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Unit Root

A unit root is a root of the autoregressive polynomial, or characteristic polynomial, of a time series model that lies on the unit circle. Its presence generally produces a nonstationary time series. The simplest example is the random walk

X_t=X_(t-1)+epsilon_t,

whose autoregressive polynomial 1-z has the unit root z=1. Applying the difference operator gives

DeltaX_t=X_t-X_(t-1)=epsilon_t,

which is stationary when the innovations are stationary.

The augmented Dickey-Fuller test is commonly used to test the null hypothesis that a time series has a unit root.

See also

Augmented Dickey-Fuller Test, Autoregressive Model, Difference Operator, Innovation, Nonstationary Time Series, Stationary Time Series

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References

Dickey, D. A. and Fuller, W. A. "Distribution of the Estimators for Autoregressive Time Series with a Unit Root." J. Amer. Statist. Assoc. 74, 427-431, 1979. https://doi.org/10.1080/01621459.1979.10482531.Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

Cite this as:

Weisstein, Eric W. "Unit Root." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UnitRoot.html

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