Dickey–Fuller test

In statistics, the Dickey–Fuller test tests the null hypothesis that a unit root is present in an autoregressive (AR) time series model. The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend-stationarity. The test is named after the statisticians David Dickey and Wayne Fuller, who developed it in 1979. A simple AR model is where is the variable of interest, is the time index, is a coefficient, and is the error term (assumed to be white noise). A unit root is present if . The model would be non-stationary in this case. The regression model can be written as where is the first difference operator and . This model can be estimated, and testing for a unit root is equivalent to testing . Since the test is done over the residual term rather than raw data, it is not possible to use standard t-distribution to provide critical values. Therefore, this statistic 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:
2. Test for a unit root with constant:
3. Test for a unit root with constant and deterministic time trend: 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, . The tests have low statistical power in that they often cannot distinguish between true unit-root processes () and near unit-root processes ( is close to zero). This is called the "near observation equivalence" problem. The intuition behind the test is as follows. If the series is stationary (or 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.