This paper extends the statistical results obtained by Gregoir and Laroque (1994, Journal of Econometrics 63, 183–214). It develops statistical tools to analyze multivariate time series that can be represented under an autoregressive equation of finite order with various polynomial error correction terms at various frequencies with possibly a non-null deterministic part as introduced by Gregoir (1999, Econometric Theory 15, 435–468). We propose an estimation procedure that proceeds through repeated applications of principal component analysis and a specification test for the omission of a polynomial relation of cointegration at each frequency.