# Trend stationary

In the statistical analysis of time series, a stochastic process is trend stationary if an underlying trend (function solely of time) can be removed, leaving a stationary process.[1]

## Formal definition

A process {Y} is said to be trend stationary if[2]

$Y_t = f(t) + e_t,$

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

## Simplest example: stationarity around a linear trend

Suppose the variable Y evolves according to

$Y_t = a \cdot t + b + e_t$

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[2][3][4]linear regression to obtain an estimate $\hat{a}$ of the true underlying trend slope $a$ and an estimate $\hat{b}$ of the underlying intercept term b; if the estimate $\hat{a}$ 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

$\hat{e}_t = Y - \hat{a} \cdot t - \hat{b}.$

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,[5] and the original series {Yt} is said to be trend stationary even though it is not stationary.

## Stationarity around other types of trend

### Exponential growth trend

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

$\text{GDP}_t = Be^{at}U_t$

with Ut being hypothesized to be a stationary error process. To estimate the parameters $a$ and B, one first takes[5] the natural logarithm (ln) of both sides of this equation:

$\ln (\text{GDP}_t) = \ln B + at + \ln (U_t).$

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 $(\ln U)_t$ as the detrended value of $(\ln \text{GDP})_t$, and hence the implied $U_t$ as the detrended value of $\text{GDP}_t$, assuming one can reject the hypothesis that $(\ln U)_t$ is non-stationary.

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

$Y_t = a \cdot t + c \cdot t^2 + b + e_t.$

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 $Y_t$.

## Other non-stationary processes that are not trend stationary but can be rendered stationary

Trend stationary processes are not the only kind of non-stationary process that can be transformed into a stationary one; another prominent such process exhibits one or more unit roots[2][3][4][5] but has all its other roots smaller than unity in magnitude.