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Temporal dependence in asset pricing models
Economics Letters 0165-1765/94/$07.00
4.5 (1994) 361-366 0 1994 Elsevier
Temporal models
361 Science
B.V. All rights
dependence
reserved
in asset pricing
Harry J. Turtle* Faculty of Management, Received Accepted
University of Manitoba, Winnipeg, Manitoba, R3T 5V4, Canada
28 June 1993 13 January 1994
Abstract Estimating and testing conditional models in an unconditional statistical context display temporal dependence. This paper presents an economic explanation disturbances of financial returns in many studies. JEL
often results in model disturbances that for temporal dependence found in
classification: G12
1. Introduction Recent work in asset pricing has stressed the use of conditional relations which admit time-varying return moments. Support for the use of evolving conditional moments has been found in studies such as Schwert (1989) and Schwert and Seguin (1991) who find support for time variability in first and second moments of stock returns and show numerous methods of parameterizing the evolution of these moments using both statistical and economic rationale. The autoregressive patterns displayed by financial returns have been considered by Poterba and Summers (1988) and Cutler et al. (1990) who suggest that the positive autocorrelation displayed by financial series over relatively short horizons cannot be adequately modelled using economic risk factors. Subsequently, Fama and French (1988) counter these arguments and suggest temporal behavior in stock prices is consistent with time-varying equilibrium expected returns. This paper demonstrates that if a conditional CAPM holds with GARCH variances, then unconditional model disturbances will display temporal dependence. ’ Empirical evidence is provided through a simple GARCH-M model of market returns over the period 1926-1988. The analysis presented may be generalized to suggest that temporal dependence in unconditional model disturbances may result solely from the use of inadequate conditioning variables. In particular, if second moment evolution involves random elements not captured in an unconditional paradigm, temporal dependence in model disturbances will result. The result is important because it demonstrates an economically meaningful alternative to previous statistical characterizations of excess return dynamics suggested in recent papers such as Akgiray (1989). The remainder of the paper is organized as follows. Section 2 presents the conditional CAPM * I am grateful to Bob Korkie, Adolf Buse, Charles Mossman and Mick Swartz for helpful comments. Freddy Delbaen for particularly beneficial discussions. Financial support was provided by SSHRC. ’ Engle (1982) and Bollerslev (1986) first introduced ARCH and GARCH models, respectively. SSDI
0165-1765(94)00423-Y
I also wish to thank
362
H.J.
model and shows the relation specification. Section 3 reports are offered in section 4.
Turtle
I Economics
Letters
45 (1994) 361-366
between a conditional GARCH-M model the empirical results and diagnostic analysis.
and an unconditional Concluding comments
2. The model Consider (GARCH-M)
the simple univariate model of conditional
Generalized asset returns
AutoRegressive Heteroscedastic for the market portfolio, rr,
in
Mean
r, = &~f + u, ,
(1)
where 6 is a market price of risk, related to aggregate The distribution of u, is assumed to be independently conditional variance given by the GARCH process:
relative risk aversion. normally distributed with zero mean
‘Tf=y+P&+(YU:~I. Unconditionally,
and
(2)
we can also estimate
the model
r,=Q+u, = E, where E, distributed
+ u, ,
(3)
is the unconditional expectation of the excess return with mean zero and constant variance: a’=E[a:]=
for (Y, y >O,
l_;_,
Temporal dependence from the unconditional
in u, can be examined model
on the market
and u, is normally
p 2 0, and (Y+ /3 < 1. ’
using pk, the kth order
autocorrelation
coefficient
E[UP-kl Pk= a(u,)a(u,m,). In particular,
it is sufficient E[u,u,_,]
is non-zero.
= 6’(a
’ Akgiray
(4)
to notice + /3)E[u;-
(1989)
suggests
that for k = 1 the numerator
of (4) is non-zero,
because
‘1-~‘(~+P)(1_;_a)2 using an AR(l)
process
rather
than
a GARCH-M
process
of 6 estimates ’ The population parameter 6 in Eqs. (1) and (2) are equal because (T’ = E[crf]; however, the distributions from (1) and (2) will clearly differ. ’ Notice that the correlation coefficient equals zero if and only if the unconditional expectation of the squared conditional variance equals the squared unconditional expectation of the conditional variance,
Evaluation of the left-hand evaluating the expectation process governing GARCH
side of the equation can proceed by substituting the definition of conditional of the squared result. Equality results only if (Y equals zero, which violates variances.
variance and the evolution
H. J. Turtle
I Economics
Letters 45 (1994) 361-366
363
to capture conditional mean dynamics. The discussion above demonstrates that conditional market variances induce serial correlation into model disturbances in an obvious manner and suggest a GARCH-M is preferred to a purely statistical description. Below we show that the result demonstrated above is consistent with the empirical evidence for Standard and Poor’s 500 (SPSOO) Index.
3. Data and empirical
analysis
Thirty day Treasury-Bill rates are subtracted from monthly returns on the SP500 index from 1926 to 1988 to generate monthly excess returns. The data are obtained from Ibbotson Associates and are described in Ibbotson Associates (1986). The market excess return series is estimated according to the conditional GARCH-M model given by (1) and (2). The unconditional model of excess return behavior is given by (3). The log of the likelihood is maximized using the method of Berndt et al. (1974) assuming disturbances are normally distributed, N(0, af). For comparison purposes both models are estimated using the same maximum likelihood procedure. The estimation results for both the conditional GARCH-M model and the unconditional model are shown in Table 1. The results show the significance of all parameters in both models at the 5% level. 4 The performance of the GARCH-M model is strongly supported by the reported likelihood ratio test; although the conditional CAPM is rejected. 5,6 Further observations can be drawn from the parameter estimates in Table 1. The conditional model of excess return behavior is based upon an underlying unconditional market variance. Specifically, the GARCH-M model implies an unconditional series variance of
?
= 0.00276
l-S-6 using the conditional model of excess return evolution under the CAPM restriction b = 0 and the property of invariance. This estimate understates the variance estimate of 0.00349 from the unconditional model. In contrast, the maximum likelihood estimate of the unconditional mean implied by the conditional model is 0.00768, which exceeds the unconditional model mean estimate of 0.00680. ’ Examination of the temporal behavior of unconditional residuals and conditional standardized residuals demonstrates that the empirical results are consistent with the theoretical analysis presented in section 2. Temporal dependence in unconditional residuals is purged from the data using the conditional market model without resorting to an autoregressive process, as shown in Table 2. Residual means show that the maximum likelihood procedure overestimates the mean of
’ All significance levels, except those for b in the conditional model, are based upon simple one-tailed I-tests of the null hypothesis that the parameter value is less than or equal to zero. Because the myopic CAPM predicts b to be zero, and there are no a priori reasons to require that b is either weakly positive or negative, the alternative hypothesis is two-tailed in this case. ’ The existence of a positive intercept may also be consistent with market imperfections such as tax or dividend effects not captured by the model. The strict conditional CAPM may be rejected according to either the likelihood ratio test statistic or the f-statistic for b = 0 in the conditional model. ‘The consistent rejection of the myopic CAPM in a large number of studies has led researchers to examine multifactor versions of the CAPM [cf. Turtle et al. (1994)]. ’ The unconditional mean and variance estimates under the restricted conditional GARCH-M model are 0.00288522 and 0.00898708, respectively.
H. J. Turtle I Economics
364 Table 1 The conditional
and unconditional
CAPM
MLE parameter
estimate
Variable Panel A. Estimation
of a conditional
r, = b +
r%r: +u,
over the period
Letters 45 (1994) 361-366
January
1926-December
1988
t-statistic
GARCH-M
model
of market
excess
return
behavior,
) 3
where
u, - N(0, uy).
Log of the likelihood = 1207.51212 Restricted log of the likelihood (b = 0) = 1205.54792
b 6
0.00499233 1.384558
;
0.8506884 0.000085069 0.1198272
2.178600 1.700261
b h
’ (Y
42.44866 3.395272 a 6.007078 d
xZ test of myopic CAPM: 2[Unrestricted likelihood Panel B. Estimation r, = E, where
of an unconditional
- Restricted model
Log of the likelihood
0.00680244 0.003492219
ES,
of market
= 3.928406
excess
return
h behavior,
+ v, ,
v, - N(0, c2).
(T-
likelihood]
xZ test for GARCH-M 2[Unrestricted
= 1065.99865 3.128283 d 46.12606 a
model versus unconditional model: likelihood - Restricted likelihood]
= 283.02694
*
Note: Maximum likelihood estimates for both the conditional GARCH-M model and the unconditional model of market excess returns are reported. The test statistic for the conditional myopic CAPM is distributed as a xZ random variable with one degree of freedom. Similarly, the reported GARCH-M statistic is asymptotically distributed as a x2 variable with three degrees of freedom. a Significant at the 1% level. b Significant at the 5% level.
market excess returns using the conditional GARCH-M model, while the unconditional model underestimates slightly. 8 The standard deviation of the standardized residuals is very close to unity, as desired. Standardized residuals are clearly left skewed, whereas unconditional residuals are right skewed. In both cases the skewness is highly significant suggesting neither the conditional nor the unconditional form of the model entirely captures excess return behavior. The kurtosis in residuals appears to be dampened by the conditional form of the model although both series are significantly leptokurtic. The reported Box-Pierce Portmanteau Q statistics summarize the ability of the model to capture the temporal behavior of means and variances. McLeod and Li (1983) demonstrate that residuals and squared residuals are asymptotically standard normal random variables, and that the associated Q statistics are distributed as x2 random variables. The reported Q, and QXX confirm ‘The
apparent
difference
in magnitude
between
means
arises largely
due to the standardization
procedure.
H. J. Turtle I Economics Table 2 Diagnostic
residual
Mean Standard deviation Skewness ’ Kurtosis ’
9x(14
~~24)
QxAW Q,,(24)
analysis
on the SP500 over the period
365
Letters 45 (1994) 361-366
January
1926-December
Standardized GARCH-M model residuals”
Unconditional residuals h
-0.03973 1.009716 -0.46495 d 1.419402 ’ 20.64 32.83 12.00 24.27
0.000011 0.05915 0.507398 d 9.58736 ’ 36.87 d 70.73 (’ 426.23 d 511.63 ”
1988
model
Note: Summary statistics based upon standardized residuals from the conditional GARCH-M tional model residuals are presented. The Q,(K) and Qrl(K) statistics are asymptotically variables to test the null hypothesis that the first K autocorrelations for the series, or the series zero. ” GARCH-M model residuals are standardized by the contemporaneous conditional standard h The unconditional model is estimated using the same maximum likelihood procedure used GARCH-M model. ’ Significance levels are based upon expected values assuming normality. ’ Significant at the 1% level.
model and the uncondidistributed as xZ random squared, respectively, are deviation. to estimate
the univariate
the significance of the initial 12 and 24 autocorrelations at the 1% level for both unconditional residuals and unconditional residuals squared, respectively. Contrarily, the Q statistics for the standardized residuals show the initial 12 and 24 autocorrelations are not significantly different from zero when considered as a group. Thus, the GARCH-M model effectively eliminates temporal dependence from standardized model residuals in a manner consistent with the analysis of section 2.
4. Conclusions Systematic prediction errors can be caused by conditioning upon a weaker information set than required by the asset pricing paradigm. Model estimates may be consistent, as in the special case considered herein; however, disturbances may display dynamics. Fortunately, explicitly modelling time variability in second moments assists in explaining temporal behavior in model disturbances.
References Akgiray. V., 1989, Conditional heteroscedasticity in time series of stock returns: Evidence and forecasts, Journal of Business 62, 55-80. Berndt, E.K., B.H. Hall, R.E. Hall and J.A. Hausmann, 1974, Estimation and inference in nonlinear structural models, Annals of Economic and Social Measurement 4, 653-665. Bollerslev, T., 1986, Generalized autoregressive conditional heteroscedasticity, Journal of Econometrics 31, 307-327. Cutler, D.. J. Poterba and L. Summers, 1990, Speculative dynamics and the role of feedback traders, American Economic Review 80, 63-68. Engle, R.F., 1982, Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation, Econometrica 20, 83-104.
366
H.J.
Turtle
I Economics
Letters
4.5 (1994) 361-366
Fama, E.F. and K.R. French, 1988, Permanent and temporary components of stock prices, Journal of Political Economy 96, 246-273. Ibbotson Associates, 1986, Stocks, bonds, bills, and inflation: 1986 yearbook (Chicago). McLeod, A.J. and W.K. Li, 1983, Diagnostic checking ARMA time series models using squared-residual autocorrelations. Journal of Time Series Analysis 4, 269-273. Poterba, J.M. and L.H. Summers, 1988, Mean reversion in stock prices: Evidence and implications, Journal of Financial Economics 22, 27-59. Schwert, G.W., 1989, Why does stock market volatility change over time. Journal of Finance 44, 111551153. Schwert, G.W. and P. Seguin, 1990, Heteroskedasticity in stock returns, Journal of Finance 45, 1129-1155. Turtle, H.J., A. Buse and B. Korkie, 1994, Tests of conditional asset pricing with time-varying moments and risk prices, Journal of Financial and Quantitative Analysis 29, 1-15.
4.5 (1994) 361-366 0 1994 Elsevier
Temporal models
361 Science
B.V. All rights
dependence
reserved
in asset pricing
Harry J. Turtle* Faculty of Management, Received Accepted
University of Manitoba, Winnipeg, Manitoba, R3T 5V4, Canada
28 June 1993 13 January 1994
Abstract Estimating and testing conditional models in an unconditional statistical context display temporal dependence. This paper presents an economic explanation disturbances of financial returns in many studies. JEL
often results in model disturbances that for temporal dependence found in
classification: G12
1. Introduction Recent work in asset pricing has stressed the use of conditional relations which admit time-varying return moments. Support for the use of evolving conditional moments has been found in studies such as Schwert (1989) and Schwert and Seguin (1991) who find support for time variability in first and second moments of stock returns and show numerous methods of parameterizing the evolution of these moments using both statistical and economic rationale. The autoregressive patterns displayed by financial returns have been considered by Poterba and Summers (1988) and Cutler et al. (1990) who suggest that the positive autocorrelation displayed by financial series over relatively short horizons cannot be adequately modelled using economic risk factors. Subsequently, Fama and French (1988) counter these arguments and suggest temporal behavior in stock prices is consistent with time-varying equilibrium expected returns. This paper demonstrates that if a conditional CAPM holds with GARCH variances, then unconditional model disturbances will display temporal dependence. ’ Empirical evidence is provided through a simple GARCH-M model of market returns over the period 1926-1988. The analysis presented may be generalized to suggest that temporal dependence in unconditional model disturbances may result solely from the use of inadequate conditioning variables. In particular, if second moment evolution involves random elements not captured in an unconditional paradigm, temporal dependence in model disturbances will result. The result is important because it demonstrates an economically meaningful alternative to previous statistical characterizations of excess return dynamics suggested in recent papers such as Akgiray (1989). The remainder of the paper is organized as follows. Section 2 presents the conditional CAPM * I am grateful to Bob Korkie, Adolf Buse, Charles Mossman and Mick Swartz for helpful comments. Freddy Delbaen for particularly beneficial discussions. Financial support was provided by SSHRC. ’ Engle (1982) and Bollerslev (1986) first introduced ARCH and GARCH models, respectively. SSDI
0165-1765(94)00423-Y
I also wish to thank
362
H.J.
model and shows the relation specification. Section 3 reports are offered in section 4.
Turtle
I Economics
Letters
45 (1994) 361-366
between a conditional GARCH-M model the empirical results and diagnostic analysis.
and an unconditional Concluding comments
2. The model Consider (GARCH-M)
the simple univariate model of conditional
Generalized asset returns
AutoRegressive Heteroscedastic for the market portfolio, rr,
in
Mean
r, = &~f + u, ,
(1)
where 6 is a market price of risk, related to aggregate The distribution of u, is assumed to be independently conditional variance given by the GARCH process:
relative risk aversion. normally distributed with zero mean
‘Tf=y+P&+(YU:~I. Unconditionally,
and
(2)
we can also estimate
the model
r,=Q+u, = E, where E, distributed
+ u, ,
(3)
is the unconditional expectation of the excess return with mean zero and constant variance: a’=E[a:]=
for (Y, y >O,
l_;_,
Temporal dependence from the unconditional
in u, can be examined model
on the market
and u, is normally
p 2 0, and (Y+ /3 < 1. ’
using pk, the kth order
autocorrelation
coefficient
E[UP-kl Pk= a(u,)a(u,m,). In particular,
it is sufficient E[u,u,_,]
is non-zero.
= 6’(a
’ Akgiray
(4)
to notice + /3)E[u;-
(1989)
suggests
that for k = 1 the numerator
of (4) is non-zero,
because
‘1-~‘(~+P)(1_;_a)2 using an AR(l)
process
rather
than
a GARCH-M
process
of 6 estimates ’ The population parameter 6 in Eqs. (1) and (2) are equal because (T’ = E[crf]; however, the distributions from (1) and (2) will clearly differ. ’ Notice that the correlation coefficient equals zero if and only if the unconditional expectation of the squared conditional variance equals the squared unconditional expectation of the conditional variance,
Evaluation of the left-hand evaluating the expectation process governing GARCH
side of the equation can proceed by substituting the definition of conditional of the squared result. Equality results only if (Y equals zero, which violates variances.
variance and the evolution
H. J. Turtle
I Economics
Letters 45 (1994) 361-366
363
to capture conditional mean dynamics. The discussion above demonstrates that conditional market variances induce serial correlation into model disturbances in an obvious manner and suggest a GARCH-M is preferred to a purely statistical description. Below we show that the result demonstrated above is consistent with the empirical evidence for Standard and Poor’s 500 (SPSOO) Index.
3. Data and empirical
analysis
Thirty day Treasury-Bill rates are subtracted from monthly returns on the SP500 index from 1926 to 1988 to generate monthly excess returns. The data are obtained from Ibbotson Associates and are described in Ibbotson Associates (1986). The market excess return series is estimated according to the conditional GARCH-M model given by (1) and (2). The unconditional model of excess return behavior is given by (3). The log of the likelihood is maximized using the method of Berndt et al. (1974) assuming disturbances are normally distributed, N(0, af). For comparison purposes both models are estimated using the same maximum likelihood procedure. The estimation results for both the conditional GARCH-M model and the unconditional model are shown in Table 1. The results show the significance of all parameters in both models at the 5% level. 4 The performance of the GARCH-M model is strongly supported by the reported likelihood ratio test; although the conditional CAPM is rejected. 5,6 Further observations can be drawn from the parameter estimates in Table 1. The conditional model of excess return behavior is based upon an underlying unconditional market variance. Specifically, the GARCH-M model implies an unconditional series variance of
?
= 0.00276
l-S-6 using the conditional model of excess return evolution under the CAPM restriction b = 0 and the property of invariance. This estimate understates the variance estimate of 0.00349 from the unconditional model. In contrast, the maximum likelihood estimate of the unconditional mean implied by the conditional model is 0.00768, which exceeds the unconditional model mean estimate of 0.00680. ’ Examination of the temporal behavior of unconditional residuals and conditional standardized residuals demonstrates that the empirical results are consistent with the theoretical analysis presented in section 2. Temporal dependence in unconditional residuals is purged from the data using the conditional market model without resorting to an autoregressive process, as shown in Table 2. Residual means show that the maximum likelihood procedure overestimates the mean of
’ All significance levels, except those for b in the conditional model, are based upon simple one-tailed I-tests of the null hypothesis that the parameter value is less than or equal to zero. Because the myopic CAPM predicts b to be zero, and there are no a priori reasons to require that b is either weakly positive or negative, the alternative hypothesis is two-tailed in this case. ’ The existence of a positive intercept may also be consistent with market imperfections such as tax or dividend effects not captured by the model. The strict conditional CAPM may be rejected according to either the likelihood ratio test statistic or the f-statistic for b = 0 in the conditional model. ‘The consistent rejection of the myopic CAPM in a large number of studies has led researchers to examine multifactor versions of the CAPM [cf. Turtle et al. (1994)]. ’ The unconditional mean and variance estimates under the restricted conditional GARCH-M model are 0.00288522 and 0.00898708, respectively.
H. J. Turtle I Economics
364 Table 1 The conditional
and unconditional
CAPM
MLE parameter
estimate
Variable Panel A. Estimation
of a conditional
r, = b +
r%r: +u,
over the period
Letters 45 (1994) 361-366
January
1926-December
1988
t-statistic
GARCH-M
model
of market
excess
return
behavior,
) 3
where
u, - N(0, uy).
Log of the likelihood = 1207.51212 Restricted log of the likelihood (b = 0) = 1205.54792
b 6
0.00499233 1.384558
;
0.8506884 0.000085069 0.1198272
2.178600 1.700261
b h
’ (Y
42.44866 3.395272 a 6.007078 d
xZ test of myopic CAPM: 2[Unrestricted likelihood Panel B. Estimation r, = E, where
of an unconditional
- Restricted model
Log of the likelihood
0.00680244 0.003492219
ES,
of market
= 3.928406
excess
return
h behavior,
+ v, ,
v, - N(0, c2).
(T-
likelihood]
xZ test for GARCH-M 2[Unrestricted
= 1065.99865 3.128283 d 46.12606 a
model versus unconditional model: likelihood - Restricted likelihood]
= 283.02694
*
Note: Maximum likelihood estimates for both the conditional GARCH-M model and the unconditional model of market excess returns are reported. The test statistic for the conditional myopic CAPM is distributed as a xZ random variable with one degree of freedom. Similarly, the reported GARCH-M statistic is asymptotically distributed as a x2 variable with three degrees of freedom. a Significant at the 1% level. b Significant at the 5% level.
market excess returns using the conditional GARCH-M model, while the unconditional model underestimates slightly. 8 The standard deviation of the standardized residuals is very close to unity, as desired. Standardized residuals are clearly left skewed, whereas unconditional residuals are right skewed. In both cases the skewness is highly significant suggesting neither the conditional nor the unconditional form of the model entirely captures excess return behavior. The kurtosis in residuals appears to be dampened by the conditional form of the model although both series are significantly leptokurtic. The reported Box-Pierce Portmanteau Q statistics summarize the ability of the model to capture the temporal behavior of means and variances. McLeod and Li (1983) demonstrate that residuals and squared residuals are asymptotically standard normal random variables, and that the associated Q statistics are distributed as x2 random variables. The reported Q, and QXX confirm ‘The
apparent
difference
in magnitude
between
means
arises largely
due to the standardization
procedure.
H. J. Turtle I Economics Table 2 Diagnostic
residual
Mean Standard deviation Skewness ’ Kurtosis ’
9x(14
~~24)
QxAW Q,,(24)
analysis
on the SP500 over the period
365
Letters 45 (1994) 361-366
January
1926-December
Standardized GARCH-M model residuals”
Unconditional residuals h
-0.03973 1.009716 -0.46495 d 1.419402 ’ 20.64 32.83 12.00 24.27
0.000011 0.05915 0.507398 d 9.58736 ’ 36.87 d 70.73 (’ 426.23 d 511.63 ”
1988
model
Note: Summary statistics based upon standardized residuals from the conditional GARCH-M tional model residuals are presented. The Q,(K) and Qrl(K) statistics are asymptotically variables to test the null hypothesis that the first K autocorrelations for the series, or the series zero. ” GARCH-M model residuals are standardized by the contemporaneous conditional standard h The unconditional model is estimated using the same maximum likelihood procedure used GARCH-M model. ’ Significance levels are based upon expected values assuming normality. ’ Significant at the 1% level.
model and the uncondidistributed as xZ random squared, respectively, are deviation. to estimate
the univariate
the significance of the initial 12 and 24 autocorrelations at the 1% level for both unconditional residuals and unconditional residuals squared, respectively. Contrarily, the Q statistics for the standardized residuals show the initial 12 and 24 autocorrelations are not significantly different from zero when considered as a group. Thus, the GARCH-M model effectively eliminates temporal dependence from standardized model residuals in a manner consistent with the analysis of section 2.
4. Conclusions Systematic prediction errors can be caused by conditioning upon a weaker information set than required by the asset pricing paradigm. Model estimates may be consistent, as in the special case considered herein; however, disturbances may display dynamics. Fortunately, explicitly modelling time variability in second moments assists in explaining temporal behavior in model disturbances.
References Akgiray. V., 1989, Conditional heteroscedasticity in time series of stock returns: Evidence and forecasts, Journal of Business 62, 55-80. Berndt, E.K., B.H. Hall, R.E. Hall and J.A. Hausmann, 1974, Estimation and inference in nonlinear structural models, Annals of Economic and Social Measurement 4, 653-665. Bollerslev, T., 1986, Generalized autoregressive conditional heteroscedasticity, Journal of Econometrics 31, 307-327. Cutler, D.. J. Poterba and L. Summers, 1990, Speculative dynamics and the role of feedback traders, American Economic Review 80, 63-68. Engle, R.F., 1982, Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation, Econometrica 20, 83-104.
366
H.J.
Turtle
I Economics
Letters
4.5 (1994) 361-366
Fama, E.F. and K.R. French, 1988, Permanent and temporary components of stock prices, Journal of Political Economy 96, 246-273. Ibbotson Associates, 1986, Stocks, bonds, bills, and inflation: 1986 yearbook (Chicago). McLeod, A.J. and W.K. Li, 1983, Diagnostic checking ARMA time series models using squared-residual autocorrelations. Journal of Time Series Analysis 4, 269-273. Poterba, J.M. and L.H. Summers, 1988, Mean reversion in stock prices: Evidence and implications, Journal of Financial Economics 22, 27-59. Schwert, G.W., 1989, Why does stock market volatility change over time. Journal of Finance 44, 111551153. Schwert, G.W. and P. Seguin, 1990, Heteroskedasticity in stock returns, Journal of Finance 45, 1129-1155. Turtle, H.J., A. Buse and B. Korkie, 1994, Tests of conditional asset pricing with time-varying moments and risk prices, Journal of Financial and Quantitative Analysis 29, 1-15.