Non-linear unit root testing with arctangent trend: Simulation and applications in finance

Figures & data

Table 1. Asymptotic critical values of KSS test

Significance (%)	Case 1 (no deterministic term)	Case 2 (with constant)	Case 3 (with constant and trend)
1	−2.81	−3.47	−3.91
5	−2.23	−2.91	−3.38
10	−1.94	−2.63	−3.13

Table 2. Asymptotic critical values of the arctangent test

Significance (%)	Case 1 (no deterministic term)	Case 2 (with constant)	Case 3 (with constant and trend)
1	−2.51	−3.44	−3.94
5	−1.97	−2.87	−3.41
10	−1.61	−2.56	−3.13

Figure 1. Histograms of the null hypothesis of arctangent test (from left to right, no deterministic term, with constant, with constant and trend).

Table 3. Size of the arctangent test

	Case 2 (with constant)			Case 3 (with constant and trend)
β = 1	Arctan	KSS	DF	Arctan	KSS	DF
T = 50	0.04	0.06	0.05	0.04	0.06	0.03
T = 100	0.04	0.05	0.06	0.05	0.07	0.08
T = 200	0.05	0.05	0.08	0.05	0.06	0.07

Figure 2. LSTAR, ESTAR, and arctangent logistic functions with θ = 1 (left panel) and with θ = 0.5 (right panel).

Table 4. Power of the arctangent test

Case 2 (with constant)
	θ = 0.01			θ = 0.05			θ = 0.1			θ = 1
	Arctan	KSS	DF	Arctan	KSS	DF	Arctan	KSS	DF	Arctan	KSS	DF
γ = −1.0
T = 50	0.876	0.749	0.955	0.927	0.829	0.983	0.953	0.869	0.992	0.995	0.991	0.998
T = 100	0.999	0.989	1.000	1.000	0.994	1.000	1.000	1.000	1.000	1.000	1.000	1.000
T = 200	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
γ = −0.5
T = 50	0.499	0.435	0.549	0.642	0.534	0.661	0.747	0.655	0.743	0.913	0.802	0.961
T = 100	0.894	0.804	0.922	0.969	0.918	0.983	0.993	0.972	0.995	1.000	0.990	1.000
T = 200	1.000	0.993	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
γ = −0.25
T = 50	0.340	0.351	0.271	0.553	0.540	0.481	0.652	0.615	0.543	0.486	0.414	0.534
T = 100	0.809	0.793	0.790	0.928	0.894	0.879	0.943	0.894	0.900	0.872	0.762	0.918
T = 200	0.990	0.980	0.986	1.000	0.996	0.996	1.000	0.999	1.000	1.000	0.993	1.000
γ = −0.15
T = 50	0.007	0.011	0.005	0.444	0.454	0.369	0.516	0.536	0.448	0.342	0.329	0.328
T = 100	0.800	0.834	0.639	0.972	0.963	0.939	0.946	0.931	0.931	0.717	0.648	0.709
T = 200	1.000	1.000	1.000	1.000	1.000	1.000	0.999	0.997	0.999	0.950	0.890	0.959

Case 3 (with constant and trend)
	θ = 0.01			θ = 0.05			θ = 0.1			θ = 1
	Arctan	KSS	DF	Arctan	KSS	DF	Arctan	KSS	DF	Arctan	KSS	DF
γ = −1.0
T = 50	0.694	0.517	0.846	0.799	0.644	0.910	0.851	0.720	0.936	0.980	0.970	0.992
T = 100	0.992	0.953	1.000	0.999	0.974	1.000	1.000	0.994	1.000	1.000	1.000	1.000
T = 200	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
γ = −0.5
T = 50	0.247	0.192	0.240	0.340	0.277	0.331	0.480	0.381	0.455	0.743	0.574	0.826
T = 100	0.666	0.563	0.645	0.841	0.733	0.873	0.950	0.879	0.940	0.999	0.973	0.998
T = 200	0.990	0.961	0.994	1.000	0.996	1.000	1.000	1.000	1.000	1.000	1.000	1.000
γ = −0.25
T = 50	0.066	0.048	0.047	0.143	0.124	0.071	0.225	0.204	0.132	0.239	0.172	0.227
T = 100	0.213	0.178	0.106	0.466	0.446	0.370	0.656	0.590	0.502	0.637	0.485	0.636
T = 200	0.663	0.615	0.547	0.944	0.911	0.920	0.994	0.975	0.978	0.985	0.928	0.991
γ = −0.15
T = 50	0.026	0.028	0.052	0.052	0.049	0.013	0.091	0.064	0.026	0.094	0.061	0.060
T = 100	0.146	0.221	0.010	0.192	0.241	0.072	0.333	0.310	0.165	0.196	0.176	0.148
T = 200	0.291	0.602	0.058	0.865	0.871	0.683	0.892	0.843	0.755	0.573	0.479	0.550

Figure 3. Logarithm of BIST 100 Index between December 2011 and December 2016.

Figure 4. Logarithm of Nikkei 225 Index between August 2016 and December 2016.

Figure 5. Logarithm of FTSE/JSE Index between May 2016 and December 2016.

Table 5. ADF test results

Case 2 (with constant)		Case 3 (with constant and trend)
Index	t-statistics	Index	t-statistics
BIST 100	−2.5556	BIST 100	−2.5591
Nikkei 225	−2.8746*	Nikkei 225	−2.9026
FTSE/JSE	−2.5753*	FTSE/JSE	−2.1980

*Denotes 10% significance level.

**Denotes 5% significance level.

***Denotes 1% significance level.

Table 6. KSS test results

Case 2 (with constant)		Case 3 (with constant and trend)
Index	t-statistics	Index	t-statistics
BIST 100	−2.5800	BIST 100—5 year	−2.5862
Nikkei 225	−2.8785*	Nikkei 225—3 month	−2.9024
FTSE/JSE	−2.5651	FTSE/JSE—6 month	−2.1890

^*Denotes 10% significance level.

**Denotes 5% significance level.

***Denotes 1% significance level.

Table 7. Arctangent test results

Case 2 (with constant)		Case 3 (with constant and trend)
Index	t-statistics	Index	t-statistics
BIST 100—5 year	−2.5678*	BIST 100—5 year	−2.5728
Nikkei 225—3 month	−2.8785**	Nikkei 225—3 month	−2.9025
FTSE/JSE—6 month	−2.5703*	FTSE/JSE—6 month	−2.1936

*Denotes 10% significance level.

**Denotes 5% significance level.

***Denotes 1% significance level.