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] The trend does not have to be linear.

Contrarily, if the process requires one or more differencing to be made stationary, then it is called difference stationary and possesses one or more unit roots.[2][3] Those two concepts may sometimes be confused, but while they share many properties, they are different in many aspects. It is possible for a time series to be non-stationary, have no unit root yet be trend-stationary. In both unit root and trend-stationary processes, the mean can be growing or decreasing over time; however, in the presence of a shock, trend-stationary processes are mean-reverting (i.e. transitory, the time serie will converge again towards the growing mean, which was not affected by the shock) while unit-root processes have a permanent impact on the mean (i.e. no convergence over time).[4]

Formal definition[edit]

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

where t is time, f is any function mapping from the reals to the reals, and {e} is a stationary process. The value 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

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[5][6][7]linear regression to obtain an estimate of the true underlying trend slope and an estimate of the underlying intercept term b; if the estimate 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

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 {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

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

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 as the detrended value of , and hence the implied as the detrended value of , assuming one can reject the hypothesis that 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:

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 .

See also[edit]

Notes[edit]

  1. ^ About.com economics Online Glossary of Research Economics
  2. ^ (PDF) http://pages.stern.nyu.edu/~churvich/Forecasting/Handouts/UnitRoot.pdf.  Missing or empty |title= (help)
  3. ^ (PDF) http://www.stats.ox.ac.uk/~burke/Autocorrelation/Non-Stationary%20Series.pdf.  Missing or empty |title= (help)
  4. ^ Heino Bohn Nielsen. "Non-Stationary Time Series and Unit Root Tests" (PDF). 
  5. ^ a b 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.
  6. ^ 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
  7. ^ Lucke, Bernd. "Is Germany‘s GDP trend-stationary? A measurement-with-theory approach." http://www.wiso.uni-hamburg.de/fileadmin/wiso_vwl_iwk/paper/gdptrend.pdf
  8. ^ a b http://www.duke.edu/~rnau/411diff.htm "Stationarity and differencing"