Abstract

We consider a nonparametric CUSUM test for change in the mean of multivariate time series with time varying covariance. We prove that under the null, the test statistic has a Kolmogorov limiting distribution. The asymptotic consistency of the test against a large class of alternatives which contains abrupt, smooth and continuous changes is established. We also perform a simulation study to analyze the size distortion and the power of the proposed test.

1. Introduction

In the statistical literature there is a vast amount of works on testing for change in the mean of univariate time series. Sen and Srivastava [1, 2], Hawkins [3], Worsley [4], and James et al. [5] considered tests for mean shifts of normal i.i.d. sequences. Extension to dependent univariate time series has been studied by many authors, see Tang and MacNeill [6], Antoch et al. [7], Shao and Zhang [8], and the references therein. Since the paper of Srivastava and Worsley [9] there are a few works on testing for change in the mean of multivariate time series. In their paper they considered the likelihood ratio tests for change in the multivariate i.i.d. normal mean. Tests for change in mean with dependent but stationary error terms have been considered by HorvΓ‘th et al. [10]. In a more general context of regression, Qu and Perron [11] considered a model where changes in the covariance matrix of the errors occur at the same time as changes in the regression coefficients, and hence the covariance matrix of the errors is a step-function of time. To our knowledge there are no results testing for change in the mean of multivariate models when the covariance matrix of the errors is time varying with unknown form. The main objective of this paper is to handle this problem. More precisely we consider the 𝑑-dimensional modelπ‘Œπ‘‘=πœ‡π‘‘+Ξ“π‘‘πœ€π‘‘,𝑑=1,…,𝑛,(1.1) where (πœ€π‘‘) is an i.i.d. sequence of random vectors (not necessary normal) with zero mean and covariance 𝐼𝑑, the identity matrix. The sequence of matrices (Γ𝑑) is deterministic with unknown form. The null and the alternative hypotheses are as follows:𝐻0βˆΆπœ‡π‘‘π»=πœ‡βˆ€π‘‘β‰₯1against1∢Thereexist𝑑≠𝑠suchthatπœ‡π‘‘β‰ πœ‡π‘ .(1.2) In practice, some particular models of (1.1) have been considered in many areas. For instance, in the univariate case (𝑑=1), Starica and Granger [12] show that an appropriate model for the logarithm of the absolute returns of the S&P500 index is given by (1.1) where πœ‡π‘‘ and Σ𝑑 are step functions, that is, πœ‡π‘‘=πœ‡(𝑗)if𝑑=π‘›π‘—βˆ’1+1,…,𝑛𝑗,𝑛𝑗=ξ€Ίπœ†π‘—π‘›ξ€»,0<πœ†1<β‹―<πœ†π‘š1<1,𝑑=𝜎(𝑗)if𝑑=π‘‘π‘—βˆ’1+1,…,𝑑𝑗,𝑑𝑗=ξ€Ίπœπ‘—π‘›ξ€»,0<𝜏1<β‹―<πœπ‘š2<1,(1.3) for some integers π‘š1 and π‘š2. They also show that model (1.1) and (1.3) gives forecasts superior to those based on a stationary GARCH(1,1) model. In the multivariate case (𝑑>1), HorvΓ‘th et al. [10] considered the model (1.1) where πœ‡π‘‘ is subject to change and Σ𝑑=Ξ£ is constant; they applied such model to temperature data to provide evidence for the global warming theory. For financial data, it is well known that assets’ returns have a time varying covariance. Therefore, for example, in portfolio management, our test can be used to indicate if the mean of one or more assets returns are subject to change. If so, then taking into account such a change is very useful in computing the portfolio risk measures such as the value at Risk (VaR) or the expected shortfall (ES) (see Artzner et al. [13] and Holton [14] for more details).

2. The Test Statistic and the Assumptions

In order to construct the test statistic let𝐡𝑛(ξΞ“πœ)=βˆ’11βˆšπ‘›[π‘›πœ]𝑑=1ξ‚€π‘Œπ‘‘βˆ’π‘Œξ‚[],βˆ€πœβˆˆ0,1(2.1) where Γ is a square root of ΓΓΣ,thatis,Ξ£=β€²,ξβˆ‘=1𝑛𝑛𝑑=1ξ‚€π‘Œπ‘‘βˆ’π‘Œπ‘Œξ‚ξ‚€π‘‘βˆ’π‘Œξ‚ξ…ž,1π‘Œ=𝑛𝑛𝑑=1π‘Œπ‘‘(2.2) are the empirical covariance and mean of the sample (π‘Œ1,…,π‘Œπ‘›)β€², respectively, [π‘₯] is the integer part of π‘₯, and π‘‹ξ…ž is the transpose of 𝑋.

The CUSUM test statistic we will consider is given byℬ𝑛=sup𝜏∈[0,1]‖‖𝐡𝑛‖‖(𝜏)∞,(2.3) whereβ€–π‘‹β€–βˆž=max1≀𝑖≀𝑑||𝑋(𝑖)||𝑋if𝑋=(1),…,𝑋(𝑑)ξ€Έβ€².(2.4)

Assumption 1. The sequence of matrices (Γ𝑑) is bounded and satisfies 1𝑛𝑛𝑑=1Ξ“π‘‘Ξ“ξ…žπ‘‘βŸΆΞ£>0asπ‘›βŸΆβˆž.(2.5)

Assumption 2. There exists 𝛿>0 such that 𝐸(β€–πœ€1β€–2+𝛿)<∞, where ‖𝑋‖ denotes the Euclidian norm of 𝑋.

3. Limiting Distribution of ℬ𝑛 under the Null

Theorem 3.1. Suppose that Assumptions 1 and 2 hold. Then, under 𝐻0, β„¬π‘›β„’βŸΆβ„¬βˆž=sup𝜏∈[0,1]‖𝐡(𝜏)β€–βˆž,(3.1)ℒ→  denotes the convergence in distribution and 𝐡(𝜏) is a multivariate Brownian Bridge with independent components.
Moreover, the cumulative distribution function of 𝐡∞ is given by πΉπ΅βˆžξƒ©(𝑧)=1+2βˆžξ“π‘˜=1(βˆ’1)π‘˜ξ€½expβˆ’2π‘˜2𝑧2ξ€Ύξƒͺ𝑑.(3.2)

To prove Theorem 3.1 we will establish first a functional central limit theorem for random sequences with time varying covariance. Such a theorem is of independent interest. Let 𝐷=𝐷[0,1] be the space of random functions that are right-continuous and have left limits, endowed with the Skorohod topology. For a given π‘‘βˆˆβ„•, let 𝐷𝑑=𝐷𝑑[0,1] be the product space. The weak convergence of a sequence of random elements 𝑋𝑛 in 𝐷𝑑 to a random element 𝑋 in 𝐷𝑑 will be denoted by 𝑋𝑛⇒𝑋.

For two random vectors 𝑋 and π‘Œ,𝑋law=π‘Œ means that 𝑋 has the same distribution as π‘Œ.

Consider an i.i.d. sequence (πœ€π‘‘) of random vectors such that 𝐸(πœ€π‘‘)=0 and var(πœ€π‘‘)=𝐼𝑑. Let (Γ𝑑) satisfy (2.5) and setπ‘Šπ‘›(Ξ“πœ)=βˆ’1βˆšπ‘›[π‘›πœ]𝑑=1Ξ“π‘‘πœ€π‘‘[],,𝜏∈0,1(3.3) where Ξ“ is a square root of Ξ£,thatis,Ξ£=ΓΓ′. Many functional central limit theorems were established for covariance stationary random sequences, see Boutahar [15] and the references therein. Note that the sequence (Ξ“π‘‘πœ€π‘‘) we consider here is not covariance stationary.

There are two sufficient conditions to prove that π‘Šπ‘›β‡’π‘Š (see Billingsley [16] and Iglehart [17]), namely,(i)the finite-dimensional distributions of π‘Šπ‘› converge to the finite-dimensional distributions of π‘Š,(ii)π‘Šπ‘›(𝑖) is tight for all 1≀𝑖≀𝑑, if π‘Šπ‘›=(π‘Šπ‘›(1),…,π‘Šπ‘›(𝑑))ξ…ž.

Theorem 3.2. Assume that (πœ€π‘‘) is an i.i.d. sequence of random vectors such that 𝐸(πœ€π‘‘)=0, var(πœ€π‘‘)=𝐼𝑑 and that Assumptions 1 and 2 hold. Then π‘Šπ‘›βŸΉπ‘Š,(3.4) where π‘Š is a standard multivariate Brownian motion.

Proof. Write 𝐹𝑑=Ξ“βˆ’1Γ𝑑, 𝐹𝑑(𝑖,𝑗) the (𝑖,𝑗)-th entry of the matrix 𝐹𝑑,πœ€π‘‘=(πœ€π‘‘(1),…,πœ€π‘‘(𝑑))β€². To prove that the finite-dimensional distributions of π‘Šπ‘› converge to those of π‘Š it is sufficient to show that for all integer π‘Ÿβ‰₯1, for all 0β‰€πœ1<β‹―<πœπ‘Ÿβ‰€1, and for all π›Όπ‘–βˆˆβ„π‘‘, 1β‰€π‘–β‰€π‘Ÿ, 𝑍𝑛=π‘Ÿξ“π‘–=1π›Όξ…žπ‘–π‘Šπ‘›ξ€·πœπ‘–ξ€Έβ„’βŸΆπ‘=π‘Ÿξ“π‘–=1π›Όξ…žπ‘–π‘Šξ€·πœπ‘–ξ€Έ.(3.5) Denote by Φ𝑍𝑛(𝑒)=𝐸(exp(𝑖𝑒𝑍𝑛)) the characteristic function of 𝑍𝑛 and by 𝐢 a generic positive constant, not necessarily the same at each occurrence. We have Ξ¦π‘π‘›βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽ(𝑒)=𝐸expπ‘–π‘’βˆšπ‘›π‘Ÿξ“π‘˜=1π›Όξ…žπ‘˜[π‘›πœπ‘˜]𝑑=1πΉπ‘‘πœ€π‘‘βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽ=𝐸expπ‘–π‘’βˆšπ‘›π‘Ÿξ“π‘˜=1ξƒ©π‘Ÿξ“π‘—=π‘˜π›Όξ…žπ‘—ξƒͺ[π‘›πœπ‘˜]𝑑=[π‘›πœπ‘˜βˆ’1]+1πΉπ‘‘πœ€π‘‘βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯⎦,𝜏0==0π‘Ÿξ‘π‘˜=1Ξ¦π‘˜,𝑛(𝑒),(3.6) where Ξ¦π‘˜,π‘›βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽ(𝑒)=𝐸expπ‘–π‘’βˆšπ‘›ξƒ©π‘Ÿξ“π‘—=π‘˜π›Όξ…žπ‘—ξƒͺ[π‘›πœπ‘˜]𝑑=[π‘›πœπ‘˜βˆ’1]+1πΉπ‘‘πœ€π‘‘βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯⎦.(3.7) Since (πœ€π‘‘) is an i.i.d. sequence of random vectors we have ξ€·πœ€[π‘›πœπ‘˜βˆ’1]+1,…,πœ€[π‘›πœπ‘˜]ξ€Έlaw=ξ€·πœ€1,…,πœ€[π‘›πœπ‘˜]βˆ’[π‘›πœπ‘˜βˆ’1]ξ€Έ.(3.8) Hence Ξ¦π‘˜,π‘›βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽ(𝑒)=𝐸expπ‘–π‘’βˆšπ‘›ξƒ©π‘Ÿξ“π‘—=π‘˜π›Όβ€²π‘—ξƒͺ[π‘›πœπ‘˜]βˆ’[π‘›πœπ‘˜βˆ’1]𝑑=1𝐹[π‘›πœπ‘˜βˆ’1]+π‘‘πœ€π‘‘βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯⎦.(3.9) Let 𝐈(𝐴)=1 if the argument 𝐴 is true and 0 otherwise, π‘˜π‘›=[π‘›πœπ‘˜]βˆ’[π‘›πœπ‘˜βˆ’1], πœ‰π‘›,𝑖=1βˆšπ‘›ξƒ©π‘Ÿξ“π‘—=π‘˜π›Όξ…žπ‘—ξƒͺ𝐹[π‘›πœπ‘˜βˆ’1]+π‘–πœ€π‘–,𝑀𝑛,π‘˜π‘›=π‘˜π‘›ξ“π‘–=1πœ‰π‘›,𝑖,(3.10)ℱ𝑛,𝑑=𝜎(πœ€1,…,πœ€π‘‘,π‘‘β‰€π‘˜π‘›) the filtration spanned by πœ€1,…,πœ€π‘‘.
Then (𝑀𝑛,𝑖,ℱ𝑛,𝑖,1β‰€π‘–β‰€π‘˜π‘›,𝑛β‰₯1) is a zero-mean square-integrable martingale array with differences πœ‰π‘›,𝑖. Observe that π‘˜π‘›ξ“π‘–=1πΈξ€·πœ‰2𝑛,π‘–βˆ£β„±π‘›,π‘–βˆ’1ξ€Έ=1π‘›ξƒ©π‘Ÿξ“π‘—=π‘˜π›Όξ…žπ‘—ξƒͺπ‘˜π‘›ξ“π‘–=1𝐹[π‘›πœπ‘˜βˆ’1]+𝑖𝐹′[π‘›πœπ‘˜βˆ’1]+π‘–ξƒ©π‘Ÿξ“π‘—=π‘˜π›Όπ‘—ξƒͺ⟢𝜎2π‘˜=ξ€·πœπ‘˜βˆ’πœπ‘˜βˆ’1ξ€Έβ€–β€–β€–β€–π‘Ÿξ“π‘—=π‘˜π›Όπ‘—β€–β€–β€–β€–2asπ‘›βŸΆβˆž.(3.11) Now using Assumption 1 we obtain that ‖Γ𝑑‖<𝐾 uniformly on 𝑑 for some positive constant 𝐾, hence Assumption 2 implies that for all πœ€>0, π‘˜π‘›ξ“π‘–=1πΈξ€·πœ‰2𝑛,π‘–πˆξ€·||πœ‰π‘›,𝑖||ξ€Έ>πœ€βˆ£β„±π‘›,π‘–βˆ’1≀1πœ€π›Ώπ‘˜π‘›ξ“π‘–=1𝐸||πœ‰π‘›,𝑖||2+π›Ώβˆ£β„±π‘›,π‘–βˆ’1ξ‚β‰€πΆπ‘˜π‘›π‘›1+𝛿/2⟢0asπ‘›βŸΆβˆž,(3.12) where πΈξ‚€β€–β€–πœ€πΆ=1β€–β€–2+π›Ώξ‚πœ€π›Ώξƒ©πΎβ€–β€–Ξ“βˆ’1β€–β€–β€–β€–β€–β€–π‘Ÿξ“π‘—=π‘˜π›Όπ‘—β€–β€–β€–β€–ξƒͺ2+𝛿,(3.13) consequently (see Hall and Heyde [18], Theorem 3.2) 𝑀𝑛,π‘˜π‘›β„’βŸΆπ‘π‘˜,(3.14) where π‘π‘˜ is a normal random variable with zero mean and variance 𝜎2π‘˜. Therefore Ξ¦π‘˜,𝑛(𝑒)=𝐸exp𝑖𝑒𝑀𝑛,π‘˜π‘›ξ‚€βˆ’1ξ€Έξ€ΈβŸΆexp2𝜎2π‘˜π‘’2asπ‘›βŸΆβˆž,(3.15) which together with (3.6) implies that Ξ¦π‘π‘›ξƒ©βˆ’1(𝑒)⟢exp2π‘Ÿξ“π‘˜=1𝜎2π‘˜π‘’2ξƒͺ=Φ𝑍(𝑒)asπ‘›βŸΆβˆž,(3.16) the last equality holds since, with 𝜏0=0, π‘Ÿξ“π‘˜=1ξ€·πœπ‘˜βˆ’πœπ‘˜βˆ’1ξ€Έβ€–β€–β€–β€–π‘Ÿξ“π‘—=π‘˜π›Όπ‘—β€–β€–β€–β€–2=1≀𝑖,π‘—β‰€π‘Ÿπ›Όβ€²π‘–π›Όπ‘—ξ€·πœmin𝑖,πœπ‘—ξ€Έ.(3.17)
For 1≀𝑖≀𝑑, fixed, in order to obtain the tightness of π‘Šπ‘›(𝑖) it suffices to show the following inequality (Billingsley [16], Theorem 15.6): 𝐸||π‘Šπ‘›(𝑖)(𝜏)βˆ’π‘Šπ‘›(𝑖)ξ€·πœ1ξ€Έ||𝛾||π‘Šπ‘›(𝑖)ξ€·πœ2ξ€Έβˆ’π‘Šπ‘›(𝑖)||(𝜏)π›Ύξ‚β‰€ξ€·πΉξ€·πœ2ξ€Έξ€·πœβˆ’πΉ1𝛼,(3.18) for some 𝛾>0, 𝛼>1, where 𝐹 is a nondecreasing continuous function on [0,1] and 0<𝜏1<𝜏<𝜏2<1.
We have 𝐸||π‘Šπ‘›(𝑖)(𝜏)βˆ’π‘Šπ‘›(𝑖)ξ€·πœ1ξ€Έ||2||π‘Šπ‘›(𝑖)(𝜏2)βˆ’π‘Šπ‘›(𝑖)||(𝜏)2=𝑇1𝑇2,(3.19) where 𝑇1=1π‘›πΈβŽ›βŽœβŽœβŽœβŽ||||||[π‘›πœ]𝑑=π‘›πœ1𝑑+1𝑗=1𝐹𝑑(𝑖,𝑗)πœ€π‘‘(𝑗)||||||2⎞⎟⎟⎟⎠,𝑇2=1π‘›πΈβŽ›βŽœβŽœβŽ|||||[π‘›πœ2][]𝑑𝑑=π‘›πœ+1𝑗=1𝐹𝑑(𝑖,𝑗)πœ€π‘‘(𝑗)|||||2⎞⎟⎟⎠.(3.20) Now observe that 𝑇1=1𝑛𝑑,𝑠cov𝑑𝑗=1𝐹𝑑(𝑖,𝑗)πœ€π‘‘(𝑗),𝑑𝑗=1𝐹𝑠(𝑖,𝑗)πœ€π‘ (𝑗)ξƒͺ=1𝑛[π‘›πœ]𝑑=[π‘›πœ1𝑑]+1𝑗=1𝐹𝑑(𝑖,𝑗)2ξ€·β‰€πΆπœβˆ’πœ1ξ€Έforsomeconstant𝐢>0.(3.21) Likewise 𝑇2≀𝐢(𝜏2βˆ’πœ). Since (πœβˆ’πœ1)(𝜏2βˆ’πœ)≀(𝜏2βˆ’πœ1)2/2, the inequality (3.18) holds with 𝛾=𝛼=2, √𝐹(𝑑)=𝐢𝑑/2.

In order to prove Theorem 3.1 we need also the following lemma.

Lemma 3.3. Assume that (π‘Œπ‘‘) is given by (1.1), where (πœ€π‘‘) is an i.i.d sequence of random vectors such that 𝐸(πœ€π‘‘)=0, var(πœ€π‘‘)=𝐼𝑑 and that (Γ𝑑) satisfies (2.5). Then under the null 𝐻0, the empirical covariance of π‘Œπ‘‘ satisfies ξβˆ‘a.s.βŸΆξ“,(3.22) where a.s.β†’ denotes the almost sure convergence.

Proof. Let π‘Šπ‘‘=Ξ“π‘‘πœ€π‘‘,ℱ𝑑=𝜎(πœ€1,…,πœ€π‘‘) and for 𝑖,𝑗 fixed, 1≀𝑖≀𝑑,1≀𝑗≀𝑑,𝑒𝑑=π‘Šπ‘‘(𝑖)π‘Šπ‘‘(𝑗)βˆ’πΈ(π‘Šπ‘‘(𝑖)π‘Šπ‘‘(𝑗)βˆ£β„±π‘‘βˆ’1).
Then (𝑒𝑑) is a martingale difference sequence with respect to ℱ𝑑. Since 𝑒𝑑=π‘Šπ‘‘(𝑖)π‘Šπ‘‘(𝑗)βˆ’βˆ‘π‘‘π‘˜=1Γ𝑑(𝑖,π‘˜)Γ𝑑(𝑗,π‘˜) and the matrix Γ𝑑 is bounded, it follows that 𝐸||𝑒𝑑||(2+𝛿)/2||π‘Šβ‰€πΆ+𝐸𝑑(𝑖)π‘Šπ‘‘(𝑗)||(2+𝛿)/2||π‘Šβ‰€πΈπ‘‘(𝑖)||2+𝛿1/2𝐸||π‘Šπ‘‘(𝑗)||2+𝛿1/2≀𝐢,(3.23) since by using Assumptions 1 and 2 we get 𝐸||π‘Šπ‘‘(𝑖)||2+π›Ώξ‚β‰€ξƒ©π‘‘ξ“π‘˜=1𝐸||Γ𝑑(𝑖,π‘˜)πœ€π‘‘(π‘˜)||2+𝛿1/(2+𝛿)ξƒͺ2+𝛿≀𝐢.(3.24) Therefore, Theorem 5 of Chow [19] implies that 𝑛𝑑=1𝑒𝑑=π‘œ(𝑛)almostsurely(3.25) or 1𝑛𝑛𝑑=1π‘Šπ‘‘(𝑖)π‘Šπ‘‘(𝑗)=1𝑛𝑛𝑑=1Γ𝑑Γ′𝑑(𝑖,𝑗)+π‘œ(1)almostsurely,(3.26) where (Γ𝑑Γ′𝑑)(𝑖,𝑗) denotes the (𝑖,𝑗)-th entry of the matrix Ξ“π‘‘Ξ“ξ…žπ‘‘. Hence 1𝑛𝑛𝑑=1π‘Šπ‘‘π‘Šξ…žπ‘‘a.s.βŸΆξ“.(3.27)

Lemma 2 of Lai and Wei [20], page 157, implies that with probability one𝑛𝑑=1Γ𝑑(𝑖,π‘˜)πœ€π‘‘(π‘˜)=π‘œπ‘›ξ“π‘‘=1Γ𝑑(𝑖,π‘˜)2ξƒͺ+𝑂(1)=π‘œ(𝑛)+𝑂(1)βˆ€1≀𝑖≀𝑑,(3.28) or1𝑛𝑛𝑑=1Γ𝑑(𝑖,π‘˜)πœ€π‘‘(π‘˜)ξ‚€1=π‘œ(1)+𝑂𝑛almostsurely,(3.29) which implies that1𝑛𝑛𝑑=1π‘Šπ‘‘=π‘‘ξ“π‘˜=11𝑛𝑛𝑑=1Γ𝑑(1,π‘˜)πœ€π‘‘(π‘˜)1,…,𝑛𝑛𝑑=1Γ𝑑(𝑑,π‘˜)πœ€π‘‘(π‘˜)ξƒͺβ€²a.s.⟢0.(3.30) Note that π‘Œπ‘‘=πœ‡+π‘Šπ‘‘, hence combining (3.27) and (3.30) we obtainξβˆ‘=1𝑛𝑛𝑑=1π‘Œπ‘‘π‘Œβ€²π‘‘βˆ’π‘Œπ‘Œξ…ž1=πœ‡π‘›π‘›ξ“π‘‘=1π‘Šπ‘‘+1𝑛𝑛𝑑=1π‘Šπ‘‘πœ‡ξ…ž+πœ‡πœ‡ξ…ž+1𝑛𝑛𝑑=1π‘Šπ‘‘π‘Šξ…žπ‘‘βˆ’π‘Œπ‘Œξ…ža.s.βŸΆξ“.(3.31)

Proof of Theorem 3.1. Under the null 𝐻0 we have π‘Œπ‘‘=πœ‡+Ξ“π‘‘πœ€π‘‘, thus recalling (3.3) we can write 𝐡𝑛(ξΞ“πœ)=βˆ’11βˆšπ‘›[π‘›πœ]𝑑=1ξ‚€π‘Œπ‘‘βˆ’π‘Œξ‚=ξΞ“βˆ’1Ξ“1βˆšπ‘›Ξ“βˆ’1[π‘›πœ]𝑑=1ξ‚ƒξ€·π‘Œπ‘‘ξ€Έβˆ’ξ‚€βˆ’πœ‡=ξΞ“π‘Œβˆ’πœ‡ξ‚ξ‚„βˆ’1Ξ“ξ‚΅π‘Šπ‘›[](𝜏)βˆ’π‘›πœπ‘›π‘Šπ‘›ξ‚Ά.(1)(3.32) Therefore the result (3.1) holds by applying Theorem 3.2, Lemma 3.3, and the continuous mapping theorem.

4. Consistency of ℬ𝑛

We assume that under the alternative 𝐻1 the means (πœ‡π‘‘) are bounded and satisfy the following.

Assumption H1. There exists a function π‘ˆ from [0,1] into ℝ𝑑 such that [],1βˆ€πœβˆˆ0,1𝑛[π‘›πœ]𝑑=1πœ‡π‘‘βŸΆπ‘ˆ(𝜏)asπ‘›βŸΆβˆž.(4.1)

Assumption H2. There exists πœβˆ—βˆˆ(0,1) such that π’°ξ€·πœβˆ—ξ€Έξ€·πœ=π‘ˆβˆ—ξ€Έβˆ’πœβˆ—π‘ˆ(1)β‰ 0.(4.2)

Assumption H3. There exists βˆ‘πœ‡ such that 1𝑛𝑛𝑑=1ξ€·πœ‡π‘‘βˆ’πœ‡πœ‡ξ€Έξ€·π‘‘βˆ’πœ‡ξ€Έβ€²βŸΆξ“πœ‡asπ‘›βŸΆβˆž,(4.3) where βˆ‘πœ‡=(1/𝑛)𝑛𝑑=1πœ‡π‘‘.

Theorem 4.1. Suppose that Assumptions 1 and 2 hold. If (π‘Œπ‘‘) is given by (1.1) and the means (πœ‡π‘‘) satisfy the Assumptions H1, H2, and H3, then the test based on ℬ𝑛 is consistent against 𝐻1, that is, β„¬π‘›π‘ƒβŸΆ+∞,(4.4) where 𝑃→ denotes the convergence in probability.

Proof. We have 𝐡𝑛(𝜏)=𝐡0𝑛(𝜏)+𝐡1𝑛(𝜏),(4.5) where 𝐡0𝑛(ξΞ“πœ)=βˆ’1βˆšπ‘›[π‘›πœ]𝑑=1ξ‚€π‘Šπ‘‘βˆ’π‘Šξ‚,π‘Šπ‘‘=Ξ“π‘‘πœ€π‘‘,1π‘Š=𝑛𝑛𝑑=1π‘Šπ‘‘,𝐡1𝑛Γ(𝜏)=βˆ’1βˆšπ‘›[π‘›πœ]𝑑=1ξ€·πœ‡π‘‘βˆ’πœ‡ξ€Έ.(4.6)
Straightforward computation leads to ξβˆ‘a.s.βŸΆξ“βˆ—=+ξ“πœ‡.(4.7) Therefore 𝐡0𝑛(𝜏)β„’βŸΆΞ“βˆ—βˆ’1Γ𝐡(𝜏),(4.8) where Ξ“βˆ— is a square root of Ξ£βˆ—, that is, Ξ£βˆ—=Ξ“βˆ—Ξ“β€²βˆ—, and 𝐡1𝑛(𝜏)βˆšπ‘›a.s.βŸΆΞ“βˆ—βˆ’1𝒰(𝜏).(4.9) Hence ||||𝐡𝑛(πœβˆ—)||||βˆžπ‘ƒβŸΆ+∞,(4.10) which implies that β„¬π‘›π‘ƒβŸΆ+∞.(4.11)

4.1. Consistency of ℬ𝑛 against Abrupt Change

Without loss of generality we assume that under the alternative hypothesis 𝐻1 there is a single break date, that is, (π‘Œπ‘‘) is given by (1.1) whereπœ‡π‘‘=ξ‚»πœ‡(1)ξ€Ίif1β‰€π‘‘β‰€π‘›πœ1ξ€»πœ‡(2)ξ€Ίifπ‘›πœ1ξ€»+1≀𝑑≀𝑛forsome𝜏1∈(0,1)andπœ‡(1)β‰ πœ‡(2).(4.12)

Corollary 4.2. Suppose that Assumptions 1 and 2 hold. If (π‘Œπ‘‘) is given by (1.1) and the means (πœ‡π‘‘) satisfy (4.12), then the test based on ℬ𝑛 is consistent against 𝐻1.

Proof. It is easy to show that (4.1)–(4.3) are satisfied with π’°ξ‚»πœξ€·(𝜏)=1βˆ’πœ1πœ‡ξ€Έξ€·(1)βˆ’πœ‡(2)ξ€Έifπœβ‰€πœ1𝜏1ξ€·1βˆ’πœ1πœ‡ξ€Έξ€·(1)βˆ’πœ‡(2)ξ€Έif𝜏>𝜏1,ξ“πœ‡=𝜏1ξ€·πœ‡(1)βˆ’πœ‡(2)πœ‡ξ€Έξ€·(1)βˆ’πœ‡(2)ξ€Έβ€².(4.13) Note that (4.2) is satisfied for all 0<πœβˆ—<𝜏1 since πœ‡(1)β‰ πœ‡(2).

Remark 4.3. The result of Corollary 4.2 remains valid if under the alternative hypothesis there are multiple breaks in the mean.

4.2. Consistency of ℬ𝑛 against Smooth Change

In this subsection we assume that the break in the mean does not happen suddenly but the transition from one value to another is continuous with slow variation. A well-known dynamic is the smooth threshold model (see TerΓ€svirta [21]), in which the mean πœ‡π‘‘ is time varying as followsπœ‡π‘‘=πœ‡(1)+ξ€·πœ‡(2)βˆ’πœ‡(1)𝐹𝑑𝑛,𝜏1,𝛾,1≀𝑑≀𝑛,πœ‡(1)β‰ πœ‡(2),(4.14) where 𝐹(π‘₯,𝜏1,𝛾) is a the smooth transition function assumed to be continuous from [0,1] into [0,1], πœ‡(1) and πœ‡(2) are the values of the mean in the two extreme regimes, that is, when 𝐹→0 and 𝐹→1. The slope parameter 𝛾 indicates how rapid the transition between two extreme regimes is. The parameter 𝜏1 is the location parameter.

Two choices for the function 𝐹 are frequently evoked, the logistic function given by𝐹𝐿π‘₯,𝜏1ξ€Έ=ξ€Ίξ€·ξ€·,𝛾1+expβˆ’π›Ύπ‘₯βˆ’πœ1ξ€Έξ€Έξ€»βˆ’1,(4.15) and the exponential one𝐹𝑒π‘₯,𝜏1ξ€Έξ‚€ξ€·,𝛾=1βˆ’expβˆ’π›Ύπ‘₯βˆ’πœ1ξ€Έ2.(4.16) For example, for the logistic function with 𝛾>0, the extreme regimes are obtained as follows:(i)if π‘₯β†’0 and 𝛾 large then 𝐹→0 and thus πœ‡π‘‘=πœ‡(1),(ii)if π‘₯β†’1 and 𝛾 large then 𝐹→1 and thus πœ‡π‘‘=πœ‡(2).

This means that at the beginning of the sample πœ‡π‘‘ is close to πœ‡(1) and then moves towards πœ‡(2) and becomes close to it at the end of the sample.

Corollary 4.4. Suppose that Assumptions 1 and 2 hold. If (π‘Œπ‘‘) is given by (1.1) and the means (πœ‡π‘‘) satisfy (4.14), then the test based on ℬ𝑛 is consistent against 𝐻1.

Proof. The assumptions (4.1) and (4.3) are satisfied with π’°ξ€·πœ‡(𝜏)=(2)βˆ’πœ‡(1)𝑇(𝜏),(4.17) where ξ€œπ‘‡(𝜏)=𝜏0𝐹π‘₯,𝜏1ξ€Έξ€œ,𝛾𝑑π‘₯βˆ’πœ10𝐹π‘₯,𝜏1,𝛾𝑑π‘₯,πœ‡=ξ€·πœ‡(2)βˆ’πœ‡(1)πœ‡ξ€Έξ€·(2)βˆ’πœ‡(1)ξ€Έξ…žξƒ―ξ€œ10𝐹2ξ€·π‘₯,𝜏1ξ€Έξ‚΅ξ€œ,𝛾𝑑π‘₯βˆ’10𝐹π‘₯,𝜏1ξ€Έξ‚Ά,𝛾𝑑π‘₯2ξƒ°.(4.18)
Since πœ‡(2)βˆ’πœ‡(1)β‰ 0, to prove (4.2), it suffices to show that there exists πœβˆ— such 𝑇(πœβˆ—)β‰ 0.
Assume that 𝑇(𝜏)=0 for all 𝜏∈(0,1) then 𝑑𝑇(𝜏)ξ€·π‘‘πœ=𝐹𝜏,𝜏1ξ€Έβˆ’ξ€œ,𝛾10𝐹π‘₯,𝜏1ξ€Έ,𝛾𝑑π‘₯=0βˆ€πœβˆˆ(0,1),(4.19) which implies that 𝐹(𝜏,𝜏1∫,𝛾)=10𝐹(π‘₯,𝜏1,𝛾)𝑑π‘₯=𝐢 for all 𝜏∈(0,1) or πœ‡π‘‘=πœ‡(1)+ξ€·πœ‡(2)βˆ’πœ‡(1)𝐢=πœ‡βˆ€π‘‘β‰₯1,(4.20) and this contradicts the alternative hypothesis 𝐻1.

4.3. Consistency of ℬ𝑛 against Continuous Change

In this subsection we will examine the behaviour ℬ𝑛 under the alternative where the mean (πœ‡π‘‘) varies at each time, and hence can take an infinite number of values. As an example we consider a polynomial evolution for πœ‡π‘‘:πœ‡π‘‘=𝑃1𝑑𝑛,…,π‘ƒπ‘‘ξ‚€π‘‘π‘›ξ‚ξ‚ξ…ž,𝑃𝑗(π‘₯)=π‘π‘—ξ“π‘˜=0𝛼𝑗,π‘˜π‘₯π‘˜,1≀𝑗≀𝑑.(4.21)

Corollary 4.5. Suppose that Assumptions 1 and 2 hold. If (π‘Œπ‘‘) is given by (1.1) and the means (πœ‡π‘‘) satisfy (4.21), then the test based on ℬ𝑛 is consistent against 𝐻1.

Proof. The assumptions H1–H3 are satisfied with 𝒰(𝜏)=𝑝1ξ“π‘˜=0𝛼1,π‘˜ξ€·πœπ‘˜+1π‘˜+1ξ€Έβˆ’πœ,…,π‘π‘‘ξ“π‘˜=0𝛼𝑑,π‘˜ξ€·πœπ‘˜+1π‘˜+1ξ€Έξƒͺβˆ’πœξ…ž,ξ“πœ‡(𝑖,𝑗)=π‘π‘–ξ“π‘π‘˜=0𝑗𝑙=01𝛼𝑙+π‘˜+1𝑖,π‘˜π›Όπ‘—,π‘™βˆ’ξƒ©π‘π‘–ξ“π‘˜=0𝛼𝑖,π‘˜π‘˜+1ξƒͺξƒ©π‘π‘—ξ“π‘˜=0𝛼𝑗,π‘˜ξƒͺ.π‘˜+1(4.22) Note that (4.2) is satisfied for all 0<πœβˆ—<1, provided that there exist 𝑖,1≀𝑖≀𝑑 and π‘˜,1β‰€π‘˜β‰€π‘π‘– such that 𝛼𝑖,π‘˜β‰ 0.

5. Finite Sample Performance

All models are driven from an i.i.d. sequences πœ€π‘‘=(πœ€π‘‘(1),…,πœ€π‘‘(𝑑))ξ…ž, where each πœ€π‘‘(𝑗),1≀𝑗≀𝑑, has a 𝑑(3) distribution, a Student distribution with 3 degrees of freedom, and πœ€π‘‘(𝑖) and πœ€π‘‘(𝑗) are independent for all 𝑖≠𝑗. Simulations were performed using the software R. We carry out an experiment of 1000 samples for seven models and we use three different sample sizes, 𝑛=30, 𝑛=100, and 𝑛=500. The empirical sizes and powers are calculated at the nominal levels 𝛼=1%, 5%, and 10%, in both cases.

5.1. Study of the Size

In order to evaluate the size distortion of the test statistic ℬ𝑛 we consider two bivariate models π‘Œπ‘‘=πœ‡π‘‘+Ξ“π‘‘πœ€π‘‘ with the following.

Model 1 (constant covariance). πœ‡π‘‘=βŽ›βŽœβŽœβŽ11⎞⎟⎟⎠,Γ𝑑=βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ 2112.(5.1)

Model 2 (time varying covariance). πœ‡π‘‘=βŽ›βŽœβŽœβŽ11⎞⎟⎟⎠,Γ𝑑=βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ‹2sin(π‘‘πœ”)βˆ’1βˆ’12cos(π‘‘πœ”),πœ”=4.(5.2)

From Table 1, we observe that for small sample size (𝑛=30) the test statistic ℬ𝑛 has a severe size distortion. But as the sample size 𝑛 increases, the distortion decreases. The empirical size becomes closer to (but always lower than) the nominal level. The distortion in the nonstationary Model 2 (time varying covariance) is a somewhat greater than the one in the stationary Model 1 (constant covariance). However the test seems to be conservative in both cases.

5.2. Study of the Power

In order to see the power of the test statistic ℬ𝑛 we consider five bivariate models π‘Œπ‘‘=πœ‡π‘‘+Ξ“π‘‘πœ€π‘‘ with the following.

5.2.1. Abrupt Change in the Mean

Model 3 (constant covariance). πœ‡π‘‘=⎧βŽͺ⎨βŽͺβŽ©ξ‚΅01𝑛if1≀𝑑≀2ξ‚„ξ‚΅10𝑛if2ξ‚„Ξ“+1≀𝑑≀𝑛,𝑑=βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ 2112.(5.3)

Model 4. In this model the mean and the covariance are subject to an abrupt change at the same time: πœ‡π‘‘=⎧βŽͺ⎨βŽͺβŽ©ξ‚΅01𝑛if1≀𝑑≀2ξ‚„ξ‚΅10𝑛if2ξ‚„Ξ“+1≀𝑑≀𝑛𝑑=⎧βŽͺ⎨βŽͺβŽ©ξ‚΅ξ‚Άξ‚ƒπ‘›1001if1≀𝑑≀2𝑛2102if2ξ‚„+1≀𝑑≀𝑛.(5.4)

Model 5. The mean is subject to an abrupt change and the covariance is time varying (see Figure 1): πœ‡π‘‘=⎧βŽͺ⎨βŽͺβŽ©ξ‚΅01𝑛if1≀𝑑≀2ξ‚„ξ‚΅10𝑛if2ξ‚„Ξ“+1≀𝑑≀𝑛,𝑑=βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ‹2sin(π‘‘πœ”)βˆ’1βˆ’12cos(π‘‘πœ”),πœ”=4.(5.5)

5.2.2. Smooth Change in the Mean

Model 6. We consider a logistic smooth transition for the mean and a time varying covariance (see Figure 1): πœ‡π‘‘=πœ‡(1)+ξ€·πœ‡(2)βˆ’πœ‡(1)ξ€Έπœ‡(1+exp(βˆ’30(𝑑/π‘›βˆ’1/2))),1≀𝑑≀𝑛,(1)=βŽ›βŽœβŽœβŽ01⎞⎟⎟⎠,πœ‡(2)=βŽ›βŽœβŽœβŽ10⎞⎟⎟⎠,Γ𝑑=βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ‹2sin(π‘‘πœ”)βˆ’1βˆ’12cos(π‘‘πœ”),πœ”=4.(5.6)

5.2.3. Continuous Change in the Mean

Model 7. In this model the mean is a polynomial of order two and the covariance matrix is also time varying as in the preceding Models 5 and 6 (see Figure 1): πœ‡π‘‘=βŽ›βŽœβŽœβŽœβŽœβŽœβŽπ‘‘π‘›ξ‚€π‘‘2βˆ’π‘›ξ‚π‘‘π‘›ξ‚€π‘‘2βˆ’π‘›ξ‚βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,Γ𝑑=βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ‹2sin(π‘‘πœ”)βˆ’1βˆ’12cos(π‘‘πœ”),πœ”=4.(5.7)

From Table 2, we observe that for small sample size (𝑛=30), the test statistic ℬ𝑛 has a low power. However, for the five models, the power becomes good as the sample size 𝑛 increases. The powers in nonstationary models are always smaller than those of stationary models. This is not surprising since, from Table 1, the test statistic ℬ𝑛 is more conservative in nonstationary models. We observe also that the power is almost the same in abrupt and logistic smooth changes (compare Models 5 and 6). However, for the polynomial change (Model 7) the power is lower than those of Models 5 and 6. To explain this underperformance we can see, in Figure 1, that in the polynomial change, the time intervals where the mean stays near the extreme values 0 and 1 are very short compared to those in abrupt and smooth changes. We have simulated other continuous changes, linear and cubic polynomial, trigonometric, and many other functions. Like in Model 7, changes are hardly detected for small values of 𝑛, and the test ℬ𝑛 has a good performance only in large samples.

Acknowledgment

The author would like to thank the anonymous referees for their constructive comments.