Table 9 Open in new tab Out-of-sample...

Panel A. Model versus historical mean: Full extrapolation
5	−7.76	−7.17	−6.99	−6.86	−7.26
10	−2.68	−2.05	−1.67	−1.55	−1.86
15	−0.02	0.35	0.81	0.92	0.99
20	0.83	1.13	1.40	1.42	1.58
25	1.01	1.35	1.61	1.61	1.75
30	1.63	2.40	2.69	2.56	2.51
35	2.03	2.71	3.00	2.93	2.95
40	2.43	2.98	3.30	3.24	3.34
45	2.69	3.11	3.39	3.35	3.42
50	3.18	3.56	3.82	3.82	3.94
55	−4.32	−4.25	−4.47	−4.62	−5.01
60	−7.38	−7.22	−7.46	−7.58	−7.95

Panel A. Model versus historical mean: Full extrapolation

R_{O S}^{2}

Rolling window length (months)

Without stabilization

With 25% stabilization

With 50% stabilization

With 75% stabilization

With 100% stabilization

−7.76

−7.17

−6.99

−6.86

−7.26

−2.68

−2.05

−1.67

−1.55

−1.86

−0.02

0.35

0.81

0.92

0.99

0.83

1.13

1.40

1.42

1.58

1.01

1.35

1.61

1.75

1.63

2.40

2.69

2.56

2.51

2.03

2.71

3.00

2.93

2.95

2.43

2.98

3.30

3.24

3.34

2.69

3.11

3.39

3.35

3.42

3.18

3.56

3.82

3.94

−4.32

−4.25

−4.47

−4.62

−5.01

−7.38

−7.22

−7.46

−7.58

−7.95

Panel A. Model versus historical mean: Full extrapolation
5	−7.76	−7.17	−6.99	−6.86	−7.26
10	−2.68	−2.05	−1.67	−1.55	−1.86
15	−0.02	0.35	0.81	0.92	0.99
20	0.83	1.13	1.40	1.42	1.58
25	1.01	1.35	1.61	1.61	1.75
30	1.63	2.40	2.69	2.56	2.51
35	2.03	2.71	3.00	2.93	2.95
40	2.43	2.98	3.30	3.24	3.34
45	2.69	3.11	3.39	3.35	3.42
50	3.18	3.56	3.82	3.82	3.94
55	−4.32	−4.25	−4.47	−4.62	−5.01
60	−7.38	−7.22	−7.46	−7.58	−7.95

Panel B. Model versus historical mean: Domain reduction (DRed) and symmetrization (DSym)
5	−7.35	−7.35	−8.21	−7.92	−7.61
10	−2.28	−2.28	−2.86	−3.22	−2.83
15	0.22	0.20	−0.15	−0.14	0.16
20	0.93	0.85	0.41	−0.12	0.14
25	1.05	0.93	0.41	0.16	0.41
30	1.45	1.23	0.65	1.56	1.92
35	1.92	1.73	1.16	2.07	2.46
40	2.23	1.98	1.34	2.63	3.12
45	2.52	2.30	1.75	3.09	3.57
50	2.97	2.72	2.07	3.63	4.10
55	−4.28	−3.97	−2.91	−4.60	−5.06
60	−7.27	−6.94	−5.87	−8.08	−8.74

Panel B. Model versus historical mean: Domain reduction (DRed) and symmetrization (DSym)

R_{O S}^{2}

Rolling window length (months)

25% DRed

50% DRed

75% DRed

DSym (d₁)

DSym (strike price)

−7.35

−8.21

−7.92

−7.61

−2.28

−2.86

−3.22

−2.83

0.22

0.20

−0.15

−0.14

0.16

0.93

0.85

0.41

−0.12

0.14

1.05

0.93

0.41

0.16

0.41

1.45

1.23

0.65

1.56

1.92

1.73

1.16

2.07

2.46

2.23

1.98

1.34

2.63

3.12

2.52

2.30

1.75

3.09

3.57

2.97

2.72

2.07

3.63

4.10

−4.28

−3.97

−2.91

−4.60

−5.06

−7.27

−6.94

−5.87

−8.08

−8.74

Panel B. Model versus historical mean: Domain reduction (DRed) and symmetrization (DSym)
5	−7.35	−7.35	−8.21	−7.92	−7.61
10	−2.28	−2.28	−2.86	−3.22	−2.83
15	0.22	0.20	−0.15	−0.14	0.16
20	0.93	0.85	0.41	−0.12	0.14
25	1.05	0.93	0.41	0.16	0.41
30	1.45	1.23	0.65	1.56	1.92
35	1.92	1.73	1.16	2.07	2.46
40	2.23	1.98	1.34	2.63	3.12
45	2.52	2.30	1.75	3.09	3.57
50	2.97	2.72	2.07	3.63	4.10
55	−4.28	−3.97	−2.91	−4.60	−5.06
60	−7.27	−6.94	−5.87	−8.08	−8.74

Table 9

Open in new tab

Out-of-sample return forecasting ability of implied moments after alternative treatments

Panel A. Model versus historical mean: Full extrapolation
	$R_{O S}^{2}$
Rolling window length (months)	Without stabilization	With 25% stabilization	With 50% stabilization	With 75% stabilization	With 100% stabilization
5	−7.76	−7.17	−6.99	−6.86	−7.26
10	−2.68	−2.05	−1.67	−1.55	−1.86
15	−0.02	0.35	0.81	0.92	0.99
20	0.83	1.13	1.40	1.42	1.58
25	1.01	1.35	1.61	1.61	1.75
30	1.63	2.40	2.69	2.56	2.51
35	2.03	2.71	3.00	2.93	2.95
40	2.43	2.98	3.30	3.24	3.34
45	2.69	3.11	3.39	3.35	3.42
50	3.18	3.56	3.82	3.82	3.94
55	−4.32	−4.25	−4.47	−4.62	−5.01
60	−7.38	−7.22	−7.46	−7.58	−7.95

Panel A. Model versus historical mean: Full extrapolation
	$R_{O S}^{2}$
Rolling window length (months)	Without stabilization	With 25% stabilization	With 50% stabilization	With 75% stabilization	With 100% stabilization
5	−7.76	−7.17	−6.99	−6.86	−7.26
10	−2.68	−2.05	−1.67	−1.55	−1.86
15	−0.02	0.35	0.81	0.92	0.99
20	0.83	1.13	1.40	1.42	1.58
25	1.01	1.35	1.61	1.61	1.75
30	1.63	2.40	2.69	2.56	2.51
35	2.03	2.71	3.00	2.93	2.95
40	2.43	2.98	3.30	3.24	3.34
45	2.69	3.11	3.39	3.35	3.42
50	3.18	3.56	3.82	3.82	3.94
55	−4.32	−4.25	−4.47	−4.62	−5.01
60	−7.38	−7.22	−7.46	−7.58	−7.95

Panel B. Model versus historical mean: Domain reduction (DRed) and symmetrization (DSym)
	$R_{O S}^{2}$
Rolling window length (months)	25% DRed	50% DRed	75% DRed	DSym (d₁)	DSym (strike price)
5	−7.35	−7.35	−8.21	−7.92	−7.61
10	−2.28	−2.28	−2.86	−3.22	−2.83
15	0.22	0.20	−0.15	−0.14	0.16
20	0.93	0.85	0.41	−0.12	0.14
25	1.05	0.93	0.41	0.16	0.41
30	1.45	1.23	0.65	1.56	1.92
35	1.92	1.73	1.16	2.07	2.46
40	2.23	1.98	1.34	2.63	3.12
45	2.52	2.30	1.75	3.09	3.57
50	2.97	2.72	2.07	3.63	4.10
55	−4.28	−3.97	−2.91	−4.60	−5.06
60	−7.27	−6.94	−5.87	−8.08	−8.74

Panel B. Model versus historical mean: Domain reduction (DRed) and symmetrization (DSym)
	$R_{O S}^{2}$
Rolling window length (months)	25% DRed	50% DRed	75% DRed	DSym (d₁)	DSym (strike price)
5	−7.35	−7.35	−8.21	−7.92	−7.61
10	−2.28	−2.28	−2.86	−3.22	−2.83
15	0.22	0.20	−0.15	−0.14	0.16
20	0.93	0.85	0.41	−0.12	0.14
25	1.05	0.93	0.41	0.16	0.41
30	1.45	1.23	0.65	1.56	1.92
35	1.92	1.73	1.16	2.07	2.46
40	2.23	1.98	1.34	2.63	3.12
45	2.52	2.30	1.75	3.09	3.57
50	2.97	2.72	2.07	3.63	4.10
55	−4.28	−3.97	−2.91	−4.60	−5.06
60	−7.27	−6.94	−5.87	−8.08	−8.74

Notes: This table presents the results of the out-of-sample return forecasting ability tests conducted while applying alternative integration domain treatments. Panel A reports the test results where flat extrapolation is applied up to strike prices equivalent to one-third and three times the underlying price. Panel B presents the test results where domain reduction and symmetrization are applied as truncation treatments. Following Campbell and Thompson (2008), we report the $R_{O S}^{2}$ statistic, which is defined as $R_{O S}^{2} = 1 - [\sum_{t = 1}^{T} {(r_{t} - \hat{r_{t}})}^{2}] / [\sum_{t = 1}^{T} {(r_{t} - \bar{r_{t}})}^{2}]$ ⁠, where $\hat{r_{t}}$ is the fitted value derived from a predictive regression estimated through the rolling window that ends at time $t - 1$ ⁠, and $\bar{r_{t}}$ is the benchmark value for the rolling window. Benchmark value is defined as the historical mean log-return. A positive value of $R_{O S}^{2}$ indicates that the predictive regression produces a lower mean squared prediction error than the benchmark value. The value of $R_{O S}^{2}$ is expressed as a percentage.

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