A process {Y} is said to be trend-stationary if^[5]

${\displaystyle Y_{t}=f(t)+e_{t},}$ 

where t is time, f is any function mapping from the reals to the reals, and {e} is a stationary process. The value ${\displaystyle f(t)}$  is said to be the trend value of the process at time t.

Suppose the variable Y evolves according to

${\displaystyle Y_{t}=a\cdot t+b+e_{t}}$ 

where t is time and e_t 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^[5][6][7] linear regression to obtain an estimate ${\displaystyle {\hat {a}}}$  of the true underlying trend slope ${\displaystyle a}$  and an estimate ${\displaystyle {\hat {b}}}$  of the underlying intercept term b; if the estimate ${\displaystyle {\hat {a}}}$  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

${\displaystyle {\hat {e}}_{t}=Y_{t}-{\hat {a}}\cdot t-{\hat {b}}.}$ 

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,^[8] and the original series {Y_t} is said to be trend-stationary even though it is not stationary.

Exponential growth trend edit

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.

Quadratic trend edit

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 ${\displaystyle Y_{t}}$ .

• Trend estimation
• Decomposition of time series
• KPSS test