journal of statistical planning

Journal of Statistical Planning and and inference ELSEVIER Inference 64 (1997) 325 340

Regression with integrated regressors

Dong Wan Shin a'*, Sahadeb Sarkar b

"Department of Statistics, Ewha Womans University, Seoul 120-750. South Korea b Department of Statistics, Oklahoma State University, Stillwater, OK, 74078, USA

Received 2 June 1995; revised 3 January 1997

Abstract

The ordinary least-squares (OLS) estimation in regression models with integrated regressors is considered. The limiting distribution of the OLS estimator is established under suitable normalization. This unifies asymptotic results for various models studied by numerous authors in the past. It is shown that the limiting distribution of the OLS estimator in the polynomial regression and that in the unstable autoregression can be expressed by the same functional defined on the set of all continuous functions on [0, l]. The functional evaluated at the standard Brownian motion gives the limiting distribution of the OLS estimator in the unstable autoreg- ression. The functional evaluated at the identity function gives the limiting distribution of the OLS estimator in the polynomial regression model. Application of our theory is also illustrated in autoregression containing a polynomial trend and stable random components. :~; 1997 Elsevier Science B.V.

A M S classification: primary 62M10; 62F12

Keywords: Integrated regressors; Ordinary least-squares estimator; Nonstationary time series: Limiting distribution; Brownian motion

1. Introduction

Cons ide r the regression mode l

Yt = ~:xXt. 1 + "'" + ~pXt,p + ut, (1.1)

for t = 1 . . . . . n, where Yt is the regressand, • = (~1, . - - , ~p)' is a vector of pa rame te r s to

be es t imated , and p is a nonnega t ive integer. The system of the in tegrated regressor

var iables Xt. Ss is defined by

X t , , i = ~ Xi , j - l , j = 1,2, . . . , p , Xt , o = Xt, i - I

* Corresponding author. Fax: 2-360-2614; e-mail: shindw@mm.ewha.ac.kr.

0378-3758/97/$17.00 ~'~ 1997 Elsevier Science B.V. All rights reserved PI1 S 0 3 7 8 - 3 7 5 8 ( 9 7 ) 0 0 0 4 2 - 6

326 D. Wan Shin, S. Sarkar / Journal of Statistical Planning and ln[erence 64 (1997) 325 340

where {xt} is a deterministic or random or mixed sequence. The ut is assumed to be a zero-mean error process satisfying the invariance principle that the asymptotic distribution of the partial sums of the ut is a Brownian motion on [-0, 1].

Model (1.1) is called an integrated-regressor model and p is called the order of integration. Park and Phillips (1988, 1989) considered vector regression models with integrated processes having order of integration p equal to 1 and 2. They considered the xt to be a zero-mean error process. In our treatment p is any nonnegative integer and the xt is allowed to be random or fixed or mixed. In this paper, we establish the limiting distribution of the ordinary least-squares estimator (OLSE) in a generalized version of model (1.1), which (model (2.1) below) contains two or more systems of integrated regressors and some stable (in the sense of (C4) (C6), Section 2.1) regressors. Applications of the asymptotics are shown in models such as unstable autoregression, polynomial regression, and autoregression with a polynomial trend and stable regressors.

There are several important examples which can be represented by the integrated- regressor model. For example, if xt = 1 model (1.1) yields the polynomial regression model

p

y, = ~ flit ~ + u,, (1.2) j = l

where flj's are linear combinations of ctj's. Also, if xt = u,_ 1 model (1.1) becomes an autoregressive model

p

y, = ~ y j y , - j + ut (1.3) j - -1

containing p unit roots, where ~j's are linear combinations of ~fs. The asymptotic distribution of the OLSE in model (1.2) can be used to derive distributional results of the OLSE considered by MacNeill (1978) for iid ut and Kulperger (1987) for stationary autoregressive process u , The limiting distribution of the OLSE in model (1.3) can be used for testing the null hypothesis that y, is an I(p) process. This problem was studied by many authors, to name a few, Dickey and Fuller (1979) for p -- 1 and independent, identically distributed (iid) ut; Hasza and Fuller (1979), Dickey and Pantula (1987) and Haldrup (1994) for p = 2 and iid ut; and Phillips (1987) for p = 1 and strong mixing ut.

The asymptotics in the general model (2.1) of Section 2.1 can be used to test the null hypothesis that y, is an l(q) process in the case when an unstable regressive model containing polynomial (in time) regressors and p( > q) lags y,_ 1 . . . . , y~ p has been fit to the data. See, for example, results of Chan and Wei (1988) and Tsay and Tiao (1990). The asymptotics under the general model (2.1) are also useful for various testing purposes in the following situations. Durlauf and Phillips (1987) investigated limiting properties of the estimator in the trend regression model Yt = )'0 + 71t + ut when the true yt is an I(1) process: Yt = Y~ 1 + u,. Dickey and Fuller (1979, 1981) studied the model Yt = ~o + ~ t + fl~y, 1 + ut and established the limiting distribution of the OLSE when 70 -- 71 = 0 and fll = 1 and {ut} is an iid error sequence. Dickey Fuller's

D. Wan Shin, S. Sarkar/Journal of Statistical Planning and Inference 64 (1997) 325.-340 327

results were extended by Fuller et al. (1981), whose model allow multiple stable

regressors and an autoregressive part with the largest root of the autoregressive polynomial equal to one. Sims et al. (1990) studied a vector autoregressive model in

which the true y, is a nonstat ionary vector process and a regression model having time trends and lags of Yt is fit to the y,.

The remainder of this paper is organized as follows. In Section 2, we establish the

asymptotic distribution of the OLSE in integrated-regressor models with one or more systems of integrated regressors. Applications of the asymptotic theory in various models are discussed in Section 3. Proof of Lemma 1 is given in the appendix.

2. A s y m p t o t i c d i s t r ibut ion o f the O L S E

In Section 2.1, we first introduce the general model with multiple systems of

regressors and discuss the assumptions imposed. Section 2.2 presents the limiting distribution of the OLSE in the general model under suitable normalization. We then discuss model (1.1) as a special case in Section 2.3.

2. l. The genera l model

Consider the following general model containing several systems of integrated regressor variables Xt as well as some stable (in the sense of (C4) and (C6) below)

variables zt:

Yt = ~:'zt + ~'X, + u,, (2.1)

where

zt = (z,~ . . . . . zt~)', ~ = (yl . . . . ,Tk)',

I t t t = (~'11 "" '~p) , ~j = (~j l . . . . , ~ j , , ) , j = 1, . . . , p ,

I ! t r X = ( X t t ,11 . . - t X t , p ) , X t, j = ( X t j l , X t j 2 , . . . , X r j m ) , j = 1, . ,p .

We assume that all the elements of X, are integrated, i.e.,

Xt, j = X I , ~ - I + "" + X , , j - 1 , j = 1 , . . . , p .

The elements ofX~. o = xt = (x t l , . . . , Xtm)' can be random or deterministic or mixed. For simplicity of analysis, in model (2.1) all m systems of integrated regressors in Xt are considered to have the same order of integration p. Analysis of models with different orders of integration in the variables is discussed after Theorem 1. Examples of stable regressor variables in z, are seasonal dummy variables and lags of differenced depen- dent variables.

In order to establish the limiting distribution of the OLSE, we make the following assumptions on xt, z, and ut. Let D = D [0, 1] denote the space of all real-valued

328 D. Wan Shin, S. Sarkar/Journal o f Statistical Planning and Inference 64 (1997) 325-340

functions that are right continuous at each point of [0, 1] and have finite left limits, equipped with the Skorohod topology.

Assumption 1. There exist sequences of nonsinyular real, symmetric matrices An and B., real number a 2 > O, a random vector !Pxu, a random matrix I2xz, a m-vector function g ~ D m, a k-vector function h ~ D k, and nonsingular matrices r,=, ~,z. such that jointly, as n ----~ oc ~

[n r] (C1) A. 1 ~ x, d ,g(r) , 0~<r~< 1,

t = l

[,, r] d (C2) (aen) -1/2 ~ u, ,W(r) , O<~r<<. 1,

t = l

(C3) (aZn) 1/2A[1 xi u, ' ~x., t = l \ i = 1 /

[n r]

d h(r), O<.r<. 1, (C4) B~ -1 Z z, , t = l

( C 6 ) n- 1 ZtZt , ( t 7 2 n ) 1/2 ZtUt ) ['~'zz, Nk(O, ~'zu)], 1 = 1 t = l

where W (.) is the standard Brownian motion on [0, 1], and Nk(O, r,z.) denotes a k- variate zero-mean normal vector with variance covariance matrix ~.~..

We assume (C3) in addition to (C1) and (C2) because, in general, (C3) does not follow from the joint convergence of (C1) and (C2). For the same reason we assume (C5). Since (C3) is a special case of (C5) we discuss only assumption (C5) in detail. Condition (C5) involves the limiting distribution of the integral process

a - lV~,,r~ ,~ and B.- lx-E.rl, respectively, S~X*(r) dZ*'(r) where X*(r) and Z*(r) are , . , /_,,= 1 ~, ~,= 1 ~,, which are sequences of cadlag (i.e., right continuous with left limits) processes converg- ing weakly in Skorohod topology to {g(r),h(r)}. The asymptotic distribution of stochastic integrals has been studied by several authors. Chan and Wei (1988) study the case when X*(r) and Z*(r) are martingale arrays with uniformly bounded condi- tional variances. Kurtz and Protter (1991) obtain very general results for martingale arrays. These authors consider various limit theorems when Z*(r) is a semimartingale. In case of martingale processes Z*(r), the weak convergence limit fax= in (C5) follows from (C1) and (C4) and is of the form

~2x= = 11 g-(r)dh'(r), jo

D. Wan Shin, S. Sarkar/Journal of Statistical Planning and lnlerence 64 (1997) 325 340 329

where g - ( .) is the left limit of g(. ). Also, when xt is a square integrable martingale difference sequence, (C5) (as well as weak convergence of A~ly~7_ l x, z',B~ 1) follows from (CI) and (C4) because we have the identity

When ZT, (r) is not a martingale the result gax= = j'olg - (r)dh'(r) is typically violated. Phillips (1988) studies strictly stationary linear processes for ZT, (r). Hansen (1992) extends this analysis to cover strong mixing (a-mixing) sequences. However, their results do not establish convergence of (C5) from the joint convergence of (C1) and

(C4). They derive weak limit g2xz in special cases such as {x, = ut-~, zt = u , } ,

Ixt = zt ® zt }, {xt is a near integration of ut, zt = xt }, and in such cases, the form of the limit 12xz is not ~ g (r)dh'(r) and does not have a general expression. For details see Phillips (1987, p. 284) and Hansen (1992, Section 4). Thus, in general (C5) does not follow from (C1) and (C4).

Usually, the components of function g in (C1) of Assumption 1 is either determinis- tic or a standard Brownian motion. The standard Brownian motion as a limit in (C2) for the ut is guaranteed by conditions such as

(I1) E(ut) = O for all t, sup, Elu, I ~+~ <0o for some r > 2 and r > O, ~ut¢¢ ~ is strong-

mixing with mixing coefficients 6m satisfying S~9~ -~2/~) < o% a 2 = lim . . . . n - 1E(Ul + • .- + u,,) 2 exists and a 2 > 0; or

(I2) u, = yq~_o~;e, ~ where {et} is an iid sequence with E(et) = O, E(e 2) = ace < ~ , oc 2 S j 2 ( o 2 < ,:3C and a 2 = aee(Ej=O4)j) > O.

Conditions (I1) and 02) are due to Herrndorf (1984) and Phillips and Solo (1992), respectively.

Consider the case k = 0 and m = 1 that include important models (1.2) and (1.3). Conditions (C1) (C3) are satisfied under some mild regularity conditions when x, = i or x, = u,_ 1. If xt = 1, then (C1)-(C3} are satisfied with

A, = n, g(r) = lim n 1 ~ x , = r , ~Px, = r d W ( r ) = W ( 1 ) - W(r)dr . n ~ o o t = l

Let p lim . . . . denote the limit in probability as n -+ a:.. Let a~, = p lim ... . . n- i~,= l u 2, which is guaranteed to be finite under assumption (I2), or under assumption (I1) (by Phillips, 1987, Lemma 2.2). If x, = u,_ 1 and (I1) and (I2) holds, then (C1)-(C3) are satisfied with

A n = ( G 2 n ) t/2, g( r )= W(r), T x . = 2 l[w2(1)--cT-2fruu],

because, by summation by parts,

~ ( ~ ) ~ ( , ~ ) 2 - ~ [ ( ~ , )z ~ u 2] X i U t = U i bl t = U t - -

t = 1 i t = l \ i = 1 t t = l

330 D. Wan Shin, S. Sarkar/Journal of Statistical Planning and Inference 64 (1997) 325-340

and, hence,

(o~n) ~"~ a21 xi u~ , % . . t = l i = 1

We next discuss condit ion (C6). The convergence of n - l y T = l z , z't to a positive

definite matrix 2 = essentially assumes linear independence of the elements of z,. Also

convergence of (aZn)-l/2 ~ _ ~ zt ut to Nk(O, 1~:,) is usually satisfied when zt is determin-

istic or a r andom sequence independent of u,, in which case the classical central limit

theorems are applicable. The assumpt ion of zero mean for the asymptot ic normal

distribution is required for consistency of the estimator. Otherwise, dependence

between zt and ut makes the O L S E inconsistent.

2.2. The asymptotic distribution

Let 0 = (?', ~')', and let 0 = (~), 01')', be the O L S E in model (2.1) defined by

where Vt = (z,,X~)'. We now present the joint limiting distributions of the cross

p roduc t terms y~_ 1X, XI, ~ _ ~ X~ut and ~ _ 1X~z~. First, we define some notation. Let

g o = 0 , , × l , an m x l vector of zeros, g1 = g , g ~ ( r ) = S o g j l(r)dr, j = 2 . . . . . p, and

let S = (g'l . . . . . g~)', $1 = (g'o,g'~, ... ,g'p-1)'. Let D, = diag(An, An n, ... ,Ann p-~) and

~kx, = ~ u - - g l ( 1 ) W ( 1 ) .

L e m m a 1. Let model (2.1) hold with Assumption 1. Then

i fl (i) n - l D 2 1 X tX;D21 d ' S(r)S ' (r)dr , t = l 0

, o S(1)W(1) -- W ( r ) S ~ ( r ) d r - C(0xu) , t = l

1 X . ' B n 1 d (iii) n J+~A~ t, jZt ,gj(1)h ' (1) -- g~_~(r)h'(r)dr, j = 2 . . . . . p, t

where C(0x,) = (~b'x,,, 01 . . . . . . . . . 0 1 × m ) ' is a (pm) x 1 vector, Om×n is an m × n matrix of zeros. Moreover, the convergence of (i) (iii) hold jointly.

Proof of L e m m a 1 is given in the appendix. Theorem 1 establishes the asymp-

totic distribution of the O L S E under suitable normal izat ion and it follows from

Lemma 1.

Theorem 1. Let model (2.1) hold with Assumption 1. Let C(t)~,), {gj, j = 0, 1,2 . . . . . p},

S, $1 and Dn be as defined above. Assume that Sl S(r)S(r)' dr is nonsingular and that

D. Wan Shin, S. Sarkar/Journal of Statistical Planning and Inference 64 (1997) 325 340 3 3 1

l i m , . ~ n - t B , exists. Le t J , = diag(D, , Ik) where I k is the k × k identity matrix. Then

the asymptot ic distribution o f n 1/2 Jn(O- O) is given by

FY, zz F' ] - l FNk(O,~zu ) 1 (2.3) g L r ~ S ( r ) S ( r ) ' d r [S(1) W(1) - ~ W ( r ) S l ( r ) d r + C(~9x,)

and

[ fo ) F S( l )h ' (1 ) Sa(r )h ' ( r )dr ' ' ~ ' ' " n 1 = - --(O~xz~0k×m~"'~0k×m) X hm B, , j \ n ~ -~c

09x_, = I2x~ - gl(1)h '(1) .

T h e o r e m 1 can easily be modif ied in the si tuat ion where orders of integrat ion in the

systems of regressors are not the same. In this situation, the number of elements in Xt,; is not the same for all j = 1, . . . , p and we modify the definition of ~, 0, S and $1

accordingly, and Theo rem 1 remains true. Fo r example, let us consider a model with

two systems of integrated regressors. Assume that the orders of integrat ion for the first

and second systems of regressors are two and one, respectively. In this case we have

X, = (Xt, 1 lX[, 2 f = (Xtl 1, X t 1 2 1 X t 2 1 f ,

• : (~'11 ~2)' : (~11, ~12] ~21)', S = ( g ' l ' ~ g ' 2 ) ' = ( g l l , g 1 2 g 2 1 ) ' .

Then the O L S E 0 is compu ted using this Xt and Theorem 1 is applicable with these and S. When zt is a ze ro -mean error sequence satisfying the invariance principle such as

lags of ut (when u, is white noise) or a ze ro -mean r a n d o m sequence independent of

(xt, u~) and having covar iance matr ix X~, then (C6) and the invariance principle in (C2) is usually satisfied under minor regulari ty condit ions on zt with B, = n l / Z Z ~ / 2 .

Then lim . . . . n - 1B, = 0 and F = 0, giving the inverted matr ix in the limiting distribu- t ion expression in (2.3) a block diagonal s t ructure and yielding

- , a S ( r ) S ' ( r ) d r S(1) W(I )

i' ] - o W ( r ) S l ( r ) d r + C(Ox,) , (2.4)

n l / 2 ( f , _ ~,) d , Nk(O,Y,L~ X~,,~L~). (2.5)

Fo r example, if Yt is integrated of order p, one can consider z , = VPy ,_ I , . . . ,

ztk = V P y t - , to incorpora te lags of VPy, in the regression (2.1). Then the z, = (z,1 . . . . . Ztk)' is stable satisfying (C4)-(C6) and in this case F in Theo rem 1 is a zero

matrix. However , when {zt} is not a ze ro -mean sequence such as the constant or the seasonal dummies , F is not a zero matr ix and we do not have a nice expression as in (2.4).

332 D. Wan Shin, S. Sarkar /Journal o f Statistical Planning and Injerence 64 (1997) 325-340

2.3. Model (1.1)

Now we discuss model (1.1) as a special case of model (2.1) with m = 1 and no z,. Under model (1.1), A. = a, for some real sequence a,, D, = a, diag(1, n . . . . , n p- i), G(r) = [91 (r) . . . . . 9p(r)]', G1 (r) = [9o(r), . . . , 9p- 1 (r)]', 9~(r) = ~o9~-1 (s) ds, j = 2 . . . . . p, go(r) = O, gl(r) = g(r),

~bx, = ~x. - gl(1)W(1), (2.6)

C(s) = (s,0 . . . . . 0)'. Then we have the following result as a corollary of Theorem 1.

Corollary 1. Let model (1.1) hold with (C1), (C2), (C3) of Assumption 1 for m = 1. Assume ~ G(r)G'(r)dr is nonsingular. Then

-- , a G(r) G'(r)dr G(1) W(1) 0

Note that if 9 and ~xu of Assumption 1 are given, the limiting distribution in Corollary 1 is completely determined. Since ~x. can be expressed in terms of 9 and 0x. as in (2.6), we use the notation

L(g, ~,x,)= [ f~ G(r)G'(r)dr I l [ G ( 1 ) W ( 1 ) - f ] W(r)Gl(r )dr + C(~x,)]. (2.7)

The limiting distribution aL(g, ~x,) is general in the sense that it can provide the limiting distribution for various {(xt, ut)} under Assumption 1 for testing various null hypotheses when data are fitted with polynomial regression, unstable autoregression and other integrated-regressor models. For example, in Section 3, the limiting distri- bution of the OLSE in the polynomial regression (1.2)is shown to be aL(g, 0) and that in the unstable autoregression (1.3) is shown to be L(gw, ~bx,), where

g(r) = r, gw(r) = W(r), ~x, = - 2-1(W2(1) + a 2a,,),

a , , = p l i m n - 1 ~ u~. n~ ~, t= 1

3. Applications

In this section we apply Theorem 1 and Corollary 1 to derive the limiting distribu- tion of the OLSE in different models. Corollary 1 is applied to give unified asymp- totics in the unstable autoregression and polynomial regression models. Theorem 1 is applied to autoregression with a polynomial trend and stable regressors.

D. Wan Shin, S. Sarkar/Journal of Statistical Planning and Inference 64 (1997) 325 -340 333

3.1. Unstable autoregression

Consider the situation when Yt is generated by the l (p) process VPyt = ut and

modeled by the autoregressive model Yt = ?xYt i + "'" + 7pYt ; + ut, the recursion

meaning that the yt evolve from initial values Yo, - . . , Y(-p +a) with the innovat ion u,. Using the difference opera tor I7, we reparameterize the above model as

yt = ~1 Vp- l y t 1 + "'" + ~ p - 1 V Y , - I + 7pYt-1 + ut (3.1)

which is of the form (1.1) with x~ = ut-a and

Xt,,i • VP-JY t -1 = ~ ( V P - j + l y i - I ) = i Xi , j 1. i -1 i-1

Since the initial values of Yo, Y- 1, . . . , Y- v + 1 do not affect the asymptotics we assume them to be zeros. The vectors ~ = (71 . . . . . )'p)' and ~t = (~1 . . . . . at,)' are related by 7 = R~t where R is a p × p matrix whose (i , j) th element is equal to ( - 1) p - j - i + 1 ~i-~P-i~

with (o-~) _ 0 i f / + j > p + 1. In Corol lary 2 below we give the asymptot ic distribu-

tion of the O L S E of 0t in model (3.1).

Corollary 2. Suppose the data Yt is generated by ~TPy t = U t and model (3.1) is f i t to the

data. Let ~ denote the O LS E of ~. Assume that {ut } satisfies (I 1) or (I2) defined in Section 2. Then

In(ill -- ~1) . . . . . n p ( @ _ ~p)], d >L(g, t~x~),

where L(g, ~ , ) is given in (2.7) with g(r) = W (r) and

G(r) = (Wl(r), W2(r) . . . . . Wp(r))', Gl(r) = (0, W l (r), W2(r), . . . , Wp-1 (r))',

~b~, = - 2-~(W2(1) + a Za,,,), Wl(r) = W(r),

W j ( r ) = f~ Wj_x(s ) ds, j = 2 . . . . . p.

In L(g, ~ , , ) , nonsingulari ty of~01 G(r) G(r)' dr is guaranteed by Lemma 3.1.1 of Chan and Wei (1988). When a,u = a 2 the asymptot ic distribution reduces to L(g, 0) which is

the same as that in Theorem 3.1.2 of Chan and Wei (1988), developed for martingale

differences {ut}. Therefore, Corol lary 2 is an extension of Theorem 3.1.2 of Chan and

Wei (1988) because the u~ in Corol lary 2 is allowed to be a more general process

satisfying (I1) or (I2).

As an application of Corol lary 2, one may construct a Phillips (1987) type

test for double unit roots when the errors are autocorrelated as in (I1). Consider the model

Yt = 0~1 Vy t -1 + ~2Yt-1 ~- ut,

334 D. Wan Shin, S. Sarkar/Journal o f Statistical Planning and Inference 64 (1997) 325-340

where 0~ 1 = ~ 2 = 1 and ut is an error process satisfying (I1). Let (c21,022) be the OLSE. Then the limiting distr ibution of the normalized O L S E (n(~2~ - I), n 2 ( ~ 2 - - 1 ) ) is given

by

- 1 I ~ W ~ ( r ) d r ~ W , ( r ) W2(r)dr 2 ~ ( W Z ( 1 ) - a - z a , , )

~ W~(r) W2(r)dr ~ W~(r)dr W2(1)Wl(1) -- ~o ~ W2(r)dr

Then one can construct the double unit roots test as in Hasza and Fuller (1979) after adjusting for a -2a .u .

3.2. Polynomial regression

Consider the polynomial regression model with an intercept

Y, = flo + flat + " + ~pt p + ut. (3.2)

We reparameterize (3.2) as

Y, = ~o + ~l t + ~2(t2/2[) + ... + ~p(tP/p!) + ut, (3.3)

which is of the form (2.1) with m = 1, x, = 1, q = 1 and zt = 1 because

X t , 1 = t, X t , 2 = ~ X i , 1 = 2- 1 t(t + 1) ~_ tz/2!, i = 1

X,. 3 ~- t3/3! . . . . . X,, p ~- tV/p!. (3.4)

Note that X,, j is t~/j! plus terms which are negligible compared to t J/j!. Hence we can

use t~/j! instead of X, , j in (3.4). An application of Theorem 1 gives the limiting

distribution of the O L S E in Corol lary 3 with A, = n, g(r) = r, and ~Px, = 0.

Corollary 3. Let model (3.3) hold. Assume that {ut} satisfies (I1) or (I2) defined in Section 2. Then

[ n l / 2 ( ~ 0 - - ~ 0 ) , n 3 / 2 ( ~ 1 - - ~ 1 ) . . . . . np+l /2 (~p - - O~p)]' d , a H _ t K ' (3.5)

where

H = fjS*(r)S*(r)'dr, K = S * ( 1 ) W ( 1 ) - f ] W(r)S*(r)dr , (3.6)

S*(r) = [1,gl(r) . . . . ,gp(r)]', S*(r) = [0, 1, gl(r) . . . . . gp-x(r)],

gl(r) = g(r) = r, gj(r) = f l g j l ( r)dr = rJ/j!, j = 2 ~ po

F r o m Corol lary 3, after evaluating H, we have

[nl/Z(flo - flo), n3/Z(f11 -- fla) . . . . . np+ l/2(flp _ tip)], a aM_ ~ V, (3.7)

D. Wan Shin, S. Sarkar/Journal o f Statistical Planning and Inference 64 (1997) 325 340 335

w h e r e M i s a (p + l ) × ( p + 1) matrix with the (i,j)th element equal to (i + j - 1) -1 .

and

V = IW(1), W(1) - f~ W(r)dr, W(1) - 2 f l rW(r)dr . . . . . W ( 1 )

_ p r p - l W ( r ) dr . 0

Note that the limiting distribution a M -1 V is the same as (2.2) of Kulperger (1987)

established for a s tat ionary AR(p) process ut. Corol lary 3 is a generalization of (2.2) of

Kulperger (1987) since the error ut in (I1) or (I2) is more general than a s tat ionary

AR(p) process considered by Kulperger. When g ( r ) = r the nonsingulari ty of

H follows from the linear independence of the set of functions {1,,ql(r), . . . . gp(r)} = {1, r, r2/2, . . . , rP/(p!)}.

In the autoregression

p

Yt = 0¢0 + 2 0 C j v P - J y t_ 1 + ut, j - 1

whose true model is V P y t = Ut, the limiting distribution of the O L S E is

[n1"2(~o - ~o) , n ( ~ l - ~1), . - - , nP(:?p - %)] a , H - 1 [ K + (0, ~'x. , 0 . . . . . 0) ' ] ,

t3 .8)

where 0x, = - 2 1(W2(1) + a-2a , , ) , and H and K are as defined in (3.6) but are

evaluated with g(r) = W(r) instead of g(r) = r.

One can construct a Phillips (1987) type test for the model with an intercept

Yt = ~0 -~- ~ l Y t - - 1 -~- /~t,

where ~o = 0 and cq = 1,u, is an error process satisfying (I1). N o w by (3.8) the

asymptot ic distr ibution of n(~l - 1) is given by

Dii ( Ir dr) ] l 'lt ] 3.3. Regression model with a polynomial trend and stable random components'

A nons ta t ionary time series Yt may be estimated by the following general regression

model

yt = ~j + ~ f l j vq -Jy , -1 + ~ 7jz,,j + ut. (3.9) j = 0 j = l j = l

Since the initial values of Yo, Y- 1, .--, Y- q + 1 do not affect the asymptot ic distribution of the parameter estimators we assume them to be zeros. Let the true model for y, be

p

Vqyt = ~ ccj(tJ/j!) + u*, (3.10) j = o

336 D. Wan Shin, S. Sarkar /Journal of Statistical Planning and Inference 64 (1997) 325-340

where u* satisfies (I 1) or (I2) defined in Section 2, and the zt. f s are r a n d o m processes satisfying condi t ions (C4) and (C6) with B, = nl/ZL h(r) being a k-dimensional zero-

mean Brownian motion. Fo r z,, f s models with lags of Vqyt (when ut is white noise) or of a r a n d o m process independent of the ut are c o m m o n and in such cases condi t ions

(I1) or (I2) for zt, f s are satisfied. If it is desirable to test the hypothesis that y, is an AR(q + k) process having q unit

roots and other k roots inside the unit circle, one m a y fit model (3.9) with zt.j = V q y t - l - j . Under the null hypothesis (C4) will be satisfied and C o r o l l a r y 4

below gives the null l imiting distribution. If in model (3.9) u, is white noise and the

zt. f s are independent of the ut, then in model (3.10) u* is ( ~ = 1 ";jzt, j + u,), which is a stable process because Vqy, 1 j is s tat ionary. When :to = :tl . . . . . :tp = 0, (3.10) becomes Vqyt = u*. When some 2j's are nonzero, the asympto t ic behavior of the )'t in (3.10) is the same as that in the po lynomia l regression model. Thus, (3.10) accom-

moda tes both model Vqyt = u* and the po lynomia l t rend model. Corol la ry 4 below states that the limiting distr ibution of the normal ized O L S E in model (3.9) is no rmal when Yt behaves like a po lynomia l t rend dis turbed by noise and is not normal when Yt is purely nons ta t ionary and random. Model (3.9) is of the form (2.1) with

xt = (1, Vqyt 1)' and, therefore, we have the following result as a direct consequence

of Theo rem 1.

Corollary 4. (i) Assume that :to = :tl . . . . . :tp = O. Then

[ n o ' ( f l a - - •1 ), " ' ' ' nqtT(flq - - • q ) , na/e(~o -- :to), n 3 / Z ( ~ l - - ~1 ), . . . , n(l/Z)+P(~p -- ~ p ) ] '

Ill 0' ]-' [0( I]' d , ~ r )Q( r ) ' d r 1) W(1) - Ql(r) W( r )d r - (Oxu,O, ... ,0 ,

where

Q(r) = [W1 (r) . . . . . Wq(r), 1, r / l! . . . . . rP/p!] ',

Qa(r) = [0, Wa(r), . . . , Wo-x(r), O, 1, r / l ! . . . . ,rP-1/(p - 1)!]',

i / / x . = - - 2 - 1 ( W 2 ( l ) + o - 26uu ).

(ii) Assume that some of the :tfs are nonzero. Let ~c = m a x { j : ~ j # 0}. Then, the asymptotic distribution of In ~ + 3 / 2 0 ( ~ 1 - i l l ) . . . . , nK +q+~l/2)tY(flq -- flq), n 3 / 2 ( ~ 0 - ~0 ) ,

n5/2(~1 - ~1 ) . . . . . n ( 3 / 2 ) + P ( ~ r - - : t p ) ] ' is

t r [ f j T(r)T(r) 'dr]+ [ T ( 1 ) W ( 1 ) - f~ T1(r)W(r)dr],

where

T(r) = [r ~+ ~/(~: + 1)!, ... ,r~+V(~: + q)!, r, r2/2! . . . . . rP/p!] ',

Tx(r) = l-O, r ~+ 1/(,~ + 1)!, . . . , r ~+°- 1/(,~ + q - 1)!, O, r, r:/2! . . . . . r p- 1/(p _ 1)!]',

D. Wan Shin, S. Sarkar/Journal of Statistical Planning and Inference 64 (1997) 325-340 337

and A + is the Moore-Penrose inverse of the matrix A. The limitin 9 distribution is

a normal distribution with a singular variance covariance matrix.

The asymptotic distribution of the OLSE of 7Ss in model (3.9) is given by (2.5). Fuller et al. (1981) studied model (3.9) for q = 1 and independent {us}. There are situations when the order of integration q ~> 2. See Dickey and Pantula (1987) and Haldrup (1994). In Corollary 4, we considered model (3.9) for any q ~>1 and a general error process {ut}. In the testing for double unit roots problem considered by Dickey and Pantula (1987) and Haldrup (1994), an alternative to the null hypothesis Ho: VZyt=ut * is the trend hypothesis HI" V2ys =

~ : o ~ j ( t J / j ! ) + u*. Note that H~ represents a polynomial regression of order (p + 2)/> 2. Since a polynomial regression model of order greater than one is an alternative to the double unit root hypothesis, H~ defines one possible alternative to H0. Now Corollary 4 gives the limiting distribution of the OLSE under both the null and alternative hypotheses.

Appendix

Proof of Lemma 1. Observe thatA~-lxE,,r],l = A , -1vEer1 ~" d z~s:l ~, ,g l ( r ) = g(r). Then

[,,r] f l n-lAnlx[,,@2 = n - l A ; 1 ~ Xt, 1 d > g z ( r ) : gx(s)ds

t = l

because of (C1) and the continuity of the map gl ~ ~og~(s)ds and the continuous mapping theorem (Billingsley 1968, Theorem 5.1). Using the same argument repeatedly we obtain

[,,r] n i+ IA; 1X[,,~],j = n- 1A; 1 ~ Xt, j -a ~d gj(r) (A.1)

t = l

for j = 2, ... ,p. Hence, proof of (i) follows from continuity of the map (g,g*} --. 5~g(r) (g* (r))' dr.

Now we prove part (ii). Let Sn, j = Y~_ iXs.jus. F o r j >~ 2, summation by parts gives

Sn, j : X i , j 1 Ht t : l i

= Xt ,~- i u, - ~, ui Xs, j I + S , , j 1. (A.2) t = l s t = l i = 1 /

By (C 1) and the continuous mapping theorem, we have

fi n-J+lA, -1 X~,j-1 ' g j _ l ( r ) d r = gj(1), j = 2, ... ,p; (A.3) t = l 0

3 3 8 D. Wan Shin, S. Sarkar/Journal of Statistical Planning and Inference 64 (1997) 325-340

and (C2) gives,

( 0 . 2 / , / ) - 1 /2 Ut ,

\ t = l

By (A.1), (C2) and the continuous mapping theorem we also have

( a 2 n ) - l / 2 n - J + l A 2 1 ui X t . ~ - I , W ( r ) g j _ l ( r ) d r , j = 2 . . . . ,p. (A.5) t = l i = 0 0

Also, (C3) implies

(¢Tzn)_1/2 A_ l S d n n, 1 ) ~ . . (A.6)

Since from (A.2) we have

Sn, j-1 ~Sn, l-~ i~_12 { ( t t l I t ' k - l ) ( t=~l bit) - - t t 1 ( i t l bli) xt'k-x }

= Ot , (nJ -2n l /ZA , ) for j = 2 . . . . . p.

Therefore,

(¢72n)_ l /Zn_an_J+ZAy1S . d , O, j = 2, ,p. (A.7) n,J 1 "'"

Result (ii) then follows from (A.3)-(A.7). We now prove part (iii). Letting P,, j = ~ = 1 I t , jz't, summation by parts yields

P , , j = ' -- It, j-1 + P , , i - a . (A.8) t t t = l i 1 I

By (A.3) we have

n J+IA21 X~, j -1 , gj_l(r)dr =gj(1), j = 2 . . . . . p; t=l and (C4) gives,

zl B.-1 d , h'(1). (A.9) t By (A.1), (C4) and the continuous mapping theorem we obtain

I tl ( ~ 1 ) ] f: n J + l A Z l Xt, j 1 Z'i B , 1 d , g2 a(r)h'(r)dr, j = 2, . . . ,p. (A.10) t = i

Also, (C5) implies

A[1p B - 1 d ,,o , ' g2xz. (A.11)

D. Wan Shin, S. Sarkar/Journal of Statistical Planning and Inference 64 (1997) 325 340 339

F r o m (A.2) we h a v e

3 n n Z '

Po,~ t = P . , l + X , , k - 1 z', - X , , k - 1 i k=2 t t t= i=1

=Op(n ~ 2A,B,) for j = 2 . . . . . p.

The re fo re ,

n_J+lA~lp, , j_lBffX d , o , j = 2 , . . . , p . (A.12)

F r o m (A.8)- (A.12) it t hen fo l lows tha t

n ' i + l A f f l P . , j B n l d f ~ , g j (1 )h ' (1 ) -- gj_l(r)h'(r)drj = 2 . . . . . p,

w h i c h is resul t (iii).

T h e j o i n t c o n v e r g e n c e is ver i f ied by (C3) and (C5) a n d c h e c k i n g the c o n t i n u i t y of

the func t iona l E(g, w, h): D m + 1 + k ~ D"P x ~ mz p2 + ( 1 + m ( p - 1)) + (p - 1)km given by E(g, w, h) 1 = [ (g l . . . . . gp), (~og,(r)gj(r)dr, i = 1 . . . . . p, j = 1 . . . . . p), (w(1), ~w(r)g2(r)dr . . . . .

~ w(r)gp(r)dr), (~gl(r)h'(r)dr, ... ,~l gp l(r)h,(r)dr)]. []

Acknowledgements

T h e r e sea rch o f the first a u t h o r was s u p p o r t e d by K o r e a Resea rch F o u n d a t i o n . T h e

r e sea rch of the s econd a u t h o r was pa r t ly s u p p o r t e d by a G r a n t f r o m the Co l l ege of

Ar t s a n d Sciences at O k l a h o m a Sta te Un ive r s i ty . T h e a u t h o r s wish to t h a n k a referee

for m a n y helpful c o m m e n t s and sugges t ions tha t led to the p resen t i m p r o v e d ve r s ion

of the paper .

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