ADF-GLS test
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). 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. Explanation. Consider a simple time series model formula_1 with formula_2 where formula_3 is the deterministic part and formula_4 is the stochastic part of formula_5. When the true value of formula_6 is close to 1, estimation of the model, i.e. formula_3 will pose  efficiency problems because the formula_5 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 formula_9 where formula_10 is the number of observations. Now consider filtering the series using formula_11 with formula_12 being a standard lag operator, i.e. formula_13. Working with formula_14 would result in power gain, as ERS show, when testing the stationarity features of formula_15 using  the augmented Dickey-Fuller test. This is a point optimal test for which formula_16 is set in such a way that the test would have a 50 percent power when the alternative is characterized by formula_17 for formula_18. Depending on the specification of formula_19, formula_16 will take different values. References. A Primer on Unit Root Tests, P.C.B. Phillips and Z. Xiao