Changepoint in dependent and non-stationary panels

Abstract

Detection procedures for a change in means of panel data are proposed. Unlike classical inference tools used for the changepoint analysis in the panel data framework, we allow for mutually dependent and generally non-stationary panels with an extremely short follow-up period. Two competitive self-normalized test statistics are employed and their asymptotic properties are derived for a large number of available panels. The bootstrap extensions are introduced in order to handle such a universal setup. The novel changepoint methods are able to detect a common break point even when the change occurs immediately after the first time point or just before the last observation period. The developed tests are proved to be consistent. Their empirical properties are investigated through a simulation study. The invented techniques are applied to option pricing and non-life insurance.

Appendix: Proofs

Proof of Theorem 1

Firstly, we show that a multivariate CLT holds for a sequence of the T-dimensional $\alpha$-mixing random vectors $\{[\sum _{s=1}^1\varepsilon _{i,s},\ldots ,\sum _{s=1}^T\varepsilon _{i,s}]^{\top }\}_{i\in \mathbb {N}}\equiv \{\varvec{\theta }_i\}_{i\in \mathbb {N}}$. According to the Cramér-Wold theorem, it is sufficient to ensure that all assumptions of the one-dimensional $\alpha$-mixing CLT (Pešta 2013, Lemma 2.1) for triangular arrays are valid for any linear combination $\varvec{b}=[b_1,\ldots ,b_T]^{\top }\in \mathbb {R}^T$ of the elements of the random vector $\varvec{\theta }_i,\,i\in \mathbb {N}$. Hence, let us consider $\vartheta _i:=\varvec{b}^{\top }\varvec{\theta }_i$ and keep Assumption $\mathcal {A}1$ in mind. Then, we get

$$\begin{aligned} \frac{\mathsf {E}\left( \sum _{i=1}^{N}\vartheta _i\right) ^2}{N}=\frac{1}{N}\mathsf {Var}\,\sum _{i=1}^N\varvec{b}^{\top }\varvec{\theta }_i=\varvec{b}^{\top }\frac{\varvec{\Lambda }_N}{N}\varvec{b}\rightarrow \varvec{b}^{\top }\varvec{\Lambda }\varvec{b},\quad N\rightarrow \infty , \end{aligned}$$

(6)

where $\varvec{\Lambda }_N=\mathsf {Var}\,\sum _{i=1}^{N}[\sum _{s=1}^1\varepsilon _{i,s},\ldots ,\sum _{s=1}^T\varepsilon _{i,s}]^{\top }$. Furthermore,

$$\begin{aligned} \sup _{i\in \mathbb {N}}\mathsf {E}\left| \vartheta _i\right| ^{2+\chi }&\le \sup _{i\in \mathbb {N}}\left( \max _{t=1,\ldots ,T}|b_t|^{2+\chi }\right) \sum _{t=1}^T\mathsf {E}\left| \theta _{t,i}\right| ^{2+\chi }\nonumber \\&\le T\left( \max _{t=1,\ldots ,T}|b_t|^{2+\chi }\right) \max _{t=1,\ldots ,T}\sup _{i\in \mathbb {N}}\mathsf {E}\left| \sum _{s=1}^t\varepsilon _{i,s}\right| ^{2+\chi }\nonumber \\&\le T^2\left( \max _{t=1,\ldots ,T}|b_t|^{2+\chi }\right) \max _{t=1,\ldots ,T}\sup _{i\in \mathbb {N}}\mathsf {E}\left| \varepsilon _{i,t}\right| ^{2+\chi }<\infty . \end{aligned}$$

(7)

The $\sigma$-algebra generated by $\vartheta _i$ is contained in the $\sigma$-algebra generated by $\varvec{\varepsilon }_i$; consequently for all $I\subseteq \mathbb {N}$ (possibly infinite), $\sigma \{\vartheta _i:\,i\in I\}\subseteq \sigma \{\varvec{\varepsilon }_i:\,i\in I\}$. For this reason, for all $i\in \mathbb {N}$, $\alpha (\vartheta _{\circ },i)\le \alpha (\varvec{\varepsilon }_{\circ },i)$. One obtains

$$\begin{aligned} \sum _{i=1}^{\infty }\alpha (\vartheta _{\circ },i)^{\chi /(2+\chi )}\le \sum _{i=1}^{\infty }\alpha (\varvec{\varepsilon }_{\circ },i)^{\chi /(2+\chi )}<\infty ,\quad \chi >0. \end{aligned}$$

(8)

Relations (6)–(8) imply

(9)

Let us define

$$\begin{aligned} U_N(t):=\frac{1}{\sqrt{N}}\sum _{i=1}^N\sum _{s=1}^t(Y_{i,s}-\mu _i). \end{aligned}$$

Under $\mathcal {H}_0$ and according to (9), we have

Moreover, let us define the reverse analogue to $U_N(t)$, i.e.,

$$\begin{aligned} V_N(t):=\frac{1}{\sqrt{N}}\sum _{i=1}^N\sum _{s=t+1}^T(Y_{i,s}-\mu _i)=U_N(T)-U_N(t). \end{aligned}$$

Hence,

$$\begin{aligned} U_{N}(s)-\frac{s}{t}U_{N}(t)&=\frac{1}{\sqrt{N}}\sum _{i=1}^N\left\{ \sum _{r=1}^s\left[ \left( Y_{i,r}-\mu _i\right) -\frac{1}{t}\sum _{v=1}^t \left( Y_{i,v}-\mu _i\right) \right] \right\} \\&=\frac{1}{\sqrt{N}}\sum _{i=1}^N\sum _{r=1}^s\left( Y_{i,r}-\bar{Y}_{i,t}\right) \end{aligned}$$

and, consequently,

$$\begin{aligned}&V_{N}(s)-\frac{T-s}{T-t}V_{N}(t)\\&\quad =\frac{1}{\sqrt{N}}\sum _{i=1}^N\left\{ \sum _{r=s+1}^T\left[ \left( Y_{i,r}-\mu _i\right) -\frac{1}{T-t}\sum _{v=t+1}^T \left( Y_{i,v}-\mu _i\right) \right] \right\} \\&\quad =\frac{1}{\sqrt{N}}\sum _{i=1}^N\sum _{r=s+1}^T\left( Y_{i,r}-\widetilde{Y}_{i,t}\right) . \end{aligned}$$

Using the continuous mapping theorem, we end up with

and

$\square$

Proof of Theorem 2

Let us define $L_N(s,t):=\sum _{i=1}^N\sum _{r=1}^s\big (\varepsilon _{i,r}-\bar{\varepsilon }_{i,t}\big )$ and $R_N(s,t):=\sum _{i=1}^N\sum _{r=s+1}^T\big (\varepsilon _{i,r}-\widetilde{\varepsilon }_{i,t}\big )$ such that $\bar{\varepsilon }_{i,t}=\frac{1}{t}\sum _{s=1}^t \varepsilon _{i,s}$ and $\widetilde{\varepsilon }_{i,t}=\frac{1}{T-t}\sum _{s=t+1}^T \varepsilon _{i,s}$. With respect to Assumption $\mathcal {A}1$ and according to the underlying proof of Theorem 1, we have

$$\begin{aligned} \frac{1}{\sqrt{N}}\max _{s=1,\ldots ,\tau }|L_N(s,\tau )|&=\mathcal {O}_{\mathsf {P}}(1);\\ \frac{1}{\sqrt{N}}\max _{s=\tau ,\ldots ,T-1}|R_N(s,\tau )|&=\mathcal {O}_{\mathsf {P}}(1);\\ \frac{1}{\sqrt{N}}|L_N(\tau ,T)|&=\mathcal {O}_{\mathsf {P}}(1), \end{aligned}$$

as $N\rightarrow \infty$. Note that there are no changes in the expectation of $Y_{i,1},\ldots ,Y_{i,\tau }$ as well as in the expectation of $Y_{i,\tau +1},\ldots ,Y_{i,T}$. Let $t=\tau$. Then, under $\mathcal {H}_A$,

according to Assumption $\mathcal {A}2$.

Similarly for $\mathcal {S}_N(T)$, we get that

$$\begin{aligned} {{\,\mathrm{plim}\,}}_{N\rightarrow \infty }\mathcal {S}_N(T)&\ge {{\,\mathrm{plim}\,}}_{N\rightarrow \infty }\frac{\sum _{t=1}^{T-1}\mathcal {L}_N^2(\tau ,T)}{\sum _{s=1}^{\tau }\mathcal {L}_N^2(s,\tau )+\sum _{s=\tau }^{T-1}\mathcal {R}_N^2(s,\tau )}\\&={{\,\mathrm{plim}\,}}_{N\rightarrow \infty }\frac{N^{-1}\sum _{t=1}^{T-1}\left( L_N(\tau ,T)-\frac{\tau }{T}(T-\tau )\sum _{i=1}^N\delta _i\right) ^2}{N^{-1}\sum _{s=1}^{\tau }L_N^2(s,\tau )+N^{-1}\sum _{s=\tau }^{T-1}R_N^2(s,\tau )}=\infty , \end{aligned}$$

where ${{\,\mathrm{plim}\,}}_{N\rightarrow \infty }$ is the probability limit operator corresponding to convergence in probability. The latter mentioned relation holds again because there are no changes in the means of $Y_{i,1},\ldots ,Y_{i,\tau }$ as well as in the means of $Y_{i,\tau +1},\ldots ,Y_{i,T}$ and due to Assumption $\mathcal {A}2$. $\square$

Proof of Theorem 3

Let us define $\widehat{\epsilon }_{i,t}:=\sum _{s=1}^t\widehat{e}_{i,s}$, $\widehat{\epsilon }_{i,t}^{(b)}:=\sum _{s=1}^t\widehat{e}_{i,s}^{(b)}$,

$$\begin{aligned} \widehat{U}_N(t):=\frac{1}{\sqrt{N}}\sum _{i=1}^N\sum _{s=1}^t\widehat{e}_{i,s}=\frac{1}{\sqrt{N}}\sum _{i=1}^N\widehat{\epsilon }_{i,t}, \end{aligned}$$

and

$$\begin{aligned} \widehat{U}_N^{(b)}(t)&:=\frac{1}{\sqrt{N}}\sum _{i=1}^N\sum _{s=1}^t\widehat{Y}_{i,s}^{(b)}=\frac{1}{\sqrt{N}}\sum _{i=1}^N\sum _{s=1}^t\left( \widehat{e}_{i,s}^{(b)}-\frac{1}{N}\sum _{i=1}^N\widehat{e}_{i,s}\right) \\&=\frac{1}{\sqrt{N}}\sum _{i=1}^N\sum _{s=1}^t\left( \widehat{e}_{i,s}^{(b)}-\widehat{e}_{i,s}\right) =\frac{1}{\sqrt{N}}\sum _{i=1}^N\left( \widehat{\epsilon }_{i,t}^{(b)}-\widehat{\epsilon }_{i,t}\right) . \end{aligned}$$

Realize that $\widehat{\epsilon }_{i,t}$ depends on $\widehat{\tau }_N$ and, hence, it depends on N. Thus, $\widehat{\epsilon }_{i,t}\equiv \widehat{\epsilon }_{i,t}(N)$. Since Assumption $\mathcal {B}1$ holds, then according to the bootstrap multivariate CLT for $\alpha$-mixing triangular arrays of T-dimensional vectors $\varvec{\xi }_{N,i}=[\widehat{\epsilon }_{i,1}(N),\ldots ,\widehat{\epsilon }_{i,T}(N)]^{\top }$ with $k_N=N$ by Pešta(2017, minor modification of Theorem A.5), we have

where $\widehat{\varvec{\Gamma }}_N=\frac{1}{N}\mathsf {Var}\,\sum _{i=1}^N[\widehat{\epsilon }_{i,1},\ldots ,\widehat{\epsilon }_{i,T}]^{\top }$.

Now, it is sufficient to realize that $[\widehat{U}_N(1),\ldots ,\widehat{U}_N(T)]^{\top }$ has approximately a multivariate normal distribution with zero mean and the covariance matrix $\lim _{N\rightarrow \infty }\widehat{\varvec{\Gamma }}_N$. Using the law of total variance,

$$\begin{aligned} \mathsf {Var}\,\widehat{\epsilon }_{i,t}&=\mathsf {E}[\mathsf {Var}\,\{\widehat{\epsilon }_{i,t}|\widehat{\tau }_N\}]+\mathsf {Var}\,[\mathsf {E}\{\widehat{\epsilon }_{i,t}|\widehat{\tau }_N\}]\\&=\sum _{\pi =1}^T\mathsf {P}[\widehat{\tau }_N=\pi ]\mathsf {Var}\,[\widehat{\epsilon }_{i,t}|\widehat{\tau }_N=\pi ]+\sum _{\pi =1}^T\mathsf {P}[\widehat{\tau }_N=\pi ]\{\mathsf {E}[\widehat{\epsilon }_{i,t}|\widehat{\tau }_N=\pi ]\}^2\\&\quad -\left\{ \sum _{\pi =1}^T\mathsf {P}[\widehat{\tau }_N=\pi ]\mathsf {E}[\widehat{\epsilon }_{i,t}|\widehat{\tau }_N=\pi ]\right\} ^2. \end{aligned}$$

Since $\lim _{N\rightarrow \infty }\mathsf {P}[\widehat{\tau }_N=\tau ]=1$ (Assumption $\mathcal {C}1$) and $\mathsf {E}[\widehat{e}_{i,t}|\widehat{\tau }_N=\tau ]=0$, then

$$\begin{aligned} \lim _{N\rightarrow \infty }\mathsf {Var}\,\widehat{\epsilon }_{i,t}=\lim _{N\rightarrow \infty }\mathsf {Var}\,[\widehat{\epsilon }_{i,t}|\widehat{\tau }_N=\tau ]. \end{aligned}$$

Similarly with the covariance, i.e., after applying the law of total covariance, we have

$$\begin{aligned} \lim _{N\rightarrow \infty }\mathsf {Cov}\,\left( \widehat{\epsilon }_{i,t},\widehat{\epsilon }_{i,v}\right) =\lim _{N\rightarrow \infty }\mathsf {Cov}\,\left( \widehat{\epsilon }_{i,t},\widehat{\epsilon }_{i,v}|\widehat{\tau }_N=\tau \right) . \end{aligned}$$

Note that

$$\begin{aligned} \left( \widehat{e}_{i,t}|\widehat{\tau }_N=\tau \right) =\left\{ \begin{array}{ll} \varepsilon _{i,t}-\bar{\varepsilon }_{i,\tau },&{} t\le \tau ;\\ \varepsilon _{i,t}-\widetilde{\varepsilon }_{i,\tau },&{} t>\tau . \end{array} \right. \end{aligned}$$

Taking into account the definitions of $e_{i,t}$’s from Assumption $\mathcal {B}1$, we get $\varvec{\Gamma }=\lim _{N\rightarrow \infty }\widehat{\varvec{\Gamma }}_N$.

Then $\mathcal {L}_N^{(b)}(t,T)$ from the numerators of $\mathcal {Q}_N^{(b)}(T)$ and $\mathcal {S}_N^{(b)}(T)$ can be alternatively rewritten as

$$\begin{aligned} \frac{1}{\sqrt{N}}\sum _{i=1}^N\sum _{r=1}^s\left( \widehat{Y}_{i,r}^{(b)}-\bar{\widehat{Y}}_{i,t}^{(b)}\right)= & {} \frac{1}{\sqrt{N}}\sum _{i=1}^N\left\{ \left[ \sum _{r=1}^s \widehat{Y}_{i,r}^{(b)}\right] -\frac{s}{t}\sum _{v=1}^t \widehat{Y}_{i,v}^{(b)}\right\} \\= & {} \widehat{U}_{N}^{(b)}(s)-\frac{s}{t}\widehat{U}_{N}^{(b)}(t). \end{aligned}$$

Concerning the denominators of $\mathcal {Q}_N^{(b)}(T)$ and $\mathcal {S}_N^{(b)}(T)$, one needs to perform a similar calculation as in the proof of Theorem 1 with $V_N(t)$, i.e., to define $\widehat{V}_N(t)$ and $\widehat{V}_N^{(b)}(t)$ analogously to $\widehat{U}_N(t)$ and $\widehat{U}_N^{(b)}(t)$ as $V_N(t)$ is to $U_N(t)$. Applying the continuous mapping theorem completes the proof. $\square$

Proof of Corollary 1

Recall the notation from the proof of Theorem 3. Under $\mathcal {H}_0$ and Assumption $\mathcal {C}1$, it holds that $\lim _{N\rightarrow \infty }\mathsf {P}[\widehat{\tau }_N=T]=1$. Then in view of (4),

$$\begin{aligned} \lim _{N\rightarrow \infty }\mathsf {P}\left[ \widehat{U}_N(s)-\frac{s}{t}\widehat{U}_N(t)=U_N(s)-\frac{s}{t}U_N(t)\right] =1,\quad 1\le s\le t\le T. \end{aligned}$$

$\square$