Mobile Edition
Print
Get the Mag
Weekly UpdatesAbstract The objective of this paper is to put forward a new autoregressive asymmetric stochastic volatility model for modeling volatility and to compare results obtained for this model with an autoregressive stochastic model and another asymmetric volatility model, such as, asymmetric generalized autoregressive conditional heteroskedasticity model. The results obtained from the estimation by maximum likelihood have shown the volatility behavior is asymmetric in the majority of cases. This fact is better shown by the ARSVA model, than the rest of alternative models. Moreover, the ARSVA model is able to reproduce other stylized facts of such series, such as high kurtosis, no autocorrelation of returns, slow decreasing of the autocorrelation function of the squared returns and high persistence.
Keywords Leverage effect * Stochastic volatility * Stock returns
JEL C22 * C51
Introduction
In the past decades, there has been a growing interest in the correct modelization of high frequency time series. The greater part of these series, such as the stock index returns, have some characteristics which cannot be explained with ARIMA models because many of these time series do not show time changes in their mean, and they fluctuate around a constant zero mean level, but they do have a changing conditional variance in time.
The existence of the changing conditional variance is appreciated in the different variations of returns with respect to their mean. There are some periods in which exists successive values with high volatility (which coincide with periods in which the variation of index stock returns are bigger) to mix with other periods where the volatility is lower (which coincide with periods in which the index stock returns have not big variation), i.e., there are volatility clusters. This fact is observed by the Fig. la, where the daily returns of Nasdaq100 index from 1/07/1987 to 30/07/2004 are depicted. These returns are given a value through the variable [y.sub.t] = 100(log [X.sub.t]-log [X.sub.t-1]), where [X.sub.t] is the index value at day t.
The volatility clusters imply, in statistical terms, that there is a positive correlation in the square returns and in their absolute values. In this way, if the volatility is high in a period, it tends to be high in the next one, and conversely, if the volatility is low in a period, it tends to be low in the next one. The autocorrelation function of square returns, Fig. 1d, due to the existence of volatility clusters, shows a strong dependence structure which is shown by significant positive correlations. In the majority of time series, these positive correlations decrease slowly to zero, which is known as volatility persistence. Therefore, the returns frequently are uncorrelated, but are not independent because non-linear transformations of them are positively correlated.
[FIGURE 1 OMITTED]
The autocorrelation function in this case, Fig. 1b, shows there is no time dependence in mean, because its coefficients are not significant. However, there are some financial time series in which exists a small correlation in mean; this dependence can be modeled with a low order autoregressive or moving average model with a low coefficient.
The kurtosis excess and the heavy tails show that these financial series do not have a normal distribution, Fig. 1c, (Mandelbrot 1963).
These usual characteristics of financial time series are known in econometric literature as stylized facts (Bollerslev et al. 1994; Ghysels et al. 1996).
To model the preceding characteristics of the financial returns, there are basically two model types according to the dependent structure for the conditional variance. A type of model assumes that conditional variance is a function of previous values. These models are on the one hand, the AutoRegresive Conditional Heteroskecasticity model, ARCH model, (Engle 1982), and on the other hand, the Generalized AutoRegresive Conditional Heteroskecasticity model, GARCH model, (Bollerslev 1986) and all its variants. In these models, the volatility is determined by linear deterministic function of past observed values of square returns and the lagged conditional variance. In this paper, the asymmetric GARCH model is used to explain the asymmetric volatility observed in financial return series (Engle and Ng 1990).
Another type of model is the stochastic volatility model, SV model. In the SV model, the conditional variance is modeled by a non-observed component which follows a latent stochastic process.
Model of Asymmetric Stochastic Volatility
The expression of general equation to model the stylized facts of financial returns series ([y.sub.t]), (when the mean is zero) is:
[y.sub.t] = [[sigma].sub.t][[epsilon].sub.t] [[epsilon].sub.t] iid(0,1), t = 1,...,T. (1)
Where [[sigma].sub.t] represents the volatility and it is expressed in different forms depending on the model used. It is assumed that [[sigma].sub.t] is generated by a stationary process but, its value at t period, depends on information set until t-1 period ([[OMEGA].sub.t-1]), which is, [[sigma].sub.t], has a dynamic structure. Moreover [[epsilon].sub.t], is a random disturbance (white noise), which is distributed with zero mean, unit variance and finite fourth order moments. Also,[[sigma].sub.t] and [[epsilon].sub.t], are two independent processes.
The proposed models to collect the changing conditional variance over time are: the AGARCH model, the ARSV model and a new variant of the last one, the asymmetric autoregressive volatility model (ARSVA model).
The AGARCH Model
The AGARCH model (Engle and Ng 1990) was proposed to capture the asymmetric answer of the volatility by the different sign of the stock market shock. In an AGARCH(p,q) model, [[sigma].sub.t.sup.2] in Eq. 1 is:
[[sigma].sub.t.sup.2] = [[alpha].sub.0] + [q.summation over (i = 1)][[alpha].sub.i][([y.sub.t-i] - [delta]).sup.2] + [p.summation over (j = 1)][[beta].sub.j][[sigma]sub.t-j.sup.2] (2)
where the asymmetric effects are captured by the [delta] parameter; q is the number of lags of [y.sub.t]; p the number of lags of [[sigma].sub.t.sup.2]. The [[alpha].sub.i] and [[beta].sub.j] parameters are bigger or equalthan zero. The condition that [[q.summation over (i = 1)] [[alpha].sub.i] + [p.summation over (j = 1)] [[beta].sub.j] < 1 is needed to ensure a positive variance and the covariance stationary of the process [y.sub.t].
The ARSV model
The ARSV model is an alternative to the GARCH model but it is not able to describe an asymmetric effect. There are a lot of studies to compare the GARCH and SV models, (Malmsten and Terasvirta 2004) but it is not the subject of this paper.
The Eq. 1 is the mean equation for a SV model (the same as in an AGARCH model.) The volatility equation in an ARSV model is different to the AGARCH model, being its expression as follows:
[[sigma].sub.t.sup.2] = [[sigma]*.sup.2]exp([h.sub.t]) (3)
The volatility is modeled as an exponential to guarantee its positivity and [[sigma].sub.t.sup.2] in an AGARCH model is deterministic but in an ARSVA model, it is stochastic. In fact, [h.sub.t] is the volatility logarithm and the log-normality hypothesis, [h.sub.t] = log [[sigma].sub.t.sup.2] is appropriate to represent the behavior of the volatility of daily financial time returns (Andersen et al. 2003).
It is assumed that the stochastic process for [h.sub.t] is modeled by a pth-order autoregressive 4, and the disturbance of mean and variance equations have normal distributions which are uncorrelated for all the lags:
[h.sub.t] = [p.summation over (i = 1)][[phi].sub.i][h.sub.t-i] + [[eta].sub.t] [[eta].sub.t] iid N (0, [[sigma].sub.[eta].sup.2]) (4)
Thus, the equations of ARSV (p) model are:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where the parameter of the model verifies the stationarity condition, that is, all the polynomial roots 1 - [p.summation over (i)] [[phi].sub.i][L.sup.i] which are out of unit circle.
The most simple case is the ARSV(l) model, (Taylor 1994) where the volatility logarithm is generated by a first order stochastic process:
[h.sub.t] = [phi][h.sub.t-1] + [[eta].sub.t] [[eta].sub.t] iid N N(0,[[sigma].sub.[eta].sup.2]) (6)
The dynamic structure of the volatility in the ARSV(l) model is defined using logarithms to permit the variance to be positive. This model is simple but it collects the main characteristics of financial returns. The equations of ARSV(l) model can be written as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
where [sigma]* is a positive scale factor in the mean equation to avoid including a constant in the Eq. 6. If [absolute value of ([phi])] is less than one, the stationarity of the [h.sub.t] process is ensured; [[sigma].sub.[eta]] is a parameter which measures the variation in the volatility process; [[eta].sup.t] is a white noise process with zero mean and [[sigma].sub.[eta].sup.2] variance. The distribution of [[epsilon].sub.t] and [[eta].sub.t] are independent [for all]t, s.
The ARSV(1) model expressed in the Eq. 7 is a non linear model, because the volatility is defined as an exponential function. But the model can be expressed in a linear model taking logarithms (Sandmann and Koopman 1998). Squaring the mean equation and taking logarithms, the expression 7 can be written as:
{log([y.sub.t.sup.2]) = log[[sigma]*.sup.2] + [h.sub.t] + [[xi].sub.t] (8a)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8b)
Where, the log ([y.sub.t.sup.2]) is the sum of the volatility, which is a stationary linear process because [absolute value of ([phi])] < 1, and [[xi].sub.t], are a non gaussian white noise. Both processes are independent; [[xi].sub.t] = log([[epsilon].sub.t.sup.2]) is a non gaussian white noise, whose properties depend on [[epsilon].sub.t]. For example, if [[epsilon].sub.t] has a normal distribution, N(0,1), then the [[epsilon].sub.t.sup.2] has a ChiSquare distribution, [[chi].sub.1sup.2], with a mean equal to (-1.27) and a variance equal to ([[pi].sup.2]/2) (Abramowitz and Stegun 1970, p.943).
FEED