# Testing for Change in Mean of Independent Multivariate Observations with Time Varying Covariance

Authors: Mohamed Boutahar
Journal: Journal of Probability and Statistics
 (2012)

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## 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.


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## Body

## 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 Horváth 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 model(1.1)Yt=μt+Γtɛt,t=1,…,n,
where (ɛt) 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 (Γt) is deterministic with unknown form. The null and the alternative hypotheses are as follows:(1.2)H0:μt=μ∀t≥1againstH1:Thereexistt≠ssuchthatμt≠μs.
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 μt and Σt are step functions, that is,
(1.3)μt=μ(j)ift=nj-1+1,…,nj,nj=[λjn],0<λ1<⋯<λm1<1,∑t=σ(j)ift=tj-1+1,…,tj,tj=[τjn],0<τ1<⋯<τm2<1,
for some integers m1 and m2. 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), Horváth et al. [10] considered the model (1.1) where μt is subject to change and Σt=Σ 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 let(2.1)Bn(τ)=Γ̂-11n∑t=1[nτ](Yt-Y¯)∀τ∈[0,1],
where Γ̂ is a square root of Σ̂,thatis,Σ̂=Γ̂Γ̂′,(2.2)∑̂=1n∑t=1n(Yt-Y¯)(Yt-Y¯)′,Y¯=1n∑t=1nYt
are the empirical covariance and mean of the sample (Y1,…,Yn)′, respectively, [x] is the integer part of x, and X′ is the transpose of X.The CUSUM test statistic we will consider is given by(2.3)Bn=sup⁡τ∈[0,1]‖Bn(τ)‖∞,
where(2.4)‖X‖∞=max⁡1≤i≤d|X(i)|ifX=(X(1),…,X(d))′.Assumption 1.
The sequence of matrices(Γt) is bounded and satisfies
(2.5)1n∑t=1nΓtΓt′⟶Σ>0asn⟶∞.Assumption 2.
There existsδ>0 such that E(∥ɛ1∥2+δ)<∞, where ∥X∥ denotes the Euclidian norm of X.


## 3. Limiting Distribution ofℬn under the Null

Theorem 3.1.
Suppose that Assumptions1 and 2 hold. Then, under  H0,
(3.1)Bn⟶LB∞=sup⁡τ∈[0,1]‖B(τ)‖∞,→ℒ  denotes the convergence in distribution and  B(τ) is a multivariate Brownian Bridge with independent components.
Moreover, the cumulative distribution function ofB∞ is given by
(3.2)FB∞(z)=(1+2∑k=1∞(-1)kexp⁡⁡{-2k2z2})d.To prove Theorem3.1 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 d∈ℕ, 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 vectorsX and Y,X=lawY means that X has the same distribution as Y.Consider an i.i.d. sequence(ɛt) of random vectors such that E(ɛt)=0 and var(ɛt)=Id. Let (Γt) satisfy (2.5) and set(3.3)Wn(τ)=Γ-1n∑t=1[nτ]Γtɛt,τ∈[0,1],
where Γ is a square root of Σ,thatis,Σ=ΓΓ′. Many functional central limit theorems were established for covariance stationary random sequences, see Boutahar [15] and the references therein. Note that the sequence (Γtɛt) we consider here is not covariance stationary.There are two sufficient conditions to prove thatWn⇒W (see Billingsley [16] and Iglehart [17]), namely,(i)
the finite-dimensional distributions ofWn converge to the finite-dimensional distributions of W,(ii)
Wn(i) is tight for all 1≤i≤d, if Wn=(Wn(1),…,Wn(d))′.Theorem 3.2.
Assume that(ɛt) is an i.i.d. sequence of random vectors such that  E(ɛt)=0,  var(ɛt)=Id and that Assumptions 1 and 2 hold. Then
(3.4)Wn⟹W,
where  W is a standard multivariate Brownian motion.Proof.
WriteFt=Γ-1Γt, Ft(i,j) the (i,j)-th entry of the matrix Ft,ɛt=(ɛt(1),…,ɛt(d))′. To prove that the finite-dimensional distributions of Wn converge to those of W it is sufficient to show that for all integer r≥1, for all 0≤τ1<⋯<τr≤1, and for all αi∈ℝd, 1≤i≤r,
(3.5)Zn=∑i=1rαi′Wn(τi)⟶LZ=∑i=1rαi′W(τi).
Denote by ΦZn(u)=E(exp(iuZn)) the characteristic function of Zn and by C a generic positive constant, not necessarily the same at each occurrence. We have
(3.6)ΦZn(u)=E[exp⁡(iun∑k=1rαk′∑t=1[nτk]Ftɛt)]=E[exp⁡(iun∑k=1r(∑j=krαj′)∑t=[nτk-1]+1[nτk]Ftɛt)],τ0=0=∏k=1rΦk,n(u),
where
(3.7)Φk,n(u)=E[exp⁡(iun(∑j=krαj′)∑t=[nτk-1]+1[nτk]Ftɛt)].
Since (ɛt) is an i.i.d. sequence of random vectors we have
(3.8)(ɛ[nτk-1]+1,…,ɛ[nτk])=law(ɛ1,…,ɛ[nτk]-[nτk-1]).
Hence
(3.9)Φk,n(u)=E[exp⁡(iun(∑j=krαj′)∑t=1[nτk]-[nτk-1]F[nτk-1]+tɛt)].
Let I(A)=1 if the argument A is true and 0 otherwise, kn=[nτk]-[nτk-1],
(3.10)ξn,i=1n(∑j=krαj′)F[nτk-1]+iɛi,Mn,kn=∑i=1knξn,i,ℱn,t=σ(ɛ1,…,ɛt,t≤kn) the filtration spanned by ɛ1,…,ɛt.
Then(Mn,i,ℱn,i,1≤i≤kn,n≥1) is a zero-mean square-integrable martingale array with differences ξn,i. Observe that
(3.11)∑i=1knE(ξn,i2∣Fn,i-1)=1n(∑j=krαj′)∑i=1knF[nτk-1]+iF′[nτk-1]+i(∑j=krαj)⟶σk2=(τk-τk-1)‖∑j=krαj‖2asn⟶∞.
Now using Assumption 1 we obtain that ∥Γt∥<K uniformly on t for some positive constant K, hence Assumption 2 implies that for all ɛ>0,
(3.12)∑i=1knE(ξn,i2I(|ξn,i|>ɛ)∣Fn,i-1)≤1ɛδ∑i=1knE(|ξn,i|2+δ∣Fn,i-1)≤Cknn1+δ/2⟶0asn⟶∞,
where
(3.13)C=E(‖ɛ1‖2+δ)ɛδ(K‖Γ-1‖‖∑j=krαj‖)2+δ,
consequently (see Hall and Heyde [18], Theorem 3.2)
(3.14)Mn,kn⟶LZk,
where Zk is a normal random variable with zero mean and variance σk2. Therefore
(3.15)Φk,n(u)=E(exp⁡(iuMn,kn))⟶exp⁡(-12σk2u2)asn⟶∞,
which together with (3.6) implies that
(3.16)ΦZn(u)⟶exp⁡(-12∑k=1rσk2u2)=ΦZ(u)asn⟶∞,
the last equality holds since, with τ0=0,
(3.17)∑k=1r(τk-τk-1)‖∑j=krαj‖2=∑1≤i,j≤rαi′αjmin⁡⁡(τi,τj).
For1≤i≤d, fixed, in order to obtain the tightness of Wn(i) it suffices to show the following inequality (Billingsley [16], Theorem 15.6):
(3.18)E(|Wn(i)(τ)-Wn(i)(τ1)|γ|Wn(i)(τ2)-Wn(i)(τ)|γ)≤(F(τ2)-F(τ1))α,
for some γ>0, α>1, where F is a nondecreasing continuous function on [0,1] and 0<τ1<τ<τ2<1.
We have(3.19)E(|Wn(i)(τ)-Wn(i)(τ1)|2|Wn(i)(τ2)-Wn(i)(τ)|2)=T1T2,
where
(3.20)T1=1nE(|∑t=[nτ1]+1[nτ]∑j=1dFt(i,j)ɛt(j)|2),T2=1nE(|∑t=[nτ]+1[nτ2]∑j=1dFt(i,j)ɛt(j)|2).
Now observe that
(3.21)T1=1n∑t,scov⁡(∑j=1dFt(i,j)ɛt(j),∑j=1dFs(i,j)ɛs(j))=1n∑t=[nτ1]+1[nτ]∑j=1d(Ft(i,j))2≤C(τ-τ1)forsomeconstantC>0.
Likewise T2≤C(τ2-τ). Since (τ-τ1)(τ2-τ)≤(τ2-τ1)2/2, the inequality (3.18) holds with γ=α=2, F(t)=Ct/2.In order to prove Theorem3.1 we need also the following lemma.Lemma 3.3.
Assume that(Yt) is given by (1.1), where  (ɛt) is an i.i.d sequence of random vectors such that  E(ɛt)=0,  var(ɛt)=Id and that  (Γt) satisfies (2.5). Then under the null  H0, the empirical covariance of  Yt satisfies
(3.22)∑̂⟶a.s.∑,
where  →a.s. denotes the almost sure convergence.Proof.
LetWt=Γtɛt,ℱt=σ(ɛ1,…,ɛt) and for i,j fixed, 1≤i≤d,1≤j≤d,et=Wt(i)Wt(j)-E(Wt(i)Wt(j)∣ℱt-1).
Then(et) is a martingale difference sequence with respect to ℱt. Since et=Wt(i)Wt(j)-∑k=1dΓt(i,k)Γt(j,k) and the matrix Γt is bounded, it follows that
(3.23)E(|et|(2+δ)/2)≤C+E(|Wt(i)Wt(j)|(2+δ)/2)≤E(|Wt(i)|2+δ)1/2E(|Wt(j)|2+δ)1/2≤C,
since by using Assumptions 1 and 2 we get
(3.24)E(|Wt(i)|2+δ)≤(∑k=1dE(|Γt(i,k)ɛt(k)|2+δ)1/(2+δ))2+δ≤C.
Therefore, Theorem 5 of Chow [19] implies that
(3.25)∑t=1net=o(n)almostsurely
or
(3.26)1n∑t=1nWt(i)Wt(j)=1n∑t=1n(ΓtΓt′)(i,j)+o(1)almostsurely,
where (ΓtΓt′)(i,j) denotes the (i,j)-th entry of the matrix ΓtΓt′. Hence
(3.27)1n∑t=1nWtWt′⟶a.s.∑.Lemma 2 of Lai and Wei [20], page 157, implies that with probability one(3.28)∑t=1nΓt(i,k)ɛt(k)=o(∑t=1n(Γt(i,k))2)+O(1)=o(n)+O(1)∀1≤i≤d,
or(3.29)1n∑t=1nΓt(i,k)ɛt(k)=o(1)+O(1n)almostsurely,
which implies that(3.30)1n∑t=1nWt=∑k=1d(1n∑t=1nΓt(1,k)ɛt(k),…,1n∑t=1nΓt(d,k)ɛt(k))′⟶a.s.0.
Note that Yt=μ+Wt, hence combining (3.27) and (3.30) we obtain(3.31)∑̂=1n∑t=1nYtYt′-Y¯Y¯′=μ1n∑t=1nWt+1n∑t=1nWtμ′+μμ′+1n∑t=1nWtWt′-Y¯Y¯′⟶a.s.∑.Proof of Theorem3.1.
Under the nullH0 we have Yt=μ+Γtɛt, thus recalling (3.3) we can write
(3.32)Bn(τ)=Γ̂-11n∑t=1[nτ](Yt-Y¯)=Γ̂-1Γ1nΓ-1∑t=1[nτ][(Yt-μ)-(Y¯-μ)]=Γ̂-1Γ(Wn(τ)-[nτ]nWn(1)).
Therefore the result (3.1) holds by applying Theorem 3.2, Lemma 3.3, and the continuous mapping theorem.


## 4. Consistency ofℬn

We assume that under the alternativeH1 the means (μt) are bounded and satisfy the following.Assumption H1.
There exists a functionU from [0,1] into ℝd such that
(4.1)∀τ∈[0,1],1n∑t=1[nτ]μt⟶U(τ)asn⟶∞.Assumption H2.
There existsτ*∈(0,1) such that
(4.2)U(τ*)=U(τ*)-τ*U(1)≠0.Assumption H3.
There exists∑μ such that
(4.3)1n∑t=1n(μt-μ¯)(μt-μ¯)′⟶∑μasn⟶∞,
where μ¯=(1/n)∑t=1nμt.Theorem 4.1.
Suppose that Assumptions1 and 2 hold. If  (Yt) is given by (1.1) and the means  (μt) satisfy the Assumptions H1, H2, and H3, then the test based on  ℬn is consistent against  H1, that is,
(4.4)Bn⟶P+∞,
where  →P denotes the convergence in probability.Proof.
We have(4.5)Bn(τ)=Bn0(τ)+Bn1(τ),
where
(4.6)Bn0(τ)=Γ̂-1n∑t=1[nτ](Wt-W¯),Wt=Γtɛt,W¯=1n∑t=1nWt,Bn1(τ)=Γ̂-1n∑t=1[nτ](μt-μ¯).
Straightforward computation leads to(4.7)∑̂⟶a.s.∑*=∑+∑μ.
Therefore
(4.8)Bn0(τ)⟶LΓ*-1ΓB(τ),
where Γ* is a square root of Σ*, that is, Σ*=Γ*Γ*′, and
(4.9)Bn1(τ)n⟶a.s.Γ*-1U(τ).
Hence
(4.10)||Bn(τ*)||∞⟶P+∞,
which implies that
(4.11)Bn⟶P+∞.


### 4.1. Consistency ofℬn against Abrupt Change

Without loss of generality we assume that under the alternative hypothesisH1 there is a single break date, that is, (Yt) is given by (1.1) where(4.12)μt={μ(1)if1≤t≤[nτ1]μ(2)if[nτ1]+1≤t≤nforsomeτ1∈(0,1)andμ(1)≠μ(2).Corollary 4.2.
Suppose that Assumptions1 and 2 hold. If  (Yt) is given by (1.1) and the means  (μt) 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
(4.13)U(τ)={τ(1-τ1)(μ(1)-μ(2))ifτ≤τ1τ1(1-τ1)(μ(1)-μ(2))ifτ>τ1,∑μ=τ1(μ(1)-μ(2))(μ(1)-μ(2))′.
Note that (4.2) is satisfied for all 0<τ*<τ1 since μ(1)≠μ(2).Remark 4.3.
The result of Corollary4.2 remains valid if under the alternative hypothesis there are multiple breaks in the mean.


### 4.2. Consistency ofℬn 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 Teräsvirta [21]), in which the mean μt is time varying as follows(4.14)μt=μ(1)+(μ(2)-μ(1))F(tn,τ1,γ),1≤t≤n,μ(1)≠μ(2),
where F(x,τ1,γ) is a the smooth transition function assumed to be continuous from [0,1] into [0,1], μ(1) and μ(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 functionF are frequently evoked, the logistic function given by(4.15)FL(x,τ1,γ)=[1+exp⁡(-γ(x-τ1))]-1,
and the exponential one(4.16)Fe(x,τ1,γ)=1-exp⁡(-γ(x-τ1)2).
For example, for the logistic function with γ>0, the extreme regimes are obtained as follows:(i)
ifx→0 and γ large then F→0 and thus μt=μ(1),(ii)
ifx→1 and γ large then F→1 and thus μt=μ(2).This means that at the beginning of the sampleμt is close to μ(1) and then moves towards μ(2) and becomes close to it at the end of the sample.Corollary 4.4.
Suppose that Assumptions1 and 2 hold. If  (Yt) is given by (1.1) and the means  (μt) satisfy (4.14), then the test based on  ℬn is consistent against  H1.Proof.
The assumptions (4.1) and (4.3) are satisfied with
(4.17)U(τ)=(μ(2)-μ(1))T(τ),
where
(4.18)T(τ)=∫0τF(x,τ1,γ)dx-τ∫01F(x,τ1,γ)dx,∑μ=(μ(2)-μ(1))(μ(2)-μ(1))′{∫01F2(x,τ1,γ)dx-(∫01F(x,τ1,γ)dx)2}.
Sinceμ(2)-μ(1)≠0, to prove (4.2), it suffices to show that there exists τ* such T(τ*)≠0.
Assume thatT(τ)=0 for all τ∈(0,1) then
(4.19)dT(τ)dτ=F(τ,τ1,γ)-∫01F(x,τ1,γ)dx=0∀τ∈(0,1),
which implies that F(τ,τ1,γ)=∫01F(x,τ1,γ)dx=C for all τ∈(0,1) or
(4.20)μt=μ(1)+(μ(2)-μ(1))C=μ∀t≥1,
and this contradicts the alternative hypothesis H1.


### 4.3. Consistency ofℬn against Continuous Change

In this subsection we will examine the behaviourℬn under the alternative where the mean (μt) varies at each time, and hence can take an infinite number of values. As an example we consider a polynomial evolution for μt:(4.21)μt=(P1(tn),…,Pd(tn))′,Pj(x)=∑k=0pjαj,kxk,1≤j≤d.Corollary 4.5.
Suppose that Assumptions1 and 2 hold. If  (Yt) is given by (1.1) and the means  (μt) satisfy (4.21), then the test based on  ℬn is consistent against  H1.Proof.
The assumptions H1–H3 are satisfied with
(4.22)U(τ)=(∑k=0p1α1,kk+1(τk+1-τ),…,∑k=0pdαd,kk+1(τk+1-τ))′,∑μ(i,j)=∑k=0pi∑l=0pj1l+k+1αi,kαj,l-(∑k=0piαi,kk+1)(∑k=0pjαj,kk+1).
Note that (4.2) is satisfied for all 0<τ*<1, provided that there exist i,1≤i≤d and k,1≤k≤pi such that αi,k≠0.


## 4.1. Consistency ofℬn against Abrupt Change

Without loss of generality we assume that under the alternative hypothesisH1 there is a single break date, that is, (Yt) is given by (1.1) where(4.12)μt={μ(1)if1≤t≤[nτ1]μ(2)if[nτ1]+1≤t≤nforsomeτ1∈(0,1)andμ(1)≠μ(2).Corollary 4.2.
Suppose that Assumptions1 and 2 hold. If  (Yt) is given by (1.1) and the means  (μt) 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
(4.13)U(τ)={τ(1-τ1)(μ(1)-μ(2))ifτ≤τ1τ1(1-τ1)(μ(1)-μ(2))ifτ>τ1,∑μ=τ1(μ(1)-μ(2))(μ(1)-μ(2))′.
Note that (4.2) is satisfied for all 0<τ*<τ1 since μ(1)≠μ(2).Remark 4.3.
The result of Corollary4.2 remains valid if under the alternative hypothesis there are multiple breaks in the mean.


## 4.2. Consistency ofℬn 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 Teräsvirta [21]), in which the mean μt is time varying as follows(4.14)μt=μ(1)+(μ(2)-μ(1))F(tn,τ1,γ),1≤t≤n,μ(1)≠μ(2),
where F(x,τ1,γ) is a the smooth transition function assumed to be continuous from [0,1] into [0,1], μ(1) and μ(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 functionF are frequently evoked, the logistic function given by(4.15)FL(x,τ1,γ)=[1+exp⁡(-γ(x-τ1))]-1,
and the exponential one(4.16)Fe(x,τ1,γ)=1-exp⁡(-γ(x-τ1)2).
For example, for the logistic function with γ>0, the extreme regimes are obtained as follows:(i)
ifx→0 and γ large then F→0 and thus μt=μ(1),(ii)
ifx→1 and γ large then F→1 and thus μt=μ(2).This means that at the beginning of the sampleμt is close to μ(1) and then moves towards μ(2) and becomes close to it at the end of the sample.Corollary 4.4.
Suppose that Assumptions1 and 2 hold. If  (Yt) is given by (1.1) and the means  (μt) satisfy (4.14), then the test based on  ℬn is consistent against  H1.Proof.
The assumptions (4.1) and (4.3) are satisfied with
(4.17)U(τ)=(μ(2)-μ(1))T(τ),
where
(4.18)T(τ)=∫0τF(x,τ1,γ)dx-τ∫01F(x,τ1,γ)dx,∑μ=(μ(2)-μ(1))(μ(2)-μ(1))′{∫01F2(x,τ1,γ)dx-(∫01F(x,τ1,γ)dx)2}.
Sinceμ(2)-μ(1)≠0, to prove (4.2), it suffices to show that there exists τ* such T(τ*)≠0.
Assume thatT(τ)=0 for all τ∈(0,1) then
(4.19)dT(τ)dτ=F(τ,τ1,γ)-∫01F(x,τ1,γ)dx=0∀τ∈(0,1),
which implies that F(τ,τ1,γ)=∫01F(x,τ1,γ)dx=C for all τ∈(0,1) or
(4.20)μt=μ(1)+(μ(2)-μ(1))C=μ∀t≥1,
and this contradicts the alternative hypothesis H1.


## 4.3. Consistency ofℬn against Continuous Change

In this subsection we will examine the behaviourℬn under the alternative where the mean (μt) varies at each time, and hence can take an infinite number of values. As an example we consider a polynomial evolution for μt:(4.21)μt=(P1(tn),…,Pd(tn))′,Pj(x)=∑k=0pjαj,kxk,1≤j≤d.Corollary 4.5.
Suppose that Assumptions1 and 2 hold. If  (Yt) is given by (1.1) and the means  (μt) satisfy (4.21), then the test based on  ℬn is consistent against  H1.Proof.
The assumptions H1–H3 are satisfied with
(4.22)U(τ)=(∑k=0p1α1,kk+1(τk+1-τ),…,∑k=0pdαd,kk+1(τk+1-τ))′,∑μ(i,j)=∑k=0pi∑l=0pj1l+k+1αi,kαj,l-(∑k=0piαi,kk+1)(∑k=0pjαj,kk+1).
Note that (4.2) is satisfied for all 0<τ*<1, provided that there exist i,1≤i≤d and k,1≤k≤pi such that αi,k≠0.


## 5. Finite Sample Performance

All models are driven from an i.i.d. sequencesɛt=(ɛt(1),…,ɛt(d))′, where each ɛt(j),1≤j≤d, has a t(3) distribution, a Student distribution with 3 degrees of freedom, and ɛt(i) and ɛt(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 statisticℬn we consider two bivariate models Yt=μt+Γtɛt with the following.Model 1 (constant covariance).
(5.1)μt=(11),Γt=(2112).Model 2 (time varying covariance).
(5.2)μt=(11),Γt=(2sin⁡(tω)-1-12cos⁡(tω)),ω=π4.From Table1, 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.Table 1
Empirical sizes (%).
αn=30n=100n=5001%0.20.30.4Model 15%2.12.9410%4.97.38.91%0.00.20.3Model 25%1.12.73.010%2.96.47.3


### 5.2. Study of the Power

In order to see the power of the test statisticℬn we consider five bivariate models Yt=μt+Γtɛt with the following.


#### 5.2.1. Abrupt Change in the Mean

Model 3 (constant covariance).
(5.3)μt={(01)if1≤t≤[n2](10)if[n2]+1≤t≤n,Γt=(2112).Model 4.
In this model the mean and the covariance are subject to an abrupt change at the same time:(5.4)μt={(01)if1≤t≤[n2](10)if[n2]+1≤t≤nΓt={(1001)if1≤t≤[n2](2102)if[n2]+1≤t≤n.Model 5.
The mean is subject to an abrupt change and the covariance is time varying (see Figure1):
(5.5)μt={(01)if1≤t≤[n2](10)if[n2]+1≤t≤n,Γt=(2sin⁡(tω)-1-12cos⁡(tω)),ω=π4.Figure 1
The three kinds of change in the mean.


#### 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 Figure1):
(5.6)μt=μ(1)+(μ(2)-μ(1))(1+exp⁡(-30(t/n-1/2))),1≤t≤n,μ(1)=(01),μ(2)=(10),Γt=(2sin⁡(tω)-1-12cos⁡(tω)),ω=π4.


#### 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 Models5 and 6 (see Figure 1):
(5.7)μt=(tn(2-tn)tn(2-tn)),Γt=(2sin⁡(tω)-1-12cos⁡(tω)),ω=π4.From Table2, 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.Table 2
Empirical powers (%).
αn=30n=100n=5001%11.481.8100Model 35%34.192.110010%46.995.11001%3.949.899.9Model 45%15.771.499.910%2980.41001%1.422.795.9Model 55%8.344.798.410%15.956.599.31%1.417.294Model 65%8.538.597.710%16.351.998.71%0.15.344.2Model 75%2.216.170.010%5.925.579.4


## 5.1. Study of the Size

In order to evaluate the size distortion of the test statisticℬn we consider two bivariate models Yt=μt+Γtɛt with the following.Model 1 (constant covariance).
(5.1)μt=(11),Γt=(2112).Model 2 (time varying covariance).
(5.2)μt=(11),Γt=(2sin⁡(tω)-1-12cos⁡(tω)),ω=π4.From Table1, 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.Table 1
Empirical sizes (%).
αn=30n=100n=5001%0.20.30.4Model 15%2.12.9410%4.97.38.91%0.00.20.3Model 25%1.12.73.010%2.96.47.3


## 5.2. Study of the Power

In order to see the power of the test statisticℬn we consider five bivariate models Yt=μt+Γtɛt with the following.


### 5.2.1. Abrupt Change in the Mean

Model 3 (constant covariance).
(5.3)μt={(01)if1≤t≤[n2](10)if[n2]+1≤t≤n,Γt=(2112).Model 4.
In this model the mean and the covariance are subject to an abrupt change at the same time:(5.4)μt={(01)if1≤t≤[n2](10)if[n2]+1≤t≤nΓt={(1001)if1≤t≤[n2](2102)if[n2]+1≤t≤n.Model 5.
The mean is subject to an abrupt change and the covariance is time varying (see Figure1):
(5.5)μt={(01)if1≤t≤[n2](10)if[n2]+1≤t≤n,Γt=(2sin⁡(tω)-1-12cos⁡(tω)),ω=π4.Figure 1
The three kinds of change in the mean.


### 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 Figure1):
(5.6)μt=μ(1)+(μ(2)-μ(1))(1+exp⁡(-30(t/n-1/2))),1≤t≤n,μ(1)=(01),μ(2)=(10),Γt=(2sin⁡(tω)-1-12cos⁡(tω)),ω=π4.


### 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 Models5 and 6 (see Figure 1):
(5.7)μt=(tn(2-tn)tn(2-tn)),Γt=(2sin⁡(tω)-1-12cos⁡(tω)),ω=π4.From Table2, 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.Table 2
Empirical powers (%).
αn=30n=100n=5001%11.481.8100Model 35%34.192.110010%46.995.11001%3.949.899.9Model 45%15.771.499.910%2980.41001%1.422.795.9Model 55%8.344.798.410%15.956.599.31%1.417.294Model 65%8.538.597.710%16.351.998.71%0.15.344.2Model 75%2.216.170.010%5.925.579.4


## 5.2.1. Abrupt Change in the Mean

Model 3 (constant covariance).
(5.3)μt={(01)if1≤t≤[n2](10)if[n2]+1≤t≤n,Γt=(2112).Model 4.
In this model the mean and the covariance are subject to an abrupt change at the same time:(5.4)μt={(01)if1≤t≤[n2](10)if[n2]+1≤t≤nΓt={(1001)if1≤t≤[n2](2102)if[n2]+1≤t≤n.Model 5.
The mean is subject to an abrupt change and the covariance is time varying (see Figure1):
(5.5)μt={(01)if1≤t≤[n2](10)if[n2]+1≤t≤n,Γt=(2sin⁡(tω)-1-12cos⁡(tω)),ω=π4.Figure 1
The three kinds of change in the mean.


## 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 Figure1):
(5.6)μt=μ(1)+(μ(2)-μ(1))(1+exp⁡(-30(t/n-1/2))),1≤t≤n,μ(1)=(01),μ(2)=(10),Γt=(2sin⁡(tω)-1-12cos⁡(tω)),ω=π4.


## 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 Models5 and 6 (see Figure 1):
(5.7)μt=(tn(2-tn)tn(2-tn)),Γt=(2sin⁡(tω)-1-12cos⁡(tω)),ω=π4.From Table2, 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.Table 2
Empirical powers (%).
αn=30n=100n=5001%11.481.8100Model 35%34.192.110010%46.995.11001%3.949.899.9Model 45%15.771.499.910%2980.41001%1.422.795.9Model 55%8.344.798.410%15.956.599.31%1.417.294Model 65%8.538.597.710%16.351.998.71%0.15.344.2Model 75%2.216.170.010%5.925.579.4



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*Source: 969753-2012-02-29.xml*