# Does autocorrelation imply temporal dependence?

It is always said that a time series should be stationary for forecasting using classical methods.

While converting a time series to stationary, we detrend and deseasonalize the time series.

Does this imply that time series does not contain any autocorrelation after being stationary?

• Hi: No. Take the AR(1) with $\phi < 1$. That series is stationary but autocorrelation exists at each lag until the lags get large. Commented Oct 16, 2021 at 16:11
• @mlofton, more precisely, in this case autocorrelation is nonzero at any and all lags. Commented Oct 16, 2021 at 16:12
• that's true. I was just saying as the lags get large, it approaches zero. Commented Oct 16, 2021 at 19:54

Removing non-stationarity just makes statistical structure of of your time series independent of absolute time-steps. It will typically reduce the auto-correlation but will not remove them. In some cases, it might happen that the time series consists of a deterministic seasonal component with some white-noise superimposed on it. In this case, removing non-stationarity will remove auto-correlations. But this is a special case.

A time series $$X_t$$ is stationary if the joint probability distribution of $${x_{t_1},x_{t_2},x_{t_3},x_{t_4},...,x_{t_n}}$$ and $${x_{t_1+c},x_{t_2+c},x_{t_3+c},x_{t_4+c},...,x_{t_n+c}}$$ is same for all $$n$$ and $$c$$. Intuitively, it means that if that the joint distribution depends upon the relative position of your time-steps, not the absolute position.

With some work, this definition translates to some important facts:

(1) Probability distribution of $$X_t$$ is same for all $$t$$.

(2) Expected value of $$X_t$$, $$E(X_t)$$, is independent of $$t$$: the mean does not change with time.

(3) In fact, the variance and all the higher moments do not change with time.

(4) The correlation between $$X_{t_1}$$ and $$X_{t_2}$$ is a function of $$t_1-t_2$$: it does not depend upon on $$t_1$$ and $$t_2$$, only on their relative positions.

Typically, a time series such as rainfall have seasonal fluctuations. Rainfall would be higher in Monsoon months than in other months. It means that the probability distribution of rainfall time series is changing with time (because the mean is changing with time). This is the reason behind removing the trends and seasonal components before using the classical time-series methods based on the assumption of stationarity.

Yes, autocorrelation implies temporal dependence. But both stationary and non-stationary time series have temporal dependence. The nature of dependence is different for stationary and non-stationary time series. In stationary time series, the autocorrelation depends on relative position of the time-steps only. In non-stationary time series, it can also depend upon the absolute value of time-steps.

Edit: (Based on a comment by Dilip Sarwate) The stationarity definition given above defines strict sense stationarity. However, for time-series analysis, what we typically requires is something called weak sense stationarity. A time series is stationary in weak sense if the expected value does not change with time and autocorrelation at time-steps $$t_1$$ and $$t_2$$ is a function of $$t_1-t_2$$ only. Weak sense stationarity does not satisfy the fact (1) mentioned above.

• In the time-series literature, "stationary" usually means what many statisticians would call"weakly stationary" or "wide-sense stationary" and it is not necessary for a weakly stationary process to satisfy your fact (1). For an example of a random process that is weakly stationary (in the sense of constant mean and autocorrelation dependent only on the difference $t_1-t_2$, and not the individual values of $t_1$ and $t_2$), but is not stationary (not even to order $1$),see this answer of mine. Commented Oct 17, 2021 at 1:57
• @DilipSarwate I just described what is stationarity means because OP seems to have a wrong notion of it. But you are right, strict sense stationarity is not usually required . I have made an edit to accommodate this. Commented Oct 17, 2021 at 2:01

Correlation does not imply causation, neither the other way around, nor when it regards time. The same applies to autocorrelation. Correlation(s) measure particular kinds of relationships between variables, while there may be many other non-linear relationships that are possible, so correlation and causation or dependence, while not unrelated, are not the same.

No, a stationary TS can still have a ACF showing a temporal dependency. Autocorrelation is the dependency of on point on the previous ones. This temporal dependency can be a drift or an oscillation and those parts will indeed be removed by making it stationary. But you can still have a dependency on the previous point, if your points are not independent one from another.

E.g. think of rain or temperature measured each hour. Lets make it stationary: we take out seasonal temperature variation and say, climate change leading to a slow temperature increase. Still, if last hour you had a certain temperature, it will be unlikely that you will have a completely different temperature now. So you still have temporal dependency, but it is dependent on only the last points.

You might want to look at ARMA models to understand that.