AR Mean reversion level


Looking at the covariance stationary rules, I do not understand how a time series would have an mean reverting level if absolute value of the lag coefficienct is less than 1.

Even if the abs value is less than 1, the line would still be going in one direction (away from the mean I suppose) how does this work otherwise in general?


The best way to see this is by modeling it in Excel.

Nope, it will eventually converge to the mean reverting level (as long as the coefficient is less than 1 in absolute value). As S2000 says, model it in excel.

Apply this simple model:

X(t) = 0.1 + 0.8*X(t-1)

If you apply the mean reverting formula 0.1 / (1 - 0.8), you will get 0.5.

Suppose variable X has been shocked and the current value is 5. So, calculate X(t+1) using X(t) = 5

You will get X(t+1) = 0.1 + 0.8*5 = 4.1

The next would be X(t+2) = 0.1 + 0.8*4.1 = 3.38

…and so on until you arrive at 0.5

Note that you can change the initial value as you wish. Shock the variable to -200, yeah negative, and you will still arrive to 0.5 positive.

Try this in excel and set a control cell for the 0.8 value. Change it for 1.2 and you will see a divergent time-series!

Hope this helps