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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

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

Suppose the innovations are generated as

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

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

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

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

Define the residual series

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

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

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

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

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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