Detecting long-range dependence with truncated ratios of periodogram ordinates

Figures & data

Table 1. Critical values for the tests based on the test statistics $T^{-}$ and $T^{+}.$

	K = 5	6	7	8	9	10	15	20	24	25	50	100	150
0.05%	−2.702	−2.881	−3.139	−3.358	−3.499	−3.712	−4.406	−5.174	−5.503	−5.802	−8.340	−11.500	−13.932
1%	−2.479	−2.697	−2.934	−3.057	−3.257	−3.402	−4.084	−4.657	−5.026	−5.344	−7.488	−10.279	−12.733
2.50%	−2.205	−2.371	−2.534	−2.661	−2.805	−2.929	−3.524	−4.041	−4.275	−4.454	−6.260	−8.748	−10.828
5%	−1.907	−2.039	−2.159	−2.273	−2.396	−2.475	−2.992	−3.438	−3.669	−3.739	−5.346	−7.425	−9.056
95%	2.159	2.342	2.450	2.590	2.789	2.854	3.240	3.815	4.095	4.050	5.503	7.673	9.507
97.50%	2.551	2.758	2.890	3.083	3.277	3.415	3.954	4.499	5.021	4.872	6.630	9.336	11.377
99%	2.957	3.225	3.429	3.719	3.860	4.035	4.654	5.322	5.904	5.740	7.827	11.250	13.925
99.50%	3.205	3.657	3.832	4.087	4.304	4.401	5.137	5.862	6.507	6.382	8.728	12.466	15.258

Table 2. Rejection rates (at the 5% level of significance) obtained by applying the tests (with $K=15$) based on the test statistics $T_a^{-}$ (if $d_A<d_0)$ and $T_a^{+}$ (if $d_A\ge d_0),$ respectively, to samples of size $n=250$ from an ARFIMA(0,d,0)-GARCH(1,1) model with $t(5)$ distributed innovations and GARCH parameters α₀ = 1, α₁ = 0.1, β₁ = 0.8999.

d0\dA	−0.4	−0.3	−0.2	−0.1	0.0	0.1	0.2	0.3	0.4
−0.4	0.054	0.093	0.142	0.238	0.355	0.480	0.626	0.753	0.842
−0.3	0.076	0.056	0.080	0.144	0.230	0.343	0.489	0.633	0.747
−0.2	0.121	0.081	0.047	0.088	0.144	0.226	0.345	0.492	0.623
−0.1	0.197	0.131	0.071	0.056	0.083	0.141	0.224	0.349	0.487
0.0	0.302	0.216	0.121	0.082	0.050	0.087	0.133	0.235	0.357
0.1	0.441	0.328	0.206	0.142	0.077	0.052	0.075	0.137	0.223
0.2	0.587	0.468	0.317	0.219	0.133	0.080	0.045	0.082	0.134
0.3	0.728	0.615	0.459	0.331	0.221	0.132	0.077	0.046	0.072
0.4	0.842	0.754	0.617	0.487	0.330	0.218	0.133	0.076	0.045

Table 3. Rejection rates (at the 5% level of significance) obtained by applying the tests (with $K=15$) based on the test statistics $T_a^{-}$ (if $d_A<d_0)$ and $T_a^{+}$ (if $d_A\ge d_0),$ respectively, to samples of size $n=250$ from an ARFIMA(p,d,q)-GARCH(1,1) model with $t(5)$ distributed innovations, GARCH parameters α₀ = 1, α₁ = 0.1, β₁ = 0.8999, and different values of (a) the AR parameter ${\varphi }_1$ and (b) the MA parameter ${\theta }_1,$ respectively.

(a) p = 1, q = 0
d0\dA		−0.4			−0.2			0			0.2			0.4
ϕ1	0.25	0.50	0.75	0.25	0.50	0.75	0.25	0.50	0.75	0.25	0.50	0.75	0.25	0.50	0.75
−0.4	0.06	0.07	0.15	0.15	0.17	0.35	0.35	0.42	0.65	0.65	0.69	0.86	0.87	0.9	0.97
−0.2	0.12	0.09	0.05	0.05	0.06	0.14	0.15	0.17	0.36	0.36	0.4	0.64	0.67	0.71	0.88
0	0.3	0.26	0.13	0.12	0.1	0.05	0.05	0.06	0.15	0.14	0.17	0.35	0.39	0.42	0.66
0.2	0.58	0.54	0.32	0.31	0.27	0.13	0.13	0.11	0.05	0.05	0.06	0.15	0.17	0.18	0.38
0.4	0.84	0.81	0.61	0.61	0.55	0.32	0.31	0.27	0.14	0.13	0.1	0.05	0.07	0.06	0.15
(b) p = 0, q = 1
d0\dA		−0.4			−0.2			0			0.2			0.4
θ1	0.25	0.50	0.75	0.25	0.50	0.75	0.25	0.50	0.75	0.25	0.50	0.75	0.25	0.50	0.75
−0.4	0.06	0.05	0.03	0.13	0.11	0.06	0.33	0.28	0.14	0.63	0.56	0.35	0.85	0.81	0.63
−0.2	0.12	0.14	0.18	0.05	0.04	0.02	0.13	0.1	0.05	0.33	0.28	0.14	0.63	0.57	0.35
0	0.31	0.33	0.39	0.13	0.17	0.3	0.04	0.04	0.02	0.13	0.1	0.05	0.34	0.29	0.15
0.2	0.59	0.61	0.66	0.34	0.4	0.57	0.14	0.17	0.34	0.05	0.04	0.02	0.14	0.11	0.05
0.4	0.85	0.85	0.87	0.64	0.69	0.83	0.35	0.4	0.63	0.14	0.18	0.34	0.05	0.04	0.02

Figure 1. Cumulative number of rejections with $n=250,$ $K=24,$ α = 0.05 by tests based on $T_a^{-}$ with $d_0=0.2$(a,c) and $T_a^{+}$ with $d_0=-0.2$(b,d): gold returns (bold in gold), DJIA returns (bold in magenta), synthetic ARFIMA(0,d,0)-series with $d=-0.2$ (red), $d=-0.1$ (orange), $d=0$ (darkgray), $d=0.1$ (green), $d=0.2$ (blue).

Figure 2. (a) Log Warsaw Stock Exchange Index, (b) Log returns, (c) Log Amsterdam Exchange Index (green) & log BEL 20 Index (red), (d) Difference, (e) 5 Year Treasury Yield Index (green) & 10 Year Treasury Yield Index (red), (f) Difference, (g) Log iShares Physical Gold ETC (green) & log iShares Silver ETC (red), (h) Difference.