Unit roots and smooth transitions

It is common practice in time series econometrics to test the null hypothesis that the generating function is integrated—i.e. that a series is stationary only after differencing—against the alternative of stationarity about either a fixed mean or a linear trend. However, there has been considerable recent interest in the possibility of stationarity around a linear trend with an abrupt break. Here we broaden this class of alternatives to allow for a smooth transition from one trend function to another. Dickey–Fuller type tests against this alternative are developed, and their properties are explored.