Distinguishing Trend-Stationary and Difference-Stationary Time Series
Summary
The document explores why a significant deterministic time trend can coexist with a negative, significant coefficient on the lagged level in a regression of first differences. The response considers a series made up of a linear trend plus a stationary error: after differencing, the trend becomes a constant, while the error contributes serially dependent changes. Under that specification, the lagged-level coefficient should be zero, so a significant negative estimate suggests the assumed data-generating process may be inadequate.
The proposed next step is to inspect residuals from the trend regression, including their autocorrelation, and consider a richer model such as an ARIMA process if dependence remains. Another response distinguishes deterministic trend from drift and notes that differencing a trend-stationary series or detrending a difference-stationary series can fail to produce stationarity. The discussion is introductory and does not provide a complete testing procedure; trend terms, residual dynamics, and test specification all matter.
Key ideas
- A deterministic linear trend with stationary errors becomes a constant after first differencing.
- Under that trend-only model, the lagged-level coefficient in a differenced regression should be zero.
- A significant coefficient may indicate that the assumed trend-only data-generating process is incomplete.
- Residual autocorrelation after detrending can motivate a richer time-series model.
- Trend-stationary and difference-stationary series require different treatments, so differencing and detrending are not interchangeable.
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# Confusion on stationarity vs deterministic trend
# Confusion on stationarity vs deterministic trend
Sorry for the newbie inquiry but I'm having a little trouble making sense of stationarity and how a the presence of a time trend impacts this. I'm working on a model for operating margins and as a first step I want to determine if the original series is stationary before proceeding. I first fitted a simple linear trend line to the data and the time regressor, while small in magnitude, registered as highly significant. I was always under the impression that this implied a non constant mean, thus non-stationary and may require a transform or differencing. I decided to regress the first differenced time series on the lag of the original time series and found the regressor of the lagged value to be negative and highly significant (t-stat greater than 9). This is where I got a little confused as these two seem to contradict my understanding of the subject. I thought a rejection of the null: g =0 (Dickey Fuller test) indicated no unit root, thus mean reverting and stationary. This seems to conflict with my initial assessment based on the deterministic time trend component. Thanks in advance!
## Answer by zsljulius (score 1)
https://quant.stackexchange.com/a/30983
Suppose the data generating process as your have suspected is as follows: $$y_t = \gamma t + \epsilon_t$$ A first difference of the series will be $$\Delta y_t = \gamma + \epsilon_t - \epsilon_{t-1}$$ Now as what you have done in your 2nd stage, regressing $\Delta y_t$ on $y_{t-1}$, what you will have estimated in the 2nd stage is $$\Delta y_t = \alpha + \beta y_{t-1} + e_t$$ Which is equivalent to $$\gamma + \epsilon_t - \epsilon_{t-1} = \alpha + \beta y_{t-1} + e_t$$ re-arranging gives you the following expression: $$\epsilon_t - \epsilon_{t-1} = (\alpha - \gamma) + \beta y_{t-1} + e_t$$ So your 2nd stage regression should yield $\beta =0$ if you have a deterministic time trend. The reason you have a negative and significant coeffcient in 2nd stage, would suggest that the DGP is wrong. I would highly recommend you to perform a residual check on 1st stage. You can fit a deterministic trend to the original model and plot the acf of the residual, I suspect you will see significant autocorrelation for many lags, indicating that you might consider fitting more complicated models such as ARIMA type models.
## Answer by GuestNo3829297 (score 0)
https://quant.stackexchange.com/a/22365
A difference-stationary series will not be stationary if it is detrended (regression) and a trend-stationary series will not be stationary if differenced. Trend is deterministic, drift is non-zero expectation of the change. I recommend Enders, Applied Econometric Time Series.
## Answer by Neeraj (score 0)
https://quant.stackexchange.com/a/22393
If there is time trend in your data then it is better to take demean data and then test for stationarity. If $$Y_t=\alpha + \beta t +e_t $$ $$e_t=Y_t-\alpha-\beta t \sim N(0,\sigma^2)$$
such that $e_t$ is independent and identical distributed.
If $e_t$ is not stationary then also try for log difference.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.