Trend-stationary process

Formal definition

A process {Y} is said to be trend-stationary if

: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 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 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_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, and the original series {Y**t} 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 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

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.

Notes

References

1. "Differencing And Unit Root Tests".
2. (2011). "Non-Stationary Series". [[University of Oxford]].
3. Heino Bohn Nielsen. "Non-Stationary Time Series and Unit Root Tests".
4. 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.
5. 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." http://www.uh.edu/~dpapell/realgdp.pdf
6. "Archived copy".
7. http://www.duke.edu/~rnau/411diff.htm "Stationarity and differencing"