Statistical Methods& Applications(2003) 12:19-39 DOI: 10.1007/s 10260-003-0048-0 SMA @ Spfinger-Verlag 2003 BL-GARCH models and asymmetries in volatility Giuseppe Storti, Cosimo Vitale Dipartimento di ScienzeEconomiche e Statistiche, Universit~di Salerno, Italy (e-mall: [email protected], [email protected]) Abstract. In this paper the class of Bilinear GARCH (BL-GARCH) models is proposed. BL-GARCH models allow to capture asymmetries in the conditional variance of financial and economic time series by means of interactions between past shocks and volatilities. The availability of likelihood based inference is an attractive feature of BL-GARCH models. Under the assumption of conditional normality, the log-likelihood function can be maximized by means of an EM type algorithm. The main reason for using the EM algorithm is that it allows to obtain parameter estimates which naturally guarantee the positive definiteness of the con- ditional variance with no need for additional parameter constraints. We also derive a robust LM test statistic which can be used for model identification. Finally, the effectiveness of BL-GARCH models in capturing asymmetric volatility patterns in financial time series is assessed by means of an application to a time series of daily returns on the NASDAQ Composite stock market index. Key words: BL-GARCH, Leverage effect, Conditional heteroskedasticity, Finan- cial time series 1 Introduction Financial analysts have often observed that markets tend to become more volatile as a consequence of negative shocks than of positive shocks. Empirical evidence in favour of this intuition is provided by the fact that a negative correlation between the current conditional variance and lagged returns is a relevant feature characterizing many financial time series. In the financial literature this has been designated as the leverage effect. Although, on a theoretical ground, an unanimous consensus on the economic reasons of the above described phenomenon has not been reached yet, several approaches to its empirical modeling have already been proposed in the literature.
Transcript
Statistical Methods & Applications (2003) 12:19-39 DOI: 10.1007/s 10260-003-0048-0 SMA
Abstract. In this paper the class of Bilinear GARCH (BL-GARCH) models is proposed. BL-GARCH models allow to capture asymmetries in the conditional variance of financial and economic time series by means of interactions between past shocks and volatilities. The availability of likelihood based inference is an attractive feature of BL-GARCH models. Under the assumption of conditional normality, the log-likelihood function can be maximized by means of an EM type algorithm. The main reason for using the EM algorithm is that it allows to obtain parameter estimates which naturally guarantee the positive definiteness of the con- ditional variance with no need for additional parameter constraints. We also derive a robust LM test statistic which can be used for model identification. Finally, the effectiveness of BL-GARCH models in capturing asymmetric volatility patterns in financial time series is assessed by means of an application to a time series of daily returns on the NASDAQ Composite stock market index.
Key words: BL-GARCH, Leverage effect, Conditional heteroskedasticity, Finan- cial time series
1 Introduction
Financial analysts have often observed that markets tend to become more volatile as a consequence of negative shocks than of positive shocks. Empirical evidence in favour of this intuition is provided by the fact that a negative correlation between the current conditional variance and lagged returns is a relevant feature characterizing many financial time series. In the financial literature this has been designated as the leverage effect. Although, on a theoretical ground, an unanimous consensus on the economic reasons of the above described phenomenon has not been reached yet, several approaches to its empirical modeling have already been proposed in the literature.
20 G. Storti, C. Vitale
In the present paper we propose an innovative approach to modelling leverage effects in financial time series based on the class of Bilinear GARCH, abbrevi- ated BL-GARCH, models which, as it is shown in the paper, can be considered as a generalization of traditional GARCH processes (Bollerslev, 1986). Also, the statistical properties of the proposed model are analysed and an algorithm for the maximum likelihood estimation of model parameters is illustrated. Namely, we give conditions for the positivity of the conditional variance and for the second order (covariance) stationarity of the general BL-GARCH model. Furthermore, for the BL-GARCH(1,1) model, the availability of analytic expressions for the uncondi- tional kurtosis coefficient of the process and for the global autocorrelation function of the squared process allows us to investigate the relations existing between these quantities and the parameters of the volatility model in different market conditions.
The structure of the paper is as follows. In Sect. 2 some of the most relevant contributions in the field of asymmetric conditional heteroskedastic models are reviewed while, in Sect. 3, the class of BL-GARCH models is introduced. The probabilistic properties of this class of models, including stationarity and moment conditions, are illustrated in Sect. 4. In this section, an analytical expression for the autocorrelation function of the squared observations is also derived. A strategy for maximum likelihood estimation of BL-GARCH models is illustrated in Sect. 5 while in Sect. 6 we derive a robust LM test statistic which can be used for model identification purposes. Finally, Sect. 7 presents the results of an application to the daily NASDAQ returns series in which the performance of the proposed BL- GARCH model is compared with that of some alternative asymmetric models. Section 8 concludes.
2 Leverage effects in financial time series
The main source of asymmetry in the volatility of financial time series is related to the so called leverage effect first noted by Black (1976). From a statistical point view, this effect can be identified with the presence of a negative correlation between changes in stock volatility and past returns. This leads the conditional variance of returns to be differently influenced by returns of the same magnitude but of differ- ent signs. In general, following a widely diffused approach, positive and negative variations in stock prices will both generate a positive variation in stock volatility but the response associated to negative ones will be higher. Hence it can be easily observed that a suitable volatility model for adequately explaining a similar effect cannot be given by a function of squared or absolute returns only, as for the standard GARCH or Taylor's model (Taylor, 1986), but should also include a term able to take into account the direction of the variation of stock prices i.e. the sign of returns. In the last decade a substantial amount of research has been devoted to extend the standard GARCH framework, in order to allow for more general model structures able to capture correlation between returns and conditional variance. One of the ear- liest contributions to the literature on asymmetric GARCH models is given by the Exponential Generalized ARCH (EGARCH) model proposed by Nelson (1991 ). In the general EGARCH(r,s) model the natural logarithm of the conditional variance
where ut = htzt and ht 2 = Var(ut l I t - l ) with I t-1 denoting past information up to time (t - 1); zt is a series of i.i.d, random variables with zero mean and unit variance, for t = 1, . . . ,T; c~0,/3j, ~ and )~i (i = 1~.. . , r , j = 1 , . . . ,s) are constant parameters. Another popular model is given by the GJR of Glosten et al. (1993) which attempts at capturing leverage effects by simply augmenting the standard GARCH model with an additive term incorporating asymmetric effects. The GJR model of order (r, s) is defined as
h2t = ao + E aiu2t-i + wiu2t-id~i + E bjh2t J (2) i = 1 i = 1 j = l
where d~- i is an indicator function that takes the value one when ut_i < 0 and zero otherwise, ai, by and wi are the model parameters. A sufficient condition for the positivity of the conditional variance is given by ao, ai, bj and (ai + wl) being non negative for i = 1 , . . . , r and j = 1 , . . . , s. In this case, under the assumption that the density of the standardized returns zt = (ut/ht) is symmetric,
r 2 s for the process to be stationary we must require }-~g=l (a~ § w~/ ) + }-~j=l bj < 1. Asymmetric effects enter into the Threshold GARCH (TGARCH) model proposed by Rabemananjara and Zakoian (1993) in a similar fashion. In the TGARCH the conditional standard deviation ht is related to past shocks and volatilities by the following equation:
r 8
ht = a; + E(a+u+_i - a i u~-_i) + E b;ht- j (3) i = 1 j = l
where a~, b~, a + and a~- are constant parameters, u + = rnax(ut~ O) and u t = min(ut, 0). It is worth noting that, the model for the conditional standard deviation can be reparameterized as
ht =- a; + ~ a;lut il + ~ w;ut-~dt--i + b;ht-j . (4) i = 1 i = 1 3=1
It follows that the conditional standard deviation in the TGARCH model has the same functional form as the conditional variance in the GJR model.
In order to account for the presence of asymmetric effects, the Generalized Quadratic ARCH model (GQARCH; Sentana, 1995) augments the conditional vari- ance equation of the standard GARCH model by linear functions of past returns and their interactions. The GQARCH(r,s) model can be written as:
with 5i and ffi,j (i, j = 1 , . . . , r) being constant coefficients. The resulting model can be interpreted as a second order Taylor approximation to the unknown condi- tional variance equation. In the (1,1) case the GQARCH model simplifies to:
ht 2 : ao + al~tt2_l -[- (~l?tt_l n t- blh2_l (6)
where (~1 is the parameter controlling asymmetry and the conditional variance is guaranteed to be positive if al > 0, bl > 0 and 5i < 4aoal. Under this assumption, the condition (al + bl) < i is required for second order stationarity.
Leverage effects are introduced into the the NonLinear GARCH (NLGARCH) model proposed by Engle (1990) by means of a linear combination of squared returns taken with their original signs. The conditional variance equation is:
8
2 2 h 2 : ao + ai(1 - 2r/isgn(ut-i) + r h )ut_ i + E bjh2t-J i=1 j = l
(7)
with I~?d < 1 and ai > 0, by > 0, for i = 1 , . . . , r and j = 1 , . . . ,s. Leverage effects are related to the r/i coefficients. By choosing r/i = 0 (Vi) the standard GARCH model is obtained. The NLGARCH can be regarded as a special case of the Asymmetric Power ARCH model (A-PARCH) of Ding et al. (1993). The statistical properties of this class of models are discussed by He and Ter~isvirta (1997). Finally, the NonLinear Asymmetric GARCH model (Engle and Ng, 1993) captures asymmetry by means of interactions between past returns and volatilities. In the simple (1,1) case the conditional variance equation is given by:
h~t = ao + a l (u t -1 + ~ l h t _ l ) 2 + blh~_ 1 (8)
with the model becoming symmetric when the coefficient ~1 is equal to zero. The BL-GARCH model, discussed in the next section, can be considered as a general- ization of the model in (8). Compared to the latter, the proposed BL-GARCH model has the advantage of being characterized by a more flexible parametric structure.
3 The Bilinear GARCH model
Let ut be an observed time series such that Urn(O, cry). A BL-GARCH model for ut is specified by the following equations
Ut = ht zt 8 7"*
h~ = ao + aiu2t-i + E bjh2-j + E c k u t - k h t - k (9) i=1 j = l k= l
where, for i = 1 , . . . , r, j = 1 , . . . , s and k = 1 , . . . , r*, ai, by and ck are constant parameters; r* = rain{r, s} and zt is defined as in the previous section. It follows that the observation process ut is serially uncorrelated with (ut II t - t ) ~ (0, h2t ), where I t-1 denotes past information up to time (t - 1). In model (9) leverage effects are explained by the interactions between past observations and volatilities.
BL-GARCH models and asymmetries in volatility 23
More precisely, for ck < 0, a positive quantity is added to ht 2 if ut-k < 0 while the same quantity is subtracted if ut-k > 0. It is interesting to note how the magnitude of the effect will depend not only on the value of ut-k but also on the value of ht-k. ff ck = 0 Yk, the conventional GARCH model is obtained. Hence, the BL-GARCH can be considered as a generalization of the basic GARCH model proposed by Bollerslev (1986). The denomination Bilinear is due to the interaction terms included into the conditional variance equation.
The following theorem estabilishes conditions for the positivity of the condi- tional variance in equation (9).
Theorem 1 (Positivity of the conditional variance) The conditional variance h~ in equation (9) can be expressed as a quadratic form and, more precisely, as:
h~ = v'~Avt
where
[1,ut- l ,h t -1 , ,Ut-r*,ht-r* Ut--(r*+l), t-r]
Vt [1, Ut-1, ht-1, , Ut--r, ht-r]'
[1,Ut-l,ht-1, , u t - r . , h t - r . , h t - ( r .+ l ) , ,ht-s]'
and A has the following block diagonal structure:
(.o o) with
(lO)
f o r t > 8
for r = s
for r < s
aoO . . . . . . 0 I R ~ 0 . . . 0
R = "'. "" . . . . 0 S =
\ 0 ::: ..'. O" R~.
s l . . . O . . . 0 ~
0 "'. 0 . . . 0 ~
i . . . . . . . . i \ o . . . . . . . . . s q /
( a~ �89 ) and, for i = 1 , . . . ,q, with q = where, for i = 1 , . . . ,r*, Ri = �89 b~
max{r, s} - r*, si = ar*+~, if r > s, or si -= br*+i, if r < s. Hence for ao > 0 and s~>0 (i = 1 , . . . , q), a sufficient condition for ht2>0 is given by
2 (11) 4aibi > c i ,
for i = 1 , . . . ,r*.
An important point to note is that the constraints in Theorem 1 are only sufficient for the positivity of the conditional variance implying that weaker conditions can be found. Also note that, for the BL-GARCH(1,1) model, the positive definiteness of R1 is a necessary and sufficient condition for the positivity of h 2.
24 G. Storti , C. Vitale
4 Statistical properties of the BL-GARCH model
In this section we analyse the statistical properties of the BL-GARCH model. In particular, general conditions for the second order stationarity of BL-GARCH processes are given.
Furthermore, for the BL-GARCH(1,1) process, expressions for the uncondi- tional fourth moment and kurtosis coefficient of ut are derived. Exploiting a result achieved by He and Ter~isvirta (1999), we also obtain a recursive formula for the calculation of higher order moments of the BL-GARCH(1,1) process. Finally, an analytical expression for the autocorrelation function of u 2 is derived.
The stationarity condition for the general BL-GARCH process is stated in the following theorem which exploits a particular random coefficient representation of the conditional variance model.
Theorem 2 (Stationarity of the BL-GARCH process) The BL-GARCHprocess in equation (9), with r = s, is second order stationary if and only if all the roots Bi of the polynomial
T
qSr(B ) = 1 - E OiB i i = 1
with B being the Backward operator and r = ai § bi, for i = 1, . . . , r, lie outside the unit circle i.e.
~ ( B ) = O f o r l B i [ > l i = l , . . . , r .
Proof The conditional variance equation in (9) can be rewritten as
2 2 bibS_ i E cizt_ih2t_i (12) h 2 = ao + E aiht-~zt-i + + i = 1 i = 1 i = 1
from which the following random coefficient autoregressive representation for ht 2 can be obtained
h 2 = ao + [(aiz2_i + bi + cizt-i)h~_i] = ao + E r (13) i = 1 i = 1
with r = (aiz2t_i + b~ + cizt-i). Then, for the assumptions on {zt}, taking the expectation of (13) gives:
r r
E(h2) E E(• 2 ) Z E( 2 2 2 2 ) = ao+ i,tht_i =ao+ aizt iht_i+biht i+cizt-~ht , = i = 1 i=J_
= ao+ Z aiE(z2t_{)E(h2t_i)+ZbiE(h2 i)+ Z c~E(zt i)E(h'2t_i) i = 1 i = 1 i = 1
BL-GARCH models and asymmetries in volatility 25
which, since h t is a function of past information up to time t - 1, I t - z , simplifies to
r r
E(h2t ) = ao + E a~E(zk i )E(h2t - i ) + E biE(h~-i) = i=1 i : 1
i=1 i ~ l r
= ao + +
i : l
Noting that E(h2t) =- E(E/u2t l I t -1) ) = E(ut2), it follows that:
Vt = a0 + ~ @Vt-~ (14) i : 1
with @ = (hi + b~), for i = 1 , . . . , r , and Vt = E(u2t). Letting t3 be the backward operator such that BkVt = Vt-k, equation (14) can be rewritten as
Y t = a0 (1 - E , \ I �9
The proof is completed by observing that V~ in (15) converges to a finite value if and only if all the roots B~ of the polynomial (1 - Y~.[=I @ B~) are such that IBm{ > 1.
For the sake of simplicity and for notational convenience we have limited our attention to the case in which r = s. This does not imply any loss of generality since the extension of this result to the more general case in which r ~ s is straightforward.
A similar argument can be now used to derive an analytic expression for the unconditional fourth moment of the BL-GARCH(1, l) process and conditions for its existence.
Theorem 3 (Fourth unconditional moment and kurtosis coefficient of the BL-GARCH(I,1) process) Let 7~,i i = E(r and "i = E(}zt]i). The fourth unconditional moment of the BL-GARCH process iz4 = E ( u 4) exists if and only if
'ffr = J~(~)2, t ) = g ( ( a l z2 + 51 q- ClZt ) 2) < 1 . (15)
I f we further assume z t ~ N ( O, 1), considering that
j~( (a lZ2t ,q_bl .+.c l z t )2) 2 4 :2 2 :2 2 3 = E ( a l z ~ + b l + c l z ~ +2alb lz t +2b lc l z t+2a lc l z t ) = 2 4 2 2 2 2 = E ( a l z t +bl+clz t +2alblZt ) =
= 3a 2 +52 + c 2 +26151
condition (15) can be rewritten as (3a~ + b~ + c21 + 2albl) < 1. Furthermore, the fourth unconditional moment of the BL-GARCH(1,1) process ut can he expressed as."
~'4ao2 (1 + "/~,1) (16)
26 G. Storti, C. Vitale
and the kurtosis coefficient of the process can be computed as:
~ 4 (1 - - ")/~,1)
/~4 - - ( /Z2)2 - - /24 (1 - - 7 r (17)
following:
3a~(1 + 7r . 4 = (1 - 7r - - 7r
3(1 7 2 _ _ - - % 1 )
1 - 7 r
Proof Note that:
3a~(1 + a l + bl)
(1 -- a l -- bl)(1 - 3a~ - b~ - c~ - 2a lb l )
311 - (al + 51) 2] ( 1 - 3 a ~ - b ~ - c ~ - 2 a l b J "
tl,4 : E ( u 4) : E ( h 4 ) E ( z 4 ) . (18)
Given the assumption that E(z~) = v4 < oc, p4 exists if and only if E(h4t) exists. The square of equation (13) gives
2 4 2a0~l, th~_ 1 (19) h 4 = a~ + r +
Taking expectation of (19) gives
E(h 4) = a~ + 7r + 2 a~7r (20) 1 - 7 r
which, setting Ht 4 = E(h4t), can be written as
Ht4(1 - % , 2 B ) = a 2 + 2 a27r (21) 1 - - 7 r
and further simplified to give
Ht 4 = a 2 + 2 a~162 1 _ 1--7r "(1-7r
a2 (1 + 7r i (22) = ~ 1 6 2
It follows that, if the process is stationary, #4 exists if and only if [7r < 1. Substituting (22) in (18) gives equation (16). Finally, dividing by (#2) 2 = a~/(1 - 7r 2, equation (17) is obtained.
In order to obtain a recursive expression for the higher order moments of the BL-GARCH(1,1) process and conditions for their existence, it is useful to con- sider the following class of Conditionally Heteroskedastic (CH) processes (He and Terasvirta, 1999)
Ut = h t z t (23)
Under the assumption of normality of &, the above expressions simplify to the
BL-GARCH models and asymmetries in volatility 27
where, for the existence of the nth moment of ut, it is necessary to assume that Elzt }n < ec. Furthermore assume that the following relation holds:
a t k = f f (Zt - -1) + C(Zt_l)hkt_l, ~ ----- 1, 2 (24)
where gt = g(zt) and ct = c(zt) are well-defined functions of zt, and:
P r { h ~ > 0} = ] . (25)
Obviously, constraints on the parameters in 9t and ct can be necessary in order to guarantee that (25) holds. The class of processes defined by equations (23)-(25) includes several widely used models for CH time series and some of these allow to account for leverage effects to be present in the estimated volatility. We recall, among the others, the standard GARCH model, the GQARCH, GJR, NLGARCH, and TGARCH models of order (1,1) (for more details the interested reader may refer to He and Ter~isvirta, 1999). As shown in equation (13), the BL-GARCH(1,1) model also belongs to this class of processes with 9t-1 ~--- a0 , ct- I : ~)l , t =
( a l z ? _ 1 @ bl "F ClZt-1) and k = 2. A key result is given by the following theorem which particularizes a more
general result given by He and Ter~isvirta (1999).
Theorem 4 (Recursive calculation and existence of higher order even moments) For the general GARCH(1,1) process (23)-(25) that started at some finite value in the infinite past, the 2ruth unconditional moment exists i f and only i f
m %,.~ = E(r < 1. (26)
Under this condition the 2mth moment of ut can be expressed recursively as
/~t2m=Elut lem={z%,~/(1-7r aoTr i=l J
Also, the condition ao > 0 needs to be verified in order to satisfy the requirement in (25).
Proof The result follows by the same argument used in Theorem 1 by He and Terasvirta (1999).
Furthermore, the General GARCH representation in Eqs. (23)-(25) allows us to derive an analytical expression for the autocorrelation function of the squared observations u~ exploiting a general result stated by He and Ter~isvirta (1999) (Theorem 2 on page 179). This has been done in the following theorem which shows how, for the BL-GARCH(I,1) model, the autocorrelation function of u~ follows an exponential decay similar to that characterizing autoregressive models. T he rate of decay depends on the persistence of the volatility model measured as (al + bl).
28 G. Storti, C. Vitale
T h e o r e m 5 ( A u t o c o r r e l a t i o n f u n c t i o n o f u~) Let ~r = E(Z~r For the BL- GAR CH(1,l ) process in (9) the autocorrelation function of { u~ } at lag one is given by:
Proof It is sufficient to note that for a BL-GARCH(1,1) process:
~/r = E ( r = al + bl
2 t/4a~ + b 2 cl 2 + 2albl 7r = E ( r = +
and then apply Theorem 2 in He and Ter~isvirta (1999) 1
In the remainder we will say that a particular model parameters combination is included in the admissible region if it simultaneously satisfies the sufficient conditions for the positivity of the conditional variance and the conditions given above for the second order stafionarity and existence of the fourth unconditional moment. As an example, assuming that the value of Cl is fixed, in Fig. 1 we report the admissible regions for three different specifications of the BL-GARCH(I ,1) characterised by different degrees of asymmetry corresponding to three different values of the cl parameter and namely 0.15, 0.25, 0.35.
Finally, it is worth noting that the values of the kurtosis coefficient t% and of the first order autocorrelation of ut 2 both depend on the module of the asymmetry parameter [Cll. In the remainder of this section the nature of this relationship will be further explored. Our analysis is performed for three different sets of parameter values:
( H ) : (ao, al,bl) = (0.01,0.09,0.9)
( M ) : (ao, al,bl) = (0.05,0.05,0.9)
( L ) : (a0, a l , b l ) - - (0 .2 , 0 .05 , 0 .75)
corresponding to three different degrees of persistence of the volatility pattern: High(H), Medium (M) and Low (L). The distribution of the innovations zt has been assumed to be Gaussian and, namely, zt,-~N(O, 1), Vt. Figure 2 shows how, for the BL-GARCH(1,1) model, the pattern of dependence which relates the kurtosis k4 to the value of the cl coefficient varies with the level of persistence in the volatility
1 In principle, using the same argument as for the BL-GARCH(1,1) model, analogous expressions can be derived for other asymmetric GARCH models, such as the GQARCH, NLGARCH, GIR and TGARCH models.
BL-GARCH models and asymmetries in volatility 29
Fig. la--c. Admissible regions for different values of the cl parameter a 0.15 b 0.25 c 0.35
I
7o
I
4~
0 0 ~ ~ 0 ~ ~ ~ 4 1 ~ 01 ~2 0~ 4 6 4 4 ~Z 0 ~ ~4
a b c
Fig. 2a--c. B L-GARCH (1,1) model (from left to fight): response profile of the kurtosis coefficient k4 to the value of ca for the three different sets of parameter values (a) H (b) M (c) L
model. The value of the kurtosis coefficient tends to increase more steadily as the persistence level increases. The range of values of cl varies from one plot to another since, for higher persistence models, it has been necessary to adjust the range in order to ensure the existence of the fourth unconditional moment of the underlying observation process {ut} 2
The same exercise has been repeated for the first order autocorrelation coef- ficient of ut 2. As for the kurtosis coefficient, the BL-GARCH(I,1) model shows a symmetric bell-shaped relationship between the autocorrelation of the squared observations at lag 1 and the value of the asymmetry parameter cl (Fig. 3).
2 Leverage effects are associated to negative values of c l , however in other fields positive values of cl could be meaningful as, for example, in the case of hydrological data such as the hourly rate of variation of fiver discharge.
30 G. Storti, C. Vitale
04, , q ] 2
^
0,15
01
a b c
Fig. 3a--c. BL-GARCH(1,1) model (from left to right): response profile of the autocorrelation coefficient of u~ at lag 1 (Pl) to the value of cl for the three different sets of parameter values a H b M c L
5 M a x i m u m l i k e l i h o o d e s t i m a t i o n
Under the assumption of normality of the standardized innovations zt, the condi- tional log-likelihood of the general BL-GARCH model can be written as:
T 1 T 1 ~ - ~ U 2 s 0) = - 7 log 27r - ~ Z log ht 2 - 2 ~ h-~ (27)
t = l t = l
where 0 is a vector containing all the unknown parameters in equation (9) i.e. {ai, bj, ck}, for i = 0, ..., r, j = 1, ..., s and k = 1, ..., r*. In order to prevent the estimated conditional variance from becoming negative, the maximization of (27) using standard numerical methods would require to impose appropriate constraints on the admissible parameter space. In order to overcome this problem, Storti and Vitale (2003) propose an indirect estimation strategy which exploits the analogies in the conditional distributions of the observation process u t (e.g. stock returns) implied by BL-GARCH models and Constrained Changing Parameters Volatility (CPV-C) models, first proposed by Storti (1999) and further discussed by Amendola and Storti (2002). A CPV-C(r, s) model is a special kind of random coefficient autoregressive model including past conditional standard deviations as endogenous explanatory variables:
Ut = a i , t u t - i + b j , t h t - j + et i=1 j = l
= C t a t (28)
where a i , t , b j , t and e t a r e sequences of i i d random variables. Also, given at = [et, a l , t , . . . , a t , t , b l , t , . . . , b s , t ] ' letting E ( a t l I t - l ) = 0 and V a r ( a t ] I t a) = Q ,
we have:
= h f =
BL-GARCH models and asymmetries in volatility 31
If Q : A, where A is the block-diagonal matrix in (10), then there is a BL-GARCH model of order (r,s) parameterised by A which is observationally equivalent to the resulting CPV-C model (28) in the sense that the two models are characterised by the equality of the first two conditional moments. Furthermore, under the assumption of conditional normality of ut, i.e. (ut 1 i t - ~) ~N(0, ht 2), the two models have the same conditional distribution and, hence, the same conditional log-likelihood function given by (27). Hence an estimate of the unknown parameters of a given BL-GARCH model can be obtained by estimating the unknown parameters of an observationally equivalent CPV-C model of the same order.
It can be easily shown that the CPV-C model in (28) can be represented in state space form with a time varying observation matrix Ct dependent on past observations and volatilities, state vector given by at and a transition matrix with all the elements identically equal to zero. Hence we can use the EM algorithm proposed by Shumway and Stoffer (1982) and further refined by Wu et al. (1996) to perform the maximum likelihood estimation of CPV-C models. The unknown parameters are given, as for the observationally equivalent BL-GARCH model, by the elements of A different from zero. A detailed description of the actual implementation of the EM algorithm for this particular estimation problem can be found in Storti and Vitale (2003).
One of the main advantages deriving from the application of the EM algorithm in this situation is that it allows to obtain estimates which naturally, i.e. by construction, satisfy the sufficient conditions for the positivity of ht 2. Furthermore, a measure of the asymptotic standard errors associated to the estimated parameters can obtained by evaluating the Observed Information Matrix at the maximum likelihood estimate. An analytic expression for this matrix and for the score vector is given in Storti and Vitale (2003) who also provide a recursive algorithm for their evaluation.
6 Model identification
In this section we illustrate how the LM test statistic proposed and discussed in Storti (2002) can be used to detect, in the conditional variance of an observed time series, the presence of asymmetric effects which can be satisfactorily explained by means of a BL-GARCH model of a predefined order. Under the null, ut follows a GARCH (r,s) process while, under the alternative hypothesis, the relevant data generating process is a BL-GARCH(r,s). The LM test statistic has the classical form:
!
where Oo is the maximum likelihood estimate of the vector of unknown model parameters 0 under the null; s(O) is the score function given by:
both to be evaluated at the restricted estimate i.e. for 0 = Oo. Under the null, LM --+ X~. for T - - ~ with r* corresponding to the number of restrictions imposed.
d Alternatively, an asymptotically equivalent test statistic is given by LM• = T • R~ where, letting y = (Yl,-- �9 , YT)' and X = (x l , . �9 �9 , XT)',
n~ = Y ' X ( X t X ) - I X ' Y y~y
is the uncentered squared multiple correlation coefficient obtained from the regres- sion of:
Yt d U ~ - l ) o n 1
= [ o o ' ] ,to (t = 1, . . . , T)
[oh l where by h~t and L~-Or ] Ho we indicate the conditional variance o fu t and the vector
of first derivatives of ht 2 with respect to the unknown elements of the parameter vector 0 evaluated under the null hypothesis. The finite sample properties of the test under the assumption of conditional normality have been investigated by means of a simulation study by Storti (2002). Here, we propose a modified version of the test, which is robust to violations of the conditional normality assumption. This robustified test statistic can be obtained following the procedure described in Woolridge (1991). The first step is to estimate a GARCH(r,s) model and then regre s s:
h02 [ 0c' ] -o o n 1 [Oht 2 ]
where 0o denotes the vector of model parameters under the null and c = (Cl , . . . , e~.)' is the vector of interaction coefficients with 0 = (0o, e). Under the null, c = 0. Letting ~/t = Qlt,1,- �9 - ,zlt,~*), for t = 1, . . . ,T, be the residuals from this regression (of dimension 1 • r*), we then further regress a variable defined to be equal to 1 Vt on the following set of regressors:
G = (ytVt,1,... ,ytW,~*)
for t = 1 , . . . , T. The test statistic is calculated as LM~ = T • Rro b2 where Rro b2 is" the uncentered squared multiple correlation coefficient from the above regression. As shown in Woolridge (1991), this test statistic asympotically converges to a X~. random variable.
BL-GARCH models and asymmetries in volatility 33
Table to Results of the ARCH-LM test
F-statistic 24.561 p-value 0.0000 T • R 2 44.807 p-value 0.0000
In order to show the effectiveness of the BL-GARCH model in modelling leverage effects in financial time series, in this section we report the results of an application of the proposed approach to the time series of daily returns relative to the NASDAQ stock market index. The returns, observed from January 5, 1999 to February 9, 2001, for a total of 531 observations, are calculated as the logarithmic first difference of the price series:
ut = v l o g ( P t ) = (1 - B)Zog(Pt)
where Pt indicates the price at time t. Only the first 481 observations have been used for model identification and estimation while the last 50 records have been left for out-of-sample forecast evaluation. The estimated autocorrelation functions of the returns do not show any significant serial correlation structure in the data as confirmed by a Ljung-Box test performed for lags from 1 to l0 3. The presence of ARCH effects in the return series is investigated by means of the usual ARCH-LM test (Engle, 1982) performed up to lag 2.
The results of the test (Table 1) provide evidence in favour of the alternative hypothesis of ARCH at any reasonable significance level. The following step is to verify the hypothesis that negative and positive returns affect the volatility pattern in a symmetric fashion against the alternative that asymmetric effects are present. More precisely, the LM test statistics illustrated in Sect. 6 are used to test the null hypothesis of a symmetric GARCH(1,1) model against the alternative of an underlying data generating process given by a BL-GARCH model of the same order. The values of the usual LM test statistics (LM, LMR) and of the robust test statistic described in the previous section (LMRR), reported in Table 2, provide substantial evidence in favour of the presence of asymmetric effects of BL-GARCH type rejecting the null hypothesis that the conditional variance of ut was generated by a symmetric GARCH model.
Hence, in order to account for asymmetry, we estimate a BL-GARCH(1,1) model. The maximum likelihood estimates, standard errors and z-statistics for this model have been reported in Table 3.
The negative sign of the cl parameter's estimate confirms the test findings implying that the impact of negative shocks on the conditional variance is higher
3 Results are available from the authors upon request
than that of posit ive ones, as it can be easily observed looking at the estimated news
impac t curves (Engle and Ng, 1993) plotted in Fig. 4a for different percentiles of the empirical distr ibution of the est imated condi t ional variance (5, 50, 95 and 99th). Differently, Fig. 4b illustrates the relationship be tween the magni tude of the leverage effect and absolute returns at t ime (t - 1). The measure of the leverage
effect in the plot is obtained from the est imated news impac t curve as h 2 ( - ] r t - i 1) - h2 ([rt-- 1 [), for 0 < [ r t - 1 [ _< 0.10, where h 2 (c~) is the value of the news impact curve impl ied by a return at t ime (t - 1) equal to r t - 1 = ol. So, given r t - 1 , the leverage effect is a l inear funct ion of the volatili ty at (t - 1) while, if we assume
that the value of ht2_l is fixed, the leverage is a l inear funct ion of Irt-al- The maximized l ikelihood value (MLL) is 1140.20, against the value of 1129.45
obtained for the nul l model. The Akaike (AIC) and the Schwarz (SC) informat ion criteria are equal to - 4 . 2 7 9 5 and - 4 . 2 4 7 3 , respectively, while the corresponding
values for the est imated G A R C H model are - 4 . 2 4 2 7 , for the AIC, and - 4 . 2 1 8 6 , for the SC (Table 4). The performance of the est imated BL-GARCH(1 ,1 ) in mod- el l ing the condi t ional variance of the t ime series of NASDAQ returns has been then compared with the results obtained by means of some alternative asymmetr ic
G A R C H models of order (1,1). More precisely, the models which have been con-
Fig. 4a,b. NASDAQ Composite stock market index: a news impact curve associated to the estimated BL-GARCH (1,1) model computed for different percentiles of the empirical distribution of the estimated conditional variance series (5,50,95 and 99th) b leverage effect vs. absolute returns
Table 4. Values of Maximized Log-Likelihood (MLL), Akaike Information Criterion (AIC), Schwarz Criterion (SC) for the five models considered
sidered for this comparison are the GQARCH, GJR and NLGARCH models. For each of these, the estimated parameters and standard errors have been reported in Table 3. In all the cases, except for the NLGARCH model, the estimates provide evidence in favour of the presence of a statistically significant leverage effect.
As a first step, in order to evaluate the performance of the models, we look at the values of the maximized log-likelihood and of the Akaike and Schwarz Information Criteria (Table 4) and note that, on the basis of these criteria, the BL-GARCH
slightly outperforms all the other asymmetric models.
Then, the out-of-sample predictive ability of each of the models in forecasting the volatility of the observed returns over the validation period (from November 28, 2000 to February 9, 2001 to) is assessed by means of some widely used loss
36 G. Storti, C. Vitale
functions (for a discussion see Lopez, 1999). Namely, the loss functions which have been here considered are:
k 1 2
M S E = -
j = l
k 1
M A E = ~ E l U 2 . + j -- h2.+j] j = l
k 1
L S E = -~ E [ l o g ( u 2 . + j ) - lo9(h2.+j)] 2 j = l
k 1 ^
L A E = ~ E [log(u~.+j) - log(h~.+j)[ j = l
k 2 1 V , [u~ .+ j 1] 2 H M S E
~ h 2 j = l T*+j
where T* is the last data point used for model identification and estimation, h~. +j is the (one-step-ahead) predicted volatility at time T* + j and k is the length of the validation period. The first two criteria (MSE and MAE) penalize errors of opposite sign and of the same magnitude in a symmetric fashion. Differently the remaining three loss functions (LSE, LAE and HMSE) penalize errors in forecasting volatility asymmetrically with the first two penalizing inaccurate forecasts more heavily when u~. +j is low. Also, LSE and LAE penalize volatility underprediction more heavily while, more weight is given to volatility overprediction when using the HMSE criterion. The results (Table 5) show that the BL-GARCH model is superior to all the other models on the basis of MAE, LSE and LAE and gives a MSE slightly higher than the value registered for GJR and NLGARCH. Finally, GJR and NLGARCH outperform all the other models on the basis of HMSE.
To sharpen our analysis, we have then evaluated the predictive performance of the different models considering the cases in which uT* + j - 1 is positive separately from those in which it assumes a negative value. Namely, throughout the validation period we have chosen, over a total of 50 observations, we observe u y . +j_ 1 < 0 for 29 times and UT*+j-1 > 0 for the remaining 21 observations. In the first case (Table 6) we obtain results quite close to those obtained for the whole validation period while (Table 7) we come to a different conclusion if we focus on the subset of observations for which UT*§ > 0. In this case, the performance of the BL- GARCH model is superior to that of all the other models in terms of each of the criteria we have considered.
In conclusion, the overall predictive performance of the BL-GARCH model can be considered satisfactory since it outperforms its competitors for three out of the five forecast evaluation criteria which have been taken into account. Also, with respect to the other models, the BL-GARCH model generates better predictions of market volatility when a positive return has been observed on the day before, i.e. when investors tend to be more optimistic.
BL-GARCH models and asymmetries in volatility 37
Table 6. Forecast evaluation of different GARCH models over the chosen validation period for u~_ i < 0 (29 daily records)
The above results show that the performance gap when comparing different asymmetric GARCH models in some cases can be quite substantial. Such large differences are mainly due to the way in which asymmetry is introduced into the conditional variance equation within different model structures. This can be graph- ically illustrated by comparing the news impact curves associated to the models considered for the present application as it has been done in Fig. 5. The first plot (Fig. 5a) shows the behaviour of the news impact curves of the different models for daily returns included in the range [ -0 .10 , 0.10] in correspondence of the 50th percentile of the empirical volatility distribution. The news impact curves relative to the N L G A R C H and GJR models are characterised by a higher degree of skewness than the other models implying higher volatilities for negative returns and lower volatilities for positive ones. Since, differently from all the other models here con- sidered, for the BL-GARCH the impact of asymmetry depends on the interaction between past returns and volatilities, for a given value of Irt[, the magnitude of the leverage effect increases as the volatility increases. Also, for all the other models, the relative weight of the asymmetric component on the overall estimated volatily decreases as h~_ 1 increases. Hence, it is also interesting to look at the shape of the news impact curve implied by the 99th percentile of the empirical volatility distribution (fig. 5b). We can note that the news impact curve associated to the GJR model now dominates all the others but with the BL-GARCH model giving more weight to negative returns than NLGARCH and GQARCH.
8 Concluding remarks
This paper discusses the statistical properties of a new class of CH time series models which allows to capture leverage effects in the conditional variance of financial time series by means of interactions between past shocks and volatilities. This way of
38 G. Storti, C. Vitale
,~o' (a)
~ 1 -o,o8 -0o6 -0.04 4.o2 o 002
returns
a
x l o ~ (b)
-~- GQARCH - ~ R
5
i L i i i i i i i
-01 008 006 -004 002 0 0.02 004 0o6 0.08 01
returns
b
Fig. 5a,b. NASDAQ Composite stock market index; news impact curves associated to the estimated asymmetric GARCH (l, 1) models computed for two different percentiles of the empirical distribution of the estimated conditional variance series and for daily returns in the range [-0.10 + 0.10]: a 50th percentile b 99th percentile
introducing asymmetr ic effects into the mode l is appeal ing for two reasons. First,
the leverage effect is ent irely expla ined by an addit ive te rm which can be easily
isolated and evaluated. Second, the impact o f asymmetry is state dependent in the
sense that its magni tude depends on the market ' s uncertainty condi t ions measured
in terms of past volatili t ies.
The results o f the applicat ion which has been here presented al low us to consider
the B L - G A R C H mode l as an useful tool for model l ing t ime varying condi t ional
var iances and leverage effects in financial t ime series.
Acknowledgements This paper was supported by the MURST COFIN 2000 project: "Modelli Stocas- tici e Metodi di Simulazione per l'Analisi di Dati Dipendenti".
References
Amendola A, Storti G (2002) A non-linear time series approach to modelling asymmetry in stock market indexes. Statistical Methods & Applications 11(2), 201-216
Black F (1976) Studies in stock price volatility changes. Proceedings of the American Statistical Asso- ciation, Business and Economics Statistics Section, American Statistical Association, 177-181
Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31,307-327
Ding Z, Granger CWJ, Engle RF (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83-106
Engle RF (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of the U.K. inflation. Econometrica 50, 987-1008
Engle RF (1990) Discussion: Stock market volatility and the crash of 87. Review of Financial Studies 3, 103-106
Engle RF, Ng KV (1993) Measuring and testing the impact of news on volatility. The Journal of Finance 48, 1749-1778
Glosten LR, Jagannathan R, Runkle DE (1993) On the relation between the expected value and the volatility of the nominal excess returns on stocks. The Journal of Finance 48, 1779-1801
Hagerud GE (1997) Specification tests for asymmetric GARCH. Working Paper Series in Economics and Finance, Stockholm School of Economics, No. 163
He C, Ter~isvirta T (1997) Statistical properties of the Asymmetric Power ARCH process. Working Paper Series in Economics and Finance, Stockholm School of Economics, No. t99
He C, Tedisvirta T (1999) Properties of moments of a family of GARCH processes. Journal of Econo- metrics 92, 173-t92
Lopez JA (1999) Evaluating the predictive accuracy of volatility models. Working paper, FederalReserve Bank of San Francisco
Nelson DB (1991) Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347-370
Rabemananjara R, Zakoian JM (1993) Threshold ARCH models and asymmetries in volatility~ Journal of Applied Econometrics 8, 31--49
Sentana E (1995) Quadratic ARCH models. Review of Economic Studies 62, 639-661 Shumway RH, Stoffer DS (1982) An approach to time series smoothing and forecasting using the EM
algorithm. Journal of Time Series Analysis 3(4), 253-264 Storti G (1999) A state space framework for forecasting non-stationary economic time series. Quaderni
di Statistica 1, 121-142 Storti G (2002) A LM specification test for BL-GARCH asymmetry. Proceedings of the XLI General
Meeting of the ltalian Statistical Society, 649-652 Storti G, Vitale C (2003) Likelihood inference in BL-GARCH models. Computational Statistics 18:
387-400 Taylor S (1986) Modelling financial time series. Wiley, New York Woolridge JM (1991) On the application of robust, regression based diagnostics to models of conditional
means and conditional variances. Journal of Econometrics 47 5-46 Wu LS, Pal JS, Hosking JRM (1996) An algorithm for estimating parameters of state space models.
