Table 7 Open in new tab Out-of-sample...

Panel A. With versus without stabilization
5	−0.37	0.30	0.27	0.27	−0.25
10	−0.37	0.43	0.54	0.52	0.14
15	−0.24	0.38	0.62	0.67	0.42
20	−0.13	0.32	0.43	0.44	0.25
25	−0.07	0.37	0.45	0.46	0.29
30	0.20	1.01	1.02	0.97	0.57
35	0.13	0.85	0.90	0.88	0.60
40	0.18	0.84	0.91	0.90	0.64
45	0.14	0.64	0.71	0.70	0.51
50	0.16	0.60	0.71	0.77	0.60
55	0.07	−0.07	−0.26	−0.46	−1.05
60	0.02	−0.08	−0.27	−0.49	−1.02

Panel A. With versus without stabilization

R_{O S}^{2}

Rolling window length (months)

No stabilization

0% stabilization

25% stabilization

50% stabilization

75% stabilization

100% stabilization

0.00

−0.37

0.30

0.27

−0.25

0.00

−0.37

0.43

0.54

0.52

0.14

0.00

−0.24

0.38

0.62

0.67

0.42

0.00

−0.13

0.32

0.43

0.44

0.25

0.00

−0.07

0.37

0.45

0.46

0.29

0.00

0.20

1.01

1.02

0.97

0.57

0.00

0.13

0.85

0.90

0.88

0.60

0.00

0.18

0.84

0.91

0.90

0.64

0.00

0.14

0.64

0.71

0.70

0.51

0.00

0.16

0.60

0.71

0.77

0.60

0.00

0.07

−0.07

−0.26

−0.46

−1.05

0.00

0.02

−0.08

−0.27

−0.49

−1.02

Panel A. With versus without stabilization
5	−0.37	0.30	0.27	0.27	−0.25
10	−0.37	0.43	0.54	0.52	0.14
15	−0.24	0.38	0.62	0.67	0.42
20	−0.13	0.32	0.43	0.44	0.25
25	−0.07	0.37	0.45	0.46	0.29
30	0.20	1.01	1.02	0.97	0.57
35	0.13	0.85	0.90	0.88	0.60
40	0.18	0.84	0.91	0.90	0.64
45	0.14	0.64	0.71	0.70	0.51
50	0.16	0.60	0.71	0.77	0.60
55	0.07	−0.07	−0.26	−0.46	−1.05
60	0.02	−0.08	−0.27	−0.49	−1.02

Panel B. Model versus historical mean
5	−7.42	−7.79	−7.12	−7.15	−7.15	−7.67
10	−2.33	−2.70	−1.90	−1.79	−1.81	−2.20
15	0.18	−0.05	0.56	0.80	0.85	0.61
20	0.94	0.81	1.26	1.37	1.38	1.19
25	1.08	1.01	1.45	1.53	1.54	1.37
30	1.54	1.74	2.55	2.56	2.51	2.11
35	2.00	2.13	2.84	2.90	2.88	2.59
40	2.33	2.51	3.18	3.25	3.23	2.98
45	2.61	2.75	3.25	3.32	3.31	3.12
50	3.08	3.24	3.68	3.79	3.85	3.68
55	−4.37	−4.30	−4.44	−4.63	−4.83	−5.42
60	−7.38	−7.36	−7.46	−7.65	−7.86	−8.40

Panel B. Model versus historical mean

R_{O S}^{2}

Rolling window length (months)

No stabilization

0% stabilization

25% stabilization

50% stabilization

75% stabilization

100% stabilization

−7.42

−7.79

−7.12

−7.15

−7.67

−2.33

−2.70

−1.90

−1.79

−1.81

−2.20

0.18

−0.05

0.56

0.80

0.85

0.61

0.94

0.81

1.26

1.37

1.38

1.19

1.08

1.01

1.45

1.53

1.54

1.37

1.54

1.74

2.55

2.56

2.51

2.11

2.00

2.13

2.84

2.90

2.88

2.59

2.33

2.51

3.18

3.25

3.23

2.98

2.61

2.75

3.25

3.32

3.31

3.12

3.08

3.24

3.68

3.79

3.85

3.68

−4.37

−4.30

−4.44

−4.63

−4.83

−5.42

−7.38

−7.36

−7.46

−7.65

−7.86

−8.40

Panel B. Model versus historical mean
5	−7.42	−7.79	−7.12	−7.15	−7.15	−7.67
10	−2.33	−2.70	−1.90	−1.79	−1.81	−2.20
15	0.18	−0.05	0.56	0.80	0.85	0.61
20	0.94	0.81	1.26	1.37	1.38	1.19
25	1.08	1.01	1.45	1.53	1.54	1.37
30	1.54	1.74	2.55	2.56	2.51	2.11
35	2.00	2.13	2.84	2.90	2.88	2.59
40	2.33	2.51	3.18	3.25	3.23	2.98
45	2.61	2.75	3.25	3.32	3.31	3.12
50	3.08	3.24	3.68	3.79	3.85	3.68
55	−4.37	−4.30	−4.44	−4.63	−4.83	−5.42
60	−7.38	−7.36	−7.46	−7.65	−7.86	−8.40

Table 7

Open in new tab

Out-of-sample return forecasting ability of implied moments after stabilization

Panel A. With versus without stabilization
	$R_{O S}^{2}$
Rolling window length (months)	No stabilization	0% stabilization	25% stabilization	50% stabilization	75% stabilization	100% stabilization
5	0.00	−0.37	0.30	0.27	0.27	−0.25
10	0.00	−0.37	0.43	0.54	0.52	0.14
15	0.00	−0.24	0.38	0.62	0.67	0.42
20	0.00	−0.13	0.32	0.43	0.44	0.25
25	0.00	−0.07	0.37	0.45	0.46	0.29
30	0.00	0.20	1.01	1.02	0.97	0.57
35	0.00	0.13	0.85	0.90	0.88	0.60
40	0.00	0.18	0.84	0.91	0.90	0.64
45	0.00	0.14	0.64	0.71	0.70	0.51
50	0.00	0.16	0.60	0.71	0.77	0.60
55	0.00	0.07	−0.07	−0.26	−0.46	−1.05
60	0.00	0.02	−0.08	−0.27	−0.49	−1.02

Panel A. With versus without stabilization
	$R_{O S}^{2}$
Rolling window length (months)	No stabilization	0% stabilization	25% stabilization	50% stabilization	75% stabilization	100% stabilization
5	0.00	−0.37	0.30	0.27	0.27	−0.25
10	0.00	−0.37	0.43	0.54	0.52	0.14
15	0.00	−0.24	0.38	0.62	0.67	0.42
20	0.00	−0.13	0.32	0.43	0.44	0.25
25	0.00	−0.07	0.37	0.45	0.46	0.29
30	0.00	0.20	1.01	1.02	0.97	0.57
35	0.00	0.13	0.85	0.90	0.88	0.60
40	0.00	0.18	0.84	0.91	0.90	0.64
45	0.00	0.14	0.64	0.71	0.70	0.51
50	0.00	0.16	0.60	0.71	0.77	0.60
55	0.00	0.07	−0.07	−0.26	−0.46	−1.05
60	0.00	0.02	−0.08	−0.27	−0.49	−1.02

Panel B. Model versus historical mean
	$R_{O S}^{2}$
Rolling window length (months)	No stabilization	0% stabilization	25% stabilization	50% stabilization	75% stabilization	100% stabilization
5	−7.42	−7.79	−7.12	−7.15	−7.15	−7.67
10	−2.33	−2.70	−1.90	−1.79	−1.81	−2.20
15	0.18	−0.05	0.56	0.80	0.85	0.61
20	0.94	0.81	1.26	1.37	1.38	1.19
25	1.08	1.01	1.45	1.53	1.54	1.37
30	1.54	1.74	2.55	2.56	2.51	2.11
35	2.00	2.13	2.84	2.90	2.88	2.59
40	2.33	2.51	3.18	3.25	3.23	2.98
45	2.61	2.75	3.25	3.32	3.31	3.12
50	3.08	3.24	3.68	3.79	3.85	3.68
55	−4.37	−4.30	−4.44	−4.63	−4.83	−5.42
60	−7.38	−7.36	−7.46	−7.65	−7.86	−8.40

Panel B. Model versus historical mean
	$R_{O S}^{2}$
Rolling window length (months)	No stabilization	0% stabilization	25% stabilization	50% stabilization	75% stabilization	100% stabilization
5	−7.42	−7.79	−7.12	−7.15	−7.15	−7.67
10	−2.33	−2.70	−1.90	−1.79	−1.81	−2.20
15	0.18	−0.05	0.56	0.80	0.85	0.61
20	0.94	0.81	1.26	1.37	1.38	1.19
25	1.08	1.01	1.45	1.53	1.54	1.37
30	1.54	1.74	2.55	2.56	2.51	2.11
35	2.00	2.13	2.84	2.90	2.88	2.59
40	2.33	2.51	3.18	3.25	3.23	2.98
45	2.61	2.75	3.25	3.32	3.31	3.12
50	3.08	3.24	3.68	3.79	3.85	3.68
55	−4.37	−4.30	−4.44	−4.63	−4.83	−5.42
60	−7.38	−7.36	−7.46	−7.65	−7.86	−8.40

Notes: This table presents the results of the out-of-sample return forecasting ability test. 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 fitted value estimated without stabilization for Panel A, and the historical mean log-return for Panel B. 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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