', ~j(q) as the (i,j)th element of ~(q), we have (noting that we are assuming ,~ = F(0)= I), d

lim nvar(b2,d)= V= ~ n--+cx~

~

[8~2(q)~2t(q) + 16~ij(q)~kl(q)~il(q)~kj(q)].

i,j,k, I q=--c~

(3.8) Hence for large n, under the null hypothesis, b2,a is asymptotically normal with mean d(d + 2) and variance V/n. The results of Mardia (1970) and Lomnicki (1961) can be deduced from the above.

4. Applications For the purpose of illustration, we consider two real data sets. The first one is the external weather data from the Silsoe Research Institute, England showing readings of temperatures and humidity levels recorded at 5 min intervals from 0.00 to 23.55 hours on 17 May 1991. The second data is the quarterly seasonally adjusted West German disposable income and expenditures from 1960 to 1978 (Liitkepohl, 1990). The German data has been analysed by Liitkepohl on the assumption that the series is Gaussian. The two data sets are plotted in Figs. 1 and 2. It is clear that the two data sets are non-stationary, and need to be transformed.

380

T. Subba Rag, W.K WonglJournal of Statistical Planning and Inference 68 (1998) 373-386 100

12.0 -Temperature ICl ...... Humidity [%1

11.5

]V~ 95

11.0 10.5

90~

E lO.O 85 ~3

9.5 ~L

~., .:

E 9.0

E

80

&5 8.0

75

7.5 7.0

50

100 150 200 T i m e in 5 m i n u t e s i n t e r v a l s

70 300

250

Fig. 1. Silsoe external weather data.

3000 -Income I ...... Consumption

"

I

2500

~2000 C3 O

1500

0 m lOOO

500

0 0

10

20

30 40 50 60 70 Tune in 3 months intervals

80

Fig. 2. West German quarterly seasonally adjusted economic data.

90

I00

T. Subba Rao, WK. Wong/Journal of Statistical Plannin9 and Inference 68 (1998) 373-386

381

Temperature

0.25 0.2 015 0.1 0.05 0.0 -0.05 -0.1 -0.15

Humidity

20 1.5 1.0 0.5

o.0 -0.5 -I.0 -1.5 -2.0

0

10

20

30

40

50

60

70

80

90

Fig. 3. First differences of Silsoe external weather data.

For the weather data, we have taken the first differences, and the differenced series is plotted in Fig. 3. For the German economic data, we followed Ltitkepohl (1990) and considered the first differences of the natural logarithms of both series, and the transformed data is given in Fig. 4. We are interested in testing individual components and also Gaussianity of the bivariate series. We note that the expressions for the asymptotic variances of bl,a and b2,a involve infinite number of autocorrelations and cross correlations of various lags. These correlations are estimated from the data, and we have used various truncation values (M) to approximate the sums. In each case, the sums have stabilised for values M>20. In Tables 3 and 4, we give the p values of these tests for both data sets. Very small values of p indicate statistical significance, i.e. departure from Gaussianity. From Table 3, we see that the individual components Xl,t (temperature), Xz, t (humidity) are Gaussian, and they are also jointly Gaussian. From Table 4, we see that from the skewness test both income and expenditure can be considered to have come

T. Subba Rao, 146K. WonglJournal of Statistical Plannino and Inference 68 (1998) 373-386

382

id>

005 0.04 0.03 0.02 0-01 0.0

Income

-0.01 -0.02 q

q

Expendilure 0.04

0.03

¢~

0.02 0.01 0.0 -4).01

0

10

20

3o

40

50

6o

70

80

90

I00

Fig. 4. First differences of logarithms of income and consumption.

Table 3 Tests of non-Gaussianity on Silsoe temperature and humidity data M

0 2 4 6 8 10 12 14 16 18 20

Skewness tests

Kurtosis tests

Xl,t

X2,t

Yt

&,t

X2,t

Xt

0.6552 0.6611 0.6704 0.6708 0.6765 0.6770 0.6771 0.6766 0.6756 0.6758 0.6764

0.8876 0.8858 0.8881 0.8872 0.8896 0.8866 0.8859 0.8858 0.8801 0.8806 0.8777

0.4832 0.5190 0.5545 0.5548 0.5692 0.5638 0.5627 0.5618 0.5474 0.5505 0.5472

0.8411 0.8418 0.8431 0.8431 0.8439 0.8439 0.8439 0.8440 0.8441 0.8441 0.8441

0.8124 0.8131 0.8141 0.8144 0.8155 0.8184 0.8186 0.8186 0.8217 0.8218 0.8235

0.8998 0.9127 0.9146 0.9152 0.9156 0.9161 0.9161 0.9162 0.9167 0.9170 0.9174

T. Subba Rao, V~K. WonolJournal of Statistical Plannin9 and Inference 68 (1998) 373-386 383

Table 4 Tests of non-Gaussianity on West German economic data M Xl,t

0 2 4 6 8 10 12 14 16 18 20

0.0384 0.0391 0.0426 0.0431 0.0423 0.0423 0.0422 0.0422 0.0423 0.0423 0.0427

Skewness tests Y2,t 0.1717 0.1767 0.1837 0.1851 0.1846 0.1858 0.1806 0.1804 0.1812 0.1818 0.1827

St

Xl,t

0.0467 0.0434 0.0481 0.0486 0.0484 0.0485 0.0476 0.0478 0.0480 0.0482 0.0484

0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

Ku~osis tests X2,t 0.8297 0.8302 0.8309 0.8310 0.8310 0.8311 0.8316 0.8316 0.8317 0.8317 0.8317

Xt

0.0003 0.0004 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005

from a symmetric distribution, whereas the Kurtosis test clearly shows that income data is non-Gaussian, expenditure is Gaussian and they are jointly non-Gaussian. I f one uses only skewness test in this case, it is possible to conclude that the data is Gaussian, and perhaps this is the reason why Liitkepohl (1990) fitted a Gaussian vector A R M A model to this data.

5. Tests for linearity So far we have considered tests for Gaussianity, and the methods we have used are extensions o f the classical tests in statistics. In time series it is important to distinguish between linearity and Gaussianity, as it is possible that a time series can be linear but not Gaussian. With this clear distinction in mind, Subba Rao and Gabr (1980) have proposed tests for Gaussianity and linearity separately. In recent years, several tests (based on time domain approach) have been proposed and most o f them are based on parametric models. Though these tests are called linearity tests, they are in fact Gaussianity tests. This is because under the null hypothesis the model reduces to a linear model with Gaussian errors and this implies the process is Gaussian. One o f the interesting aspects o f Gaussianity o f a vector is that the vector is Gaussian if and only if every linear combination is Gaussian. In other words, tests for multivariate Gaussianity can be approached through linear combinations. Therefore, it is interesting to know whether a similar characterisation holds for linearity. W e will show that a linear combination o f a vector linear stationary process is not necessarily linear. Let {xt} be an univariate stationary stochastic process. W e say xt is linear if it can be written in the form xt = ~ auet-u, u

(5.1)

384

T Subba Rao, W.K. WonolJournal of Statistical Plannin9 and Inference 68 (1998) 373-386

where {et} is a sequence of independent, identically distributed random variables with mean zero and the nth-order cumulant k,(e). If {et} is Gaussian, k , ( e ) = 0 for n > 2 . It is well known that the nth-order cumulant of xt satisfying (5.1) is given by

cum(xt,xt+,l .... ,xt+~._, ) = k,( e ) ~ auau+,, . . . au+,._,

(5.2)

u

and the nth order cumulant spectral density function is given by: 1 fn((Ol, (92 . . . . . O9,-1 ) - (2~),_ l X..Scum(xt,xt+~ . . . . . xt+~._, ) x e x p [ - - i ( m f f C l + (D2Z 2 -q- • • • + O)n_l'Cn_1)]

(~-~-sI.A(Col)A(~o2)...A

--

wj

(5.3)

,

where A(~o)= Xa,e -i"~° and the second-order spectral density function f ( o g ) = (k2(e)/ 2n)lA(o~)[ 2. When k , ( e ) # O , ( n >2), from the relations (5.2) and (5.3) it follows that the ratio Ifn((-01, ( D 2 , . . . , (Dn--1)l 2

f (- Ej=-I l (.Oj)

(5.4)

is a constant. This ratio can be used for testing linearity (see Suhba Rao and Gabr, 1990). We say the d-dimensional stationary random process Xt is linear if

Xk = XAjet_j,

(5.5)

where {et} is a sequence of independent, identically distributed random vectors such that

cum( et ) = E( et ) = 0 and the nth-order cumulant vector of et is given by

cure(eta, et2. . . . . et.) =

{

Cn 0

if t l = t 2 . . . . . otherwise,

in,

where Cn is a column vector of order d" × 1. We can show that the nth-order cumulants and the nth-order cumulant spectral density function of the process {Xt} are given by (Wong, 1993) (5.6)

cum(Xt, Xt+~, . . . . . Xt+~. t ) = z~{Aj @ Aj+z, @ ' ' ' @ Aj+z._, }On,

l l {A(COl)®A(co2)® " " @A ( g,(o91,o92 ..... ~o,-1)-- (21t),_

2o j

j=l

c,, (5.7)

T. Subba Rag, W..K. WonglJournal of Statistical Planning and Inference 68 (1998) 373-386

385

where A(¢o) = Zaje -ij°'. Here gn is a column vector of order d n x 1. It is well known that the second-order covariance matrix and the second-order spectral density matrix of Xt are given by (Harman, 1970)

r(s) = ~ A:+sz.G J f(¢o) = ~-~A(oo)~,A*(¢o),

(5.8)

where ~,=E(ete~) and C2 = v e c ( Z ) . From (5.8) we can show that the column vector is

g2(cO)

g2(co) = vec(f(og)), and further (Wong, 1993), the scalar product

{

g*(e)l,e)2 . . . . . ~on-l) f ( e ) l ) ® f ( e ) 2 ) ® ... ® f

( )/-I - ~ o)j j=l

(5.9)

x gn(o91, CO2,.--, (D.--I)

is a constant (independent of frequencies ~o1,~o2.... ~on) when Xt is linear and given by (5.5). The relation (5.9) is a generalisation of (5.4). Let us define the scalar random process Yt = ~'Xt, where ~ is a column vector of order d x 1 and Xt is given by (5.5). We can show that the second-order spectral density function of Yt and the nth-order cumulant spectral density function of Yt are given by 9,.(¢o) = ~'f(~o)~t,

gn,y((-Ol, 0)2 . . . . .

Ca)n--1)

(5.10) = [o¢[k]]'gn(¢Ot,032 . . . . .

(Dn--1 ),

(5.11 )

where a[k] = a ® = ® a . . . ® a and gn(o91,~o2..... ~on-i) is given by (5.7). From the relations (5.10) and (5.11), it is clear that the ratio

[gn,y( (.Ol,(-02. . . . . ('On-1 )l 2 gV((Dl )Oy(O.)2)...Oy(-- ~

O)j)

is not a constant, and hence the random process of Yt is not necessarily linear in the sense we defined earlier. This result shows that the test for linearity of the vector stationary process Xt cannot be approached through linear combinations (see Wong, 1993).

Acknowledgements We wish to thank Dr. Z. Cholabi of Silsoe Research Institute for providing us with the weather data. We thank the referee for reading this paper carefully and making many helpful suggestions. The research of one of the authors (Wong) was supported by an ORS award and R.A. Needham Hall (University of Manchester) postgraduate research scholarship.

386

T. Subba Rao, W.K. WonolJournal of Statistical Plannin 9 and Inference 68 (1998) 373-386

References Hannan, E.J., 1970. Multiple Time Series. John Wiley, New York. Lomnicki, Z.A., 1961. Tests for departure from normality in the case of linear stochastic process. Metrika 9, 37-62. Liitkepohl, H., 1990. Introduction to Multiple Time Series Analysis. Springer, New York. Mardia, K.V., 1970. Measures of multivariate skewness and kurtosis with applications. Biometrika 57, 519-530. McCullagh, P., 1987. Tensor Methods in Statistics. Chapman & Hall, London. Subba Rao, T., Gabr, M.M., 1980. A test for linearity of stationary time series. J. Time Series Anal. 1 (2) 145-158. Subba Rao, T., Gabr, M.M., 1984. An introduction to bispectral analysis and bilinear time series models, Lecture Notes in Statistics, vol. 24. Springer, New York. Wong, W.K., 1993. Some contributions to multivariate stationary nonlinear time series. Unpublished Ph.D. Thesis submitted to UMIST, Manchester. Wong, W.K., 1997. Frequency domain tests of multivariate Guassianity and linearity. Journal of Time Series Anal., to appear.