Invariance principles for fractionally integrated nonlinear processes

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1 IMS Lecture Notes Monograph Series Invariance principles for fractionally integrated nonlinear processes Xiaofeng Shao, Wei Biao Wu University of Chicago Abstract: We obtain invariance principles for a wide class of fractionally integrated nonlinear processes. The limiting distributions are shown to be fractional Brownian motions. Under very mild conditions, we extend earlier ones on long memory linear processes to a more general setting. The invariance principles are applied to the popular R/S and KPSS tests. 1. Introduction Invariance principles (or functional central limit theorems) play an important role in econometrics and statistics. For example, to obtain asymptotic distributions of unitroot test statistics, researchers have applied invariance principles of various forms; see Phillips (1987), Sowell (1990) and Wu (006) among others. The primary goal of the paper is to establish invariance principles for a class of fractionally integrated nonlinear processes. Let the process u t = F (, ε t 1, ε t ), t Z, (1.1) where ε t are independent and identically distributed (iid) random variables and F is a measurable function such that u t is well-defined. Then u t is stationary and causal. Let d ( 1/, 1/) and define Type I fractional I(d) process X t by (1 B) d (X t µ) = u t, t Z, (1.) where µ is the mean and B is the backward shift operator: BX t = X t 1. The Type II I(d) fractional process is defined as (1 B) d (Y t Y 0 ) = u t 1(t 1). (1.3) where Y 0 is a random variable whose distribution is independent of t. There is a recent surge of interest in Type II processes [Robinson and Marinucci (001), Phillips and Shimotsu (004)] and it arises naturally when the processes start at a given time point. The framework (1.1) includes a very wide class of processes [Wiener (1958), Rosenblatt (1971), Priestley (1988), Tong (1990), Wu (005a), Tsay (005)]. It includes linear processes u t = j=0 b jε t j as a special case. It also includes a large class of nonlinear time series models, such as bilinear models, threshold models and GARCH type models [Wu and Min (005), Shao and Wu (005)]. Recently, fractionally integrated autoregressive and moving average models (FARIMA) with GARCH innovations have attracted much attention in financial time series modeling [see Baillie et al. (1996)]. In financial time series analysis, the The work is supported in part by NSF grant DMS AMS 000 subject classifications: Primary 60F17; secondary 6M10 Keywords and phrases: Fractional integration, Long memory, Nonlinear time series, Weak convergence 1

2 /Fractional Invariance Principles conditional heteroscedasticity and long memory are commonly seen [Hauser and Kunst (1998), Lien and Tse (1999)]. The FARIMA-GARCH model naturally fits into our framework. Most of the results in the literature assume {u t } to be either iid or linear processes. Recently, Wu and Min (005) established an invariance principle under (1.) when d [0, 1/). The literature seems more concentrated on the case d (0, 1/). Part of the reason is that this case corresponds to long memory and it appears in various areas such as finance, hydrology and telecommunication. When d (1/, 1), the process is non-stationary and it can be defined as t s=1 X s or t s=1 Y s, where X s and Y s are Type I and Type II I(d 1) processes, respectively. Empirical evidence of d (1/, 1) has been found by Byers et al. (1997) in the poll data modeling and Kim (000) in macroeconomics time series. Therefore the study of partial sums of I(d), d ( 1/, 0) is also of interest since it naturally leads to I(d) processes, d (1/, 1). In fact, our results can be easily extended to the process with fractional index p + d, p N, d ( 1/, 0) (0, 1/) [cf. Corollary.1]. The study of invariance principle has a long history. Here we only mention some representatives: Davydov (1970), Mcleich (1977), Gorodetskii (1977), Hall and Heyde (1980), Phillips and Solo (199), Davidson and De Jong (000), De Jong and Davidson (000) and the references cited therein. Most of them deal with Type I processes. Recent developments for Type II processes can be found in Marinucci and Robinson (1999a), Wang et al. (00) and Hosoya (005) among others. The paper is structured as follows. Section presents invariance principles for both types of processes. Section 3 considers limit distributions of tests of long memory under mild moment conditions. Technical details are given in the appendix.. Main Results We first define two types of fractional Brownian motions. For Type I fractional Brownian motion, let d ( 1/, 1/) and IB d (t) = 1 A(d) [{(t s) + } d {( s) + } d ]dib(s), t R, where (t) + = max(t, 0), IB(s) is a standard Brownian motion and A(d) = { 1 d [(1 + s) d s d ] ds} 1/. Type II fractional Brownian motion {W d (t), t 0}, d > 1/, is defined as W d (0) = 0, t W d (t) = (d + 1) 1/ (t s) d dib(s). 0 The main difference of IB d (t) and W d (t) lies in the prehistoric treatment. See Marinucci and Robinson (1999b) for a detailed discussion of the difference between them. Here we are interested in the weak convergence of the partial sums T m = m i=1 X i and Tm = m i=1 Y i. Let D[0, 1] be the space of functions on [0, 1] which are right continuous and have left-hand limits, endowed with the Skorohod topology (Billingsley, 1968). Denote weak convergence by. For a random variable X, write X L p (p > 0) if X p := [E( X p )] 1/p < and =. Let F t = (, ε t 1, ε t ) be the shift process. Define the projections

3 /Fractional Invariance Principles 3 P k by P k X = E(X F k ) E(X F k 1 ), X L 1. For two sequences (a n ), (b n ), denote by a n b n if a n /b n 1 as n. The symbols D and P stand for convergence in distribution and in probability, respectively. The symbols O P (1) and o P (1) signify being bounded in probability and convergence to zero in probability. Let N(µ, σ ) be a normal distribution with mean µ and variance σ. Hereafter we assume without loss of generality that E(u t ) = 0, µ = 0 and Y 0 = 0. Let {ε k, k Z} be an iid copy of {ε k, k Z} and F k = (F 1, ε 0, ε 1,..., ε k ). Theorems.1 and. concern Type I and II processes respectively. Using the continuous mapping theorem, Theorems.1 and. imply Corollary.1 which deals with general fractional processes with higher orders. For d ( 1/, 0), an undesirable feature of our results is that the moment condition depends on d. However, this seems to be necessary; see Remark 4.1. Similar conditions were imposed in Sowell (1990) and Wang et al. (003). Theorem. extends early results by Akonom and Gourieroux (1987), Tanaka (1999) and Wang et al. (00), who assumed u t to be either iid or linear processes. See Marinucci and Robinson (1999a) and Hosoya (005) for a multivariate extension. Theorem.1. Let u t L q, where q > /(d + 1) if d ( 1/, 0) and q = if d (0, 1/). Assume P 0 u k q <. (.1) k=0 Then ζ j := k=j P ju k L q and, if ζ 0 > 0, T nt n d+1/ κ 1(d)IB d (t) in D[0, 1], where κ 1 (d) = A(d) ζ 0 Γ(d + 1). (.) Remark.1. Note that ζ 0 = πf u (0), where f u ( ) is the spectral density function of {u t }; see Wu (005b) and Wu and Min (005) for the details. Theorem.. Under the conditions of Theorem.1, for type II processes, we have T nt n κ (d)w d+1/ d (t) in D[0, 1], where κ (d) = ζ 0 (d + 1) 1/ Γ(d + 1). (.3) By the continuous mapping theorem and the standard argument in Wang et al. (003), we have the following corollary. Corollary.1. Let u t satisfy conditions in Theorem.1; let d (0, 1/) ( 1/, 0) and p N. [a] Define the process X t by (1 B) p+d Xt = u t. Then (i). n (d+p 1/) X nt κ 1 (d)ib d,p (t) in D[0, 1]; (ii). n (d+p+1/) nt X j κ 1 (d)ib d,p+1 (t) in D[0, 1]; (iii). n (d+p) nt X j κ 1(d) t 0 [IB d,p(s)] ds in D[0, 1]. Here IB d,p (t) is defined as { IB d (t), p = 1, IB d,p (t) = tp 1 t 0 0 IB d(t 1 )dt 1 dt dt p 1, p. t 0 [b] Define the process Ỹt by (1 B) p+d Ỹ t = u t 1(t 1). Then similarly (i). n (d+p 1/) Ỹ nt κ (d)w d+p 1 (t) in D[0, 1]; (ii). n (d+p+1/) nt Ỹj κ (d)w d+p (t) in D[0, 1]; (iii). n (d+p) nt Ỹ j κ (d) t 0 [W d+p 1(s)] ds in D[0, 1].

4 /Fractional Invariance Principles 4 We now discuss condition (.1). Let g k (F 0 ) = E(u k F 0 ) and δ q (k) = g k (F 0 ) g k (F0 ) q. Then δ q (k) measures the contribution of ε 0 in predicting u k. In Wu (005a) it is called the predictive dependence measure. Since P 0 u k q δ q (k) P 0 u k q (Wu, 005a), (.1) is equivalent to the q-stability condition (Wu, 005a) k=0 δ q(k) <, which suggests short-range dependence in that the cumulative contribution of ε 0 in predicting future values of u k is finite. For a variety of nonlinear time series, δ q (k) = O(ρ k ) for some ρ (0, 1). The latter is equivalent to the geometric moment contraction (GMC) [Wu and Shao (004), Wu and Min (005)]. Shao and Wu (005) verified GMC for GARCH(r, s) model and its asymmetric variants and showed that the GMC property is preserved under ARMA filter. In the special case u t = k=0 b kε t k, (.1) holds if k=0 b k < and ε 1 L q. 3. Applications There have been a large amount of work on test of long memory under short memory null hypothesis, i.e. I(0) versus I(d), d (0, 1/). For example, Lo (1991) introduced modified R/S test statistics, which admits the following form: Q n = 1 w n,l max 1 k n (X j X n ) min 1 k n (X j X n ), where X n = n 1 n X j is the sample mean and w n,l is the long run variance estimator of X t. Following Lo (1991), w n,l = 1 n n (X j X n ) + l ( 1 j ) ˆγ j, (3.1) l + 1 where ˆγ j = 1 n j n i=1 (X i X n )(X i+j X n ), 0 j < n. The form (3.1) is equivalent to the nonparametric spectral density estimator of {X t } evaluated at zero frequency with Bartlett window (up to a constant factor). Here the bandwidth satisfies l = l(n) and l/n 0, as n. (3.) Lee and Schmidt (1996) applied the KPSS test (Kwiatkowski et al. 199) for I(0) versus I(d), d ( 1/, 0) (0, 1/). The test statistics has the form: K n = 1 n wn,l (X j X n ) n k=1 with wn,l given by (3.1). Lee and Schmidt showed that the test is consistent against fractional alternatives and derived its asymptotic distribution under the assumption that u t are iid normal random variables. Giraitis et al. (003) investigated the theoretical performance of various forms of nonparametric tests under both short memory hypotheses and long memory alternatives. In a quite general setting, we obtain asymptotic distributions of R/S and KPSS test statistics under fractional alternatives. Theorem 3.1. Suppose that X t is generated from (1.) and u t satisfies (.1) with some q > max(, /(d + 1)). Assume (3.). Then l d w n,l P κ 1(d). (3.3)

5 /Fractional Invariance Principles 5 Consequently, we have l d n d+1/ Q n D sup 0 t 1 IB d (t) inf 0 t 1 IB d (t), (3.4) where IB d (t) is the fractional Brownian bridge, i.e. IB d (t) = IB d (t) tib d (1), and l d n d K n D 1 0 ( B d (t)) dt. (3.5) Remark 3.1. For d (0, 1/), Giraitis et al. (003) obtained (3.4) under the joint cumulant summability condition sup h n r,s= n cum(x 0, X h, X r, X s ) = O(n d ). (3.6) For linear processes, (3.6) can be verified. But for nonlinear fractional processes (1.), it seems hard to directly verify (3.6). In contrast, we only need to impose q-th (q > ) moment condition when d (0, 1/). Our dependence condition (.1) can be easily verified for various nonlinear time series models [cf. Wu and Min (005) and Shao and Wu (005)]. 4. Appendix Lemma 4.1. Let a i = i β l(i), i 1, where l is a slowly varying function and β > 1/; let q > (3/ β) 1 if 1 < β < 3/ and q = if 1/ < β < 1; let σ n = A(1 β)n 3/ β l(n)/(1 β) and σ n = (3 β) 1/ n 3/ β l(n)/(1 β). Assume either (1 ) 1 < β < 3/, i=0 a i = 0 or ( ) 1/ < β < 1. Let η j = G(..., ε j 1, ε j ), j Z, (4.1) be a martingale difference sequence with respect to σ(, ε j 1, ε j ). Assume η j L q. Let Y j = i=0 a iη j i, Yj = j 1 i=0 a iη j i, S i = i Y j, Si = i Y j. Then we have [a] σn 1 S nt η 0 IB 3/ β (t) in D[0, 1] and [b] σ n 1 S nt η 0 W 3/ β (t) in D[0, 1]. Proof of Lemma 4.1. [a] Consider (1 ) first. For the finite dimensional convergence, we shall apply the Cramer-Wold device. Fix 0 t 1 < t 1 and let m 1 = nt 1 and m = nt. Let A i = i j=0 a j if i 0 and A i = 0 if i < 0. For λ, µ R let c n,l = λ(a m 1 m +l A l m ) + µ(a l A l m ), σ n σλµ = [λ t 3 β 1 + µ t 3 β + λµ(t 3 β 1 + t 3 β (t t 1 ) 3 β )]. Then (λs m1 + µs m )/σ n = l=0 c n,lη m l has martingale difference summands and we can apply the martingale central limit theorem. By Karamata s Theorem, A n = j=n+1 a j n 1 β l(n)/(β 1). Elementary calculations show that l=0 c n,l σλµ and sup c n,l 0 as n. (4.) l 0

6 /Fractional Invariance Principles 6 Let V l = E(ηm F l m l 1). By the argument in the proof of Theorem 1 in Hannan (1973), (4.) implies that l=0 c n,l V l σλµ in L1. For completeness we prove it here. Let ω > 0 be fixed, V l = V l 1 Vl ω and V l = V l V l. By (4.), lim sup c n n,l(v l EV l ) lim sup c n,lc n,l cov(v 0, V l l ) = 0 n l=0 l,l =0 since lim k cov(v 0, V k ) = 0. Therefore, using V l = V l + V l, again by (4.), lim sup E c n n,l(v l EV l ) lim sup E c n n,l(v l EV l ) l=0 l=0 lim sup c n,lev l 0 as ω. n Under (4.), for any δ > 0, l=0 E{ c n,l η m 1 l c n,l η m l δ} 0. So the finite dimensional convergence holds. By Proposition 4 of Dedecker and Doukhan (003), S n q q η 0 q (A j A j n ) = O(σn). By Theorem.1 of Taqqu (1975), the tightness follows. ( ) Note that S n η 0 σ n, the conclusion similarly follows. [b] The finite dimensional convergence follows in the same manner as [a]. For the tightness, let 1 m 1 < m n, by Proposition 4 in Dedecker and Doukhan (003), S m S m 1 q q η 0 q m 1 l=0 (A j A j (m m 1)) = O( σ m m 1 ). j=0 With the above inequality, using the same argument as in Theorem.1 of Taqqu (1975), we have for any 0 t 1 t t 1, there exists a generic constant C (independent of n, t 1, t and t ), such that for β (1/, 1), and for β (1, 3/), E( S nt S nt 1 S nt S nt ) C σ n(t t 1 ) 3 β E S nt S nt 1 q/ S nt S nt q/ C σ q n(t t 1 ) q(3/ β). Thus the tightness follows from Theorem 15.6 in Billingsley (1968). Remark 4.1. Under (1 ) of Lemma 4.1, the moment condition η j L q, q > (3/ β) 1, is optimal. and it can not be reduced to η j L q0, q 0 = (3/ β) 1. Consider the case in which η i are iid symmetric random variables and P( η 0 q0 g) g 1 (log g) as g. Then η j L q0. Let l(n) = 1/ log n, n > 3. Elementary calculations show that σn 1 max 1 j n η j in probability. Let Y j = Y j + η j η j 1 and S i = i Y j. Then the coefficients a j of Y j also satisfy the conditions in Lemma 4.1. The two processes σn 1 S nt and σn 1 S nt, 0 t 1, cannot both converge weakly to fractional Brownian motions. If so, since max j n η j η 0 max j n S j + max j n S j, we have max j n η j = O P (σ n ), contradicting σn 1 max 1 j n η j in probability. Similar examples are given in Wu and Min (005) and Wu and Woodroofe (004).

7 /Fractional Invariance Principles 7 Proof of Theorem.1. Let a j = Γ(j + d)/{γ(d)γ(j + 1)}, j 0, and A k = k i=0 a i if k 0 and 0 if k < 0. Note that β = 1 d. Then X t = j=0 a ju t j. By (.1), ζ j = i=j P ju i L q. Let M n = n i=1 ζ i, S n = n Y j, Y j = i=0 a iζ j i, U n = n i=1 u i and R n = T n S n. By Theorem 1 in Wu (005b), U n M n q = o( n). By Karamata s theorem and summation by parts, we have 3m R m q (A i A i m )(u m i ζ m i ) i=0 q = + i=3m+1 3m i=1 + (A i A i m )(u m i ζ m i ) q (A i A i m ) (A i 1 A i 1 m ) o( i) i=3m+1 By Proposition 1 in Wu (005b), max R m (k r)/q R r q = m k q (A i A i m ) (A i 1 A i 1 m ) o( i) = o(σ m ). r=0 (k r)/q o(σ r) = o(σ k), since q > /(d + 1). So the limit of {T nt /σ n, 0 t 1}, if exists, is equal to the limit of {S nt /σ n, 0 t 1}. By Lemma 4.1, the latter has a weak limit. So (.) follows. Proof of Theorem.. As in the proof of Theorem.1, let Si Rm = Tm Sm. By Karamata s theorem, m Rm+l Rl q A j A j 1 o( j) = o( σ m ). Again by the maximal inequality [Proposition 1 in Wu (005b)], max k r Rm R m k r j R r (j 1) q q q = r=0 r=0 (k r)/q o( σ r) = o( σ k), r=0 which proves the theorem in view of Lemma 4.1. = i Y j Proof of Theorem 3.1. If (3.3) holds, by the continuous mapping theorem, Theorem.1 entails (3.4) and (3.5). In the sequel we shall prove (3.3). Note that wn,l = 1 n Xj + l ( 1 j ) n j X i X i+j n n l + 1 X n X n n l =: I 1n X n + I n. ( 1 j l + 1 ) n j i=1 i=1 (X i + X i+j ) + X n n l 1/q and ( 1 j ) (n j) l + 1

8 /Fractional Invariance Principles 8 Since X n = O P (n d 1/ ), l d ( X n + I n ) = O P ((l/n) 1 d ) = o P (1). Thus it suffices to show that l d I 1n P κ 1(d). Let V j = j i=1 X i, Ṽ j = n i=n j+1 X i, 1 j l, then a straightforward calculation shows that I 1n = J 1n + J n, where 1 n i J 1n := X j 1 l, J n := (Vj + (l + 1)n n(l + 1) Ṽ j ). i=l+1 j=i l Corollary.1 implies that l V j = O P (l d+ ). Since l Ṽ j distribution as l V j, we have J n = O P (l d+ /(ln)) = o P (l d ). has the same It remains to show l d J 1n P κ 1(d). Let ζ j = i=j P ju i L q, M n = n i=1 ζ i, U n = n i=1 u i and r n = sup j n U j M j / j. Then r n 0 and it is nonincreasing. Let L = min{ nl, l(r l )1/( 1 d) }. Then l = o(l) and L = o(n). Let W j,l = L (A i A i l 1 )u j i, Q j,l = i=0 L (A i A i l 1 )ζ j i and S j,l = j i=j l X i. Then J 1n = 1 n n(l+1) j=l+1 S j,l. Since l/l 0, S j,l W j,l = (A i A i l 1 )u j i i=l+1 [ ] 1/ (A i A i l 1 ) P 0 u t = o(σ l ). (4.3) i=l+1 i=0 t=0 By the definition of L, using summation by parts, we have W j,l Q j,l L (A i A i l 1 ) (A i 1 A i l ) r i i i=1 r l L l a i i + a i r i i = o(σl ). (4.4) i=1+ l i=1 Now we shall show n E {Q j,l E(Q j,l)} = o(nσ l ). (4.5) Since τ m := E E(ζ 0 F m ) E(ζ 0) 0 as m, E E(Q Lk,l F Lk L ) E(Q Lk,l) L (A i A i l 1 ) τ L i = o(σl ).(4.6) i=0 Let D k = Q Lk,l E(Q Lk,l F Lk L), k = 1,..., b = n/(l). Set C q = 18q 3/ (q 1) 1/ and q = min(q, 4). By Burkholder s inequality, b D k C q/ b /q D k q/ C q/ b /q Q Lk,l q k=1 q/

9 /Fractional Invariance Principles 9 = b /q O(σl ) = o(bσl ). (4.7) Thus (4.5) follows from (4.6) and (4.7). By (4.3), (4.4) and (4.5), n n E {Sj,l E(Q j,l)} E Sj,l Q j,l + o(nσl ) = o(nσl ) which completes the proof since E(Q j,l ) σ l ζ 0. REFERENCES Akonom, J. and Gourieroux, C. (1987). A functional central limit theorem for fractional processes. Discussion paper #8801. Paris: CEPREMAP. Baillie, R. T., Chung, C. F. and Tieslau, M. A. (1996). Analyzing inflation by the fractionally integrated ARFIMA-GARCH model. J. Appl. Econom Billingsley, P. (1968). Convergence of Probability Measures. Wiley. Byers, D., Davidson, J. and Peel, D. (1997). Modelling political popularity: an analysis of long-range dependence in opinion poll series. J. Roy. Statist. Soc. Ser. A Chung, K. L. (1968). A Course in Probability Theory. Harcourt, Brace and World, New York. Davidson, J. and De Jong, R. M. (000). The functional central limit theorem and weak convergence to stochastic integrals, II: Fractionally integrated processes. Econom. Theory De Jong, R. M. and Davidson, J. (000). The functional central limit theorem and weak convergence to stochastic integrals, I: Weakly dependent processes. Econom. Theory Davydov, Y. A. (1970). The invariance principle for stationary processes. Theory Probab. Appl Dedecker, J. and Doukhan, P. (003). A new covariance inequality and applications. Stochast. Process. Appl Giraitis, L., Kokoszka, P., Leipus, R. and Teyssiére, G. (003). Rescaled variance and related tests for long memory in volatility and levels. J. Econometrics Gorodetskii, V. V. (1977). On convergence to semi-stable Gaussian process. Theory Probab. Appl Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and its Applications. New York: Academic Press.

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