Time-Varying Markov Models for Binary Temperature Series in Agrorisk Management

Abstract

This paper uses high-order categorical non-stationary Markov chains to model the occurrence of extreme temperature events, in particular frost days. These models can be applied to estimate: the probability that a given day in the future is a frost day (below zero); the probability that a given period is frost-free; the distribution of the length of the frost-free period. These quantities then can be used for pricing of weather derivatives. Several stationary and non-stationary high-order (yet parsimonious) Markov models are proposed and compared using AIC and BIC. Partial likelihood theory is used to estimate the parameters of these models. We show that optimal (in terms of AIC/BIC) non-stationary Markov models that have constant “Markov coefficients” (across the year) are not adequate to estimate the aforementioned probabilities. Therefore this paper develops Markov models with a time-varying periodic structure across the year. A challenge in fitting these models (by maximizing the partial likelihood) is the large number of parameters. The paper presents a method for overcoming this challenge; one that uses parametric fits to the logit of the nonparametric estimates of the seasonal transition probability curves to initialize the optim function in the R package. Satisfactory results are shown to obtain from this approach. The work is applied to temperature records for the Province of Alberta, Canada.