The time series data used are generally sums over time of data generated more frequently than the reporting interval. In this research, we focused on the effect of temporal aggregation on a vector autoregressive moving average (VARMA) model structure, a cointegration relationship, the causality, and multiplicative seasonal VARMA processes.; First, we worked on the cointegration problem and showed that while the cointegrating matrix remains unchanged, temporal aggregation changes the model form and affects the results of the cointegration trace test. We derived a modified test statistic and proved that the limiting distribution of the new statistic is the same as that of Johansen's trace test statistic. We can use Johansen's table of critical values but we have to use the modified test statistic that incorporates the effect of aggregation in computing the test statistic when aggregate data are used for the test. The use of aggregate data for causal inference is common in practice. Since the form of the vector time series model changes after aggregation, non-causality conditions for the basic model and for the model of aggregates are different. Temporal aggregation often deduces a causal relationship between aggregate variables. Because the standard test fails to detect cointegration in aggregate series, we developed a modified testing procedure to test the Granger non-causality in cointegrated systems for aggregates.; Many business and economic time series show seasonality. The best way to present seasonality is by using multiplicative models. We studied the representation problem in multiplicative seasonal VARMA models and showed that the correct order of non-seasonal and seasonal parameters in the representation improves parameter estimation and forecasts. We recommend fitting a multiplicative model by using different representations and making selection with information criteria. We also derived the model for aggregates of multiplicative processes.
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