Abstract: Common factors for seasonal multivariate time series are usually obtained by first filtering the series to eliminate the seasonal component and then extracting the nonseasonal common factors. This approach has two drawbacks. First, we cannot detect common factors with seasonal structure; second, it is well known that a deseasonalized time series may exhibit spurious cycles that the original data do not contain, which can make more difficult the detection of the nonseasonal factors. In this paper we propose a procedure using the original data to estimate the dynamic common factors when some, or all, of the time series are seasonal. We assume that the factor may be stationary or nonstationary and seasonal or not. The procedure is based on the asymptotic behavior of the sequence of the so-called sample generalized autocovariance matrices and of the sequence of canonical correlation matrices, and it includes a statistical test for detecting the total number of common factors. The model is estimated by the Kalman Filter. The procedure is illustrated with an environmental example where two interesting seasonal common factors are found.

Key words and phrases: Common seasonality, dynamic common factors, multivariate time series.