ROBUST DICKEY-FULLER TESTS BASED ON RANKS FOR TIME SERIES WITH ADDITIVE OUTLIERS

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1 ROBUST DICKEY-FULLER TESTS BASED ON RANKS FOR TIME SERIES WITH ADDITIVE OUTLIERS V. A. REISEN, C. LÉVY-LEDUC, M. BOURGUIGNON, AND H. BOISTARD Abstract. In this paper the unit root tests propose by Dickey an Fuller DF) an their rank counterpart suggeste by Breitung an Gouriéroux 1997) BG) are analytically investigate uner the presence of aitive outlier AO) contaminations. The results show that the limiting istribution of the former test is outlier epenent, while the latter one is outlier free. The finite sample size properties of these tests are also investigate uner ifferent scenarios of testing contaminate unit root processes. In the empirical stuy, the alternative DF rank test suggeste in Granger an Hallman 1991) GH) is also consiere. In Fotopoulos an Ahn 003), these unit root rank tests were analytically an empirically investigate an compare to the DF test, but with outlier-free processes. Thus, the results provie in this paper complement the stuies of the previous works, but in the context of time series with aitive outliers. Equivalently to DF an Granger an Hallman 1991) unit root tests, the BG test shows to be sensitive to AO contaminations, but with less severity. In practical situations where there woul be a suspicion of aitive outlier, the general conclusion is that the DF an Granger an Hallman 1991) unit root tests shoul be avoie, however, the BG approach can still be use. Keywors: Time Series; robust tests; ranks; outliers; unit root. 1. Introuction An important issue in time series analysis is the etermination of the egree of integration. In the stanar ARIMAp,,q) moel, the egree of integration is efine as the number of times the series must be ifference to yiel a stationary time series with invertible MA representation Breitung 1994)). Then, the orer of integration of a time series is a crucial eterminant of the properties exhibite by the series an, therefore, unit root tests play an important role in the analysis of non-stationary time series. Since the seminal works by Dickey an Fuller 1979, 1981) an Sai an Dickey 1984), usually referre as DF an ADF tests respectively, testing unit root has become routine in empirical stuies, specially in the economic area. Since the early 90 s, various generalizations of DF an ADF tests have been propose for hanling unit root processes, see for example, Elliot et al. 1996), Aparicio et al. 006), Demetrescu et al. 008) an Astill et al. 013) an Westerlun 014) for a recent review. Among all the research eicate to unit root processes, a special attention has been pai to Date: September 3,

2 V. A. REISEN, C. LÉVY-LEDUC, M. BOURGUIGNON, AND H. BOISTARD testing unit root against fractional alternatives, an the stuies have inicate that the power of DF test is low in such circumstances see Hassler an Wolters 1994); Kramer1998); Wang 011) an references therein). To improve the power an the size of unit root tests, bootstrap techniques have been consiere in a large number of papers as an alternative way to istinguish long-memory processes from more common I0) an I1) processes, see for example, Franco an Reisen 007) an Palm et al. 008). Demetrescu et al. 008) propose an integration test against fractional alternatives which is base on the pioneering Lagrange multiplier LM) test suggeste by Robinson 1991) an further stuie by Breitung an Hassler 00) among others. An alternative nonparametric unit root test, the DF test base on ranks DF-rank test), to eal with non-linear transformations was firstly iscusse in Granger an Hallman 1991), further stuie by Breitung an Gouriéroux 1997) an by Fotopoulos an Ahn 003). The DF-rank tests o not use the observations but rather the ranks of the observations which makes the tests relatively insensitive to outlying observations. A iscussion of the avantage of the nonparametric unit root test over the parametric one is given in the introuction section of Breitung an Gouriéroux 1997). Uner some assumptions, Breitung an Gouriéroux 1997) erive the asymptotic behavior of their test uner the null hypothesis. In the same irection, Fotopoulos an Ahn 003) consiere comparisons between Breitung an Gouriéroux 1997) an Granger an Hallman 1991) rank statistics for the unit root test. The authors also provie the limiting istribution of the Breitung an Gouriéroux 1997) test uner the null hypothesis an give small sample properties. In these works, the tests were built an empirically teste uner the alternative of AR processes. The empirical stuy unertaken by Fotopoulos an Ahn 003) inicate that the DF test showe higher power than the rank test counterparts. An alternative robust approach base on ranks to test unit root was recently propose by Hallin et al. 011), an they showe that their metho is robust with respect to the innovation ensity. An avantage of the DF-rank test over the parametric is that the nonparametric approach is, in general, outlier-resistant. As is well-known, the outliers may seriously estroy the statistical property of an estimator Chan1995)). See, for example, Martin an Yohai 1986) in the case of the least square estimator an the recent works by Molinares et al. 009) an by Lévy-Leuc et al. 011b,a) an Reisen an Molinares 01) in the context of time series with short an long-memory properties. Franses an Halrup 1994) investigate the size of the DF test in the presence of aitive outliers. These authors provie analytical as well as empirical eviences that AO may prouce spurious stationarity, that is, the aitive outliers provoke the rejection of unit root too often. The recent work Astill et al. 013) suggests a bootstrap test for etecting aitive AO in a univariate unit root process. As an alternative way to testing unit root with ata anomalies an, also, with ifferent error istributions, M-estimators have been invoke in some stuies, for example, Lucas 1995) an Carstensen 003).

3 ROBUST DICKEY-FULLER TESTS BASED ON RANKS 3 Although the M-robust unit root tests are in some way better protecte against some eviations from gaussianity, however, uner aitive outliers they fail to prouce stable sizes an goo power properties. Their size istortions are similar to the DF test. The work reporte here complements the research of Granger an Hallman 1991), Breitung an Gouriéroux 1997), Fotopoulos an Ahn 003) an Hallin et al. 011) by proviing the asymptotic istribution of the rank tests for unit root processes with aitive outliers. Uner outliers contaminations, the size an the power of the tests are investigate empirically for finite sample sizes. The rest of this paper is organize as follows. Section iscusses the analytical results of the unit root testings whether the process is contaminate with aitive outliers or not. In Section 3, the tests are investigate for small sample sizes in ifferent scenarios of the unit root processes. Conclusions are given in Section 4.. Theoretical results In this section some theoretical results are erive an iscusse. Let y t = ρy t 1 +ε t, t = 1,...,T, 1) where ρ 1, y 0 = 0 an ε t s are i.i.. ranom variables with mean zero an variance σ. If ρ = 1, the process 1) is nonstationary in the mean an it is known as ranom walk or integrate process of orer 1 I1)). On the other han, the boun ρ < 1 implies stability an asymptotic stationarity properties of the process y t. The type of outliers has quite ifferent impact an properties for moels an robust estimates. In general, the innovation IO) an aitive outliers are the most common type of atypical observations iscusse in the literature see, for example, Chen an Liu 1993)). Uner the assumption of IO, the least squares estimates of ρ are consistent even when the error in the moel is not Gaussian but with a finite variance. As is well-known, AO are known to eteriorate more the estimator than IO. In the presence of AO, the least squares estimates not only lack robustness in terms of variability but also suffer from bias problems see, for example, Denby an Martin 1979) an Franses an Halrup 1994)). In the case of unit root moel I1)), the AO provokes a spurious moel estimation by inicating that the process has stationary roots. The moel contaminate by aitive outliers is efine here as z t = y t +θδ t, t = 1,...,T, ) whereθ is the magnitueof the outliers fixe an unknownparameter), δ t s are i.i.. ranom variables, with variance σ δ, such that Pδ t = 1) = Pδ t = 1) = π/ an Pδ t = 0) = 1 π, where π is in 0,1). It is assume here that the ranom variables δ t s are inepenent of y t s. The terms θδ t in ) are calle effects rather than errors since it causes an immeiate an one-shot effect on the observe series. As is iscusse in the literature, the AO seems to be

4 4 V. A. REISEN, C. LÉVY-LEDUC, M. BOURGUIGNON, AND H. BOISTARD a more appropriate escriptor of non repetitive events see, for example, Denby an Martin 1979) an Franses an Halrup 1994)). In the sequel, the asymptotic behavior of two ifferent unit-root tests uner the null hypothesis ρ = 1 is iscusse when the process is contaminate with AO..1. Dickey-Fuller test uner outliers. Following the regression efine in ), the least square estimates is ˆρ DF 1 = T t=1 z t 1 z t T t=1 z t 1, 3) where z t = z t z t 1. Uner the null hypothesis H 0 ): ρ = 1, the DF test statistics an their limiting istributions are given as follows T t=1 Tˆρ DF 1) = T z t 1 z t T t=1 z t 1, 4) an ˆτ DF = ˆρ DF 1 ˆσ DF T t=1 z t 1 where ˆσ DF = T 1 T t=1 z t ˆρ DF 1)z t 1 ). Theorem 1. Assume that z t ) 1 t T are given by ), then when ρ = 1, ˆτ DF Tˆρ DF 1) W1) Wr) r [1+θ/σ) π] 1/ W1) Wr) r ) 1/, 5) ) θ π σ 1, as T, 6) 0 Wr) r θ/σ) π ) 1/ 1 0 Wr) r ) 1/, as T, 7) where enotes the convergence in istribution an where {Wr), r [0,1]} enotes the stanar Brownian motion. The proof of Theorem 1 is given in Franses an Halrup 1994). We have also written the proof with full etails an it is available upon request. The stanar DF tests are a particular case of 6) an 7) when either θ = 0 or π = 0. As argue by Franses an Halrup 1994) an also iscusse here, for θ 0, the limiting istributions of the tests epen on the parameter θ an, therefore, the behavior of the test istributions will be shifte to the left sie accoring to the size of θ an, also, to the probability of the occurrence of outliers. This translation will inflate the size of the test by leaing to reject H 0 ) too often in favor of a spurious stationary solution. Base on these properties, the DF test uner a suspicion of outliers in the ata shoul be avoie see Franses an Halrup 1994)).

5 ROBUST DICKEY-FULLER TESTS BASED ON RANKS 5.. Breitung-Gouriéroux test. The Breitung-Gouriéroux s statistics Breitung an Gouriéroux 1997)) consists in replacing the first ifference of the observations y t ) 1 t T by their ranks r T,t ) in the Dickey-Fuller test statistics. Then, by using the same computations as those use in Fotopoulos an Ahn 003), the unit root tests can be written as follows where Tˆρ BG 1) = T ˆσ BG = 1 T ˆτ BG = ˆρ BG 1 ˆσ BG T t t= T t= rt,t T+1 ) { t 1 t= T t={ t 1 rt,j T+1 T t 1 r T,j T +1 ) r T,j T +1 t 1 ) ˆρ BG rt,j T+1 ) } ) }, 8) 1/ r T,j T +1, 9) ). 10) Contrary to the limiting istributions of DF tests, it is prove in the following theorem that the limiting istribution of BG test statistics are outliers-free. Theorem. Assume that z t ) 1 t T are given by ), then when ρ = 1, 1 Tˆρ BG 1) 1 0 Br) r ˆτ BG, as T, 11) 1 { } 1 1/, as T, 1) 0 Br) r where {Br), r [0,1]} enotes the stanar Brownian brige, ˆρ BG an ˆτ BG are efine in 8) an 9), respectively, r T,t enoting the rank of z t ) 1 t T. The proof of Theorem is given in Section 5. From Theorem, it can be seen that the statistic 11) has the same orer as in 6). It can also be observe that the limiting istributions of the BG statistics are always negative. This implies that the estimate of ρ is asymptotically smaller than 1 an this particular problem was seen with some negative criticism by Fotopoulos an Ahn 003). As an alternative form to overcome this, the authors investigate an suggeste the statistics given by Granger an Hallman 1991)GH). However, as will be seen in the simulation section 3, the empirical istribution of this statistic carries the same problem as the DF tests when the ata has outliers, that is, the empirical results inicate that its limiting istribution also epens on the parameter θ. This problem motivate the eication of this paper to the stuy of the theoretical properties of the BG statistics uner the presence of aitive outliers.

6 6 V. A. REISEN, C. LÉVY-LEDUC, M. BOURGUIGNON, AND H. BOISTARD 3. Small-sample properties The BG test along with the stanar DF an GH tests are applie to unit root processes uner the presence of AO with the aim to analyze their performances for finite sample sizes. This empirical investigation will give a better unerstaning of whether or not the outliers can affect the size of the tests an if it oes, how. For this objective, samples following Equations 1 an, where the ε t s are i.i.. Gaussian ranom variables with unit variance, were simulate accoring to the following scenarios: time series with small an large sample sizes an ifferent outlier magnitues with fixe probability occurrences of π. The empirical rejection rates of the null hypothesis of unit root are base on 10,000 replications. From now on, the nominal level of the tests is 5%. Table 1 isplays the size of the tests without θ = 0) an uner θ 0) outliers in the ata with π = The results for π = 0.05 are available upon request. The sizes of the tests are also investigate uner a fixe number of outliers Tπ = 1 an n = 50,400 the results are in Table ). Other cases such as Tπ = an n = 100,00,1000 isplaye similar conclusions an are available upon request. Table 1. Rejection rates of H 0 : ρ = 1, with π = 0.01 sample sizes Tests Empirical critical points θ = 0 θ = 3 θ = 5 θ = 7 θ = 10 T = 50 ˆτ DF Tˆρ DF 1) ˆτ GH Tˆρ GH 1) ˆτ BG Tˆρ BG 1) T = 100 ˆτ DF Tˆρ DF 1) ˆτ GH Tˆρ GH 1) ˆτ BG Tˆρ BG 1) T = 00 ˆτ DF Tˆρ DF 1) ˆτ GH Tˆρ GH 1) ˆτ BG Tˆρ BG 1) T = 400 ˆτ DF Tˆρ DF 1) ˆτ GH Tˆρ GH 1) ˆτ BG Tˆρ BG 1) T = 1000 ˆτ DF Tˆρ DF 1) ˆτ GH Tˆρ GH 1) ˆτ BG Tˆρ BG 1)

7 ROBUST DICKEY-FULLER TESTS BASED ON RANKS 7 It is well-known that the Orinary Least Square estimator is sensitive to the occurrence of outliers in the ata. This sensitivity is inherite by the DF-test as is observe in the table. The empirical size of DF test is inflate ue to the sizes of θ an T, an this leas to reject a unit root too often inicating a spurious stationary. This confirms the analytical results given in Theorem 1 an also in Franses an Halrup 1994). In contrast to the DF test, the rank tests are much less sensitive to aitive outliers. Regarless of the sample size an magnitue of the outliers, the results in Table 1 show that the sizes of the tests base on the BG metho o not fluctuate too much. They are always near to 5%. However, the GH approach oes not always guarantee this empirical property. This test is also sensitive to outliers, but with less impact from the effect of the magnitue an the sample size compare with DF test θ = 0 θ = 3 θ = 5 θ = 7 θ = θ = 0 θ = 3 θ = 5 θ = 7 θ = a) ˆτ GH b) Tˆρ GH 1) Figure 1. Empirical ensities of ˆτ GH an Tˆρ GH 1) for ifferent values of θ, T = 00 an π = Figures 1 an give the empirical ensities of the GH an BG tests, respectively, for T = 00. The ensity of GH test shifts to the left with respect to the istribution of uncontaminate ata accoring to the sizes of θ. The ensity of DF isplays similar behaviour an it is available upon request. On the other han, the empirical istributions of BG shows to be very similar regarless of the size of θ which confirm the results in Table 1 that this test has an approximate constant size uner a variety of contaminations. As previously state, Theorem shows that the asymptotical istributions of BG tests o not epen on the size of θ. However, this outlier-free property oes not always hol for finite sample sizes. As an example of the size istortions of the tests, these quantities are plotte in Figure 3 a) for T an θ fixe an π varying from 00.01) to It turns out that BG test also shows to have the same problems as DF an GH tests, but it is less severe. It is obvious that the sizes of DF an GH tests eteriorate even more with growing θ an T, which were expecte results.

8 8 V. A. REISEN, C. LÉVY-LEDUC, M. BOURGUIGNON, AND H. BOISTARD θ = 0 θ = 3 θ = 5 θ = 7 θ = θ = 0 θ = 3 θ = 5 θ = 7 θ = a) ˆτ BG b) Tˆρ BG 1) Figure. Empirical ensities of ˆτ BG an Tˆρ BG 1) for ifferent values of θ, T = 00 an π = Rejection rates τ^df τ^gh τ^bg Power function τ^df τ^gh τ^bg π ρ a) b) Figure 3. a) Size istortions of the tests for ifferent π, T = 00 an θ = 5; b) Empirical size-ajuste powers, T = 100, θ = 3 an π = 0.01 Table isplays the sizes of the tests for fixe outlier numbers Tπ = 1, n = 50,400. The effects on the size of the tests o not become serious by the increasing of the sample size, that is, size istortions are less pronounce for all tests. Results for n = 100, 00, 1000 presente similar conclusions an are available upon request. Base on the analytical an empirical properties of the tests iscusse here, a general conclusion is that the DF test uner a suspicion of outliers in the ata shoul be avoie, see also Franses an Halrup 1994). The rank test base on GH is somewhat sensitive to

9 ROBUST DICKEY-FULLER TESTS BASED ON RANKS 9 Table. Rejection rates of H 0 : ρ = 1, setting Tπ = 1 sample sizes θ = 0 θ = 3 θ = 5 θ = 7 θ = 10 T = T = outliers as the DF test is. Hence, caution also has to be taken when using this test uner ata contaminations for small sample sizes. Although the BG test also presente some size istortions this test seems to be an alternative unit root test to be use in most of scenarios iscusse in this paper. The empirical powers of the tests were also consiere in the scenario of unit root against the alternative of stationary AR processes, but with outlier contaminations. Since the unit root tests presente size istortions uner outliers in the ata, their empirical powers properties were obtaine uner size-ajuste power which is not a realistic power in practical situations see also Lucas 1995), page 165). Figure 3 b) gives some insights of the lower size-ajuste power of the rank tests against DF test. Other cases are available upon request. In all case consiere, the results were not surprising by showing that the powers of the rank tests were inferior to the DF test which are somewhat in accorance with the one iscusse in Fotopoulos an Ahn 003), for no outlier contamination cases see also Granger an Hallman, 1991, p. 17)). 4. Conclusions In this paper robust unit root tests base on ranks, initially suggeste in Granger& Hallman 1991) an consiere in Fotopoulos & Ahn 003), are stuie theoretically an empirically unerlying aitive outliers in the level of non stationary time series by complementing the finings of the previous works. In this context, the analytical results show that the limiting istribution of the Dickey-Fuller rank test base on the ranks of Y t is outlier free in contrast with the stanar DF test which epens on the magnitue of outliers. The limiting istribution of this test will be shifte to the left an, in consequence, the null hypothesis of unit root will be rejecte too often. These theoretical finings were closely mimicke by the Monte Carlo experiments. Samples from non stationary processes were generate uner ifferent scenarios of outlier contaminations. In the simulation stuy, the Granger an Hallman 1991) test was also consiere an, equivalently to DF unit root tests, it showe to be sensitive to

10 10 V. A. REISEN, C. LÉVY-LEDUC, M. BOURGUIGNON, AND H. BOISTARD AO contaminations, but with less severity. The general conclusion is that, in practical situations where there woul be a suspicion of aitive outlier, the DF an Granger an Hallman 1991) unit root tests shoul be avoie, however, the BG approach can still be use. Acknowlegements V. A. Reisen an M. Bourguignon gratefully acknowlege partial financial support from CAPES-CNPq/Brazil an FAPES-ES-Brazil. The authors woul like to thank the referee for his valuable suggestions. 5. Proofs 5.1. Proof of Theorem. Letusstartbyproving11). Letusfirstprovethatthenumerator of 8) is not ranom but is just a fixe function of T. T r T,t T +1 t= = 1 T ) t 1 = r T,j T +1 ) = r T,t T +1 ) r T,j T +1 ) 1 j<t T [ T r T,j + T +1 rt,j T+1 ) ] T T r T,j T rt,j T+1 T +1 ) ) TT +1)T 1) =, 13) 4 where we use that T r T,j T +1)/) = 0 an that T r T,j = TT +1)T +1)/6. By 13), stuying the asymptotic behavior of 11) amounts to stuying the following quantity 1 T 4 T t 1 r T,j T +1 ) t= = 1 T 4 [ T t 1 T t= = 1 T t= k=1 1 { zk z j } ) T +1 ] T 1 t 1 F T ζ j ) T +1 ) T T, 14) where F T x) = T 1 T t=1 1 {ζ t x}, where ζ t = z t = ε t +θδ t δ t 1 ). 15) Let us now prove that 1 F T ζ i ) T +1 ), u [0,1] = {Bu), u [0,1]}, as T, 16) T T

11 ROBUST DICKEY-FULLER TESTS BASED ON RANKS 11 where [x] enotes the integer part of x, {Bu), u [0,1]} enotes the Brownian brige an = enotes here the weak convergence in the space of calag functions of [0, 1] enote D[0,1]) an equippe with the topology of uniform convergence. Splitting T into an T j=+1, we get 1 F T ζ i ) T +1 ) = 1 T T = 1 T 3/ = 1 T 3/ T 3/ T T 1 {ζj ζ i } 1 ) + 1 T 3/ 1 {ζj ζ i } T +1 ) T 1 {ζj ζ i } 1 T ) j=[t u]+1 T 3/ 1 {ζj ζ i } 1 ). 17) T3/ Let fx,y) = 1 {x y} 1/. Since fy,x) = fx,y), when x y, we get that the first term in the r.h.s of 17) is equal to 1 T 3/ 1 {ζj ζ i } 1 ) = 1 T 3/ 1 {ζi ζ i } 1 ) =. 18) T3/ Thus, 1 F T ζ i ) T +1 ) = 1 T T T 3/ T j=[t u]+1 Let hx,y) = 1 {x y}. By using the Hoeffing s ecomposition, we get that 1 T 3/ T j=[t u]+1 1 T 3/ hζ j,ζ i ) 1 ) = 1 T 3/ T j=[t u]+1 Fζ j ) 1 ) + 1 T 3/ T j=[t u]+1 T j=[t u]+1 1 {ζj ζ i } 1 ). 19) Fζ i ) 1 ) hζ j,ζ i ) Fζ i )+Fζ j ) 1 ) F enoting the c..f of ζ t ). Observe that since ε t has a continuous c..f it is also the case of the c..f of ζ t. Let us now stuy the behavior of the first two terms in the r.h.s of 0): T T 3/ Fζ i ) 1 ) T 3/ T j=[t u]+1 Fζ j ) 1 ) = X T,u +Y T,u. Observe first that ζ i ) is a strictly stationary process since it is efine as the sum of two strictly stationary processes. It is also a 1-epenent process in the sense of Example 1 p. 167 ofbillingsley 1968)anFζ i ))hasthesameproperties. ByusingTheorem0.1ofBillingsley, 0)

12 1 V. A. REISEN, C. LÉVY-LEDUC, M. BOURGUIGNON, AND H. BOISTARD 1968), we get that, as T tens to infinity, {X T,u } = {1 u)wu)}, an that {Y T,u } = {uw1) Wu))}, where {Wu)} enotes the Wiener process. Since the process X T,u,Y T,u ) is tight an since for fixe u an s, CovX T,u,Y T,s ) Cov1 u)wu),sw1) Ws))), as T, {X T,u,Y T,u )} = {1 u)wu),uw1) Wu))}. We euce from this an Lemma 3 that 1 T 3/ T j=[t u]+1 hζ j,ζ i ) 1 ) 1 u)wu) uw1) Wu)) = Wu) uw1), 1) which with 19) gives 16). Let w i = F T ζ i ) T +1)/T). Using similar arguments as those use in Hamilton, 1994, p. 483) 1 T T 1 t 1 F T ζ j ) T +1 ) t= T T [ = 1 ) ) w1 w1 +w ) ] w1 + +w + T T T T T = T w i t by the continuous mapping theorem an 16), which conclues the proof of 11). Lemma 3. Uner the assumptions of Theorem, sup 1 T ) u [0,1] T 3/ 1 {ζj ζ i } Fζ i )+Fζ j ) 1 j=[t u]+1 Proof of Lemma 3. Observe that Z T u) = 1 T 3/ T = sup u [0,1] 1 {ζj ζ i } Fζ i )+Fζ j ) 1 ) T 3/. Thus it is enough to prove ) when Z T u) is replace by R T u) = 1 T 3/ T 1 {ζj ζ i } Fζ i )+Fζ j ) 1 ) 1 0 Bu) u, Z T u) = o p 1), as T. We want to apply Lemma 5. of Borovkova et al. 001) to {R T u), t [0,1]}. Let us first prove that for s an u such that 0 s u < s+δ 1, R T u) R T s) R T s+δ) R T s) + W T s+δ) W T s) + [Ts+δ)] [Ts] T,. )

13 where Observe that Thus, W T u) = T T 3/ ROBUST DICKEY-FULLER TESTS BASED ON RANKS 13 R T u) R T s) = 1 T 3/ Fζ i ) 1 ) T 3/ T i=[t s]+1 R T u) R T s)) R T s+δ) R T s)) = 1 T 3/ Using 3), + T T 3/ = 1 T 3/ [Ts+δ)] i=[t u]+1 W T s+δ) W T s) = 1 [Ts+δ)] T an thus, we get that i=[t s]+1 [Ts+δ)] i=[t u]+1 T j=[t u]+1 Fζ j ) 1 ) 1 {ζj ζ i } Fζ i )+Fζ j ) 1 ) T Fζ i ) 1 ) [Ts+δ)] T i=[t u]+1 1 {ζj ζ i } 1 ). 3). 4) 1 {ζj ζ i } Fζ i )+Fζ j ) 1 ) + [Ts+δ)] T 3/ Fζ i ) 1 ) + [Ts] [Ts+δ)] T 3/ R T u) R T s)) R T s+δ) R T s)) W T s+δ) W T s)) = 1 T 3/ [Ts+δ)] T i=[t u]+1 1 {ζj ζ i } 1 ) 1 T + [Ts] T 3/ T i=[t s]+1 Fζ j ) 1 ) T T Fζ i ) 1 ) Fζ j ) 1 ) Fζ j ) 1 ) 3 [Ts+δ)] [Ts]. T Since we can follow the same line of reasoning for R T s) R T u), we get Conition 5.10) of Lemma 5. in Borovkova et al. 001) with α = 1/. Let us now check Conition 5.9) of Lemma 5. in Borovkova et al. 001). Observe that for 0 s u 1, W T u) W T s) = T [Ts]) T 3/ i=[t s]+1 + [Ts] T 3/ [Ts] Fζ i ) 1 ) Fζ i ) 1 ) + T i=[t u]+1 Fζ i ) 1 ).,.

14 14 V. A. REISEN, C. LÉVY-LEDUC, M. BOURGUIGNON, AND H. BOISTARD Thus, for some positive constant C 1 E[W T u) W T s)) 4 T [Ts])) 4 ] C 1 E T 6 +C 1 [Ts]) 4 T 6 [Ts] E +C 1 [Ts]) 4 T 6 i=[t s]+1 Fζ i ) 1 E Fζ i ) 1 ) 4 ) 4 T i=[t u]+1 Fζ i ) 1 ) 4. Using that ζ i ) is a 1-epenent process an that E[Fζ i ) 1/)] = 0, Fζ i ) 1/ 1/ for all i 1, there exist positive constants C an C 3 such that E[W T u) W T s)) 4 ] C T [Ts])) 4 T 6 { [Ts])+ [Ts]) } +C [Ts]) 4 T 6 { [Ts]+[Ts] +T )+T ) } C 3 { [Ts]) T + [Ts]) } T which is Conition 5.10) of Lemma 5. in Borovkova et al. 001) with h = 1, gt) = t an r = 4. Let us now check Conition 5.8) of Lemma 5. in Borovkova et al. 001). Using 4), we get that for 0 s u 1, E[R T u) R T s)) ] = E 3 T 3E [Ts] A i,j i=[t s]+1 1 [Ts] T 3/ i=[t s] T 3E A i,j + 1 T 3/ i=[t s]+1 j=[t u]+1 T i=[t s]+1 j=[t u]+1 T A i,j A i,j + [Ts] T 3/ + 3 [Ts]) T 3, where A i,j = 1 {ζj ζ i } Fζ i ) +Fζ j ) 1/. Using that ζ i ) is a 1-epenent process, that EA i,j ) = 0 for all i j an that EA i,j A k,l ) = 0 as soon as the istances between the ifferent inices i,j,k,l is larger than 1, there exist positive constants C 4, C 5 an C 6 such that E[R T u) R T s)) [Ts])[Ts] [Ts])T ) [Ts]) ] C 4 T 3 +C 4 T 3 +C 5 T 3 u s u s 1 ν C 6 C 6, T T for some positive ν which gives Conition 5.8) of Lemma 5. in Borovkova et al. 001) with β = 1 an γ = 1 ν an thus conclues the proof of Lemma 3.,

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16 16 V. A. REISEN, C. LÉVY-LEDUC, M. BOURGUIGNON, AND H. BOISTARD Departamento e Estatística, Universiae Feeral o Espírito Santo, Vitória, Brazil AgroParisTech/INRA MIA 518, Paris, France Departamento e Estatística, Universiae Feeral o Rio Grane o Norte, Natal, Brazil Toulouse School of Economics, GREMAQ, Université Toulouse 1, Toulouse, France