Trend-stationary process

Many economic time series are characterized by exponential growth. For example, suppose that one hypothesizes that gross domestic product is characterized by stationary deviations from a trend involving a constant growth rate. Then it could be modeled as

${\displaystyle {\text{GDP}}_{t}=Be^{at}U_{t}}$

with U_t being hypothesized to be a stationary error process. To estimate the parameters ${\displaystyle a}$ and B, one first takes[8] the natural logarithm (ln) of both sides of this equation:

${\displaystyle \ln({\text{GDP}}_{t})=\ln B+at+\ln(U_{t}).}$

This log-linear equation is in the same form as the previous linear trend equation and can be detrended in the same way, giving the estimated ${\displaystyle (\ln U)_{t}}$ as the detrended value of ${\displaystyle (\ln {\text{GDP}})_{t}}$, and hence the implied ${\displaystyle U_{t}}$ as the detrended value of ${\displaystyle {\text{GDP}}_{t}}$, assuming one can reject the hypothesis that ${\displaystyle (\ln U)_{t}}$ is non-stationary.

Trends do not have to be linear or log-linear. For example, a variable could have a quadratic trend:

${\displaystyle Y_{t}=a\cdot t+c\cdot t^{2}+b+e_{t}.}$

This can be regressed linearly in the coefficients using t and t² as regressors; again, if the residuals are shown to be stationary then they are the detrended values of ${\displaystyle Y_{t}}$.