Modeling the changing asymmetry of conditional variances

economics letters ELSEVIER

Economics Letters 50 (1996) 197-203

Modeling the changing asymmetry of conditional variances Fabio Fornari a'*, Antonio Mele b aBanca d'Italia-Research Department, Via Nazionale 91, 00184 Roma, Italy bUniversity of Paris X, 201 Avenue de la Republique, 92001 Nanterre, France

Received 10 March 1995; accepted 19 July 1995

Abstract We analyze the asymmetric response of the volatility to the arrival of shocks of opposite sign. After revising the major formulations developed so far to capture the phenomenon, a more general model is proposed; it cannot be rejected against three competing specifications when fitted to stock exchange indices returns. Keywords: Asymmetric variance; Conditional heteroskedasticity JEL classification: C51

1. Introduction Conditional heteroskedasticity has been the focus of m a n y theoretical and applied econometric papers in the last decade. Since Engle's (1982) and Bollerslev's (1986) pathbreaking works, generalized autoregressive conditional heteroskedastic ( G A R C H ) models have b e e n successfully applied to a variety of financial variables, including interest rates, exchange rates, commodities and stocks (see Bollerslev et al. (1992) for a review). New schemes able to detect and m o d e l particular features of the different series have been formulated: the a s y m m e t r y in the reaction of the conditional volatility to the arrival of n e w s - with shocks of opposite sign inducing different levels of volatility - and long m e m o r y - i.e. variances g e n e r a t e d by fractionally integrated stochastic p r o c e s s e s - are the most investigated topics. The present paper deals with asymmetric conditional variances.

2. Asymmetric GARCH models T h e standard G A R C H ( 1 , 1 ) model assumes that the rates of change of an asset price be conditionally normally distributed with zero m e a n and time-varying conditional variance, after * Corresponding author. Tel.: +39-6-4792-3189; fax: +39-6-474-7820; e-mail: [email protected]. 0165-1765/96/$12.00 © 1996 Elsevier Science S.A. All rights reserved S S D I 0165-1765(95)00736-9

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removing the presence of autocorrelation. If e, denotes such uncorrelated rates of change of an asset price, then a GARCH(1,1) model assumes that (1)

e, l I , _ a ~ N ( O , o ' ~ ) , O" 2t ~ tO ÷ a E t _2 1

+/300~_1

(2)

,

where to > 0 and a, /3 t> 0. In this model the variance reacts symmetrically to shocks of different sign, since only the squares of the latter are mapped onto 0°2. Because of this, the information contained in the sign of the last price variation (i.e. sign(e,_l) ) is lost. With concern for stock returns, Black (1976) noted that volatilities tend to be higher after negative shocks than after positive shocks of a similar size; such a phenomenon has come to be referred to as the leverage effect, since it links the equity value of a firm to the risk of the market. Various attempts have been made to include asymmetries into the modeling of the conditional variance; among the proposed models, that by Glosten et al. (1993; hereafter GJR), the exponential scheme ( E G A R C H ) of Nelson (1991), the threshold A R C H ( T A R C H ) of Zakoian (1994) and, finally, the volatility switching of Fornari and Mele (1995) are worth mentioning. We omit the power A R C H of Ding et al. (1993) from our short review, since it embodies both asymmetries and long memory features. The GJR(1,1) equation for the conditional variance is specified as follows: O ' t2 = 60 + 0 / E t2- 1

2

+ /300 - 1

(3)

+ ~lSt_lgt_ 1

with to > 0, a , / 3 / > 0, where s t is a dummy variable that equals one when et_ 1 is negative and is zero elsewhere. Thus, if Y is negative, the conditional variance at time t will be higher after a negative shock (G-~ < 0). The E G A R C H relaxes the constraints on the parameters of the conditional variance generating process by modeling the logarithm of the latter, i.e. 1og(00~) =

60+a((~t_,/00,_1)+(rle,_,/00,_,l

_ (2/7r)0.5)) +/3 log(00,_a) .2

(4)

In this case the presence of asymmetry is captured by the sign of a. The T A R C H ( p ) assumes that the following equation generates 00,: 00t = 60 + O/l(Ig,_ll + "Yle,_l) + oe2(let_2l + ~/2e,_2) + . . . ÷ ,,Ale,_pl + ~ j , _ p ) ,

(5)

with w, a l , . . . , % all greater than zero. Unlike the previous models, the volatility switching (VS) model is able to capture a phenomenon that has not been modeled before, i.e. the reversion of the asymmetric behavior of the variances. It implies that asymmetries can become inverted, with positive errors inducing more volatility than negative errors of the same size when the observed value of the conditional variance is lower than expected at t - 1 . The equation for the conditional variance of a VS(1,1) model is assumed to be (6)

O't2 = 0 . ) ÷ ae~-i ÷ /300~-1 ÷ ")lSt-lOt-l ' 2

2

where s,_ 1 is a dummy that equals one if e,_ 1 > 0 and minus one if st_ 1 < 0 and v t = (e,/00t-1) measures the difference between the forecast of the volatility at time t, 00~, on the basis of the information set dated t - 1, and the realized value, e~. As already seen, a negative shock can

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give rise to a level of volatility at time t that is below the value expected at t - 1; such a circumstance may be considered as good news and lead to lower expected volatility in the next period. Engle and Ng (1993) proposed specific tests for the presence of asymmetries, the most general one being the T R 2 of the following regression (with T being the sample size): 2 E t : Ol ÷ /318t_1 ÷ /32St_1Et_1 ÷ / 3 3 ( 1 -

S t _ l ) E t _ 1 ÷ ~'~t,

(7)

where s,_ 1 is a dummy variable that equals one if e,_ 1 is negative and is zero elsewhere. The test is asymptotically distributed as a chi-square with three degrees of freedom. Of course, one could simply detect asymmetries by estimating a symmetric heteroskedastic model and comparing the likelihood of the latter with that of an asymmetric one. Engle and Ng fitted different asymmetric models (not the VS) to stock returns and found the GJR to be the best parametric scheme to model asymmetries. Fornari and Mele (1995) fitted the GJR and VS models to seven international stock exchanges, and showed that the performance of the models may vary considerably as a function of the particular country being analyzed.

3. Results based o n a new asymmetric G A R C H model

According to the results in Engle and Ng, and Fornari and Mele, a mixture of the G JR and the VS models should provide a better fit for the asymmetric response of the conditional volatility to news, if compared with each of the two models alone. In fact, while the GJR captures the higher influence of negative shocks over positive ones (or vice versa), the VS is able to detect whether the direction of such a phenomenon could be reversed. In the remainder of the paper we link the two models, allowing for asymmetry and reversals in the following way" O" t : (JO ÷ OlF,t_ 1 ÷ ~3Oft_ 1 ÷ ~ / S t _ I E , _ 1

_l/Oft_l)

-- k}s,_ 1 .

(8)

The above equation can be re-written as 2 2 2 2 2 O" t = 09 ÷ Old,t_ 1 "~- /30"t_ 1 ÷ ~ / S t _ l e t _ 1 ÷ ~ ( E t _ 1 / o r t1_ l )2S , _

+

~kst

1

•

(9)

The first three terms on the RHS are the standard GARCH(1,1) model. These, together with the fourth term, define a GJR(1,1) model (except that s t is now a dummy that equals one or minus one instead of zero and one). The last two terms capture the reversal of asymmetry that is observed when ( e t2_ l / O ' t2_ l ) reaches k, the threshold value. The unconditional moments of the model are important to understand why Eq. (9) should be able to capture a well-known characteristic of returns distributions, i.e. the fat tails. Since E ( s , ) = 0, the first unconditional moment of cr~ is given by E(tr 2) = to + o~Z(e2) +/3E(tr 2) = oo/(1 - o~ - / 3 ) ,

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and coincides with the corresponding moment of a GARCH(1,1), implying that asymmetries do not affect 0.2 in the long run. The fourth unconditional moment is given by E(e4) = 3E(w2

+ O/2E4 _ 1820.4 q_ ,yEa _ 6 2 k 2 + 62(82/0.2) q. 2oeoge2 + 218o90.2 + 2oe18820.2) .

Collecting the terms in E(e 4) on the left-hand side and solving yields: E ( e 4 ) = [(3w 2 + 62(1 - k2))(1 - a - 18) + 6to2(a + 18)1

[(1 - a - 18)(1 - y __182 - 3a 2 - 218a)] Hence, the kurtosis of the model is given by k = 3E(e4)/E(e2)

2 -

3(1 - a - 18)[(3 + 62(1 - k))(1 - a - 18) + 2(a + 18)] (1 -- y -- 182 _ 3Ot2 _ 2a18)

(10)

The latter expression shows that both the fourth moment and the kurtosis are functions of 6, k and 3,; thus asymmetries and reversals affect the unconditional moments of the conditional variance. Because of this, the proposed model is able to generate high coefficients of kurtosis, as shown in (10), for particular values of the parameters of interest. We assess the ability of Eq. (9), called AARCH(1,1), to outperform other asymmetric heteroskedastic formulations, such as the GJR(1,1) or the VS(1,1), by means of the logarithmic rates of change of three stock exchange indices, namely New York's SP500, Paris's CAC40, and Milan's MIB, observed daily between January 1990 and September 1994. Table 1 reports the estimates of the GARCH(1,1), and GJR(1,1), the VS(1,1) and finally the A A R C H ( 1 , 1 ) for the three stock indices. Before the estimation was carried out, a test for the presence of heteroskedasticity was run. We chose Engle's (1982) T R 2 with five lags, which is asymptotically a chi-square with five degrees of freedom, under the null of homoskedasticity; it equalled 124.87, 267.15 and 266.82 for the United States, France and Italy, respectively, and clearly revealed the presence of heteroskedasticity. The test reported in Eq. (7) equalled 100.51, 218.71 and 244.86 for the United States, France and Italy, respectively, which unambiguously indicated an asymmetric behavior of the conditional variances. As evidenced by the results in Table 1, the presence of asymmetry is supported also by the likelihood ratio test, which compares a symmetric G A R C H model with an asymmetric one. With regard to the estimates, the AARCH(1,1) outperformed the three competing models for the CAC40: its likelihood was more than 20 above that for the VS(1,1), so that a likelihood ratio test, distributed as a chi-square with two degrees of freedom, would prefer the former at any level of confidence. The sum a +/3 is usually referred to as the persistence of the conditional volatility; when it equals one, 0.2 is generated by a random walk process. It is worth noting that while the GARCH(1,1) and the GJR(1,1) variances are generated by an integrated process, they tend towards stationarity under the VS(1,1) or A A R C H ( 1 , 1 ) models. This occurrence, which can be detected also for the United States and Italy, may indicate that the integration of the conditional variances is a phenomenon induced by neglecting to model the reversal of asymmetry: once it has been taken into consideration, in fact, we obtain a better specification of the model in terms of likelihood and less evidence of integration. For

a

Underlined

Italy AARCH(1,1 GARCH(1,1 GJR(1,1) VS(1,1)

3.06 4.70 9.59 2.40

x 10 - 4 × 10- s x 10-5 X 10--4

x 10-4 × 10-5

- 4 . 8 2 x 10 2 -0.056 -0.055 - - 2 . 4 9 X 10--2

-3.30 x 10-3 -0.0108 0.0112 - 6 . 6 3 x 1 0 -3

- 2 . 4 5 X 10 -3 -0.142 -0.117 -0.017

0

× X x x

10 - s 10 -8 10 -8 10 -6

4.57 9.06 9.79 4.56 x 10 -8 x 10 8 X 10 -6

X 10 6

7.81 x 10 6 5 . 4 0 x 10 -7 9 . 8 4 x 10 -7 1.87x10 _s

1.49 7.63 8.09 3.57

to

c o e f f i c i e n t s a r e n o t s i g n i f i c a n t at t h e 9 5 % l e v e l .

) )

1.86 x 1 0 - 4 - 7 . 5 6 x 10 - 4 8 . 8 3 x 10 6 1.87x10 -4

) )

France AARCH(1,1 GARCH(1,1 GJR(1,1) VS(1,1)

x 10 - 4

X 10 - 4

1.75 7.46 5.95 -7.25

United States AARCH(1,1) GARCH(1,1) GJR(1,1) VS(1,1)

/~

models a r, = tz + Oe,_l + e, 2 2 2 2 2 2 ~r, = ~0 + o z , _ 1 + t~r,_~ + 4 , s , _ ~e,_, + S s , _ , ( e , _ J o ' , _ , ) + S K ( s , _ l )

Table 1 GARCH

0.010 0.144 0.091 0.047

0.048 0.172 0.003 0.070

0.036 0.219 0.278 0.043

a

0.871 0.873 0.891 0.924

0.897 0.863 0.958 0.792

0.857 0.859 0.846 0.918

fl

--0.044 0.0608 -

-0.137 0.0654 -

-0.162 0.0897 -

4~

2 . 2 2 X 10 -6

1.821 x 10 -5

--2.55 X 10 -9 --3.49 X 10-6

1.25 x 1 0 - 5 -3.81 xl0 6

-

~

5 . 0 4 X 10 7 k = 1 _

- 8 . 2 8 x 10 -5 k = 1 _

- 1 . 7 5 x 10 -5 k = 1 -

~

4975.52 4657.58 4709.26 4988.96

5060.05 4938.66 4973.87 5037.85

5196.7 4617.34 4631.69 5197.32

Log-likelihood

tO

-,4

~" ~

~" "" r~

a

202

F. Fornari, A. Mele / Economics Letters 50 (1996) 197-203

Table 2 Pagan and Sabau's goodness-of-fit test a United States GARCH(1,1) GJR(1,1) VS(1,1) AARCH(1,1)

6.045 × 10-5 (4.46) 6.230 × 10-5 (4.55) 3.650 × 10-5 (2.12) -8.135 × 10-5 (-3.18)

0.269 0.266 0.673 1.480

France GARCH(1,1) GJR(1,1) VS(1,1) AARCH(I,1)

-5.490 × 10 -5 (4.29) 1.480 × 10-5 (0.90) -5.035 × 10-5 (-2.21) -3.000 × 10 -5 (-1.82)

0.461 (8.37) 0.862 (8.49) 1.363 (8.78) 1.243 (11.62)

Italy GARCH(1,1) GJR(1,1) VS(1,1) AARCH(1,1)

7.080 × 6.660 × 2.119 × 5.190 x

10 5 10 5 10 5 10-5

(5.56) (5.20) (5.20) (8.09)

(5.01) (4.68)

0.461 (7.03) 0.493 (7.30)

(1.03)

0.838 (6.81)

(3.28)

0.644 (7.33)

" Under the null that the GARCH variance is an optimal predictor of the realized variance, a and/3 should equal zero and one, respectively. The coefficients reported in bold are not significantly different from one. Student's t values are in parentheses.

the U n i t e d States, the VS(1,1) and A A R C H ( 1 , 1 ) are the best models in terms of the likelihood function. We n o w come to the last country under consideration, Italy. H e r e the VS(1,1) is the best parametric m o d e l to capture asymmetry of the conditional volatility; the A A R C H ( 1 , 1 ) ranks second, while the G A R C H ( 1 , 1 ) and G J R ( 1 , 1 ) are well below them. Table 2 shows the results of Pagan and Sabau's (1988) goodness-of-fit test, obtained by regressing the squares of the logarithmic rates of change of the stock indices, filtered with an M A ( 1 ) process, on a constant and the conditional variance. U n d e r the null that the G A R C H variance is an optimal predictor for the observed variance, the intercept should be zero and the slope one. According to this criterion (Table 2) we cannot unambiguously identify which is the best model: for the U n i t e d States, the VS and the A A R C H models show a better fit, although the slope (/3) is far from one. The unit restriction for/3 can be accepted in only two cases: for the C A C 4 0 in the G J R estimation and for the M I B in the case of the VS model. H o w e v e r , in the case of the A A R C H , the divergences b e t w e e n the slope and unity are never large enough to reject its validity (1.05 for France and 0.80 for Italy), if c o m p a r e d with the competing models.

References

Black, F., 1976, Studies of stock market volatility changes, Proceedings of the American Statistical Association, Business and Economic Statistics Section, Chicago, IL.

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Bollerslev, T., 1986, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31, 307-327. Bollerslev, T., R. Chou and K. Kroner, 1992, ARCH models in finance: A review of the theory and empirical evidence, Journal of Econometrics 52, 5-59. Ding, Z., R.F. Engle and C.W.J. Granger, 1993, A long memory property of stock returns and a new model, Journal of Empirical Finance 1, 83-106. Engle, R.F., 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50, 987-1008. Engle, R.F. and V. Ng, 1993, Measuring and testing the impact of news on volatility, Journal of Finance 45, 1749-1777. Fornari, F. and A. Mele, 1995, Sign- and volatility-switching ARCH models: Theory and applications to international stock markets, Banca d'Italia, Discussion Paper 251. Glosten, L., R. Jagannathan and D. Runkle, 1993, On the relation between the expected value and the volatility on the nominal excess returns on stocks, Journal of Finance 48, 1779-1801. Nelson, D., 1991, Conditional heteroskedasticity in asset returns: A new approach, Econometrica 59, 347-370. Pagan, A. and H. Sabau, 1988, Consistency tests for heteroskedasticity and risk models, University of Rochester, Mimeo. Zakoian, J.M., 1994, Threshold heteroskedastic models, Journal of Economic Dynamics and Control 18,931-955.