Trend-stationary process

In the statistical analysis of time series, a trend-stationary process is a stochastic process from which an underlying trend (function solely of time) can be removed, leaving a stationary process. The trend does not have to be linear. Conversely, if the process requires differencing to be made stationary, then it is called difference stationary and possesses one or more unit roots. Those two concepts may sometimes be confused, but while they share many properties, they are different in many aspects. It is possible for a time series to be non-stationary, yet have no unit root and be trend-stationary. In both unit root and trend-stationary processes, the mean can be growing or decreasing over time; however, in the presence of a shock, trend-stationary processes are mean-reverting (i.e. transitory, the time series will converge again towards the growing mean, which was not affected by the shock) while unit-root processes have a permanent impact on the mean (i.e. no convergence over time). A process {Y} is said to be trend-stationary if where t is time, f is any function mapping from the reals to the reals, and {e} is a stationary process. The value is said to be the trend value of the process at time t. Suppose the variable Y evolves according to where t is time and et is the error term, which is hypothesized to be white noise or more generally to have been generated by any stationary process. Then one can use linear regression to obtain an estimate of the true underlying trend slope and an estimate of the underlying intercept term b; if the estimate is significantly different from zero, this is sufficient to show with high confidence that the variable Y is non-stationary. The residuals from this regression are given by If these estimated residuals can be statistically shown to be stationary (more precisely, if one can reject the hypothesis that the true underlying errors are non-stationary), then the residuals are referred to as the detrended data, and the original series {Yt} is said to be trend-stationary even though it is not stationary.

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