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 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. [ [http://econterms.com/glossary.cgi?action=++Search++&query=augmented+dickey-fuller 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

:Delta y_t = alpha + eta t + gamma y_{t-1} + delta_1 Delta y_{t-1} + cdots + delta_p Delta y_{t-p} + varepsilon_t,

where alpha is a constant, eta the coefficient on a time trend and p the lag order of the autoregressive process. Imposing the contraints alpha = 0 and eta = 0 corresponds to modelling a random walk and using the constraint eta = 0 corresponds to modelling a random walk with a drift.

By including lags of the order p the ADF formulation allows for higher-order autoregressive processes. This means that the lag length p 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 gamma = 0 against the alternative hypothesis of gamma < 0. Once a value for the test statistic

:DF_ au = frac{hat{gamma{SE(hat{gamma})}

is computed it can be compared to the relevant critical value for the Dickey-Fuller Test. If the test statistic is greater (in absolute value) than the critical value, then the null hypothesis of gamma = 0 is rejected and no unit root is present.

Intuition

The intuition behind the test is that if the series is integrated then the lagged level of the series ( y_{t-1}) will provide no relevant information in predicting the change in y_{t} besides the one obtained in the lagged changes ( Delta y_{t-k} ). In that case the gamma = 0 null hypothesis is not rejected.

Examples

A model that includes a constant and a time trend is estimated using sample of 50 observations and yields the DF_ au statistic of -4.57. This is more negative than the tabulated critical value of -3.50, so at the 95 per cent level the null hypothesis of a unit root will 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. [http://links.jstor.org/sici?sici=0012-9682%28199607%2964%3A4%3C813%3AETFAAU%3E2.0.CO%3B2-8 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.

ee also

* Phillips-Perron test
* Unit root


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