Spurious regressions between stationary generalized long memory processes

Economics Letters 90 (2006) 446 – 454 www.elsevier.com/locate/econbase

Spurious regressions between stationary generalized long memory processes Yixiao Sun * Department of Economics, 0508, University of California, San Diego, La Jolla, CA 92093-0508, USA Received 16 January 2005; received in revised form 6 August 2005; accepted 6 October 2005

Abstract This paper shows that a spurious regression can occur between two stationary generalized fractional processes, as long as their generalized fractional differencing parameters sum up to a value greater than 0.5 and their spectral densities have poles at the same location. This theoretical finding is supported by simulations. D 2005 Elsevier B.V. All rights reserved. Keywords: Cyclical co-movement; Gegenbauer process; Spurious regression JEL classification: C22

1. Introduction Since the first Monte Carlo study by Granger and Newbold (1974), much effort has been taken to understand the nature of spurious regressions; see for example, Phillips (1986) and Marmol (1995, 1998). It has long been believed that it is the deterministic or stochastic trending behavior that causes spurious effects. However, Tsay and Chung (2000) found that a spurious effect can arise in a regression between two stationary I(d) processes, as long as their orders of integration sum up to a value greater than 0.5. Based on this, they suggested that it is long memory or strong dependence, rather than deterministic or stochastic trending, that causes the spurious effects. This paper explores the possible existence of a spurious relationship between two stationary Gegenbauer processes. This class of processes generalizes the popular fractionally integrated processes * Tel.: +1 858 534 4692; fax: +1 858 534 7040. E-mail address: [email protected]. 0165-1765/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2005.10.009

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and is capable of capturing long-range dependence, as well as a periodic cyclical pattern. It has been shown that the autocovariances of a Gegenbauer process decay hyperbolically and sinusoidally, a feature that is manifested in a number of financial and economic time series. We find that the spurious regression can occur between two stationary Gegenbauer processes. This provides further evidence that the spurious effect is not unique to nonstationary processes, as suggested by Tsay and Chung (2000). We show that strong dependence is not enough to give rise to the spurious effect. Another indispensable condition is that the two processes have spectral densities with poles at the same location. In other words, the two processes are required to have common cyclical movements. The cyclical co-movement of economic variables is the rule rather than exception. As suggested by Lucas (1977), the main regularities observed in cyclical fluctuations of economic time series are in their co-movement. The strong persistence and cyclical co-movement in many economic time series suggest that the spurious effect may be present more often than we previously believed. Our theoretical result is supported by Monte Carlo simulations. The rejection probability for testing the null of a zero slope coefficient based on the usual t-test increases with the sample size and the persistence of underlying processes.

2. Spurious regression between Gegenbauer processes A Gegenbauer process is a generalized fractionally integrated process defined by
d 1
2cosðuÞL þ L2 xt ¼ et ;

ð1Þ

where d is the generalized fractional differencing parameter and e t is a stationary process with continuous spectral density (e.g., Gray et al., 1989, 1994; Chung 1996a,b). When 0 b u b p, the Gegenbauer process is stationary if d b 1/2. When u = 0, p, the Gegenbauer process is stationary if d b 1/ 4. In this paper, we focus on the case that 0 b u b p so that the Gegenbauer process exhibits some periodic cyclical behavior. In this case, the autocovariance function for a stationary Gegenbauer process can be approximated by cð jÞfKj2d
1 cosð ujÞ

ð2Þ

for some constant K (Chung, 1996a), where the symbol ~ means that the ratio of the two sides tends to 1, as j Y l. The autocovariance function thus resembles a hyperbolically damped cosine wave. It is easy to show that the spectral density of a Gegenbauer process is     kþu k
u
2d ð3Þ f ðkÞ ¼ fe ðkÞj4sin sin j ; 2 2 where f e (k) is the spectral density of {e t }. Therefore, when 0 b d b 1/2, we have f ðkÞfCjk2
u2 j
2d as kYu; for some positive constant C. This implies that the spectral density has a pole at k = u. Now suppose that x t and y t follow independent Gegenbauer processes such that
dx 1
2cosux L þ L2 xt ¼ ext ;

ð4Þ

ð5Þ

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Y. Sun / Economics Letters 90 (2006) 446–454


dy 1
2cosuy L þ L2 yt ¼ eyt ;

ð6Þ

where 0 b u x , u y b p, |d x | b 1/2, |d y | b 1/2, e xt and e yt are independent stationary processes with continuous spectral densities. Consider regressing y t on a constant and x t , yt ¼ aˆ þ bˆ xt þ uˆ t ; t ¼ 1; . . . ; T:

ð7Þ

The ordinary least-squares estimate of b is given by T X

bˆ ¼

where ¯x =

ðxt
x¯Þðyt
y¯ Þ

t¼1 T X

ðxt
x¯ Þ

t¼1

PT

ð8Þ

; 2

t=1 x t /T

rˆ b;OLS ¼ rˆ u

and ¯y = T X

PT

t=1 y t /T.

The t-statistic is ˆt b,OLS = bˆ/rˆ b,OLS where

!
1=2 ðxt
x¯ Þ

2

ð9Þ

;

t¼1

P ˆ u2 = t=1Tuˆ2t /T and uˆ t = ( y t
¯y)
bˆ (x t
¯x). The R 2 and the DW (Durbin–Waston) statistic are defined in r the usual way, i.e., bˆ 2 R2 ¼

T X

t¼1 T X

T X

ðxt
x¯ Þ2 ; 2

ðyt
y¯ Þ

t¼1

and DW ¼

ðuˆ t
uˆ t
1 Þ2

t¼2 T X

:

ð10Þ

uˆ 2t

t¼1

The next theorem presents the asymptotic behaviors of bˆ, ˆtb,OLS , R 2 and the DW statistic. Theorem 1. Let xt and yt be the time series defined by (5) and (6), respectively. If dx, dy a (0, 1/2) such that dx + dy N 1/2 and ux = uy = u a (0, p), then (a) bˆ = Op(T dx+dy
1), (b) ˆtb,OLS = Op(T dx+dy
1/2), (c) R2 = Op(T 2dx+2dy
2), (d) DW = 2
2qy(1) + op(1), where qy(1) is the first order autocorrelation of yt. The most significant result in the above theorem is that the t-statistic diverges at the rate of T d x +d y
1/2. The larger the sum of d x and d y is, the faster the t-statistic diverges. The divergence of the tstatistic implies that the rejection probability for testing the null of b = 0 based on the usual t-test approaches 1 as the sample size increases. This result reflects the spurious effect in the t-test. Since both the dependent and independent variables are stationary and ergodic, the spurious effect is not expected. This

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finding suggests that it is strong dependence, instead of nonstationarity, that is potentially responsible for the spurious effect. If we replace rˆ u given in (9) by V where 2

V ¼

T X

ðxt


x¯ Þ2 uˆ 2t

þ2

ST  X j¼1

t¼1

j 1
ST þ 1

 X T


 ðxt
x¯ Þuˆ t uˆ t
j xt
j
x¯ ;

ð11Þ

t¼jþ1

for some bandwidth S T, then we get the heteroscedasticity-autocorrelation consistent (HAC) standard error. In this case, the t-statistic will still diverge but at a slower rate. It is expected that ˆt b,HAC = O p ((T/ S T )dx+dy
1/2). Therefore, the t-statistic will converge only if S T is set proportional to T. Sun (2004) investigates this specification for the usual spurious regressions. Although the t-statistic diverges as in the spurious regression studied before, the behaviors of other statistics resemble those derived in the conventional case of no spurious effect. For example, the OLS estimator bˆ of the slope coefficient and the R 2 converge to zero in probability. The DW statistic converges to 2
2q y (1), a limit that is obtained in the case of AR(1) errors. However, the convergence
1/2 rate. The speed of convergence depends on the order of rate of bˆ is much PT slower than the usual T magnitude of t=1(x t
¯x)( y t
¯y); which in turn depends on the persistence of x t and y t . The more persistent x t and y t are, the more slowly bˆ converges to zero, and the faster the t-statistic diverges. An important condition in Theorem 1 is that u x = u y. If u x p u y, then there is no spurious effect even if both processes are persistent P enough such that d x + d y z 1/2. In fact, the proofPof Theorem 1 relies crucially on the fact that Tt=1(x t
¯x)( y t
¯y) = O p (T dx+dy ). Thus, the order of Tt=1(x t
¯x) ( y t
¯y) is quite different both x t and y t are weakly dependent processes. However, when PT from the case when 1/2 u x p u y, t=1(x t
¯x) ( y t
¯y) = O p (T ), which can be proved as follows: var

T X t¼1

! xt yt

¼

T X

Ex2t y2t þ 2

t¼1

T X t
1 X

Ext xt
s yt yt
s fT r2x r2y

t¼1 s¼1

þ 2Kx Ky

T X t
1 X

s2dx þ2dy
2 cosux scosuy sfT r2x r2y

t¼1 s¼1

þ Kx Ky

T X t
1 X






  s2dx þ2dy
2 cos ux þ uy s þ cos ux
uy s ¼ OðT Þ; ð12Þ

t¼1 s¼1

P a where the last line follows from l s=1 s cosbs b l for any a a (
l,0) and b a (0,2p): Using this result and following the argument similar to the proof of Theorem 1, we can establish the following theorem. Details are omitted. Theorem 2. Let xt and yt be the time series defined by (5) and (6),respectively. If dx, dy a (0, 1/2) such that dx + dy N 1/2 and ux, uy a (0, p) with ux p uy , then (a) bˆ = Op(T
1/2), (b) ˆtb,OLS = Op(1), (c) R2 = Op(T
1), (d) DW = 2
2qy (1) + op(1).

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Y. Sun / Economics Letters 90 (2006) 446–454

Theorem 2 shows all the statistics behave as if x t and y t are weakly dependent processes. Therefore, regressions between two stationary Gegenbauer processes with different pole locations do not suffer from the spurious effect. This suggests that the strong persistence is not sufficient for the existence of a spurious relationship. Both the strong dependence and cyclical co-movement are required to generate a spurious regression.

3. A simulation study In this section, we examine the validity of our theoretical results by simulations. To simulate a and truncate the infinite sum after M Gegenbauer process z t , we use the moving average representationP iterations. More specifically, we generate z t according to z t = M j=0 c j (d,u)e t
j , where c j (d,u) are Gegenbauer coefficients defined by l X

d ¼ cj ðd; uÞLj 1
2cosðuÞL þ L2 j¼0

and e t is iid N(0,1). We take M to be 2T, where T is the sample size. To reduce the initialization effect, we generate a time series of length 2T and trim the first T observations to get the simulated time series. We consider sample sizes T = 100, 500, 1000, 2000, 5000. For each simulated sample, we calculate the OLS estimate bˆ and construct the usual t-statistic. We report the percentage of rejections in 1000 replications, i.e., the percentage of t such that |t| N 1.96. Since x t uˆ t is autocorrelated, we also construct the HAC standard error for bˆ. We take S T to be T 1/3 in computing the robust standard error. Table 1 presents the results when the parameters satisfy d x = d y = d and u x = u y = u = p/4. It is apparent that for both ˆt b,OLS and ˆt b,HAC statistics, the rejection rate increases with the sample size and the sum of d x and d y . The table also shows that the increase is not very fast, which is attributable to the slow divergence of the t-statistic. Therefore, the asymptotic results are rejected in this finite sample scenario. It should be emphasized that the spurious effect is obvious even for a small sample size, such as 100. The HAC estimate reduces the rejection rate, but the spurious effect is still apparent. We also consider the cases that u takes other values and d x and d y take different values. To save space, we do not present those results because the qualitative observations for the case d x = d y = d, u = p/4 apply. Simulation results not reported also support the absence of the spurious effect when u x is not equal to u y. Table 1 Spurious regression between two stationary Gegenbauer processes: percentage of |t| N 1.96 ˆt b,OLS ˆt b,HAC NOBS 100 500 1000 2000 5000

d = 0.15

d = 0.25

d = 0.35

d = 0.45

d = 0.15

d = 0.25

d = 0.35

d = 0.45

28.90 34.10 37.30 41.70 44.30

35.50 50.90 53.80 56.00 57.90

46.60 63.50 65.30 70.60 77.00

58.00 72.60 79.20 83.80 87.40

25.60 22.40 22.80 22.30 24.10

30.70 37.00 35.10 37.50 34.40

38.80 47.20 47.80 51.10 54.60

51.00 58.50 65.30 67.50 71.20

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4. Conclusion This paper shows that a spurious regression can arise between two stationary Gegenbauer processes, as long as the sum of their generalized fractional differencing parameters is greater than 0.5 and their power spectra have poles at the same location. This finding helps deepen the understanding of spurious regressions. Previous studies have led us to believe that it is the nonstationarity or the unboundedness of the spectral density at the zero frequency that causes the spurious effect. Our analysis shows that the unboundedness at a nonzero frequency can also give rise to the spurious effect. Similar to the spurious effect between an I(1) process and a linear trend, the spurious effect may exist between a stationary Gegenbauer process and a deterministic trigonometric series (see Sun, 2003). This paper considers only the stationary Gegenbauer processes. It is not surprising that the spurious effects are present and more dramatic among nonstationary Gegenbauer processes.

5. Proof of Theorem 1 (a) Since x t and y t are ergodic, we have plim

T T 1 X 1 X ðxt
x¯ Þ2 ¼ r2x and plim ðyt
y¯ Þ2 ¼ r2y ; T t¼1 T t¼1

ð13Þ

where r x2 and r y2 are the variances of x t and y t , respectively. To prove (a), it suffices to calculate the order of T X

ðxt
x¯ Þðyt
y¯ Þ ¼

t¼1

T X

xt yt
Tx¯ y¯ :

ð14Þ

t¼1

Using (2), we have, for some positive constants K x and K y, ! T T T X t
1 X X X var xt yt ¼ Ex2t y2t þ 2 Ext xt
s yt yt
s fTr2x r2y t¼1

t¼1

t¼1 s¼1

þ 2Kx Ky

T X t
1 X

s2dx þ2dy
2 cos2 usfTr2x r2y

t¼1 s¼1

þ Kx Ky

T X t
1 X

s2dx þ2dy
2 ðcos2us þ 1ÞfT r2x r2y

t¼1 s¼1 T T X t
1 X

1 X þ Kx Ky 2dx þ2dy
1 t 2dx þ2dy
1 þ Kx Ky s2dx þ2dy
2 cos2usfTr2x r2y t¼1

t¼1 s¼1

T X t
1 X

1

1 þKx Ky 2dx þ2dy
1 2dx þ 2dy T 2dx þ2dy þKx Ky s2dx þ2dy
2 cos2us: t¼1 s¼1

ð15Þ

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Y. Sun / Economics Letters 90 (2006) 446–454

P t
1 2d x +2d y
2 Since 2d x + 2d y
2 b 0, we have s=1 s cos2us V C for some constant C that is independent of P a t. This is because the infinite sum l s=1 s cosbs is finite for any a a (
l,0) and b a (0,2p). See Zygmund (1959, p. 70). Therefore, the last term in (15) is O(T): As a consequence, ! T X

1

1 var xt yt fKx Ky 2dx þ 2dy
1 2dx þ 2dy T 2dx þ2dy : ð16Þ t¼1

This implies that T X


 xt yt ¼ Op T dx þdy :

ð17Þ

t¼1

Following a similar argument, we have



x¯ ¼ Op T
1=2 and y¯ ¼ Op T
1=2 :

ð18Þ

Combining (17) with (18) yields T X


 ðxt
x¯Þðyt
y¯ Þ ¼ Op T dx þdy :

ð19Þ

t¼1

Hence, bˆ= O p (T d x +d y
1) using (13) and (19). (b) We calculate the order of rˆ u2: rˆ 2u

¼T

T X


1

2

ðyt
y¯ Þ þ b

t¼1

¼ T
1

T X

ˆ2

T X

ðxt
x¯ Þ
2bˆ

t¼1

2

T X

! ððyt
y¯ Þðxt
x¯ ÞÞ

t¼1



 ðyt
y¯ Þ2 þ Op T 2dx þ2dy
2
2Op T 2dx þ2dy
2

t¼1

¼ T
1

T X


 ðyt
y¯ Þ2 1 þ op ð1Þ :

ð20Þ

t¼1

ˆ u2 = r y2. Combine this with (13) and (19), we have t bˆ,OLS = O p (T d x +d y
1/2). Therefore, plimT Y lj (c) Using (13) and part (a), we have bˆ 2 R2 ¼

T X

t¼1 T X

ðxt
x¯Þ2 2

ðyt
y¯Þ

t¼1


 ¼ Op T 2d1 þ2d2
2 :

ð21Þ

Y. Sun / Economics Letters 90 (2006) 446–454

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(d) For the DW statistic, we have T X

DW ¼

ðuˆ t
uˆ t
1 Þ2

t¼2 T X

2

XT

¼

T X

uˆ 2t

t¼1

T X

2uˆ t uˆ t
1

t¼2

þ op ð1Þ ¼ 2
2

uˆ 2t

t¼1

T X

¼2
2

T X

uˆ 2
t¼2 t

uˆ t uˆ t
1 t¼2 T X uˆ 2t t¼1

þ op ð1Þ



yt
y¯
bˆ ðxt
x¯Þ yt
1
y¯
bˆ ðxt
1
x¯Þ

t¼2 T X

uˆ 2t

t¼1 T X

þ op ð1Þ ¼ 2
2

T X

ðyt
y¯Þðyt
1
y¯Þ

t¼2 T X

ˆ 2 t¼2


2b uˆ 2t

t¼1 T X

þ 2bˆ

T X t¼1

T X

uˆ 2t

t¼1 T X

ðyt
y¯Þðxt
1
x¯Þ

t¼2

ðxt
x¯Þðxt
1
x¯Þ

þ 2bˆ

ðxt
x¯Þðyt
1
y¯Þ

t¼2

uˆ 2t


 þ Op T 2dx þ2dy
2 ¼ 2
2qy ð1Þ þ op ð1Þ;

T X

¼ 2
2qy ð1Þ uˆ 2t

t¼1

ð22Þ

as desired.

Acknowledgements I thank James Hamilton and a referee for helpful comments. References Chung, C.-F., 1996a. Estimating a generalized long memory process. Journal of Econometrics 73, 237 – 259. Chung, C.-F., 1996b. A generalized fractionally integrated autoregressive moving-average process. Journal of Time Series Analysis 17, 111 – 140. Granger, C.W.J., Newbold, P., 1974. Spurious regressions in econometrics. Journal of Econometrics 2, 111 – 120. Gray, H.L., Zhang, N., Woodward, W., 1989. On generalized fractional processes. Journal of Time Series Analysis 10 (3), 233 – 257. Gray, H.L., Zhang, N., Woodward, W., 1994. On generalized fractional processes, a correction. Journal of Time Series Analysis 15, 561 – 562. Lucas, R.E., 1977. Understanding business cycles. Carnegie-Rochester Conference Series on Public Policy 5, 7 – 29. Marmol, F., 1995. Spurious regressions between I(d) processes. Journal of Time Series Analysis 16, 313 – 321.

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Marmol, F., 1998. Spurious regression theory with nonstationary fractionally integrated processes. Journal of Econometrics 84, 233 – 250. Phillips, P.C.B., 1986. Understanding spurious regressions in econometrics. Journal of Econometrics 33, 311 – 340. Sun, Y., 2003. Spurious regressions with stationary Gegenbauer processes and harmonic processes. Working paper. Department of Economics, University of California, San Diego. Sun, Y., 2004. A convergent t-statistic in spurious regressions. Econometric Theory 20, 943 – 962. Tsay, W.-J., Chung, C.-F., 2000. The spurious regression of fractionally integrated processes. Journal of Econometrics 96 (1), 155 – 182. Zygmund, A., 1959. Trigonometric Series, vol. I. Cambridge University Press, London.