Augmented Dickey-Fuller test

In statistics and econometrics, an augmented Dickey-Fuller test (ADF) is a test for a unit root in a time series sample. It is an augmented version of the Dickey-Fuller test to accommodate some forms of serial correlation. "Definition of Augmented Dickey-Fuller Test" From Econterms Definition: An augmented Dickey-Fuller test is a test for a unit root in a time series sample. An augmented Dickey-Fuller test is a version of the Dickey-Fuller test for a larger and more complicated set of time series models. The augmented Dickey-Fuller (ADF) statistic, used in the test, is a negative number. The more negative it is, the stronger the rejection of the hypothesis that there is a unit root at some level of confidence. In one example, with three lags, a value of -3.17 constituted rejection at the p-value of .10. (Econterms)

Testing Procedure

The testing procedure for the ADF test is the same as for the Dickey-Fuller test but it is applied to the model

<math>\Delta y_t = \mu + \beta t + \gamma y_{t-1} + \gamma_1 \Delta y_{t-1} + ... + \gamma_p \Delta y_{t-p} + \varepsilon_t, </math>

where <math>\mu</math> is a constant, <math>\beta</math> the coefficient on a time trend and <math>p</math> the lag order of the autoregressive process. Imposing the contraints <math>\mu = 0</math> and <math>\beta = 0</math> corresponds to modelling a random walk and using the constraint <math>\beta = 0</math> corresponds to modelling a random walk with a drift. By including lags of the order <math>p</math> the ADF formulation allows for higher-order autoregressive processes. This means that the lag length <math>p</math> has to be determined when applying the test. One possible approach is to test down from high orders and examine the t-values on coefficients. An alternative approach is to examine information criteria such as the Akaike information criterion, Bayesian information criterion or the Hannon Quinn criterion. The unit root test is then carried out under the null hypothesis <math>\gamma = 1</math> against the alternative hypothesis of <math>\gamma < 1.</math> Once a value for the test statistic

<math>DF_\tau = \frac{(\hat{\gamma} - 1)}{SE(\hat{\gamma})}</math>

is computed it can be compared to the relevant critical value for the Dickey-Fuller Test. If the test statistic is less than the critical value then the null hypothesis of <math>\gamma = 1</math> is rejected and no unit root is present.

Examples

A sample of 50 observations and a model which includes a constant and a time trend yields the <math>DF_\tau</math> statistic of -4.57. This is greater in absolute value than the tabulated critical value of -3.50 at the 95 per cent level the null hypothesis of a unit root cannot be rejected.

Alternatives

There are alternative unit root tests such as the Phillips-Perron test or the ADF-GLS procedure developed by Elliot, Rothenberg and Stock (1996).

References

• Elliott, G., Rothenberg, T. J. & J.H. Stock (1996) 'Efficient Tests for an Autoregressive Unit Root,' Econometrica, Vol. 64, No. 4., pp. 813-836. Stable URL
• Greene, W. H. (2003) Econometric Analysis, Fifth Edition Prentice Hall: New Jersey.
• Said E. and David A. Dickey (1984), 'Testing for Unit Roots in Autoregressive Moving Average Models of Unknown Order', Biometrika, 71, p 599–607.

See also

• Phillips-Perron test
• Unit root