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augmented Dickey-Fuller test: Information from Answers.com

  • ️Wed Jul 01 2015

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.

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

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

where μ is a constant, β the coefficient on a time trend and p the lag order of the autoregressive process. Imposing the contraints μ = 0 and β = 0 corresponds to modelling a random walk and using the constraint β = 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 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 γ = 1 against the alternative hypothesis of γ < 1. Once a value for the test statistic

DF_\tau = \frac{(\hat{\gamma} - 1)}{SE(\hat{\gamma})}


is computed it can be compared to the relvant critical value for the Dickey-Fuller Test. If the test statistic is less than the critical value then the null hypothesis of γ = 1 is rejected and no unit root is present.

Examples

A sample of 50 observations and a model which includes a contant and a time trend yields the DFτ statistic of -2.57. This is greater 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).

Reference

  • 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

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