Trend Stationary - Simplest Example: Stationarity Around A Linear Trend

Simplest Example: Stationarity Around A Linear Trend

Suppose the variable Y evolves according to

where t is time and e_t is an error term that is hypothesized to be white noise or more generally to have been generated by any stationary process. Then one can uselinear 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 {Y_t} is said to be trend stationary even though it is not stationary.