Analysis of Variance with Time Series Data

A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA OECONOMICA 152, 2000

B r o n isla w C e r a n k a *, K r y s t y n a K a t u ls k a * *

A N A LY SIS O F V A R IA N C E W IT H T IM E S E R IE S DATA

Abstract. Successive observations over time are made on individual subjects classified into different groups. It is assumed that the mean response may vary between groups, that there is a random effect for each individual, and that successive observations on each individual follow a AR model. The likelihood-ratio criteria for testing the hypothesis of equality of the group means is considered.

1. INTRODUCTION

T h e usual analysis o f variance ľ test is considered in th e case when the variance-covariancc m atrix o f errors is o f the from er2I n. In m any practical situations this assum ption is not satisfied, for example, if observations are correlated, bu t we arc interested o f testing the hypothesis o f equ ality o f the g roup m eans. In this p ap er, we assum e, th a t successive o b serv atio n s over tim e are m ade on individual subjects classified in to different groups. T his problem was considered also by Y o n g and C a r t e r (1983). T hey proposed to use the usual analysis o f variance F test for nested designs.

T h e p u rp o se o f this p aper is considered th e likelihood ra tio fo r testing h ypothesis o f equality o f the g roup m eans for co rrelated ob serv atio ns.

2. LIKELIHOOD RATIO TEST

C onsider the experim ent in w hich m y . n experim ental units are divided in m groups each o f n units. Let Yijt be the o b servation a t tim e t o n the j -th individual from g roup i. Suppose th a t Yijt can be m odelled by

* Agricultural University of Poznań, Department of Mathematical and statistical Methods. ** Adam Mickiewicz University, Faculty of Mathematics and Computer Science.

Y(j{ — Ч- ot i j “f* E[jt i — 1, ..., w, j 1, n, t 1, 2, •••) T, (1)

w here p, are fixed g roup m eans, a y represent individual effects, and e.lJt are the errors. F u rth e r, we assum e th a t F.iJt and arc ind epen dent, are independent n orm al ran d o m variables with m ean zero and variance ct2 and F.ijt are identical statio n ary G au ssian tim e scries for fixed i and j .

W e in tro d u c e th e follow ing n o ta tio n s. Let Yy = (Yy i , ..., YtJT)' be T-dim cnsional vector o f the observations on the y-th individual from g ro up i an d Eij = (е щ, ..., EljTy d en o te the ra n d o m vecto r, such th a t is distrib u ted according to N T(O , c-,2A r ), where o f \ T is the au to co v arian ce m a trix o f a statio n ary G aussian tim e series, where Np (p, <r2V) is a p-variate n o rm al d istrib u tio n w ith m ean vector p and the covariance m atrix <x2V.

T h en , the m odel (1) can be w ritten as

where l p denotes p x 1 vector o f ones. It im plies th a t is distrib u ted acco rd ing to N r ( / ^ l r , o f(y 2J r + AT)), w here J r = l r l'r an d y l = «т2/<т2.

F u rth erm o re , let Y = (Y'n , ..., Y ^ ) ' p = (p'i, ..., ц'т)' e = (e'n. •••. Cn)' and a = (a u , ..., am„)'. T h en (2) is o f the form

w here (g) denotes K ro n eck er p ro d u ct. U nder these n o tatio n s Y is distrib u ted according to N mnT ( р ® 1 nT, oflm n® ( y l J r + A r))* where I p is the p x p iden tity m atrix.

In this p ap e r we assum e th a t A r is the k now n m atrix, while o f and y l are un k n o w n param eters.

T h e p u rp o s e o f th e d a ta analy sis is to te st th e h y p o th e sis H 0: p l = p 2 = ... = pm against the general alternative H^. H 0 is n o t tru e.

Lem m a 1. If Y is given by (3) then the m axim um likelihood estim ates o f p t, y2, er2 can be w ritten as

Yjj = /x,lr + a y l r + ey, i = l , ..., m, y = l , ..., n (

2

(3)

Ô‘ m n (T -where b = l TA j - l l T, cy = Yy \ T 1 J T \ T 1 Yy, d tj = Y ý A j-1 Yy. (b) I f H 0 is true, then m n mnh C m n t m n "4 , < T - t ) ľ ľ c U- - ^ ( I I Y Í A í ‘ l T)‘ l _ Í _J________ U = i j = i ___ J= LL=J_______ J L / ш n m It \ I Z ^ y - I I c u ) \ (=U=1 ł = lj=l / ď.2 =

í.2-P ro o f, (a) I he likelihood function o f the vector Y given in (3) can be w ritten as

-mA ľ . _mn _mn

p(Yy) = (2n) 2 {(] + y2b ) a f T} 2 (det \ T) 2 x

(

- 2 а ; 1 - A r 1 J r A f 1)(Yy - f t l r ) J .

F ro m the definition o f the m axim um likelihood estim ates o f ц„ у 2, o f and the properties o f the function p(Yy) it follows th a t ß h у*, a 2 are the solu tio n o f the system

» W . o í 1 ~ d f j^ ~ ~ , = 1’ -> w ’ 3y.2 " ’ W n do 2

(b) I f Я 0 is tru e then the likelihood function is

mnT

p \ Yy) = (2 л )- * {(1 + y2b)a2T}~ 2 (det A T)~ 2 x

x exp( — 2 er“ 2 Ž Ž ^ y ^ r X Í A r 1 - j - ^ 2^ А г * J j - A r ^ Y y - z í l , . ) ].

T h e fu rth e r p art o f the p ro o f is analogou s to the p ro o f o f (a).

T h e p u rp o se o f th e d a ta an aly sis is to test th e h y p o th e sis H 0: ц = fi2 = " , — iim against the general alternative H t : H 0 is n o t true. T h e likelihood - ra tio principle ( S c h e f f e , 1959, p. 33) m ay be used to derive the statistical test.

T h e lik elih ood-ratio test consists in rejecting H 0 if A < A0, w here the c o n s ta n t X0 is ch osen to give th e desired sign ifican ce level a i.e. R = {Y: X < A0}, where P(X < A0) = a and

X =P* (fi,fl, ° 2) Pifii, fa, <íf) m n mr Yj X (^U АД тУ^т 1J т^т l ( Yi j ~ £ Д г)\ 2 i = u = 1 Yj X O^y — Д1гУА г '(У ц ~ P ^t) \ i = U = i Since ! m n mn . I ( Yj Yj (Уи ~ А г ^ г а г Ч ^ ,7 — £,1Г)\ 2 H U l L Ż Yj (Уц ~ l (YlJ ~ M^t) \ \ i = U = i < A0 / = P j nb2 £ ß f — mnbzfi i=i Г.2 > A0 mn - 1 Z E Y Ú A r 1 J r A r 1 Y.j - nfc2 I P-f \i=U=i (= i

th e n a = P(A < A0) = P(X* > AJ) and

A* = nb2 £ flf — mnb2ß i= i /7 2 X Z ^ 0 Аг 1^ г А г 1 Yy — nb2 YjP? i=l7=l i=l (4)

T heorem 1. I f Y is d is trib u te d a c c o rd in g to N mnT(n ® 1пГ, ® ( y ? J r + A r )) then ľ given in (4) has a n o n ce n tral F d istrib u tio n w ith m - I , m ( n - 1) degrees o f freedom and n o n cen trality p a ram eter

nb "

(5) *,2(1 + by2) A '

m _{i = 1}

Proof. T h e statistic Л* can be w ritten as the ra tio o f tw o q u a d ra tic form s д* _ _CY (wl m ® ® A r 1 Jj-Aj- 1 — Jmn ® A ]■1 Jj-Aj' *)Y

" cY '(m n ® l mn ® A f1 J rA f П Г mlm ® J n ® A f1 J j A f 1 )Y ’

w here с = ( т т г е2(1 -f Ьу*)Ь)~1. F ro m T h eo rem 1 ( R a o , M i t r a , 1971, p. 171) we conclude th a t if Y is distrib u ted accordin g to ® 1bT, °V2Imn ® (ľ2J r + A r )) then the statistic Y 'B Y has the %2(k, Ö) d istrib u tio n if an d on ly if B ( a f lm„ ® (y2J r - A r ) is id e m p o te n t, in w hich case к = tr(B(<rt2Imn ® (ľ 2J r + A r )) and ö = I nT)' В {р. ® l„ r ).

C o n sid er the q u a d ra tic form

cY '(m Im ® J„ ® A f M j-A f1 - J m„ ® A f ‘ J ^ A f J)Y = Y 'B Y (6) Since В(ст,21тл ® (y2J T -f A r )) = (m n b ) ~ l ( m lm ® J„ - J m„) ® A f xJ r it is easy to verify th a t this m atrix is idem po ten t. H ence th e q u a d ra tic form (6) h a s th e x 2(fc<5) d is trib u tio n , in w h ich case к = (mnb) - *tr{mlm ® J„ ® A f 1J T - J mn ® A f 1 J r } = m - 1,

<5 = (ß ® 1ят) (c(mlm ® J„ ® A t 1J TAT1 — J m„ ® Ar 1 J-fAj- 1))(ji ® l„j-) =

- е т

( t * -

M И

=е т = Ь

-F u rth e r, we investigate the d istrib u tio n o f the q u a d ra tic form

cY'(rnnlmn ® A f 1 J r A f 1 - m lm ® J„ ® A f xJ r A f X)Y = Y'BY (7) Since B(ff2I mn ® (ľ2J r + A r )) = (n b )“ 1 (n lm„ - l m® J n ® A f xJ r , it can be easily seen th a t this m a trix is idem potent. T herefo re, th e q u a d ra tic form (7) has th e x 2(k, S) d istrib u tio n , in which case

к = (nb)~ Нг(и1тл ® A f 1 J r - lm ® J„ ® A f 4J r = m(n - 1),

It im plies, th a t the q u a d ra tic form (7) has the central x 2 d istrib u tio n on m ( n — 1) degrees o f freedom .

M oreover, c(mlm ® J„ ® Лг IJ TA f 1 — J m„ ® \ Ť XJ r A r 'YpHm* ® (ľ* J r + A r) ) x c(m nlm„ ® A f ‘J j - A f 1 — m lm ® ® A f 1 J r A f *) = 0.

It im plies (T heorem 9.4.1, R a o , M i t r a , 1971, p. 178) th a t the q u a d ra tic form s are independently d istrib u ted . T h e p ro o f is com plete.

Corollary 1. If H 0 is true then the statistic X* has the central F distribution w ith m — 1, m(n — 1) degrees o f freedom .

F ro m T heorem 1 it follows th a t the optim al test is to rcject H 0 a t the a level o f significance if

In app licatio n s o f the statistic X* for testing H 0: щ = ... = Pm ag ain st altern ativ e Я х: H 0 is no t

3. STATISTIC X* IN THE CASE AR(p)

P PT - l P PГ - 3 ( G o l d b e r g e r , 1972, p. 203). H ence ( G o l d b e r g e r , 1972, p. 305) A - 1 = _______ ' ЛТ 1 - p 2 ... 0 0 0 1 - p о ... 1 - p l + p 2 - p . . . о 0 о 0 - p and 1 1 — p . . . 1 — p 1 1 - p (1 - p f . . . (1 - p ) 2 1 - p

I f r-ij is th e p o rd e r a u to re g re ssiv e p ro c e ss A R (p ), i.e. flofyi + a i eif i- i + ... + apľ‘ijt- p = z„ where z, is a w hite noise, th en to calculate A f 1 the recu rrin g algorithm given by S i d d i q u i (1958) ca n be used.

T h e testing o f hypothesis o f equality o f the g ro u p m ean s considered in this p ap e r can be used for econom ical records, which o ften c o n ta in repeated m easurem ents o f one o r m ore variables over tim e. In these situ atio n s, we have correlated observations.

REFERENCES

G o l d b e r g e r A. S. (1972), Teoria ekonometrii, PWE, Warszawa.

R a o C. R, M i t r a S. K. (1971), Generalized Inverse o f Matrices and its Applications, John Wiley and Sons, New York.

S c h e f f e H. (1959), The Analysis o f Variance, Wiley, New York.

S i d d i q u i M. M. (1958), On the Inversion o f the Sample Covariance Matrix in a Stationary

Autoregressive Process, Ann. Math. Stat., 29, 585-588.

Y a n g M. C. K., C a r t e r R. L. (1983), One Way Analysis o f Variance with Time Series