Time Series > Unit Root

## What is “Unit Root”?

A **unit root** (also called a unit root process or a difference stationary process) is a stochastic trend in a time series, sometimes called a “random walk with drift”; If a time series has a unit root, it shows a systematic pattern that is unpredictable.

## The Mathematics Behind Unit Roots

The reason why it’s called a unit root is because of the mathematics behind the process. At a basic level, a process can be written as a series of monomials (expressions with a single term). Each monomial corresponds to a root. If one of these roots is equal to 1, then that’s a unit root.

If you’re analyzing time series, these roots can cause your analysis to have serious issues like:

**Spurious regressions**: you could get high r-squared values even if the data is uncorrelated.**Errant behavior**due to assumptions for analysis not being valid. For example, t-ratios will not follow a t-distribution.

## What is a Unit Root Test?

These are tests for stationarity in a time series. A time series has stationarity if a shift in time doesn’t cause a change in the shape of the distribution; unit roots are one cause for non-stationarity.

These tests are known for having low statistical power. Many tests exist, in part, because none stand out as having the *most* power. Tests include:

- The
**Dickey Fuller Test**(sometimes called a Dickey Pantula test), which is based on linear regression. Serial correlation can be an issue, in which case the**Augmented Dickey-Fuller (ADF) test**can be used. The ADF handles bigger, more complex models. It does have the downside of a fairly high Type I error rate. - The
**Elliott–Rothenberg–Stock Test**, which has two subtypes:- The
*P-test*takes the error term’s serial correlation into account, - The
*DF-GLS test*can be applied to detrended data without intercept.

- The
- The
**Schmidt–Phillips Test**includes the coefficients of the deterministic variables in the null and alternate hypotheses. Subtypes are the*rho-test*and the*tau-test*. - The
**Phillips–Perron (PP) Test**is a modification of the Dickey Fuller test, and corrects for autocorrelation and heteroscedasticity in the errors. - The
**Zivot-Andrews test**allows a break at an unknown point in the intercept or linear trend.

## References

Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences, Wiley.

Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics, Cambridge University Press.

Vogt, W.P. (2005). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences. SAGE.