Seasonality with Trend and Cycle Interactions in Unobserved Components Models

Abstract

Summary. Unobserved components time series models decompose a time series into a trend, a season, a cycle, an irregular disturbance, and possibly other components. These models have been successfully applied to many economic time series. The standard assumption of a linear model, often appropriate after a logarithmic transformation of the data, facilitates estimation, testing, forecasting and interpretation. However, in some settings the linear-additive framework may be too restrictive. In this paper, we formulate a non-linear unobserved components time series model which allows interactions between the trend-cycle component and the seasonal component. The resulting model is cast into a non-linear state space form and estimated by the extended Kalman filter, adapted for models with diffuse initial conditions. We apply our model to UK travel data and US unemployment and production series, and show that it can capture increasing seasonal variation and cycle dependent seasonal fluctuations.

Trends, Cycles and Convergence in U.S. Regional House Prices

“…A non-linear unobserved components time series model which allows interactions between the trendcycle component and the seasonal component was recently presented inKoopman and Lee (2008).…”

A Bayesian Semiparametric Approach for Trend–Seasonal Interaction: an Application to Migration Forecasts

“…Second, the interaction can impact on the number of incomers only linearly. This last restriction can be particularly limiting since there is no empirical evidence supporting the idea that the interaction is linear or of any other specific functional form like exp {trend t *( sin it + cos it )} (Koopman and Lee, ). The combination of these two limitations might translate into estimates that are far in probability from the true unobserved parameters.…”

“…As it is particularly malleable, the latter can conveniently model multiple seasonal spikes by simply increasing its order. These two components become the arguments of an unknown bivariate smooth function, which relaxes the hypothesis that trend and seasonality evolve independently (Koopman and Lee, ; Hindrayanto et al ., ). The non‐parametric nature of the interaction does not impose a rigid structure to the trend–seasonal comovements, returning an additive model with interaction (AMI).…”

Under-/Over-Valuation of the Stock Market and Cyclically Adjusted Earnings*

“…Watson 1986; Perron and Wada 2009), while in another popular specification (e.g. Harvey 1985;Clark 1987;Koopman and Lee 2009), m follows a random walk:…”