Working Paper 98-20 Departamento de Estadística y Econometría Statistics and Econometrics Series 13 Universidad Carlos III de Madrid January 1998 Calle Madrid, 126 28903 Getafe (Spain) Fax (341) 624-9849 GAUSSIAN SEMIPARAMETRIC ESTIMATION OF NON-STATIONARY TIME SERIES Carlos Velasco* Abstract ---------------------------- Generalizing the definition of the memory parameter d in terms of the differentiated series, we showed in Velasco (1997a) that it is possible to estimate consistently the memory of non-stationary processes using methods designed for stationary long range dependent time series. In this paper we consider the Gaussian semi-parametyric estimate anlyzed in Robinson (1995b) for stationary processes. Without a priori knowledge about the possible non-stationarity of the observed process, we obtain that this estimate is consistent for d E(-l/z, 1) and asymptotically normal for d E e/z ,3/4) under a similar set of assumptions to Robinson's paper. Tapering the observations, we can estimate any degree of non-stationarity, even in the presence of deterministic polynomial trends of time. The semi-parametric efficiency of this estimate for stationary sequences also extends to the non-stationary framework. Keywords: Non-stationary time series, semiparametric inference, tapering. *Departamento de Estadística y Econometría, Universidad Carlos III de Madrid. CI Madrid, 126 28903 Madrid. Spain. Ph: 34-1-624.98.87, Fax: 34-1-624.98.49, e­ mail: [email protected]. Research funded by the Spanish Dirección General de Enseñanza Superior, Ref. n. PB95-0292. ~-~~------------------------------------------- Gaussian Semiparametric Estimation of Non-Stationary Time Series Carlos Velasco Department 01 Statistics and Econometrics \ \ \ Universidad Carlos JI! de Madrid \ 28903 Getale (Madrid) Spain January 22, 1998 1 Gaussian Semiparametric Estimation of Non-Stationary Time Series Carlos Velasco Abstract Generalizing the definition of the memory parameter d in terms of the differentiated series, we slio\\'ed in Velasco (1997a) tliat it is possible to estimate consistently the memory of non-stationary processes using methods designed for stationary long range dependent time series. In this paper \\'e consider the Gaussian semi-parametric estimate analyzed in Robinson (1995b) for stationary processes. \Vithout a priori knowledge about the possible non-stationarity of the obser\'(~d process, \\"e obtain that this estimate is consistent for d E (-~, 1) and asymptotically normal for d E (-~, ~) under a similar set of assumptions to Robinson's paper. Tapering the observations, we can estimate an)" degree of non-stationarity, even in the presence of deterministic polynomial trends of time. Tite scmi-parametric efficiency of this estimatc for stationary sequences also extends to the non­ stationary framework. Key words: Non-stationary time series; semiparamctric inference; tapering. 2 1 Introduction Statistical inference for stationary long range dependent time series is often based on semiparametric estimates that avoid parameterization of the short run behaviour. Frequently, it is assumed that the spectral density f(>') of the observed stationary sequence satisfies for one O< G < 00, as >. -+ 0+, (1) where d E (-~, ~) is the parameter that governs the degree of memory of the series. This is the interval of values of d for which the process is stationary and invertible. If d E (O, ~) then we say that the series exhibits long memory 01' long range dependence. When d < Othe spectral density satisfies f(O) = Oand :s - if d ~ the process is not invertible.Many non-stationary time series are transformed into stationary ones after taking enough number of differences. In this case it is straightforward to generalize the definition of the memory parameter d in terms of the properties of the spectral density of the stationary increments of the observed process and the unit root filter(s). Robinson (1995a) recommended an initial, possibly repeated, differentiation (integration) of the observed time series when non-stationarity (non­ invertibility) is suspected, to obtain a value of d in the stationary and invertible interval (-~, ~) and then perform stationary procedures on the transformed series, adjusting the estimate with the number of differences (integrations) taken. However in many empirical applications values of d outside the stationary range are found when the estimates are not constrained to the stationary range, d < ~, as is the case of explicit form estimates, like the log-periodogram regression (e.g. Agiakloglou et al. (1993), Bloomfield (1991)). In Velasco (1997a) \ve considered the application of the log-periodogram regression estimate (see Robinson (1995a) and Ge\veke and Porter-Hudak (1983)) to the raw non-stationary processes, following sorne previous ideas in Hassler (1993) and Hurvich and Ray (1995). The last reference considered the expectation of the periüclogram at low Fourier frequencies for non-stationary and non-invertible fractionally integrated processes. They showed that the normalized periodogram has bounded expectation for d E [~, ~) but it is hiased (for a function f satisfying (1)) in this case. Robillson (1995b) found that in the stationary and invertible case an estimate of el minimizing an approximation to a Gaussian likelihood for frequencies close to the origin had better efficiency properties than rival semiparametric estimates, in the sense of having smaller asymptotic variance after proper normalization when using the same amount of sample information. Using Velasco's (1997a) results for the periodogram of non-stationary time series, we address in this papel' whether it is possible to extend tlle range of allowed values of d in this implicitly defined estimate to cover sorne non-stationary situations and what are the properties of the estimates when the series is non stationary, including sorne possible efficiency gains. Dndcr similar conditions to those assumed by Robinson we find that the Gaussian semiparametric estimate is consistent for d E (-~, 1), asymptotically normal for d < ~, with the same variance as 3 in the stationary situation, being more efficient that the log-periodogram regression estímator. If we taper the observations adequately we can estimate higher degrees of nonstationarity, as was found for the log-periodogram estímate in Velasco (1997a). Finally, we perform a limited numerical study with simulated and real data of these theoretical results. We give all the proofs at the end of the papel' in several appendices together with sorne technical lemmas. We do not discuss the non-invertible case here, d ~ - ~, but this could be done using similar methods to those of Velasco (1997a) for the log-periodogram estímate (see Theorems 9 and 10 in that paper). 2 Assumptions and definitions In the first two sections we consider the original estimate analyzed by Robinson (1995b) and concentrate in the illterval - ~ < d < ~. When the observed time series is stationary with spectral density fx (>") satisfying (1), d < ~, we say that the process has memory d and we define the function f, as f(>") = fx(>"). t Whell {Xt } is a non-stationary process, we say that it has memory parameter d ( ~ d < ~) if the zero mean stationary process Ut = .6.Xt has spectral density fu(>") = 11 - exp(i>..)/-2(d-l) j*(>"), where 1* (>") is a spectral density on [-71",71"] which is bounded aboye and away from zero and is continuous at>.. = n. Thus fu(>") satisfies (1) with sorne - ~ ~ du < ~, but we do not restrict its form for frequencies away tlw origino Then we assume, following Hurvich and R.ay (1995), that for any t ~ 1, t =¿:Uk Xt +Xo k=l where -Yo is a random variable not depending on time t. Next, define the function f(>") for d ~ ~, f(>") = 11 - exp(i>..)1-2fu(>") = 11 - exp(i>")1-2dj*(>") = 12sin(>../2)1-2d j*(>"). Note that f satisfies (1), but when 2d ~ 1 it is not integrable in [-71",71"] and is not a spectral density. We do llot assume that 1* is the spectral density of an stationary and invertible AR.MA process as would be the case if Ut followed a fractional ARIMA model. Here 1* may have (integrable) poles 01' zeroes at frequencies beyond the origino 'Ve want to give a unified theory for semiparametric estimates of d E (-~, 1), including stationary (\vith f x (O) equal to zero, a constant 01' infinity) and non-stationary processes. We introduce now the following assumptions about the behaviour of the spectral densities fx(>") (d < ~) and fu(>") (d ~ ~) (alld thus of the functions f(>") and 1*(>")) at the origin: 4 t, t), Assumption 1 When d E (- the spectral density f x (>..) satisfies, for O< G < 00, as >.. -+ 0+ and when d E [t, !), the spectral density fu(>") satisfies, fu(>") rv G. >..-2(d-l) as >.. -+ 0+. A slightly stronger version of this assumption, and that we will use to obtain the asymptotic nor­ mality of our estimates is t, t), Assumption 2 When d E (- the spectral density f x (>..) satisfies for numbers O < f3 S; 2, O < G < 00, as >.. -+ 0+, [i,!), anrlwhcn d E the spectral density fu(>") satisfies fu(>") = G· >..-2(d-l) + 0(>..-2(d-1l+13 ) as >.. -+ 0+. Dndcr Assumption 2 we write, defining the function g(>..) = G>.. -2d, O< f3 S; 2, as >.. -+ 0+. (2) This is ec¡uivalent to Assumption 1 in Robinson (1995a) when f is the spectral density of X (stationary) t and d E (-i, ~). See also Remarl< 3.1 in Giraitis et al. (1995). Also, Assumption 2 implies that 1* (>..) is bounded aboye ancl away from zero ancl is continuous in an inten'al (O, e), e> O. i), Assumption 3 In a neighbourhood (O, e) of the origin, if d E (-~, fx (>..) is differentiablc and d~ I fx(>") I= 0(>"-1-2d) as>" -+ 0+, anrl íf el;:::~, fu(>") is differentiable and Id~ fu(>") I= 0(>"-1-2(d-l)) as >.. -+ 0+. Then f(>") has first derivative satisfying (d. Assumption 2 of Robinson (1995a) in the stationary case el < i), (3) Thesc assumptions could have been formulated in terms of the functions 1* and/or f, since we are interested in the implications they have on the function f, (2) and (3). However, we did not find appropriate to make assumptions directly on f 01' 1*, since these functions have not immediate and clear statistical interpretation as fu 01' f x have. No\\' \Ve make the following assumptions about the series Ut when d ;::: ~, 01' for Xt when d < ~, paralleling Robinson (1995b), 5 Assumption 4 We have, for -~ < d < ~, Yt = X t or for ~ S d < 1, Yt = Ut, with L00 "00'2 = Yt a{Et_{, ~ae < 00, {=o (=o where = = = E[EtlFt-¡J O, E[EFIFt-1] 1, a.s., t O, ±1, ... , in which Ft is the u-fleld of events generated by Et, S S t, and there exists a randorn variable E, such that EE 2 < 00 and for all r¡ > O and sorne C > O, P(IEtI > r¡) S CP(/E/ > r¡). Then \Ve obtain that, d :2: ~, f(A) =11-exp(iA)I-2 fU(A) =11-exp(iA) 1-2 la(A)1 2 , 27f where L00 a(A) = aee W' e=o amIIO:(AW 127f = fU(A), thc spectral density of Ut. Define the discrete Fourier transform of Xt, t =1,..., n, Aj = 27fj In, j integer, t, amI when d:2: \Ve obtain, 1 n Lt W(Aj) = ~L Uk exp(iAjt), v 27fn t=1 k=1 so W(Aj) is a complex linear combination of the (non observable) stationary variables Uk . Thc Fourier transfonn at any frequcncy Aj, O<j < n, of a non-stationary sequence X t allows the elimination of t,he rauc!om yariable Xo, so W(Aj) is not, depending on the values of Uk for k < 1. Define the periodogram of XI as Because the estimate is not defined in closed form, we denote by Go and do the true parameter values, and by G and d any admissible values. Consider t,he objective function (see Robinson (1995b) ancl Künsch (1987)), ~ ~ Q(G d) = {IOgGA-:2d + I(Aj,) } , m ~ J GA-: 2d ' j=1 J e and define the closed interval of admissible estimates of do, = [\71,\72]' where \71and \72are numbers t sueh that. - < \71 < \72 < 1. Note that we cover part of the range of values of d for which Xt is (t non-st,ationary. As in Robinson (1995b) \71 and \72 can be chosen arbitrarny close to - ~ and 1 in t) his case), respectively, or reflecting sorne prior knowledge on do' When do E (-~, the asymptotics for I(Aj) are exactly the same as in Robinson's discussion, but when do :2: ~, we have to resort to the results of Velaseo (1997a), weaker in general. Robinson used notation in terms of the parameter H = d + ~, 6 but we find more natural to use the number of differences parameter d in a possibly non-stationary contexto We define the estimates (G,d)=arg min Q(G,d), O<G<oo,dE8 which always exist and also d= argminR(d), dE8 where f 1 m ~ R(d) = logG(d) - 2d- ¿IOgAj, G(d) = A;dI(Aj). m j=l 1 Vsing the discussion in Velasco (1997a), the main idea to show that Robinson (1995b) results go through in the non-stationary case (do ~ ~) is to analyse the asymptotic behaviour of the discrete Fourier transform of X t for frequencies Aj, 1 ~ j ~ m, with l/m+m/n -+ Oas n -+ oo. Therefore, assuming the .'lame c:ouditions for Ek'S, we could repeat the steps in Robinson (1995b) to obtain the consistency and asymptotic distribution of the estimate of the parameter d for non-stationary processes. However, due to a bias problem, the .'lame results as in Robinson (1995b) can only be obtained for do < ~, consistency holding for do < 1. 'Ve stress the point that the discrete surn in the previous definitions cannot be substituted by an integral forrn as is considered for related estirnates in a full pararnetric context (.'lee Fox and Taqqu (1986), ¡[nd Giraitis and Surgailis (1990)), .'lince the properties of the periodograrn for non-stationary proccsses are only equivalent to the stationary case when evaluated at frequencies Aj, 1 ~ j ~ n - 1. 3 Consistency In this section we obtain the consistency of das defined previously for values do E (-~, 1). Vnder Assumptions 2 and 3, the conditions on the behaviour of the fuuction f(A) at the origin of Theorern 1 in Robinson (1995b) hold now also for do E [~,~) (we do not need the integrability of 1). Iu thc stationary case, the analysis of the asyrnptotic properties of W(Aj) is done in Robinson (1995a). For tlIe uon-stationary situation, d ~ ~, we can obtain following sorne ideas of Hurvich and Ray (1995) that ¡:¡r = E[I(Aj)] f(A)K(A - Aj)dA, = where K(A) (21l"n)-11I:7exp{iAt}12 is Fejér kernel. Frorn this expression it is possible to .'lee that when Xi. is non stationary, f(A) plays exactly the .'lame role as a spectral density in the asyrnptotics for the ciiscrete Fourier transforrn at frequencies Aj, j 1:- O rnod(n), and Velasco (1997a) showed that the periodograrn is (asyrnptotically) unbiased for f if j is growing slowly with n and d < 1. This is done in the next theorern, which is Theorern 1 in Velasco (1997a). Defining V(A) = W(A)/ fl/2(A), 7 Theorem 1 Under Assumptions 1 and 3, d E [~, 1), for any sequenees of positive integers j = j(n) :s and k = k(n) sueh that 1 k < j and j /n -+ O as n -+ 00, defining Ók,j = (jk)d-1log(j + 1), (b) E[v(>'j)v(>'j)] = O (Ój,j) , = (e) E[v(>'j)V(>'d] O (k-1log(j) +Ók,j), (d) E[v(>'j)V(>'k)] =O (k-1log(j) + Ók,j). The next two results hold in a similar way for the log periodogram estimate of d fol' non-stationary Gallssian time series. Here we do not need to assume Gaussianity in any formo Fil'st we show the consistency of dwhen d < 1: Theorem 2 Under Assumptions 1 (do E (-t, 1)), 3, 4 and 1 m -+--+0 as n -+ 00, m n we obtain d -+1' d. 4 Asymptotic N ormality Fol' yalllCS do :::: 1 the periodogl'am at frequencies >'j is not unbiased fol' the function f as j increases, and thcrefon~ (1 can not be consistent. Unlike for stationary processes, we can only obtain the asymptotic distl'ill11tion for d in the non-stationary case for a smaller range of values of do. (do < i) than the intcrnll \\'hel'e the estimate is consistent, do < 1 . This is due to the fact that the propel'ties of the pel'ioc1ogl'am depend on convolutions of the function f(>'), which deteriorate rapidly as .f becomes more "non-integrable" ,i.e. as do increases (see Theorem 1 aboye and Theorem 1 in Velasco (1997a), and the sullseqllC'llt discussion). \\'e introduce two new assumptions that will be needed in the proofs. Assumption 5 In a neighbourhood (0,10) of the origin, a(>.) is differentiable and as>' -+ 0+. Clcal']y Assumptíon 5 implies Assumption 3, since f(>') = la(>.)¡2/27r when -~ < do < ~ and f(>') = 2 (2 sin >./2) -2Ia(>.) 1 /27r when do :::: ~. Assumption 6 Assumption 4 holds and also = t O, ±1, ... , for finite eonstants J.L3 and J.L4' 8