Monitoring memory parameter change-points in long-memory time series

Abstract

In this paper, we propose two ratio-type statistics to sequentially detect the memory parameter change-points in the long-memory time series. The limiting distributions of monitoring statistics under the no-change-point null hypothesis as well as their consistency under the alternative hypothesis are proved. In particular, a sieve bootstrap approximation method is proposed to determine the critical values. Extensive simulations indicate that the new monitoring procedures perform well in finite samples. Finally, we illustrate our monitoring procedures by two sets of real data.

Appendix

Proof of Theorem 3.1

Let $t=[Tr],\ n=[Ts],\ m=[T\tau ]$, then if $\gamma _t=1$, we have ${\hat{\varepsilon }}_{0,i}=X_i-[T\tau ]^{-1}\sum \nolimits _{j=1}^{[T\tau ]}X_j$, and

$$\begin{aligned} {\hat{\varepsilon }}_{1,i}=\left\{ \begin{array}{ll} X_i-[T\tau ]^{-1}\sum \limits _{j=[Ts]-[T\tau ]+1}^{[Ts]}X_j, &{}\quad [Ts]\le 2[T\tau ],\\ X_i-([Ts]-[T\tau ])^{-1}\sum \limits _{j=[T\tau ]+1}^{[Ts]}X_j,&{}\quad [Ts]> 2[T\tau ]. \end{array} \right. \end{aligned}$$

Then,

$$\begin{aligned} T^{-\frac{1}{2}-d_0}\sum \limits _{i=1}^t{\hat{\varepsilon }}_{0,i}= & {} T^{-\frac{1}{2}-d_0}\sum \limits _{i=1}^{[Tr]} X_i-\frac{[Tr]T^{-\frac{1}{2}-d_0}}{[T\tau ]} \sum \limits _{j=1}^{[T\tau ]}X_j\nonumber \\\Rightarrow & {} \kappa (B_{d_0}(r)-r\tau ^{-1} B_{d_0}(\tau )) \nonumber \\:= & {} \kappa U_{1,0}(d_0,r). \end{aligned}$$

(14)

$$\begin{aligned} T^{-\frac{1}{2}-d_0}\sum \limits _{i=t+1}^{n}{\hat{\varepsilon }}_{1,i}= & {} \left\{ \begin{array}{ll} T^{-\frac{1}{2}-d_0}\sum \limits _{i=[Tr]+1}^{[Ts]} X_i-\frac{[Ts]-[Tr]}{[T\tau ]T^{\frac{1}{2}+d_0}} \sum \limits _{j=[Ts]-[T\tau ]+1}^{[Ts]}X_j,&{}\quad n\le 2m,\nonumber \\ T^{-\frac{1}{2}-d_0}\sum \limits _{i=[Tr]+1}^{[Ts]} X_i-\frac{([Ts]-[Tr])T^{-\frac{1}{2}-d_0}}{[Ts]-[T\tau ]} \sum \limits _{j=[T\tau ]+1}^{[Ts]}X_j,&{}\quad n> 2m \end{array} \right. \nonumber \\\Rightarrow & {} \left\{ \begin{array}{ll} \kappa \left( B_{d_0}(s)-B_{d_0}(r)-\frac{s-r}{\tau }( B_{d_0}(s)-B_{d_0}(s-\tau ))\right) ,&{}\quad s\le 2\tau \\ \kappa \left( B_{d_0}(s)-B_{d_0}(r)-\frac{s-r}{s-\tau }( B_{d_0}(s)-B_{d_0}(\tau ))\right) ,&{}\quad s> 2\tau \end{array} \right. \nonumber \\:= & {} \left\{ \begin{array}{ll} \kappa U_{1,1}(d_0,r),\\ \kappa U_{1,2}(d_0,r). \end{array} \right. \end{aligned}$$

(15)

When $\gamma _t=(1,t)'$, let $\delta =(\alpha ,\beta )'$, then by the definition of LS we have

$$\begin{aligned} \left( \begin{array}{c} {\hat{\alpha }}-\alpha \\ {\hat{\beta }}-\beta \end{array}\right)= & {} \left( \begin{array}{ll} \sum 1 &{}\quad \sum t\\ \sum t &{}\quad \sum t^2 \end{array}\right) ^{-1} \left( \begin{array}{l} \sum X_t\\ \sum tX_t \end{array}\right) \\= & {} \left| \begin{array}{ll} \sum 1 &{}\quad \sum t\\ \sum t &{}\quad \sum t^2 \end{array}\right| ^{-1} \left( \begin{array}{l} \left( \sum t^2\right) \left( \sum X_t\right) -\left( \sum t\right) \left( \sum tX_t\right) \\ -\left( \sum t\right) \left( \sum X_t\right) +\left( \sum 1\right) \left( \sum tX_t\right) \end{array}\right) \end{aligned}$$

where $\sum =\sum \nolimits _{t=1}^{m}$, if we estimate $\delta$ using the samples $y_{1},\ldots ,y_{m}$, $\sum =\sum \nolimits _{t=n-m+1}^{n}$, if we estimate $\delta$ using the samples $y_{n-m+1},\ldots ,y_{n}$, and $\sum =\sum \nolimits _{t=m+1}^{n}$, if we estimate $\delta$ using the samples $y_{m+1},\ldots ,y_{n}$. Since

$$\begin{aligned}&12T^{-4}\left| \begin{array}{ll} \sum \limits _{t=1}^m 1 &{}\sum \limits _{t=1}^m t\\ \sum \limits _{t=1}^m t &{}\sum \limits _{t=1}^m t^2 \end{array}\right| = 12T^{-4}\left( \frac{1}{6}m^2(m+1)(2m+1)-\frac{1}{4}m^2(m+1)^2\right) \\&\quad \rightarrow \tau ^4, \\&\qquad 12T^{-7/2-d_0}\left( \left( \sum \limits _{t=1}^m t^2\right) \left( \sum \limits _{t=1}^m X_t\right) -\left( \sum \limits _{t=1}^m t\right) \left( \sum \limits _{t=1}^m tX_t\right) \right) \\&\quad =2T^{-3}m(m+1)(2m+1)T^{-1/2-d_0}\sum \limits _{t=1}^m X_t -\,6T^{-2}m(m+1)T^{-3/2-d_0}\sum \limits _{t=1}^m tX_t\\&\quad \Rightarrow 4\tau ^3\kappa B_{d_0}(\tau )\\&\qquad -\,6\tau ^{2}\kappa \left( \tau B_{d_0}(\tau )-\int _{0}^\tau B_{d_0}(v)\mathrm{d}v\right) \\&\qquad 12T^{-5/2-d_0}\left( -\left( \sum \limits _{t=1}^m t\right) \left( \sum \limits _{t=1}^m X_t\right) +\left( \sum \limits _{t=1}^m 1\right) \left( \sum \limits _{t=1}^m tX_t\right) \right) \\&\quad =-\,6T^{-2}m(m+1)T^{-1/2-d_0}\sum \limits _{t=1}^m X_t +12T^{-1}mT^{-3/2-d_0}\sum \limits _{t=1}^m tX_t\\&\quad \Rightarrow -\,6\tau ^2\kappa B_{d_0}(\tau )+12\tau \kappa \left( \tau B_{d_0}(\tau )-\int _{0}^\tau B_{d_0}(v)\mathrm{d}v\right) \end{aligned}$$

Hence, by a tedious calculation, we have

$$\begin{aligned}&T^{-\frac{1}{2}-d_0}\sum \limits _{i=1}^t{\hat{\varepsilon }}_{0,i}\nonumber \\&\quad =T^{-\frac{1}{2}-d_0}\sum \limits _{i=1}^{[Tr]}\left( X_i -({\hat{\alpha }}-\alpha )-({\hat{\beta }}-\beta )i\right) \nonumber \\&\quad \Rightarrow \kappa \left( B_{d_0}(r)-(2r\tau ^{-1}-3r^2\tau ^{-2})B_{d_0}(\tau ) +6r\tau ^{-3}(r-\tau )\int _{0}^\tau B_{d_0}(v)\mathrm{d}v \right) \nonumber \\&\quad := \kappa U_{2,0}(d_0,r). \end{aligned}$$

(16)

Let

$$\begin{aligned} K_1= & {} \frac{(s-r)}{4\left( s^3-(s-\tau )^3\right) -3\tau (2s-\tau )^2},\\ K_2= & {} \frac{(s-r)}{(s-\tau )^3}. \end{aligned}$$

Then, a similar arguments gives that if $n\le 2m$,

$$\begin{aligned}&T^{-\frac{1}{2}-d_0}\sum \limits _{i=t}^{n}{\hat{\varepsilon }}_{1,i}\nonumber \\&\quad \Rightarrow \kappa \left\{ B_{d_0}(s)-B_{d_0}(r)- K_1(4\tau ^2+3\tau r-3\tau s)(B_{d_0}(s)-B_{d_0}(r))\right. \nonumber \\&\left. \qquad -6K_1(r+\tau -s)\left( \tau B_{d_0}(s-\tau )-\int _{s-\tau }^sB_{d_0}(v)\mathrm{d}v\right) \right\} \nonumber \\&\quad := \kappa U_{2,1}(d_0,r). \end{aligned}$$

(17)

If $n> 2m$,

$$\begin{aligned}&T^{-\frac{1}{2}-d_0}\sum \limits _{i=t}^{[Ts]}{\hat{\varepsilon }}_{1,i}\nonumber \\&\quad \Rightarrow \kappa \left\{ B_{d_0}(s)-B_{d_0}(r)-K_2(4\tau ^2+(s+\tau )(s-3r))(B_{d_0}(s)-B_{d_0}(\tau ))\right. \nonumber \\&\left. \qquad - 6K_2(r-s)\left( sB_{d_0}(s)-\tau B_{d_0}(\tau )-\int _{\tau }^sB_{d_0}(v)\mathrm{d}v\right) \right\} \nonumber \\&\quad := \kappa U_{2,2}(d_0,r). \end{aligned}$$

(18)

Combining (14)–(18), together with the result $\frac{T^k}{m^k} \rightarrow \tau ^{-k},\ \frac{T^k}{(n-m)^k} \rightarrow (s-\tau )^{-k}, \ k=2,3$, as $T\rightarrow \infty$, Theorem 3.1 can be proved similar to Lavancier et al. (2013).

In the remainder of this section, we omit the proof for the case of $\gamma _t = (1, t)'$; this follows the same logical development as those presented for the case of $\gamma _t = 1$. In addition, we omit the proofs of Theorems 3.3 and 3.4, for these are similar to those of Theorems 3.1 and 3.2, respectively. $\square$

Proof of Theorem 3.2

From the proof of Theorem 3.1, we have the denominator of $\varGamma _T(s)$ is $O_p(T^{2+2d_0})$. If $2\tau \ge s> \tau ^*$, then

$$\begin{aligned} T^{-\frac{1}{2}-d_1}\sum \limits _{i=[Ts]-[T\tau ]+1}^{[Ts]}X_i= & {} T^{-\frac{1}{2}-d_1}\sum \limits _{i=[Ts]-[T\tau ]+1}^{[T\tau ^*]}X_i +T^{-\frac{1}{2}-d_1}\sum \limits _{i=[T\tau ^*]+1}^{[Ts]}X_i\\\Rightarrow & {} O_p(T^{d_0-d_1})+\kappa (B_{d_1}(s)-B_{d_1}(\tau ^*)).\\ T^{-\frac{1}{2}-d_1}\sum \limits _{i=[Tr]+1}^{[Ts]}X_i= & {} \left\{ \begin{array}{ll} \sum \limits _{i=[Tr]}^{[T\tau ^*]}X_i+\sum \limits _{i=[T\tau ^*]+1}^{[Ts]}X_i,&{}\quad r\le \tau ^*\\ \sum \limits _{i=[Tr]}^{[Ts]}X_i,&{}\quad r> \tau ^* \end{array} \right. \\\Rightarrow & {} \left\{ \begin{array}{ll} O_p(T^{d_0-d_1})+\kappa (B_{d_1}(s)-B_{d_1}(\tau ^*)),&{}\quad r\le \tau ^*\\ \kappa (B_{d_1}(s)-B_{d_1}(r)),&{}\quad r> \tau ^*. \end{array} \right. \end{aligned}$$

Note that $d_1>d_0$, so

$$\begin{aligned} T^{-\frac{1}{2}-d_1}\sum \limits _{i=t}^{[Ts]}{{\hat{\varepsilon }}}_{1,i}= & {} T^{-\frac{1}{2}-d_1}\sum \limits _{i=t}^{[Ts]}X_i-\frac{[Ts]-[Tr]}{[T\tau ]T^{\frac{1}{2}+d_1}}\sum \limits _{i=t}^{[Ts]}X_i\\\Rightarrow & {} \left\{ \begin{array}{ll} \kappa (B_{d_1}(s)-B_{d_1}(\tau ^*))-\frac{s-r}{\tau }\kappa (B_{d_1}(s)-B_{d_1}(\tau ^*)),&{}\quad r\le \tau ^*\\ \kappa (B_{d_1}(s)-B_{d_1}(r))-\frac{s-r}{\tau }\kappa (B_{d_1}(s)-B_{d_1}(\tau ^*)),&{}\quad r> \tau ^*. \end{array} \right. \end{aligned}$$

This indicates the numerator of $\varGamma _T(s)$ is $O_p(T^{2+2d_1})$ if $2\tau \ge s> \tau ^*$. Similar calculation gives that this conclusion is still hold if $s>2\tau$. Hence,

$$\begin{aligned} \varGamma _T(s)= O_p(T^{2(d_1-d_0)}),\,\,\, s\in (\tau +\tau ^*,1]. \end{aligned}$$