䉷 2009 by The University of Chicago. All rights reserved.
Appendix D from Liu et al., “Investment-Based Expected Stock
Returns”
(JPE, vol. 117, no. 6, p. 1105)
A. Unabridged Tables
Tables D1–D4 in this appendix report the unabridged versions of tables 1, 3, 4, and 5 in the paper.
B. Robustness
This section reports detailed results from a long list of robustness tests that are described only briefly in the
paper.
1. Second-Stage GMM
The estimation and test results reported in the paper are based on one-stage GMM. Table D5 reports that the
parameter estimates from the second-stage GMM are similar to the first-stage estimates. Table D6 and figures D1
and D2 show that the inferences about the expected return errors and variance errors from the second-stage
GMM are largely similar to those from the one-stage GMM.
2. The Market Value of Debt
The estimation and test results reported in the paper are based on the book value of debt as the proxy for the
market value of debt, Bit, in the model. In this appendix we use the Bernanke and Campbell (1988) algorithm to
convert the book value of debt into the market value of debt. The detailed algorithm is described in Whited
(1992, 1457–58).
The imputed market values of debt are highly correlated with the book values of debt, and their use makes
little difference for the Euler equation estimation results. Table D7 reports that the parameter estimates with the
market value of debt are similar to the benchmark estimates. Table D8 and figures D3 and D4 show that the
inferences about the expected return errors and variance errors with the market value of debt are largely similar
to those with the book value of debt.
3. Value-Weighted Returns
The benchmark estimation results reported in the paper are based on equal-weighted portfolio returns. Table D9
shows that the estimates of the adjustment cost parameter, a, are somewhat smaller than those in the benchmark
estimation when we use the q-theory model to match expected returns only. The estimates of a are similar to
those in the benchmark estimation when the model is used to match both expected returns and variances.
Table D10 shows that the q-theory model performs reasonably well in accounting for the expected returns and
variances of the testing portfolios. Although the model produces an expected return error of ⫺2.23 percent per
year (t p ⫺1.91) for the high-minus-low SUE portfolio when matching expected returns alone, this error is only
1.74 percent per year (t p 0.89) when matching means and variances of returns of 10 SUE portfolios
simultaneously. This error is smaller than the corresponding error of 12.37 percent per year (t p 2.51) in the
benchmark estimation. The expected return error for the high-minus-low B/M portfolio is only ⫺0.53 percent per
year (t p ⫺0.18) when matching both expected returns and stock return variances. In contrast, this error in the
benchmark estimation with equal-weighted returns is much larger, 5.89 percent (t p 1.08). The other aspects of
the estimation and tests are similar to the benchmark specification. More details are in the scatter plots of figures
D5 and D6.
1
App. D from Liu et al., “Investment-Based Expected Stock Returns”
4. Alternative Window Length in the Standard Bartlett Kernel
We consider two alternative cases with window length in the standard Bartlett kernel different from five (the
benchmark specification). The first case has the window length of one, and the second case has the window
length of 10. Changing the window length in the Bartlett kernel does not affect the parameter estimates. Only
their standard errors, x 2, and p-values of various tests are affected. As such, the expected return errors and the
variance errors are identical to those in the benchmark estimation, and the scatter plots also remain the same.
Even with standard errors affected, tables D11–D14 show that the basic inferences are largely similar to those in
the benchmark estimation and tests.
5. Alternative Measures of Capital and Investment
Our basic results are robust to an alternative measure of the capital stock as the net property, plant, and
equipment (Compustat annual item 8) and to an alternative definition of investment as capital expenditures (item
128). Tables D15–D18 report the parameter estimates and Euler equation errors and figures D7–D10 report the
related scatter plots.
6. Time-Invariant Tax Rates
The benchmark specification uses time-varying tax rates. Using time-invariant tax rates measured at the sample
mean of 42.3 percent from 1963 to 2005 yields largely similar results. See table D19 for the parameter estimates
and tests of overidentification, table D20 for Euler equation errors, and figures D11 and D12 for the scatter plots
in the alternative specification with time-invariant tax rates.
7. Portfolio-Specific Tax Rates
The benchmark specification uses time-varying but portfolio-invariant tax rates. Using time-varying and
portfolio-specific corporate tax rates yields largely similar results. To measure the portfolio-specific corporate tax
i
rate, tt⫹1
, we first construct firm-specific tax rates using the trichotomous variable approach of Graham (1996)
and then take the value-weighted tax rates across all firms within a given portfolio i. In estimating the model
with firm-specific tax rates, we assume that firms take these tax rates as exogenous. Table D21 reports the details
for the parameter estimates and tests of overidentification, table D22 for Euler equation errors, and figures D13
and D14 for the scatter plots in the alternative specification with time-varying and portfolio-specific tax rates.
Table D1
Descriptive Statistics of Testing Portfolio Returns
Low
2
3
4
5
6
r̄Si
10.89
12.04
14.95
15.43
18.95
19.39
jiS
ei
22.35
20.50
22.01
21.42
22.51
23.50
⫺1.69
[⫺.67]
⫺.18
[⫺.07]
3.28
[1.08]
6.56
[2.17]
6.43
[2.43]
7
8
9
High
H⫺L
m.a.e.
[p]
20.34
20.43
22.53
23.39
12.50
22.59
21.87
23.09
21.13
8.46
7.61
[3.09]
7.72
[3.46]
9.78
[3.37]
10.86
[4.96]
12.55
[12.69]
5.67
[.00]
A. 10 SUE Portfolios
2.59
[.98]
eiFF
⫺4.59
⫺2.78
⫺.47
eiC
[⫺2.24]
⫺8.07
[⫺1.34]
[⫺1.46]
⫺4.56
[⫺1.04]
[⫺.37]
⫺1.80
[⫺.41]
.56
[.24]
⫺2.42
[⫺.49]
1.96
3.05
4.26
6.07
6.83
9.47
14.06
4.01
[.00]
[1.02]
⫺.04
[.01]
[1.86]
⫺1.88
[.40]
[3.76]
⫺1.58
[.38]
[4.99]
4.13
[1.10]
[3.52]
6.39
[1.63]
[6.73]
5.31
[1.37]
[8.15]
13.38
[.58]
3.62
[.00]
6.34
[.00]
B. 10 B/M Portfolios
S
i
r̄
jiS
ei
8.65
14.14
15.68
15.54
17.93
18.47
19.50
19.94
22.81
25.78
17.13
27.93
⫺4.91
26.44
.81
24.94
2.88
23.64
3.02
24.92
5.19
23.12
6.10
23.69
7.41
22.49
8.20
22.93
11.26
26.97
13.65
20.54
18.56
[⫺1.85]
[.34]
[1.20]
[1.26]
[2.29]
[2.69]
[2.54]
[3.06]
[4.06]
[3.77]
[6.00]
eiFF
⫺.54
[⫺.18]
2.08
[1.26]
1.77
[1.24]
⫺1.06
[⫺1.02]
1.80
[1.81]
2.42
[2.40]
2.26
[1.98]
3.58
[4.05]
5.63
[5.81]
6.76
[2.57]
7.30
[2.55]
2.79
[.00]
eiC
⫺5.43
⫺1.86
⫺1.56
⫺1.60
.27
.56
.82
2.48
2.12
6.88
12.31
2.36
[.00]
[⫺.72]
[⫺.27]
[⫺.26]
[⫺.32]
[.06]
[.13]
[.19]
[1.06]
[.65]
[2.40]
2
[.21]
App. D from Liu et al., “Investment-Based Expected Stock Returns”
C. 10 CI Portfolios
⫺6.96
r̄Si
22.12
19.67
18.85
18.80
18.10
18.31
16.97
17.32
17.66
15.16
jiS
ei
32.42
26.17
23.73
23.31
22.26
22.25
21.91
21.44
25.32
26.73
8.21
[2.44]
7.05
[2.26]
6.25
[2.51]
6.57
[2.34]
5.89
[2.59]
6.26
[2.52]
4.73
[2.10]
5.39
[2.20]
4.84
[1.88]
1.91
[.67]
⫺6.30
[⫺4.49]
5.71
[.01]
eiFF
6.45
[2.90]
3.01
[1.72]
2.85
[2.22]
2.58
[2.18]
1.54
[1.60]
1.41
[1.52]
1.22
[1.27]
1.91
[1.79]
1.29
[1.06]
.11
[.06]
⫺6.34
[⫺6.51]
2.24
[.01]
eiC
4.03
[.75]
3.76
[.92]
1.66
[.42]
.12
[.03]
.46
[.12]
.04
[.01]
⫺1.46
[⫺.29]
⫺.57
[⫺.12]
⫺1.09
[⫺.22]
⫺4.35
[⫺.82]
⫺8.38
[⫺.39]
1.75
[.00]
11.37
Note.— For each testing portfolio i, we report in annualized percent the average stock return, r̄Si , the stock return volatility, jiS , the intercept from the CAPM
regression, ei, the intercept from the Fama-French three-factor regression, eiFF , and the model error from the standard consumption-CAPM, eiC . The H⫺L portfolio
is long in the high portfolio and short in the low portfolio. The heteroscedasticity- and autocorrelation-consistent t-statistics for the model errors are reported in
brackets beneath the corresponding errors. m.a.e. is the mean absolute error in annual percent for a given set of 10 testing portfolios. For the CAPM and the FamaFrench model, the p-values in brackets in the last column are for the Gibbons, Ross, and Shanken (1989) tests of the null hypothesis that the intercepts for a given
set of portfolios are jointly zero. For the standard consumption-CAPM the p-values are for the x 2 test from one-stage GMM that the moment restrictions are jointly
zero. In panel A for the standard consumption-CAPM the estimate of the time preference coefficient is b p 2.76 (standard error 0.93) and the estimate of risk
aversion is g p 127.59 (54.86). In panel B b p 3.31 (1.23) and g p 142.08 (58.51). In panel C b p 3.30 (1.23) and g p 143.28 (57.59).
Table D2
Euler Equation Errors
Low
2
3
4
5
6
7
8
9
High
H⫺L
⫺1.25
[⫺1.12]
⫺.50
[⫺.59]
⫺.15
[⫺.14]
⫺.40
[⫺.41]
1.62
[.59]
.05
[.03]
⫺2.73
[⫺1.37]
1.21
[.79]
.46
[.78]
⫺1.02
[⫺.94]
⫺1.45
[⫺1.24]
⫺.49
[⫺.41]
A. Euler Equation Errors from Matching Expected Returns
10 SUE Portfolios
eqi
.26
[.61]
⫺1.72
[⫺1.75]
⫺.05
[⫺.07]
.72
[.98]
1.66
[1.70]
.51
[.69]
.61
[1.07]
10 B/M Portfolios
q
i
e
⫺3.94
[⫺1.76]
⫺3.20
[⫺1.38]
⫺1.02
[⫺.66]
2.74
[1.39]
2.35
[1.37]
3.07
[1.11]
2.51
[1.31]
10 CI Portfolios
q
i
e
⫺.97
[⫺.51]
⫺2.71
[⫺1.95]
⫺.50
[⫺.61]
.93
[.96]
2.72
[1.74]
3.37
[2.19]
.94
[.75]
B. Euler Equation Errors from Matching Expected Returns and Variances
10 SUE Portfolios
eji
2
⫺.04
[⫺1.93]
⫺.04
[⫺1.85]
.01
[.76]
⫺.01
[⫺.40]
.02
[.95]
.03
[1.57]
.04
[1.66]
.01
[.92]
.02
[.80]
.03
[1.47]
.08
[1.83]
eqi
⫺6.99
[⫺2.24]
⫺6.50
[⫺2.27]
⫺2.12
[⫺1.49]
⫺1.62
[⫺1.06]
2.60
[1.91]
1.79
[1.32]
2.27
[1.82]
1.48
[.83]
3.75
[1.74]
5.38
[2.01]
12.37
[2.51]
eji
2
.10
[2.35]
.07
[2.19]
.06
[2.07]
.01
[.60]
.01
[.50]
.02
[.85]
.01
[.28]
⫺.01
[⫺.31]
⫺.02
[⫺1.19]
⫺.10
[⫺1.99]
⫺.20
[⫺2.39]
eqi
⫺6.46
[⫺1.89]
⫺3.83
[⫺1.73]
⫺2.11
[⫺1.02]
⫺.04
[⫺.02]
1.71
[.94]
2.60
[1.21]
3.54
[1.78]
3.11
[1.47]
1.85
[1.14]
⫺.58
[⫺.15]
5.89
[1.08]
10 B/M Portfolios
10 CI Portfolios
eji
.01
⫺.00
eqi
[.34]
1.29
[.49]
[⫺.17]
⫺2.51
[⫺1.56]
2
.02
[1.13]
⫺.11
[⫺.09]
.01
.03
.02
.02
.02
⫺.02
⫺.06
⫺.07
[.55]
1.86
[1.15]
[1.33]
3.47
[1.97]
[1.06]
3.48
[1.80]
[.75]
1.12
[.88]
[1.05]
.28
[.21]
[⫺1.13]
⫺2.82
[⫺1.53]
[⫺1.77]
⫺5.32
[⫺1.97]
[⫺1.36]
⫺6.60
[⫺2.04]
3
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Note.—Euler equation errors and t-statistics are from one-stage GMM estimation with an identity weighting matrix. In panel A the moment
S
Iw
S
Iw
conditions are E[rit⫹1
⫺ rit⫹1
] p 0. The expected return errors are defined as eqi { E T[rit⫹1
⫺ rit⫹1
] , in which E T[7] is the sample mean of the series
S
Iw
S
S
Iw
Iw
in brackets. In panel B the moment conditions are E[rit⫹1
⫺ rit⫹1
] p 0 and E[(rit⫹1
⫺ E[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E[rit⫹1
]) 2 ] p 0. The variance errors are
j2
S
S
2
Iw
Iw
2
defined as ei { E T[(rit⫹1 ⫺ E T[rit⫹1 ]) ⫺ (rit⫹1 ⫺ E T[rit⫹1 ]) ]. The expected return errors are defined as in panel A. In the last column we report
the difference in the expected return errors and the difference in the variance errors between the high and low portfolios, as well as their tstatistics (in brackets). Expected return errors are in annual percent, and variance errors are in annual decimals.
Table D3
Expected Returns Accounting
A. Characteristics in Levered Investment Returns
Low
2
3
4
5
6
7
8
9
High
H⫺L
[tH⫺L]
10 SUE Portfolios
I it⫹1/K it⫹1
(I it⫹1/K it⫹1)/(I it/K it)
Yit⫹1/K it⫹1
dit⫹1
wit
B
rit⫹1
.12
.89
.12
.93
.11
.96
.11
.96
.11
1.00
.11
1.01
.11
1.02
.11
1.05
.11
1.07
.12
1.06
.00
.17
[.70]
[4.06]
1.52
1.52
1.48
1.47
1.50
1.58
1.61
1.62
1.65
1.83
.31
[5.16]
.08
.07
.08
.07
.08
.08
.08
.08
.08
.08
.00
[.63]
.30
9.44
.29
9.48
.32
9.61
.28
9.81
.28
9.76
.26
9.53
.27
9.69
.27
9.45
.27
9.50
.21
9.38
⫺.10
⫺.06
[⫺5.83]
[⫺.27]
[⫺7.95]
10 B/M Portfolios
Iit/K it
(Iit⫹1/K it⫹1)/(Iit/K it)
Yit⫹1/K it⫹1
dit⫹1
wit
B
rit⫹1
.18
.14
.13
.13
.11
.11
.10
.10
.09
.08
⫺.10
.98
1.00
.99
.98
1.00
1.01
1.01
1.01
1.03
1.02
.04
[.68]
1.95
.10
1.88
.08
1.70
.08
1.58
.08
1.45
.07
1.37
.07
1.30
.07
1.32
.07
1.39
.07
1.38
.07
⫺.57
⫺.03
[⫺6.77]
[⫺5.01]
.08
.17
.25
.24
.27
.31
.34
.42
.45
.53
.44
[12.44]
8.17
8.01
8.04
8.12
8.09
8.24
8.32
8.29
8.33
8.52
.35
[1.05]
10 CI Portfolios
Iit/K it
(Iit⫹1/K it⫹1)/(Iit/K it)
.09
.09
.10
.10
.11
.12
.12
.13
.14
.16
.07
[11.06]
1.25
1.20
1.10
1.08
1.04
1.01
.99
.93
.91
.81
⫺.44
[⫺7.23]
Yit⫹1/K it⫹1
1.84
1.94
1.85
1.75
1.58
1.58
1.72
1.81
1.91
1.89
.05
[.38]
.08
.35
.07
.27
.07
.24
.07
.23
.07
.25
.07
.24
.08
.23
.07
.23
.07
.26
.08
.28
.00
⫺.07
[⫺.46]
[⫺2.59]
8.47
8.50
8.33
8.27
8.27
8.23
8.14
8.13
8.27
8.44
⫺.03
[⫺.15]
m.a.e.
dit⫹1
wit
B
rit⫹1
B. Expected Return Errors from Comparative Static Experiments
Low
2
3
4
5
6
7
8
9
High
H⫺L
10 SUE Portfolios
Iit/K it
⫺2.48
⫺4.45
.17
1.99
4.45
1.80
1.80
1.01
1.13
⫺4.26
⫺1.78
2.35
qit⫹1/qit
Yit⫹1/K it⫹1
⫺5.23
⫺.78
⫺5.09
⫺2.60
⫺1.83
⫺1.90
⫺1.10
⫺1.10
1.76
.39
1.18
.70
1.80
1.21
1.33
⫺.52
3.36
.62
3.62
3.53
8.85
4.31
2.62
1.34
.13
⫺1.35
.41
.83
1.89
.41
.44
⫺1.23
⫺.65
⫺1.46
⫺1.58
.88
21.25
wit
10 B/M Portfolios
⫺42.06
⫺21.23
⫺12.22
⫺4.31
4.69
10.98
17.07
21.19
30.53
48.17
90.23
qit⫹1/qit
⫺1.92
⫺.88
⫺.26
1.56
2.11
2.77
2.91
1.34
⫺.96
⫺4.06
⫺2.14
1.87
Yit⫹1/K it⫹1
wit
.16
⫺6.00
.76
⫺5.20
1.26
⫺1.68
2.93
1.66
.92
2.19
.14
2.63
⫺1.63
3.37
⫺2.63
4.03
⫺2.65
3.34
⫺6.33
5.58
⫺6.49
11.58
1.94
3.57
Iit/K it
10 CI Portfolios
Iit/K it
qit⫹1/qit
Yit⫹1/K it⫹1
wit
2.86
⫺.40
1.03
2.06
3.50
3.08
.07
⫺.89
⫺3.01
⫺5.67
⫺8.53
2.26
.73
⫺1.50
.10
1.39
2.97
3.41
.82
⫺.29
⫺2.01
⫺3.87
⫺4.60
1.71
.57
⫺.25
.17
.54
⫺.44
.40
⫺.15
.67
.69
.09
⫺.48
.40
1.80
⫺2.32
⫺.93
.36
2.61
3.26
.62
⫺.02
⫺.71
⫺.91
⫺2.71
1.35
4
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Note.— Panel A reports the averages of investment-to-capital, Iit/K it , future investment growth, (Iit⫹1/K it⫹1)/(Iit/K it) , sales-to-capital, Yit⫹1/K it⫹1 , the depreciation
B
rate, dit⫹1, market leverage, wit , and corporate bond returns in annual percent, rit⫹1
, for all the testing portfolios. The column H⫺L reports the average differences
between high and low portfolios and the column [tH⫺L] reports the heteroscedasticity- and autocorrelation-consistent t-statistics for the test that the differences equal
zero. Panel B performs four comparative static experiments denoted Iit/K it , qit⫹1/qit , Yit⫹1/K it⫹1 , and wit, in which qit⫹1/qit p [1 ⫹ (1 ⫺ tt⫹1 )a(Iit⫹1 /K it⫹1 )]/[1 ⫹ (1 ⫺
tt )a(Iit /K it )]. In the experiment denoted Yit⫹1 /K it⫹1, we set Yit⫹1 /K it⫹1 for a given set of 10 portfolios, indexed by i, to be its cross-sectional average in t ⫹ 1. We
then use the parameters reported in panel A of table 2 in the paper to reconstruct the levered investment returns, while keeping all the other characteristics unchanged.
S
Iw
The other three experiments are designed analogously. We report the expected return errors defined as eiq { E T[rit⫹1
⫺ rit⫹1
] for the testing portfolios, the highminus-low portfolios and the mean absolute value of eiq (m.a.e.).
Table D4
Correlations
Low
2
3
4
5
6
7
8
9
High
All
A. 10 SUE Portfolios
S
Iw
r(rit⫹1
, rit⫹1
)
S
it
Iw
it⫹1
r(r , r
S
it⫹1
r(r
)
, Iit⫹1 /Iit )
r(ritS, Iit⫹1 /Iit )
⫺.28
⫺.19
⫺.19
⫺.15
⫺.21
⫺.26
⫺.23
⫺.22
⫺.18
⫺.26
⫺.11**
.22
.15
.11
.12
.01
.13
.21
.11
.03
.14
⫺.29
⫺.13
⫺.14
⫺.14
⫺.24
⫺.16
⫺.14
⫺.19
⫺.15
⫺.23
.18
.11
.07
.04
.01
.06
.13
.08
⫺.03
⫺.00
.14**
⫺.12
⫺.13
⫺.05
⫺.12**
.19***
⫺.08
B. 10 B/M Portfolios
S
it⫹1
r(r
S
it
Iw
it⫹1
,r
)
Iw
it⫹1
r(r , r )
S
r(rit⫹1
, Iit⫹1 /Iit )
r(ritS, Iit⫹1 /Iit )
⫺.17
⫺.35***
⫺.23
.06
⫺.14
.04
⫺.19
.35***
⫺.23
⫺.17
⫺.12
⫺.26*
.25
⫺.14
.23
⫺.06
.20
⫺.07
.12
.12
.31*
.20
.17
.04
.00
⫺.30*
.07
⫺.34**
⫺.24
.24
⫺.28*
.12
.08
.00
.23
⫺.16
⫺.04
.26
.33***
⫺.13
.29*
.22***
⫺.15***
.14***
C. 10 CI Portfolios
S
it⫹1
r(r
S
it
Iw
it⫹1
,r
Iw
it⫹1
r(r , r
S
it⫹1
r(r
)
)
, Iit⫹1 /Iit )
r(ritS, Iit⫹1 /Iit )
.22
.44***
.28*
.20
⫺.19
.13
.19
⫺.07
⫺.26
.05
.23
.10
⫺.11
⫺.37**
⫺.31*
⫺.30*
⫺.31*
.26
.04
.16
.26
.27*
.33**
.34**
.30*
.30*
.10
.21
.14
.25
⫺.30*
.30*
⫺.25
⫺.12
.03
.26
⫺.06
.21***
⫺.04
.16***
S
Note.—For each testing portfolio we report time-series correlations of stock returns (contemporaneous, rit⫹1
, and one-period-lagged, ritS ) with levered
Iw
investment returns, rit⫹1
, and with investment growth, Iit⫹1 /Iit . r(7, 7) denotes the correlation between the two series in the parentheses. In the last column,
denoted all, we report the correlations and their significance by pooling all the observations for a given set of 10 testing portfolios (SUE, B/M, or CI).
The levered investment returns are constructed using the parameters in panel A of table 2 in the paper.
∗
Significant at the 10 percent level.
∗∗
Significant at the 5 percent level.
∗∗∗
Significant at the 1 percent level.
Table D5
Parameter Estimates and Tests of
Overidentification from the Second-Stage GMM
SUE
B/M
CI
A. Matching Expected Returns
a
a
8.66
[1.20]
.30
23.17
[16.49]
.54
[.03]
[.20]
1.12
[.31]
.20
[.01]
x2
d.f.
4.45
8
293.31
8
7.39
8
p
m.a.e.
.82
2.87
.00
2.42
.50
1.47
B. Matching Expected Returns and
Variances
a
29.37
[4.17]
11.20
[1.10]
16.13
[2.06]
a
.61
[.06]
.34
[.01]
.36
[.03]
5
App. D from Liu et al., “Investment-Based Expected Stock Returns”
x2(2)
d.f.(2)
6.85
8
7.53
8
7.46
8
.55
.02
.48
.04
.49
.02
x2(1)
d.f.(1)
6.98
8
6.60
8
7.19
8
p(1)
m.a.e.(1)
.54
3.46
.58
2.59
.52
2.21
7.27
18
8.25
18
8.50
18
.99
.98
.97
p(2)
m.a.e.(2)
x2
d.f.
p
Note.—Estimates and tests are from the second-stage GMM estimation
with the two-step weighting matrix. In panel A the moment conditions
S
Iw
are E[rit⫹1
⫺ rit⫹1
] p 0. a is the adjustment cost parameter and a is
capital’s share. Their standard errors are reported in brackets beneath
2
the estimates. x is the statistic from the second-stage GMM that the
moment conditions are jointly zero. d.f. is the degrees of freedom, and
p is the p-value associated with the test. m.a.e. is the mean absolute
S
Iw
error, E T[rit⫹1
⫺ rit⫹1
], in which E T[7] is the sample mean of the series
in brackets, in annual percent across a given set of testing portfolios. In
S
Iw
S
panel B the moment conditions are E[rit⫹1
⫺ rit⫹1
] p 0 and E[(rit⫹1
⫺
S
Iw
Iw
2
E[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E[rit⫹1
]) 2 ] p 0. x (2)
, d.f.(2), and p(2) are the statistic,
degrees of freedom, and p-value for the x 2 test that the variance errors,
S
S
Iw
Iw
defined as E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ]), are jointly zero.
2
m.a.e.(2) is the mean absolute error in annual decimals. x (1)
, d.f.(1), and
p(1) are the statistic, degrees of freedom, and p-value for the x 2 test
that the expected return errors, defined in the same way as in panel A,
are jointly zero. m.a.e.(1) is the mean absolute expected return error in
annual percent. x 2, d.f., and p are the statistic, degrees of freedom, and
p-value of the test that both the expected return errors and the variance
errors are jointly zero.
Table D6
Euler Equation Errors from the Second-Stage GMM
Low
2
3
4
5
6
7
8
9
High
H⫺L
2.59
[1.71]
⫺.98
[⫺.51]
A. Euler Equation Errors from Matching Expected Returns
10 SUE Portfolios
eqi
3.57
[1.35]
1.33
[.70]
2.92
[1.21]
3.55
[1.46]
4.40
[1.66]
3.28
[1.31]
3.39
[1.55]
1.45
[.68]
2.18
[1.34]
10 B/M Portfolios
q
i
e
⫺4.64
⫺4.09
⫺1.96
1.90
1.44
2.11
1.48
.36
⫺1.48
⫺4.75
⫺.12
[⫺1.38]
[⫺.98]
[⫺.92]
[.70]
[.61]
[.99]
[.65]
[.07]
[⫺.48]
[⫺.75]
[⫺.02]
1.33
[.85]
⫺.02
[⫺.01]
⫺.15
[⫺.07]
.17
[.12]
10 CI Portfolios
⫺.33
[⫺.14]
q
i
e
⫺2.09
[⫺.76]
.15
[.06]
1.56
[.67]
3.33
[1.21]
4.03
[1.60]
1.68
[1.06]
B. Euler Equation Errors from Matching Expected Returns and Variances
10 SUE Portfolios
e ij
⫺.05
⫺.04
⫺.01
.02
.03
.04
.01
.02
.03
eqi
[⫺1.33]
⫺6.98
[⫺1.47]
[⫺1.34]
⫺6.49
[⫺1.74]
[.25]
⫺2.10
[⫺.62]
[⫺.28]
⫺1.61
[⫺.45]
[.92]
2.61
[.72]
[1.13]
1.81
[.47]
[1.11]
2.28
[.71]
[.37]
1.50
[.51]
[.42]
3.77
[.99]
[.84]
5.41
[1.41]
[2.18]
12.39
[2.45]
.10
[2.24]
.08
[1.70]
.06
[1.58]
.02
[.41]
.02
[.35]
.02
[.52]
.01
[.22]
⫺.01
[⫺.20]
⫺.02
[⫺.57]
⫺.10
[⫺1.43]
⫺.20
[⫺2.35]
1.73
2.63
3.58
3.15
1.92
⫺.44
2
.01
.08
10 B/M Portfolios
2
e ij
eqi
⫺6.49
⫺3.84
⫺2.11
⫺.02
6
6.04
App. D from Liu et al., “Investment-Based Expected Stock Returns”
[⫺1.77]
[⫺1.19]
[⫺.71]
[⫺.01]
[.54]
[.90]
[1.40]
[.83]
[.62]
[⫺.07]
[1.03]
10 CI Portfolios
e
j2
i
eqi
.01
[.57]
.00
[⫺.13]
.02
[.76]
.01
[.37]
.03
[.99]
.02
[.59]
.02
[.50]
.02
[.77]
⫺.02
[⫺.64]
⫺.06
[⫺1.17]
⫺.07
[⫺1.45]
1.14
[.22]
⫺2.63
[⫺.76]
⫺.22
[⫺.06]
1.76
[.45]
3.37
[.95]
3.39
[1.10]
1.02
[.46]
.18
[.09]
⫺2.92
[⫺.81]
⫺5.42
[⫺1.34]
⫺6.56
[⫺1.56]
Note.—Euler equation errors and t-statistics (in brackets) are from the second-stage GMM estimation with the two-step weighting matrix.
S
q
S
Iw
In panel A the moment conditions are E[rit⫹1
⫺ rIw
it⫹1] p 0. The expected return errors are defined as ei { E T[rit⫹1 ⫺ rit⫹1 ] , in which E T[7] is
S
Iw
S
S
Iw
the sample mean of the series in brackets. In panel B the moment conditions are E[rit⫹1
⫺ rit⫹1
] p 0 and E[(rit⫹1
⫺ E[rit⫹1
]) 2 ⫺ (rit⫹1
⫺
2
Iw
S
S
Iw
Iw
E[rit⫹1
]) 2 ] p 0. The variance errors are defined as eij { E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ]. The expected return errors are defined
as in panel A. In the last column we report the difference in the expected return errors and the difference in the variance errors between
the high and low portfolios, as well as their t-statistics. Expected return errors are in annual percent, and variance errors are in annual
decimals.
Table D7
Parameter Estimates and Tests of
Overidentification, the Market Value of Debt per
the Bernanke and Campbell (1988) Algorithm
SUE
B/M
CI
A. Matching Expected Returns
a
a
x2
d.f.
p
m.a.e.
7.91
[1.85]
.33
[.03]
23.31
[30.52]
.51
[.37]
1.03
[.30]
.21
[.02]
4.85
8
.77
.74
5.66
8
.69
2.49
6.81
8
.56
1.42
B. Matching Expected Returns and
Variances
a
a
2
x (2)
d.f.(2)
p(2)
m.a.e.(2)
30.99
[18.48]
.64
[.30]
10.27
[4.13]
.32
[.06]
17.93
[6.90]
.38
[.09]
5.58
8
.70
6.50
8
.59
6.38
8
.60
.02
.05
.02
2
x (1)
d.f.(1)
5.68
8
5.85
8
6.32
8
p(1)
m.a.e.(1)
.68
3.50
.66
3.30
.61
2.03
x2
5.77
d.f.
p
18
1.00
7.07
18
.99
7
6.91
18
.99
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Note.—We measure the market value of debt, Bit, per the Bernanke
and Campbell (1988) algorithm. Estimates and tests are from the firststage GMM estimation with an identity weighting matrix. In panel A
S
Iw
the moment conditions are E[rit⫹1
⫺ rit⫹1
] p 0. a is the adjustment cost
parameter and a is capital’s share. Their standard errors are reported in
brackets beneath the estimates. x 2 is the statistic from the first-stage
GMM that the moment conditions are jointly zero. d.f. is the degrees of
freedom, and p is the p-value associated with the test. m.a.e. is the mean
S
Iw
absolute error, E T [rit⫹1
⫺ rit⫹1
], in which E T[7] is the sample mean of the
series in brackets, in annual percent across a given set of testing
S
Iw
portfolios. In panel B the moment conditions are E[rit⫹1
⫺ rit⫹1
] p 0 and
S
S
Iw
Iw
2
E[(rit⫹1
⫺ E[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E[rit⫹1
]) 2 ] p 0. x (2)
, d.f.(2), and p(2) are the
statistic, degrees of freedom, and p-value for the x 2 test that the variance
S
S
Iw
Iw
errors, defined as E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ], are jointly
zero. m.a.e.(2) is the mean absolute variance error in annual decimals.
2
, d.f.(1), and p(1) are the statistic, degrees of freedom, and p-value
x (1)
for the x 2 test that the expected return errors, defined in the same way
as in panel A, are jointly zero. m.a.e.(1) is the mean absolute expected
return error in annual percent. x 2 , d.f., and p are the statistic, degrees of
freedom, and p-value of the test that both the expected return errors and
the variance errors are jointly zero.
Table D8
Euler Equation Errors, the Market Value of Debt per the Bernanke and Campbell (1988)
Algorithm
Low
2
3
4
5
6
7
8
9
High
H⫺L
⫺1.24
[⫺1.21]
⫺.51
[⫺.74]
⫺.08
[⫺.08]
⫺.38
[⫺.46]
2.32
[.83]
1.00
[.50]
⫺3.47
[⫺1.43]
.67
[.37]
.14
[.21]
⫺.72
[⫺.75]
⫺1.55
[⫺1.63]
⫺.99
[⫺.92]
A. Euler Equation Errors from Matching Expected Returns
10 SUE Portfolios
eqi
.30
[.78]
⫺1.81
[⫺1.85]
⫺.06
[⫺.10]
.80
[1.12]
1.70
[1.68]
.43
[.61]
.52
[1.19]
10 B/M Portfolios
⫺4.13
[⫺1.81]
q
i
e
⫺3.00
[⫺1.35]
⫺.59
[⫺.44]
2.54
[1.25]
2.50
[1.52]
3.47
[1.24]
1.88
[1.10]
10 CI Portfolios
⫺.57
[⫺.39]
q
i
e
⫺2.88
[⫺2.11]
⫺.57
[⫺.83]
.74
[.83]
2.71
[1.98]
3.22
[2.21]
1.12
[.86]
B. Euler Equation Errors from Matching Expected Returns and Variances
10 SUE Portfolios
e ij
⫺.04
[⫺1.84]
⫺.03
[⫺1.79]
.01
[.84]
⫺.01
[⫺.57]
.01
[.53]
.03
[1.54]
.04
[1.71]
.01
[.63]
.02
[.92]
.03
[1.34]
.06
[1.75]
eqi
⫺6.67
[⫺2.24]
⫺6.76
[⫺2.30]
⫺2.46
[⫺1.67]
⫺1.63
[⫺1.10]
2.60
[1.99]
1.74
[1.37]
2.27
[1.83]
1.56
[.84]
3.71
[1.77]
5.56
[2.05]
12.23
[2.29]
2
10 B/M Portfolios
e
j2
i
eqi
.10
.07
.07
.04
.03
.02
.05
.02
⫺.13
⫺.23
[2.36]
[2.00]
[1.96]
[1.24]
[.83]
[.99]
[1.41]
[.64]
[⫺.03]
[⫺1.87]
[⫺2.18]
⫺6.18
[⫺1.69]
⫺3.14
[⫺1.31]
⫺1.48
[⫺.74]
.95
[.47]
2.06
[1.20]
3.62
[1.72]
4.09
[1.73]
4.32
[1.69]
4.11
[1.70]
⫺3.02
[⫺.52]
3.16
[.40]
.01
[.58]
⫺.02
[⫺1.13]
⫺.05
[⫺1.75]
⫺.07
[⫺1.45]
.00
10 CI Portfolios
e
j2
i
eqi
.02
[.81]
⫺.01
[⫺.62]
.02
[1.07]
.01
[.42]
.03
[1.25]
.01
[.62]
.02
[.84]
⫺.02
⫺2.50
.05
1.93
3.46
3.69
1.51
.05
⫺2.49
⫺4.63
⫺4.60
[⫺.01]
[⫺1.53]
[.04]
[1.09]
[2.10]
[1.81]
[1.12]
[.04]
[⫺1.46]
[⫺1.95]
[⫺1.28]
8
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Note.—We measure the market value of debt, Bit , per the Bernanke and Campbell (1988) algorithm. Euler equation errors and t-statistics (in
S
brackets) are from the first-stage GMM estimation with an identity weighting matrix. In panel A the moment conditions are E[rit⫹1
⫺ rIw
it⫹1] p
S
Iw
0. The expected return errors are defined as eiq { E T[rit⫹1
⫺ rit⫹1
], in which E T[7] is the sample mean of the series in brackets. In panel B the
2
S
Iw
S
S
2
Iw
Iw
2
moment conditions are E[rit⫹1 ⫺ rit⫹1 ] p 0 and E[(rit⫹1 ⫺ E[rit⫹1 ]) ⫺ (rit⫹1 ⫺ E[rit⫹1 ]) ] p 0. The variance errors are defined as eij {
S
S
Iw
Iw
E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ]. The expected return errors are defined as in panel A. In the last column we report the difference in
the expected return errors and the difference in the variance errors between the high and low portfolios, as well as their t-statistics. Expected
return errors are in annual percent, and variance errors are in annual decimals.
Table D9
Parameter Estimates and Tests of
Overidentification, Value-Weighted Returns
SUE
B/M
CI
A. Matching Expected Returns
a
.50
[.41]
8.11
[5.44]
1.18
[.40]
a
.18
[.03]
.27
[.06]
.16
[.02]
x2
d.f.
p
m.a.e.
5.36
8
.72
1.03
6.16
8
.63
1.37
5.85
8
.66
1.11
B. Matching Expected Returns and
Variances
a
28.42
[10.88]
.44
[.16]
8.33
[2.06]
.25
[.04]
13.05
[3.41]
.25
[.05]
2
x (2)
d.f.(2)
p(2)
m.a.e.(2)
3.59
8
.89
.02
4.69
8
.79
.03
6.49
8
.59
.03
2
x (1)
d.f.(1)
p(1)
m.a.e.(1)
3.30
8
.91
1.30
3.11
8
.93
1.69
3.44
8
.90
2.23
3.88
18
1.00
6.39
18
.99
6.01
18
1.00
a
x2
d.f.
p
Note.—We value-weight all portfolio stock returns, ritS , and corporate
bond returns, ritB. Estimates and tests are from the one-stage GMM
estimation with an identity weighting matrix. In panel A the moment
S
Iw
conditions are E[rit⫹1
⫺ rit⫹1
] p 0. a is the adjustment cost parameter
and a is capital’s share. Their standard errors are reported in brackets
beneath the estimates. x 2 is the statistic from the second-stage GMM
that the moment conditions are jointly zero. d.f. is the degrees of
freedom, and p is the p-value associated with the test. m.a.e. is the mean
S
Iw
absolute error, E T [rit⫹1
⫺ rit⫹1
], in which E T[7] is the sample mean of
the series in brackets, in annual percent across a given set of testing
S
Iw
portfolios. In panel B the moment conditions are E[rit⫹1
⫺ rit⫹1
] p 0 and
S
S
Iw
Iw
2
E[(rit⫹1
⫺ E[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E[rit⫹1
]) 2 ] p 0. x (2)
, d.f.(2), and p(2) are
the statistic, degrees of freedom, and p-value for the x 2 test that the
S
S
Iw
Iw
variance errors, defined as E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ],
are jointly zero. m.a.e.(2) is the mean absolute variance error in annual
2
decimals. x (1)
, d.f.(1), and p(1) are the statistic, degrees of freedom, and
p-value for the x 2 test that the expected return errors, defined in the
same way as in Panel A, are jointly zero. m.a.e.(1) is the mean absolute
expected return error in annual percent. x 2 , d.f., and p are the statistic,
degrees of freedom, and p-value of the test that both the expected return
errors and the variance errors are jointly zero.
9
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Table D10
Euler Equation Errors, Value-Weighted Returns
Low
2
3
4
5
6
7
8
9
High
H⫺L
A. Euler Equation Errors from Matching Expected Returns
10 SUE Portfolios
eqi
.88
[1.66]
⫺1.29
[⫺1.38]
.38
[.57]
⫺.34
[⫺.38]
⫺.64
[⫺.60]
.32
[.35]
1.67
[1.84]
⫺1.52
[⫺1.75]
1.87
[1.97]
⫺1.35
[⫺1.27]
⫺2.23
[⫺1.91]
1.39
[.63]
.71
[.38]
⫺2.86
[⫺1.65]
⫺2.09
[⫺1.27]
.01
[.01]
⫺.41
[⫺.32]
⫺1.07
[⫺.80]
.38
[.27]
10 B/M Portfolios
⫺.77
[⫺.42]
q
i
e
⫺1.41
[⫺.94]
⫺.96
[⫺.97]
.39
[.31]
.09
[.07]
1.88
[.90]
3.21
[2.01]
10 CI Portfolios
⫺1.45
[⫺.70]
q
i
e
⫺2.00
[⫺1.64]
1.43
[1.01]
1.50
[1.04]
1.23
[1.04]
2.01
[1.04]
.02
[.02]
B. Euler Equation Errors from Matching Expected Returns and Variances
10 SUE Portfolios
2
e ij
eqi
⫺.03
[⫺1.39]
⫺.64
[⫺.47]
⫺.02
[⫺1.26]
⫺2.78
[⫺1.65]
.01
[.80]
⫺.18
[⫺.21]
⫺.01
[⫺.87]
⫺1.33
[⫺.80]
.01
[.53]
.70
[.57]
.02
[.80]
.00
[.00]
.03
[1.84]
2.18
[1.76]
.00
[⫺.50]
⫺1.55
[⫺1.13]
.01
[.51]
2.49
[1.52]
.04
[1.89]
1.09
[.66]
.07
[1.86]
1.74
[.89]
10 B/M Portfolios
e
j2
i
eqi
.06
[2.11]
.05
[2.14]
.05
[2.26]
.02
[.93]
.02
[.95]
.02
[1.09]
.01
[.66]
⫺.01
[⫺.52]
⫺.02
[⫺1.38]
⫺.08
[⫺1.93]
⫺.14
[⫺2.13]
⫺1.81
[⫺.84]
⫺1.36
[⫺.79]
⫺1.10
[⫺.77]
⫺1.17
[⫺.65]
.02
[.01]
1.58
[.81]
3.67
[1.94]
2.02
[1.05]
1.80
[.92]
⫺2.34
[⫺1.05]
⫺.53
[⫺.18]
10 CI Portfolios
e
j2
i
eqi
⫺.01
[⫺.32]
.00
[.09]
.04
[1.72]
.03
[1.12]
.03
[1.32]
.03
[1.76]
.02
[1.19]
.02
[1.26]
⫺.03
[⫺1.80]
⫺.06
[⫺1.73]
⫺.05
[⫺1.54]
1.98
[.60]
⫺.91
[⫺.55]
2.32
[1.17]
2.79
[1.45]
1.94
[1.40]
1.77
[.86]
⫺.32
[⫺.20]
⫺.96
[⫺.52]
⫺2.96
[⫺1.45]
⫺6.33
[⫺2.13]
⫺8.31
[⫺2.11]
Note.— We value-weight all portfolio stock returns, ritS , and corporate bond returns, ritB . Estimates and tests are from the one-stage GMM
S
estimation with an identity weighting matrix. In panel A the moment conditions are E[rit⫹1
⫺ rIw
it⫹1] p 0. The expected return errors are defined
S
Iw
S
Iw
as eiq { E T[rit⫹1
⫺ rit⫹1
], in which E T[7] is the sample mean of the series in brackets. In panel B the moment conditions are E[rit⫹1
⫺ rit⫹1
]p0
S
S
2
Iw
Iw
2
j2
S
S
Iw
Iw
and E[(rit⫹1 ⫺ E[rit⫹1 ]) ⫺ (rit⫹1 ⫺ E[rit⫹1 ]) ] p 0. The variance errors are defined as ei { E T[(rit⫹1 ⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ]. The expected
return errors are defined as in panel A. In the last column we report the difference in the expected return errors and the difference in the variance
errors between the high and low portfolios, as well as their t-statistics. Expected return errors are in annual percent, and variance errors are in
annual decimals.
Table D11
Parameter Estimates and Tests of
Overidentification, Window Length of One in the
Standard Bartlett Kernel
SUE
B/M
CI
A. Matching Expected Returns
a
7.68
[2.35]
22.34
[25.74]
.97
[.42]
a
.32
[.04]
.50
[.29]
.21
[.02]
x2
d.f.
6.57
8
7.65
8
10
13.94
8
App. D from Liu et al., “Investment-Based Expected Stock Returns”
p
.58
.47
.08
m.a.e.
.74
2.32
1.51
B. Matching Expected Returns and
Variances
a
28.88
[17.52]
11.48
[4.30]
16.23
[6.07]
a
.61
[.27]
9.18
8
.35
[.07]
14.20
8
.36
[.09]
13.08
8
2
x (2)
d.f.(2)
p(2)
m.a.e.(2)
.33
.03
.08
.04
.11
.02
2
x (1)
d.f.(1)
7.50
8
9.09
8
6.88
8
p(1)
m.a.e.(1)
.48
3.45
.33
2.58
.55
2.22
10.55
18
.91
15.52
18
.63
14.03
18
.73
x2
d.f.
p
Note.—Estimates and tests are from the one-stage GMM estimation
with an identity weighting matrix. We use a standard Bartlett kernel with
a window length of one to calculate the optimal weighting matrix when
S
conducting inferences. In panel A the moment conditions are E[rit⫹1
⫺
Iw
rit⫹1
] p 0. a is the adjustment cost parameter and a is capital’s share.
Their standard errors are reported in brackets beneath the estimates.
x 2 is the statistic from the second-stage GMM that the moment conditions
are jointly zero. d.f. is the degrees of freedom, and p is the p-value
S
associated with the test. m.a.e. is the mean absolute error, E T [rit⫹1
⫺
Iw
rit⫹1
], in which E T[7] is the sample mean of the series in brackets, in
annual percent across a given set of testing portfolios. In panel B the
S
Iw
S
S
moment conditions are E[rit⫹1
⫺ rit⫹1
] p 0 and E[(rit⫹1
⫺ E[rit⫹1
]) 2 ⫺
Iw
Iw
2
(rit⫹1
⫺ E[rit⫹1
]) 2 ] p 0. x (2)
, d.f.(2), and p(2) are the statistic, degrees of
freedom, and p-value for the x 2 test that the variance errors, defined as
S
S
Iw
Iw
E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ], are jointly zero. m.a.e.(2) is
2
the mean absolute variance error in annual decimals. x (1)
, d.f.(1), and
p(1) are the statistic, degrees of freedom, and p-value for the x 2 test that
the expected return errors, defined in the same way as in panel A, are
jointly zero. m.a.e.(1) is mean absolute expected return error in annual
percent. x 2, d.f., and p are the statistic, degrees of freedom, and p-value
of the test that both the expected return errors and the variance errors
are jointly zero.
Table D12
Euler Equation Errors, Window Length of One in the Standard Bartlett Kernel
Low
2
3
4
5
6
7
8
9
High
H⫺L
⫺1.25
[⫺1.39]
⫺.50
[⫺.52]
⫺.15
[⫺.14]
⫺.40
[⫺.44]
1.62
[.56]
.05
[.03]
⫺2.73
[⫺1.34]
1.21
[.75]
A. Euler Equation Errors from Matching Expected Returns
10 SUE Portfolios
eqi
.26
[.56]
⫺1.72
[⫺2.14]
⫺.05
[⫺.07]
.72
[.99]
1.66
[1.68]
.51
[.58]
.61
[.73]
10 B/M Portfolios
q
i
e
⫺3.94
[⫺1.87]
⫺3.20
[⫺1.72]
⫺1.02
[⫺.70]
2.74
[1.75]
2.35
[1.45]
3.07
[1.30]
2.51
[1.22]
10 CI Portfolios
q
i
e
⫺.97
⫺2.71
⫺.50
.93
2.72
3.37
.94
.46
⫺1.02
⫺1.45
⫺.49
[⫺.62]
[⫺2.58]
[⫺.60]
[1.01]
[2.20]
[2.94]
[.99]
[.56]
[⫺.83]
[⫺1.42]
[⫺.46]
11
App. D from Liu et al., “Investment-Based Expected Stock Returns”
B. Euler Equation Errors from Matching Expected Returns and Variances
10 SUE Portfolios
e ij
⫺.04
[⫺2.04]
⫺.04
[⫺2.00]
.01
[.60]
⫺.01
[⫺.39]
.02
[1.18]
.03
[1.49]
.04
[1.84]
.01
[.66]
.02
[.79]
.03
[1.66]
.08
[2.13]
eqi
⫺6.99
[⫺2.04]
⫺6.50
[⫺2.54]
⫺2.12
[⫺1.44]
⫺1.62
[⫺.75]
2.60
[1.62]
1.79
[.86]
2.27
[1.29]
1.48
[.77]
3.75
[1.56]
5.38
[1.94]
12.37
[2.40]
2
10 B/M Portfolios
e
j2
i
eqi
.10
[3.45]
.07
[2.84]
.06
[2.58]
.01
[.61]
.01
[.59]
.02
[.88]
.01
[.27]
⫺.01
[⫺.33]
⫺.02
[⫺1.00]
⫺.10
[⫺2.58]
⫺.20
[⫺3.40]
⫺6.46
[⫺2.14]
⫺3.83
[⫺1.62]
⫺2.11
[⫺.93]
⫺.04
[⫺.02]
1.71
[.74]
2.60
[1.13]
3.54
[1.49]
3.11
[1.19]
1.85
[.73]
⫺.58
[⫺.13]
5.89
[1.00]
10 CI Portfolios
e
j2
i
eqi
.01
[.34]
.00
[⫺.18]
.02
[1.26]
.01
[.56]
.03
[1.64]
.02
[1.22]
.02
[.84]
.02
[1.01]
⫺.02
[⫺.98]
⫺.06
[⫺2.01]
⫺.07
[⫺1.41]
1.29
[.42]
⫺2.51
[⫺1.12]
⫺.11
[⫺.05]
1.86
[1.04]
3.47
[1.93]
3.48
[1.91]
1.12
[.75]
.28
[.16]
⫺2.82
[⫺1.11]
⫺5.32
[⫺1.49]
⫺6.60
[⫺1.31]
Note.— Estimates and tests are from the one-stage GMM estimation with an identity weighting matrix. We use a standard Bartlett kernel
S
with a window length of one to calculate the optimal weighting matrix when conducting inferences. In panel A the moments are E[rit⫹1
⫺
q
S
Iw
rIw
it⫹1] p 0. The expected return errors are defined as ei { E T[rit⫹1 ⫺ rit⫹1 ] , in which E T[7] is the sample mean of the series in brackets. In panel
S
Iw
S
S
2
Iw
Iw
2
j2
B the moment conditions are E[rit⫹1 ⫺ rit⫹1 ] p 0 and E[(rit⫹1 ⫺ E[rit⫹1 ]) ⫺ (rit⫹1 ⫺ E[rit⫹1 ]) ] p 0. The variance errors are defined as ei {
S
S
Iw
Iw
E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ]. The expected return errors are defined as in panel A. In the last column we report the difference in
the expected return errors and the difference in the variance errors between the high and low portfolios, and their t-statistics. Expected return
errors are in annual percent, and variance errors are in annual decimals.
Table D13
Parameter Estimates and Tests of
Overidentification, Window Length of 10 in the
Standard Bartlett Kernel
SUE
B/M
CI
A. Matching Expected Returns
a
a
x2
d.f.
p
m.a.e.
7.68
[1.51]
.32
[.03]
22.34
[24.41]
.50
[.29]
.70
4.18
3.99
8
8
.84
2.32
.86
1.51
8
1.00
.74
.97
[.30]
.21
[.01]
B. Matching Expected Returns and
Variances
a
28.88
[11.54]
11.48
[4.24]
16.23
[4.22]
a
.61
[.20]
.35
[.06]
.36
[.06]
2
x (2)
3.08
3.44
3.38
d.f.(2)
p(2)
m.a.e.(2)
8
8
8
2
x (1)
3.23
3.16
2.83
d.f.(1)
p(1)
m.a.e.(1)
8
8
8
.92
3.45
.92
2.58
.95
2.22
.93
.03
.90
.04
12
.91
.02
App. D from Liu et al., “Investment-Based Expected Stock Returns”
x2
d.f.
p
2.79
18
3.42
18
3.96
18
1.00
1.00
1.00
Note.—Estimates and tests are from the one-stage GMM estimation
with an identity weighting matrix. We use a standard Bartlett kernel with
a window length of 10 to calculate the optimal weighting matrix when
S
conducting inferences. In panel A the moment conditions are E[rit⫹1
⫺
Iw
rit⫹1
] p 0. a is the adjustment cost parameter and a is capital’s share.
Their standard errors are reported in brackets beneath the estimates.
x 2 is the statistic from the second-stage GMM that the moment conditions
are jointly zero. d.f. is the degrees of freedom, and p is the p-value
S
associated with the test. m.a.e. is the mean absolute error, E[rit⫹1
⫺
Iw
rit⫹1
], in which E T[7] is the sample mean of the series in brackets, in
annual percent across a given set of testing portfolios. In panel B the
S
Iw
S
S
moment conditions are E[rit⫹1
⫺ rit⫹1
] p 0 and E[(rit⫹1
⫺ E[rit⫹1
]) 2 ⫺
Iw
Iw
2
(rit⫹1
⫺ E[rit⫹1
]) 2 ] p 0. x (2)
, d.f.(2), and p(2) are the statistic, degrees of
freedom, and p-value for the x 2 test that the variance errors, defined as
S
S
Iw
Iw
E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ], are jointly zero. m.a.e.(2) is
2
the mean absolute variance error in annual decimals. x (1)
, d.f.(1), and
p(1) are the statistic, degrees of freedom, and p-value for the x 2 test that
the expected return errors, defined in the same way as in panel A, are
jointly zero. m.a.e.(1) is the mean absolute expected return error in annual
percent. x 2, d.f., and p are the statistic, degrees of freedom, and p-value
of the test that both the expected return errors and the variance errors
are jointly zero.
Table D14
Euler Equation Errors, Window Length of 10 in the Standard Bartlett Kernel
Low
2
3
4
5
6
7
8
9
High
H⫺L
⫺1.25
[⫺1.07]
⫺.50
[⫺.59]
⫺.15
[⫺.20]
⫺.40
[⫺.49]
1.62
[.59]
.05
[.03]
⫺2.73
[⫺1.20]
1.21
[1.01]
.46
[.90]
⫺1.02
[⫺1.01]
⫺1.45
[⫺1.10]
⫺.49
[⫺.46]
A. Euler Equation Errors from Matching Expected Returns
10 SUE Portfolios
eqi
.26
[.82]
⫺1.72
[⫺1.53]
⫺.05
[⫺.12]
.72
[.96]
1.66
[1.43]
.51
[.86]
.61
[.90]
10 B/M Portfolios
⫺3.94
[⫺1.51]
q
i
e
⫺3.20
[⫺1.39]
⫺1.02
[⫺.78]
2.74
[1.15]
2.35
[1.34]
3.07
[1.16]
2.51
[1.28]
10 CI Portfolios
⫺.97
[⫺.50]
q
i
e
⫺2.71
[⫺1.60]
⫺.50
[⫺.59]
.93
[1.04]
2.72
[1.58]
3.37
[1.69]
.94
[.76]
B. Euler Equation Errors from Matching Expected Returns and Variances
10 SUE Portfolios
e ij
⫺.04
[⫺1.65]
⫺.04
[⫺1.57]
.01
[.88]
⫺.01
[⫺.36]
.02
[.85]
.03
[1.41]
.04
[1.43]
.01
[.95]
.02
[.69]
.03
[1.34]
.08
[1.52]
eqi
⫺6.99
[⫺1.78]
⫺6.50
[⫺1.76]
⫺2.12
[⫺1.25]
⫺1.62
[⫺1.12]
2.60
[1.66]
1.79
[1.25]
2.27
[1.73]
1.48
[.87]
3.75
[1.53]
5.38
[1.64]
12.37
[1.80]
2
10 B/M Portfolios
e
j2
i
eqi
.10
[1.83]
.07
[1.83]
.06
[1.73]
.01
[.81]
.01
[.47]
.02
[.90]
.01
[.31]
⫺.01
[⫺.47]
⫺.02
[⫺1.36]
⫺.10
[⫺1.69]
⫺.20
[⫺1.83]
⫺6.46
[⫺1.55]
⫺3.83
[⫺1.58]
⫺2.11
[⫺1.06]
⫺.04
[⫺.02]
1.71
[.91]
2.60
[1.37]
3.54
[1.65]
3.11
[1.40]
1.85
[1.27]
⫺.58
[⫺.14]
5.89
[.98]
10 CI Portfolios
e
j2
i
eqi
.01
[.34]
.00
[⫺.15]
.02
[1.09]
.01
[.51]
.03
[1.47]
.02
[1.04]
.02
[.67]
.02
[1.06]
⫺.02
[⫺1.46]
⫺.06
[⫺1.42]
⫺.07
[⫺1.11]
1.29
[.63]
⫺2.51
[⫺1.57]
⫺.11
[⫺.10]
1.86
[1.36]
3.47
[1.66]
3.48
[1.67]
1.12
[.97]
.28
[.28]
⫺2.82
[⫺1.63]
⫺5.32
[⫺1.69]
⫺6.60
[⫺1.97]
13
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Note.—Estimates and tests are from the one-stage GMM estimation with an identity weighting matrix. We use a standard Bartlett kernel with
S
a window length of 10 to calculate the optimal weighting matrix when conducting inferences. In panel A the moments are E[rit⫹1
⫺ rIw
it⫹1] p 0.
S
Iw
The expected return errors are defined as eiq { E T[rit⫹1
⫺ rit⫹1
], in which E T[7] is the sample mean of the series in brackets. In panel B the
2
S
Iw
S
S
2
Iw
Iw
2
moment conditions are E[rit⫹1 ⫺ rit⫹1 ] p 0 and E[(rit⫹1 ⫺ E[rit⫹1 ]) ⫺ (rit⫹1 ⫺ E[rit⫹1 ]) ] p 0. The variance errors are defined as eij {
S
S
Iw
Iw
E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ]. The expected return errors are defined as in panel A. In the last column we report the difference in
the expected return errors and the difference in the variance errors between the high and low portfolios, and their t-statistics. Expected return
errors are in annual percent, and variance errors are in annual decimals.
Table D15
Parameter Estimates and Tests of
Overidentification, an Alternative Measure of
Capital
SUE
B/M
CI
A. Matching Expected Returns
a
3.68
[.92]
8.21
[7.88]
.41
[.16]
a
.20
[.02]
3.94
8
.86
.67
.26
[.10]
6.33
8
.61
2.48
.14
[.01]
6.20
8
.63
1.46
x2
d.f.
p
m.a.e.
B. Matching Expected Returns and
Variances
a
13.32
[7.05]
.34
[.13]
5.58
[2.27]
.22
[.04]
8.10
[2.48]
.23
[.04]
2
x (2)
d.f.(2)
p(2)
m.a.e.(2)
4.99
8
.76
.03
6.25
8
.62
.04
5.05
8
.75
.02
2
x (1)
d.f.(1)
p(1)
m.a.e.(1)
5.24
8
.73
3.35
5.11
8
.75
3.08
6.00
8
.65
2.45
a
x2
6.28
6.80
6.09
d.f.
p
18
1.00
18
.99
18
1.00
14
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Note.—Estimates and tests are from the one-stage GMM estimation
with an identity weighting matrix. We measure the capital stock, K it,
as net property, plant, and equipment (Compustat annual item 7). In
S
Iw
panel A the moment conditions are E[rit⫹1
⫺ rit⫹1
] p 0. a is the
adjustment cost parameter and a is capital’s share. Their standard errors
are reported in brackets beneath the estimates. x 2 is the statistic from
the second-stage GMM that the moment conditions are jointly zero.
d.f. is the degrees of freedom, and p is the p-value associated with the
S
Iw
test. m.a.e. is the mean absolute error, E T [rit⫹1
⫺ rit⫹1
], in which
E T[7] is the sample mean of the series in brackets, in annual percent
across a given set of testing portfolios. In panel B the moment
S
Iw
S
S
Iw
conditions are E[rit⫹1
⫺ rit⫹1
] p 0 and E[(rit⫹1
⫺ E[rit⫹1
]) 2 ⫺ (rit⫹1
⫺
Iw
2
E[rit⫹1
]) 2 ] p 0. x (2)
, d.f.(2), and p(2) are the statistic, degrees of
freedom, and p-value for the x 2 test that the variance errors, defined
S
S
Iw
Iw
as E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ], are jointly zero. m.a.e.(2)
2
is the mean absolute variance error in annual decimals. x (1)
, d.f.(1),
and p(1) are the statistic, degrees of freedom, and p-value for the x 2
test that the expected return errors, defined in the same way as in panel
A, are jointly zero. m.a.e.(1) is the mean absolute expected return error
in annual percent. x 2, d.f., and p are the statistic, degrees of freedom,
and p-value of the test that both the expected return errors and the
variance errors are jointly zero.
Table D16
Euler Equation Errors, an Alternative Measure of Capital
Low
2
3
4
5
6
7
8
9
High
H⫺L
⫺1.19
[⫺1.07]
⫺.50
[⫺.55]
⫺.18
[⫺.18]
⫺.27
[⫺.28]
1.79
[.64]
⫺.58
[⫺.32]
⫺2.85
[⫺1.27]
1.14
[.59]
.14
[.23]
⫺1.25
[⫺1.00]
⫺.80
[⫺.59]
⫺1.60
[⫺1.13]
.02
[.91]
.08
[1.56]
A. Euler Equation Errors from Matching Expected Returns
10 SUE Portfolios
eqi
.09
[.21]
⫺1.47
[⫺1.63]
.24
[.30]
.52
[.72]
1.50
[1.50]
.61
[.79]
.41
[.65]
10 B/M Portfolios
⫺3.99
[⫺1.92]
q
i
e
⫺3.24
[⫺1.64]
⫺.82
[⫺.54]
2.81
[1.53]
2.55
[1.52]
3.47
[1.25]
2.72
[1.38]
10 CI Portfolios
q
i
e
.81
[.38]
⫺3.39
[⫺2.02]
⫺1.28
[⫺1.46]
.04
[.04]
2.53
[1.39]
3.54
[1.85]
.83
[.55]
B. Euler Equation Errors from Matching Expected Returns and Variances
10 SUE Portfolios
2
e ij
eqi
⫺.05
[⫺1.80]
⫺.03
[⫺1.82]
.00
[⫺.17]
.01
[.34]
.01
[.86]
.03
[1.61]
.04
[1.53]
.03
[1.79]
.03
[1.46]
⫺6.95
⫺6.60
⫺2.16
⫺1.22
2.81
1.53
2.10
1.48
3.91
4.77
11.72
[⫺2.25]
[⫺2.27]
[⫺1.49]
[⫺.89]
[1.99]
[1.14]
[1.67]
[.82]
[1.79]
[1.97]
[2.57]
10 B/M Portfolios
e
j2
i
eqi
.10
[2.29]
.07
[1.93]
.06
[2.07]
.00
[.11]
.00
[.11]
.02
[.87]
.03
[.91]
⫺.02
[⫺.57]
⫺.01
[⫺.74]
⫺.11
[⫺2.01]
⫺.21
[⫺2.37]
⫺6.90
[⫺2.03]
⫺4.69
[⫺1.92]
⫺3.28
[⫺1.41]
⫺.56
[⫺.29]
2.05
[.99]
2.84
[1.27]
3.53
[1.80]
3.77
[1.66]
2.91
[1.57]
.26
[.06]
7.16
[1.17]
10 CI Portfolios
e
j2
i
eqi
.05
⫺.01
.01
⫺.01
⫺.01
⫺.01
.01
.02
⫺.02
⫺.04
⫺.09
[1.63]
[⫺.73]
[.37]
[⫺.28]
[⫺.20]
[⫺.35]
[.65]
[1.33]
[⫺1.17]
[⫺1.46]
[⫺1.74]
3.37
[1.21]
⫺2.30
[⫺1.43]
⫺.49
[⫺.36]
1.40
[.83]
3.64
[1.71]
3.10
[1.62]
1.15
[.82]
⫺.47
[⫺.36]
⫺3.65
[⫺1.85]
⫺4.92
[⫺1.99]
⫺8.29
[⫺2.24]
15
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Note.—Estimates and tests are from the one-stage GMM estimation with an identity weighting matrix. We measure the capital stock, K it , as
S
Iw
net property, plant, and equipment (Compustat annual item 7). In panel A the moments are E[rit⫹1
⫺ rit⫹1
] p 0. The expected return errors are
S
Iw
S
defined as eiq { E T[rit⫹1
⫺ rit⫹1
], in which E T[7] is the sample mean of the series in brackets. In panel B the moment conditions are E[rit⫹1
⫺
Iw
S
S
2
Iw
Iw
2
j2
S
S
Iw
Iw
rit⫹1 ] p 0 and E[(rit⫹1 ⫺ E[rit⫹1 ]) ⫺ (rit⫹1 ⫺ E[rit⫹1 ]) ] p 0. The variance errors are defined as ei { E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ].
The expected return errors are defined as in panel A. In the last column we report the difference in the expected return errors and the difference
in the variance errors between the high and low portfolios, and their t-statistics. Expected return errors are in annual percent, and variance errors
are in annual decimals.
Table D17
Parameter Estimates and Tests of
Overidentification, an Alternative Measure of
Investment
SUE
B/M
CI
A. Matching Expected Returns
a
8.50
[1.84]
47.10
[93.66]
1.01
[.29]
a
.34
[.04]
4.86
8
.77
.76
.84
[1.19]
5.78
8
.67
1.99
.21
[.02]
6.56
8
.59
1.50
x2
d.f.
p
m.a.e.
B. Matching Expected Returns and
Variances
a
27.42
[16.54]
.60
[.28]
11.50
[5.10]
.35
[.08]
16.94
[6.67]
.38
[.09]
2
x (2)
d.f.(2)
p(2)
m.a.e.(2)
4.56
8
.80
.02
6.09
8
.64
.04
6.40
8
.60
.02
2
x (1)
d.f.(1)
p(1)
m.a.e.(1)
5.22
8
.73
3.39
4.48
8
.81
2.50
4.74
8
.79
2.16
a
x2
5.12
6.22
d.f.
p
18
1.00
18
1.00
16
22.61
18
.21
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Note.—Estimates and tests are from the one-stage GMM estimation
with an identity weighting matrix. We measure investment, I it , as capital
expenditures (Compustat annual item 128). In panel A the moment
S
Iw
conditions are E[rit⫹1
⫺ rit⫹1
] p 0. a is the adjustment cost parameter and
a is capital’s share. Their standard errors are reported in brackets beneath
the estimates. x 2 is the statistic from the second-stage GMM that the
moment conditions are jointly zero. d.f. is the degrees of freedom, and
p is the p-value associated with the test. m.a.e. is the mean absolute
S
Iw
error, E T [rit⫹1
⫺ rit⫹1
], in which E T[7] is the sample mean of the series
in brackets, in annual percent across a given set of testing portfolios. In
S
Iw
S
panel B the moment conditions are E[rit⫹1
⫺ rit⫹1
] p 0 and E[(rit⫹1
⫺
S
Iw
Iw
2
E[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E[rit⫹1
]) 2 ] p 0. x (2)
, d.f.(2), and p(2) are the statistic,
degrees of freedom, and p-value for the x 2 test that the variance errors,
S
S
Iw
Iw
defined as E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ], are jointly zero.
2
m.a.e.(2) is the mean absolute variance error in annual decimals. x (1)
,
d.f.(1), and p(1) are the statistic, degrees of freedom, and p-value for
the x 2 test that the expected return errors, defined in the same way as
in panel A, are jointly zero. m.a.e.(1) is the mean absolute expected
return error in annual percent. x 2 , d.f., and p are the statistic, degrees of
freedom, and p-value of the test that both the expected return errors and
the variance errors are jointly zero.
Table D18
Euler Equation Errors, an Alternative Measure of Investment
Low
2
3
4
5
6
7
8
9
High
H⫺L
⫺1.57
[⫺1.33]
⫺.40
[⫺.49]
⫺.10
[⫺.11]
⫺.28
[⫺.30]
2.47
[.93]
⫺.47
[⫺.21]
⫺2.95
[⫺1.19]
.12
[.07]
.44
[.74]
⫺.93
[⫺.84]
⫺1.46
[⫺1.24]
⫺.55
[⫺.45]
A. Euler Equation Errors from Matching Expected Returns
10 SUE Portfolios
eqi
.18
[.46]
⫺1.66
[⫺1.87]
⫺.04
[⫺.04]
.49
[.76]
1.72
[1.74]
.79
[1.04]
.65
[1.22]
10 B/M Portfolios
⫺3.06
[⫺1.64]
q
i
e
⫺2.60
[⫺1.34]
.05
[.04]
2.06
[1.25]
1.29
[.66]
2.22
[.73]
2.72
[1.29]
10 CI Portfolios
⫺.91
[⫺.48]
q
i
e
⫺2.71
[⫺1.95]
⫺.59
[⫺.72]
.87
[.90]
2.75
[1.77]
3.35
[2.21]
.94
[.76]
B. Euler Equation Errors from Matching Expected Returns and Variances
10 SUE Portfolios
e ij
⫺.04
[⫺1.86]
⫺.04
[⫺1.94]
.01
[.79]
.00
[⫺.15]
.02
[.96]
.02
[1.05]
.02
[1.04]
.02
[1.13]
.01
[.25]
.04
[1.58]
.08
[1.76]
eqi
⫺7.09
[⫺2.23]
⫺6.40
[⫺2.26]
⫺1.91
[⫺1.33]
⫺1.56
[⫺1.03]
2.61
[1.91]
1.75
[1.12]
2.13
[1.68]
1.72
[1.00]
3.45
[1.66]
5.28
[2.01]
12.37
[2.50]
2
10 B/M Portfolios
e
j2
i
eqi
.10
[1.83]
⫺6.46
.07
[1.83]
⫺3.83
.06
[1.73]
⫺2.11
.01
[.81]
⫺.04
.01
[.47]
1.71
.02
[.90]
2.60
.01
[.31]
3.54
⫺.01
[⫺.47]
3.11
⫺.02
[⫺1.36]
1.85
⫺.10
[⫺1.69]
⫺.58
⫺.20
[⫺1.83]
5.89
[⫺1.55]
[⫺1.58]
[⫺1.06]
[⫺.02]
[.91]
[1.37]
[1.65]
[1.40]
[1.27]
[⫺.14]
[.98]
10 CI Portfolios
e
j2
i
eqi
.01
[.32]
.01
[.63]
.03
[1.49]
.02
[.79]
.02
[.79]
.01
[.50]
.01
[.61]
.02
[1.16]
⫺.03
[⫺1.94]
⫺.06
[⫺1.83]
⫺.07
[⫺1.44]
1.69
[.62]
⫺2.09
[⫺1.49]
.02
[.02]
1.96
[1.23]
3.22
[1.85]
3.17
[1.73]
.96
[.77]
.10
[.09]
⫺3.13
[⫺1.63]
⫺5.28
[⫺2.00]
⫺6.98
[⫺2.00]
17
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Note.—Estimates and tests are from the one-stage GMM estimation with an identity weighting matrix. We measure investment, I it , as capital
S
Iw
expenditures (Compustat annual item 128). In panel A the moments are E[rit⫹1
⫺ rit⫹1
] p 0. The expected return errors are defined as eiq {
S
Iw
S
Iw
E T[rit⫹1
⫺ rit⫹1
], in which E T[7] is the sample mean of the series in brackets. In panel B the moment conditions are E[rit⫹1
⫺ rit⫹1
] p 0 and
2
S
S
Iw
Iw
S
S
Iw
Iw
E[(rit⫹1
⫺ E[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E[rit⫹1
]) 2 ] p 0. The variance errors are defined as eij { E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ]. The expected
return errors are defined as in panel A. In the last column we report the difference in the expected return errors and the difference in the variance
errors between the high and low portfolios, and their t-statistics. Expected return errors are in annual percent, and variance errors are in annual
decimals.
Table D19
Parameter Estimates and Tests of
Overidentification, Time-Invariant Tax Rates
SUE
B/M
CI
A. Matching Expected Returns
a
3.52
[.80]
10.31
[11.76]
.44
[.12]
a
.29
[.03]
.44
[.29]
.17
[.02]
x2
d.f.
p
m.a.e.
4.32
8
.83
.75
5.79
8
.67
2.27
6.57
8
.58
1.59
B. Matching Expected Returns and
Variances
a
12.44
[6.95]
.52
[.22]
5.17
[2.13]
.30
[.06]
7.09
[2.40]
.31
[.06]
2
x (2)
d.f.(2)
p(2)
m.a.e.(2)
5.12
8
.75
.02
6.22
8
.62
.04
6.05
8
.64
.02
2
x (1)
d.f.(1)
p(1)
m.a.e.(1)
5.33
8
.72
3.46
4.41
8
.82
2.63
4.91
8
.77
2.26
97.96
18
.00
6.17
18
1.00
6.45
18
.99
a
x2
d.f.
p
Note.—Estimates and tests are from the one-stage GMM estimation
with an identity weighting matrix. We measure the corporate tax rate,
tt⫹1, as its sample mean of 42.3 percent from 1963 to 2005. In panel
S
Iw
A the moment conditions are E[rit⫹1
⫺ rit⫹1
] p 0. a is the adjustment
cost parameter and a is capital’s share. Their standard errors are reported
in brackets beneath the estimates. x 2 is the statistic from the secondstage GMM that the moment conditions are jointly zero. d.f. is the
degrees of freedom, and p is the p-value associated with the test. m.a.e.
S
Iw
is the mean absolute error, E T [rit⫹1
⫺ rit⫹1
] , in which E T[7] is the sample
mean of the series in brackets, in annual percent across a given set of
S
testing portfolios. In panel B the moment conditions are E[rit⫹1
⫺
Iw
S
S
Iw
Iw
2
rit⫹1
] p 0 and E[(rit⫹1
⫺ E[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E[rit⫹1
]) 2 ] p 0. x (2)
, d.f.(2),
and p(2) are the statistic, degrees of freedom, and p-value for the x 2
S
S
Iw
test that the variance errors, defined as E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺
Iw
E T[rit⫹1
]) 2 ], are jointly zero. m.a.e.(2) is the mean absolute variance
2
error in annual decimals. x (1), d.f.(1), and p(1) are the statistic, degrees
of freedom, and p-value for the x 2 test that the expected return errors,
defined in the same way as in panel A, are jointly zero. m.a.e.(1) is the
mean absolute expected return error in annual percent. x 2, d.f., and p
are the statistic, degrees of freedom, and p-value of the test that both
the expected return errors and the variance errors are jointly zero.
18
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Table D20
Euler Equation Errors, Time-Invariant Tax Rates
Low
2
3
4
5
6
7
8
9
High
H⫺L
⫺1.24
[⫺1.13]
⫺.52
[⫺.63]
⫺.09
[⫺.09]
⫺.40
[⫺.40]
1.75
[.64]
⫺.07
[⫺.04]
⫺2.50
[⫺1.30]
1.40
[.89]
.49
[.83]
⫺1.04
[⫺.99]
⫺1.58
[⫺1.47]
⫺.41
[⫺.37]
A. Euler Equation Errors from Matching Expected Returns
10 SUE Portfolios
eqi
.31
[.76]
⫺1.75
[⫺1.79]
⫺.12
[⫺.15]
.72
[.96]
1.64
[1.66]
.53
[.74]
.61
[1.08]
10 B/M Portfolios
⫺3.90
[⫺1.73]
q
i
e
⫺3.32
[⫺1.42]
⫺.97
[⫺.63]
2.64
[1.33]
2.27
[1.36]
2.90
[1.04]
2.41
[1.25]
10 CI Portfolios
⫺1.17
[⫺.68]
q
i
e
⫺2.66
[⫺1.98]
⫺.48
[⫺.59]
1.03
[1.08]
2.85
[1.80]
3.55
[2.32]
1.02
[.85]
B. Euler Equation Errors from Matching Expected Returns and Variances
10 SUE Portfolios
2
e ij
eqi
⫺.04
[⫺1.93]
⫺6.89
[⫺2.23]
⫺.04
[⫺1.91]
⫺6.53
[⫺2.28]
.00
[.36]
⫺2.17
[⫺1.54]
⫺.01
[⫺.46]
⫺1.58
[⫺1.05]
.02
[.81]
2.62
[1.92]
.03
[1.40]
1.85
[1.37]
.04
[1.65]
2.27
[1.84]
.01
[1.05]
1.53
[.85]
.02
[.82]
3.77
[1.75]
.04
[1.61]
5.40
[2.02]
.08
[1.87]
12.29
[2.51]
10 B/M Portfolios
e
j2
i
eqi
.10
[2.36]
.07
[2.18]
.06
[2.08]
.01
[.52]
.01
[.51]
.02
[.80]
.01
[.36]
⫺.01
[⫺.26]
⫺.03
[⫺1.39]
⫺.10
[⫺1.88]
⫺.19
[⫺2.35]
⫺6.54
[⫺1.90]
⫺4.11
[⫺1.79]
⫺2.21
[⫺1.06]
⫺.34
[⫺.18]
1.57
[.89]
2.48
[1.24]
3.38
[1.71]
3.28
[1.52]
1.93
[1.16]
⫺.44
[⫺.12]
6.10
[1.14]
10 CI Portfolios
e
j2
i
eqi
.00
[.11]
.00
[.02]
.02
[.91]
.01
[.56]
.04
[1.44]
.02
[1.33]
.02
[.89]
.02
[1.05]
⫺.02
[⫺1.36]
⫺.06
[⫺1.70]
⫺.06
[⫺1.15]
1.11
[.45]
⫺2.42
[⫺1.53]
⫺.16
[⫺.13]
1.99
[1.20]
3.67
[2.02]
3.67
[1.89]
1.23
[.98]
.24
[.18]
⫺2.82
[⫺1.56]
⫺5.27
[⫺1.99]
⫺6.38
[⫺2.04]
Note.—Estimates and tests are from the one-stage GMM estimation with an identity weighting matrix. We measure the corporate tax rate,
S
Iw
tt⫹1, as its sample mean of 42.3 percent from 1963 to 2005. In panel A the moments are E[rit⫹1
⫺ rit⫹1
] p 0. The expected return errors are
S
Iw
S
defined as eiq { E T[rit⫹1
⫺ rit⫹1
], in which E T[7] is the sample mean of the series in brackets. In panel B the moment conditions are E[rit⫹1
⫺
Iw
S
S
2
Iw
Iw
2
j2
S
S
Iw
Iw
rit⫹1 ] p 0 and E[(rit⫹1 ⫺ E[rit⫹1 ]) ⫺ (rit⫹1 ⫺ E[rit⫹1 ]) ] p 0. The variance errors are defined as ei { E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ].
The expected return errors are defined as in panel A. In the last column we report the difference in the expected return errors and the difference
in the variance errors between the high and low portfolios, and their t-statistics. Expected return errors are in annual percent, and variance errors
are in annual decimals.
Table D21
Parameter Estimates and Tests of
Overidentification, Portfolio-Specific Tax Rates
SUE
B/M
CI
A. Matching Expected Returns
a
8.22
[1.79]
15.48
[17.35]
.85
[.30]
a
.31
[.03]
.38
[.19]
.19
[.02]
x2
d.f.
p
4.56
8
.80
6.41
8
.60
6.60
8
.58
19
App. D from Liu et al., “Investment-Based Expected Stock Returns”
m.a.e.
.73
2.44
1.99
B. Matching Expected Returns and
Variances
a
27.57
[15.50]
10.86
[4.50]
15.57
[5.39]
a
.54
[.23]
.31
[.06]
.32
[.07]
5.17
8
6.11
8
5.78
8
.74
.03
.64
.04
.67
.02
2
x (1)
d.f.(1)
5.25
8
4.28
8
5.01
8
p(1)
m.a.e.(1)
.73
3.58
.83
2.62
.76
2.43
5.35
18
1.00
6.15
18
1.00
7.22
18
.99
2
x (2)
d.f.(2)
p(2)
m.a.e.(2)
x2
d.f.
p
Note.—Estimates and tests are from the one-stage GMM estimation
with an identity weighting matrix. To measure the portfolio-specific
i
corporate tax rate, tt⫹1
, we first construct firm-specific tax rates using
the trichotomous variable approach of Graham (1996) and then take the
value-weighted tax rates across all firms within a given portfolio i. In
S
Iw
panel A the moment conditions are E[rit⫹1
⫺ rit⫹1
] p 0. a is the
adjustment cost parameter and a is capital’s share. Their standard errors
2
are reported in brackets beneath the estimates. x is the statistic from
the second-stage GMM that the moment conditions are jointly zero. d.f.
is the degrees of freedom, and p is the p-value associated with the test.
S
Iw
m.a.e. is the mean absolute error, E T [rit⫹1
⫺ rit⫹1
] , in which E T[7] is the
sample mean of the series in brackets, in annual percent across a given
set of testing portfolios. In panel B the moment conditions are
S
Iw
S
S
Iw
Iw
E[rit⫹1
⫺ rit⫹1
] p 0 and E[(rit⫹1
⫺ E[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E[rit⫹1
]) 2 ] p 0.
2
x (2)
, d.f.(2), and p(2) are the statistic, degrees of freedom, and p-value
S
for the x 2 test that the variance errors, defined as E T[(rit⫹1
⫺
S
Iw
Iw
E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E T[rit⫹1
]) 2 ], are jointly zero. m.a.e.(2) is the mean
2
absolute variance error in annual decimals. x (1)
, d.f.(1), and p(1) are the
statistic, degrees of freedom, and p-value for the x 2 test that the expected
return errors, defined in the same way as in panel A, are jointly zero.
m.a.e.(1) is the mean absolute expected return error in annual percent.
x 2, d.f., and p are the statistic, degrees of freedom, and p-value of the
test that both the expected return errors and the variance errors are jointly
zero.
Table D22
Euler Equation Errors, Portfolio-Specific Tax Rates
Low
2
3
4
5
6
7
8
9
High
H⫺L
A. Euler Equation Errors from Matching Expected Returns
10 SUE Portfolios
eqi
[t]
.25
[.61]
⫺1.58
[⫺1.76]
⫺.13
[⫺.16]
.58
[.80]
1.46
[1.64]
eqi
⫺4.13
[⫺1.85]
⫺2.97
[⫺1.37]
⫺.77
[⫺.58]
2.52
[1.30]
2.27
[1.43]
eqi
⫺1.93
[⫺.96]
⫺3.17
[⫺1.94]
⫺.15
[⫺.19]
1.46
[1.37]
3.35
[1.91]
.58
[.76]
.73
[1.22]
⫺1.20
[⫺1.10]
⫺.71
[⫺.82]
.08
[.08]
⫺.17
[⫺.18]
2.18
[.82]
⫺.16
[⫺.12]
⫺3.13
[⫺1.56]
1.00
[.69]
1.03
[1.53]
⫺.80
[⫺.70]
⫺2.38
[⫺1.76]
⫺.45
[⫺.36]
10 B/M Portfolios
3.26
[1.26]
2.95
[1.54]
10 CI Portfolios
4.09
[2.34]
20
1.52
[1.10]
App. D from Liu et al., “Investment-Based Expected Stock Returns”
B. Euler Equation Errors from Matching Expected Returns and Variances
10 SUE Portfolios
e ij
⫺.05
[⫺1.96]
⫺.04
[⫺1.92]
.01
[.59]
⫺.01
[⫺.49]
.02
[1.04]
.03
[1.67]
.04
[1.70]
.01
[.90]
.01
[.67]
.03
[1.60]
.08
[1.90]
eqi
⫺7.21
[⫺2.25]
⫺6.49
[⫺2.28]
⫺2.21
[⫺1.51]
⫺1.77
[⫺1.14]
2.47
[1.90]
1.97
[1.40]
2.52
[1.94]
1.72
[.96]
3.74
[1.74]
5.69
[2.04]
12.89
[2.50]
2
10 B/M Portfolios
e
j2
i
eqi
.10
[2.34]
.08
[2.24]
.06
[2.10]
.02
[.74]
.02
[.64]
.02
[.97]
.01
[.34]
⫺.01
[⫺.31]
⫺.02
[⫺1.08]
⫺.11
[⫺2.04]
⫺.21
[⫺2.41]
⫺6.03
[⫺1.80]
⫺3.30
[⫺1.50]
⫺1.45
[⫺.71]
.30
[.16]
1.86
[.97]
2.95
[1.35]
3.79
[1.82]
3.45
[1.52]
1.42
[.87]
⫺1.64
[⫺.40]
4.38
[.79]
10 CI Portfolios
e
j2
i
eqi
.01
[.39]
.00
[⫺.10]
.02
[1.15]
.01
[.55]
.03
[1.36]
.02
[1.16]
.02
[.75]
.02
[1.12]
⫺.02
[⫺1.07]
⫺.07
[⫺1.90]
⫺.08
[⫺1.48]
.10
[.04]
⫺2.94
[⫺1.69]
.22
[.17]
2.29
[1.37]
4.04
[2.08]
4.14
[1.98]
1.66
[1.21]
.89
[.65]
⫺2.40
[⫺1.36]
⫺5.66
[⫺1.99]
⫺5.77
[⫺1.92]
Note.— Estimates and tests are from the one-stage GMM estimation with an identity weighting matrix. To measure the portfolio-specific
i
corporate tax rate, tt⫹1
, we first construct firm-specific tax rates using the trichotomous variable approach of Graham (1996) and then take the
S
Iw
value-weighted tax rates across all firms within a given portfolio i. In panel A the moments are E[rit⫹1
⫺ rit⫹1
] p 0. The expected return errors
q
S
Iw
are defined as ei { E T[rit⫹1
⫺ rit⫹1
], in which E T[7] is the sample mean of the series in brackets. In panel B the moment conditions are
2
S
Iw
S
S
Iw
Iw
S
S
Iw
E[rit⫹1
⫺ rit⫹1
] p 0 and E[(rit⫹1
⫺ E[rit⫹1
]) 2 ⫺ (rit⫹1
⫺ E[rit⫹1
]) 2 ] p 0. The variance errors are defined as eij { E T[(rit⫹1
⫺ E T[rit⫹1
]) 2 ⫺ (rit⫹1
⫺
Iw
E T[rit⫹1
]) 2 ]. The expected return errors are defined as in panel A. In the last column we report the difference in the expected return errors and
the difference in the variance errors between the high and low portfolios and their t-statistics. Expected return errors are in annual percent, and
variance errors are in annual decimals.
21
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D1.— Average predicted stock returns versus average realized stock returns, the q-theory model, matching
only expected stock returns, second-stage GMM. Figures D1a, D1b, and D1c report the results for the 10 SUE
portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively. High denotes the high decile and low
denotes the low decile.
22
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D2.— Predicted stock return volatilities versus realized stock return volatilities, average predicted stock
returns versus average realized stock returns, the q-theory model, matching expected returns and variances
simultaneously, second-stage GMM. Figures D2a, D2c, and D2e report the volatility plots for the 10 SUE
portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively. Figures D2b, D2d, and D2f report the
expected return plots for the 10 SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively.
23
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D3.— Average predicted stock returns versus average realized stock returns, the q-theory model, matching
only expected stock returns, the market value of debt per the Bernanke and Campbell (1988) algorithm. Figures
D3a, D3b, and D3c report the results for the 10 SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios,
respectively. High denotes the high decile and low denotes the low decile.
24
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D4.— Predicted stock return volatilities versus realized stock return volatilities, average predicted stock
returns versus average realized stock returns, the q-theory model, matching expected returns and variances
simultaneously, the market value of debt per the Bernanke and Campbell (1988) algorithm. Figures D4a, D4c,
and D4e report the volatility plots for the 10 SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios,
respectively. Figures D4b, D4d, and D4f report the expected return plots for the 10 SUE portfolios, the 10 B/M
portfolios, and the 10 CI portfolios, respectively.
25
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D5.— Average predicted stock returns versus average realized stock returns, the q-theory model, matching
only expected stock returns, value-weighted returns. Figures D5a, D5b, and D5c report the results for the 10
SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively. High denotes the high decile and
low denotes the low decile.
26
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D6.— Predicted stock return volatilities versus realized stock return volatilities, average predicted stock
returns versus average realized stock returns, the q-theory model, matching expected returns and variances
simultaneously, value-weighted returns. Figures D6a, D6c, and D6e report the volatility plots for the 10 SUE
portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively. Figures D6b, D6d, and D6f report the
expected return plots for the 10 SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively.
27
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D7.— Average predicted stock returns versus average realized stock returns, the q-theory model, matching
only expected stock returns, an alternative measure of capital. Figures D7a, D7b, and D7c report the results for
the 10 SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively. High denotes the high
decile and low denotes the low decile.
28
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D8.— Predicted stock return volatilities versus realized stock return volatilities, average predicted stock
returns versus average realized stock returns, the q-theory model, matching expected returns and variances
simultaneously, an alternative measure of capital. Figures D8a, D8c, and D8e report the volatility plots for the
10 SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively. Figures D8b, D8d, and D8f
report the expected return plots for the 10 SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios,
respectively.
29
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D9.— Average predicted stock returns versus average realized stock returns, the q-theory model, matching
only expected stock returns, an alternative measure of investment. Figures D9a, D9b, and D9c report the results
for the 10 SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively. High denotes the high
decile and low denotes the low decile.
30
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D10.— Predicted stock return volatilities versus realized stock return volatilities, average predicted stock
returns versus average realized stock returns, the q-theory model, matching expected returns and variances
simultaneously, an alternative measure of investment. Figures D10a, D10c, and D10e report the volatility plots
for the 10 SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively. Figures D10b, D10d,
and D10f report the expected return plots for the 10 SUE portfolios, the 10 B/M portfolios, and the 10 CI
portfolios, respectively.
31
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D11.— Average predicted stock returns versus average realized stock returns, the q-theory model, matching
only expected stock returns, time-invariant tax rates. Figures D11a, D11b, and D11c report the results for the 10
SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively. High denotes the high decile and
low denotes the low decile.
32
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D12.— Predicted stock return volatilities versus realized stock return volatilities, average predicted stock
returns versus average realized stock returns, the q-theory model, matching expected returns and variances
simultaneously, time-invariant tax rates. Figures D12a, D12c, and D12e report the volatility plots for the 10 SUE
portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively. Figures D12b, D12d, and D12f report
the expected return plots for the 10 SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively.
33
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D13.— Average predicted stock returns versus average realized stock returns, the q-theory model, matching
only expected stock returns, portfolio-specific tax rates. Figures D13a, D13b, and D13c report the results for the
10 SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively. High denotes the high decile
and low denotes the low decile.
34
App. D from Liu et al., “Investment-Based Expected Stock Returns”
Fig. D14.— Predicted stock return volatilities versus realized stock return volatilities, average predicted stock
returns versus average realized stock returns, the q-theory model, matching expected returns and variances
simultaneously, portfolio-specific tax rates. Figures D14a, D14c, and D14e report the volatility plots for the 10
SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios, respectively. Figures D14b, D14d, and D14f
report the expected return plots for the 10 SUE portfolios, the 10 B/M portfolios, and the 10 CI portfolios,
respectively.
35