Univariate Time Series: Volatility Models
In Chapter 3 we have considered approaches to modelling conditional mean of a univariate time series. However, many areas of financial theory are concerned with the second moment of time series - conditional volatility as a proxy for risk.
In this chapter we introduce time series models that represent the dynamics of conditional variances. In particular we consider ARCH, GARCH model as well as their extensions.
The reader is also referred to Engle (1982), Bollerslev (1986), Nelson (1991), Hamilton (1994), Enders (2004), Zivot and Wang (2006).
The ARCH Model
Besides a time varying conditional mean of financial time series, most of them also exhibit changes in volatility regimes. This is especially applicable to many high frequency macroeconomic and financial time series.
While modelling such time series, we cannot use homoscedastic models. The simplest way to allow volatility to vary is to model conditional variance using a simple autoregressive (AR) process.
Let Yt denote a stationary time series, then Yt can be expressed as its mean plus a white noise:
where c is the mean of Yt, and ut is i.i.d. with mean zero. To allow for conditional
heteroscedasticity, assume that
Here vart- denotes the variance conditional on information at time t — 1, and is modelled in the following way:
In order to show that this specification is equivalent to AR representation of squared residuals, note that vart-i[ut] = E [u2^] = a2 since E[ut] = 0. Thus, equation (5.2.2) can be rewritten as:
where et = u2 — Et-i [u2] is a zero mean white noise process. The model in (5.2.1) and (5.2.3) is known as the autoregressive conditional heteroscedasticity (ARCH) model of Engle (1982), which is usually referred to as the ARCH(p) model. More generally, ARCH model can be rewritten as
where rjt is an iid normal random variable.
Example: Simulating an ARCH(p) model in EViews
It is relatively easy to simulate ARCH process in EViews. Let us consider as example the following ARCH(2) model
with rjt being independent random variables following N(0 1)distribution. Similarly to ARMA process we need to generate error term process jt and first two initial values of Yt after which the whole process can be simulated. Create a new workfile and in the command line enter
The last statement is included to ensure that we come back the whole data range. The plot of the simulated series is given in the following figure.
Figure 5.1: Plot of simulated ARCH process
Visually, the process looks stationary, mean reverting and with zero mean as expected from the equation (5.2.4).
Testing for ARCH Effects In order to test for the presence of ARCH effects in the residuals, we can use AR representation of squared residuals in the following way. Based on equation (5.2.2), construct an auxiliary regression
. The significance of parameters a would indicate the presence of conditional volatility. Under the null hypothesis that there are no ARCH effects:
the test statistic LM = TR2 ~ xp where T is the sample size and R2 is computed from the regression (5.2.5).
The GARCH Model
More general form of conditional volatility is based on ARMA specification as an extension of AR process of squared residuals. Bollerslev (1986) introduces GARCH model (which stands for generalized ARCH) where he replaces the AR model in
where the coefficients a and [3j are positive to ensure that the conditional variance a"2 is always positive. In order to emphasize the number of lags used in (5.3.1) we denote the model by GARCH(p q).
When q = 0, the GARCH model reduces to the ARCH model. Under the GARCH(p q) model, the conditional variance of ut, a"2, depends on the squared residuals in the previous p periods, and the conditional variance in the previous q periods. The most commonly used model is a GARCH(1 1) model with only three parameters in the conditional variance equation.
A GARCH model can be expressed as an ARMA model of squared residuals. For example, for a GARCH(1 1) model:
Since Et_i [u2] = the above equation can be rewritten as:
which is an ARMA(1 1) model. Here et = u2 — Et-1 [u2] is the white noise error term.
Given the ARMA representation of the GARCH model, we conclude that stationarity of the GARCH(1 1) model requires a1 + f31 < 1. The unconditional variance of ut is given by
Indeed, from (5.3.2)
and thus E [u2] = a0 + (a1 + [31)E[u^] since u2 is stationary. For the general GARCH(p q) model (5.3.2), the squared residuals u2 behave like an ARMA(max(p q) q) process.
One can identify the orders of the GARCH model using the correlogram of the squared residuals. They will coincide with ARMA orders of the squared residuals of the time series.
GARCH Model and Stylized Facts In practice, researchers have uncovered many so-called stylized facts about the volatility of financial time series; Bollerslev, Engle and Nelson (1994) give a complete account of these facts. Using the ARMA representation of GARCH models shows that the GARCH model is capable of explaining many of those stylized facts. This section will focus on three important ones: volatility clustering, fat tails, and volatility mean reversion. Other stylized facts are illustrated and explained in later sections.
Volatility Clustering Usually the GARCH coefficient f1 is found to be around 0.9 for many weekly or daily financial time series. Given this value of f31, it is obvious that large values of a2_1 will be followed by large values of a2, and small values of a2_1 will be followed by small values of. The same reasoning can be obtained from the ARMA representation in (5.3.2), where large/small changes in u^_1 will be followed by large/small changes in of.
Fat Tails It is well known that the distribution of many high frequency financial time series usually have fatter tails than a normal distribution. This means that large changes are more often to occur than under a normal distribution. Thus a GARCH model can replicate the fat tails usually observed in financial time series.
Volatility Mean Reversion Although financial markets may experience excessive volatility from time to time, it appears that volatility will eventually settle down to a long run level. The previous subsection showed that the long run variance of ut for the stationary GARCH(1 1) model is a0 (1 — a1 — In this case, the volatility is always pulled toward this long run level.
Example: Simulating an GARCH (p, q) model in EViews
It is slightly trickier to simulate GARCH process than the ARCH one in EViews. Since it is necessary simultaneously to generate Yt and ot processes, we will need to use loop to accomplish it. Therefore, it is more convenient to use program object rather than doing it in the command line. Consider as an example GARCH(2 1) series
We start the program with the same commands as in the ARCH case; the only difference is that we generate a conditional variance process s.
The next part of the program creates the loop where both series Yt and o"l are generated observation after observation.
The graph of the simulated GARCH process is given on Figure?
Figure 5.2: Plot of the simulated GARCH process
We can see on the graph a clear effect of volatility clustering. In most cases volatility stays low but there are several spikes with high volatility which persist for a number of periods. Another stylized fact can be seen from the histogram of the simulated observations (click on View/Descriptive Statistic and Tests/Histogram and Stats). Jarque-Bera test strongly rejects the null hypothesis of normality and the kurtosis is extremely high indicating fat tails of the generated distribution.