BUS606: Operations and Supply Chain Management

Autoregressive Conditional Heteroskedasticity (ARCH) models are used to characterize and model observed time series. They are used whenever there is reason to believe that, at any point in a series, the terms will have a characteristic size, or variance. In particular ARCH models assume the variance of the current error term or innovation to be a function of the actual sizes of the previous time periods' error terms: often the variance is related to the squares of the previous innovations. Such models are often called ARCH models, although a variety of other acronyms is applied to particular structures of model which have a similar basis. ARCH models are employed commonly in modeling financial time series that exhibit time-varying volatility clustering, i.e. periods of swings followed by periods of relative calm.

Suppose one wishes to model a time series using an ARCH process of order q. Let $\varepsilon_{t}$ denote the error terms (return residuals, with respect to a mean process) i.e. the series terms. These $\varepsilon_{t}$ are split into a stochastic part $z_{t}$ and a timedependent standard deviation $\sigma_{t}$ characterizing the typical size of the terms so that

$\varepsilon_{t}=\sigma_{t} Z_{t}$                                                                                                         (26)

where $z_{t}$ is a random variable drawn from a Normal distribution centered at 0 with standard deviation equal to 1. $\text { (i.e. } \left.z_{t} \stackrel{\text { iid }}{\longrightarrow} N(0,1)\right)$ and where the series $\sigma_{t}{ }^{2}$ are modeled by

$\sigma_{t}^{2}=-\alpha_{0}+\alpha_{1} \varepsilon_{t}^{2}+\ldots+\alpha_{q} \varepsilon_{t-q}^{2}=-\alpha_{0}+\sum_{i=1}^{q} \alpha_{i} \varepsilon_{t-i}^{2}$                       (27)

and where $\alpha_{0} \geq 0, \alpha_{i} \geq 0, i>0$. An ARCH(q) model can be estimated using ordinary least squares. A methodology to test for the lag length of ARCH errors using the Lagrange multiplier test was proposed by Engle.