# 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!

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.

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.

• Thanks for the reply I'll be sure to check out the resource mentioned. I'm not quite following at the moment though - could you elaborate perhaps? I guess I'm trying to understand what is meant and not meant by the fact that the margin time series: 1) shows a highly significant time trend component when just regressed against time AND 2) rejects the H_0: g =0 when the differenced series is regressed on a lag of the original series. I thought that absence of a unit root via the DF test (#2) indicated stationarity and mean reversion. But how can this be mean reverting given the time coefficient? Dec 16, 2015 at 16:58
• I'm not sure your point about detrending a difference stationary process is that clear.
– John
Dec 16, 2015 at 17:42

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.