Dickey-Fuller test

In statistics, the Dickey-Fuller test tests whether a unit root is present in an autoregressive model. It is named after the statisticians D. A. Dickey and W. A. Fuller, who developed the test in the 1970s.

Explanation

A simple AR(1) model is

: $y\_\{t\}=\; ho\; y\_\{t-1\}+u\_\{t\},$

where "y"_{"t" is the variable of interest, "t" is the time index, "ρ" is a coefficient, and "u""t" is the error term. A unit root is present if |"ρ"| = 1. The model would be non-stationary in this case. Naturally it would be even more non-stationary if |"ρ"| ≥ 1.}

The regression model can be written as

: $Delta\; y\_\{t\}=(\; ho-1)y\_\{t-1\}+u\_\{t\}=delta\; y\_\{t-1\}+u\_\{t\},$

where Δ is the first difference operator. This model can be estimated and testing for a unit root is equivalent to testing "δ" = 0 (where "δ" = "ρ" − 1). Since the test is done over the residual term rather than raw data, it is not possible to use standard t-distribution to as critical values. Therefore this statistic "τ" has a specific distribution simply known as the Dickey-Fuller table.

There are three main versions of the test:

1. Test for a unit root:

:: $Delta\; y\_\{t\}=delta\; y\_\{t-1\}+u\_\{t\}$

2. Test for a unit root with drift:

:: $Delta\; y\_\{t\}=a\_0+delta\; y\_\{t-1\}+u\_\{t\}$

3. Test for a unit root with drift and deterministic time trend:

: $Delta\; y\_\{t\}=a\_0+a\_1t+delta\; y\_\{t-1\}+u\_\{t\}$

Each version of the test has its own critical value which depends on the size of the sample. In each case, the null hypothesis is that there is a unit root, "δ" = 0. The tests have low Statistical power in that they often cannot distinguish between true unit-root processes ("δ" = 0)and near unit-root processes ("δ" is close to zero). This is called the "near observation equivalence" problem.

The intuition behind the test is as follows. If the series "y" is (trend-)stationary, then it has a tendency to return to a constant (or deterministically trending) mean. Therefore large values will tend to be followed by smaller values (negative changes), and small values by larger values (positive changes). Accordingly, the level of the series will be a significant predictor of next period's change, and will have a negative coefficient. If, on the other hand, the series is integrated, then positive changes and negative changes will occur with probabilities that do not depend on the current level of the series; in a random walk, where you are now does not affect which way you will go next.

There is also an extension called the augmented Dickey-Fuller test (ADF), which removes all the structural effects (autocorrelation) in the time series and then tests using the same procedure.

References

Dickey, D.A. and W.A. Fuller (1979), “Distribution of the Estimators for Autoregressive Time Series with a Unit Root,” "Journal of the American Statistical Association", 74, p. 427–431.

ee also

* Augmented Dickey-Fuller test
* Phillips-Perron test
* Unit root