In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
and
econometrics
Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of ...
, the ADF-GLS test (or DF-GLS test) is a test for a
unit root
In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if 1 is ...
in an economic
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
sample. It was developed by Elliott, Rothenberg and Stock (ERS) in 1992 as a modification of the
augmented Dickey–Fuller test (ADF).
A unit root test determines whether a time series variable is non-stationary using an autoregressive model. For series featuring deterministic components in the form of a constant or a linear trend then ERS developed an asymptotically point optimal test to detect a unit root. This testing procedure dominates other existing unit root tests in terms of power. It locally de-trends (de-means) data series to efficiently estimate the deterministic parameters of the series, and use the transformed data to perform a usual ADF unit root test. This procedure helps to remove the means and linear trends for series that are not far from the non-stationary region.
Efficient Tests for an Autoregressive Unit Root, Elliott, Rothenberg and Stock, Econometrica Vol. 64, No. 4, pp. 813-836, Jul., 1996
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Explanation
Consider a simple time series model with where is the deterministic part and is the stochastic part of . When the true value of is close to 1, estimation of the model, i.e. will pose efficiency problems because the will be close to nonstationary. In this setting, testing for the stationarity features of the given times series will also be subject to general statistical problems. To overcome such problems ERS suggested to locally difference the time series.
Consider the case where closeness to 1 for the autoregressive parameter is modelled as where is the number of observations. Now consider filtering the series using with being a standard lag operator, i.e. . Working with would result in power gain, as ERS show, when testing the stationarity features of using the augmented Dickey-Fuller test. This is a point optimal test for which is set in such a way that the test would have a 50 percent power when the alternative is characterized by for . Depending on the specification of , will take different values.
References
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A Primer on Unit Root Tests, P.C.B. Phillips and Z. Xiao
Time series statistical tests