Estimation When A Unit Root May Be Present
Often, ordinary least squares (OLS) is used to estimate the slope coefficients of the autoregressive model. Use of OLS relies on the stochastic process being stationary. When the stochastic process is non-stationary, the use of OLS can produce invalid estimates. Granger and Newbold called such estimates 'spurious regression' results: high R2 values and high t-ratios yielding results with no economic meaning.
To estimate the slope coefficients, one should first conduct a unit root test, whose null hypothesis is that a unit root is present. If that hypothesis is rejected, one can use OLS. However, if the presence of a unit root is not rejected, then one should apply the difference operator to the series. If another unit root test shows the differenced time series to be stationary, OLS can then be applied to this series to estimate the slope coefficients.
For example, in the AR(1) case, is stationary.
In the AR(2) case, can be written as 
where L is a lag operator that decreases the time index of a variable by one period: . If, the model has a unit root and we can define ; then
is stationary if . OLS can be used to estimate the slope coefficient, .
If the process has multiple unit roots, the difference operator can be applied multiple times.
Read more about this topic: Unit Root
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