Dickey–Fuller test

As Dickey - Fuller test is referred to in the statistics developed by D. Dickey and W. Fuller in the 1970s, unit root tests, which test the null hypothesis of a stochastic process with unit root against the alternative of a unit root process without. Such tests are designed to determine whether an integrated process exists.

Idea and implementation

For a stochastic process of the form

With a white noise to the null hypothesis

Against the alternative

Be tested. If, now, we can write:

Null and alternative hypothesis are now:

We regressed by now and the constant. Depending on the estimation method (OLS, ML ) is then obtained estimates. Then, it forms a test statistic

However, the no - distribution, but one of Dickey and Fuller tabulated distribution follows. Because the test is on the left side, the null hypothesis is rejected if the value of the test statistic is less than the threshold value corresponding to the selected level of significance.

Field of application

In the cointegration analysis of time series, such as GDP, inflation, interest, etc., will determine whether inpatient differences follow a common stochastic trend, ie a real relationship exists. Since by regression of the time series, which are higher than of degree 0 integrated, there is a possibility that the regression analysis shows a high coefficient of determination and significance of the regressors, although except the simultaneous occurrence at time t no correlation between these time series is, you run the risk interpreted spurious correlations as true correlations. The ADF / DF test now checks whether the difference of a variable is stationary or not. A time series is stationary, if it has a constant mean and a variance not dependent on the time t, it will also be integrated zero-order named. If a time series is non-stationary, the question arises, what order there is nonstationarity. Is their first difference stationary, it has the property of the integration of the first order. So there is a unit root exists. If the first difference is not stationary, one tests the second differences with analogue reasoning.

The ADF test can test for cointegration by Engle and Granger in the context of the static tests on existence of a common stochastic trend. This is the long-term growth path of the rows. Long term, the variables can not move independently. If a variable is changed, for example by an external shock, then the other adapt over time to bring the system back into balance. For this, the ADF test is applied to the estimated residuals regression of the time series. So he checks whether the residuals are stationary.

DF test

There are three cases:

The pair of hypotheses is:

ADF test

With k, so that the empirical residuals rushing white.

The pair of hypotheses is:

Problems

If the data-generating process trend stationary, but you falsely leads the unit root test with the model through without trend variable, the tests have asymptotically to zero continuous power, because the null hypothesis of random walk is then falsely seldom or never refused.

Alternative Approaches

• Phillips -Perron test ( 1988)
• KPSS - Test ( 1992) - Null hypothesis: stationarity
• HEGY Test ( 1990)
• ADF - GLS method ( 1996)