Forecasting unstable and nonstationary time series

Many time series are asymptotically unstable and intrinsically nonstationary, i.e. satisfy difference equations with roots greater than one (in modulus) and with time-varying parameters. Models developed by Box–Jenkins solve these problems by imposing on data two transformations: differencing (unit-roots) and exponential (Box–Cox). Owing to the Jensen inequality, these techniques are not optimal for forecasting and sometimes may be arbitrary. This paper develops a method for modeling time series with unstable roots and changing parameters. In particular, the effectiveness of recursive estimators in tracking time-varying unstable parameters is shown with applications to data-sets of Box–Jenkins. The method is useful for forecasting time series with trends and cycles whose pattern changes over time.