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
Famous quotes containing the words estimation, unit, root and/or present:
“... it would be impossible for women to stand in higher estimation than they do here. The deference that is paid to them at all times and in all places has often occasioned me as much surprise as pleasure.”
—Frances Wright (1795–1852)
“During the Suffragette revolt of 1913 I ... [urged] that what was needed was not the vote, but a constitutional amendment enacting that all representative bodies shall consist of women and men in equal numbers, whether elected or nominated or coopted or registered or picked up in the street like a coroner’s jury. In the case of elected bodies the only way of effecting this is by the Coupled Vote. The representative unit must not be a man or a woman but a man and a woman.”
—George Bernard Shaw (1856–1950)
“To those who despair of everything reason cannot provide a faith, but only passion, and in this case it must be the same passion that lay at the root of the despair, namely humiliation and hatred.”
—Albert Camus (1913–1960)
“The past is of no importance. The present is of no importance. It is with the future that we have to deal. For the past is what man should not have been. The present is what man ought not to be. The future is what artists are.
The facts: nothing matters but the facts: worship of the facts leads to everything, to happiness first of all and then to wealth.”
—Edmond De Goncourt (1822–1896)