aTSA (version 3.1.2)

arch.test: 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)


an object from arima model estimated by arima or estimate function.
a logical value indicating to print the results in R console, including the plots. The default is 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.


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.


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))