6.1 Introduction

The linear models presented so far in cross-section data and in time series assume a constant variance and covariance function, that is to say, the homoscedasticity is assumed.

The possibility of having dependence between higher conditional moments, most notably variances, implies to examine non-linear stochastic processes from a more realistic perspective in time series data. Moreover, as we have seen in previous chapter, some financial time series are approximated by random walk models, but the usual random walk assumptions are too restrictive because the asset price volatility changes over time and consequently, violating the conditional stationarity of the process.

From an empirical point of view, financial time series present various forms of non linear dynamics, the crucial one being the strong dependence of the variability of the series on its own past and furthermore, with the fitting of the standard linear models being poor in these series. Some non-linearities of these series are a non-constant conditional variance and, generally, they are characterised by the clustering of large shocks to the dependent variable.

That is to say, variance changes over time, and large (small) changes tend to be followed by large (small) changes of either sign and, furthermore, unconditional distributions have tails heavier than the normal distribution. The volume of data and the high-frequency sampling of data in financial and currency markets are increasing day to day, in such a way that we almost have a time-continuous information process and consequently, we need to adopt ad-hoc methods to allow shifts in the variance or may be a better alternative using a new class of models with non-constant variance conditional on the past. An additional advantage of these models is to take into account the conditional variance in the stochastic error to improve the estimation of forecasting intervals. A general study of the problem of heteroscedasticity and their consequences in regression and autocorrelation models can be seen in Mills (1993) and Johnston and DiNardo (1997).

Models which present some non-linearities can be modelled by conditional specifications, in both conditional mean and variance. The standard approach to heteroscedasticity is to introduce an exogenous variable $ x_t$ which takes into account the incertitude of the model. In figure 6.1 we can see a simulated time series $ \{y_t\} $ and moreover, we can observe the existence of a non-constant variance in the right scatter plot of $ \{y_t\} $ over the exogenous variable $ x_t$. This behaviour is very often in time series data of high-frequency.

Figure 6.1: Time plot and scatter plot with the exogenous variable causing heteroscedasticity.
\includegraphics[width=0.75\defpicwidth]{heteros.ps}

One simple model which captures these features, might be

$\displaystyle y_t=\varepsilon_t x_{t-1}
$

where, as usual $ \textrm{E}(\varepsilon)=0$ and $ V(\varepsilon) =
\sigma^2$. The conditional variance of $ y_t $ is $ \sigma^2x^2_{t-1}$ and, therefore, the prediction interval depends on an exogenous variable changing over time.

However, the theory of finance does not usually provide adequate exogenous variables to explain changes in expected mean and variance of the rates of returns in asset prices. As consequence, a preferable model is

$\displaystyle y_t=\varepsilon_t y_{t-1},
$

now the conditional variance is $ \sigma^2y^2_{t-1}$ but, unfortunately the unconditional variance is zero or infinity.

The basic models presented in this chapter are able to capture all observed phenomena and to give a satisfactory solution to most statistical requirements, such as asymptotic moments or stationarity, etc. The clustering volatility of these markets over time is a function of the most recent news, and in the next sections, we will describe the statistical properties of a basic autoregressive conditional heteroscedastic model and its generalisations.

In order to illustrate these ideas, we offer several examples. These are, respectively time series from the Spanish Stock Market, daily Spanish Peseta/US dollar exchange rate and several data sets simulated.


6.1.0.0.1 Ibex35 index

Ibex35 index is a weighted mean of the 35 firms with the largest trading volumes. The series analysed corresponds to daily returns of the Ibex35 index, which is constructed as the logarithmic difference of the closing price, that is to say,

$\displaystyle y_t = 100*\log\left(\frac{S_t}{S_{t-1}}\right)
$

where $ S_t$ is the above mentioned closing price at time $ t$, thus excluding dividend yields. We have a total of 499 data available to us. The time period analysed runs from 1st January 1999 to 29th December 2000. The return time series is plotted in 6.2

Figure 6.2: Daily returns of the Ibex35 Stock Market Index.
\includegraphics[width=0.59\defpicwidth]{ibex35.ps}

25646 XEGarch02.xpl


6.1.0.0.2 Spanish peseta/US dollar data

In this example, the data series is the daily spot exchange rate from January 1990 to December 1991. There are 500 observations. Let $ S_t$ denote the spot price of the one US dollar to Spanish pesetas. We then analyse the continuously compounded percentage rate of return,

$\displaystyle y_t = 100*\log \left(\frac{S_t}{S_{t-1}}\right).
$

These are the data plotted in figure 6.3

Figure 6.3: Daily rates of returns on Spanish peseta/US dollar exchange rate.
\includegraphics[width=0.59\defpicwidth]{tspesdol.ps}

6.1.0.0.3 Simulated Data

The third example is a data set simulated in such a way that the mean is zero and the unconditional variance of the model is 1. The corresponding time plot is seen in figure 6.4,

Figure 6.4: simulated ARCH(1) data.
\includegraphics[width=0.59\defpicwidth]{archsimu.ps}

The first aspect to note is that for all series the means appear to be constant, while the variances change over time.

Many researchers have introduced informal and ad-hoc procedures to take account of the changes in the variance. One of the first authors to incorporate variance changing over time was Mandelbrot (1963), who used recursive estimates of the variance for modelling volatility. More recently, Klein (1977) used rolling estimates of quadratic residuals.
Engle's ARCH model (see Engle; 1982) was the first formal model which seemed to capture the stylised facts mentioned above. The ARCH model (autoregressive conditional heteroscedastic models) has become one of the most important models in financial applications. These models are non-constant variances conditioned on the past, which are a linear function on recent past perturbations. This means that the more recent news will be the fundamental information that is relevant for modelling the present volatility.

Moreover, the accuracy of the forecast over time improves when some additional information from the past is considered. Specifically, the conditional variance of the innovations will be used to calculate the percentile of the forecasting intervals, instead of the homoscedastic formula used in standard time series models.

The simplest version of the ARCH disturbance model is the first-order, which can be generalised to a $ p$-order model. The financial models do not use exogenous variables, but in another economic context a regression model with ARCH perturbations could be considered as in section 1.4. In what follows, we will describe the basic statistical properties of these proposed models, as well as the most appropriate estimation methods. Finally, we will present the usual hypothesis tests for detecting the structure and the order of the model. In addition, we present the extension of the ARCH model to a more general model in which the lagged conditional variance is included in the present conditional variance.