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An uncorrelated time series can still be serially dependent
due to a dynamic conditional variance process. A time series exhibiting
conditional heteroscedasticity—or autocorrelation in the squared
series—is said to have *autoregressive conditional
heteroscedastic* (ARCH) effects. Engle’s ARCH test
is a Lagrange multiplier test to assess the significance of ARCH effects [1].

Consider a time series

$${y}_{t}={\mu}_{t}+{\epsilon}_{t},$$

where$${\mu}_{t}$$ is the conditional mean of the process, and$${\epsilon}_{t}$$ is an innovation process with mean zero.

Suppose the innovations are generated as

$${\epsilon}_{t}={\sigma}_{t}{z}_{t},$$

where *z _{t}* is
an independent and identically distributed process with mean 0 and
variance 1. Thus,

$$E({\epsilon}_{t}{\epsilon}_{t+h})=0$$

for all lags $$h\ne 0$$ and the innovations are uncorrelated.

Let *H _{t}* denote the
history of the process available at time

$$Var({y}_{t}|{H}_{t-1})=Var({\epsilon}_{t}|{H}_{t-1})=E({\epsilon}_{t}^{2}|{H}_{t-1})={\sigma}_{t}^{2}.$$

Thus, conditional heteroscedasticity in the variance process is equivalent to autocorrelation in the squared innovation process.

Define the residual series

$${e}_{t}={y}_{t}-{\widehat{\mu}}_{t}.$$

If all autocorrelation
in the original series, *y _{t}*,
is accounted for in the conditional mean model, then the residuals
are uncorrelated with mean zero. However, the residuals can still
be serially dependent.

The alternative hypothesis for Engle’s ARCH test is autocorrelation in the squared residuals, given by the regression

$${H}_{a}:{e}_{t}^{2}={\alpha}_{0}+{\alpha}_{1}{e}_{t-1}^{2}+\dots +{\alpha}_{m}{e}_{t-m}^{2}+{u}_{t},$$

where *u _{t}* is
a white noise error process. The null hypothesis is

$${H}_{0}:{\alpha}_{0}={\alpha}_{1}=\dots ={\alpha}_{m}=0.$$

To conduct Engle’s ARCH test using `archtest`

,
you need to specify the lag *m* in the alternative
hypothesis. One way to choose *m* is to compare loglikelihood
values for different choices of *m*. You can use
the likelihood ratio test (`lratiotest`

) or information
criteria (`aicbic`

) to compare loglikelihood values.

To generalize to a GARCH alternative, note that a GARCH(*P*,*Q*)
model is locally equivalent to an ARCH(*P* + *Q*)
model. This suggests also considering values *m* = *P* + *Q* for
reasonable choices of *P* and *Q*.

The test statistic for Engle’s ARCH test is the usual *F* statistic
for the regression on the squared residuals. Under the null hypothesis,
the *F* statistic follows a$${\chi}^{2}$$ distribution with *m* degrees
of freedom. A large critical value indicates rejection of the null
hypothesis in favor of the alternative.

As an alternative to Engle’s ARCH test, you can check
for serial dependence (ARCH effects) in a residual series by conducting
a Ljung-Box Q-test on the first *m* lags of the squared
residual series with `lbqtest`

. Similarly, you can
explore the sample autocorrelation and partial autocorrelation functions
of the squared residual series for evidence of significant autocorrelation.

[1] Engle, Robert F. “Autoregressive
Conditional Heteroskedasticity with Estimates of the Variance of United
Kingdom Inflation.” *Econometrica*. Vol.
50, 1982, pp. 987–1007.

`aicbic`

| `archtest`

| `lbqtest`

| `lratiotest`

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