Estimation methods for linear, nonlinear, and multivariate time series: Applications of state-space modeling

Burman and Shumway (2004) use penalized-least squares to generate trend estimates for the trend-only linear time series model, Y(t) = T(t) + e(t), where T(t) is called the trend and e(t) is random error. We extend their method and apply it to the trend plus seasonal linear time series model, Y(t) = T(t) + S(t) + e(t), where S(t) is called the seasonal and has period p. We obtain closed-form expressions for the trend and seasonal estimators. Next, we extend the method further and consider the class of time series where the distribution of the observation is a member of the exponential family of distributions. We focus on Poisson and Bernoulli time series and present an estimation procedure based on the penalized log-likelihood. Last, we extend the method even further and develop an estimation procedure for multivariate time series observations. Using a principal components analysis, we reduce the dimension of the estimation problem. This gives rise to a co-integration model. We provide heuristic asymptotic results for all of the proposed estimators and we provide examples showing how the procedures can be applied to real data.