Continuous‐time threshold autoregressions with jumps: Properties, estimation, and application to electricity markets

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AbstractContinuous‐time autoregressive processes have been applied successfully in many fields and are particularly advantageous in the modeling of irregularly spaced or high‐frequency time series data. A convenient nonlinear extension of this model are continuous‐time threshold autoregressions (CTAR). CTAR allow for greater flexibility in model parameters and can represent a regime switching behavior. However, so far only Gaussian CTAR processes have been defined, so that this model class could not be used for data with jumps, as frequently observed in financial applications. Hence, as a novelty, we construct CTAR processes with jumps in this paper. Existence of a unique weak solution and weak consistency of an Euler approximation scheme is proven. As a closed form expression of the likelihood is not available, we use kernel‐based particle filtering for estimation. We fit our model to the Physical Electricity Index and show that it describes the data better than other comparable approaches.