The estimated conditional market variance (equation (14)) is ˆ
σt2 = ω0+ ω1Zt (A1)
where the state variables follow the process
Zt+1 = B0+ B1Zt+ εZ,t+1 (A2)
as in equation (15). The revision of the risk component is defined as:
ησ,t+1 = Et+1
The process for the state variables implies
EtZt+j = (I − B1j)(I − B1)−1B0+ B1jZt (A4)
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Footnotes
1The expected stock market return was long considered to be constant until relatively recent work documenting the predictability of market returns (e.g., Fama and French (1989)). It is now well un-derstood that time-varying expected returns are consistent with rational expectations. See Camp-bell and Cochrane (1999) and Guo (2003) for recent examples of this literature.
2Strictly speaking, equation (1) is the discrete time version of Merton’s ICAPM (see Long [1974]).
In addition, the equation holds for the aggregate wealth portfolio for which we use the market portfolio as a proxy.
3Specifically, if the optimal consumption function of the representative agent is much less sensitive to the state variables that describe investment opportunities than to wealth or if the variance of the change in wealth is much larger than the variance of the change in the state variables, then this approximation will be reasonable (see footnote 12 in his paper).
4French, Schwert and Stambaugh (1987) argue that full information maximum likelihood estimators such as GARCH are generally more sensitive to model misspecification than instrumental variable estimators.
5Pindyck (1984, 1988), Poterba and Summers (1986), French, Schwert and Stambaugh (1987), Campbell and Hentschel (1992) and Wu (2001) all emphasize the importance of the volatility feed-back effect in detecting the risk-return relation.
6Part of the implied volatility data are constructed by Christensen and Prabhala (1998) and are kindly provided to us by N. Prabhala.
7This type of specification has a long history in the literature. For example, French, Schwert and Stambaugh (1987) use a time series model of realized variance to model the conditional variance.
Numerous papers since then have employed predetermined financial variables as additional predic-tors.
8Extending the VAR to longer lags is conceptually straightforward, but it adds nothing to the intuition from the model.
9Since Xt is a subset of Zt, equation (18) is inconsistent with equation (15) unless the additional variables do not enter equation (18). In implementing this procedure we guarantee consistency by zeroing out the statistically insignificant coefficients throughout. Consequently, equation (18) is
subsumed by equation (15)
10See the CBOE web site (http://www.cboe.com) for details and the historical data series.
11Restricting the sample to the November 1983 to May 1995 subperiod spanned by the Christensen and Prabhala series also produces qualitatively similar results.
12See Anderson, Bollerslev, Diebold and Labys (2003) and the references therein for a discussion of the theory of quadratic variation and the conditions under which realized variance is an unbiased and efficient predictor of return volatility.
13In our sample, the first order autocorrelation of daily returns is 13.9% and all higher order auto-correlations are essentially zero.
14See his web site, http://pages.stern.nyu.edu/ ˜mlettau/, for details.
15An earlier version of the paper used a somewhat different set of four instrumental variables for both the variance and the hedge component of returns: (1) RREL, (2) the yield spread between Baa-rated and Aaa-rated bonds, (3) the yield spread between 6-month commercial paper and 3-month Treasury bills, and (4) the dividend yield. The results are qualitatively similar, and the new specification is more parsimonious.
16Other variables such as the commercial paper–Treasury spread and longer lags of realized vari-ance have a small amount of marginal predictive power over and above CAY and RREL in our sample. However, the results are invariant to including these additional variables and we restrict the set of variables for parsimony.
17Note that throughout the paper significance levels are based on two-sided tests. To the extent that one-sided tests are appropriate in some cases, the relevant significance level is half of that reported.
18The standard errors are computed via GMM and are asymptotic. Given the sample size, there is some question as to whether asymptotic standard errors are appropriate. This question is im-possible to answer definitively, but we also compute small sample standard errors using a block bootstrap methodology. The resulting standard errors on the state variables are actually slightly lower than those reported in Table II, but the standard errors on γ are somewhat higher.
19Campbell et al (2000) suggest this type of adjustment to the crash variance, although a similar result could be achieved by dropping this observation from the sample. There may be other reasons to exclude the October 19, 1987 market crash from the sample. Schwert (1990b) shows it is unusual
in many ways, and Seyhun (1990) argues that it is not explained by the fundamentals. It is not predicted by the option data used in this paper.
20Aiyagari and Gertler (1999) and Allen and Gale (1994) emphasize the liquidity effect on stock market volatility.
Table I: The Efficiency of Implied Volatility
We estimate equation (30) and restricted versions thereof over the period from January 1984 to May 2001.
Heteroscedasticity consistent standard errors are reported in parentheses. Coefficients that are significant at the 1%, 5% and 10% levels are marked by ***, ** and *, respectively. aThe constant has been scaled by a factor of 100 for presentation purposes.
Const. CAYt−1 RRELt−1 vt−12 Vt2 R2 vt2 0.020*** -0.043*** 0.246 0.394*** 0.274
(0.006) (0.013) (0.186) (0.091)
vt2 0.011* -0.023* 0.045 0.105 0.420*** 0.386 (0.006) (0.014) (0.163) (0.122) (0.101)
vt2 0.057***a 0.530*** 0.365
(0.019) (0.073)
Table II: Estimation Results with Implied Volatility
For Panel A we estimate the processes in equation (23) with statistically insignificant coefficients zeroed out. For Panel B we estimate the model in equation (24) and various restricted versions thereof (see Section III.A). In each case, the equations are estimated jointly via GMM over the period from January 1984 to May 2001. Heteroscedasticity consistent standard errors are reported in parentheses. Coefficients that are significant at the 1%, 5% and 10% levels are marked by ***, ** and *, respectively. The last two columns provide the degrees of freedom, test statistic, and p-value (in brackets) for the overidentifying restrictions test (see Section III.A for details). aThe constant has been scaled by a factor of 100 for presentation purposes.
Panel A: The Conditional Variance Process and the Processes for the State Variables
Const. vt Vt2 CAYt RRELt R2
RRELt+1 -0.002a 0.846*** 0.702
(0.003) (0.045)
Panel B: The Model
Const. γ CAY RREL R2 DF χ2
Model 1 0.003 2.485 0.008 0
(0.004) (1.909)
Model 1 0.004 3.842** 0.008 2 9.411
(0.004) (1.918) [0.009]
Model 2 0.003 2.485 0.203 0
(0.004) (1.909)
Model 2 0.001 4.571*** 0.203 2 4.974
(0.003) (1.705) [0.083]
Model 3 -0.211** 0.493** -4.706 0.034 0
(0.097) (0.215) (3.500)
Model 3 -0.221** 0.513** -4.395 0.034 1 4.778
(0.097) (0.214) (3.505) [0.029]
Model 4 -0.334*** 5.623*** 0.744*** -5.767 0.068 0 (0.103) (1.885) (0.266) (3.503)
Model 4 -0.327*** 5.919*** 0.727*** -5.616* 0.068 3 0.298
(0.101) (1.731) (0.222) (3.382) [0.960]
Model 5 -0.191** 4.350*** 0.426** -5.161 0.230 0 (0.084) (1.623) (0.188) (3.426)
Model 5 -0.201** 3.656*** 0.453** -4.279 0.227 3 1.038
(0.083) (1.352) (0.184) (3.221) [0.792]
Model 6 -0.189** 4.311 0.424** -2.875 0.248 0 (0.085) (2.692) (0.193) (3.794)
Model 6 -0.144** 4.928** 0.322** -3.462 0.278 3 0.831
(0.060) (2.142) (0.133) (3.066) [0.842]
Table III: Variance Predictability Regressions
We estimate equation (31) for various subsamples. Heteroscedasticity consistent standard errors are reported in parentheses. Coefficients that are significant at the 1%, 5% and 10% levels are marked by ***, ** and *, respectively. bThe market variance of October 1987 is adjusted as discussed in Section III.C.
Subperiod Const. CAYt−1 RRELt−1 vt−12 v2t−2 R2 9/62-9/87 vt2 0.004 -0.009 -0.009 0.441*** 0.172** 0.298
(0.003) (0.006) (0.087) (0.117) (0.067)
1/88-5/01 vt2 0.015** -0.032** 0.141 0.268*** 0.213** 0.278 (0.007) (0.015) (0.185) (0.094) (0.098)
9/62-5/01 vt2 0.012*** -0.023** 0.016 0.151* 0.113** 0.048 (0.004) (0.009) (0.093) (0.077) (0.056)
9/62-5/01b vt2 0.007*** -0.015** 0.015 0.360*** 0.207*** 0.252 (0.002) (0.006) (0.078) (0.066) (0.047)
9/62-5/01b vt2 0.008*** -0.015*** -0.014 0.453*** 0.219 (0.003) (0.006) (0.084) (0.067)
Table IV: Estimation Results with Financial State Variables
For Panel A we estimate the processes in equation (23) with statistically insignificant coefficients zeroed out. For Panel B we estimate the model in equation (24) and various restricted versions thereof (see Section III.A). In each case, the equations are estimated jointly via GMM over the period from September 1962 to May 2001. Heteroscedasticity consistent standard errors are reported in parentheses. Coefficients that are significant at the 1%, 5% and 10% levels are marked by ***, ** and *, respectively. The last two columns provide the degrees of freedom, test statistic, and p-value (in brackets) for the overidentifying restrictions test (see Section III.A for details).
Panel A: The Conditional Variance Process and the Processes for the State Variables
Const. vt2 CAYt RRELt R2
vt+12 0.008*** 0.454*** -0.015*** 0.219 (0.003) (0.067) (0.006)
CAYt+1 0.066*** 0.851*** 0.726
(0.011) (0.026)
RRELt+1 0.002** -0.004** 0.813*** 0.672
(0.001) (0.002) (0.056)
Panel B: The Model
Const. γ CAY RREL R2 DF χ2
Model 1 0.008 -1.459 0.001 0
(0.006) (3.039)
Model 1 0.011* -2.917 0.001 2 15.130
(0.006) (3.233) [0.001]
Model 2 0.008 -1.468 0.001 0
(0.006) (3.066)
Model 2 0.004 1.455 0.030 2 14.500
(0.004) (2.182) [0.001]
Model 3 -0.166*** 0.390*** -5.827*** 0.036 0
(0.058) (0.133) (2.117)
Model 3 -0.166*** 0.391*** -5.812*** 0.036 1 0.003
(0.058) (0.132) (2.097) [0.958]
Model 4 -0.164** -0.160 0.387** -5.832*** 0.036 0 (0.068) (3.051) (0.148) (2.132)
Model 4 -0.175*** 0.657 0.408*** -5.637*** 0.035 3 0.925
(0.065) (2.777) (0.142) (2.105) [0.819]
Model 5 -0.164** -0.161 0.387*** -5.827*** 0.035 0 (0.067) (3.070) (0.147) (2.115)
Model 5 -0.167** -0.084 0.392*** -5.716*** 0.035 3 0.978
(0.065) (2.898) (0.143) (2.110) [0.807]
Model 6 -0.182** 1.597 0.421*** -5.474** 0.040 0 (0.070) (3.482) (0.153) (2.326)
Model 6 -0.179*** 2.053 0.412*** -2.816 0.067 3 4.272
(0.066) (2.979) (0.145) (1.851) [0.234]
Table V: Estimation Results with Financial State Variables–11/83-5/01 Subsample
For Panel A we estimate the processes in equation (23) with statistically insignificant coefficients zeroed out. For Panel B we estimate the model in equation (24) and various restricted versions thereof (see Section III.A). In each case, the equations are estimated jointly via GMM over the period from January 1984 to May 2001. Heteroscedasticity consistent standard errors are reported in parentheses. Coefficients that are significant at the 1%, 5% and 10% levels are marked by ***, ** and *, respectively. The last two columns provide the degrees of freedom, test statistic, and p-value (in brackets) for the overidentifying restrictions test (see Section III.A for details). aThe constant has been scaled by a factor of 100 for presentation purposes.
Panel A: The Conditional Variance Process and the Processes for the State Variables
Const. vt2 CAYt RRELt R2
vt+12 0.020*** 0.390*** -0.043*** 0.269 (0.006) (0.090) (0.013)
CAYt+1 0.063*** 0.859*** 0.738
(0.016) (0.035)
RRELt+1 -0.002a 0.846*** 0.702
(0.003) (0.045)
Panel B: The Model
Const. γ CAY RREL R2 DF χ2
Model 1 0.013** -2.029 0.004 0
(0.005) (2.639)
Model 1 0.014*** -1.564 0.003 2 5.851
(0.005) (2.448) [0.054]
Model 2 0.013** -2.083 0.032 0
(0.005) (2.730)
Model 2 0.019*** -4.984* 0.052 2 5.008
(0.005) (2.470) [0.082]
Model 3 -0.211** 0.493** -4.706 0.034 0
(0.097) (0.215) (3.500)
Model 3 -0.207** 0.485** -4.941 0.034 1 0.900
(0.097) (0.214) (3.493) [0.343]
Model 4 -0.303** 3.070 0.687** -4.486 0.039 0 (0.146) (3.247) (0.316) (3.516)
Model 4 -0.305** 2.950 0.690** -3.706 0.039 3 1.047 (0.143) (3.272) (0.308) (3.342) [0.790]
Model 5 -0.308* 3.202 0.697** -5.183 0.001 0 (0.156) (3.616) (0.336) (3.773)
Model 5 -0.194* -0.566 0.457* -4.368 0.047 3 1.801 (0.114) (1.993) (0.249) (3.314) [0.615]
Model 6 -0.311** 3.419 0.704** -1.428 0.122 0 (0.143) (3.390) (0.309) (3.683)
Model 6 -0.287** 4.116 0.647** -1.856 0.131 3 0.495 (0.134) (3.037) (0.289) (3.064) [0.920]
Table VI: Estimation Results with Financial State Variables–Quarterly Data
For Panel A we estimate the processes in equation (23) with statistically insignificant coefficients zeroed out. For Panel B we estimate the model in equation (24) and various restricted versions thereof (see Section III.A). In each case, the equations are estimated jointly via GMM over the period 1952Q4-2002Q3. Heteroscedasticity consistent standard errors are reported in parentheses.
Coefficients that are significant at the 1%, 5% and 10% levels are marked by ***, ** and *, respectively. The last two columns provide the degrees of freedom, test statistic, and p-value (in brackets) for the overidentifying restrictions test (see Section III.A for details). aThe constant has been scaled by a factor of 100 for presentation purposes.
Panel A: The Conditional Variance Process and the Processes for the State Variables
Const. vt2 CAYt RRELt R2
vt+12 0.003*** 0.423*** -0.095*** 0.231 (0.000) (0.088) (0.027)
CAYt+1 -0.002a 0.833*** 0.693
(0.048) (0.038)
RRELt+1 -0.001a -0.022** 0.711*** 0.527
(0.014) (0.009) (0.081)
Panel B: The Model
Const. γ CAY RREL R2 DF χ2
Model 1 0.024 -1.379 0.002 0
(0.016) (3.107)
Model 1 0.057*** -8.712*** 0.004 2 16.893
(0.013) (2.420) [0.000]
Model 2 0.025 -1.547 0.080 0
(0.018) (3.435)
Model 2 0.071*** -10.658*** 0.184 2 10.931
(0.013) (2.382) [0.004]
Model 3 0.017*** 1.945*** -5.599*** 0.131 0
(0.005) (0.399) (1.994)
Model 3 0.014*** 1.946*** -5.240*** 0.131 1 3.947
(0.005) (0.399) (1.989) [0.047]
Model 4 -0.018 6.273** 2.709*** -5.158** 0.154 0 (0.017) (3.082) (0.573) (1.962)
Model 4 -0.014 6.187** 2.446*** -6.210*** 0.152 3 4.615
(0.016) (2.883) (0.523) (1.904) [0.202]
Model 5 -0.024 7.375 2.822*** -6.091** 0.001 0 (0.027) (4.880) (0.727) (2.402)
Model 5 -0.018 6.433** 2.522*** -6.902** 0.006 3 2.563
(0.019) (3.235) (0.565) (2.177) [0.464]
Model 6 -0.019 6.515** 2.779*** -3.216 0.429 0 (0.019) (3.258) (0.586) (2.107)
Model 6 -0.027 7.746** 2.970*** -3.091* 0.447 3 2.903
(0.017) (2.795) (0.587) (1.740) [0.407]
-0.02 0 0.02 0.04 0.06
Mar-84 Dec-86 Sep-89 Jun-92 Mar-95 Dec-97 Sep-00 Date
Monthly Expected Excess Return
Figure 1: The Components of Expected Returns using Implied Volatility
The risk (dashed line) and hedge (solid line) components of monthly expected excess returns for the period 3/84 to 5/01, using the implied volatility data. The estimation results are reported in Table II, Panel C, Model 6, overidentified specification. Recessions are marked as shaded bars.
-0.15 -0.1 -0.05 0 0.05 0.1
Dec-52 Dec-62 Dec-72 Dec-82 Dec-92 Dec-02 Date
Quarterly Expected Excess Return
Figure 2: The Components of Expected Returns using Financial State Variables
The risk (dashed line) and hedge (solid line) components of quarterly expected excess returns for the period 1952Q2 to 2002Q3, using instrumental variables to estimate the conditional variance. The estimation results are reported in Table VI, Panel B, Model 6, overidentified specification. Recessions are marked as shaded bars.