Modeling Multivariate Time Series with Uneven Spacing on Multiple Time Scales to Estimate Large Scale Time Trend

In certain situations, observations may be made on a multivariate time series on a given temporal scale. However, there may be an underlying, unobserved time series on a larger temporal scale that is of greater interest. Often times, identifying the behavior of the data over the course of the larger scale is a key objective. Because this large scale trend is not being directly observed, describing the trends of the data can be more difficult. To further complicate matters, the observed data on the smaller time scale can be unevenly spaced from one larger scale time point to the next. This means it may be more appropriate to view the observations as coming from multiple, shorter multivariate time series occurring at each large scale time point as opposed to a single, long multivariate time series. We discuss the process of modeling each of these smaller scale multivariate time series and obtain estimates of the corresponding parameters. We then introduce a method to use these parameter estimates to estimate the unobserved values of the larger scale multivariate time series at each large scale time point. A model is then fit to the estimated unobserved values of the larger scale multivariate time series in order to estimate the parameters that describe the large scale temporal trends of the data.