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 \(y_t=(x_{t}-x_{t-1})/x_{t-1}\), the proportion gained or lost since the last time, or \(log (x_t / x_{t-1})=log(x_t)-log(x_{t-1})\), 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.

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) \(\text{Var}(y_t | y_{t-1}) = \sigma^2_t = \alpha_0 + \alpha_1 y^2_{t-1}\)

We impose the constraints \(\alpha_0\) ≥ 0 and \(\alpha_1\) ≥ 0 to avoid negative variance.

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

(2) \( y_t = \sigma_t \epsilon_t, \)

with \( \sigma_t = \sqrt{\alpha_0 + \alpha_1y^2_{t-1}}, \)

and \( \epsilon_t \overset{iid}{\sim} (\mu=0,\sigma^2=1)\)

For inference (and maximum likelihood estimation) we would also assume that the \(\epsilon_t\) are 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:

• \(y^2_t\) has the AR(1) model \( y^2_t = \alpha_0 +\alpha_1y^2_{t-1}+\) error.

  This model will be causal, meaning it can be converted to a legitimate infinite order MA only when \(\alpha^2_1 < \frac{1}{3}\) 

• \(y_t\) is white noise when 0 ≤ \(\alpha_1\) ≤ 1.

An ARCH(m) process is one for which the variance at time \(t\) is conditional on observations at the previous m times, and the relationship is

\(\text{Var}(y_t | y_{t-1}, \dots, y_{t-m}) = \sigma^2_t = \alpha_0 + \alpha_1y^2_{t-1} + \dots + \alpha_my^2_{t-m}.\)

With certain constraints imposed on the coefficients, the y_t series squared will theoretically be AR(m).

A GARCH (generalized autoregressive conditionally heteroscedastic) model uses values of the past squared observations and past variances to model the variance at time \(t\). As an example, a GARCH(1,1) is

\(\sigma^2_t = \alpha_0 + \alpha_1 y^2_{t-1} + \beta_1\sigma^2_{t-1}\)

In the GARCH notation, the first subscript refers to the order of the y² terms on the right side, and the second subscript refers to the order of the \(\sigma^2\) terms.

The best identification tool may be a time series plot of the series. It’s usually easy to spot periods of increased variation sprinkled through the series. It can be fruitful to look at the ACF and PACF of both y_t and \(y^2_t\). For instance, if y_t appears to be white noise and \(y^2_t\) appears to be AR(1), then an ARCH(1) model for the variance is suggested.  If the PACF of the \(y^2_t\) suggests AR(m), then ARCH(m) may work. GARCH models may be suggested by an ARMA type look to the ACF and PACF of \(y^2_t\). In practice, things won’t always fall into place as nicely as they did for the simulated example in this lesson. You might have to experiment with various ARCH and GARCH structures after spotting the need in the time series plot of the series.

In the book, read Example 5.4 (an AR(1)-ARCH(1) on p. 257-middle of p. 259), and Example 5.5 (GARCH(1,1) on p. 259-p.260).  R code for will also be given in the homework for this week.