# Dickey-Fuller test

In statistics, Dickey-Fuller tests are the test class of unit root tests developed by D. Dickey and W. Fuller , which test the null hypothesis of a stochastic process with a unit root against the alternative of a process without a unit root. Such tests are used to determine whether there is an integrated process.

## Idea and implementation

For a stochastic process of form ${\ displaystyle X}$ ${\ displaystyle X_ {t} = \ alpha _ {0} + \ varphi X_ {t-1} + \ varepsilon _ {t}}$ with a white noise , the null hypothesis should be${\ displaystyle \ varepsilon}$ ${\ displaystyle H_ {0}: \ \ varphi = 1}$ (Random walk with drift)

against the alternative

${\ displaystyle H_ {1}: \ \ varphi <1}$ (AR (1) process)

getting tested. If you now sit , you can write: ${\ displaystyle \ delta: = \ varphi -1}$ ${\ displaystyle \ Delta X_ {t} = X_ {t} -X_ {t-1} = \ alpha _ {0} + (\ varphi -1) X_ {t-1} + \ varepsilon _ {t} = \ alpha _ {0} + \ delta X_ {t-1} + \ varepsilon _ {t}.}$ Null and alternative hypotheses are now:

${\ displaystyle H_ {0}: \ \ delta = 0, \ quad H_ {1}: \ \ delta <0.}$ You regress through and the constant . Depending on the estimation method ( least squares method , maximum likelihood estimation ), estimated values ​​are then obtained . Then a test statistic is created ${\ displaystyle \ Delta X_ {t}}$ ${\ displaystyle X_ {t-1}}$ ${\ displaystyle \ alpha _ {0}}$ ${\ displaystyle {\ hat {\ alpha}} _ {0}, {\ hat {\ delta}}}$ ${\ displaystyle \ tau: = {\ dfrac {\ hat {\ delta}} {\ sqrt {{\ widehat {\ operatorname {Var}}} ({\ hat {\ delta}})}}},}$ which, however, does not follow a distribution, but a distribution tabulated by Dickey and Fuller. Since the test is left-sided, the null hypothesis is rejected if the value of the test statistic is smaller than the threshold value corresponding to the selected significance level. ${\ displaystyle t}$ ## field of use

In the cointegration analysis of time series, for example that of GDP , inflation , interest, etc., it is checked whether stationary differences follow a common stochastic trend , i.e. whether there is a real connection. Since the regression of the time series that are integrated higher than grade 0, there is the possibility that the regression analysis yields a high degree of determination and significance of the regressors , although there is no connection between these time series apart from the simultaneous occurrence at time t , one runs the risk of To understand spurious correlations as true relationships. The ADF / DF test now checks whether the difference in a variable is stationary or not. A time series is stationary if it has a constant expected value and a variance that does not depend on the time t ; it is also called integrated of order zero. If a time series is non-stationary, the question arises, what order instationarity present. If its first difference is stationary, it has the property of first-order integration . So there is a root of unity. If the first difference is not stationary, the second differences are tested with analogous inference.

As part of the static test for cointegration according to Engle and Granger, the ADF test can also test for the existence of a common stochastic trend. This is the long-term growth path of the ranks. In the long term, the variables cannot move independently of one another. If one variable is changed, for example by an external shock, the others adjust over time in order to bring the system back into equilibrium. For this purpose, the ADF test is applied to the residuals of a regression of the time series. So it checks whether the residuals are stationary.

## DF test

The Dickey-Fuller test tests the equation of the DF test in the case without a deterministic trend and without a constant

${\ displaystyle \ Delta y_ {t} = (\ rho -1) y_ {t-1} + u_ {t} = \ delta y_ {t-1} + u_ {t}.}$ There are three cases:

1. Test on Random Walk :${\ displaystyle \ Delta y_ {t} = \ delta y_ {t-1} + u_ {t}}$ 2. Test on random walk with drift ${\ displaystyle \ Delta y_ {t} = a_ {0} + \ delta y_ {t-1} + u_ {t}}$ 3. Test on random walk with drift and deterministic trend ${\ displaystyle \ Delta y_ {t} = a_ {0} + a_ {1} t + \ delta y_ {t-1} + u_ {t}}$ The pair of hypotheses is:

${\ displaystyle H_ {0}: \, \ rho = 1}$ , d. i.e., the AR part has a unit root
${\ displaystyle H_ {1}: \, - 1 <\ rho <1}$ The extended Dickey-Fuller test ( English augmented Dickey-Fuller test , or ADF test ) generalizes the test equation of the DF test in the case with a deterministic trend

${\ displaystyle \ Delta y_ {t} = \ alpha + \ beta t + (\ rho -1) y_ {t-1} + \ theta _ {1} \ Delta y_ {t-1} + ... + \ theta _ {k} \ Delta y_ {tk} + u_ {t}}$ ,

with k, so that the empirical residuals have white noise .

The pair of hypotheses is:

${\ displaystyle H_ {0}: \, \ rho = 1}$ , d. that is, the AR part has a unit root, and so the variable is not stationary
${\ displaystyle H_ {1}: \, - 1 <\ rho <1}$ There is no stochastic instationarity, but possibly deterministic, then one speaks of a trend-stationary time series.

## Problems

If the data generating process is trend stationary, but the unit root test is incorrectly carried out with the model without a trend variable, the tests have an asymptotically tending towards zero power , because the null hypothesis of the random walk is then incorrectly rejected too rarely or never.

## Individual evidence

1. Peter Hackl : Introduction to Econometrics. 2nd updated edition, Pearson Deutschland GmbH, 2008., ISBN 978-3-86894-156-2 , p. 257.

## literature

• G. Elliott, TJ Rothenberg & JH Stock: Efficient Tests for an Autoregressive Unit Root , Econometrica, 1996, Vol. 64, No. 4., pp. 813-836. doi : 10.3386 / t0130 JSTOR 2171846
• WH Greene: Econometric Analysis, Fifth Edition , 2003, Prentice Hall, New Jersey.
• Said E. and David A. Dickey: Testing for Unit Roots in Autoregressive Moving Average Models of Unknown Order , Biometrika , 1984, 71, pp. 599-607. doi : 10.1093 / biomet / 71.3.599 JSTOR 2336570
• Dickey, DA and WA Fuller: Distribution of the Estimators for Autoregressive Time Series with a Unit Root , Journal of the American Statistical Association, 1979, 74, pp. 427-431. doi : 10.1080 / 01621459.1979.10482531 JSTOR 2286348