ARCH Engle's Test for Residual Heteroscedasticity
Performs Portmanteau Q and Lagrange Multiplier tests for the null hypothesis that the residuals of a ARIMA model are homoscedastic.
arch.test(object, output = TRUE)
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.
A matrix with the following five columns:
- the lag parameter to calculate the test statistics.
- the Portmanteau Q test statistic.
- the p.value for PQ test.
- the Lagrange Multiplier test statistic.
- the p.value for LM test.
Missing values are removed before analysis.
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.
x <- rnorm(100) mod <- estimate(x,p = 1) # or mod <- arima(x,order = c(1,0,0)) arch.test(mod)