ADF-GLS test explained

In statistics and econometrics, the ADF-GLS test (or DF-GLS test) is a test for a unit root in an economic time series sample. It was developed by Elliott, Rothenberg and Stock (ERS) in 1992 as a modification of the augmented Dickey–Fuller test (ADF).^[1]

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.^[2]

Explanation

Consider a simple time series model

y_t=d_t+u_t

with

u_t=\rhou_t-1+e_t

where

d_t

is the deterministic part and

u_t

is the stochastic part of

y_t

. When the true value of

\rho

is close to 1, estimation of the model, i.e.

d_t

will pose efficiency problems because the

y_t

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

\rho=1-	c
	T

where

is the number of observations. Now consider filtering the series using

1-	\bar{c

}L \, with

being a standard lag operator, i.e.

\bar{y}_t=y_{t-(\bar{c}/T)y}_t-1

. Working with

\bar{y}_t

would result in power gain, as ERS show, when testing the stationarity features of

y_t

using the augmented Dickey-Fuller test. This is a point optimal test for which

\bar{c}

is set in such a way that the test would have a 50 percent power when the alternative is characterized by

\rho=1-c/T

for

c=\bar{c}

. Depending on the specification of

d_t

\bar{c}

will take different values.

References

A Primer on Unit Root Tests, P.C.B. Phillips and Z. Xiao

Notes and References

Cheung . Yin-Wong . Kon S. Lai . Practitioner's Corner: Lag Order and Critical Values of a Modified Dickey-Fuller Test . Oxford Bulletin of Economics and Statistics . 57 . 3 . 411–419 . 1995 . 10.1111/j.1468-0084.1995.mp57003008.x.
http://ideas.repec.org/a/ecm/emetrp/v64y1996i4p813-36.html Efficient Tests for an Autoregressive Unit Root, Elliott, Rothenberg and Stock, Econometrica Vol. 64, No. 4, pp. 813-836, Jul., 1996