Maximum likelihood estimation of a generalized threshold stochastic regression model

There is hardly any literature on modelling nonlinear dynamic relations involving nonnormal time series data. This is a serious lacuna because nonnormal data are far more abundant than normal ones, for example, time series of counts and positive time series. While there are various forms of nonlinearities, the class of piecewise-linear models is particularly appealing for its relative ease of tractability and interpretation. We propose to study the generalized threshold model which specifies that the conditional probability distribution of the response variable belongs to an exponential family, and the conditional mean response is linked to some piecewise-linear stochastic regression function. We introduce a likelihood-based estimation scheme, and the consistency and limiting distribution of the maximum likelihood estimator are derived. We illustrate the proposed approach with an analysis of a hare abundance time series, which gives new insights on how phase-dependent predator-prey-climate interactions shaped the ten-year hare population cycle. A simulation study is conducted to examine the finite-sample performance of the proposed estimation method.