## Tuesday, 27 December 2011

### Seasonality in Time Series - Level II Quantitative Methods

When a time series variable exhibit a repeating patterns at regular intervals over time, it is known as seasonality e.g. sales in Dec > sales in Jan. A time series with seasonality also has a non-constant mean and thus is not covariance stationary.

Detecting seasonality:
In case of seasonality in the data, autocorrelation in the model differ by season. For example, in case of quarterly sales data of a company, if the fourth autocorrelations of the error term differ significantly from 0  → This is a sign of seasonality in the model.

Decision Rule:
When t-statistic of the fourth lag of autocorrelations of the error > critical t-value → reject null hypothesis that fourth autocorrelations is 0.  Thus, there is seasonality problem.

Correcting Seasonality: This problem can be solved by adding seasonal lags in an AR model i.e. after including a seasonal lag in case of quarterly sales data, the AR model becomes:

xt = b0 + b1x (t-1) + b2x(t-4) + et

NOTE: R(square) of the model without seasonal lag will be less than the R(square) of the model with seasonal lag. This implies that when time series exhibit seasonality, including a seasonal lag in the model improves the accuracy of the model.