A self-updating model driven by a higher-order hidden Markov chain for temperature dynamics

We develop a model for the evolution of daily average temperatures (DATs) that could benefit the analysis of weather derivatives in finance and economics as well as the modelling of time series data in meteorology, hydrology and other branches of the sciences and engineering. Our focus is to capture the mean-reverting, seasonality, memory and stochastic properties of the temperature movement and other time series exhibiting such properties. To model both mean-reversion and stochasticity, a deseasonalised component is assumed to follow an Ornstein-Uhlenbeck (OU) process modulated by a higher-order hidden Markov chain, which takes into account short/long range dependence in the data. The seasonality part is modelled through a combination of linear and sinusoidal functions with appropriate coefficients and arguments. Furthermore, we put forward a parameter estimation approach that establishes recursive higher-order hidden Markov model (HOHMM) filtering algorithms customised for the regime-switching evolution of model parameters. Quantities that are functions of HOHMC characterise these filters. Utilising the EM method in conjunction with the change of measure technique, optimal and self-updating parameter estimates are obtained. We illustrate the numerical implementation of our model and estimation technique using a 4-year Toronto DATs data set compiled by the National Climatic Data Center. We perform pertinent model selection and validation diagnostics to assess the performance of our methodology. It is shown that a 2-state HOHMM-based model best captures the empirical characteristics of the temperature data under examination on the basis of various error-based and information-criterion metrics.