Spurious regressions when stationary regressors are included

economics letters EI~SEVIER

Economics Letters 50 (1996) 25-31

.....

Spurious regressions when stationary regressors are included U w e Hassler Institute of Statistics and Econometrics, Free University of Berlin, Boltzmannstrasse 20, 14195 Berlin, Germany Received 14 October 1994; accepted 15 April 1995

Abstract

Spurious regressions of I(1) variables without drift are analyzed when additional I(0) regressors are included. The asymptotic distributions of Phillips (Journal of Econometrics, 1986, 33,311-340) are embedded in our results.

Keywords: Unbalanced regressors; Integrated and stationary time series JEL classification: C22

I. Introduction

Since the Monte Carlo evidence on spurious regressions by Granger and Newbold (1974), this phenomenon has troubled applied time series econometricians. Only the work by Phillips (1986) provided a theoretical understanding of spurious regressions. Phillips (1986) considers variables Yt and x2, integrated of order oneA I(1), without drift and finds that, even if they are independent, the least-squares estimator /32, where Yt =& + ~2X2t '~ Ult,

(1)

does not converge to zero and that the t-statistics diverge. Moreover, the coefficient of determination has a non-degenerate limiting distribution, while the Durbin-Watson statistic approaches zero asymptotically. These results have been extended recently. Choi (1994) investigates spurious regressions when the m2-dimensional I(1) vector of regressors is cointegrated in the sense of Engle and Granger (1987): y'x2, is stationary, or integrated of order zero, I(0), where 7' = (71,- • •, %,2) is a vector of constants. If 7 / ~ 0, j = 1 . . . . , m2, Choi (1994) derives that the t-statistics of /32 in (1) do not diverge, but he states that the t-statistic of the jth component will diverge if ~/j = 0. Haldrup (1994) introduces an additional vector of variables x~t integrated of order two, i.e. Y , = a +/3'1xl, +/32x2, +/~2t' 0165-1765/96/$12.00 © 1996 Elsevier Science S.A. All rights reserved SSDI 0165-1765(95)00728-8

(2)

26

U. Hassler / Economics Letters 50 (1996) 25-31

and studies spurious regressions, a m o n g other things. In this paper, it is assumed that x~, in (2) is I(0). This situation is also considered by Park and Phillips (1989), provided y, and x2t are cointegrated. Given cointegration, they find in their t h e o r e m 3.1 that /3~ from (2) is T~/Z-consistent, where T denotes the n u m b e r of observations. This contrasts with our results in the case of no cointegration, where the true p a r a m e t e r values of/3~ and 82 are zero. We find that/3~ has a non-degenerate limiting distribution. Nevertheless, the t-statistics associated with /3~ converge, which parallels the findings by Choi (1994) just m e n t i o n e d . T h e asymptotic distributions of/32 and the associated t-statistics in (2), however, are identical to those in (1), as originally derived by Phillips (1986).

2. Preliminaries

Let vit , i = 0, 1, 2, be vectors of zero m e a n variables of dimensions m 0 = 1, m 1 1> 1, m 2 ~ 1 respectively. Let us define Y, = Y , - 1 + Vo, ,

x2, = x z t - ~ + % , ,

l

m2

as I(1) variables, while X l t "~- V l t m I

is s u p p o s e d to be I(0). With w~ = (vo,, v'~,, v'2,), we consider the partial sum process: S,=~wj, j=l

t=172,...,T,

and assume the following multivariate invariance principle after normalization:

T-O'5S[Tr]~

B(r),

r E [0, 1],

(3)

w h e r e B'(r) = (Bo(r), B'~(r), B'z(r)) is a vector of Brownian m o t i o n partitioned according to v~t with covariance matrix = lim T - 1 E ( S r S ~ ) . T--~ oc

A s s u m p t i o n s on vi, that guarantee (3) are given in Phillips and Durlauf (1986, t h e o r e m 2.1). F o r the sake of simplicity, let us assume that vit, i = 0, 1, 2, are i n d e p e n d e n t vectors, so that ~O = (Oi) is block diagonal) F u r t h e r m o r e , we simplify matters by supposing that vi, are stationary and ergodic with covariance matrices: T

~i.r = p l i m T -~ ~ vitu~t_r, T~*c

i = 0, 1, 2,

~- = 0, 1.

(4)

t= 1

To avoid cointegration (i.e. to obtain spurious regressions) of y, and x2,, we only had to assume that 12 is regular (cf. Phillips, 1986, p. 321).

U. Hassler /

EconomicsLetters50 (1996) 25-31

27

Finally, the following lemma collects some results from Phillips (1986, lemma 1) and Park and Phillips (1989, lemma 2.1) needed in the next section. For the rest of the paper, let x, denote -1 T the mean corrected series, x, = x t - £ with £ = T E,= l X t .

Lemma. With the notations and the assumptions introduced above the following terms converge in distribution as T---~~:

'

r -2 ZT ~,2~fB~o(r)dr/=1

(j

Bo(r) dr

=:

-B2o ,

0 1

T

1

' fB2(r)B;(r)drf

r-2Zx_~_~,~ /=1

B~(r)dr

0

1

"

1

f B'~(r)dr="f _B~_B~, '

0

0

I

I

f B2(r)Bo(r ) dr - f B2(r) dr f Bo(r) dr =" f _B2_Bo,

T - 2 ~ x2,2, ~ t=l

0

0

0

1

f

T-1Z Xlt~.t~ t=l

1

Bo(r) dB,(r) - B 1 ( 1 )

0

f

1

f

T - ' ~'~ X ~ a t ~ t=l

Bo(r) dr=:

0

f

_BodB~,

1

f

B'z(r) dr =:

(Bo(r),B'2(r))dr="

(B o , B ; ) .

B'z(r) dB,(r) - B,(1)

0

0

f

p

_B2 d B , ,

1

t

T -15 ~, (Y,,X2,)~ t=l

0

3. Spurious regressions Let us rewrite the regression equation (2) as ¢

A t

,v,=/3,xl,+/32x2,+u2,,

t=l,2,...,T.

(5)

Applying the formula for inverting partitioned matrices, the least-squares estimators from (5) are 2 (~2)=(

g

FH)(Zxlt~tlZX2t'~t,/

with 2 For the rest of the paper, all summations run from t = 1 to T if not indicated otherwise.

(6)

28

U. Hassler / Economics Letters 50 (1996) 25-31 g

=

(Z

t ' X p Xl,X2,(X _x2,_x2,)-' X_2g,,)

, X_ltx_mt- Z

'(2 F = - E £ _x1,~2, g = (•

,)-1 Xzt~_Zt

,

-1

,

G = F' ,

I],

_x2rzc_'2,)-'[ 1 + ~2 x_2~;,E 2 x,tx_'2,(2 x2t_x'2t)

w h e r e I denotes the m2-dimensional identity matrix. With (4) and the above l e m m a , it holds as T--> ~ : -1

plim T E =

,

1,0

'

T2F ~

-

T2H~

(f

1,0

~ 2

,

dB1

2B- 2

'

,) 1

B_2B_ 2

(7)

Now (6) and (7) provide the limiting distributions of/3~ a n d J~2 by the continuous mapping t h e o r e m . Moreover, we are interested in the limits of the coefficient of determination:

R2

/~; X-----XltX'lt/31-]- 2/~'2 X X2tXlt~31 ' * + J~2, g X2tX2tJ[3 , *2

of the residual variance:

V - ' s 2 : v -2 3". ( ~ , - f i ,!x . -

^t 2 ~2_x2,) .

and of the D u r b i n - W a t s o n statistic: T dw =

T -1 xtT=2 (Oo,- j~;(Xlt--Xlt_l ) -- fi'202,) 2 T-1s 2

M o r e o v e r , let us consider the t-statistics to test for the significance of the kth c o m p o n e n t of fil and the jth c o m p o n e n t of fi2:

[fil]k t~-,-.~/2, SP_. kk

[fi2lj b-..1/2,

k=l,2,.-.,mt,

SI-I jj

j=l,2

....

,m 2 .

(8)

Finally, we look at the F-statistics testing for c o m m o n significance, H0:/3; = 0, i = 1, 2: F1-

j~, -1 ^ 1E J~l S2ml

'

F2

-

fi2H-lfi2

(9)

$2m2

The desired asymptotic results are again readily available from the continuous mapping t h e o r e m . To simplify the presentation of the theorem, we adopt below the following notation from Phillips (1986, eq. (7)): all=

f _a2 ,

A22 =

f _B2_B2, ,

a21 =

f _B2_B0 .

U. Hassler / Economics Letters 50 (1996) 25-31

29

Theorem• With the assumptions o f Section 2 and the notation above, it holds f o r the leastsquares estimation o f (5) or (2) as T---->~:

(a) A s in Phillips (1986, theorem 2)3:

2 -

~

0.2

-1

T lsZ~all-azlAzza21=:

T IF2~

, -1 a21 a21A22 2

T

,

,

R2~

or

--

0.

all

[b2] j ~ / ~ -1

-0 5. -tj ~

Or m 2

all

22

]lj

.

(b) For the I(0) variables xl,:

'If

~1~Z

B_odB , -

1,0

F1 ~

b'lXl.o bl 2 , 0- m 1

l

B_;dBtAzzla2t

"J

tk :ff

]

=:b,,

[b,]k -1 O'V[Y] 1AI]kk

(c) For the Durbin-Watson statistic" Tdw~

Zo,,, + 2b; ( ~ 1 . o - Xl,l)bl -~- b¢2X2,ob2 2 0-

It is intuitively plausible that, in (a), the distributions of spurious regressions without additional stationary regressors are reproduced. However, it is somehow surprising that /31 does not converge to zero. Nevertheless, we observe no spurious significance of /31 in (5), meaning that the t- and F-statistics do not diverge. Finally, we find that the covariances of x~, = vl, affect the distribution of T dw, so that the limit differs from that in Phillips (1986, theorem 2). However, this does not have severe impacts for residual-based tests against cointegration as they have been analyzed, for example, by Engle and Granger (1987) and Phillips and Ouliaris (1990). To check y, and x2t for cointegration, the residuals of (1) instead of (2) or (5) may be considered. To understand why the results in part (b) of the theorem arise, let us consider the regression of an I(1) variable on a vector of I(0) series: £,

=

~' ~ /31_x1,+ u3~ ,

3 Please note that b 2 is the limit of/32 and not the vector defined in Phillips (1986, theorem 2).

(10)

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u. Hassler / Economics Letters 50 (1996) 25-31

with

L = (Y-,

Z,,,s,

•

Regression (10) is unbalanced or not 'consistent' in the sense of Granger (1981), in that it unsuccessfully tries to explain the dynamics of a non-stationary series by means of stationary regressors. T h e distribution of/31 is obvious from the lemma. Just as well the limits of TR 2, T dw and T - i s 2 from (10) are derived by simply letting /32 = 0 in the formulae above the t h e o r e m . Finally, let us define the t- and F-statistics referring to (10) instead of (8) and (9) as

[~l]k

i'k = SIE Xlt~'lt]~-~/2'

k=

1,2,

""'

mI

L -

'

T- mI- 1

R2

m,

1 -- g 2"

(11)

T h e asymptotic distributions are obtained from the l e m m a by the continuous m a p p i n g theorem. Corollary. With the assumptions o f Section 2 and the notation above, it holds f o r the least-squares estimation o f (10) as T--->~: /31~

P, =>

1,0

-B0 dB1 =:/~1 ,

, tr m I

TR2~ b

T-ls2~

B-o

,

6V'[E-lo],,

'

1~1,0bl 62

f

,

T dw ~

~0 0 + 2b 1(~ 1,0 -- ~ 1,1)bl ' 62

T h e analogous structure of the distributions of/31 and/31 as well as of the statistics testing for significance of xl, and of the D u r b i n - W a t s o n statistics from (5) and (10) is obvious. H o w e v e r , in contrast to the t h e o r e m , the coefficient of d e t e r m i n a t i o n from the u n b a l a n c e d regression (10) tends to zero, of course.

4. Summarizing remarks T h e distributions involved with spurious regressions with additional stationary regressors, such as in (2) or (5), c o m b i n e asymptotic results of spurious regressions of I(1) regressors only and of unbalanced regressions, such as in (10).

U. Hassler / Economics Letters 50 (1996) 25-31

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References Choi, I., 1994, Spurious regressions and residual-based tests for cointegration when regressors are cointegrated, Journal of Econometrics 60, 313-320. Engle, R.F. and C.W.J. Granger, 1987, Co-integration and error correction: Representation, estimation, and testing, Econometrica 55, 251-276. Granger, C.W.J., 1981, Some properties of time series data and their use in econometric model specification, Journal of Econometrics 16, 121-130. Granger, C.W.J. and P. Newbold, 1974, Spurious regressions in econometrics, Journal of Econometrics 2, 111-120. Haldrup, N., 1994, The asymptotics of single-equation cointegration regressions with I(1) and I(2) variables, Journal of Econometrics 63, 153-181. Park, J.Y. and P.C.B. Phillips, 1989, Statistical inference in regressions with integrated processes: Part 2, Econometric Theory 5, 95-131. Phillips, P.C.B., 1986, Understanding spurious regressions in econometrics, Journal of Econometrics 33,311-340. Phillips, P.C.B. and S.N. Durlauf, 1986, Multiple time series regression with integrated processes, Review of Economic Studies Vol. LIII, 473-495. Phillips, P.C.B. and S. Ouliaris, 1990, Asymptotic properties of residual based tests for cointegration, Econometrica 58, 165-193.