Rank tests for short memory stationarity

Transcription

1 Rank tests for short memory stationarity Pranab K. Sen jointly with Matteo M. Pelagatti University of North Carolina at Chapel Hill Università degli Studi di Milano-Bicocca 50th Anniversary of the Department of Statistics University of Connecticut, Storrs MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

2 Outline 1 Motivation 2 The KPSS test in two slides 3 The Rank KPSS test 4 Asymptotic relative efficiency 5 Rank KPSS for trend-stationarity 6 Monte Carlo 7 Empirical Example 8 Conclusions MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

3 Motivation Motivation One of us was working with time series of electricity prices and found that: in many paper prices were found (or held as) stationary and this is quite strange as they depend on gas and oil prices which are usually well approximated by integrated processes (in logs); due to technical reasons electricity prices are extremely volatile and so the nonstationary signal is buried into a volatile and leptokurtic noise; most unit-root and the KPSS stationarity tests are optimal under Gaussianity and fail to find nonstationarity when data are leptokurtic and second moments may not exist. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

4 Motivation The Index KPSS test An article inspired our idea for robust stationarity tests de Jong et al. (2007, J.Econometrics) prove that the KPSS test applied to the sign of the median-centered observations (IKPSS) has the same asymptotic distribution under the null as the standard KPSS. IKPSS PRO: existence of moments not required, good power under extremely fat-tailed distribution. IKPSS CON: under Gaussianity or moderate excess kurtosis significant loss of power when compared to KPSS. de Jong et al. (2007) do not provide a test for trend-stationarity (stationarity on a linear trend), whereas time series analysts are usually interested in this hypothesis. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

5 The KPSS test in two slides The KPSS test in two slides Suppose that for t = 1,..., T X t = µ t + ε t µ t = µ t 1 + ζ t with ε t and ζ t i.i.d. zero-mean processes with variances σε 2 > 0 and σζ 2 0. Under Gaussianity, the locally best invariant (LBI) test for the hypothesis σζ 2 = 0 is (Nabeya, Tanaka 1988 Annals of Statistics) where LBI T := 1ˆσ 2 ε e t := X t X T, S t := T t=1 S 2 t t e s, ˆσ ε 2 := 1 T S T 2. Under the null LBI T /T 2 V (r) 2 dr, where V is a standard Brownian bridge on [0, 1]. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35 s=1

6 The KPSS test in two slides Kwiatowski, Phillips, Schmidt & Shin (KPSS) show that if we relax the assumption of normality of ε t and ζ t to the existence of second moments and the i.i.d.-ness of ε t to (strong) mixing stationarity and the existence of the long-run variance [ T ] 2 σ 2 1 := lim t T E ε t t=1 then where ˆσ 2 T η T := 1 T ˆσ 2 T T St 2 t=1 1 0 V (r) 2 dr, is the consistent estimator of the long-run variance ˆσ 2 T := 1 T T s=1 t=1 T ( ) s t k e s e t, with k kernel function with bandwidth γ T such that γ T as T diverges and γ T = o( T ). MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35 γ T

7 The Rank KPSS test The Rank KPSS test Let the observed time series be a sample path of the real random sequence {X 1,..., X T } and let R T,t = T I {Xi X t}, for t = 1,..., T, (1) i=1 with I A indicator function of the set A, be the rank of X t among {X 1,..., X T }. Notice that the arithmetic mean of the rank sequence {R T,1,..., R T,T } is (T + 1)/2 and does not depend on the data. The test statistic we propose in this paper is the KPSS applied to the ranks of the observations. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

8 Partial sums of ranks The Rank KPSS test So, let S T,t be the sequence of demeaned partial sums: S T,t = t i=1 ( RT,i T T + 1 ). (2) 2T Notice that the KPSS statistic is invariant to scale transformations, so working with R T,i /T rather than R T,i turns out to generate the same statistic. We chose to work on the former form since under stationarity this makes our partial sum process diverge at the same rate as the analogous quantity defined in Kwiatowski et al. (1992). MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

9 The RKPSS statistic The Rank KPSS test In complete analogy with Kwiatowski et al. (1992), define η R T = T 2 T i=t and the rank KPSS (RKPSS) test statistic as S 2 T,t (3) ˆη T R = ηr T ˆσ T 2, (4) where ˆσ 2 T is a kernel estimator of the long-run variance of {R T,t/T }: ˆσ 2 T = 1 T T s=1 t=1 T ( s t k γ T ) [ RT,s T T + 1 ] [ RT,t 2T T T + 1 ], (5) 2T with k( ) symmetric kernel function and γ T bandwidth parameter. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

10 The Rank KPSS test Null Hypothesis & Kernel function Assumption 1. (Short memory stationarity) 1 {X 1,..., X T } is a strictly stationary random sequence. 2 {X 1,..., X T } is strong mixing with α(t ) = O(T v ), v > 2. 3 For all i {1,..., T } and T N, X i has non-degenerate absolutely continuous distribution function F ( ) defined on R with density f ( ). Assumption 2. (Regularity of the kernel function) 1 k( ) satisfies 1 ψ(z) dz <, ψ(z) = 2π k(x) exp( izx) dx. 2 k( ) is continuous at all but a finite number of points, k(x) = k( x), k(x) < l(x) where l(x) is non-increasing and 0 l(x) dx <, and k(0) = 1. 3 γ T / T 0 and γ T as T. Spearman s rank autocorrelation coefficient { [F ρ i,j = 12E (XT,i ) 1/2 ][ F (X T,j ) 1/2 ]} MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

11 The Rank KPSS test Asymptotics under the Null Theorem (Distribution under short-memory stationarity) Under Assumption 1, 1 ηµ,t R σ2 V (r) 2 dr, with V standard Brownian bridge and σ 2 = 1 12 [1 + 2 k=2 ρ 1,k] ; furthermore { t } T 1/2 S T,t = T 1/2 F (X i ) t T F (X i ) + O p (T 1/2 ). T i=1 i=1 Under Assumptions 1 and 2, 1 ˆη µ,t R V (r) 2 dr. 0 0 MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

12 The Rank KPSS test Asymptotics under the alternative of integration Theorem (Distrib. under integration possibly after monotone transform) Suppose there exists a strictly monotone (Borel) function g : R R such that T 1/2 g(x rt,t ) ωw (r), where ω is a strictly positive real number and W is standard Brownian motion on [0, 1], then η R µ,t T 1 [ s 2 R 0 (r) dr] ds, 0 0 with R 0 (r) = 1 0 I {W (u)<w (r)} du 1 2, and ˆσ 2 T 1 T T s=1 t=1 T ( ) s t k = O(γ T ). γ T MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

13 Remarks The Rank KPSS test Corollary The RKPSS statistic ˆη R µ,t is consistent against I(1)-ness. The alternative hypothesis we used is much weaker than the corresponding hypothesis for the KPSS statistic. While for the KPSS test, the process X T,t must be I(1), in the RKPSS case the I(1) process can be any strictly monotonic transformation of X T,t. Theorem 2 suggests that the statistic ηt R /T can be used to test the hypothesis g(x T,t ) I(1) against stationarity. Indeed, ηt R /T is asymptotically free of nuisance parameters and converges weakly to a proper distribution under the null and to the Dirac (point mass) measure concentrated at zero under the alternative. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

14 Asymptotic relative efficiency Asymptotic relative efficiency Consider the local alternative Y t = σ z T t s=1 Z t }{{} integrated + X }{{} t, t = 1, 2,..., T, stationary where σ z and σ x are positive real numbers, and Z t and X t are mutually independent stationary processes such that, for r [0, 1], rt T 1/2 i=1 rt Z t W z (r) and T 1/2 i=1 with W z and W x independent standard Brownian motions. X t σ x W x (r). MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

15 Asymptotic relative efficiency Asymptotic relative efficiency (cont.) Define the partial sum processes of the KPSS and RKPSS statistic as S K T,t := S R T,t := t (Y s Ȳ T ), s=1 t s=1 ( ) R y T,t T T + 1 2T. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

16 Asymptotic relative efficiency Asymptotics under local alternatives Theorem (A.R.E. of the RKPSS with respect to the KPSS) Assume that Y t is generated by the above local alternative, where X t satisfies Assumption 1, then, for r [0, 1], 1 ST K, rt V (r) + σ z K(r) T σx σ x 1 ST R, rt V (r) + f 2(0) σ z T σ σ K(r) where f 2 (0) := E f (X ) and V (r) is a standard Brownian bridge independent of K(r) := r 0 W z(u) du r 1 0 W z(u) du. The asymptotic relative efficiency of the RKPSS test with respect to the KPSS is e R,K = f 2 (0) σ x σ. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

17 Asymptotic relative efficiency ARE under serial independence of X t and Z t Distribution f 2 (0) e R,K 1 Normal 2 π Uniform π Logistic Student π 1 Laplace Student3 4π For Student s t, the ARE diverges as the df approach 2. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

18 Asymptotic relative efficiency ARE for the Generalized Error or Exponential Power Distr r is the shape parameter (r = 2 Gauss, r = 1 Laplace, r = Uniform) The function reaches its minimum at point (4.193, 0.934). MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

19 Asymptotic relative efficiency The bounds of the ARE? The ARE is unbounded from above. The ARE is bounded from below by The least favorable density was found by Hodges and Lehman (1956): { 3 f (x) = 20 (5 x 2 ), for x , for x 2 > MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

20 Asymptotic relative efficiency ARE under dependence Under dependence, the ARE values must be multiplied by ( ) / κ := ρ 1,i (1 + 2 ϱ 1,i ), i=1 i=1 with ρ i,j Spearman s and ϱ i,j Pearson s correlation. If X t is Gaussian AR(1) then ς MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

21 Rank KPSS for trend stationarity KPSS for (linear) trend stationarity Suppose the data generating process is Y t = α + βt + X t, with X t short memory stationary sequence. The KPSS applied to least-squares detrended data converges to V2 (r) 2 dr under the null, with V 2 second-level (or detrended) Brownian bridge. Is there a robust rank-based way to detrend data so that our RKPSS converges to V 2 (r) 2 dr as well? R-estimates or the asymptotically equivalent Theil-Sen regression estimator: { } Yj Y i β T := median ; 1 i < j T. j i MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

22 Rank KPSS for trend stationarity Theil-Sen estimator (TSE) asymptotics under dependence Theorem (Asymptotic distribution of the TSE) Let the linear model hold with X t strictly stationary and ergodic having a continuous distribution with density f. Then, β T is consistent for β. Furthermore, if the regression errors {X t } are strong mixing with mixing coefficients n=1 α(n) <, and f 2(0) := f (x) 2 dx <, then Q T ( β T β) N ( 0, σ 2 f 2 (0) 2 ), where Q T := T (T 2 1)/12. Finally, under the same conditions and t = (T + 1)/2, Q T ( β T β) = 12 T 3/2 f 2 (0) T [F (X t ) 1/2] (t t) + o p (1). t=1 MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

23 Rank KPSS for trend stationarity Rank KPSS test after Theil-Sen detrending Theorem (Distribution under trend-stationarity) Let η R τ,t and ˆηR τ,t the RKPSS statistics applied to Y t β T t. Under the linear model, Assumption 1 for the regression errors and f 2 (0) := f (x) 2 dx <, η R τ,t σ V 2 (r) 2 dr, where V 2 (r) is a second-level Brownian bridge. Under the above assumptions and Assumptions 2, ˆη τ,t R V 2 (r) 2 dr. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

24 Rank KPSS for trend stationarity Rank KPSS test after Theil-Sen detrending Theorem (Distribution of TSE and RKPSS under integration) Let Y t = βt + X t with T 1/2 X rt ωw (r), r [0, 1], where ω is a positive real number and W a standard Brownian motion on [0, 1], then i) β T p β; ii) T 1/2 ( β T β) H, where H is a random variable with an absolutely continuous distribution symmetric about zero; iii) iv) η R τ,t T 1 0 [ s 2 R 0 (r) dr] ds, with R 0 (r) = 1 0 I {W (u) Hu W (r) Hr} du 1 2 ˆσ 2 T 1 T T 0 and r [0, 1]; T ( ) s t k = O(γ T ). γ s=1 t=1 T MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

25 Monte Carlo Results for finite samples Monte-Carlo experiments design We reproduce the design of de Jong et al. (2007). 20,000 replications. Sample sizes ranging from T = 50 to T = Distributions: Normal, Student s t 5, t 3, t 2, t 1 (Cauchy) and local to finite variance x t = x 1,t + T 1/2 x 2,t, where x 1 is normal and x 2 Cauchy. Under I(0): white noise and AR(1) processes with φ = 0.5. Under I(1): µ 0 = 0, µ t = µ t 1 + λ η t, x t = µ t + ε t, where λ is the signal-to-noise ratio and ranges between and 1. Kernel: Bartlett with γ T = 4(T /100) 1/4 (white noise case), γ T = ( T ) 1/3 (AR(1) case, Andrews 1991). MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

26 Monte Carlo Results for finite samples Size: (a) no Kernel, (b) with Kernel. (a) Normal t5 t3 t2 Local Cauchy T KPSS RKPSS IKPSS KPSS RKPSS IKPSS KPSS RKPSS IKPSS KPSS RKPSS IKPSS KPSS RKPSS IKPSS KPSS RKPSS IKPSS (b) Normal, ρ =.5 t3, ρ =.5 Cauchy, ρ =.5 Normal, ρ =.9 t3, ρ =.9 Cauchy, ρ =.9 T KPSS RKPSS IKPSS KPSS RKPSS IKPSS KPSS RKPSS IKPSS KPSS RKPSS IKPSS KPSS RKPSS IKPSS KPSS RKPSS IKPSS MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

27 Monte Carlo Results for finite samples Size-adjusted power IID POWER Normal t5 t3 t2 Local Cauchy λ RKPSS KPSS IKPSS MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

28 Monte Carlo Results for finite samples Size-adjusted power AR(1) POWER Normal (phi=0.5) t3 (phi=0.5) Cauchy (phi=0.5) Normal (phi=0.9) t3 (phi=0.9) Cauchy (phi=0.9) λ RKPSS KPSS IKPSS MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

29 Monte Carlo Results for finite samples Size for detrended tests T Normal t 5 t 2 Cauchy KPSS RKPSS IKPSS KPSS RKPSS IKPSS KPSS RKPSS IKPSS KPSS RKPSS IKPSS (a) i.i.d. data T Normal t 5 t 2 Cauchy KPSS RKPSS IKPSS KPSS RKPSS IKPSS KPSS RKPSS IKPSS KPSS RKPSS IKPSS (b) AR(1) with φ = MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

30 Monte Carlo Results for finite samples SA power detrended POWER RKPSS KPSS IKPSS MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35 λ Normal t5 t2 Cauchy Normal (phi=0.5) t5 (phi=0.5) t2 (phi=0.5) Cauchy (phi=0.5)

31 Empirical Example Empirical Example The three tests are applied to eight time series of European wholesale electricity prices, named after their respective market makers: EEX-DE (Germany), GME-IT (Italy), APX-NL (Netherlands), APX-UK (United Kingdom), NordPool-NO (Norway), Omel-PT (Portugal), Omel-ES (Spain), Powernext-FR (France). Each observation represents the daily (working day) price at noon. The starting date is different for each series, ranging from the 4th May 1992 of NordPool to the 2nd July 2007 of Omel-PT, while the last observation dates 25th May 2012 for all markets. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

32 Empirical Example Test statistics as functions of the bandwidth parameter γ T EEX DE 4 2 GME IT APX NL APX UK γ NordPool NO γ Omel PT 10 5 γ Omel ES γ Powernext FR γ γ γ γ RKPSS Bandwidth KPSS Bandwidth IKPSS Bandwidth The horizontal line indicates the 5% critical value and the vertical line denotes the optimal bandwidth for an AR(1) with φ = 0.9. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

33 Empirical Example Detrended test statistics as functions of the bandwidth EEX DE GME IT APX NL 0.5 APX UK 1.5 γ NordPool NO 1.5 γ Omel PT 1.5 γ Omel ES 1.5 γ Powernext FR γ γ γ γ RKPSS Bandwidth KPSS Bandwidth IKPSS Bandwidth The horizontal line indicates the 5% critical value and the vertical line denotes the optimal bandwidth for an AR(1) with φ = 0.9. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

34 Empirical Example Remarks on the empirical application KPSS fails to find nonstationarity for APX-NL (both on level and trend), the RKPSS statistic is never the closest to zero, the RKPSS and IKPSS statistics behave similarly for those cases in which the series are extremely leptokurtic (Powernext-FR, EEX-DE, APX-NL, NordPool-NO, APX-UK). MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35

35 Conclusions New short-memory stationarity test: invariant to monotonic transformations; no moments required; robust to fat-tailed distributions; very good size (typical of rank tests due to distribution freeness); better power than IKPSS and KPSS in many empirically relevant situations (leptokurtosis and positive dependence); maximum loss of ARE w/r to KPSS: 7%. maximum gain of ARE w/r to KPSS: unbounded. Extensions: asymptotics under long memory; cointegration rank test of Nyblom & Harvey (2000) type; general score functions of the type used for linear rank statistics: for example Van der Waerden scores ( ) a T,t = Φ 1 RT,t, T + 1 with Φ 1 standard normal quantile function and optimality analysis. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November / 35