Change-point methods for multivariate time-series: paired vectorial observations

Abstract

We consider paired and two-sample break-detection procedures for vectorial observations and multivariate time series. The new methods involve L2-type criteria based on empirical characteristic functions and are easy to compute regardless of dimension. We obtain asymptotic results that allow for application of the methods to a wide range of settings involving on-line as well as retrospective circumstances with dependence between the two time series as well as with dependence within each series. In the ensuing Monte Carlo study the new detection methods are implemented by means of resampling procedures which are properly adapted to the type of data at hand, be it independent or paired, autoregressive or GARCH structured, medium or heavy-tailed. The new methods are also applied on a real dataset from the financial sector over a time period which includes the Brexit referendum.

Appendix

This section contains the proofs of the assertions formulated in Sect. 3. We start with a useful auxiliary relationship and auxiliary lemmas. D denotes a generic constant which may vary from line to line.

Direct calculations give for $t=1,\ldots ,n$

$$\begin{aligned}&\int _{{\mathbb {R}}^p}|{\widehat{\varphi }}_{x,t}({\varvec{u}})- {\widehat{\varphi }}_{y,t}({\varvec{u}})|^2 W({\varvec{u}}) d {\varvec{u}} \\&\quad = \frac{1}{t^2} \int _{{\mathbb {R}}^p}\Big |\sum _{\tau =1}^t\{\exp (i {\varvec{u}}^\top {\varvec{x}}_{\tau })- \exp (i {\varvec{u}}^\top {\varvec{y}}_{\tau })\}\Big |^2 W({\varvec{u}}) d{\varvec{u}} \\&\quad = \frac{1}{t^2} \int _{{\mathbb {R}}^p}\Big ( Z_T(t/N, {\varvec{u}}) \Big )^2 W({\varvec{u}}) d {\varvec{u}}, \end{aligned}$$

where

$$\begin{aligned} Z_T(s, {\varvec{u}})&= \frac{1}{\sqrt{T}}\sum _{\tau =1}^{\lfloor sT\rfloor } h_\tau ({\varvec{u}}),\quad s \in (0,1],\, {\varvec{u}}\in {\mathbb {R}}^p,\\ h_\tau ({\varvec{u}})&= \cos ({\varvec{u}}^\top \varvec{x}_{\tau }) + \sin ({\varvec{u}}^\top {\varvec{x}}_{\tau }) -\{\cos ({\varvec{u}}^\top {\varvec{y}}_{\tau }) + \sin (\varvec{u}^\top {\varvec{y}}_{\tau })\} \end{aligned}$$

Next lemma summarizes needed assertions on partial sums of $\alpha$-mixing random variables.

Lemma 1

Let sequence $\{({\varvec{x}}_t,{\varvec{y}}_t),\, t=1,\ldots \}$ be 2p-dimensional strictly stationary $\alpha$-mixing with coefficients $\alpha (j)$ and let $E h_t({\varvec{u}})=0$ then for arbitrary ${\varvec{u}}\in {\mathbb {R}}^p$ and any $\xi >0$

$$\begin{aligned} E\Big | \sum _{t=1}^T h_t({\varvec{u}}) \Big |^2\le D T \Big ( E| h_1({\varvec{u}})|^{2+\xi }\Big )^{2/(2+\xi )} \sum _{j=1}^{\infty }(\alpha (j))^{\xi /(2+\xi )} \end{aligned}$$

and for any $b_1\ge \cdots \ge b_T>0$

$$\begin{aligned}&E\Big | \max _{1\le j\le T} b_j\sum _{t=1}^j h_t({\varvec{u}}) \Big |^{2}\le \\&D (\log T)^2 T^{(2+\delta )2} \Big ( E| h_1(\varvec{u})|^{2+\xi }\Big )^{2/(2+\xi )} \sum _{j=1}^{\infty }(\alpha (j))^{\xi /(2+\xi )} \end{aligned}$$

The inequalities remain true if $h_t({\varvec{u}})$’s is replaced by $h_t({\varvec{u}}_1)- h_t({\varvec{u}}_2)$. If additionally

$$\begin{aligned} 0<\sigma ^2= \mathrm{{var}}\{ h_1({\varvec{u}})\} + 2\sum _{j=1}^{\infty }\mathrm{{cov}}\{ h_1({\varvec{u}}),\, h_{j+1}({\varvec{u}})\}<\infty \end{aligned}$$

(25)

then for $T\rightarrow \infty$ the limit distribution of $\frac{1}{\sqrt{T}} \sum _{t=1}^T h_t({\varvec{u}})$ is asymptotically normal with zero mean and variance $\sigma ^2$.

(b) Let assumption (A.1) and (25) be satisfied then

$$\begin{aligned} E\Big | \sum _{t=1}^T h_t({\varvec{u}}) \Big |^{2+\delta }\le D T^{2+\delta } \Big ( E| h_1(\varvec{u})|^{2+\xi }\Big )^{2/(2+\delta )} \end{aligned}$$

and

$$\begin{aligned} E\Big |\max _{1\le t\le T} \sum _{\ell =1}^t h_t({\varvec{u}}) \Big |^{2+\delta }\le D T^{(2+\delta )2} \Big ( E| h_1(\varvec{u})|^{2+\xi }\Big )^{2/(2+\delta )} \end{aligned}$$

Proof

It follows from classical results on $\alpha$-mixing sequences of random variables, see e.g. Yokoyama (1980) and Doukhan (1994). $\square$

The following Lemma is substantial for the proof of Theorem 1.

Lemma 2

Let assumptions (A.1)–(A.2) and (25) be satisfied and let the null hypothesis hold true. Then

(a) for any compact subset F of ${\mathbb {R}}^p$ it holds

$$\begin{aligned} \sup _T E \int _F Z^2_T(s, {\varvec{u}}) W ({\varvec{u}}) d{\varvec{u}} < \infty , \end{aligned}$$

(b) there exist $a>0$ and $0<C<\infty$ such that for any $0\le s\le 1$ and any ${\varvec{u}}_1$ and ${\varvec{u}}_2$ it holds

$$\begin{aligned} \sup _T E \Big | Z_T(s, {\varvec{u}}_1)^2 -Z_T(s, \varvec{u}_2)^2\Big |\le Cx \Vert {\varvec{u}}_1-{\varvec{u}}_2\Vert ^a, \end{aligned}$$

(c) the marginal distributions of $\{Z_T(s, {\varvec{u}})\}$ converge to the marginal distributions of the Gaussian process $\{Z(s, {\varvec{u}})\}$ with zero mean and covariance structure $(0\le s_1<s_2\le 1)$

$$\begin{aligned} { \mathrm{{cov}}\{ Z(s_1, {\varvec{u}}_1), Z(s_2, {\varvec{u}}_2)\}= } s_1 E\Big ( h_\tau ({\varvec{u}}_1), h_\tau ({\varvec{u}}_2) + \sum _{\ell =1}^{ \infty }2 E\big ( h_\tau ({\varvec{u}}_1) h_{\tau +\ell }({\varvec{u}}_2)\Big ). \end{aligned}$$

Proof

It follows the line of Lemma 7.1 in Hlávka et al. (2017), where the basic difference is the present paper works with functionals of partial sums of $\alpha$-mixing random vectors while in Hlávka et al. (2017) functionals of martingale differences are considered. In other words, instead of assertions on partial sums of martingale differences analogous results on partial sums of $\alpha$-mixing random vectors are applied.

By the assumptions for each ${\varvec{u}}$ the sequence $\{ h_\tau ({\varvec{u}})\}_{\tau }$ is $\alpha$-mixing with the same coefficient as the sequence $\{({\varvec{x}}_\tau , \varvec{y}_{\tau }\}_{\tau }$. Moreover,

$$\begin{aligned} E Z(s, {\varvec{u}})&=0, \\ E Z^2(s, {\varvec{u}})&=\frac{\lfloor Ts\rfloor }{T} \Big (E h^2_1({\varvec{u}}) + 2 \sum _{\ell =1}^{\infty } E h_1({\varvec{u}}) h_{\ell +1}({\varvec{u}})\Big ) \end{aligned}$$

By the assumptions $E Z^2(s, {\varvec{u}})$ is uniformly bounded and therefore (a) holds true.

Concerning assertion (b) notice that

$$\begin{aligned} E\big |&Z^2_T(s, {\varvec{u}}_1) -Z^2_T(s, {\varvec{u}}_2)\big | \le D \left\{ E\left| Z_T(s, {\varvec{u}}_1) -Z_T(s, \varvec{u}_2)\right| ^2\right\} ^{1/2} \\&\le D\left[ E \left\{ h_1({\varvec{u}}_1)- h_1(\varvec{u}_2)\right\} ^2\right] ^{1/2} \le D \Vert {\varvec{u}}_1- \varvec{u}_2\Vert ^a \end{aligned}$$

for some $a>0$. We used the moment inequalities for $\alpha$-mixing sequences. Particularly, we used inequalities for $\alpha$-mixing.

Concerning (c) it is a consequence of Lemma 1. $\square$

Proof

(Theorem 1) From Lemma 7.1 in Hlávka et al. (2017) in addition to Theorem 22 in Ibragimov and Chasminskij (1981, pp. 380, 381) and Lemma 2 we get that for each $s\in (0,1]$ as $T\rightarrow \infty$

$$\begin{aligned} \int _F Z_T^2 (s, {\varvec{u}}) W({\varvec{u}}) d{\varvec{u}} {\mathop {\rightarrow }\limits ^{{\mathcal {L}}}} \int _F V^2 (s, {\varvec{u}}) W({\varvec{u}}) d{\varvec{u}} \end{aligned}$$

for any compact $F\subset {\mathbb {R}}^p$. Since $W(\cdot )$ is integrable and $EZ^2_T(s,{\varvec{u}})\le D$ then

$$\begin{aligned} \int _{{\mathbb {R}}^p} Z_T^2 (s, {\varvec{u}}) W({\varvec{u}}) d{\varvec{u}} {\mathop {\rightarrow }\limits ^{{\mathcal {L}}}} \int _{{\mathbb {R}}^p} V^2 (s, {\varvec{u}}) W({\varvec{u}}) d{\varvec{u}} \end{aligned}$$

which proves (a).

Next we have to study limit behavior of the process

$$\begin{aligned} S_T(s)=\sqrt{\int _{{\mathbb {R}}^p} Z_T^2 (s, {\varvec{u}}) W({\varvec{u}}) d{\varvec{u}}}\quad s\in (0,1] \end{aligned}$$

Convergence finite dimensional distributions follow from Lemma 1 in combination with continuous mapping theorem. To obtain tightness by the Minkowski inequality we have

$$\begin{aligned} |S_T(s_1) - S_T(s_2) |\le D\sqrt{\int |Z_T(s_1,{\varvec{u}})- Z_T(s_2,{\varvec{u}})|^2 W({\varvec{u}}) d {\varvec{u}}} \end{aligned}$$

and further by the Jensen inequality for any $\varepsilon >0$

$$\begin{aligned}&P(|S_T(s_1) - S_T(s_1) |\ge \varepsilon )\\&\quad \le D E\Big (\int |Z_T(s_,{\varvec{u}})- Z_T(s_2,{\varvec{u}})|^2 W({\varvec{u}}) d{\varvec{u}}\Big )^{1+\delta /2} \varepsilon ^{-(1+\delta /2)}\\&\quad \le D \int E|Z_T(s_,{\varvec{u}})- Z_T(s_2,\varvec{u})|^{2+\delta } W({\varvec{u}}) d{\varvec{u}} \varepsilon ^{-(1+\delta /2)} \\&\quad \le D|s_1-s_2|^a \end{aligned}$$

Then by Billingsley (2013, Theorem 15.6) we conclude that

$$\begin{aligned} S_T(\cdot ) {\mathop {\rightarrow }\limits ^{D[0,1]}} \int _{{\mathbb {R}}^p} V^2 (\cdot , {\varvec{u}}) W({\varvec{u}}) d{\varvec{u}} \end{aligned}$$

which further implies that

$$\begin{aligned} \max _{s_1<s\le 1} S_T(s)s^{\eta -2}{\mathop {\rightarrow }\limits ^{{\mathcal {L}}}} \max _{s_1<s\le 1} \int _{{\mathbb {R}}^p} V^2 (\cdot , {\varvec{u}}) W({\varvec{u}}) d{\varvec{u}}. \end{aligned}$$

$\square$

Proof

(Theorem 2) It follows the line of Theorem 1 and therefore is omitted. $\square$

Proof

(Theorem 3) By inequalities in Lemma 2 we notice that for any $s\in (s_0,1]$

$$\begin{aligned} \int _{{\mathbb {R}}^p} \Big (Z_T (s, {\varvec{u}})-EZ_T (s, \varvec{u})\Big )^2 W({\varvec{u}}) d{\varvec{u}} =O_P(1) \end{aligned}$$

and by assumptions

$$\begin{aligned} \int _{{\mathbb {R}}^p} \Big (EZ_T (s, {\varvec{u}})\Big )^2 W({\varvec{u}}) d{\varvec{u}} \rightarrow \infty . \end{aligned}$$

The proof can be straightforwardly finished. $\square$

Proof

(Theorem 4): It is omitted since it is done in the same way as Theorem 3. $\square$