# Testing for Seasonality

When testing for seasonality, you look at the significance of the t-stats of the autocorrelations. Do you only observe seasonality if the 4th (for quarterly) or 12th (for monthly) residual lag is significant? In other words, say you have monthly data, the 12th residual lag’s t-stat is NOT significant but the 11 month t-stat is: is there seasonality? My “working theory” is that if you have 12 residuals/t-stats and monthly data, if the 12th one is significant then you have seasonality but if any others are significant then you have autocorrelation.

I think your interpretation is pretty spot on. I havn’t seen a question with autocorrelation in other then t-1 or t-2 though. Here’s a tip… A model with a lagged coefficient and a seasonality term is still and AR(1) model

Yes. xt = b0 +b1xt-1 +b2xt-4 is an AR1 model. thanks.

I think we should follow justinkc’s rule of thumb but I would say that if lag two is sta sig and it makes economical sense it should be added to the model as a lag. If the lag does not make economic sense call it autocorrelation.

i think you might have mixed up some terms there. You can of course have an AR(2) model with t-1, t-2, and still have a seasonal term at t-4 or t-12. Any of the autorcorrelations that are significant should be added as a lag

autocorrelation will not be statistically significant. If they are - they would need to be included as lags… if you see e.g. in a twelve month period regression that the 11th lag has a seasonal issue - the t-stat on its autocorrelation would be significant. YOu might find more significant autocorrelations, but you start with including the 11th period as a lag element. Then repeat the process. You should find that the autocorrelations now are no longer significant.

Ok but the original question was about statistically significant t-stats at “off-seasonal time periods.” Let’s say I start with xt = b0 +b1xt-1 So if I have quarterly sales data and t = 3 has significant residual, I say there’s serial correlation. I add a lagged dependent variable and get xt = b0 + b1xt-1 + b2xt-2 But if t= 4 is significant, I say that this shows seasonality and my new model becomes xt = b0 + b1xt-1 + b2xt-4. That is what i am trying to confirm.