Exponential tilted likelihood for stationary time series models

Figures & data

Table 1. Coverage probabilities for ARMA(1,1) models.

		${ϵ}_t\sim N(0,1)$			${ϵ}_t\sim t\left(5\right)$
Method	$(\varphi ,\theta )$	T = 100	200	300	T = 100	200	300
	(0.1,0.7)	0.898	0.917	0.927	0.897	0.920	0.924
ET	(0.7,0.3)	0.903	0.922	0.933	0.906	0.922	0.941
	(0.6,0.8)	0.850	0.885	0.903	0.853	0.894	0.911
	(0.2,0.4)	0.925	0.934	0.936	0.919	0.934	0.938
	(0.1,0.7)	0.914	0.924	0.934	0.911	0.927	0.930
AET	(0.7,0.3)	0.919	0.930	0.936	0.923	0.929	0.948
	(0.6,0.8)	0.866	0.894	0.908	0.872	0.902	0.917
	(0.2,0.4)	0.937	0.942	0.943	0.940	0.944	0.946
		${ϵ}_t\sim {\chi }^2\left(5\right)$			${ϵ}_t\sim \exp \left(1\right)$
Method	$(\varphi ,\theta )$	T = 100	200	300	T = 100	200	300
	(0.1,0.7)	0.896	0.917	0.920	0.892	0.917	0.940
ET	(0.7,0.3)	0.911	0.927	0.930	0.913	0.924	0.933
	(0.6,0.8)	0.850	0.889	0.907	0.855	0.894	0.917
	(0.2,0.4)	0.916	0.930	0.942	0.918	0.932	0.939
	(0.1,0.7)	0.909	0.926	0.935	0.911	0.925	0.945
AET	(0.7,0.3)	0.929	0.935	0.937	0.927	0.933	0.939
	(0.6,0.8)	0.869	0.898	0.912	0.871	0.901	0.923
	(0.2,0.4)	0.933	0.939	0.949	0.935	0.942	0.944

Table 2. Coverage probabilities for ARFIMA(1,d,0) models.

		${ϵ}_t\sim N(0,1)$			${ϵ}_t\sim t\left(5\right)$
Method	$(\varphi ,d)$	T = 100	200	300	T = 100	200	300
	(0.3,0.1)	0.867	0.898	0.908	0.866	0.896	0.909
ET	(0.9,0.1)	0.779	0.848	0.878	0.789	0.845	0.879
	(0.3,0.4)	0.864	0.888	0.911	0.866	0.894	0.917
	(0.8,0.4)	0.792	0.847	0.868	0.798	0.849	0.876
	(0.3,0.1)	0.884	0.908	0.914	0.882	0.904	0.915
	(0.9,0.1)	0.796	0.858	0.884	0.807	0.857	0.884
AET	(0.3,0.4)	0.879	0.897	0.916	0.881	0.905	0.923
	(0.8,0.4)	0.808	0.859	0.875	0.821	0.861	0.883
		${ϵ}_t\sim {\chi }^2\left(5\right)$			${ϵ}_t\sim \exp \left(1\right)$
Method	$(\varphi ,d)$	T = 100	200	300	T = 100	200	300
	(0.3,0.1)	0.868	0.903	0.908	0.866	0.895	0.905
ET	(0.9,0.1)	0.783	0.858	0.884	0.792	0.850	0.880
	(0.3,0.4)	0.873	0.891	0.908	0.878	0.903	0.911
	(0.8,0.4)	0.796	0.855	0.874	0.809	0.852	0.874
	(0.3,0.1)	0.884	0.910	0.909	0.883	0.906	0.910
AET	(0.9,0.1)	0.801	0.866	0.892	0.808	0.858	0.886
	(0.3,0.4)	0.890	0.901	0.914	0.894	0.913	0.915
	(0.8,0.4)	0.821	0.866	0.879	0.828	0.862	0.883

Table 3. Coverage probabilities for ARFIMA(2,d,0) models.

		${ϵ}_t\sim N(0,1)$			${ϵ}_t\sim t\left(5\right)$
Method	$({\varphi }_1,{\varphi }_2,d)$	T = 100	200	300	T = 100	200	300
	(0.1,0.2,0.1)	0.848	0.879	0.887	0.842	0.876	0.893
ET	(0.1,0.2,0.4)	0.877	0.907	0.913	0.880	0.904	0.915
	(0.2,0.7,0.2)	0.855	0.894	0.916	0.866	0.900	0.905
	(0.5,0.3,0.1)	0.829	0.884	0.902	0.837	0.875	0.900
	(0.1,0.2,0.1)	0.866	0.890	0.896	0.866	0.894	0.900
AET	(0.1,0.2,0.4)	0.899	0.919	0.921	0.900	0.920	0.921
	(0.2,0.7,0.2)	0.884	0.906	0.925	0.890	0.911	0.913
	(0.5,0.3,0.1)	0.852	0.899	0.911	0.858	0.896	0.911
		${ϵ}_t\sim {\chi }^2\left(5\right)$			${ϵ}_t\sim \exp \left(1\right)$
Method	$({\varphi }_1,{\varphi }_2,d)$	T = 100	200	300	T = 100	200	300
	(0.1,0.2,0.1)	0.853	0.873	0.886	0.839	0.883	0.900
ET	(0.1,0.2,0.4)	0.882	0.906	0.912	0.876	0.907	0.913
	(0.2,0.7,0.2)	0.855	0.901	0.917	0.863	0.906	0.914
	(0.5,0.3,0.1)	0.846	0.892	0.911	0.838	0.888	0.910
	(0.1,0.2,0.1)	0.856	0.885	0.894	0.860	0.894	0.906
AET	(0.1,0.2,0.4)	0.906	0.914	0.920	0.902	0.918	0.920
	(0.2,0.7,0.2)	0.882	0.912	0.925	0.886	0.919	0.921
	(0.5,0.3,0.1)	0.871	0.901	0.916	0.859	0.900	0.919

Table 4. Coverage probabilities for ARFIMA(0,d,1) models.

		${ϵ}_t\sim N(0,1)$			${ϵ}_t\sim t\left(5\right)$
Method	$(d,\theta )$	T = 100	200	300	T = 100	200	300
	(0.1,0.3)	0.870	0.897	0.906	0.871	0.898	0.905
	(0.1,0.9)	0.861	0.882	0.902	0.853	0.890	0.899
ET	(0.3,0.5)	0.837	0.876	0.891	0.848	0.881	0.898
	(0.4,0.3)	0.876	0.891	0.910	0.873	0.900	0.908
	(0.4,0.9)	0.860	0.887	0.907	0.849	0.894	0.913
	(0.1,0.3)	0.887	0.905	0.912	0.889	0.905	0.909
	(0.1,0.9)	0.881	0.894	0.908	0.875	0.899	0.905
AET	(0.3,0.5)	0.858	0.887	0.898	0.861	0.888	0.903
	(0.4,0.3)	0.891	0.899	0.916	0.889	0.907	0.914
	(0.4,0.9)	0.874	0.897	0.914	0.868	0.904	0.919
		${ϵ}_t\sim {\chi }^2\left(5\right)$			${ϵ}_t\sim \exp \left(1\right)$
Method	$(d,\theta )$	T = 100	200	300	T = 100	200	300
	(0.1,0.3)	0.874	0.894	0.910	0.879	0.894	0.912
	(0.1,0.9)	0.858	0.878	0.894	0.842	0.872	0.900
ET	(0.3,0.5)	0.855	0.875	0.898	0.862	0.875	0.892
	(0.4,0.3)	0.880	0.905	0.909	0.879	0.905	0.914
	(0.4,0.9)	0.862	0.894	0.912	0.863	0.892	0.909
	(0.1,0.3)	0.890	0.902	0.917	0.898	0.904	0.918
	(0.1,0.9)	0.882	0.888	0.901	0.864	0.880	0.905
AET	(0.3,0.5)	0.873	0.887	0.904	0.874	0.885	0.898
	(0.4,0.3)	0.897	0.914	0.918	0.895	0.914	0.918
	(0.4,0.9)	0.877	0.900	0.918	0.877	0.902	0.915

Table 5. Coverage probabilities for ARFIMA(0,d,2) models.

		${ϵ}_t\sim N(0,1)$			${ϵ}_t\sim t\left(5\right)$
Method	$(d,{\theta }_1,{\theta }_2)$	T = 100	200	300	T = 100	200	300
	(0.1,0.1,0.2)	0.825	0.860	0.890	0.827	0.860	0.881
ET	(0.2,0.2,0.7)	0.822	0.860	0.878	0.820	0.862	0.878
	(0.4,0.2,0.7)	0.802	0.857	0.877	0.802	0.850	0.885
	(0.4,0.8,0.1)	0.842	0.887	0.898	0.841	0.886	0.903
	(0.1,0.1,0.2)	0.847	0.872	0.898	0.848	0.875	0.887
AET	(0.2,0.2,0.7)	0.842	0.874	0.890	0.843	0.874	0.890
	(0.4,0.2,0.7)	0.825	0.870	0.885	0.822	0.860	0.892
	(0.4,0.8,0.1)	0.864	0.898	0.905	0.862	0.894	0.910
		${ϵ}_t\sim {\chi }^2\left(5\right)$			${ϵ}_t\sim \exp \left(1\right)$
Method	$(d,{\theta }_1,{\theta }_2)$	T = 100	200	300	T = 100	200	300
	(0.1,0.1,0.2)	0.821	0.865	0.886	0.836	0.874	0.895
ET	(0.2,0.2,0.7)	0.813	0.863	0.883	0.812	0.862	0.886
	(0.4,0.2,0.7)	0.804	0.847	0.875	0.800	0.859	0.882
	(0.4,0.8,0.1)	0.829	0.887	0.900	0.844	0.884	0.900
	(0.1,0.1,0.2)	0.841	0.878	0.893	0.858	0.886	0.903
AET	(0.2,0.2,0.7)	0.839	0.876	0.890	0.835	0.874	0.895
	(0.4,0.2,0.7)	0.828	0.863	0.883	0.822	0.872	0.891
	(0.4,0.8,0.1)	0.849	0.899	0.910	0.864	0.896	0.906

Table 6. Coverage probabilities for ARFIMA(1,d,1) models.

		${ϵ}_t\sim N(0,1)$			${ϵ}_t\sim t\left(5\right)$
Method	$(\varphi ,d,\theta )$	T = 100	200	300	T = 100	200	300
	(0.3,0.1,0.2)	0.840	0.888	0.900	0.860	0.890	0.900
ET	(0.6,0.3,0.4)	0.824	0.857	0.888	0.832	0.864	0.882
	(0.3,0.4,0.6)	0.813	0.864	0.883	0.812	0.869	0.881
	(0.5,0.4,0.2)	0.835	0.871	0.883	0.835	0.878	0.899
	(0.3,0.1,0.2)	0.869	0.900	0.905	0.882	0.900	0.909
AET	(0.6,0.3,0.4)	0.852	0.870	0.892	0.853	0.877	0.890
	(0.3,0.4,0.6)	0.839	0.876	0.891	0.837	0.881	0.888
	(0.5,0.4,0.2)	0.856	0.886	0.890	0.856	0.889	0.907
		${ϵ}_t\sim {\chi }^2\left(5\right)$			${ϵ}_t\sim \exp \left(1\right)$
Method	$(\varphi ,d,\theta )$	T = 100	200	300	T = 100	200	300
	(0.3,0.1,0.2)	0.859	0.889	0.902	0.849	0.885	0.896
ET	(0.6,0.3,0.4)	0.817	0.866	0.878	0.817	0.870	0.881
	(0.3,0.4,0.6)	0.830	0.860	0.891	0.836	0.865	0.887
	(0.5,0.4,0.2)	0.842	0.875	0.892	0.836	0.875	0.888
	(0.3,0.1,0.2)	0.899	0.911	0.960	0.875	0.897	0.903
AET	(0.6,0.3,0.4)	0.878	0.885	0.941	0.839	0.882	0.891
	(0.3,0.4,0.6)	0.877	0.899	0.947	0.860	0.879	0.898
	(0.5,0.4,0.2)	0.887	0.903	0.949	0.858	0.888	0.895

Table 7. Performance of point estimates of the autoregression parameter ϕ in AR(1) under model misspecification.

${\varphi }_0$	T		C			M
			EL	ET	ETEL	EL	ET	ETEL
0.3	50	Bias	−0.0409	−0.0453	−0.0405	−0.0150	−0.0109	−0.0109
		SD	0.1330	0.1326	0.1325	0.0911	0.0913	0.0913
		RMSE	0.1391	0.1401	0.1385	0.0923	0.0920	0.0920
	100	Bias	−0.0208	−0.0242	−0.0206	−0.0019	0.0040	0.0040
		SD	0.0961	0.0960	0.0959	0.0727	0.0721	0.0721
		RMSE	0.0983	0.0990	0.0981	0.0727	0.0722	0.0722
	200	Bias	−0.0105	−0.0125	−0.0104	0.0044	0.0126	0.0126
		SD	0.0697	0.0697	0.0697	0.0551	0.0530	0.0530
		RMSE	0.0705	0.0708	0.0704	0.0553	0.0545	0.0545
	500	Bias	−0.0027	−0.0026	−0.0026	0.0050	0.0175	0.0175
		SD	0.0431	0.0431	0.0431	0.0413	0.0372	0.0372
		RMSE	0.0432	0.0432	0.0431	0.0416	0.0411	0.0411
0.7	50	Bias	−0.0764	−0.0759	−0.0762	−0.1764	−0.1655	−0.1794
		SD	0.1165	0.1177	0.1163	0.0627	0.0528	0.0566
		RMSE	0.1392	0.1401	0.1390	0.1872	0.1737	0.1881
	100	Bias	−0.0355	−0.0353	−0.0355	−0.1459	−0.1386	−0.1500
		SD	0.0784	0.0788	0.0782	0.0407	0.0295	0.0342
		RMSE	0.0860	0.0864	0.0859	0.1515	0.1418	0.1538
	200	Bias	−0.0172	−0.0170	−0.0171	−0.1314	−0.1272	−0.1370
		SD	0.0521	0.0522	0.0521	0.0275	0.0169	0.0217
		RMSE	0.0548	0.0549	0.0548	0.1342	0.1284	0.1387
	500	Bias	−0.0067	−0.0066	−0.0067	−0.1224	−0.1198	−0.1297
		SD	0.0327	0.0327	0.0327	0.0187	0.0136	0.0136
		RMSE	0.0334	0.0334	0.0334	0.1238	0.1205	0.1305

Note: EL, ET and ETEL stand for empirical likelihood, exponential tilted and their hybird, respectively.

Figure 1. The S$\mathrm{\&}$P 500 VIX series.

Figure 2. 95% confidence region of the parameters for fitted ARFIMA(1,d,0) of the S&P 500 VIX series.

Table 8. Point estimates of the parameter vector $(\varphi ,d)$ by the three methods.

EL	ET	ETEL
(0.0815,0.7961)	(0.7952,0.2856)	(0.0812,0.7963)