Trend stationary

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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[edit]

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[edit]

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[edit]

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

\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.

Quadratic trend[edit]

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[edit]

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.

See also[edit]


  1. ^ economics Online Glossary of Research Economics
  2. ^ a b c Nelson, Charles R. and Plosser, Charles I. (1982), "Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications," Journal of Monetary Economics, 10, 139-162.
  3. ^ a b Hegwood, Natalie, and Papell,David H. "Are real GDP levels trend, difference, or regime-wise trend stationary? Evidence from panel data tests incorporating structural change."
  4. ^ a b Lucke, Bernd. "Is Germany‘s GDP trend-stationary? A measurement-with-theory approach."
  5. ^ a b c "Stationarity and differencing"