# Standardizing vs differencing to remove non stationarity

To remove non stationarity in a time series, we can standardize the time series by subtracting the mean and dividing by the standard deviation. We can also keep differencing the time series until the correlogram indicates that the non stationarity has been removed.

Intuitively, I am not able to follow how these two methods that are so completely different are able to produce the same end result. Can these methods be used interchangeably ? Which method to use under which situation ? The differencing method seems better in that we can keep taking differences until the non stationarity is removed, but for standardization , what to do if standardizing does not remove the non stationarity?

• Standardizing a time series will not render the data stationary. Removing a time-varying mean or time-varying standard deviation may achieve it if that's what makes the data non-sationary, but the common approach of standardization will not achieve it, as shown in one of the answers below. – javlacalle Jan 3 '15 at 18:40
• Standardization merely changes the units of measurement of the values. It cannot affect any intrinsic properties of the series. – whuber Jan 3 '15 at 18:41

To remove non stationarity in a time series, we can standardize the time series by subtracting the mean and dividing by the standard deviation.

This is not true. Where did you get this from?

Consider this: $y_t=t+\varepsilon_t$, where $\varepsilon_t\sim\mathcal{N}(0,1)$ and $t\in [1..T]$.

$$E[y_t]=\sum_ty_t=\frac{(1+T)T}{2T}=\frac{1+T}{2}$$ $$Var[y_t]=Var[t]+1=\frac{T^2+11}{12}$$

Transform the series: $$z_t=\frac{y_t-E[t]}{\sqrt{Var[y_t]}} =\frac{t+\varepsilon_t-\frac{1+T}{2}}{\sqrt{\frac{T^2+11}{12}}} =-\frac{\frac{1+T}{2}}{\sqrt{\frac{T^2+11}{12}}}+\frac{1}{\sqrt{\frac{T^2+11}{12}}}t +\frac{1}{\sqrt{\frac{T^2+11}{12}}}\varepsilon_t$$

The transformed series $z_t$ is still non-stationary series.

• I got that standardizing removes periodicity from this lecture at 52 min....bullet point#2 on whiteboard.m.youtube.com/watch?v=24aUI4RUElo – Victor Jan 3 '15 at 19:05
• @Victor At minute 24 the video defines standardization as the removal of seasonal means. If the seasonal pattern is deterministic, removing the seasonal means (or including seasonal dummies in the model) might be a better alternative to taking seasonal differences in the data. – javlacalle Jan 3 '15 at 23:10