The Spurious Regression of Fractionally Integrated Processes

Transcription

1 he Spurious Regression of Fractionally Integrated Processes Wen-Jen say Institute of Economics Academia Sinica Ching-Fan Chung Institute of Economics Academia Sinica January, 999 Abstract his paper extends the theoretical analysis of the spurious regression and spurious detrending from the usual I () processes to the long memory fractionally integrated processes. It is found that when we regress a long memory fractionally integrated process on another unrelated long memory fractionally integrated process, no matter whether these processes are stationary or not, as long as their orders of integration sum up to a value greater than.5, the t ratios become divergent and spurious effects occur. Our finding suggests that it is the long memory, instead of nonstationarity or lack of ergodicity, that causes such spurious effects. As a result, spurious effects might happen more often than we previously believed as they can arise even between stationary series while the usual first-differencing procedure may not completely eliminate spurious effects when data possess strong long memory. JEL No.: C Key words: fractionally integrated processes, long memory, spurious regression, spurious detrending Ching-Fan Chung, Institute of Economics, Academia Sinica, Nankang, aipei, aiwan. el: (886) , Fax: (886) , chungcf@ccms.edu.tw We are very grateful for two referees and an associate editor for their valuable suggestions.

2 Introduction he spurious regression was first studied by Granger and Newbold (974) using simulation. hey show that when unrelated data series are close to the integrated processes of order or the I () processes, then running a regression with this type of data will yield spurious effects. hat is, the null hypothesis of no relationship among the unrelated I () processes will be rejected much too often. Furthermore, the spurious regression tends to yield a high coefficient of determination (R ) as well as highly autocorrelated residuals, indicated by a very low value of Durbin-Watson (DW ) statistic. Granger and Newbold s simulation results are later supported by Phillips (986) theoretical analysis. Phillips proves that the usual t test statistic in a spurious regression does not have a limiting distribution but diverges as the sample size approaches infinity. He also shows that R has a non-degenerate limiting distribution while the DW statistic converges in probability to zero. Phillips results has been generalized by Marmol (995) to cases with integrated processes of higher orders. he history of the research on spurious detrending follows a similar thread. Nelson and Kang (98, 984) first employ simulation to demonstrate that the regression of a driftless I () process on a time trend produces an incorrect result of a significant trend. Extending the Phillips (986) approach, Durlauf and Phillips (988) derive the asymptotic distributions for the least squares estimators in such a regression. In particular, the latter authors show that the t test statistics diverge and there are no correct critical values for the conventional significance tests. Most studies of the spurious regression concentrate on the nonstationary I () processes. It reflects the widely held belief that many data series in economics are I () processes, or near I () processes, as argued by Nelson and Plosser (98). Against this backdrop, we also witness in recent years fast growing studies on fractionally integrate processes, or the I (d) processes with the differencing parameter d being a fractional number. he I (d) processes are natural generalization of the I () processes that exhibit a broader long-run characteristics. More specifically, the I (d) processes can be either stationary or nonstationary, depending on the value of the fractional differencing parameter. he major characteristic of a stationary I (d) process is its long memory which is reflected by the hyperbolic decay in its autocorrelations. A number of economic and financial series have been shown to possess long memory. See Baillie (996) for an updated survey on the applications of the I (d) processes in economics and finance. he objective of this paper is to extend the theoretical analysis of the spurious regression from I () processes to the class of long memory I (d) processes. We establish and analyze conditions on the I (d)

3 processes that inflict the spurious effect in a simple linear regression model. he nonstandard asymptotic distributions of various coefficient estimators and test statistics are then derived. he main finding from our study is that the spurious regression can arise among a wide range of long memory I (d) processes, even in cases where both dependent variable and regressor are stationary. A few conclusions may then be drawn. First, different from what Phillips (986) and Durlauf and Phillips (988) have suggested, the cause for spurious effects seems to be neither nonstationarity nor lack of ergodicity but the strong long memory in the data series. As a result, spurious effects might occur more often than we previously believed as they can arise even among stationary series. Furthermore, the usual first-differencing procedure may not be able to completely eliminate spurious effects if the data series are not only nonstationary but possess strong long memory (such as in the case where they are I (d) processes with d > ). A General heory of Spurious Effects Our analysis of the spurious effects are based on several simple linear regression models in which the dependent variable and the single non-constant regressor are independent I (d) processes with d lying in different ranges. Before presenting these models, let s first briefly review some basic properties of the I (d) processes. A process Y t is said to be a fractionally integrated process of order d, denoted as I (d), if it is defined by ( L) d Y t = ε t, where L is the usual lag operator, d is the differencing parameter which can be a fractional number, and the innovation sequence ε t is white noise with a zero mean and finite variance. he fractional differencing operator ( L) d is defined as follows: ( L) d = j= ψ j L j, where ψ j = Ɣ( j d)/ Ɣ( j + )Ɣ( d) ] and Ɣ( ) is the gamma function. his process is first introduced by Granger (98, 98), Granger and Joyeux (98), and Hosking (98). hey show that Y t is stationary when d <.5 and is invertible when d >.5. he main feature of the I (d) process is that its autocovariance function declines at a slower hyperbolic rate (instead of the geometric rate found in the conventional ARMA models): γ ( j) = O( j d ), where γ ( j) is the autocovariance function at lag j. When d >, the I (d) process is said to have long memory since it exhibits long range dependence in the sense that j= γ ( j) =. When d <, then j= γ ( j) < and the process is sometimes referred to as an intermediate memory process. See

4 Chung (994) for other long memory properties of the I (d) process. Our analysis focuses on the class of long memory I (d) processes with d >. We are particularly interested in the distinction between the nonstationary subclass of I (d) processes with d.5 and the stationary subclass with d <.5. o examine potentially different types of spurious effects, we propose six regression models for different classes of I (d) processes, mainly based on whether the fractional differencing parameter d is greater than.5 or not. he exact specifications of these models can be conveniently expressed with four I (d) processes. Let s first define two stationary ones with different differencing parameters d and d whose values lie between.5 and.5: ( L) d v t = a t and ( L) d w t = b t, where a t and b t are two white noises with zero mean and finite variances σa and σ b, respectively; that is, v t and w t are I (d ) and I (d ) processes, respectively, and both of them are stationary and invertible. When these two processes are employed in our later analysis, the values of their differencing parameters are mostly assumed to be in (,.5); i.e., the stationary processes v t and w t are often assumed to have long memory. We can also define two nonstationary I ( + d ) and I ( + d ) processes by integrating v t and w t : y t = y t + v t and x t = x t + w t. Obviously, the orders of integration of these two nonstationary fractionally integrated processes lie between.5 and.5. Given these four fractionally integrated processes, we consider the following six simple linear regression models: Model : y t is regressed on an intercept and x t, Model : v t is regressed on an intercept and w t, where d + d >.5, Model 3: y t is regressed on an intercept and w t, where d >, Model 4: v t is regressed on an intercept and x t, where d >, Model 5: y t is regressed on an intercept and the trend t, Model 6: v t is regressed on an intercept and the trend t, where d >. In Model the orders of integration of both the dependent variable and the regressor lie between.5 and.5, and can be equal to. So Model may be considered a generalization of Phillips (986) spurious 3

5 regression to the case of fractionally integrated processes. Model presents the most interesting case in our analysis. In it both the dependent variable and the regressor are assumed to be stationary, ergodic, and strongly persistent in the sense that their fractional differencing parameters sum up to a value greater than.5. Following Phillips arguments, we tend to think no spurious effect should occur in such a model where variables are ergodic. But our analysis of Model presents a result to the contrary. he analysis of Model seems to go beyond the previous study of spurious effects and allows us to gain new insight into the problem. Models 3 to 4 differ from Model in that the order of integration in one of the dependent variable and the regressor is reduced to the stationary range between and.5. We can conveniently view Models 3 and 4 as two intermediate cases between Model of nonstationary fractionally integrated processes and Model of stationary fractionally integrated processes. We thus expect the analysis of these two new models to be a mixture of those of Models and. In Models 5 and 6 we consider the effect of detrending the nonstationary and stationary fractionally integrated processes, respectively. hrough these two models, we generalize the results of Durlauf and Phillips (988). Also, Models 5 and 6 can be regarded as variants of Models and 4, respectively, with the nonstationary regressor x t replaced by the time trend. his similarity in the model specifications will also be reflected in their analytic results. he following assumption on the two white noise processes a t and b t are made throughout this paper. Assumption Each of the two processes a t and b t is independently and identically distributed with a zero mean; and their moments satisfy the following conditions: E a t p <, with p max{4, 8d /(+d )}; and E b t q < with q max{4, 8d /( + d )}. Moreover, a t and b t are independent of each other. We also assume, without loss of generality, that the initial values of the fractionally integrated processes v, w, y, and x are all zero. Hence, y t and x t can be considered as the partial sums of v t and w t, respectively; i.e., y = v t and x = w t. he independent and identical distribution assumption is made to simplify our analysis and could be relaxed, say, to the case where a t and b t are short memory processes. See Chung (995). Before presenting Lemma, which is the cornerstone of our analysis, let s summarize two important asymptotic results on the partial sums y t and x t. First, given the variances σ y = Var(y ) and σ x = Var(x ), Sowell (99, heorem ) proves that σ y σ a Ɣ( d ) ( + d )Ɣ( + d )Ɣ( d ) +d and σ x σ b 4 Ɣ( d ) ( + d )Ɣ( + d )Ɣ( d ) +d,

6 where z w means z /w as. Furthermore, Davydov (97) shows that as, y r] B d (r) and x r] B d (r), for r, ], where r] denotes the integer part of r, the notation denotes weak convergence, and B d (t) is the normalized fractional Brownian motion which is defined by the following stochastic integral { ( + d)ɣ( d) t B d (t) (t s) d d B (s) + (t s) d ( t) d] } d B, Ɣ( + d)ɣ( d) for d (.5,.5), where B (t) is the standard Brownian motion. See Mandelbrot and Van Ness (968). Our notation for the standard and the fractional versions of Brownian motions suggests that the former is a special case of the latter with d =. he independence between v t and w t and between y t and x t implies joint weak convergence of y r] and x r]. More specifically, let z t = ], σy y t σx x t then z r] B d (r), where B d (r) = B d B d ] is a two-dimensional normalized fractional Brownian motion, of which the two elements B d and B d are independent. his result implies the following lemma: Lemma Given that Assumption holds, then, as, we have the following results:. y t σ y B d (s) ds and x t σ x B d (s) ds.. y t σ y Bd (s) ] ds and x t σ x Bd (s) ] ds. 3. (y t y) σ y Bd (s) ] ds B d (s) ds] and (x t x) σ x Bd (s) ] ds B d (s) ds]. 4. v t B d () and w t B d (). 5

7 5. v t p γ v () = Ɣ( d ) Ɣ( d ) σ a and w t p γ w () = Ɣ( d ) Ɣ( d ) σ b. 6. (v t v) p γ v () and (w t w) p γ w (). 7. v t wt = O p (), if d + d >.5, = O ] p (ln ).5, if d + d =.5, = O p (.5 d d ), otherwise, 8. y t xt B d (s) B d (s) ds. 9. y t wt = O p (), if d > ; and v t xt = O p (), if d >.. t vt B d () B d (s) ds and t wt B d () B d (s) ds.. t yt s B d (s) ds and t xt s B d (s) ds. Moreover, joint weak convergence of through 4, 8,, and also applies. Here, B d (t) and B d (t) are two independent normalized fractional Brownian motions, γ v ( j) and γ w ( j) are the autocovariance functions of v t and w t, respectively, at lag j, and σ a and σ b are the variances of the underlying white noises a t and b t, respectively. he notation p means convergence in probability. All the theorem proofs are in the appendix. Note that, while the weak convergence of z t is in the space of functions, the weak convergence established in the above lemma is in the real line, which is equivalent to convergence in distribution. Following the convention in the literature, we use the same notation for both types of weak convergence. In the rest of this section the results of Lemma will be used to develop the theory of spurious effects, presented in a series of theorems and corollaries, for the proposed six models. he first two models will 6

8 be discussed separately in Subsections. and.. hese two models provide us with a framework which facilitates the explanations of the other four models in Subsections.3 and.5. One subsection Subsection.4 will be devoted to the analysis of an important issue about how the orders of fractional integration are directly related to the spurious effects. he results in Lemma have also been used in deriving the limiting distributions of the modified Durbin-Watson statistics by say (998). We will adopt the following notation for the various statistics from the Ordinary Least Squares (OLS) estimation. Let α and β denote the usual OLS estimators of the intercept and the slope. heir respective variances are estimated by sβ and s α, from which we have the t ratios t β = β/s β and t α = α/s α. Also, let s denote the estimated variance of the OLS residuals, R the coefficient of determination, and DW the Durbin-Watson statistic. Finally, in addition to the autocovariance functions γ v ( j) and γ w ( j) of v t and w t, let ρ v ( j) and ρ w ( j) be their respective autocorrelations at lag j.. Model of Nonstationary Fractionally Integrated Processes In Model a nonstationary I( + d ) process y t is regressed on another independent and nonstationary I( + d ) process x t. Since the permissible range for the values of the fractional differencing parameters d and d is (.5,.5), Model generalizes Phillips (986) model of integrated processes in which d = d =. Not surprisingly, all the results we derive for Model are straightforward generalization of Phillips theory of the spurious effects. he results for Model are presented in the following theorem: heorem Given that Assumption holds, then, as, we have the following results:. β B d (s) B d (s) ds ] B d (s) ds ] B d (s) ds Bd (s) ] ] β. ds B d (s) ds Note that / = O( d d ).. α B d (s) ds β B d (s) ds α, where β is defined in. Note that = O(.5+d ). 3. σ y s Bd (s) ] ds ] B d (s) ds 7

9 where β is defined in. Note that σ y = O( +d ). { β Bd (s) ] ds ] } B d (s) ds σ, 4. σ x σ y s β σ Bd (s) ] ds ] σ β, B d (s) ds where σ is defined in 3. Note that σ y / σ x = O( d d ). 5. σ y sα σ + ] B d (s) ds Bd (s) ] ds B d (s) ds ] σ α, where σ is defined in 3. Note that σ y / = O( d ). 6. t β β σ β, where β is defined in and σ β is defined in t α α σ α, where α is defined in and σ α is defined in R { Bd (s) ] ds β Bd (s) ] ds ] } B d (s) ds ], B d (s) ds where β is defined in. 9. DW p. Here, B d (t) and B d (t) are two independent normalized fractional Brownian motions. 8

10 he most important result in heorem is that, as the sample size increases, the two t ratios t β and t α diverge at the same rate of, which is independent of the magnitudes of the fractional differencing parameters d and d. his result is exactly the same as what Phillips (986) has obtained for the case where d = d =. So even when the orders of integration in the dependent variable and the regressor differ from by as much as.5, the usual problem in using the t tests remains: the probability of rejecting the null hypothesis of β = or α = based on t tests increases monotonically as the sample size increases. Also note that Marmol (995) generalizes Phillips theory to cases where both y t and x t are integrated processes of the same integer orders that are higher than one. he limiting distributions in heorem for the special case where d = d are also very similar to Marmol s results. he limiting distributions of the t ratios, after normalized by, are direct generalization of those derived by Phillips (986). he same conclusion also holds for R and the DW statistics. In other words, when we compare our results with Phillips, we observe a common feature in these four statistics; namely, the nonzero values of d and d do not affect their convergence rates while the effects on their limiting distributions are quite straightforward: all the standard Brownian motions in Phillips theory are replaced by fractional Brownian motions. hat the fractional differencing parameters d and d play a relative minor role here is mainly because the four statistics are all ratios so that the effects of d and d are cancelled out. In contrast, the results on the OLS estimators β and α are a different story. In Phillips theory both β and α/ converge to some non-normal nondegenerate limiting distributions. But for the present model of the fractionally integrated processes, the orders of β and α are d d and d+.5, respectively. So while α always diverges (though the rate can be slow if d is close to.5), β can be either divergent or convergent, depending on the relative magnitudes of d and d. For example, if the order of integration in the dependent variable y t is smaller than that of the regressor x t ; i.e., d < d, then β converges to zero, just like the conventional case of no spurious effects. Moreover, if d d =.5, then, similar to the case of no spurious effects, β has a limiting distribution, though its limiting distribution is not normal.. Model of Stationary Fractionally Integrated Processes In this section we consider Model in which a stationary fractionally integrated process v t is regressed on an independent and stationary fractionally integrated process w t. We show that, although both v t and w t are stationary, the spurious effect in terms of the t tests could still exist under an additional condition on the fractional differencing parameters: d + d >.5. Loosely speaking, this condition implies that the two 9

11 processes v t and w t are both strongly persistent. Our analysis begins with a special case where we assume a set of more stringent conditions which helps deriving the limiting distribution of the OLS estimator. his theory is based on an important result of Fox and aqqu (987) who show that the product of two highly persistent but stationary Gaussian processes, if adequately normalized, can converge. After examining this special case, we then show how the spurious effects may still exist in a more general framework. Let s first reproduce Fox and aqqu s (987) heorem 6. here as Lemma. Lemma Let (X t, Y t ) be a stationary jointly Gaussian sequence with E(X t ) = E(Y t ) =, E(X t ) = E(Y t ) =, and E(X ty t ) = r. Suppose that σ and σ are two arbitrary real numbers and that there exist < δ, δ <.5, such that as j E(X t X t+ j ) σ j δ, E(X t Y t+ j ) ρσ σ b a a j (δ +δ )/, E(Y t Y t+ j ) σ j δ, E(Y t X t+ j ) ρσ σ b a a j (δ +δ )/, where ρ is a constant between and, while a = A(δ, δ ), a = A(δ, δ ), b = A(δ, δ ), and b = A(δ, δ ) are four constants with A(δ, δ ) being defined by where Z(s) = σ σ a a R (δ +δ )/ s s] (X t Y t r) Z(s), x (δ +)/ (x + ) (δ +)/ dx, then ] (u x i ) (δ i +)/ I {xi <u} du d M (x ) d M (x ). i= Here, M and M are two Gaussian random measures with respect to Lebesgue measure, having unit variances and covariance ρ. Note that the two processes X t and Y t are not only strongly persistent, as indicated by the hyperbolic convergence rates δ and δ in their autocorrelations, but also highly correlated with each other, as indicated by the hyperbolic convergence rates in their covariances. However, in our application we are only interested in the case where X t and Y t are independent so that r and ρ in the above lemma are both zero. above lemma offers us the convergence rate of X ty t and its limiting process Z(t) given the Gaussian assumption and a narrower range for the parameters δ and δ. In order to apply this lemma, we make the following assumption in addition to Assumption made earlier. he

12 Assumption he two fractionally integrated processes v t and w t are both Gaussian and the corresponding fractional differencing parameters d and d are both in the range of (.5,.5). Given the facts that ρ v ( j) Ɣ( d ) Ɣ(d ) j d and ρ w ( j) Ɣ( d ) Ɣ(d ) j d, it is straightforward to prove the following corollary in which X t and Y t in Lemma are replaced by v t / γ v () and w t / γ w (), respectively. Corollary Given that Assumptions and hold, then, as, d +d v t γv () w t γw () Z(), where the limiting random variable Z() is defined in Lemma with δ = d, δ = d, σ = Ɣ( d )/ Ɣ(d ), and σ = Ɣ( d )/ Ɣ(d ). Consequently, we have v t wt C γ v () γ w () Z(), where C = Ɣ( d )Ɣ( d )/( + d )Ɣ( + d )Ɣ( d )( + d )Ɣ( + d )Ɣ( d ). he result of Corollary supplements that of item 7 in Lemma. From these results, we can then establish the following theorem about the spurious effect in Model. heorem Given that Assumptions and hold, then, as, we have the following results:. β = O p (), Note that / = O( d +d ).. α B d (). Note that / = O( d.5 ). 3. s p γ v ().

13 4. s β p γ v() γ w (). 5. s α p γ v (). 6. 3/ t β = O p (), Note that / 3/ = O( d +d.5 ). 7. t α γv () B d (). Note that / = O( d ) σ y σ x R = O p (), Note that σ y σ x / 4 = O( d +d ). 9. DW p ρ v () = ( d ) d. Here B d (t) is a normalized fractional Brownian motions. he most important result from this theorem is the divergence rates of the two t ratios t β and t α, which are d +d.5 and d, respectively. Recall that d + d.5 is necessarily greater than (and smaller than.5) under Assumption. his result reflects the spurious effect in the t tests. Since both the dependent variable and the regressor are stationary and ergodic, the spurious effect is not really expected (see Phillips 986, p.38). he surprising results we get here suggest that the cause for the spurious effect has more to do with the strong persistence than stationarity and ergodicity of the variables involved. It is interesting to compare the divergence rates of the t ratios here with the rate we observe in Model. We note that the divergence rates in the present model depend on the magnitudes of the two fractional differencing parameters d and d while those in Model do not. Furthermore, the t ratios diverge more slowly in the present model than in Model. In particular, the divergence rate of t β can become very slow when both d and d approach to their lower boundary.5. Let s turn to the OLS estimators β and α. We note that both of them converge in probability to zero as in the conventional case of no spurious effects. However, their convergence rates are much slower than

14 the usual / rate. In contrast to these irregular convergence rates of the OLS estimators, the estimated variances sβ and s α nevertheless converge at the standard rate. It is such disparity in the convergence rates between the OLS estimators, which converge at rates slower than /, and their standard errors, which converge at the standard / rates, that causes the resulting t ratios to diverge and hence the spurious effect. R in the present model converges to as in the case of no spurious effects. It is different from what we observe in Model where R converges to a random variable. Consequently, as the sample size increases, the declining R in the present model will correctly reflect the fact that the regressor does not help explain the variations in the dependent variable. he DW statistic does not converge in probability to zero and this result is also different from that of Model. Its limit ρ v () is similar to the one we find in the conventional AR() case. his limit depends on the fractional differencing parameter d of the dependent variable v t and can only take value in the range of (, 4/3), which is to the left of the value. here is one technical detail that calls for some explanations: Unlike in heorem, we do not give an explicit expression for the limiting distribution of β in heorem because such an expression requires the joint weak convergence of the sample averages of v t, w t, and v t w t, while the proof of the joint weak convergence is beyond the scope of the present paper. However, lacking in an explicit expression for the limiting distribution of β does not hinder us from evaluating the convergence rate of β, which is all we need to show the spurious effects in the t ratios. his same argument applies to some of the later analyses, including the following one where we consider a less restricted specification of Model that is defined by the following assumption. Assumption 3 he sum of the two fractional differencing parameters d and d is greater than.5. Since the Gaussian distribution is not assumed while one of the fractional differencing parameters d and d can be smaller than.5, Assumption 3 is thus less stringent than Assumption. Corollary If Assumption is replaced by Assumption 3 in heorem, then all the conclusions there remain true. he analysis of Model can thus be summarized as follows. he OLS estimators β and α (as well as R ) do converge in probability to zero, correctly reflecting the lack of a relationship between the dependent 3

15 variable and the regressor. But the convergence rates of β and α are too slow in comparison with those of their standard errors. Consequently, the t ratios diverge and the t tests fail. he upshot is that the usual t tests can become invalid even when the dependent variable and the regressor are both stationary and ergodic (so long as they are sufficiently persistent). A profound implication from Model is as follows: If we begin with Model where both the dependent variable and the regressor are nonstationary fractionally integrated processes with the orders of integration +d and +d, respectively, where d +d >.5, then first-differencing both variables cannot completely eliminate the spurious effects. While R may be reduced and the DW statistic may be increased, the t ratios may still be so large that we cannot avoid making a spurious inference. his is a fairly serious problem with the regression for the fractionally integrated processes. It implies that even the popular first-differencing procedure might not prevent us from finding a spurious relationship among highly persistent data series. One lesson we learn from this discussion is that it is very important to check individual data series for possible long memory before regression can be applied..3 wo Intermediate Cases: Model 3 and Model 4 Model 3 and Model 4 can be considered as two intermediate models between Model and Model in that one of the dependent variable and the regressor is stationary while the other is not. We expect the asymptotic results for these two new models to be hybrid of those of Model and Model. In Model 3 a nonstationary I( + d ) process y t is regressed on an independent and stationary I(d ) process w t. Note that the fractional differencing parameter d for the regressor w t here is assumed to be positive; i.e., w t has long memory. he asymptotic properties of the OLS estimators for Model 3 are given in the following theorem: heorem 3 Given that Assumption holds, then, as, we have the following results.. β = O p (). Note that / = O( d +d ).. α B d (s) ds α. Note that = O(.5+d ). 4

16 3. σ y s ] B d (s)] ds B d (s) ds σ. Note that σ y = O( +d ). 4. σ y s β σ γ w (), where σ is defined in 3. Note that σ y / = O( d ). 5. σ y s α σ, where σ is defined in 3. Note that σ y / = O( d ). 6. t β = O p (). Note that / = O( d ). 7. t α α σ, where α is defined in and σ is defined in σx R = O p (). Note that σ x / = O( d ). 9. DW p. Here B d (t) is a normalized fractional Brownian motions. Since both t ratios diverge, Model 3 also suffers from the spurious effect in terms of the t tests. Moreover, we find the results that the OLS estimator α diverge and that DW converges in probability to are close to what we get in Model, while the result of converging R is the same as that of Model. So Model 3 is indeed a mixture of Models and. 5

17 It should be pointed out that in heorem 3 the range of the fractional differencing parameter d of the regressor w t is restricted to the positive half of the original range (.5,.5). For the case of a negative d, it is quite straightforward to show that the t ratios are convergent and there is no spurious effect. In Model 4 a stationary I(d ) process v t is regressed on an independent and nonstationary I( + d ) process x t. Similar to the restriction imposed on Model 3, the fractional differencing parameter d of the dependent variable v t is assumed to be positive so that v t has long memory. he asymptotic theory for Model 4 is presented in the following theorem. heorem 4 Given that Assumption holds, then, as, we have the following results.. β = O p (). Note that / = O( d d ).. α = O p (). Note that / = O( d.5 ). 3. s p γ v (). 4. σ x s β γ v () Bd (s) ] ds ] σ β. B d (s) ds Note that / σ x = O( d ). 5. sα γ v () + ] B d (s) ds Bd (s) ] ds B d (s) ds ] σ α. 6. t β = O p (). Note that / = O( d ). 7. t α = O p (). 6

18 Note that / = O( d ). 8. σy R = O p (). Note that σ y / = O( d ). 9. DW p ρ v () = ( d ) d. Here B d (t) is a normalized fractional Brownian motions. Since both t ratios diverge (at the same rate), the spurious effect in terms of failing t tests again exists in Model 4. But contrary to the results in Model 3, the OLS estimators β and α, together with R, all converge in probability to zero, while the DW statistic converges to ρ v (). hese findings obviously bring Model 4 closer to Model than to Model..4 he Relationship between the Orders of Integration and the Divergence Rates he divergent t ratios in the above four models and the resulting failure of the t tests are referred to as the spurious effects. In this section we compare the divergence rates of t ratios across the four models and investigate how they are related to the respective model specifications. First note that the divergence rates of the t ratio t β are.5, d +d.5, d, and d, respectively, in Models to 4. Let s also compare the specifications of the four models using Model as the benchmark: Model 3 differs from Model in that the order of integration in the regressor is reduced from above.5 to below.5 (but above ); Model 4 differs from Model in that the order of integration in the dependent variable is reduced from above.5 to below.5 (but above ); and, finally, Model differs from Model in that the orders of integration in both the dependent variable and the regressor are reduced from above.5 to below.5 (but their sum is assumed to be greater than.5). By associating these changes in the orders of integration with the changes in the divergence rates of t β, we can conclude that reducing the order of integration in the dependent variable causes the divergence rate of t β to decrease by the order of d.5 and reducing the order of integration in the regressor causes the divergence rate of t β to decline by the order of d.5, while these two effects are cumulative as in Model. Recall that in Models, 3 and 4 restrictions have been imposed on the usual ranges (.5,.5) of the fractional differencing parameters d and d. In Model 3 the range of d is restricted to be (,.5), which 7

19 is also the range of d in Model 4, while the sum of d and d must be greater than.5 in Model. From the analysis in the previous paragraph, particularly the fact that the divergence rates are directly related to the magnitudes of d and d, we come to realize that the restricted ranges of d and d in Models, 3, and 4 ensure the reduction in the divergence rates of t β from the.5 level is not too great so that t β remains divergent (in which case the spurious effects occur). Although we did not explicitly consider the asymptotic theory for cases where the fractional differencing parameters lie outside their prescribed ranges, it is readily seen that the conditions we impose on the ranges are not only sufficient but also necessary for the existence of the spurious effect in terms of divergent t β. From a similar analysis for the divergence rates of the t ratio t α we also find that reducing the order of integration in the dependent variable causes the divergence rate of t α to decrease by the order of d.5, while reducing the order of integration in the regressor does not causes the divergence rate of t α to change, as we probably should have expected. It is also interesting to see how the changes in the orders of integration of the dependent variable and the regressor affect the large-sample property of R. Recall that in Model R converges to a random variable and such asymptotic behavior of R is considered part of the spurious effect by Phillips (986). But when we examine Models, 3, and 4, we note that reducing the order of integration in the dependent variable helps to increase its convergence rate by the order of d while reducing the order of integration in the regressor helps to increase the convergence rate by the order of d. As a result, in Models, 3, and 4, R all converge to, correctly reflecting the fact that there is no relationship between the regressor and the dependent variable. his finding implies that the spurious effects in Models, 3, and 4 are confined to the two t ratios while the asymptotic tendency of R toward zero is not affected by the spurious effects (though the convergence rates are). he sharp difference in the asymptotic behavior between the t ratios and R in Models, 3, and 4 actually offers us an opportunity to diagnose the spurious effect in these models. hat is, when we find two highly significant t ratios coexisting with a completely contradictory near-zero R, we are effectively reminded of the possibilities that one of the Models, 3, and 4 may be at work and that the dependent variable and the regressor may possess strong long memory, while one of them may even be nonstationary. With the possibility of such an informal diagnosis, it seems that the spurious effects in Models, 3, and 4 are less damaging than those in Model in the sense that in Model there is no internal inconsistency among the OLS estimates to indicate the spurious effects. 8

20 Finally, let s briefly state a few more results about the asymptotic tendency of the OLS estimators β and α and the DW statistic. First, we note that β will converge unless the dependent variable is nonstationary and its order of integration is sufficiently large. Secondly, whether α diverges or not and whether the DW statistic converges in probability to zero or not depend entirely on whether the dependent variable is nonstationary or not. Note that, as mentioned earlier, even though the OLS estimators β and α can converge in probability to zero in the four proposed models, the corresponding t ratios always diverge and it is these divergent t ratios that are referred to as the spurious effects..5 Model 5 and Model 6: Detrending Fractionally Integrated Processes As has been pointed out by Nelson and Kang (98, 984) and Durlauf and Phillips (988), detrending integrated processes results in the spurious effect of finding a significant trend. In this section we extend their analysis by considering the potential problems in detrending fractionally integrated processes. It turns out that the spurious effect of divergent t ratios exists as long as the fractional differencing parameter is larger than zero. he implication is that whenever there is long memory in the process, the routine procedure of detrending can produce misleading results. It appears that the spurious effect in detrending occurs more often than we previously thought. In our analysis of detrending fractionally integrated processes, we separate the nonstationary case from the stationary case. In Model 5 we examine the regression of a nonstationary I( + d ) process y t on a time trend t. he asymptotic theory for the OLS estimation is given in the following theory. heorem 5 Given that Assumption holds, then, as, we have the following results.. β s B d (s) ds 6 B d (s) ds β. Note that / = O( d.5 ).. α 4 B d (s) ds 6 s B d (s) ds α. 3. σ y s Bd (s) ] ds ] B d (s) ds s B d (s) ds ] B d (s) ds σ. 9

21 4. 3 σy s β σ, where σ is defined in 3. Note that σ y / 3 = O( d ). 5. σ y s α 4σ, where σ is defined in 3. Note that σ y / = O( d ). 6. t β β σ, where β is defined in and σ is defined in t α α σ, where α is defined in and σ is defined in R β Bd (s) ] ds ], B d (s) ds where β is defined in. 9. DW p and σy DW γ v(). Here, B d (t) is a normalized fractional Brownian motion. σ he results on detrending a stationary long memory I(d ) process v t, which is our Model 6, are presented in the following theorem. heorem 6 Given that Assumption holds, then, as, we have the following results.. β 6 B d () B d (s) ds β. Note that / = O( d.5 ).. α 6 B d (s) ds B d () α.

22 Note that / = O( d.5 ). 3. s p γ v () s β p γ v (). 5. s α p 4γ v (). 6. t β β γv (), where β is defined in. Note that / = O( d ). 7. t α α γ v (), where α is defined in. Note that / = O( d ). 8. σy R β γ v (), where β is defined in. Note that σ y / = O( d ). 9. DW p ρ v () = ( d ) d. Here, B d (t) is a normalized fractional Brownian motion. In terms of the convergence (or divergence) rates of the various OLS estimators, Models 5 and 6 can be conveniently viewed as special cases of Models and 4, respectively. More specifically, if we replace the term by in those normalizing factors in heorems and 4, then we immediately get all the normalizing factors in heorems 5 and 6. For example, while the normalizing factor for β in heorem is /, the one in heorem 5 is /. Similarly, while the normalizing factor for β in heorem 4 is /, the one in heorem 6 is /. (Also note that the fractional Brownian motion B d (t) does not appear in heorems 5 and 6 since the regressors in Models 5 and 6 are the time trend and are unrelated to the I (d ) process w t.) Given these observations, we then conclude that all the analyses about Models and 4 can be readily extended to Models 5 and 6. In particular, the divergence rates of the t ratios, which respectively are in the

23 orders of.5 and d in Models and 4, are also the rates in Models 5 and 6. (Note that in both Model 4 and Model 6 the same condition d > is imposed on the stationary dependent variable v t so that the resulting t ratios are divergent.) As a result, the type of spurious effects we observe in Models and 4 occur again in Models 5 and 6. hat is, detrending a fractionally integrated process with a positive fractional differencing parameter, certainly including the usual case of the I () process, will result in the spurious finding of a significant trend. One important inference we draw from Models 5 and 6 is that the cause for the spurious effect in detrending a process is neither nonstationarity nor lack of ergodicity but long memory in the process. From Models 5 and 6 we also note the following result: If the data series are nonstationary with the order of integration greater than, then the spurious effect can happen to the detrending procedure even after the series are first-differenced. What first-differencing does to the detrending procedure in such a case is simply reducing R, increasing the value of the DW statistic, and slowing down the divergence of the two t ratios from the.5 rate to the d rate. Based on this observation, it seems that the spurious effects in detrending may occur more often than we previously thought. 3 Conclusion In our analysis of spurious regressions for the long memory fractionally integrated processes, we find that no matter whether the dependent variable and the regressor are stationary or not, as long as their orders of integration sum up to a value greater than.5, the t ratios become divergent. So it is the long memory, instead of nonstationarity or lack of ergodicity, that causes the spurious effects in terms of failing t tests. Nonstationarity in one or both of the dependent variable and the regressor only helps to accelerate the divergence rates of the t ratios. We thus learn that spurious effects might occur more often than we previously believed as they can arise even among stationary series and the usual first-differencing procedure may not be able to completely eliminate spurious effects when data possess strong long memory. It is interesting to note Phillips (995) recently has offered some contrast thoughts about spurious regressions (and he argues that spurious regressions may not be as serious as many researchers have been led to believe). In Subsection.4 we have carefully examined the exact relationships between the orders of integration in the fractionally integrated processes and the divergence rates in the t ratios. From this analysis we gain many insights into the problem of spurious effects which are not available in Phillips (986) classical study

24 of I () processes. In short, it is found that the extents of spurious effects are directly related to the degrees of long memory in the data. Our results on detrending fractionally integrated processes also greatly broaden Durlauf and Phillips (988) theory of spurious detrending in which the relationship between the orders of integration and the divergence rates of the t ratios again plays a useful role in the analysis. A fairly extensive Monte Carlo study has also been conducted to verify the theoretical results, especially those of convergence rates, we have established in the paper. We do not report the simulation results here other than pointing out the fact that almost all our theoretical results are well supported by simulation. A few generalizations of our study are worthy of further consideration. A natural extension is to consider the multiple regression where there are more than one non-constant regressor. Another one is to allow the fractionally integrated processes to have non-zero means. Based on Phillips (986) work, we expect most, if not all, of the asymptotic results we obtain from the simple regression case to hold in the multiple regression of fractionally integrated processes with drifts. hese issues have been examined by Chung (995). One aspect of our study that is slightly more restricted than Phillips (986) and Durlauf and Phillips (988) analysis is that the fractionally integrated processes we consider are built on white noises a t and b t that are required to satisfy the relatively stringent conditions as specified in Assumption. hese conditions effectively rule out the possibility of allowing short-run dynamics such as the ARMA components in the fractionally integrated processes we have studied. Chung (995) has studied the case where Assumption is relaxed to incorporate the short-run dynamics and found no substantial changes in the analysis of spurious effects. Finally, our study of spurious regression can serve as the basis for the analyses of fractional cointegration where the dependent variable and regressors are related I (d) processes. his line of the work appears to be quite important and has attracted a lot attention in the literature recently. One of the pioneer works in this area is by Cheung and Lai (993). 3

25 Appendix A Proof of Lemma he proofs of items,, and 3 are straightforward applications of the continuous mapping theorem to the Davydov s results. hey are omitted here. Item 4 follows directly from Davydov s result, while items 5 and 6 are due to ergodicity of the two stationary processes v t and w t. o prove item 7, we note, since v t and w t are assumed to be independent and have zero means, the autocovariance of the product v t w t at lag j is the product of their respective autocovariance at lag j: γ v ( j)γ w ( j). Also, it is well-known that γ v ( j) = O( j d ) and γ w ( j) = O( j d ) if d = and d =. Consequently, we have ( ) Var v t w t = j= ( ) ( j ) γ v ( j)γ w ( j) = j= ( ) O( j d +d ) = O( d +d ) = O( d +d ), if d + d >.5, = O(ln ) = O( ln ), if d + d =.5, = O( ), otherwise. Using Proposition 6..3 from Brockwell and Davies (99), we thus have = O p ( d +d ), if d + d >.5, v t w t = O ] p ( ln ).5, if d + d =.5, = O p (.5 ), otherwise, and, by the facts that σ y = O( +d ) and σ x = O( +d ), we also have o prove item 8, we note y t xt = y t wt = O p (), if d + d.5, = O ] p (ln ).5, if d + d =.5, = O p (.5 d d ), otherwise. y t xt + 4 (v t x t + w t y t + v t w t )

26 = t/ (t )/ y s] x s] y ds + o p () = s] x s] ds + o p () B d (s) B d (s) ds. o show the second term at the end of the first line is o p (), we note that v t x t y t w t ( vt ( yt ) / ( ) / xt = O p (.5 ) O p ( +d ) = O p (.5+d ), ) / ( ) / wt = O p ( +d ) O p (.5 ) = O p (.5+d ). he orders of the four sums of squares are based on the results of items and 5. We also note that = O( +d +d ). hese results, together with item 7, imply v t x t O p (.5 d ), y t w t O p (.5 d ), and v t w t = o p (). So the above three terms all approach to and the proof of item 8 is completed. o prove item 9, we first note ( ) Var y t w t = = Var(y t w t ) + j= t= j+ Var(y t ) Var(w t ) + Cov(y t w t, y t j w t j ) j= t= j+ Cov(y t, y t j ) Cov(w t, w t j ) = γ w () = γ w () Var(y t ) + γ w ( j) t γ w ( j) j= t k= (t ) t= j+ { t j= t= j+ ( k ) ] γ v (k) + t t k= ( Cov(y t, y t j ) ) t max{k, j} j ( γ v (k) + k + j ) ]} γ v (k). t t k= 5

27 Note that the second equality results from the facts that y t and w t are independent and that E(y t ) = E(w t ) =. Given the assumption that d > and the approximations that γ v ( j) = O( j d ) and γ w ( j) = O( j d ) when d = and d =, we then have ( ) Var y t w t = γ w () t = γ w () t k= (t ) O( k d ) + γ w ( j) j= ] t= j+ O(t d+ ) + γ w ( j) = γ w ( j) j= j= t= j+ j= { t t= j+ t k= O(t d + ) t j O(k d ) + O(t d+ ) = γ w ( j) ( j) O( j d+ ) j= k= O(k d ) ( = j ) O( j d +d ) = O( d +d + ) = O( d +d + ). Note that the fifth equality is ensured by the assumption that d >. Given this approximation, we have ]} y t w t = O p ( d +d + ) or y t wt = O p (). he result for (v t/ )(x t / ) can be proved in a similar fashion. Hence, o prove item, we first note that y t = t vt = ( t)v t = v t v t t v t. y t B d () B d (s) ds, by applying the results of items 4 and. he weak limit of (t/ )(w t/ ) can be derived using a similar argument. o prove item, we note t yt = t yt + y t + t vt = t/ (t )/ s] y s] s] ds + o p () = y s] ds + o p () 6

28 s B d (s) ds. he orders of the last two terms at the end of the first line are based on the results of items and. he result for (/ ) (t/ )(x t/ ) can be proved by a similar argument. he joint weak convergence of 4 and 8,, can be established by writing the vector of sample moments as a functional of z r] = σ y y r] σ x x r] ] up to an error of op (). See Park and Phillips (988, p49). But regarding the conclusion in item 9 where we only have results on convergence rates instead of limiting distributions, the reason is that we cannot apply the usual stochastic calculus technique due to the fact that B d (s) and B d (s) are not martingale (see De Jong and Davidson,997). B Proof of heorem Let s first summarize the formulas for all relevant statistics in the simple linear regression model of y t on x t : β = s = y t x t y t x t, α = (x t x) û t = (y t α βx t ) = sβ = s, s (x t x) R = o prove item, we note β β = (x t x) y t β (y t y) β α = s, DW = (y t y) y t xt ( y t σ y (x t x) 7 σ x )( x t, (x t x), + (x) (x t x), (û t û t ) t= x t σ x ) û t β,.

29 where the weak convergence is due to items, 3, 8 of Lemma, and the joint weak convergence of the relevant sample moments. o prove item, we note α = y t β x t α, where the weak convergence is based on item above and item of Lemma. o prove item 3, we have σ y s = (y t y) σ y ( ) σx β (x t x) σ x σ, where the weak convergence is based on item above and item 3 of Lemma. o prove item 4, we note σ x σ y s β = σ y s (x t x) σ x σ β, where the weak convergence is based on item 3 above and item 3 of Lemma. o prove item 5, we see ( ) x t s σy α = s σ x σy + σ (x t x) α, where the weak convergence is based on item 3 above and items and 3 of Lemma. Items 6 and 7 are direct results from items,, 4, and 5. o prove item 8, we note ( ) { σx (x t x) β β σ R x Bd (s) ] ds = (y t y) Bd (s) ] ds σ y σ x ] } B d (s) ds ], B d (s) ds where the weak convergence is based on item above and item 3 of Lemma. o prove item 9, we note (û t û t ) = (y t α βx t ) (y t α βx t )] = (v t βw t ) = v t βv t w t + β w t, and, based on items 5 and 7 of Lemma, we have σ y vt = o p (), t= σ x wt = o p (), t= t= v t wt = o p (). 8