Unit Root Test of Nonstationarity

Hello there! 

An AR(1) model equation should be Xt=b0 + b1 X(t-1) + Et

For an AR(1) to have an adequate forecasting utility, b1 < 1

A unit root means that the coefficient (b1) of the AR(1) model is equal to 1

In other words, the slope of the coefficient is one (b1=1). Picture a perfectly straight line in a 45 degree angle in an X and Y graph. 

When there is a “unit root” b1=1 and a random walk occurs.

There are two types of random walks: Without a drift and with a drift

No drift – b0=0 and b1=1 (no intercept) 

Xt = b0 + b1 X(t-1) + Et
Xt = 0 + (1) X(t-1) +Et
Xt = X(t-1) + Et

With a drift b0>0 and b1=1(there is an intercept)

Xt = b0 + b1 X(t-1) + Et
Xt = b0 + (1) X(t-1) + Et
Xt = b0 + X(t-1) + Et

Why b1=1 – unit roots a problem? Because the mean reverting level is undefined

Mean Reverting Level = b0/(1-b1), if b1=1, then the Mean Reverting Level is b0/0 which is undefined. 
When the Mean Reverting Level is undefined, the time series is non-stationary. 

I hope this helps! 

• When b1 is less than 1, the impact of past values (X(t-1)) on the current value (Xt) gradually weakens over time. This means the series eventually stabilizes around a certain mean, making it predictable for forecasting.
• If b1 is equal to 1 (unit root), the past value has a full impact on the current value, and this effect continues indefinitely. The series keeps moving in the direction of the previous change, making it difficult to predict future values.

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