Asymptotic and Bootstrap Tests for a Change in Autoregression Omitting Variability Estimation

Abstract

A sequence of time-ordered observations follows an autoregressive model of order one and its parameter is possibly subject to change at most once at some unknown time point. The aim is to test whether such an unknown change has occurred or not. A change-point method presented here rely on a ratio type test statistic based on the maxima of cumulative sums. The main advantage of the developed approach is that the variance of the observations neither has to be known nor estimated. Asymptotic distribution of the test statistic under the no-change null hypothesis is derived. Moreover, we prove the consistency of the test under the alternative. A bootstrap procedure is proposed in the way of a completely data-driven technique without any tuning parameters. The results are illustrated through a simulation study, which demonstrates the computational efficiency of the procedure. A practical application to real data is presented as well.

Appendix: Proofs

Proof

(of Theorem 1) Let us consider an array

$$ U_{n,i}=\frac{\sqrt{1-\beta ^2}}{\sigma ^2\sqrt{n-1}}Y_{i-1}\varepsilon _i,\quad i=2,\ldots ,n $$

and a filtration $\mathcal {F}_{n,i}=\sigma \{\varepsilon _j,\,j\le i\}$, $i=2,\ldots ,n$ and $n\in \mathbb {N}$. Then, $\{U_{n,i},\mathcal {F}_{n,i}\}$ is a martingale difference array such that

$$ {\mathsf {E}}U_{n,i}^2=\frac{1-\beta ^2}{\sigma ^4(n-1)}{\mathsf {E}}Y_{i-1}^2\varepsilon _i^2=\frac{1}{n-1}. $$

Moreover,

$$ \sum _{i=2}^nU_{n,i}^2-\sum _{i=2}^n{\mathsf {E}}U_{n,i}^2=\frac{1-\beta ^2}{\sigma ^4(n-1)}\sum _{i=2}^n(Y_{i-1}^2\varepsilon _i^2-{\mathsf {E}}Y_{i-1}^2\varepsilon _i^2). $$

Furthermore,

$$\begin{aligned} \frac{1}{n-1}\sum _{i=2}^n(Y_{i-1}^2\varepsilon _i^2&-{\mathsf {E}}Y_{i-1}^2\varepsilon _i^2)\\&=\frac{1}{n-1}\sum _{i=2}^n[Y_{i-1}^2(\varepsilon _i^2-\sigma ^2)]+\frac{1}{n-1}\sum _{i=2}^n(Y_{i-1}^2-{\mathsf {E}}Y_{i-1}^2)\sigma ^2. \end{aligned}$$

Since $\{Y_{i-1}^2(\varepsilon _i^2-\sigma ^2)\}$ is a martingale difference array again with respect to $\mathcal {F}_{n,i}$, we have under Assumption 3 from the Chebyshev’s inequality that

$$ \frac{1}{n-1}\sum _{i=2}^n[Y_{i-1}^2(\varepsilon _i^2-\sigma ^2)]\xrightarrow [n\rightarrow \infty ]{\mathsf {P}}0. $$

Similarly, as a consequence of Lemma 4.2 in [1],

$$ \frac{1}{n-1}\sum _{i=2}^n(Y_{i-1}^2-{\mathsf {E}}Y_{i-1}^2)\xrightarrow [n\rightarrow \infty ]{\mathsf {P}}0. $$

Thus,

$$\begin{aligned} \sum _{i=2}^n U_{n,i}^2\xrightarrow [n\rightarrow \infty ]{\mathsf {P}}1. \end{aligned}$$

(11)

Next, for any $\epsilon >0$,

$$\begin{aligned} \mathsf {P}\left( \max _{2\le i\le n}U_{n,i}^2>\epsilon \right)&\le \sum _{i=2}^n\mathsf {P}\left( \frac{1-\beta ^2}{\sigma ^4(n-1)}Y_{i-1}^2\varepsilon _i^2>\epsilon \right) \nonumber \\&\le \frac{(1-\beta ^2)^2}{\epsilon ^2\sigma ^8(n-1)^2}\sum _{i=2}^n{\mathsf {E}}Y_{i-1}^4{\mathsf {E}}\varepsilon _i^4\xrightarrow [n\rightarrow \infty ]{}0. \end{aligned}$$

(12)

Additionally,

$$\begin{aligned} \lim _{n\rightarrow \infty }\sum _{i=2}^{[nt]}{\mathsf {E}}U_{n,i}^2=\lim _{n\rightarrow \infty }\frac{[nt]-1}{n-1}=t \end{aligned}$$

(13)

for all $t\in [0,1]$.

According to Theorem 27.14 from [7] for the martingale difference array $\{U_{n,i}, \mathcal {F}_{n,i}\}$, where the assumptions of this theorem are satisfied due to (11), (12), and (13), we get

$$\begin{aligned} \sum _{i=2}^{[nt]}U_{n,i}\xrightarrow [n\rightarrow \infty ]{\mathscr {D}[0,1]}\mathcal {W}(t). \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{1}{\sqrt{n-1}}\left( \sum _{i=2}^{[nt]}Y_{i-1}\varepsilon _i,\sum _{i=[nt]+2}^{n}Y_{i-1}\varepsilon _i\right) \xrightarrow [n\rightarrow \infty ]{\mathscr {D}^2[0,1]}\frac{\sigma ^2}{\sqrt{1-\beta ^2}}\left( \mathcal {W}(t),\widetilde{\mathcal {W}}(t)\right) , \end{aligned}$$

(14)

where $\widetilde{\mathcal {W}}(t)=\mathcal {W}(1)-\mathcal {W}(t)$.

Let us define

$$ \mathbf{Y}_{j,l}=(Y_j,\ldots ,Y_l)^{\top }\quad \text{ and }\quad {\varvec{\varepsilon }}_{j,l}=(\varepsilon _j,\ldots ,\varepsilon _l)^{\top }. $$

Hence, for the expression in the numerator of $\mathcal {V}_n$, it holds

$$\begin{aligned}&\sum _{j=1}^{i-1}Y_{j}(Y_{j+1}-\widehat{\beta }_{1k}Y_j)=\mathbf{Y}_{1,i-1}^{\top }\left( \mathbf{Y}_{2,i}-\mathbf{Y}_{1,i-1}\widehat{\beta }_{1k}\right) \nonumber \\&=\mathbf{Y}_{1,i-1}^{\top }\left( \mathbf{Y}_{1,i-1}\beta +{\varvec{\varepsilon }}_{2,i}-\mathbf{Y}_{1,i-1}\beta -\mathbf{Y}_{1,i-1}\left( \mathbf{Y}_{1,k-1}^{\top }{} \mathbf{Y}_{1,k-1}\right) ^{-1}{} \mathbf{Y}_{1,k-1}^{\top }{\varvec{\varepsilon }}_{2,k}\right) \nonumber \\&=\mathbf{Y}_{1,i-1}^{\top }{\varvec{\varepsilon }}_{2,i}-\mathbf{Y}_{1,i-1}^{\top }{} \mathbf{Y}_{1,i-1}\left( \mathbf{Y}_{1,k-1}^{\top }{} \mathbf{Y}_{1,k-1}\right) ^{-1}{} \mathbf{Y}_{1,k-1}^{\top }{\varvec{\varepsilon }}_{2,k}. \end{aligned}$$

(15)

Similarly, for the expression in the denominator of $\mathcal {V}_n$,

$$\begin{aligned}&\sum _{j=i}^{n-1}Y_{j}(Y_{j+1}-\widehat{\beta }_{2k}Y_j)\nonumber \\&=\mathbf{Y}_{i,n-1}^{\top }{\varvec{\varepsilon }}_{i+1,n}-\mathbf{Y}_{i,n-1}^{\top }{} \mathbf{Y}_{i,n-1}\left( \mathbf{Y}_{k+1,n-1}^{\top }{} \mathbf{Y}_{k+1,n-1}\right) ^{-1}{} \mathbf{Y}_{k+1,n-1}^{\top }{\varvec{\varepsilon }}_{k+2,n}. \end{aligned}$$

(16)

Lemma 4.2 in [1] gives

$$\begin{aligned} \sup _{\gamma \le t<1}\frac{1}{[nt]}\left| \sum _{s=1}^{[nt]}(Y_s^2-{\mathsf {E}}Y_s^2)\right| =o_{\mathsf {P}}(1) \end{aligned}$$

(17)

and

$$\begin{aligned} \sup _{0<t\le 1-\gamma }\frac{1}{[n(1-t)]}\left| \sum _{s=[nt]+1}^{n-1}(Y_s^2-{\mathsf {E}}Y_s^2)\right| =o_{\mathsf {P}}(1), \end{aligned}$$

(18)

as $n\rightarrow \infty$, where [nt] and $[n(1-t)]$ mean truncated number to zero decimal digits. Finally, (14) together with (15), (16), (17), and (18) implies

$$\begin{aligned} \frac{1}{\sqrt{n-1}}&\left( \begin{array}{c} \underset{0\le u\le t}{\sup }\left| \sum \limits _{j=1}^{[nu]-1} Y_{j}(Y_{j+1}-\widehat{\beta }_{1[nt]}Y_j)\right| \\ \underset{t\le u\le 1}{\sup }\left| \sum \limits _{j=[nu]+1}^{n-1} Y_{j}(Y_{j+1}-\widehat{\beta }_{2[nt]}Y_j)\right| \end{array}\right) \\&\quad \xrightarrow [n\rightarrow \infty ]{\mathscr {D}^2[\gamma ,1-\gamma ]}\frac{\sigma ^2}{\sqrt{1-\beta ^2}}\left( \begin{array}{c} \sup \limits _{0\le u\le t}\left| \mathcal {W}(u)-u/t\mathcal {W}(t)\right| \\ \sup \limits _{t\le u\le 1}\left| \widetilde{\mathcal {W}}(u)-(1-u)/(1-t)\widetilde{\mathcal {W}}(t)\right| \end{array}\right) . \end{aligned}$$

Then, the assertion of the theorem directly follows.   $\square$

Proof

(of Theorem 2) Let us take $k=\tau +2$, $k=[\xi n]$ for some $\zeta<\xi <1-\gamma$ and $i=\tau +1$. Then,

$$\begin{aligned}&\sum _{j=1}^{\tau }Y_{j}(Y_{j+1}-\widehat{\beta }_{1(\tau +2)}Y_j)\\&=\mathbf{Y}_{1,\tau }^{\top }{\varvec{\varepsilon }}_{2,\tau +1}-\mathbf{Y}_{1,\tau }^{\top }{} \mathbf{Y}_{1,\tau }\left( \mathbf{Y}_{1,\tau +1}^{\top }{} \mathbf{Y}_{1,\tau +1}\right) ^{-1}{} \mathbf{Y}_{1,\tau +1}^{\top }{\varvec{\varepsilon }}_{2,\tau +2}-\mathbf{Y}_{1,\tau }^{\top }{} \mathbf{Y}_{1,\tau }\delta . \end{aligned}$$

According to the proof of Theorem 1, as $n\rightarrow \infty$,

$$ \frac{1}{\sqrt{n-1}}\left( \mathbf{Y}_{1,\tau }^{\top }{\varvec{\varepsilon }}_{2,\tau +1}-\mathbf{Y}_{1,\tau }^{\top }{} \mathbf{Y}_{1,\tau }\left( \mathbf{Y}_{1,\tau +1}^{\top }{} \mathbf{Y}_{1,\tau +1}\right) ^{-1}{} \mathbf{Y}_{1,\tau +1}^{\top }{\varvec{\varepsilon }}_{2,\tau +2}\right) =\mathcal {O}_{\mathsf {P}}(1). $$

Lemma 4.2 from [1] gives

$$ \frac{1}{\sqrt{n-1}}\left| \mathbf{Y}_{1,\tau }^{\top }{} \mathbf{Y}_{1,\tau }\delta \right| \xrightarrow [n\rightarrow \infty ]{\mathsf {P}}\infty . $$

Now,

$$ \frac{1}{\sqrt{n-1}}\underset{2\le i \le k}{\max }\left| \sum _{j=1}^{i-1} Y_{j}(Y_{j+1}-\widehat{\beta }_{1k}Y_j)\right| \xrightarrow [n\rightarrow \infty ]{\mathsf {P}}\infty . $$

For $\tau <k=[\xi n]$, the denominator in (4) divided by $\sqrt{n-1}$ has the same distribution as under the null hypothesis and it is, therefore, bounded in probability. It follows that the maximum of the ratio has to tend in probability to infinity as well, while $n\rightarrow \infty$.   $\square$