Non-stationary time series and ADF Test

Non-stationary time series or Shocks

If time series is dynamic / not still and the observations will be visibly dynamic, then there is shocks present in the series and this will be non-stationary time series.

There are two types of non-stationary series or data:

1.    Temporary non-stationary time series: Where the shocks are temporary and die out.

2.    Permanent non-stationary time series: Where the shocks are permanent.

Partial Auto Correlation:

By the help of partial auto correlation shocks can be investigated but it cannot define whether it is a temporary or a permanent shock. Therefore, to identify the type of shocks present in our data, we need to apply Augmented dickey fuller unit root tests.

Augmented dickey fuller unit root tests (ADF Test):

It can investigate shocks as well and can also define whether it is a permanent or a temporary shock.

Mathematical expression that represents ADF model is mentioned below:

Et = This term is known as error term or unexplainable variable or noise. When taking a sample out of the whole population, this error term is not included because the sample data will then become unexplainable as ‘Et’ has the property of being unexplainable. If ‘Et” becomes explainable, this means that there is no error in data which is again not possible.

When ‘Et’ is somehow reduced in such a way in order to minimize the error by may be through the increase of sample size, then ‘Et’ tends to ( Et→x’ t ) and therefore,  x’ t  then becomes explainable.

x’ t   is also known as wide noise or aggressor.

To check whether ∆yt has shocks, that is, visible variation from normal trend is done through unit root test.

∆y (t-1) will be predicting that variation in ∆yt has visible shocks or not. If the t-statistic value of β is greater than 1.5 (t>1.5) then ∆y (t-1) predicts that shocks are there in ∆y.

If α=1 this means shocks are permanent shocks. This is the reason, it is known as unit root test, as this unit came from the value of α, which when equals to 1, indicates that shocks are permanent shocks. If the value of α<1 then the shocks are temporary shocks.