Rank tests for short memory stationarity Pranab K. Sen jointly with Matteo M. Pelagatti University of North Carolina at Chapel Hill Universit` a 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 2012 1 / 35
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 2012 2 / 35
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 2012 3 / 35
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 2012 4 / 35
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 σ 2 ε > 0 and σ 2 ζ ≥ 0. Under Gaussianity, the locally best invariant (LBI) test for the hypothesis σ 2 ζ = 0 is (Nabeya, Tanaka 1988 Annals of Statistics) T LBI T := 1 � S 2 t σ 2 ˆ ε t =1 where t ε := 1 � e t := X t − ¯ σ 2 T S 2 X T , S t := e s , ˆ T . s =1 � Under the null LBI T / T 2 ⇒ V ( r ) 2 d r , where V is a standard Brownian bridge on [0 , 1]. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November 2012 5 / 35
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 1 � σ 2 := lim ε t T E t →∞ t =1 then � 1 T 1 � V ( r ) 2 d r , S 2 η T := t ⇒ σ 2 T ˆ 0 T t =1 σ 2 where ˆ T is the consistent estimator of the long-run variance T T � s − t � T := 1 � � σ 2 ˆ k e s e t , T γ T s =1 t =1 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 2012 6 / 35
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 T � R T , t = I { X i ≤ 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 2012 7 / 35
The Rank KPSS test Partial sums of ranks So, let S T , t be the sequence of demeaned partial sums: t � R T , i � − T + 1 � S T , t = . (2) T 2 T i =1 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 2012 8 / 35
The Rank KPSS test The RKPSS statistic In complete analogy with Kwiatowski et al. (1992), define T � η R T = T − 2 S 2 (3) T , t i = t and the rank KPSS (RKPSS) test statistic as T = η R η R T ˆ , (4) σ 2 ˆ T σ 2 where ˆ T is a kernel estimator of the long-run variance of { R T , t / T } : T T � s − t � � R T , s � � R T , t � T = 1 − T + 1 − T + 1 � � σ 2 ˆ k , (5) T γ T T 2 T T 2 T s =1 t =1 with k ( · ) symmetric kernel function and γ T bandwidth parameter. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November 2012 9 / 35
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 ) | d z < ∞ , ψ ( z ) = −∞ k ( x ) exp( − i zx ) d x . 2 π 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 ) | d x < ∞ , and k (0) = 1. √ 3 γ T / T → 0 and γ T → ∞ as T → ∞ . Spearman’s rank autocorrelation coefficient �� �� �� ρ i , j = 12 E F ( X T , i ) − 1 / 2 F ( X T , j ) − 1 / 2 MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November 2012 10 / 35
The Rank KPSS test Asymptotics under the Null Theorem (Distribution under short-memory stationarity) Under Assumption 1, � 1 V ( r ) 2 d r , η R µ, T ⇒ σ 2 0 with V standard Brownian bridge and σ 2 = 1 12 [1 + 2 � ∞ k =2 ρ 1 , k ] ; furthermore � t � T F ( X i ) − t � � T − 1 / 2 S T , t = T − 1 / 2 + O p ( T − 1 / 2 ) . F ( X i ) T i =1 i =1 Under Assumptions 1 and 2, � 1 V ( r ) 2 d r . η R ˆ µ, T ⇒ 0 MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November 2012 11 / 35
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 ) ⇒ ω W ( r ) , where ω is a strictly positive real number and W is standard Brownian motion on [0 , 1] , then � 1 �� s η R � 2 µ, T ⇒ R 0 ( r ) d r d s , T 0 0 � 1 0 I { W ( u ) < W ( r ) } d u − 1 with R 0 ( r ) = 2 , and T T � s − t � T ≤ 1 � � σ 2 ˆ k = O ( γ T ) . T γ T s =1 t =1 MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November 2012 12 / 35
The Rank KPSS test Remarks Corollary η R The RKPSS statistic ˆ 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 η R T / T can be used to test the hypothesis g ( X T , t ) ∼ I (1) against stationarity. Indeed, η R T / 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 2012 13 / 35
Asymptotic relative efficiency Asymptotic relative efficiency Consider the local alternative t Y t = σ z � Z t + X t , t = 1 , 2 , . . . , T , T ���� s =1 stationary � �� � integrated 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 ⌋ ⌊ rT ⌋ � � T − 1 / 2 T − 1 / 2 Z t ⇒ W z ( r ) and X t ⇒ σ x W x ( r ) . i =1 i =1 with W z and W x independent standard Brownian motions. MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November 2012 14 / 35
Asymptotic relative efficiency Asymptotic relative efficiency (cont.) Define the partial sum processes of the KPSS and RKPSS statistic as t � ( Y s − ¯ S K T , t := Y T ) , s =1 � � R y t − T + 1 � T , t S R T , t := . T 2 T s =1 MM Pelagatti & PK Sen (Bicocca & UNC) Rank tests for stationarity 3 November 2012 15 / 35
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