# arch.test: ARCH Engle's Test for Residual Heteroscedasticity

## Description

Performs Portmanteau Q and Lagrange Multiplier tests for the null
hypothesis that the residuals of a ARIMA model are homoscedastic.
## Usage

arch.test(object, output = TRUE)

## Arguments

object

an object from arima model estimated by
`arima`

or `estimate`

function. output

a logical value indicating to print the results in R console, including the
plots. The default is `TRUE`

.

## Value

A matrix with the following five columns:
`order`

- the lag parameter to calculate the test statistics.
`PQ`

- the Portmanteau Q test statistic.
`p.value`

- the p.value for PQ test.
`LM`

- the Lagrange Multiplier test statistic.
`p.value`

- the p.value for LM test.

## Details

The ARCH Engle's test is constructed based on the fact that if the residuals
(defined as $e[t]$) are heteroscedastic, the squared residuals ($e^2[t]$) are
autocorrelated. The
first type of test is to examine whether the squares of residuals are a sequence of white
noise, which is called Portmanteau Q test and similar to the Ljung-Box test on the squared
residuals. The second type of test proposed by Engle (1982) is the Lagrange Multiplier
test which is to fit a linear regression model for the squared residuals and examine
whether the fitted model
is significant. So the null hypothesis is that the squared residuals are a sequence
of white noise, namely, the residuals are homoscedastic. The lag parameter
to calculate the test statistics is taken from an integer sequence of $1:min(24,n)$ with
step 4 if $n > 25$, otherwise 2, where $n$ is the number of nonmissing observations.The plots of residuals, squared residuals, p.values of PQ and LM tests will be drawn if
`output = TRUE`

.

## References

Engle, Robert F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates
of the Variance of United Kingdom Inflation. *Econometrica*, 50 (4): 987-1007.McLeod, A. I. and W. K. Li. Diagnostic Checking ARMA Time Series Models Using
Squared-Residual Autocorrelations. *Journal of Time Series Analysis*.
Vol. 4, 1983, pp. 269-273.

## Examples

x <- rnorm(100)
mod <- estimate(x,p = 1) # or mod <- arima(x,order = c(1,0,0))
arch.test(mod)