Question

interprete an Arch model in Timeseries.with an example

Answer 1

An ARCH (autoregressive conditionally heteroscedastic) model is a model for the variance of a time series. ARCH models are used to describe a changing, possibly volatile variance. Although an ARCH model could possibly be used to describe a gradually increasing variance over time, most often it is used in situations in which there may be short periods of increased variation. (Gradually increasing variance connected to a gradually increasing mean level might be better handled by transforming the variable.)

ARCH models were created in the context of econometric and finance problems having to do with the amount that investments or stocks increase (or decrease) per time period, so there’s a tendency to describe them as models for that type of variable. For that reason, the authors of our text suggest that the variable of interest in these problems might either be yt=(xt?xt?1)/xt?1yt=(xt?xt?1)/xt?1, the proportion gained or lost since the last time, orlog(xt/xt?1)=log(xt)?log(xt?1)log(xt/xt?1)=log(xt)?log(xt?1), the logarithm of the ratio of this time’s value to last time’s value. It’s not necessary that one of these be the primary variable of interest. An ARCH model could be used for any series that has periods of increased or decreased variance. This might, for example, be a property of residuals after an ARIMA model has been fit to the data.

The ARCH(1) Variance Model

Suppose that we are modeling the variance of a series y_t. The ARCH(1) model for the variance of model y_t is that conditional on y_t-1, the variance at time t is

(1)                Var(yt|yt?1)=?2t=?0+?1y2t?1Var(yt|yt?1)=?t2=?0+?1yt?12

We impose the constraints ?₀ ? 0 and ?₁ ? 0 to avoid negative variance.

Note that the variance at time t is connected to the value of the series at time t – 1. A relatively large value of y2t?1yt?12 gives a relatively large value of the variance at time t. This means that the value of y_t is less predictable at time t ?1 than at times after a relatively small value of y2t?1yt?12.

If we assume that the series has mean = 0 (this can always be done by centering), the ARCH model could be written as

(2)                 yt=?t?t, with ?t=??0+?1y2t?1, and ?t?iid(0,1)yt=?t?t, with ?t=?0+?1yt?12, and ?t?iid(0,1)

For inference (and maximum likelihood estimation) we would also assume that the ?_tare normally distributed.

Possibly Useful Results

Two potentially useful properties of the useful theoretical property of the ARCH(1) model as written in equation line (2) above are the following:

• y2t(yt squared ) has the AR(1) model y2t=?0+?1y2t?1+error.yt2(yt squared ) has the AR(1) model yt2=?0+?1yt?12+error.

This model will be causal, meaning it can be converted to a legitimate infinite order MA only when ?21<13?12<13

• y_tis white noise when 0 ? ?₁ ? 1.

Thanks