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A stochastic variance model for absolute returns
economics letters Economics
Letters
46 (1994) 211-214
A stochastic variance model for absolute returns Fabio
Fornaria
’
*, Antonio
Mele b
“Research Department (IC), Banca d’ltalia, Via Nazionale, 91, Rome 00184, Italy bUniversity of Paris X G.A.M.A., 200 Avenue de la Rippublique, 92001 Nanterre, France Received 14 December
1993; accepted
17 March
1994
Abstract Empirical research has shown that the autocorrelation function of a stationary series is maximised when the latter is raised to a positive power 6, which rarely coincides with two. Thus, both the GARCH and stochastic variance (SV) models fail to capture all the non-linear dependence of the data. The proposed model, in which 6 equals one, improves the specification for the SV models. JEL
classification: C51
1. Introduction In Engle’s (1982) and Bollerslev’s (1986) ARCH and GARCH models, the conditional variance, h,, of a stationary series, E,, is modelled according to restricted AR(p) or ARMA(p, 4) models. Both formulations are based on information sets which map past squared forecast errors into the conditional variance. To express this more formally, suppose variance as given by a E, to be a stationary return of a financial asset; then, its conditional GARCH(l, 1) model is h,=“o+cr,.r:_,+p,.h,_,
,
a,>O,
at
20,
p1 a0 )
(1)
where the information content of previous returns is assumed to be maximised when they are squared. Unfortunately, this does not turn out to be the best choice. Ding et al. (1993) have shown that the autocorrelation function of the New York Stock Exchange (SPSOO) index returns, rt, observed daily between 1925 and 1993, is maximised when Y, is raised to the power 6, with 6 being towards the lower end of the [l, 21 interval. To make use of this additional information, the authors propose the Power ARCH model (PARCH), in which the conditional standard deviation, say at, is modelled similarly to the GARCH framework, but is raised to the power * Corresponding
author.
0165-1765/94/$07.00 0 1994 Elsevier SSDI 0165-1765(94)00471-D
Science
B.V. All rights reserved
212
F. Fornari,
6. The PARCH(p, aB=(Y”+
C
A. Mele I Economics
q) model
is
(Y,‘(IE,-ll
-Y’&t~l)s
+ C
r=l.p
Letters 46 (1994) 211-214
pi'"P-j.
(2)
j=l.q
To overcome the limits imposed by the deterministic framework under which h, (h, = af) is generated in (1) and (2), Harvey et al. (1992) have proposed the so-called stochastic variance returns, is (SV) model, where h,, the logarithm of the variance of T,, a series of stationary given by rr = e, * CT,)
with E, - iid(O, 1) ,
h, = 13,+ 0, . h,_, + 77,, h, = log(a:)
(3)
with nt - iid(O,a*q)
,
(4)
,
thus being stochastic. The estimation of the model can be carried out by casting it in state-space form and applying the Kalman filter to yield minimum mean squared errors; these are subsequently employed to maximise the prediction error decomposition-based likelihood (see Harvey and Shepard, 1993).
2. The absolute SV model Building on Ding et al.‘s (1993) results we propose a stochastic variance model for absolute returns (ASV). In the next section we present empirical support for higher autocorrelations in absolute returns with respect to squared returns. As in Harvey et al. (1992) let us denote by Y, the return of a financial asset and by h, the logarithm of its conditional standard deviation, a,. Our model can be written as Y, = E, . exp(h,)
,
with E, - iid(O, 1) ,
h, = 0, + 8, -h,_, +
7, ,
with 7, - NID(0,
(5)
CJ’,) ,
(6)
h, = log@,) . Squaring
(5), taking
the positive
root
and logs, we obtain:
log14 = log@?) + logl% 3
(7)
log(gr) = h, = 0, + 8, . log@_ r) + n, .
(8)
If E, is normally
distributed,
loglr,l - E(loglE,l) log(q)
where 5, = E(loglr,l) = However, ingredients (see Harvey
N(0, l), then (9)
= log@,) + 5, >
= 0” + 8, . lo&--
1) + 771>
with log(e,I - E(logl&,I) IS . a zero mean variable, -0.6352 (see the appendix for the last two results). since 5, is not normally distributed, the Kalman to carry out quasi-maximum likelihood maximisation and Shepard, 1993).
variance
(6,) = 1.2337
and
filter will provide only the of the model in (8) and (9)
F. Fornari, A. Mele I Economics Table 1 Box and Pierce’s
Q test for absolute
Australia Germany Hong Kong Italy United Kingdom
3. Autocorrelations
and squared
QL
lQx,l
77.7 74.2 95.7 135.8 127.6
77.2 340.6 156.9 270.9 126.1
Letters 46 (1994) 211-214
213
returns
of absolute returns and estimates of the model
The ASV model has been applied to five time series of stock indices’ returns (Financial Times indices, taken from D.R.I.) observed daily between 3 January 1990 and 28 October 1993 for Australia, Germany, Hong Kong, Italy and the United Kingdom. In three cases out of five (Germany, Hong Kong and Italy) the autocorrelations of absolute returns (up to the tenth lag) exceeded those of squared returns (see Table 1) as evidenced by the Box and Pierce’s & test. In the remaining cases the autocorrelations were almost similar. However, the likelihood of the five ASV models was always greater than the corresponding value of the SV models, thus confirming the richer information content of absolute returns. The superiority of the ASV, compared with the SV framework, is evidenced also by the smaller values of its one-step-ahead forecast errors. The t-statistics for the level, h,, and error component, T,, were significant for both models and are reported in Table 2. To conclude, we estimated again the models leaving 100 observations outside the estimation sample. These have subsequently been used for forecasting purposes and the percentage root mean squared errors (PRMSE) have been computed; the latter are defined as the ratio of the root of the average squared deviations between fitted and actual values and the sample mean of the actual observations. In Table 2 t-Statistics Model
for the estimated
with absolute
with squared
Australia Germany Hong Kong Italy United Kingdom
(testing
values
Australia Germany Hong Kong Italy United Kingdom Model
coefficients
t trend
terrOr
1.70 1.80 2.06 1.86 2.02
21.00 20.93 20.80 20.90 21.04
1.60 1.73 2.05 1.86 1.99
21.00 20.93 20.80 20.90 20.83
values
of the hypothesis
ui = 0, at = 0)
F. Fornari,
214
A. Mele
I Economics
Letters
46 (1994) 211-214
all cases except Australia, for which the autocorrelation of squared returns was similar to that of absolute returns, the PRMSE have been lower for the SV models estimated in terms of absolute values.
4. Conclusions Following the empirical findings of higher autocorrelations in absolute returns, as opposed to squared returns, we developed a stochastic variance model to capture the feature. When applied to five stock indices’ returns it outperforms the squared-return-based models.
Appendix Recall freedom. 2 =
that if E is a normally distributed variable, .s2 is a chi-square We are interested in finding out the mean and variance of log)&\
= 10&‘)E2)o’5 =
0.5
-lOg(E*)
of
.
Since w = log(e2) has mean G(0.5) - log(0.5) an d variance G’(0.5), the digamma and trigamma functions (Abramowitz and Stegun, z =0.5 * w, we obtain: E(z) = 0.5 * [G(0.5)
with one degree
where G(o) and G’(e) are 1970, p. 943) and since
- log(O.5)] = -0.6352
and Var(z) = 0.25. [3. S(2)] , where S(e) is the Riemann equals 1.2337.
zeta function
(Abramowitz
and Stegun,
1970, formula
6.4.4),
which
References Abramowitz, M. and I. Stegun, 1970, Handbook of mathematical functions (Dover Publication, New York). Bollerslev, T., 1986, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31, 307-327. Ding, Z., R. Engle and C. Granger, 1993, A long memory property of stock market returns and a new model, Journal of Empirical Finance 1, 83-106. Engle, R., 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50, 987-1008. Harvey, A. and N. Shephard, 1993, The econometrics of stochastic volatility, London School of Economics, Discussion Paper 166. Harvey, A., E. Ruiz and N. Shepard, 1992, Multivariate stochastic variance models, London School of Economics, Discussion Paper 132.
Letters
46 (1994) 211-214
A stochastic variance model for absolute returns Fabio
Fornaria
’
*, Antonio
Mele b
“Research Department (IC), Banca d’ltalia, Via Nazionale, 91, Rome 00184, Italy bUniversity of Paris X G.A.M.A., 200 Avenue de la Rippublique, 92001 Nanterre, France Received 14 December
1993; accepted
17 March
1994
Abstract Empirical research has shown that the autocorrelation function of a stationary series is maximised when the latter is raised to a positive power 6, which rarely coincides with two. Thus, both the GARCH and stochastic variance (SV) models fail to capture all the non-linear dependence of the data. The proposed model, in which 6 equals one, improves the specification for the SV models. JEL
classification: C51
1. Introduction In Engle’s (1982) and Bollerslev’s (1986) ARCH and GARCH models, the conditional variance, h,, of a stationary series, E,, is modelled according to restricted AR(p) or ARMA(p, 4) models. Both formulations are based on information sets which map past squared forecast errors into the conditional variance. To express this more formally, suppose variance as given by a E, to be a stationary return of a financial asset; then, its conditional GARCH(l, 1) model is h,=“o+cr,.r:_,+p,.h,_,
,
a,>O,
at
20,
p1 a0 )
(1)
where the information content of previous returns is assumed to be maximised when they are squared. Unfortunately, this does not turn out to be the best choice. Ding et al. (1993) have shown that the autocorrelation function of the New York Stock Exchange (SPSOO) index returns, rt, observed daily between 1925 and 1993, is maximised when Y, is raised to the power 6, with 6 being towards the lower end of the [l, 21 interval. To make use of this additional information, the authors propose the Power ARCH model (PARCH), in which the conditional standard deviation, say at, is modelled similarly to the GARCH framework, but is raised to the power * Corresponding
author.
0165-1765/94/$07.00 0 1994 Elsevier SSDI 0165-1765(94)00471-D
Science
B.V. All rights reserved
212
F. Fornari,
6. The PARCH(p, aB=(Y”+
C
A. Mele I Economics
q) model
is
(Y,‘(IE,-ll
-Y’&t~l)s
+ C
r=l.p
Letters 46 (1994) 211-214
pi'"P-j.
(2)
j=l.q
To overcome the limits imposed by the deterministic framework under which h, (h, = af) is generated in (1) and (2), Harvey et al. (1992) have proposed the so-called stochastic variance returns, is (SV) model, where h,, the logarithm of the variance of T,, a series of stationary given by rr = e, * CT,)
with E, - iid(O, 1) ,
h, = 13,+ 0, . h,_, + 77,, h, = log(a:)
(3)
with nt - iid(O,a*q)
,
(4)
,
thus being stochastic. The estimation of the model can be carried out by casting it in state-space form and applying the Kalman filter to yield minimum mean squared errors; these are subsequently employed to maximise the prediction error decomposition-based likelihood (see Harvey and Shepard, 1993).
2. The absolute SV model Building on Ding et al.‘s (1993) results we propose a stochastic variance model for absolute returns (ASV). In the next section we present empirical support for higher autocorrelations in absolute returns with respect to squared returns. As in Harvey et al. (1992) let us denote by Y, the return of a financial asset and by h, the logarithm of its conditional standard deviation, a,. Our model can be written as Y, = E, . exp(h,)
,
with E, - iid(O, 1) ,
h, = 0, + 8, -h,_, +
7, ,
with 7, - NID(0,
(5)
CJ’,) ,
(6)
h, = log@,) . Squaring
(5), taking
the positive
root
and logs, we obtain:
log14 = log@?) + logl% 3
(7)
log(gr) = h, = 0, + 8, . log@_ r) + n, .
(8)
If E, is normally
distributed,
loglr,l - E(loglE,l) log(q)
where 5, = E(loglr,l) = However, ingredients (see Harvey
N(0, l), then (9)
= log@,) + 5, >
= 0” + 8, . lo&--
1) + 771>
with log(e,I - E(logl&,I) IS . a zero mean variable, -0.6352 (see the appendix for the last two results). since 5, is not normally distributed, the Kalman to carry out quasi-maximum likelihood maximisation and Shepard, 1993).
variance
(6,) = 1.2337
and
filter will provide only the of the model in (8) and (9)
F. Fornari, A. Mele I Economics Table 1 Box and Pierce’s
Q test for absolute
Australia Germany Hong Kong Italy United Kingdom
3. Autocorrelations
and squared
QL
lQx,l
77.7 74.2 95.7 135.8 127.6
77.2 340.6 156.9 270.9 126.1
Letters 46 (1994) 211-214
213
returns
of absolute returns and estimates of the model
The ASV model has been applied to five time series of stock indices’ returns (Financial Times indices, taken from D.R.I.) observed daily between 3 January 1990 and 28 October 1993 for Australia, Germany, Hong Kong, Italy and the United Kingdom. In three cases out of five (Germany, Hong Kong and Italy) the autocorrelations of absolute returns (up to the tenth lag) exceeded those of squared returns (see Table 1) as evidenced by the Box and Pierce’s & test. In the remaining cases the autocorrelations were almost similar. However, the likelihood of the five ASV models was always greater than the corresponding value of the SV models, thus confirming the richer information content of absolute returns. The superiority of the ASV, compared with the SV framework, is evidenced also by the smaller values of its one-step-ahead forecast errors. The t-statistics for the level, h,, and error component, T,, were significant for both models and are reported in Table 2. To conclude, we estimated again the models leaving 100 observations outside the estimation sample. These have subsequently been used for forecasting purposes and the percentage root mean squared errors (PRMSE) have been computed; the latter are defined as the ratio of the root of the average squared deviations between fitted and actual values and the sample mean of the actual observations. In Table 2 t-Statistics Model
for the estimated
with absolute
with squared
Australia Germany Hong Kong Italy United Kingdom
(testing
values
Australia Germany Hong Kong Italy United Kingdom Model
coefficients
t trend
terrOr
1.70 1.80 2.06 1.86 2.02
21.00 20.93 20.80 20.90 21.04
1.60 1.73 2.05 1.86 1.99
21.00 20.93 20.80 20.90 20.83
values
of the hypothesis
ui = 0, at = 0)
F. Fornari,
214
A. Mele
I Economics
Letters
46 (1994) 211-214
all cases except Australia, for which the autocorrelation of squared returns was similar to that of absolute returns, the PRMSE have been lower for the SV models estimated in terms of absolute values.
4. Conclusions Following the empirical findings of higher autocorrelations in absolute returns, as opposed to squared returns, we developed a stochastic variance model to capture the feature. When applied to five stock indices’ returns it outperforms the squared-return-based models.
Appendix Recall freedom. 2 =
that if E is a normally distributed variable, .s2 is a chi-square We are interested in finding out the mean and variance of log)&\
= 10&‘)E2)o’5 =
0.5
-lOg(E*)
of
.
Since w = log(e2) has mean G(0.5) - log(0.5) an d variance G’(0.5), the digamma and trigamma functions (Abramowitz and Stegun, z =0.5 * w, we obtain: E(z) = 0.5 * [G(0.5)
with one degree
where G(o) and G’(e) are 1970, p. 943) and since
- log(O.5)] = -0.6352
and Var(z) = 0.25. [3. S(2)] , where S(e) is the Riemann equals 1.2337.
zeta function
(Abramowitz
and Stegun,
1970, formula
6.4.4),
which
References Abramowitz, M. and I. Stegun, 1970, Handbook of mathematical functions (Dover Publication, New York). Bollerslev, T., 1986, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31, 307-327. Ding, Z., R. Engle and C. Granger, 1993, A long memory property of stock market returns and a new model, Journal of Empirical Finance 1, 83-106. Engle, R., 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50, 987-1008. Harvey, A. and N. Shephard, 1993, The econometrics of stochastic volatility, London School of Economics, Discussion Paper 166. Harvey, A., E. Ruiz and N. Shepard, 1992, Multivariate stochastic variance models, London School of Economics, Discussion Paper 132.