The Effect of Aggregation on Seasonal Cointegration in Mixed Frequency data

Economic time series often show a strong persistency as well as seasonal variations that are appropri ately modelled using seasonal unit root models in addition to deterministic components. In many cases di¤erent variables within a vector time series are driven by identical common trends and cycles leading to cointegration. This paper investigates the consequences for the properties of vector processes when some components are aggregated in time. This may involve moving from a fully observed system that is seasonally cointegrated at a frequency !k = 2 k=S with k = 1;:::;(S 1)=2 where S is the number of seasons per year, to a system with time series sampled at high sampling rate (HSR) observed for S seasons per year and time series with low sampling rate (LSR) observed SA seasons per year, such that SA = S=Q and Q is an integer. The (partial) aggregation has implications on the unit root and cointegration properties: Aggregation potentially shifts the frequency of the unit roots. This may lead to an aliasing e¤ect wherein common cycles to di¤erent unit roots become aligned and cannot be separated any more, in turn impacting cointegrating relations. This paper uses the triangular systems representations in the bivariate case as well as the state space framework (in a general setting) to investigate the e¤ect of aggregation on the unit root properties of multivariate time series. The main results indicate under which assumptions and in which situations the analysis of the integration and cointegration properties of time series with mixed sampling rate relates to the same properties of the underyling data generating process. The results also discuss full aggregation of all components. These results lead to the proposal of an e¤ective econometric strategy for detecting cointegration at the various sampling rates, as is demonstrated in a simulation exercise. Finally an empirical application with monthly data of arrivals and departures of the Mallorca Airport, also illustrate the ndings collected in the present work.