Testing for Change in Stochastic Volatility with Long Range Dependence

TL;DR

This paper develops and analyzes change-point tests for long memory stochastic volatility models, revealing differences in asymptotic behavior of CUSUM and Wilcoxon tests, and proposing self-normalized methods for practical application.

Contribution

It introduces new theoretical results on the limiting behavior of change-point tests in long memory stochastic volatility models and proposes self-normalized tests to avoid nuisance parameter estimation.

Findings

CUSUM tests are unaffected by long memory asymptotically.

Wilcoxon tests are influenced by long-range dependence.

Simulation studies demonstrate finite sample performance of proposed tests.

Abstract

In this paper, change-point problems for long memory stochastic volatility models are considered. A general testing problem which includes various alternative hypotheses is discussed. Under the hypothesis of stationarity the limiting behavior of CUSUM- and Wilcoxon-type test statistics is derived. In this context, a limit theorem for the two-parameter empirical process of long memory stochastic volatility time series is proved. In particular, it is shown that the asymptotic distribution of CUSUM test statistics may not be affected by long memory, unlike Wilcoxon test statistics which are typically influenced by long range dependence. To avoid the estimation of nuisance parameters in applications, the usage of self-normalized test statistics is proposed. The theoretical results are accompanied by simulation studies which characterize the finite sample behavior of the considered testing procedures when testing for changes in mean, in variance, and in the tail index.

Tables

Table 1. Table 1: Rejection rates of the CUSUM tests for LMSV time series (standard normal ε j , subscript 𝜀 𝑗 \varepsilon_{j}, j ≥ 1 𝑗 1 j\geq 1 ) of length n 𝑛 n with Hurst parameter H 𝐻 H and a shift in the mean of height h ℎ h after a proportion τ = 0.5 𝜏 0.5 \tau=0.5 . The calculations are based on 5,000 simulation runs.

			CUSUM						self-norm. CUSUM
$H$	$n$		$h=0$		$h=0.5$	$h=1$	$h=2$		$h=0$		$h=0.5$	$h=1$	$h=2$
0.6	500		0.046		0.384	0.958	1.000		0.043		0.372	0.853	0.998
	1000		0.045		0.728	1.000	1.000		0.047		0.627	0.975	1.000
	2000		0.043		0.958	1.000	1.000		0.044		0.856	0.998	1.000
0.7	500		0.064		0.392	0.962	1.000		0.044		0.395	0.858	0.995
	1000		0.056		0.736	0.999	1.000		0.044		0.629	0.974	1.000
	2000		0.050		0.964	1.000	1.000		0.046		0.869	0.997	1.000
0.8	500		0.074		0.369	0.969	1.000		0.039		0.462	0.860	0.993
	1000		0.074		0.734	1.000	1.000		0.044		0.670	0.963	0.999
	2000		0.072		0.966	1.000	1.000		0.048		0.851	0.996	1.000
0.9	500		0.095		0.311	0.977	1.000		0.045		0.583	0.875	0.980
	1000		0.101		0.739	0.998	1.000		0.045		0.736	0.946	0.996
	2000		0.101		0.979	1.000	1.000		0.043		0.860	0.987	1.000

Table 2. Table 2: Rejection rates of CUSUM and Wilcoxon tests for LMSV time series (Pareto distributed ε j , subscript 𝜀 𝑗 \varepsilon_{j}, j ≥ 1 𝑗 1 j\geq 1 ) of length n 𝑛 n with Hurst parameter H 𝐻 H , tail index α 𝛼 \alpha and a shift in the mean of height h ℎ h after a proportion τ = 0.5 𝜏 0.5 \tau=0.5 . The calculations are based on 5,000 simulation runs.

			CUSUM							Wilcoxon
					$\alpha =2.5$			$\alpha =4$				$\alpha =2.5$			$\alpha =4$
	$H$	$n$		$h=0$	$h=0.5$	$h=1$	$h=0$	$h=0.5$	$h=1$		$h=0$	$h=0.5$	$h=1$	$h=0$	$h=0.5$	$h=1$
non-self-normalized	0.6	500		0.035	0.048	0.213	0.047	0.866	1.000		0.628	1.000	1.000	0.802	1.000	1.000
		1000		0.034	0.085	0.745	0.051	0.993	1.000		0.578	1.000	1.000	0.751	1.000	1.000
		2000		0.034	0.288	0.986	0.048	1.000	1.000		0.524	1.000	1.000	0.713	1.000	1.000
	0.7	500		0.035	0.050	0.209	0.053	0.864	1.000		0.331	1.000	1.000	0.475	1.000	1.000
		1000		0.037	0.087	0.752	0.058	0.994	1.000		0.270	1.000	1.000	0.384	1.000	1.000
		2000		0.039	0.285	0.983	0.051	1.000	1.000		0.207	1.000	1.000	0.300	1.000	1.000
	0.8	500		0.045	0.055	0.207	0.073	0.879	1.000		0.191	0.974	1.000	0.273	1.000	1.000
		1000		0.041	0.108	0.757	0.066	0.994	1.000		0.144	0.994	1.000	0.187	1.000	1.000
		2000		0.042	0.280	0.983	0.066	1.000	1.000		0.108	1.000	1.000	0.132	1.000	1.000
	0.9	500		0.057	0.069	0.191	0.080	0.901	1.000		0.188	0.863	0.984	0.232	0.995	1.000
		1000		0.059	0.111	0.783	0.090	0.992	1.000		0.139	0.888	0.990	0.165	0.996	1.000
		2000		0.064	0.238	0.984	0.092	1.000	1.000		0.108	0.917	0.996	0.121	0.998	1.000
self-normalized	0.6	500		0.046	0.384	0.801	0.042	0.904	0.990		0.032	0.994	1.000	0.030	1.000	1.000
		1000		0.049	0.564	0.909	0.044	0.973	0.998		0.032	1.000	1.000	0.030	1.000	1.000
		2000		0.053	0.744	0.962	0.026	0.993	1.000		0.034	1.000	1.000	0.028	1.000	1.000
	0.7	500		0.051	0.401	0.801	0.042	0.902	0.989		0.029	0.950	1.000	0.021	1.000	1.000
		1000		0.050	0.565	0.904	0.046	0.972	0.998		0.032	0.993	1.000	0.027	1.000	1.000
		2000		0.049	0.744	0.966	0.042	0.994	0.999		0.037	1.000	1.000	0.030	1.000	1.000
	0.8	500		0.045	0.424	0.804	0.044	0.899	0.990		0.031	0.776	0.977	0.021	0.987	0.999
		1000		0.042	0.589	0.905	0.040	0.966	0.997		0.033	0.896	0.995	0.024	0.998	1.000
		2000		0.050	0.761	0.959	0.052	0.994	0.999		0.034	0.963	0.999	0.023	1.000	1.000
	0.9	500		0.044	0.527	0.815	0.041	0.893	0.982		0.031	0.622	0.890	0.020	0.912	0.990
		1000		0.051	0.650	0.898	0.042	0.959	0.997		0.039	0.708	0.936	0.034	0.956	0.996
		2000		0.041	0.781	0.954	0.048	0.989	0.999		0.049	0.779	0.960	0.039	0.976	0.998

Table 3. Table 3: Rejection rates of the CUSUM and Wilcoxon tests for LMSV time series of length n 𝑛 n with Hurst parameter H 𝐻 H , tail index α 𝛼 \alpha and a shift in the variance of height h 2 superscript ℎ 2 h^{2} after a proportion τ = 0.5 𝜏 0.5 \tau=0.5 . The calculations are based on 5,000 simulation runs.

			CUSUM							Wilcoxon
					$\alpha =4.5$			$\alpha =6$				$\alpha =4.5$			$\alpha =6$
	$H$	$n$		$h=1$	$h=0.5$	$h=2$	$h=1$	$h=0.5$	$h=2$		$h=1$	$h=0.5$	$h=2$	$h=1$	$h=0.5$	$h=2$
non-self-normalized	0.6	500		0.464	0.377	0.954	0.467	0.417	0.967		0.119	0.988	0.990	0.130	0.990	0.990
		1000		0.589	0.565	0.992	0.596	0.594	0.996		0.112	1.000	1.000	0.118	1.000	1.000
		2000		0.708	0.774	0.999	0.694	0.814	0.999		0.108	1.000	1.000	0.104	1.000	1.000
	0.7	500		0.330	0.252	0.821	0.330	0.252	0.839		0.078	0.808	0.808	0.078	0.806	0.809
		1000		0.374	0.313	0.932	0.404	0.333	0.946		0.066	0.934	0.937	0.068	0.936	0.935
		2000		0.431	0.416	0.984	0.443	0.436	0.987		0.060	0.992	0.992	0.061	0.990	0.991
	0.8	500		0.235	0.170	0.623	0.244	0.166	0.651		0.074	0.529	0.525	0.074	0.515	0.528
		1000		0.258	0.196	0.733	0.256	0.199	0.757		0.067	0.605	0.615	0.067	0.624	0.631
		2000		0.275	0.226	0.838	0.271	0.230	0.848		0.060	0.741	0.738	0.059	0.746	0.734
	0.9	500		0.179	0.127	0.440	0.170	0.132	0.445		0.088	0.410	0.419	0.088	0.404	0.406
		1000		0.184	0.137	0.487	0.177	0.140	0.506		0.078	0.442	0.440	0.079	0.452	0.424
		2000		0.191	0.135	0.542	0.191	0.148	0.540		0.072	0.466	0.467	0.065	0.468	0.473
self-normalized	0.6	500		0.035	0.145	0.152	0.038	0.158	0.164		0.040	0.816	0.831	0.041	0.823	0.816
		1000		0.033	0.206	0.193	0.035	0.227	0.221		0.043	0.965	0.963	0.047	0.962	0.960
		2000		0.033	0.277	0.270	0.034	0.324	0.318		0.044	0.998	0.998	0.042	0.996	0.996
	0.7	500		0.022	0.090	0.090	0.024	0.100	0.093		0.046	0.524	0.530	0.048	0.534	0.521
		1000		0.022	0.125	0.121	0.025	0.130	0.135		0.044	0.701	0.695	0.046	0.698	0.699
		2000		0.020	0.165	0.168	0.021	0.196	0.193		0.049	0.859	0.863	0.050	0.854	0.863
	0.8	500		0.015	0.045	0.046	0.019	0.053	0.054		0.038	0.270	0.274	0.042	0.270	0.278
		1000		0.015	0.063	0.064	0.016	0.062	0.070		0.049	0.357	0.361	0.045	0.350	0.361
		2000		0.015	0.073	0.086	0.017	0.099	0.083		0.047	0.439	0.443	0.046	0.454	0.454
	0.9	500		0.019	0.044	0.037	0.023	0.047	0.040		0.048	0.200	0.205	0.049	0.195	0.198
		1000		0.019	0.053	0.052	0.021	0.064	0.058		0.053	0.228	0.224	0.050	0.246	0.228
		2000		0.020	0.066	0.065	0.023	0.076	0.074		0.056	0.263	0.248	0.047	0.248	0.264

Table 4. Table 4: Rejection rates of the CUSUM tests for LMSV time series of length n 𝑛 n with Hurst parameter H 𝐻 H and a change in the tail index α 𝛼 \alpha of height h ℎ h after a proportion τ = 0.5 𝜏 0.5 \tau=0.5 . The calculations are based on 5,000 simulation runs.

		CUSUM							self-norm. CUSUM
				$\alpha =0.5$			$\alpha =1$				$\alpha =0.5$			$\alpha =1$
$H$	$n$		$h=0$	$h=0.25$	$h=0.5$	$h=0$	$h=0.25$	$h=0.5$		$h=0$	$h=0.25$	$h=0.5$	$h=0$	$h=0.25$	$h=0.5$
0.6	500		0.457	0.971	1.000	0.139	0.312	0.597		0.035	0.603	0.907	0.040	0.137	0.318
	1000		0.419	0.997	1.000	0.121	0.446	0.812		0.033	0.843	0.989	0.045	0.227	0.517
	2000		0.388	1.000	1.000	0.113	0.628	0.953		0.038	0.974	1.000	0.045	0.363	0.732
0.7	500		0.213	0.794	0.976	0.084	0.161	0.296		0.026	0.364	0.692	0.041	0.085	0.167
	1000		0.177	0.916	0.998	0.071	0.188	0.402		0.038	0.551	0.869	0.043	0.120	0.248
	2000		0.141	0.982	1.000	0.071	0.241	0.531		0.040	0.750	0.969	0.047	0.157	0.344
0.8	500		0.131	0.509	0.812	0.064	0.107	0.164		0.028	0.195	0.414	0.041	0.057	0.091
	1000		0.107	0.611	0.902	0.064	0.109	0.192		0.034	0.296	0.539	0.047	0.065	0.106
	2000		0.087	0.709	0.957	0.054	0.123	0.227		0.040	0.383	0.671	0.046	0.082	0.135
0.9	500		0.087	0.339	0.611	0.056	0.070	0.106		0.030	0.135	0.288	0.047	0.059	0.083
	1000		0.075	0.360	0.656	0.054	0.068	0.121		0.038	0.181	0.350	0.050	0.059	0.092
	2000		0.061	0.391	0.716	0.051	0.071	0.127		0.046	0.216	0.410	0.052	0.060	0.085

Table 5. Table 5: Rejection rates of the CUSUM tests for LMSV time series (standard normal ε j , subscript 𝜀 𝑗 \varepsilon_{j}, j ≥ 1 𝑗 1 j\geq 1 ) of length n 𝑛 n with Hurst parameter H 𝐻 H and a shift in the mean of height h ℎ h after a proportion τ 𝜏 \tau . The calculations are based on 5,000 simulation runs.

				CUSUM						self-norm. CUSUM
	$H$	$n$		$h=0$		$h=0.5$	$h=1$	$h=2$		$h=0$		$h=0.5$	$h=1$	$h=2$
$\tau =0.25$	0.6	500		0.046		0.214	0.803	1.000		0.043		0.201	0.597	0.951
		1000		0.045		0.452	0.987	1.000		0.047		0.352	0.818	0.996
		2000		0.043		0.816	1.000	1.000		0.044		0.575	0.957	1.000
	0.7	500		0.064		0.222	0.792	1.000		0.044		0.223	0.594	0.951
		1000		0.056		0.461	0.989	1.000		0.044		0.361	0.823	0.993
		2000		0.050		0.815	1.000	1.000		0.046		0.583	0.960	1.000
	0.8	500		0.074		0.204	0.822	1.000		0.039		0.264	0.643	0.944
		1000		0.074		0.435	0.987	1.000		0.044		0.403	0.827	0.989
		2000		0.072		0.814	1.000	1.000		0.048		0.621	0.948	0.999
	0.9	500		0.095		0.190	0.849	1.000		0.045		0.391	0.720	0.939
		1000		0.101		0.370	0.991	1.000		0.045		0.550	0.858	0.976
		2000		0.101		0.847	1.000	1.000		0.043		0.694	0.931	0.997
$\tau =0.5$	0.6	500				0.384	0.958	1.000				0.372	0.853	0.998
		1000				0.728	1.000	1.000				0.627	0.975	1.000
		2000				0.958	1.000	1.000				0.856	0.998	1.000
	0.7	500				0.392	0.962	1.000				0.395	0.858	0.995
		1000				0.736	0.999	1.000				0.629	0.974	1.000
		2000				0.964	1.000	1.000				0.869	0.997	1.000
	0.8	500				0.369	0.969	1.000				0.462	0.860	0.993
		1000				0.734	1.000	1.000				0.670	0.963	0.999
		2000				0.966	1.000	1.000				0.851	0.996	1.000
	0.9	500				0.311	0.977	1.000				0.583	0.875	0.980
		1000				0.739	0.998	1.000				0.736	0.946	0.996
		2000				0.979	1.000	1.000				0.860	0.987	1.000

Table 6. Table 6: Rejection rates of the CUSUM and Wilcoxon test for LMSV time series (Pareto distributed ε j , subscript 𝜀 𝑗 \varepsilon_{j}, j ≥ 1 𝑗 1 j\geq 1 ) of length n 𝑛 n with Hurst parameter H 𝐻 H , tail index α 𝛼 \alpha and a shift in the mean of height h ℎ h after a proportion τ 𝜏 \tau . The calculations are based on 5,000 simulation runs.

			CUSUM							Wilcoxon
					$\alpha =2.5$			$\alpha =4$				$\alpha =2.5$			$\alpha =4$
	$H$	$n$		$h=0$	$h=0.5$	$h=1$	$h=0$	$h=0.5$	$h=1$		$h=0$	$h=0.5$	$h=1$	$h=0$	$h=0.5$	$h=1$
$\tau =0.25$	0.6	500		0.035	0.081	0.625	0.047	0.986	1.000		0.628	1.000	1.000	0.802	1.000	1.000
		1000		0.034	0.239	0.969	0.051	1.000	1.000		0.578	1.000	1.000	0.751	1.000	1.000
		2000		0.034	0.638	1.000	0.048	1.000	1.000		0.524	1.000	1.000	0.713	1.000	1.000
	$0.7$	500		0.035	0.092	0.621	0.053	0.986	1.000		0.331	1.000	1.000	0.475	1.000	1.000
		1000		0.037	0.239	0.961	0.058	1.000	1.000		0.270	1.000	1.000	0.384	1.000	1.000
		2000		0.039	0.648	0.999	0.051	1.000	1.000		0.207	1.000	1.000	0.300	1.000	1.000
	0.8	500		0.045	0.096	0.622	0.073	0.984	1.000		0.191	0.987	1.000	0.273	1.000	1.000
		1000		0.041	0.224	0.966	0.066	1.000	1.000		0.144	0.997	1.000	0.187	1.000	1.000
		2000		0.042	0.637	0.999	0.066	1.000	1.000		0.108	1.000	1.000	0.132	1.000	1.000
	0.9	500		0.057	0.100	0.627	0.080	0.986	1.000		0.188	0.919	0.993	0.232	0.994	1.000
		1000		0.059	0.207	0.973	0.090	1.000	1.000		0.139	0.929	0.995	0.165	0.999	1.000
		2000		0.064	0.635	1.000	0.092	1.000	1.000		0.108	0.954	0.997	0.121	0.999	1.000
$\tau =0.5$	0.6	500			0.048	0.213		0.866	1.000			1.000	1.000		1.000	1.000
		1000			0.085	0.745		0.993	1.000			1.000	1.000		1.000	1.000
		2000			0.288	0.986		1.000	1.000			1.000	1.000		1.000	1.000
	0.7	500			0.050	0.209		0.864	1.000			1.000	1.000		1.000	1.000
		1000			0.087	0.752		0.994	1.000			1.000	1.000		1.000	1.000
		2000			0.285	0.983		1.000	1.000			1.000	1.000		1.000	1.000
	0.8	500			0.055	0.207		0.879	1.000			0.974	1.000		1.000	1.000
		1000			0.108	0.757		0.994	1.000			0.994	1.000		1.000	1.000
		2000			0.280	0.983		1.000	1.000			1.000	1.000		1.000	1.000
	0.9	500			0.069	0.191		0.901	1.000			0.863	0.984		0.995	1.000
		1000			0.111	0.783		0.992	1.000			0.888	0.990		0.996	1.000
		2000			0.238	0.984		1.000	1.000			0.917	0.996		0.998	1.000

Table 7. Table 7: Rejection rates of the self-normalized CUSUM and the self-normalized Wilcoxon test for LMSV time series (Pareto distributed ε j , subscript 𝜀 𝑗 \varepsilon_{j}, j ≥ 1 𝑗 1 j\geq 1 ) of length n 𝑛 n with Hurst parameter H 𝐻 H , tail index α 𝛼 \alpha and a shift in the mean of height h ℎ h after a proportion τ 𝜏 \tau . The calculations are based on 5,000 simulation runs.

			self-norm. CUSUM							self-norm. Wilcoxon
					$\alpha =2.5$			$\alpha =4$				$\alpha =2.5$			$\alpha =4$
	$H$	$n$		$h=0$	$h=0.5$	$h=1$	$h=0$	$h=0.5$	$h=1$		$h=0$	$h=0.5$	$h=1$	$h=0$	$h=0.5$	$h=1$
$\tau =0.25$	0.6	500		0.046	0.181	0.539	0.042	0.688	0.958		0.032	0.879	0.991	0.030	0.995	1.000
		1000		0.049	0.290	0.722	0.044	0.862	0.990		0.032	0.973	1.000	0.030	1.000	1.000
		2000		0.053	0.458	0.875	0.026	0.967	0.999		0.034	0.999	1.000	0.028	1.000	1.000
	0.7	500		0.051	0.204	0.552	0.042	0.697	0.954		0.029	0.680	0.938	0.021	0.960	0.997
		1000		0.050	0.295	0.727	0.046	0.866	0.990		0.032	0.856	0.988	0.027	0.993	1.000
		2000		0.049	0.455	0.868	0.042	0.966	0.998		0.037	0.948	0.999	0.030	0.999	1.000
	0.8	500		0.045	0.226	0.580	0.044	0.720	0.951		0.031	0.424	0.772	0.021	0.815	0.964
		1000		0.042	0.338	0.736	0.040	0.870	0.989		0.033	0.559	0.862	0.024	0.915	0.984
		2000		0.050	0.498	0.881	0.052	0.960	0.998		0.034	0.673	0.938	0.023	0.958	0.998
	0.9	500		0.044	0.329	0.645	0.041	0.760	0.947		0.031	0.309	0.582	0.020	0.640	0.861
		1000		0.051	0.446	0.761	0.042	0.871	0.980		0.039	0.369	0.650	0.034	0.734	0.912
		2000		0.041	0.585	0.869	0.048	0.949	0.996		0.049	0.422	0.719	0.039	0.791	0.947
$\tau =0.5$	0.6	500			0.384	0.801		0.904	0.990			0.994	1.000		1.000	1.000
		1000			0.564	0.909		0.973	0.998			1.000	1.000		1.000	1.000
		2000			0.744	0.962		0.993	1.000			1.000	1.000		1.000	1.000
	0.7	500			0.401	0.801		0.902	0.989			0.950	1.000		1.000	1.000
		1000			0.565	0.904		0.972	0.998			0.993	1.000		1.000	1.000
		2000			0.744	0.966		0.994	0.999			1.000	1.000		1.000	1.000
	0.8	500			0.424	0.804		0.899	0.990			0.776	0.977		0.987	0.999
		1000			0.589	0.905		0.966	0.997			0.896	0.995		0.998	1.000
		2000			0.761	0.959		0.994	0.999			0.963	0.999		1.000	1.000
	0.9	500			0.527	0.815		0.893	0.982			0.622	0.890		0.912	0.990
		1000			0.650	0.898		0.959	0.997			0.708	0.936		0.956	0.996
		2000			0.781	0.954		0.989	0.999			0.779	0.960		0.976	0.998

Table 8. Table 8: Rejection rates of the CUSUM and Wilcoxon test for LMSV time series of length n 𝑛 n with Hurst parameter H 𝐻 H , tail index α 𝛼 \alpha and a shift in the variance of height h 2 superscript ℎ 2 h^{2} after a proportion τ 𝜏 \tau . The calculations are based on 5,000 simulation runs.

			CUSUM							Wilcoxon
					$\alpha =4.5$			$\alpha =6$				$\alpha =4.5$			$\alpha =6$
	$H$	$n$		$h=1$	$h=0.5$	$h=2$	$h=1$	$h=0.5$	$h=2$		$h=1$	$h=0.5$	$h=2$	$h=1$	$h=0.5$	$h=2$
$\tau =0.25$	0.6	500		0.464	0.252	0.963	0.467	0.270	0.978		0.119	0.925	0.937	0.130	0.929	0.929
		1000		0.589	0.383	0.995	0.596	0.385	0.997		0.112	0.994	0.995	0.118	0.995	0.996
		2000		0.708	0.529	1.000	0.694	0.574	1.000		0.108	1.000	1.000	0.104	1.000	1.000
	0.7	500		0.330	0.164	0.852	0.330	0.174	0.882		0.078	0.584	0.587	0.078	0.606	0.591
		1000		0.374	0.197	0.937	0.404	0.207	0.961		0.066	0.781	0.784	0.068	0.781	0.780
		2000		0.431	0.263	0.983	0.443	0.273	0.991		0.060	0.932	0.929	0.061	0.934	0.936
	0.8	500		0.235	0.116	0.670	0.244	0.111	0.686		0.074	0.314	0.328	0.074	0.332	0.319
		1000		0.258	0.131	0.770	0.256	0.132	0.786		0.067	0.398	0.386	0.067	0.398	0.382
		2000		0.275	0.139	0.837	0.271	0.137	0.861		0.060	0.502	0.499	0.059	0.505	0.501
	0.9	500		0.179	0.088	0.470	0.170	0.084	0.486		0.088	0.251	0.254	0.088	0.252	0.262
		1000		0.184	0.100	0.513	0.177	0.097	0.523		0.078	0.267	0.277	0.079	0.272	0.272
		2000		0.191	0.101	0.564	0.191	0.099	0.566		0.072	0.283	0.283	0.065	0.279	0.277
$\tau =0.5$	0.6	500			0.377	0.954		0.417	0.967			0.988	0.990		0.990	0.990
		1000			0.565	0.992		0.594	0.996			1.000	1.000		1.000	1.000
		2000			0.774	0.999		0.814	0.999			1.000	1.000		1.000	1.000
	0.7	500			0.252	0.821		0.252	0.839			0.808	0.808		0.806	0.809
		1000			0.313	0.932		0.333	0.946			0.934	0.937		0.936	0.935
		2000			0.416	0.984		0.436	0.987			0.992	0.992		0.990	0.991
	0.8	500			0.170	0.623		0.166	0.651			0.529	0.525		0.515	0.528
		1000			0.196	0.733		0.199	0.757			0.605	0.615		0.624	0.631
		2000			0.226	0.838		0.230	0.848			0.741	0.738		0.746	0.734
	0.9	500			0.127	0.440		0.132	0.445			0.410	0.419		0.404	0.406
		1000			0.137	0.487		0.140	0.506			0.442	0.440		0.452	0.424
		2000			0.135	0.542		0.148	0.540			0.466	0.467		0.468	0.473

Table 9. Table 9: Rejection rates of the self-normalized CUSUM and the self-normalized Wilcoxon test for LMSV time series of length n 𝑛 n with Hurst parameter H 𝐻 H , tail index α 𝛼 \alpha and a shift in the variance of height h 2 superscript ℎ 2 h^{2} after a proportion τ 𝜏 \tau . The calculations are based on 5,000 simulation runs.

			self-norm. CUSUM							self-norm. Wilcoxon
					$\alpha =4.5$			$\alpha =6$				$\alpha =4.5$			$\alpha =6$
	$H$	$n$		$h=1$	$h=0.5$	$h=2$	$h=1$	$h=0.5$	$h=2$		$h=1$	$h=0.5$	$h=2$	$h=1$	$h=0.5$	$h=2$
$\tau =0.25$	0.6	500		0.035	0.229	0.040	0.038	0.237	0.043		0.040	0.518	0.495	0.041	0.511	0.494
		1000		0.033	0.291	0.043	0.035	0.317	0.050		0.043	0.736	0.739	0.047	0.731	0.745
		2000		0.033	0.383	0.048	0.034	0.423	0.060		0.044	0.901	0.897	0.042	0.908	0.905
	0.7	500		0.022	0.146	0.022	0.024	0.161	0.025		0.046	0.252	0.248	0.048	0.263	0.259
		1000		0.022	0.192	0.023	0.025	0.225	0.027		0.044	0.380	0.381	0.046	0.380	0.378
		2000		0.020	0.263	0.030	0.021	0.296	0.034		0.049	0.526	0.528	0.050	0.531	0.536
	0.8	500		0.015	0.086	0.013	0.019	0.098	0.015		0.038	0.120	0.119	0.042	0.136	0.127
		1000		0.015	0.107	0.017	0.016	0.128	0.017		0.049	0.159	0.155	0.045	0.165	0.156
		2000		0.015	0.140	0.014	0.017	0.171	0.016		0.047	0.198	0.192	0.046	0.197	0.198
	0.9	500		0.019	0.079	0.018	0.023	0.088	0.021		0.048	0.100	0.101	0.049	0.097	0.096
		1000		0.019	0.099	0.015	0.021	0.112	0.020		0.053	0.109	0.107	0.050	0.110	0.105
		2000		0.020	0.123	0.020	0.023	0.133	0.023		0.056	0.118	0.114	0.047	0.104	0.123
$\tau =0.5$	0.6	500			0.145	0.152		0.158	0.164			0.816	0.831		0.823	0.816
		1000			0.206	0.193		0.227	0.221			0.965	0.963		0.962	0.960
		2000			0.277	0.270		0.324	0.318			0.998	0.998		0.996	0.996
	0.7	500			0.090	0.090		0.100	0.093			0.524	0.530		0.534	0.521
		1000			0.125	0.121		0.130	0.135			0.701	0.695		0.698	0.699
		2000			0.165	0.168		0.196	0.193			0.859	0.863		0.854	0.863
	0.8	500			0.045	0.046		0.053	0.054			0.270	0.274		0.270	0.278
		1000			0.063	0.064		0.062	0.070			0.357	0.361		0.350	0.361
		2000			0.073	0.086		0.099	0.083			0.439	0.443		0.454	0.454
	0.9	500			0.044	0.037		0.047	0.040			0.200	0.205		0.195	0.198
		1000			0.053	0.052		0.064	0.058			0.228	0.224		0.246	0.228
		2000			0.066	0.065		0.076	0.074			0.263	0.248		0.248	0.264

Table 10. Table 10: Rejection rates of the CUSUM tests for LMSV time series of length n 𝑛 n with Hurst parameter H 𝐻 H and a change in the tail index α 𝛼 \alpha of height h ℎ h after a proportion τ 𝜏 \tau . The calculations are based on 5,000 simulation runs.

			CUSUM							self-norm. CUSUM
					$\alpha =0.5$			$\alpha =1$				$\alpha =0.5$			$\alpha =1$
	$H$	$n$		$h=0$	$h=0.25$	$h=0.5$	$h=0$	$h=0.25$	$h=0.5$		$h=0$	$h=0.25$	$h=0.5$	$h=0$	$h=0.25$	$h=0.5$
$\tau =0.25$	0.6	500		0.457	0.884	0.994	0.139	0.215	0.393		0.035	0.373	0.725	0.040	0.087	0.181
		1000		0.419	0.982	1.000	0.121	0.298	0.588		0.033	0.596	0.912	0.045	0.124	0.277
		2000		0.388	0.999	1.000	0.113	0.436	0.825		0.038	0.819	0.985	0.045	0.206	0.453
	0.7	500		0.213	0.601	0.879	0.084	0.108	0.184		0.026	0.189	0.435	0.041	0.065	0.100
		1000		0.177	0.761	0.977	0.071	0.134	0.249		0.038	0.294	0.596	0.043	0.079	0.137
		2000		0.141	0.907	0.998	0.071	0.151	0.350		0.040	0.452	0.774	0.047	0.098	0.191
	0.8	500		0.131	0.329	0.590	0.064	0.081	0.105		0.028	0.092	0.194	0.041	0.046	0.056
		1000		0.107	0.379	0.716	0.064	0.081	0.115		0.034	0.130	0.276	0.047	0.053	0.064
		2000		0.087	0.491	0.836	0.054	0.083	0.138		0.040	0.177	0.350	0.046	0.050	0.078
	0.9	500		0.087	0.201	0.376	0.056	0.058	0.079		0.030	0.073	0.133	0.047	0.044	0.058
		1000		0.075	0.216	0.425	0.054	0.068	0.080		0.038	0.083	0.157	0.050	0.055	0.060
		2000		0.061	0.221	0.480	0.051	0.061	0.074		0.046	0.101	0.189	0.052	0.055	0.058
$\tau =0.5$	0.6	500			0.971	1.000		0.312	0.597			0.603	0.907		0.137	0.318
		1000			0.997	1.000		0.446	0.812			0.843	0.989		0.227	0.517
		2000			1.000	1.000		0.628	0.953			0.974	1.000		0.363	0.732
	0.7	500			0.794	0.976		0.161	0.296			0.364	0.692		0.085	0.167
		1000			0.916	0.998		0.188	0.402			0.551	0.869		0.120	0.248
		2000			0.982	1.000		0.241	0.531			0.750	0.969		0.157	0.344
	0.8	500			0.509	0.812		0.107	0.164			0.195	0.414		0.057	0.091
		1000			0.611	0.902		0.109	0.192			0.296	0.539		0.065	0.106
		2000			0.709	0.957		0.123	0.227			0.383	0.671		0.082	0.135
	0.9	500			0.339	0.611		0.070	0.106			0.135	0.288		0.059	0.083
		1000			0.360	0.656		0.068	0.121			0.181	0.350		0.059	0.092
		2000			0.391	0.716		0.071	0.127			0.216	0.410		0.060	0.085

Equations

$$X_j=\sigma (Y_j){\epsilon }_j,j\ge 1,$$

$$\bullet$$

$$Y_j=\sum _{k=1}^{\infty }{c}_k{\eta }_{j-k},j\ge 1,$$

$${\gamma }_Y(k)=\mathrm{Cov}(Y_j,Y_{j+k})=k^{-D}{L}_{\gamma }(k),$$

$$\underset{x\to \infty }{\lim }\frac{P({\epsilon }_1>x)}{P(|{\epsilon }_1|>x)}=p=1-\underset{x\to \infty }{\lim }\frac{P({\epsilon }_1<-x)}{P(|{\epsilon }_1|>x)}$$

$$P(X_1>x)\sim E[{\sigma }^{\alpha }(Y_1)]P({\epsilon }_1>x),$$

$$g(Y_1)-E\left(g(Y_1)\right)=\sum _{q=m}^{\infty }\frac{J_q(g)}{q!}{H}_q(Y_1),$$

$$J_q(g)=E\left(g(Y_1)H_q(Y_1)\right)$$

$$m=\inf \left\{q\ge 1|J_q(g)\ne 0\right\}.$$

$$1_{\left\{g(Y_1)\le x\right\}}-F_{g(Y_1)}(x)=\sum _{q=m}^{\infty }\frac{J_q(g;x)}{q!}{H}_q(Y_1)$$

$$\frac{1}{d_{n,m}}\sum _{j=1}^{\lfloor nt\rfloor }g(Y_j)\Rightarrow \frac{J_m(g)}{m!}{Z}_m(t),0\le t\le 1,$$

$$d_{n,m}^2=\mathrm{Var}\left(\sum _{j=1}^n{H}_m(Y_j)\right)\sim c_m{n}^{2-mD}{L}^m(n),c_m=\frac{2m!}{(1-Dm)(2-Dm)},$$

$$J_m(g)\frac{1}{d_{n,m}}\sum _{j=1}^{\lfloor nt\rfloor }{H}_m(Y_j)\Rightarrow \frac{J_m(g)}{m!}{Z}_m(t),0\le t\le 1.$$

$$\underset{-\infty \le x\le \infty }{\sup }\underset{0\le t\le 1}{\sup }{d}_{n,m}^{-1}\left\{\lfloor nt\rfloor \left(G_{\lfloor nt\rfloor }(x)-E\left(G_{\lfloor nt\rfloor }(x)\right)\right)-J_m(g;x)\sum _{j=1}^{\lfloor nt\rfloor }{H}_m(Y_j)\right\}\stackrel{P}{⟶}0,$$

$$d_{n,m}^{-1}\lfloor nt\rfloor \left(G_{\lfloor nt\rfloor }(x)-E\left(G_{\lfloor nt\rfloor }(x)\right)\right),x\in \left[-\infty ,\infty \right],t\in [0,1],$$

$$\displaystyle H_{0}:\operatorname{E}(\xi_{1})=\cdots=\operatorname{E}(\xi_{n})\;,\ \$$

$$H_1:\exists k\in \left\{1,\dots ,n-1\right\}\text{mbox}\mathrm{suchthat}E({\xi }_1)=\cdots =E({\xi }_k)\ne E({\xi }_{k+1})=\cdots =E({\xi }_n).$$

$$\frac{1}{n}\sum _{j=1}^n\log \left(|X_j|/c\right).$$

$$C_n(\lambda )=\left|\sum _{j=1}^{\lfloor n\lambda \rfloor }\psi (X_j)-\frac{\lfloor n\lambda \rfloor }{n}\sum _{j=1}^n\psi (X_j)\right|.$$

$$W_n(\lambda )=\left|\sum _{i=1}^{\lfloor n\lambda \rfloor }\sum _{j=\lfloor n\lambda \rfloor +1}^n\left(1_{\left\{\psi (X_i)\le \psi (X_j)\right\}}-\frac{1}{2}\right)\right|.$$

$${\mathcal{F}}_j=\sigma \left({\epsilon }_j,{\epsilon }_{j-1},\dots ,{\eta }_j,{\eta }_{j-1},\dots \right),$$

$$\frac{1}{d_{n,m}}\sum _{j=1}^{\lfloor nt\rfloor }\left(\psi (X_j)-E(\psi (X_j))\right)\Rightarrow \frac{J_m(\Psi )}{m!}{Z}_m(t),t\in [0,1],$$

$$\frac{1}{\sqrt{n}}\sum _{j=1}^{\lfloor nt\rfloor }\psi (X_j)\Rightarrow \sigma B(t),t\in [0,1],$$

$$\frac{1}{d_{n,m}}\underset{0\le \lambda \le 1}{\sup }{C}_n(\lambda )\Rightarrow \frac{\left|J_m(\Psi )\right|}{m!}\underset{0\le t\le 1}{\sup }\left|Z_m(t)-t{Z}_m(1)\right|.$$

$$\frac{1}{\sqrt{n}}\underset{0\le \lambda \le 1}{\sup }{C}_n(\lambda )\Rightarrow \sigma \underset{0\le t\le 1}{\sup }\left|B(t)-tB(1)\right|,$$

$$\sum _{j=1}^{\lfloor nt\rfloor }\left(1_{\left\{g(Y_j)\le x\right\}}-F_{g(Y_1)}(x)\right),x\in \left[-\infty ,\infty \right],t\in [0,1],$$

$$G_n(x,t)=\sum _{j=1}^{\lfloor nt\rfloor }\left(1_{\left\{\psi (X_j)\le x\right\}}-F_{\psi (X_1)}(x)\right),$$

$${\Psi }_x(y)=P\left(\psi (y{\epsilon }_1)\le x\right).$$

$$\int \frac{d}{du}P\left(\psi (u{\epsilon }_1)\le x\right)du<\infty .$$

$$\frac{1}{d_{n,m}}{G}_n(x,t)\Rightarrow \frac{J_m({\Psi }_x\circ \sigma )}{m!}{Z}_m(t),x\in \left[-\infty ,\infty \right],t\in \left[0,1\right],$$

$$\frac{1}{n{d}_{n,m}}\underset{\lambda \in [0,1]}{\sup }{W}_n(\lambda )\Rightarrow \left|\int J_m({\Psi }_x\circ \sigma )d{F}_{\psi (X_1)}(x)\right|\frac{1}{m!}\underset{\lambda \in [0,1]}{\sup }\left|Z_m(\lambda )-\lambda Z_m(1)\right|.$$

$$G_n(x,t)$$

Full text

Keywords: long memory stochastic volatility; change point tests; empirical process limit theorem; self-normalization

1 Introduction

One of the most often observed phenomena in financial data is the fact that the log-returns of stock-market prices appear to be uncorrelated, whereas the absolute log-returns or squared log-returns tend to be correlated or even exhibit long range dependence. Another characteristic of financial time series is the existence of heavy tails in the sense that the marginal tail distribution behaves like a regularly varying function. Both of these features of empirical data can be covered by the so-called long memory stochastic volatility model (LMSV, in short), with its original version introduced in Breidt et al., (1998). For this we assume that the data generating process $\{X_{j},j\geq 1\}$ satisfies

[TABLE]

where $D\in\left(0,1\right)$ and $L_{\gamma}$ is slowly varying at infinity. Also, we assume that

• $\bullet$

$\{(\varepsilon_{j},\eta_{j}),j\geq 1\}$ is a sequence of i.i.d. vectors.

The above set of assumptions we will call collectively LMSV model.

The tail behavior of the sequence $\{X_{j},j\geq 1\}$ can be related to the tail behavior of $\{\varepsilon_{j},j\geq 1\}$ or $\{\sigma(Y_{j}),j\geq 1\}$ or both. Here, we will specifically assume that

• •

the random variables $\{\varepsilon_{j},j\geq 1\}$ have a marginal distribution with regularly varying right tail, i.e. $\bar{F}_{\varepsilon}(x):=P(\varepsilon_{1}>x)=x^{-\alpha}L(x)$ for some $\alpha>0$ and a slowly varying function $L$, such that the following tail balance condition holds:

[TABLE]

for some $p\in(0,1]$;

• •

we have $E\left[\sigma^{\alpha+\delta}(Y_{1})\right]<\infty$ for some $\delta>0$.

Under these conditions, it follows by Breiman’s Lemma (see (Resnick, , 2007, Proposition 7.5)) that

[TABLE]

i.e. the process $\{X_{j},j\geq 1\}$ inherits the tail behavior from $\{\varepsilon_{j},j\geq 1\}$.

We would like to point out here that in the literature the usage of the term LMSV often presupposes that the sequences $\{Y_{j},j\geq 1\}$ and $\{\varepsilon_{j},j\geq 1\}$ are independent. In this paper, we will consider a more general model: instead of claiming mutual independence of $\{Y_{j},j\geq 1\}$ and $\{\varepsilon_{j},j\geq 1\}$, we only assume that $\{{\left(\eta_{j},\varepsilon_{j}\right)},j\geq 1\}$ is a sequence of independent random vectors. Especially, this implies that for a fixed index $j$ the random variables $\varepsilon_{j}$ and $Y_{j}$ are independent, while $Y_{j}$ may depend on $\left\{\varepsilon_{i},i<j\right\}$. In many cases, this version of the LMSV model is referred to as LMSV with leverage.

1.1 Change-point detection under long memory

One of the problems related to financial data is to detect structural changes in a time series $\{X_{j},j\geq 1\}$. Although the problem has been extensively studied for independent random variables (see an excellent book Csörgö and Horvath, (1997)) or in case of weakly dependent data, the issue has not been fully resolved for time series with long memory. The researchers focused rather on justifying that the observed long range dependence is real or that it is due to changes in weakly dependent sequences (so-called spurious long memory); see e.g. Berkes et al., (2006) and Section 7.9.1 of Beran et al., (2013) for further references on the latter issue.

As for the testing changes in long memory sequences, one of the first paper seems to be Horváth and Kokoszka, (1997), where the authors showed that long range dependence affects the asymptotic behavior of the CUSUM statistics for changes in the mean. For the general testing problem with a change in the marginal distribution under the alternative hypothesis, the paper Giraitis et al., (1996) considers Kolmogorov - Smirnov type change-point tests and change-point estimators for long memory moving average processes. For changes in the mean of long range dependent time series Horváth and Kokoszka, (1997) and Betken, 2016a consider estimators for the change-point location based on CUSUM and Wilcoxon statistics. Under the assumption of converging change-point alternatives in LRD time series, the asymptotic behavior of Kolmogorov-Smirnov and Cramér-von Mises type test statistics has also been investigated by Tewes, (2015). Likewise, in Dehling et al., (2013) the authors show that the Wilcoxon test is always affected by long memory. In fact, in the case of Gaussian long memory data, the asymptotic relative efficiency of the Wilcoxon test and the CUSUM test is 1.

In case of long range dependence, the normalization and the limiting distribution of test statistics typically depend on unknown multiplicative factors or parameters related to the dependence structure of the data generating processes. To bypass estimation of these quantities, the concept of self-normalization has recently been applied to several testing procedures in change-point analysis: Shao and Zhang, (2010) define a self-normalized Kolmogorov-Smirnov test statistic that serves to identify changes in the mean of short range dependent time series. Shao, (2011) adopted the same approach to define an alternative normalization for the CUSUM test; Betken, 2016b considers a self-normalized version of the Wilcoxon change-point test proposed by Dehling et al., (2013).

The main message from the above discussion is that long memory typically affects the limiting behaviour of different statistics used in change-point testing. We refer also to Section 7.9 of Beran et al., (2013) for further results on change-point detection for long memory processes.

1.2 Change-point detection for LMSV

In this paper, we study CUSUM and Wilcoxon tests for the LMSV model and discuss particular cases of testing for changes in the mean, in the variance and in the tail index. Although the variance and the tail index can be regarded as the mean of transformed random variables, we observe different effects for each of the three quantities. In particular, the main findings of our paper are as follows:

• A-1:

CUSUM tests for a change in the mean in LMSV time series are typically not affected by long memory (see Corollary 3.2 and Example 4.1). This is different than the findings in Horváth and Kokoszka, (1997) for subordinated Gaussian processes;

• A-2:

Wilcoxon tests for a change in the mean of LMSV models are typically affected by long memory (see Corollary 3.4 and Example 4.1). This is in line with the findings for subordinated Gaussian processes; cf. Dehling et al., (2013).

• B:

CUSUM and Wilcoxon tests for a change in the variance of LMSV models are typically affected by long memory; see Section 4.2.

• C:

CUSUM and Wilcoxon tests for a change in the tail index of LMSV models are typically affected by long memory; see Section 4.3

The paper is structured as follows: in Section 2 we collect some results on subordinated Gaussian processes. In Section 3 we discuss the change-point problem. In particular, we consider CUSUM and Wilcoxon test. The asymptotic behavior of the former is a direct consequence of the existing results, while the latter requires a new theorem (see Theorem 3.3) on the limiting behavior of empirical processes based on LMSV data. Section 4 is devoted to examples in case of testing for changes in the mean, in the variance and in the tail. Since the test statistics and/or limiting distributions involve unknown quantities, self-normalization is considered in Section 5. In fact, in Section S2 we perform simulation studies and indicate that self-normalization provides robustness.

2 Preliminaries

2.1 Some properties of subordinated Gaussian sequences

The main tool for studying the asymptotic behavior of subordinated Gaussian sequences is the Hermite expansion. For a stationary, long range dependent Gaussian process $\{Y_{j},j\geq 1\}$ and a measurable function $g$ such that $E\left(g^{2}(Y_{1})\right)<\infty$ the corresponding Hermite expansion is defined by

[TABLE]

where $H_{q}$ is the $q$-th Hermite polynomial,

[TABLE]

and

[TABLE]

The integer $m$ is called the Hermite rank of $g$ and we refer to $J_{q}(g)$ as the $q$-th Hermite coefficient of $g$.

We will also consider the Hermite expansion of the function class $\left\{1_{\left\{g(Y_{1})\leq x\right\}}-F_{g(Y_{1})}(x),\ x\in\mathbb{R}\right\}\;,$ where $F_{g(Y_{1})}$ denotes the distribution function of $g(Y_{1})$. For fixed $x$, we have

[TABLE]

with $J_{q}(g;x)=\operatorname{E}\left(1_{\left\{g(Y_{1})\leq x\right\}}H_{q}(Y_{1})\right)\;.$ The Hermite rank corresponding to this function class is defined by $m=\inf_{x}m(x)$, where $m(x)$ denotes the Hermite rank of $1_{\left\{g(Y_{1})\leq x\right\}}-F_{g(Y_{1})}(x)$. We refer to Beran et al., (2013) for further details.

The asymptotic behavior of partial sums of subordinated Gaussian sequences is characterized in Taqqu, (1979). Due to the functional non-central limit theorem in that paper,

[TABLE]

where $Z_{m}(t)$, $0\leq t\leq 1$, is an $m$-th order Hermite process,

[TABLE]

and the convergence holds in ${\mathbb{D}}([0,1])$, the space of functions that are right-continuous with left limits. In fact, the limiting behavior in (5) is the same as that of the corresponding partial sums based on $\{H_{m}(Y_{j}),j\geq 1\}$:

[TABLE]

Moreover, the functional central limit theorem for the empirical processes was established in Dehling and Taqqu, (1989). Specifically,

[TABLE]

where $G_{l}(x)=\frac{1}{l}\sum_{j=1}^{l}1_{\left\{g(Y_{j})\leq x\right\}}$ is the empirical distribution function of the sequence $\{g(Y_{j}),j\geq 1\}$ and “$\overset{P}{\longrightarrow}$” denotes convergence in probability. Thus, the empirical process

[TABLE]

converges in ${\mathbb{D}}([-\infty,\infty]\times[0,1])$ to $J_{m}(g;x)Z_{m}(t)$.

We refer the reader to Dehling and Taqqu, (1989), Taqqu, (1979) and Beran et al., (2013) for more details.

3 Change-point problem

Given the observations $X_{1},\ldots,X_{n}$ and a function $\psi$, we define $\xi_{j}=\psi(X_{j})$, $j=1,\ldots,n$, and we consider the testing problem:

[TABLE]

We choose $\psi$ according to the specific change-point problem considered. Possible choices include:

• •

$\psi(x)=x$ in order to detect changes in the mean of the observations $X_{1},\ldots,X_{n}$ (change in location);

• •

$\psi(x)=x^{2}$ in order to detect changes in the variance of the observations $X_{1},\ldots,X_{n}$ (change in volatility);

• •

$\psi(x)=\log(x^{2})$ or $\psi(x)=\log(|x|)$ in order to detect changes in the index $\alpha$ of heavy-tailed observations (change in the tail index).

It is obvious that $\psi(x)=x$ and $\psi(x)=x^{2}$ lead to testing for a change in the mean and a change in the variance, respectively. The choice $\psi(x)=\log(|x|)$ requires an additional comment. We note that (4) describes only the asymptotic tail behavior of $X_{1}$. For the purpose of this paper, we shall pretend that $P\left(|X_{1}|>x\right)=c^{\alpha}x^{-\alpha}$, $x>c$, for some $c>0$. Then the maximum likelihood estimator of $(1/\alpha)$, the reciprocal of the tail index, is

[TABLE]

This estimator is used in the CUSUM test statistic. To resolve the problem of change-points in the tail index in full generality, we need to employ a completely different technique, based on the so-called tail empirical processes (see Kulik and Soulier, (2011)). This will be done in a subsequent paper.

In any case, the following test statistics may be applied in order to decide on the change-point problem $(H_{0},H_{1})$:

• •

The CUSUM test rejects the hypothesis for large values of the test statistic $C_{n}=\sup\limits_{0\leq\lambda\leq 1}C_{n}(\lambda)$, where

[TABLE]

• •

The Wilcoxon test rejects the hypothesis for large values of the test statistic $W_{n}=\sup\limits_{0\leq\lambda\leq 1}W_{n}(\lambda)$, where

[TABLE]

The goal of this paper is to obtain limiting distributions for the CUSUM and the Wilcoxon test statistic in case of time series that follow the LMSV model.

3.1 CUSUM Test for LMSV

In order to determine the asymptotic behavior of the CUSUM test statistic computed with respect to the observations $\psi(X_{1}),\ldots,\psi(X_{n})$, we have to consider the partial sum process $\sum_{j=1}^{\lfloor nt\rfloor}\left(\psi(X_{j})-\operatorname{E}\left(\psi(X_{j})\right)\right)$.

For the observations $X_{1},\ldots,X_{n}$ that satisfy the LMSV model, the asymptotic behavior of the partial sum process is described by Theorem 4.10 in Beran et al., (2013) and hence is stated without proof. In order to formulate the result, we introduce the following notation:

[TABLE]

i.e. $\mathcal{F}_{j}$ denotes the $\sigma$-field generated by the random variables $\varepsilon_{j},\varepsilon_{j-1},\ldots,\eta_{j},\eta_{j-1},\ldots$. Due to this construction, $\varepsilon_{j}$ is independent of $\mathcal{F}_{j-1}$ and $Y_{j}$ is $\mathcal{F}_{j-1}$-measurable.

Theorem 3.1**.**

Assume that $\{X_{j},j\geq 1\}$ follows the LMSV model. Furthermore, assume that $\operatorname{E}(\psi^{2}(X_{1}))<\infty$. Define the function $\Psi$ by $\Psi(y)=\operatorname{E}\left(\psi(\sigma(y)\varepsilon_{1})\right)$. Denote by $m$ the Hermite rank of $\Psi$ and by $J_{m}(\Psi)$ the corresponding Hermite coefficient.

If $\operatorname{E}(\psi(X_{1})\left|\right.\mathcal{F}_{0})\neq 0$ and $mD<1$, then

[TABLE]

in ${\mathbb{D}}([0,1])$. 2. 2.

If $\operatorname{E}(\psi(X_{1})\left|\right.\mathcal{F}_{0})=0$, then

[TABLE]

in ${\mathbb{D}}([0,1])$, where $B$ denotes a Brownian motion process and $\sigma^{2}=\operatorname{E}(\psi^{2}(X_{1}))$.

As an immediate consequence of Theorem 3.1, we obtain the asymptotic distribution of the CUSUM statistic.

Corollary 3.2**.**

Assume that the assumptions of Theorem 3.1 hold.

If $\operatorname{E}(\psi(X_{1})\left|\right.\mathcal{F}_{0})\neq 0$ and $mD<1$, then

[TABLE] 2. 2.

If $\operatorname{E}(\psi(X_{1})\left|\right.\mathcal{F}_{0})=0$

[TABLE]

where $B$ denotes a Brownian motion process and $\sigma^{2}=\operatorname{E}(\psi^{2}(X_{1}))$.

It is important to note that the Hermite rank of $\Psi$ does not necessarily correspond to the Hermite rank of $\sigma$; see Section 4.

3.2 Wilcoxon test for LMSV

For subordinated Gaussian time series $\{g(Y_{j}),j\geq 1\}$, where $\{Y_{j},j\geq 1\}$ is a stationary Gaussian LRD process and $g$ is a measurable function, the asymptotic distribution of the Wilcoxon test statistic $W_{n}$ is derived from the limiting behavior of the two-parameter empirical process

[TABLE]

where $F_{g(Y_{1})}$ denotes the distribution function of $g(Y_{1})$; see Dehling et al., (2013).

In order to determine the asymptotic distribution of the Wilcoxon test statistic for the LMSV model, we need to establish an analogous result for the stochastic volatility process $\{X_{j},j\geq 1\}$, i.e. our preliminary goal is to prove a limit theorem for the two-parameter empirical process

[TABLE]

where now $F_{\psi(X_{1})}$ denotes the distribution function of $\psi(X_{1})$ with $X_{1}=\sigma(Y_{1})\varepsilon_{1}$. To state the weak convergence, we introduce the following notation:

[TABLE]

Theorem 3.3**.**

Assume that $\{X_{j},j\geq 1\}$ follows the LMSV model. Moreover, assume that

[TABLE]

Let $m$ denote the Hermite rank of the class $\left\{1_{\left\{\sigma(Y_{1})\leq x\right\}}-F_{\sigma(Y_{1})}(x),\ x\in\mathbb{R}\right\}$. If $mD<1$, then

[TABLE]

in ${\mathbb{D}}\left([-\infty,\infty]\times[0,1]\right)$ .

The proof of this theorem is given in Section 3.3. At this moment, we derive the asymptotic distribution of the Wilcoxon statistics.

Corollary 3.4**.**

Under the conditions of Theorem 3.3

[TABLE]

Proof of Corollary 3.4.

According to Dehling et al., (2013), the asymptotic distribution of the Wilcoxon test statistic can be derived directly from the limit of the two-parameter empirical process if the sequence $\{X_{j},j\geq 1\}$ is ergodic. Ergodicity is obvious, since $X_{j}$ can be represented as a measurable function of the i.i.d. vectors $\{(\eta_{j},\varepsilon_{j}),j\geq 1\}$. ∎

3.3 Proof of Theorem 3.3

To prove Theorem 3.3, we consider the following decomposition:

[TABLE]

It will be shown that $n^{-1/2}M_{n}(x,t)=\mathcal{O}_{P}(1)$ uniformly in $x,t$, while $d_{n,m}^{-1}R_{n}(x,t)$ converges in distribution to the limit process in formula (13). Theorem 3.3 then follows because $\sqrt{n}=\text{o}(d_{n,m})$.

Martingale part.

For fixed $x$, the following lemma characterizes the asymptotic behavior of the martingale part $M_{n}(x,t)$. We write

[TABLE]

with $\zeta_{j}(x)=1_{\left\{\psi(X_{j})\leq x\right\}}-\operatorname{E}\left(1_{\left\{\psi(X_{j})\leq x\right\}}\left|\right.\mathcal{F}_{j-1}\right)$.

Lemma 3.5**.**

Under the conditions of Theorem 3.3, we have for every $x$,

[TABLE]

in ${\mathbb{D}}([0,1])$, where $B$ denotes a Brownian motion process and $\beta^{2}(x)=\operatorname{E}(\zeta_{1}^{2}(x))$.

Proof.

Define

[TABLE]

with $X_{nj}(x)=n^{-\frac{1}{2}}1_{\left\{\psi(X_{j})\leq x\right\}}$. In order to show convergence in ${\mathbb{D}}([0,1])$, we apply the functional martingale central limit theorem as stated in Theorem 18.2 of Billingsley, (1999). Therefore, we have to show that

[TABLE]

for every $t$ and that

[TABLE]

for every $t$ and $\epsilon>0$ (Lindeberg condition). In order to show that the Lindeberg condition holds, it suffices to show that

[TABLE]

due to Lemma 3.3 in Dvoretzky, (1972). As the indicator function is bounded, the above summands vanish for sufficiently large $n$ and hence (14) follows.

Furthermore, the random variable $\operatorname{E}\left(\zeta_{j}^{2}(x)\left|\right.\mathcal{F}_{j-1}\right)$ can be considered as a measurable function of the random variable $Y_{j}$ and therefore as a function of $\eta_{j-1},\eta_{j-2},\ldots$. As a result, $\operatorname{E}\left(\zeta_{j}^{2}(x)\left|\right.\mathcal{F}_{j-1}\right)$ is an ergodic sequence and it follows by the ergodic theorem that

[TABLE]

for every $t$. ∎

The next lemma establishes uniform boundedness of the two-parameter process.

Lemma 3.6**.**

Under the conditions of Theorem 3.3, we have

[TABLE]

in ${\mathbb{D}}(\left[-\infty,\infty\right]\times[0,1])$.

The (technical) proof of this lemma can be found in Section S1 of the supplementary material.

Long memory part.

Finally, we prove weak convergence of the long memory part $R_{n}(x,t)$.

Lemma 3.7**.**

Under the conditions of Theorem 3.3,

[TABLE]

in ${\mathbb{D}}\left([-\infty,\infty]\times[0,1]\right)$ .

Proof.

Note that

[TABLE]

because $Y_{j}$ is $\mathcal{F}_{j-1}$-measurable and $\varepsilon_{j}$ is independent of $\mathcal{F}_{j-1}$. Furthermore, $\operatorname{E}\left(\Psi_{x}(\sigma(Y_{j}))\right)=F_{\psi(X_{1})}(x)$, where $F_{\psi(X_{1})}$ denotes the distribution function of $\psi(X_{1})=\psi(\sigma(Y_{1})\varepsilon_{1})$. Hence,

[TABLE]

where $G_{l}(u)=\frac{1}{l}\sum_{j=1}^{l}1_{\left\{\sigma(Y_{j})\leq u\right\}}$ is the empirical distribution function of the sequence $\{\sigma(Y_{j}),j\geq 1\}$. We have

[TABLE]

where $m$ denotes the Hermite rank of the class $\left\{1_{\left\{\sigma(Y_{1})\leq x\right\}}-F_{\sigma(Y_{1})}(x),\ x\in\mathbb{R}\right\}$ and

[TABLE]

Using the reduction principle (7) with $g=\sigma$ and the integrability condition (12), we conclude that the first summand converges to [math] in probability, uniformly in $x,t$. Furthermore,

[TABLE]

Denoting by $\varphi$ the standard normal density, integration by parts yields

[TABLE]

The proof is concluded by (6). ∎

4 Examples

4.1 Change in the mean

To test for a change in the mean, we choose $\psi(x)=x$.

CUSUM: Recall that the function $\Psi$ in Theorem 3.1 is defined as $\Psi(y)=\operatorname{E}\left(\psi(\sigma(y)\varepsilon_{1})\right)$. In this case, $\operatorname{E}(\psi(X_{1})\left|\right.\mathcal{F}_{0})=0$. Therefore, the CUSUM statistic converges to a Brownian bridge. Hence, long memory in LMSV does not influence the asymptotic behavior of the CUSUM statistic when testing for a change in the mean.

Wilcoxon: Recall that $\Psi_{x}(y)=P\left(\psi(y\varepsilon_{1})\leq x\right)$. Using integration by parts and noting that $(d/dz)\varphi(z)=-z\varphi(z)$, we conclude

[TABLE]

where $f_{\varepsilon}$ is the density of $\varepsilon_{1}$ (if it exists). Here, different scenarios are possible: if $\sigma(y)=y^{2}$, then the integrand is antisymmetric in $z$ for any choice of $f_{\varepsilon}$. Hence, $J_{1}(\Psi_{x}\circ\sigma)=0$ and some calculations show that the Hermite rank of $\Psi_{x}\circ\sigma$ is $2$. If $\sigma(y)=\exp(y)$ and $\varepsilon_{1}$ is Pareto distributed, i.e. for some $\alpha,c>0$ $f_{\varepsilon}(x)=\frac{\alpha c^{\alpha}}{x^{\alpha+1}}1_{\left\{x\geq c\right\}}$, then, as a result,

[TABLE]

Hence, $J_{1}(\Psi_{x}\circ\sigma)\not=0$. In any case, long memory in LMSV influences the asymptotic behavior of the Wilcoxon statistic.

4.2 Change in the variance

To test for a change in the variance, we choose $\psi(x)=x^{2}$.

CUSUM: Recall again that the function $\Psi$ in Theorem 3.1 is defined as $\Psi(y)=\operatorname{E}\left(\psi(\sigma(y)\varepsilon_{1})\right)$. Then $\operatorname{E}(\psi(X_{1})\left|\right.\mathcal{F}_{0})\not=0$ and hence long memory affects the limiting behavior of the CUSUM statistic. Moreover,

[TABLE]

i.e. the Hermite rank of $\Psi$ equals the Hermite rank of $\sigma^{2}$. If $mD<1$, then the limiting behavior of the CUSUM statistic is described by (11). Hence, long memory in LMSV influences the asymptotic behavior of the CUSUM statistic when testing for a change in the variance.

Wilcoxon: Recall again that $\Psi_{x}(y)=P\left(\psi(y\varepsilon_{1})\leq x\right)$. For $x\geq 0$, we have

[TABLE]

If $\sigma(y)=\exp(y)$, then we can conclude by the same argument as in the case of testing for a change in the mean that $J_{1}(\Psi_{x}\circ\sigma)\not=0$. Hence, long memory in LMSV time series influences the asymptotic behavior of the Wilcoxon statistic when testing for a change in the variance.

4.3 Change in the tail index

To test for a change in the tail index, we choose $\psi(x)=\log(x^{2})$.

CUSUM: In this case, $\operatorname{E}(\psi(X_{1})\left|\right.\mathcal{F}_{0})\not=0$ and hence long memory affects the limiting distribution of the CUSUM statistic. Moreover,

[TABLE]

so that the Hermite rank of $\Psi$ equals the Hermite rank of $h=\log\circ\sigma$.

We note further that if $\psi(x)=\log(x^{2})$, we have

[TABLE]

The first summand converges to $2(J_{m}(\log\circ\sigma)/m!)Z_{m}(t)$ as a consequence of the functional non-central limit theorem discussed in Section 2.1. The second term is $o_{P}(1)$ uniformly in $t$ by Donsker’s theorem, if $\operatorname{Var}(\log\varepsilon_{1}^{2})<\infty$. As a result, this observation is consistent with Corollary 3.2.

Wilcoxon: Due to monotonicity of the logarithm, choosing $\psi(x)=\log(x^{2})$ yields the same test statistic as if testing for a change in variance, i.e. if we choose $\psi(x)=x^{2}$. Moreover, the Wilcoxon test statistic is robust, i.e. it is designed to reduce the effect of heavy tails, so that it does not seem advisable to apply Wilcoxon-like tests when testing for a change in the tail index.

5 Self-normalization

An application of the CUSUM test presupposes knowledge of the normalizing sequence $d_{n,m}$ (if $\operatorname{E}(\psi(X_{1})\left|\right.\mathcal{F}_{0})\neq 0$) and of the coefficients $J_{m}(\Psi)$ or $\sigma$ that appear in the limit of the test statistic. Usually, these quantities are unknown. In order to avoid estimation of the normalization and the unknown coefficients in the limit, we consider the self-normalized CUSUM test statistic with respect to the observations $\xi_{j}=\psi(X_{j})$, $j=1,\ldots,n$. For $0<\tau_{1}<\tau_{2}<1$ it is defined by

[TABLE]

where

[TABLE]

with $S_{t}(j,k)=\sum_{h=j}^{t}\left(\xi_{h}-\bar{\xi}_{j,k}\right),\quad\bar{\xi}_{j,k}=\frac{1}{k-j+1}\sum_{t=j}^{k}\xi_{t}$. The self-normalized CUSUM test rejects the hypothesis for large values of the test statistic $T_{n}(\tau_{1},\tau_{2})$. Note that the proportion of the data that is included in the calculation of the supremum is restricted by $\tau_{1}$ and $\tau_{2}$. A common choice is $\tau_{1}=1-\tau_{2}=0.15$; see Andrews, (1993).

In order to detect changes in the mean of (possibly) long range dependent time series, a similar test statistic has been proposed in Shao, (2011). For long memory stochastic volatility sequences the limit of the test statistic can be derived in the same way as in Shao, (2011). Under the assumptions of Theorem 3.1 (and if $\operatorname{E}(\psi(X_{1})\left|\right.\mathcal{F}_{0})\neq 0$ and $mD<1$), an application of the continuous mapping theorem to the partial sum process $\frac{1}{d_{n,m}}\sum_{j=1}^{\lfloor nt\rfloor}\left(\psi(X_{j})-\operatorname{E}\psi(X_{j})\right)$ yields $T_{n}(\tau_{1},\tau_{2})~{}\overset{\mathcal{D}}{\longrightarrow}~{}T(m,\tau_{1},\tau_{2})$, where

[TABLE]

for $r\in\left[r_{1},r_{2}\right]$, $0<r_{1}<r_{2}<1$.

If $\operatorname{E}(\psi(X_{1})\left|\right.\mathcal{F}_{0})=0$, it follows by an application of the continuous mapping theorem to the partial sum process $\frac{1}{\sqrt{n}}\sum_{j=1}^{\lfloor nt\rfloor}\psi(X_{j})$ that $T_{n}(\tau_{1},\tau_{2})~{}\overset{\mathcal{D}}{\longrightarrow}~{}T(1,\tau_{1},\tau_{2})$, where $T(1,\tau_{1},\tau_{2})$ corresponds to (15) with $Z_{1}$ denoting a fractional Brownian motion process with Hurst parameter $1/2$, i.e. under these assumptions $Z_{1}$ denotes a Brownian motion. Note that in this case the limit does not depend on any unknown parameters. The factor $\sigma$ that appeared in the limit of the partial sum process is canceled out by self-normalization.

As an alternative to the self-normalized CUSUM test, Betken, 2016b proposes to use a self-normalized Wilcoxon test for the detection of changes in the mean of long range dependent time series. For the definition of the corresponding test statistic, we consider the ranks defined by $R_{i}:=\operatorname{rank}(X_{i})=\sum_{j=1}^{n}1_{\{X_{j}\leq X_{i}\}}$ for $i=1,\ldots,n$. The self-normalized two-sample Wilcoxon test statistic is then defined by

[TABLE]

where $S_{t}(j,k):=\sum_{h=j}^{t}\left(R_{h}-\bar{R}_{j,k}\right)\ \ \text{with}\ \ \bar{R}_{j,k}:=\frac{1}{k-j+1}\sum_{t=j}^{k}R_{t}$. The self-normalized Wilcoxon change-point test rejects the hypothesis for large values of $\max_{k\in\left\{\lfloor n\tau_{1}\rfloor,\ldots,\lfloor n\tau_{2}\rfloor\right\}}\left|G_{n}(k)\right|$, where $0<\tau_{1}<\tau_{2}<1$.

Under the hypothesis of stationarity, the asymptotic distribution of the test statistic is derived in Betken, 2016b for long range dependent subordinated Gaussian processes. Basically, the asymptotics of the test statistic follow from an application of the continuous mapping theorem to the process

[TABLE]

Theorem 3.3 implies that the above process converges in distribution to the following limit:

[TABLE]

Under the LMSV model, the limit of the test statistic can therefore be derived by the same argument that proves Theorem 1 in Betken, 2016b . As a result, $T_{n}(\tau_{1},\tau_{2})$ converges in distribution to $T(m,\tau_{1},\tau_{2})$, i.e. the self-normalized Wilcoxon test statistic and the self-normalized CUSUM test statistic converge to the same limit.

6 Simulations

For all simulations we make the following specifications:

[TABLE]

where

• •

$\{\varepsilon_{j},j\geq 1\}$ is an i.i.d. sequence;

• •

$\{Y_{j},j\geq 1\}$ is a fractional Gaussian noise sequence generated by the function fgnSim (fArma package in R) with the Hurst parameter $H$ (note that $H$ and the memory parameter $D$ are linked by $H=1-D/2$);

• •

$\sigma(y)=\exp(y)$.

In the following, we give an overview over the main simulation results for the three testing situations described in Section 4. For all scenarios, we computed the rejection rates of the different tests under the hypothesis and different change point alternatives.

6.1 Change in the mean

We investigate the finite sample performance of the CUSUM and Wilcoxon change-point tests for detecting changes in the mean of LMSV time series $\{X_{j},j\geq 1\}$, i.e. we choose $\psi(x)=x$ for the test statistics described in Section 3 and Section 5.

We consider two different scenarios:

$\{\varepsilon_{j},j\geq 1\}$ are standard normal generated by the function rnorm in R; 2. 2.

$\{\varepsilon_{j},j\geq 1\}$ follow a Pareto distribution, centered to mean zero, i.e. $\varepsilon_{j}=\tilde{\varepsilon}_{j}-\operatorname{E}\tilde{\varepsilon}_{j}$, where $\{\tilde{\varepsilon}_{j},j\geq 1\}$ are Pareto distributed with parameter $\alpha$ generated by the function rgpd (fExtremes package in R);

In order to compare the finite sample behavior of the change-point tests, we computed the empirical size and the empirical power of the testing procedures. To determine the finite sample performance under the alternative, we simulated time series with a change-point of height $h$ after a proportion $\tau$ of the data, i.e. we consider random variables $X_{j}$, $j=1,\ldots,n$ with expected value $\mu_{j}$, $j=1\ldots,n$ such that $\mu_{j}=0$ for $j=1,\ldots,\lfloor n\tau\rfloor$ while $\mu_{j}=h$ for $j=\lfloor n\tau\rfloor+1,\ldots,n$. The calculations are based on $5,000$ realizations of time series with sample sizes $500$, $1,000$ and $2,000$. The simulation results for $\tau=0.5$ are reported in Tables 5 and 7; additional simulation results for $\tau=0.25$ can be found in Section S2 of the supplementary material. The frequency of a type 1 error, i.e. the rejection rate under the hypothesis, corresponds to the values in the columns that are superscribed by “h = 0”.

Recall that the function $\Psi$ in Theorem 3.1 is defined as $\Psi(y)=\operatorname{E}\left(\psi(\sigma(y)\varepsilon_{1})\right)$. Since $\psi(x)=x$, it follows that $\operatorname{E}(\psi(X_{1})\left|\right.\mathcal{F}_{0})=0$. Due to Theorem 3.1,

[TABLE]

where $\sigma^{2}=\operatorname{E}(\psi^{2}(X_{1}))=\operatorname{E}(\varepsilon_{1}^{2}\exp(2Y_{1}))=\operatorname{Var}(\varepsilon_{1})\operatorname{E}(\exp(2Y_{1}))$. Hence, we expect long memory in the data not to influence the asymptotic behavior of the CUSUM statistic.

If $\{\varepsilon_{j},j\geq 1\}$ are standard normal, it can be shown that

[TABLE]

Therefore, the Wilcoxon test statistic converges to [math] in probability.

If $\{\varepsilon_{j},j\geq 1\}$ are centered two-sided Pareto distributed with parameter $\alpha$, it follows by some basic calculations that

[TABLE]

where $\mu$ is the expected value of $\tilde{\varepsilon}_{1}$, i.e. $\mu:=\frac{\alpha}{\alpha-1}$, and

[TABLE]

As a result, we have

[TABLE]

Since our theoretical results do not provide a non-degenerate asymptotic distribution for the Wilcoxon test statistics, we only consider CUSUM and self-normalized CUSUM test for the first scenario (when the $\varepsilon_{j}$, $j\geq 1$, are standard normally distributed).

Note that in this case the size of the non-self-normalized CUSUM change point test tends to increase when the value of the Hurst parameter increases. Aside from that, the simulation results confirm the theoretical inference that long memory does not influence the asymptotic behavior of the CUSUM statistics when testing for a change in the mean in this situation.

A comparison of the finite sample behavior of both CUSUM tests shows that:

• •

The self-normalized test has better size properties than the non-self-normalized test.

• •

The empirical power of the non-self-normalized CUSUM test exceeds the empirical power of the self-normalized CUSUM test for most parameter combinations that have been considered.

Therefore, it is not possible to definitely conclude that one of the tests outreaches the other.

For the second scenario (when the $\varepsilon_{j}$, $j\geq 1$, are centered Pareto distributed), we considered all four testing procedures: CUSUM and self-normalized CUSUM; Wilcoxon and self-normalized Wilcoxon. In this case, it is notable that for all four testing procedures, heavier tails, i.e. a decrease of the tail parameter, go along with an decrease of rejections. A change in the Hurst parameter seems to have a remarkable effect on the finite sample performance of the Wilcoxon test only: An increase of dependence goes along with a decrease of rejections.

A comparison of the finite sample performance of the testing procedures shows that:

• •

In most cases the size of the CUSUM test does not deviate much from the level of significance whereas the Wilcoxon change-point test suffers from size distortions: the empirical size of the Wilcoxon test is considerably higher than the level of significance.

• •

The self-normalized change-point tests have considerably better size properties than the non-self-normalized tests. The self-normalized Wilcoxon test is rather conservative while the rejection rates of the self-normalized CUSUM test are close to the level of significance.

• •

The power of the self-normalized Wilcoxon test is higher than the power of the self-normalized CUSUM test for almost every combination of parameters.

All in all, our simulation results give rise to choosing the self-normalized Wilcoxon test over the other testing procedures when testing for a change in the mean.

6.2 Change in the variance

We will now investigate the finite sample performance of the CUSUM and Wilcoxon change-point tests for detecting changes in the variance of LMSV time series $\{X_{j},j\geq 1\}$, i.e. we choose $\psi(x)=x^{2}$ for the test statistics described in Section 3 and Section 5.

For this purpose, we simulate observations $X_{1},\ldots,X_{n}$ which satisfy (16) with $\{\varepsilon_{j},j\geq 1\}$ that follow a two-sided Pareto distribution, centered to mean zero, i.e. $\varepsilon_{j}=\tilde{\varepsilon}_{j}-\operatorname{E}\tilde{\varepsilon}_{0}$, where $\{\tilde{\varepsilon}_{j},j\geq 1\}$ are Pareto distributed with parameter $\alpha>4$. Since $\operatorname{Var}(\varepsilon_{1})=\operatorname{Var}(\tilde{\varepsilon}_{1})=\frac{\alpha}{\alpha-2}\frac{1}{(\alpha-1)^{2}}$, we have

[TABLE]

Since $\operatorname{E}(\psi(X_{1})\left|\right.\mathcal{F}_{0})\neq 0$, the limits of the CUSUM statistics depend on the Hurst parameter $H$.

For the determination of the critical values of the non-self-normalized CUSUM test, note that

[TABLE]

In order to apply the Wilcoxon test, we have to compute the value of the multiplicative factor $\left|\int J_{1}(\Psi_{x}\circ\sigma)dF_{\psi(X_{1})}(x)\right|$. For notational convenience define $\mu:=\frac{\alpha}{\alpha-1}$. For $x>0$, it follows by some calculations that

[TABLE]

Furthermore, it follows that the density function of $\psi(X_{1})$ is given by

[TABLE]

We conclude that

[TABLE]

Therefore, the critical values for the Wilcoxon change-point test are based on an approximation of the above expression by numerical integration. In order to determine the finite sample performance of the testing procedures under the hypothesis, we simulate observations $X_{1},\ldots,X_{n}$ with variance $\omega^{2}$. For the computation of the empirical power, we added a change-point of height $h^{2}$ after $\tau$ percent of the data, i.e. we multiply $X_{\lfloor n\tau\rfloor+1},\ldots,X_{n}$ by $h$ such that $\operatorname{Var}(X_{j})=\omega^{2}$ for $j=1,\ldots,\lfloor n\tau\rfloor$ while $\operatorname{Var}(X_{j})=h^{2}\omega^{2}$ for $j=\lfloor n\tau\rfloor+1,\ldots,n$. We computed the rejection rates of the testing procedures on the basis of $5,000$ realizations of time series with sample sizes $500$, $1,000$ and $2,000$. The simulation results for $\tau=0.5$ are reported in Table 9; additional simulation results for $\tau=0.25$ can be found in Section S2 of the supplementary material. The frequency of a type 1 error corresponds to the values in the columns that are superscribed by “h = 1”.

In general, the finite sample behavior of the testing procedures does not seem to be strongly influenced by the tail parameter. An increase of dependence, i.e. increasing values of $H$, are usually followed by a decrease of the rejection rates though.

We make the following observations with respect to the behavior of the different testing procedures:

• •

The CUSUM test suffers from severe size distortions. It rejects the hypothesis too frequently for every combination of the considered parameters $\alpha$ and $H$. In particular, the number of rejections increases as the number of observations increases. Apparently, unrealistically large sample sizes are required for the asymptotics to apply. The same observation has been made in De Pooter and Van Dijk, (2004), where the CUSUM statistic is used to test for breaks in the variance of GARCH(1, 1) time series.

• •

The self-normalized CUSUM test tends to be undersized.

• •

The Wilcoxon test rejects the hypothesis too often. Yet, the rejection rates seem to converge to the level of significance.

• •

The empirical size of the self-normalized Wilcoxon test is close to the level of significance.

• •

The power of both Wilcoxon-based testing procedures does not depend on $\alpha$. Contrary to the behavior of the CUSUM test, the increase and decrease of the variance are almost equally well detected by both Wilcoxon tests.

• •

The power of the Wilcoxon test exceeds the power of the self-normalized Wilcoxon test for every parameter combination that has been considered.

All in all, the simulation results show that both CUSUM tests are outperformed by the Wilcoxon-based testing procedures: Although the CUSUM test has high size distortions, its power can only compete with the power of the Wilcoxon test if the shift height is relatively high. Even though the self-normalized CUSUM test does not suffer from severe size distortions, its power cannot compete with the power of any other test. Obviously, CUSUM-based tests are highly unreliable when testing for a change in the variance.

6.3 Change in the tail

We will now investigate the finite sample performance of the CUSUM change-point tests for detecting changes in the tail parameter $\alpha$ of LMSV time series $\{X_{j},j\geq 1\}$, i.e. we choose $\psi(x)=\log(|x|)$ for the test statistics described in Section 3 and Section 5. In this case, the Hermite rank of $\Psi(y)=\operatorname{E}\left(\psi(\sigma(y)\varepsilon_{1})\right)$ equals $m=1$ and $J_{m}(\Psi)=1$ such that the limit of the CUSUM test statistic does not depend on $\alpha$.

We simulate observations $X_{1},\ldots,X_{n}$ which satisfy (16) for $\{\varepsilon_{j},j\geq 1\}$ that follow a (non-centered) Pareto distribution with parameter $\alpha$. For the computation of the empirical power, we consider LMSV time series with a change-point of height $h$ after a proportion $\tau$ of the data, i.e. we consider Pareto distributed random variables $\varepsilon_{j}$, $j=1,\ldots,n$ with parameters $\alpha_{j}$, $j=1,\ldots,n$ such that $\alpha_{j}=\alpha$ for $j=1,\ldots,\lfloor n\tau\rfloor$ while $\alpha_{j}=\alpha+h$ for $j=\lfloor n\tau\rfloor+1,\ldots,n$. We computed the rejection rates of the testing procedures on the basis of $5,000$ realizations of time series with sample sizes $500$, $1,000$ and $2,000$. The simulation results for $\tau=0.5$ are reported in Table 10; additional simulation results for $\tau=0.25$ can be found in Section S2 of the supplementary material. The frequency of a type 1 error, i.e. the rejection rate under the hypothesis, corresponds to the values in the columns that are superscribed by “h = 0”.

The finite sample performance of the testing procedures is affected by a change of the tail parameter in the following way: lighter tails, i.e. an increase in the value of the tail parameter, go along with a decrease in power, but an empirical size which is closer to the level of significance. An increase of dependence, i.e. an increase of the Hurst parameter, results in a decrease of rejection rates.

We make the following observations with respect to the behavior of the testing procedures:

• •

The non-self-normalized test suffers from size distortions in so far as the rejection rates under the hypothesis are considerably higher than the nominal level (especially for small values of $\alpha$ and $H$).

• •

All in all, the self-normalized test tends to be slightly undersized.

• •

The non-self-normalized test has considerably better power than the self-normalized test for every parameter combination that has been considered.

Due to the heavy size distortions of the non-self-normalized CUSUM test, it seems advisable to choose the self-normalized test over the non-self-normalized test in the case of heavy tails characterized by small values of $\alpha$. However, for relatively light tails, the discrepancy between the test performances under the alternative suggests to choose the non-self-normalized test over the self-normalized CUSUM test.

In all three test situations that have been considered, we observe that under the hypothesis the rejection rates of the self-normalized tests yield better approximations of the nominal level than the non-self-normalized tests. On the other hand, considering self-normalized tests instead of applying a deterministic normalization leads to a loss of power. In the context of change-point tests, the so-called “better size, less power” phenomenon has also been observed by Shao, (2011), Shao and Zhang, (2010) and Betken, 2016b . Moreover, it is consistent with observations made in Lobato, (2001) and Sun et al., (2008). For a comparison of the testing procedures, it is important to note that the finite sample results are based on simulations which were executed under the assumption that the normalization of the non-self-normalized tests and the multiplicative quantities that appear in the limits of the corresponding test statistics are known. In particular, normalization and limit of the non-self-normalized statistics usually depend on $H$, $m$, the slowly-varying function $L_{\gamma}$ that characterizes the autocovariances of the Gaussian random variables $Y_{j}$, $j\geq 1$, the distribution of $\varepsilon_{j}$, $j\geq 1$, (or at least the tail parameter $\alpha$), and the function $\sigma$. For all practical purposes these quantities are unknown, and for this reason have to be estimated. In contrast, the self-normalized test statistic can be computed from the given data while its limit depends on the parameters $m$ and $H$ only. For an adequate comparison of the testing procedures this has to be taken into consideration.

Moreover, the simulation results show that the choice of the change-point test depends on the particular test situation that is considered. In general, an application of Wilcoxon-type tests reduces the influence of heavy tails in data generating processes on test decisions. In fact, we have seen that Wilcoxon-based testing procedures yield better results than CUSUM tests when testing for changes in mean and variance.

Supplementary Material

This section contains supplementary material. In Section S1 the proof of Lemma 3.6 is presented; Section S2 comprises the full set of simulation results, i.e. in addition to the results presented in the main document, the simulation tables in Section S2 contain the empirical power of the testing procedures for changes located after a proportion $\tau=0.25$ of the data.

S1 Proof of Lemma 3.6

Assume that $\{X_{j},j\geq 1\}$ follows the LMSV model. Define

[TABLE]

Lemma 3.6 states that

[TABLE]

in ${\mathbb{D}}(\left[-\infty,\infty\right]\times[0,1])$

Proof of Lemma 3.6.

In order to prove Lemma 3.6 we have to verify tightness in ${\mathbb{D}}(\left[-\infty,\infty\right]\times[0,1])$. For this we quote Theorem 3.1 in Ivanoff, (1983).

Theorem S1.1**.**

Let $\left(X_{n},\mathcal{F}_{n},P_{n}\right)$ be a sequence of probability spaces such that $X_{n}$ is a random element of ${\mathbb{D}}\left([0,1]\times\left[0,1\right]\right)$ for each $n$, and the process $X_{n}(x,t)$ is adapted to a complete, right continuous filtration $\mathcal{F}_{n}(x,t)\subseteq\mathcal{F}_{n}$. If $X_{n}(x,t)$ is tight for each $(x,t)\in[0,1]\times\left[0,1\right]$ and if for all $\delta_{n}\searrow 0$ and each canonical 1-stopping time $S_{n}$ and each canonical 2-stopping time $T_{n}$

[TABLE]

then $X_{n}$ is tight in ${\mathbb{D}}\left(\left[0,1\right]\times\left[0,1\right]\right)$.

In the following we will show that the conditions of the theorem hold for $X_{n}(x,t):=n^{-1/2}M_{n}\circ H$, where $H:\left[0,1\right]\times\left[0,1\right]\longrightarrow\left[-\infty,\infty\right]\times\left[0,1\right]$ with $H(x,t)=(h(x),t)$ for some increasing isomorphism $h:\left[0,1\right]\longrightarrow\left[-\infty,\infty\right]$. Then, Lemma 3.6 immediately follows from an application of Theorem S1.1. Indeed, due to Lemma 3.5, $n^{-1/2}M_{n}\circ H(x,t)$ is tight for each $(x,t)\in\left[0,1\right]\times\left[0,1\right]$.

Recall that $\ \mathcal{F}_{j}=\sigma\left(\varepsilon_{j},\varepsilon_{j-1},\ldots,\eta_{j},\eta_{j-1},\ldots\right)$. Define

[TABLE]

for all $x\in[0,1]$. Then, $X_{n}(x,t)$ is adapted to $\mathcal{F}_{n}(x,t)$. Moreover, the corresponding filtration is right-continuous.

Let $T_{n}$ denote a canonical $2$-stopping time for $X_{n}(x,t)$. In particular, we require $\left\{T_{n}\leq t\right\}$ to be measurable with respect to $\mathcal{F}_{n}(1,t)=\mathcal{F}_{\lfloor nt\rfloor}$. Define $\tau_{n}:=\lfloor nT_{n}\rfloor$.

Note that

[TABLE]

with

[TABLE]

For $0\leq x\leq 1$ define

[TABLE]

In order to show (S1.2), we have to prove that

[TABLE]

Prior to the proof, we establish the following result:

Lemma S1.2**.**

We have $\operatorname{E}\left[\tilde{M}_{n}(x)\right]=0$ for all $x\in\mathbb{R}$. For $x<y$

[TABLE]

where $F_{\tilde{\psi}(X_{j+\tau_{n}})}(x):=P\left(\tilde{\psi}(X_{j+\tau_{n}})\leq x\right)$ and $\tilde{\psi}:=h^{-1}\circ\psi$.

Proof.

Define

[TABLE]

Then

[TABLE]

Since $\left\{\tau_{n}=k\right\}$ is measurable with respect to $\mathcal{F}_{j}$ for all $j\geq k$, it follows that

[TABLE]

where $\tilde{\psi}:=h^{-1}\circ\psi$. Furthermore,

[TABLE]

For $i<j$

[TABLE]

As a result,

[TABLE]

Moreover, we have

[TABLE]

It follows that

[TABLE]

∎

Due to Lemma S1.2 $\operatorname{E}(\tilde{M}_{n}(x))=0$ and

[TABLE]

Hence, $\tilde{M}_{n}(x)$ converges to [math] in probability. This implies fidi-convergence of $\tilde{M}_{n}$ as a process with values in ${\mathbb{D}}\left[0,1\right]$.

Proving (S1.2).

In order to establish (S1.2), it remains to show tightness of $\tilde{M}_{n}$. For this, we adopt the argument that proves Theorem 15.6 in Billingsley, (1968). For any function $v$ in ${\mathbb{D}}[0,1]$ define the modulus $\omega^{{}^{\prime\prime}}_{v}(\delta)$ by

[TABLE]

where the supremum extends over $x_{1}$, $x$, and $x_{2}$ satisfying

[TABLE]

Under the assumption of fidi-convergence it suffices to show that for each positive $\varepsilon$ and $\eta$, there exists a $\delta$, $0<\delta<1$, and an integer $n_{0}$ such that

[TABLE]

(see Theorem 15.4 in Billingsley, (1968)).

Define

[TABLE]

where $S_{i}=\tilde{M}_{n}(\tau+\frac{i}{m}\delta)$. Following the proof of Theorem 12.5 in Billingsley, (1968), there exists an $n_{0}\in\mathbb{N}$ such that

[TABLE]

holds for all positive $\lambda$, some constant $K$ and $n\geq n_{0}$, if

[TABLE]

for some sequence $\varepsilon_{n}$, $n\in\mathbb{N}$, converging to [math]. For $x_{1}\leq x\leq x_{2}$ it follows by the Cauchy - Schwarz inequality for expected values and Lemma S1.2 that

[TABLE]

The Markov inequality yields

[TABLE]

Therefore, it follows by (S1.6) that for some $\gamma\in(0,1)$

[TABLE]

where

[TABLE]

As a result, we get

[TABLE]

Define

[TABLE]

where the supremum extends over $x_{1}$, $x$, $x_{2}$ satisfying $\tau\leq x_{1}\leq x\leq x_{2}\leq\tau+\delta$. Letting $m\longrightarrow\infty$ in (S1.7) yields

[TABLE]

due to right-continuity of $\tilde{M}_{n}$. Suppose that $\delta=\frac{1}{u}$ for some integer $u$ and assume that

[TABLE]

If $x_{1}\leq x\leq x_{2}$ and $x_{2}-x_{1}\leq\delta$, then $x_{1}$ and $x_{2}$ both lie in one of the $2u-1$ intervals $[2i\delta,(2i+2)\delta]$, $0\leq i\leq u-1$, $[(2i+1)\delta,(2i+3)\delta]$, $0\leq i\leq u-2$, so that

[TABLE]

Thus, (S1.9) and (S1.10) together imply $\omega^{{}^{\prime\prime}}_{\tilde{M}_{n}}(\delta)\leq\varepsilon$. It now follows by (S1.8) that

[TABLE]

where each of $\Sigma^{\prime}$ and $\Sigma^{\prime\prime}$ is a sum of the form

[TABLE]

with $0\leq x_{1}\leq\ldots\leq x_{r}\leq 1$ and $x_{k}-x_{k-1}\leq 2\delta$. Hence, we may conclude that

[TABLE]

Since the right-hand side of the above inequality converges to [math], (S1.2) has been proved.

Proving (S1.1).

It remains to show (S1.1). For this, let $S_{n}$ denote a canonical $1$-stopping time for $X_{n}(x,t)$. Note that

[TABLE]

with

[TABLE]

where $\tilde{\psi}:=h^{-1}\circ\psi$.

Prior to the proof of (S1.1), we establish the following result:

Lemma S1.3**.**

We have $\operatorname{E}(M_{n}^{*}(t))=0$ for all $t\in\left[0,1\right]$. For $s<t$

[TABLE]

where $F_{\tilde{\psi}(X_{i})-S_{n}}(x):=P\left(\tilde{\psi}(X_{i})-S_{n}\leq x\right)$.

Proof.

Note that

[TABLE]

where

[TABLE]

Obviously,

[TABLE]

Furthermore,

[TABLE]

The canonical $1$-stopping time $S_{n}$ takes on only countably many values $s_{i}$, $i\in\mathbb{N}$; see Ivanoff, (1983). Note that for $i<j$

[TABLE]

As a result, $\operatorname{E}\left(\alpha_{i}(\delta_{n},0)\alpha_{j}(\delta_{n},0)\right)=0$. Moreover,

[TABLE]

As a result,

[TABLE]

∎

Lemma S1.3 yields $\operatorname{E}(M^{*}_{n}(t))=0$ and

[TABLE]

where $F_{\lfloor nt\rfloor}$ denotes the empirical distribution function of $\tilde{\psi}(X_{i})$, $i\geq 1$, i.e.

[TABLE]

Note that $\sup_{0\leq x\leq 1}\left(F_{\lfloor nt\rfloor}(\delta_{n}+x)-F_{\lfloor nt\rfloor}(x)\right)$ converges to [math] almost surely due to the Glivenko-Cantelli theorem for stationary, ergodic sequences and since $F_{\tilde{\psi}(X_{1})}$ is a continuous distribution function. It follows that its expected value converges to [math], as well. Therefore, the right-hand side of (S1.12) converges to [math]. Thus, $M^{*}_{n}(t)$ converges to [math] in probability. This implies fidi-convergence of $M^{*}_{n}$ as a process with values in ${\mathbb{D}}\left[0,1\right]$. Again, we make use of Theorem 15.4 in Billingsley, (1968), i.e. we verify that for each positive $\varepsilon$ and $\eta$, there exists a $\delta$, $0<\delta<1$, and an integer $n_{0}$ such that

[TABLE]

Define

[TABLE]

where $S_{i}=M^{*}_{n}(\tau+\frac{i}{m}\delta)$.

For $t_{1}\leq t\leq t_{2}$ we have

[TABLE]

The Cauchy-Schwarz inequality yields

[TABLE]

By the same argument as in the proof of (S1.2) it follows that

[TABLE]

for any sequence $\gamma_{n}$, $n\in\mathbb{N}$, converging to [math]. Taking the right-continuity of $M_{n}^{*}$ into consideration, we may, as before, conclude that

[TABLE]

If we choose $\gamma_{n}$, $n\in\mathbb{N}$, such that

[TABLE]

converges to [math], the right-hand side of the above inequality vanishes as $n$ tends to $\infty$ due to the choice of $\gamma_{n}$. This concludes the proof of (S1.1) as well as the proof of Lemma 3.6. ∎

S2 Simulations

We computed the empirical size and the empirical power of the testing procedures on the basis of simulated time series according to the corresponding specifications in Section 6 of the main document. In addition to the simulation results referred to in the main document, the simulation tables contain the empirical power of the testing procedures when the change point is located closer to the testing region ($\tau=0.25$). The rejection rates are reported in Tables 5 to 10.