Table 8 Open in new tab In-sample return...

Panel A. Full extrapolation (FE)
r(t−1)	−0.040	−0.028	−0.008	−0.012	0.012
(−0.37)	(−0.26)	(−0.07)	(−0.11)	(0.11)
ΔVOL(t−1)	4.995	5.495	6.167	6.115	6.968^*
(1.36)	(1.47)	(1.58)	(1.59)	(1.71)
ΔSKEW(t−1)	0.443^**	0.664^***	0.937^**	0.782^**	0.831^*
(2.38)	(2.62)	(2.03)	(1.99)	(1.77)
ΔKURT(t−1)	0.020	0.066^**	0.086	0.064	0.093
(1.46)	(1.96)	(1.32)	(1.13)	(0.76)
Intercept	0.059^*	0.059^*	0.058^*	0.058^*	0.056^*
(1.85)	(1.82)	(1.78)	(1.79)	(1.74)
# of obs.	1,526	1,526	1,526	1,526	1,526
R²	0.0430	0.0444	0.0460	0.0467	0.0482

Panel A. Full extrapolation (FE)

FE with 0% stabilization

FE with 25% stabilization

FE with 50% stabilization

FE with 75% stabilization

FE with 100% stabilization

r(t−1)

−0.040

−0.028

−0.008

−0.012

0.012

(−0.37)

(−0.26)

(−0.07)

(−0.11)

(0.11)

ΔVOL(t−1)

4.995

5.495

6.167

6.115

6.968^*

(1.36)

(1.47)

(1.58)

(1.59)

(1.71)

ΔSKEW(t−1)

0.443^**

0.664^***

0.937^**

0.782^**

0.831^*

(2.38)

(2.62)

(2.03)

(1.99)

(1.77)

ΔKURT(t−1)

0.020

0.066^**

0.086

0.064

0.093

(1.46)

(1.96)

(1.32)

(1.13)

(0.76)

Intercept

0.059^*

0.058^*

0.056^*

(1.85)

(1.82)

(1.78)

(1.79)

(1.74)

# of obs.

1,526

R²

0.0430

0.0444

0.0460

0.0467

0.0482

Panel A. Full extrapolation (FE)
r(t−1)	−0.040	−0.028	−0.008	−0.012	0.012
(−0.37)	(−0.26)	(−0.07)	(−0.11)	(0.11)
ΔVOL(t−1)	4.995	5.495	6.167	6.115	6.968^*
(1.36)	(1.47)	(1.58)	(1.59)	(1.71)
ΔSKEW(t−1)	0.443^**	0.664^***	0.937^**	0.782^**	0.831^*
(2.38)	(2.62)	(2.03)	(1.99)	(1.77)
ΔKURT(t−1)	0.020	0.066^**	0.086	0.064	0.093
(1.46)	(1.96)	(1.32)	(1.13)	(0.76)
Intercept	0.059^*	0.059^*	0.058^*	0.058^*	0.056^*
(1.85)	(1.82)	(1.78)	(1.79)	(1.74)
# of obs.	1,526	1,526	1,526	1,526	1,526
R²	0.0430	0.0444	0.0460	0.0467	0.0482

Panel B. Domain reduction (DRed) and symmetrization (DSym)
r(t−1)	−0.042	−0.051	−0.080	−0.001	0.024
(−0.38)	(−0.47)	(−0.73)	(0.02)	(0.21)
ΔVOL(t−1)	4.972	4.667	3.592	6.745	7.848^*
(1.31)	(1.21)	(0.92)	(1.53)	(1.78)
ΔSKEW(t−1)	0.471^**	0.481^***	0.504^**	1.629^***	1.600^***
(2.26)	(2.30)	(2.39)	(2.65)	(2.83)
ΔKURT(t−1)	0.025	0.027	0.032	0.072	0.031
(1.37)	(1.41)	(1.52)	(0.91)	(0.41)
Intercept	0.060^*	0.060^*	0.062^**	0.057^*	0.056^*
(1.85)	(1.86)	(1.91)	(1.78)	(1.74)
# of obs.	1,526	1,526	1,526	1,526	1,526
R²	0.0422	0.0413	0.0393	0.0519	0.0561

Panel B. Domain reduction (DRed) and symmetrization (DSym)

25% DRed

50% DRed

75% DRed

DSym (d₁)

DSym (strike price)

r(t−1)

−0.042

−0.051

−0.080

−0.001

0.024

(−0.38)

(−0.47)

(−0.73)

(0.02)

(0.21)

ΔVOL(t−1)

4.972

4.667

3.592

6.745

7.848^*

(1.31)

(1.21)

(0.92)

(1.53)

(1.78)

ΔSKEW(t−1)

0.471^**

0.481^***

0.504^**

1.629^***

1.600^***

(2.26)

(2.30)

(2.39)

(2.65)

(2.83)

ΔKURT(t−1)

0.025

0.027

0.032

0.072

0.031

(1.37)

(1.41)

(1.52)

(0.91)

(0.41)

Intercept

0.060^*

0.062^**

0.057^*

0.056^*

(1.85)

(1.86)

(1.91)

(1.78)

(1.74)

# of obs.

1,526

R²

0.0422

0.0413

0.0393

0.0519

0.0561

Panel B. Domain reduction (DRed) and symmetrization (DSym)
r(t−1)	−0.042	−0.051	−0.080	−0.001	0.024
(−0.38)	(−0.47)	(−0.73)	(0.02)	(0.21)
ΔVOL(t−1)	4.972	4.667	3.592	6.745	7.848^*
(1.31)	(1.21)	(0.92)	(1.53)	(1.78)
ΔSKEW(t−1)	0.471^**	0.481^***	0.504^**	1.629^***	1.600^***
(2.26)	(2.30)	(2.39)	(2.65)	(2.83)
ΔKURT(t−1)	0.025	0.027	0.032	0.072	0.031
(1.37)	(1.41)	(1.52)	(0.91)	(0.41)
Intercept	0.060^*	0.060^*	0.062^**	0.057^*	0.056^*
(1.85)	(1.86)	(1.91)	(1.78)	(1.74)
# of obs.	1,526	1,526	1,526	1,526	1,526
R²	0.0422	0.0413	0.0393	0.0519	0.0561

Table 8

Open in new tab

In-sample return predictive ability of implied moments after alternative treatments

Panel A. Full extrapolation (FE)
	FE with 0% stabilization	FE with 25% stabilization	FE with 50% stabilization	FE with 75% stabilization	FE with 100% stabilization
r(t−1)	−0.040	−0.028	−0.008	−0.012	0.012
r(t−1)	(−0.37)	(−0.26)	(−0.07)	(−0.11)	(0.11)
ΔVOL(t−1)	4.995	5.495	6.167	6.115	6.968^*
ΔVOL(t−1)	(1.36)	(1.47)	(1.58)	(1.59)	(1.71)
ΔSKEW(t−1)	0.443^**	0.664^***	0.937^**	0.782^**	0.831^*
ΔSKEW(t−1)	(2.38)	(2.62)	(2.03)	(1.99)	(1.77)
ΔKURT(t−1)	0.020	0.066^**	0.086	0.064	0.093
ΔKURT(t−1)	(1.46)	(1.96)	(1.32)	(1.13)	(0.76)
Intercept	0.059^*	0.059^*	0.058^*	0.058^*	0.056^*
Intercept	(1.85)	(1.82)	(1.78)	(1.79)	(1.74)
# of obs.	1,526	1,526	1,526	1,526	1,526
R²	0.0430	0.0444	0.0460	0.0467	0.0482

Panel A. Full extrapolation (FE)
	FE with 0% stabilization	FE with 25% stabilization	FE with 50% stabilization	FE with 75% stabilization	FE with 100% stabilization
r(t−1)	−0.040	−0.028	−0.008	−0.012	0.012
r(t−1)	(−0.37)	(−0.26)	(−0.07)	(−0.11)	(0.11)
ΔVOL(t−1)	4.995	5.495	6.167	6.115	6.968^*
ΔVOL(t−1)	(1.36)	(1.47)	(1.58)	(1.59)	(1.71)
ΔSKEW(t−1)	0.443^**	0.664^***	0.937^**	0.782^**	0.831^*
ΔSKEW(t−1)	(2.38)	(2.62)	(2.03)	(1.99)	(1.77)
ΔKURT(t−1)	0.020	0.066^**	0.086	0.064	0.093
ΔKURT(t−1)	(1.46)	(1.96)	(1.32)	(1.13)	(0.76)
Intercept	0.059^*	0.059^*	0.058^*	0.058^*	0.056^*
Intercept	(1.85)	(1.82)	(1.78)	(1.79)	(1.74)
# of obs.	1,526	1,526	1,526	1,526	1,526
R²	0.0430	0.0444	0.0460	0.0467	0.0482

Panel B. Domain reduction (DRed) and symmetrization (DSym)
	25% DRed	50% DRed	75% DRed	DSym (d₁)	DSym (strike price)
r(t−1)	−0.042	−0.051	−0.080	−0.001	0.024
r(t−1)	(−0.38)	(−0.47)	(−0.73)	(0.02)	(0.21)
ΔVOL(t−1)	4.972	4.667	3.592	6.745	7.848^*
ΔVOL(t−1)	(1.31)	(1.21)	(0.92)	(1.53)	(1.78)
ΔSKEW(t−1)	0.471^**	0.481^***	0.504^**	1.629^***	1.600^***
ΔSKEW(t−1)	(2.26)	(2.30)	(2.39)	(2.65)	(2.83)
ΔKURT(t−1)	0.025	0.027	0.032	0.072	0.031
ΔKURT(t−1)	(1.37)	(1.41)	(1.52)	(0.91)	(0.41)
Intercept	0.060^*	0.060^*	0.062^**	0.057^*	0.056^*
Intercept	(1.85)	(1.86)	(1.91)	(1.78)	(1.74)
# of obs.	1,526	1,526	1,526	1,526	1,526
R²	0.0422	0.0413	0.0393	0.0519	0.0561

Panel B. Domain reduction (DRed) and symmetrization (DSym)
	25% DRed	50% DRed	75% DRed	DSym (d₁)	DSym (strike price)
r(t−1)	−0.042	−0.051	−0.080	−0.001	0.024
r(t−1)	(−0.38)	(−0.47)	(−0.73)	(0.02)	(0.21)
ΔVOL(t−1)	4.972	4.667	3.592	6.745	7.848^*
ΔVOL(t−1)	(1.31)	(1.21)	(0.92)	(1.53)	(1.78)
ΔSKEW(t−1)	0.471^**	0.481^***	0.504^**	1.629^***	1.600^***
ΔSKEW(t−1)	(2.26)	(2.30)	(2.39)	(2.65)	(2.83)
ΔKURT(t−1)	0.025	0.027	0.032	0.072	0.031
ΔKURT(t−1)	(1.37)	(1.41)	(1.52)	(0.91)	(0.41)
Intercept	0.060^*	0.060^*	0.062^**	0.057^*	0.056^*
Intercept	(1.85)	(1.86)	(1.91)	(1.78)	(1.74)
# of obs.	1,526	1,526	1,526	1,526	1,526
R²	0.0422	0.0413	0.0393	0.0519	0.0561

Notes: This table presents the results of in-sample return prediction tests conducted under 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. The dependent variable, $r (t)$ ⁠, represents the S&P 500 index log-return on day $t$ ⁠, expressed as a percentage. $Δ VOL (t)$ ⁠, $Δ SKEW (t)$ ⁠, and $Δ KURT (t)$ denote the daily first-order differences of implied volatility, skewness, and kurtosis estimates on day t, respectively. The Huber–White sandwich estimator is used to estimate standard errors; therefore, the unadjusted $R^{2}$ is reported. $t$ -statistics are presented in parentheses. ***, **, and * indicate statistical significance at the 1%, 5%, and 10% levels, respectively.

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