aTSA (version 3.1.2)

# 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)
```