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Article Excerpt
Abstract 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;...
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