Seasonality in a time series is a pattern that tends to repeat from year to year. One example is monthly sales data for a retailer. Given that sales data normally vary accordingly to the time of year, we might expect this month's sales (x_t) to be related to sales for the same month last year (x_t-12).

To adjust for seasonality in an AR model, an additional lag of the dependent variable (corresponding to the same period in the previous year) is added to the original model as another independent variable. For example, if quarterly data are used, the seasonal lag is 4; if monthly data is used, the seasonal lag is 12.

Suppose for example, we model a particular quarterly time series using an AR (1) model, x_t = b₀ + b₁ x_t-1 + ε_t. If the time series had significant seasonality, this model would not be correctly specified. The seasonality would be easy to detect because the seasonal autocorrelation in the case of quarterly data, the 4th autocorrelation of the error term would differ significantly from 0.

Suppose this quarterly model has significant seasonality. In this case, we might include a seasonal lag in the autoregressive model and estimate x_t = b₀ + b₁ x_t-1 + b₂ x_t-1 + ε_t, to test whether including the seasonal lag in the autoregressive model would eliminate statistically significant autocorrelation in the error term.