# Diagnosing ARCH in time series

Hi all How do you look for presence of ARCH in time series? Many thanks S

Seasonality seems to be the one that comes up a lot, right… so if you see a chart with blips in the errors every 4th quarter, you probably need to run an ARCH regression on four quarter lagged squared errors. If the coefficient of this is significant it means the model is misspecified and you need to add a seasonal lag. Maybe more generally, any cyclicality in the error terms would suggest ARCH.

plotting the squared residuals against time usually does a good job of revealing heteroskedasticity. Beyond that, there are a bunch of tests you can run, one of which being a regression of squared error terms on their own lagged values.

“Seasonality seems to be the one that comes up a lot, right… so if you see a chart with blips in the errors every 4th quarter, you probably need to run an ARCH regression on four quarter lagged squared errors. If the coefficient of this is significant it means the model is misspecified and you need to add a seasonal lag. Maybe more generally, any cyclicality in the error terms would suggest ARCH” Testing for serial correlation/autocorrelaton using t-tests would reveal seasonality. I understand the theory - ARCH refers to variance of residuals dependant on prior period’s variance of residuals while autocorellation refer correlation of residuals with prior period residuals. I can visualize autocorrelation but not ARCH. Appreciate if someone can point out the clear difference.

Do just as I said above. There are a number of ways that you can do this graphically. For autocorrelation, generate the residuals from the regression, and then plot them. One way is to simply plot against time and look for persistence. Another is to plot against its own lag. Do so with the time t residuals on the x-axis, and time t-1 residuals on the y-axis. If you see a random blob, autocorrelation is likely not an issue. If you see a clear pattern, you have a problem. For heteroskedasticity, we cannot actually know the true variance, but the squared residuals are a good proxy for it. Therefore, plotting squared residuals against time should reveal any potential heteroskedasticity. If the plot has relatively constant volatility, it is likely not present. If you have periods of peakedness, and the periods of calm, you have an issue.

Thanks wyantjs- that explantion helps.