# Time Series Analysis and ARIMA

Why it is important to convert any time series problem data to stationary before applying ARIMA .can anyone please tell me intuition behind it ? ( I know if i dont convert it to stationary it will follow same trend but i want to understand how making stationary series is suitable for ARIMA )

Hi: All of the theory behind arima time series depends on weak stationarity ( there are a few different types of stationarity and weak is the least restrictive. I tend to get confused with the names so check what I say here ) which means that the first two moments of the response, the mean and the variance, are constant. So, if you have a trending series, then, by definition, the mean of the series isn't constant which implies that all of the arima theory cannot be applied. By differencing, the hope is that the mean becomes constant and, of course, you still need the constant variance also. Note that there are some cons to differencing that other time series fields attempt to deal with. The main negative is that, even if differencing does seem to induce stationarity , it can cause a loss of information.

i want to understand how making stationary series is suitable for ARIMA

Here are two somewhat simplified but intuitive explanations of what ARIMA needs stationary data. I will stick to the AR component of ARIMA, because it is easier to explain, but in more general terms, the same considerations are applicable to the MA component as well.

1 - First explanation - What does AR stand for? It stands for Auto-Regressive, meaning that you are performing a regression against the same variable, against passed values of itself, instead of against independent variables. Now remember one of the conditions for being able to perform linear regression: The variance has to remain the same all over the data set. In the case of an auto-regressive process, this means that the variance of the past have to remain the same in the future.

2 - Second explanation - Consider the most basic AR process possible, $$X_{t+1} = aX_t + \epsilon(t)$$: what will happen if $$a=2$$?

After only ten time steps, $$X_{t+10} = 1024 × X_t$$, and after 20 forecast steps $$X_{t+20} = 1048576 × X_t$$ (forecasting 20 steps ahead is not uncommon at all, the business use case I deal with most often involves forecasting 26 or 52 steps ahead). So you see the times series model growing very quickly out of bounds.

The only way to keep the model from growing out of bounds so quickly is to ensure that $$|a| < 1$$ in $$X_{t+1} = aX_t + \epsilon(t)$$. Stationarity requirements ensure that this conditions is satisfied before modeling.

The conditions become a little bit more complex, when multiple lags are included, but the overall intuition is still the same: Stationarity insures that the time series never goes out of bands.

Check the Z-Transform and the concept of Unit Circle to better understand the second explanation.

• You say "The variance has to remain the same all over the data set" .... more precisely "The variance of the error process has to remain the same all over the data set" . The assumptions are all about the error process not the observed Y. for example see autobox.com/cms/index.php/blog/entry/u-didnt-need-logs where a few (3) anomalies cleaned up the variance of the error process. – IrishStat Nov 10 at 14:16