DF-GLS Detrending And Power Improvements

Why Change the Detrending Step?

A Dickey–Fuller test asks whether a series has a unit root, but its power depends on how deterministic components are handled. Removing a constant or a linear trend by ordinary least squares before testing can use information inefficiently when the process is highly persistent. The Elliott–Rothenberg–Stock procedure, usually called DF-GLS, replaces that ordinary detrending step with a generalized least-squares (GLS) transformation designed for alternatives close to a unit root.

The procedure still ends with a Dickey–Fuller-type regression. Its distinctive feature is that the series is detrended first using a quasi-difference, and the test is then applied to the detrended series. This tutorial develops that construction and explains the local-power rationale; it does not make the resulting statistic an ordinary \(t\)-statistic with standard critical values.

Definition: In DF-GLS, GLS detrending means transforming the observed series and its deterministic regressors by the same quasi-difference, regressing the transformed series on the transformed deterministic regressors, and subtracting the fitted deterministic component from the original series. The quasi-difference uses \(\bar\alpha\), a value close to one chosen for the deterministic specification and sample size.

The GLS Quasi-Difference

Write the deterministic component as \(x_t'\delta\), where \(x_t\) is a vector of known regressors and \(\delta\) is an unknown coefficient vector. For a constant-only specification, \(x_t=1\). For a constant and linear trend, \(x_t=(1,t)'\). Given a prespecified \(\bar\alpha\), transform both the response and the regressors:

y_1^\alpha=y_1,\qquad y_t^\alpha=y_t-\bar\alpha y_{t-1}\quad (t\geq 2),

x_1^\alpha=x_1,\qquad x_t^\alpha=x_t-\bar\alpha x_{t-1}\quad (t\geq 2).

The first observation is left unfiltered. Estimate \(\delta\) by least squares in the transformed regression \(y_t^\alpha=(x_t^\alpha)'\delta+\text{error}\), and define the detrended series as \(\tilde y_t=y_t-x_t'\tilde\delta\). The regression is least squares in transformed data; this is GLS for the corresponding level regression under the specified covariance structure. The parameter \(\bar\alpha\) is not estimated from the observed series in this step.

Worked Example: Transforming and Detrending a Hypothetical Series

Suppose a hypothetical series has ten observations \(2,3,2,4,5,4,6,7,6,8\). To make the arithmetic transparent, use \(\bar\alpha=0.3\) and a constant-only transformed regression. This value is for illustrating the calculation, not the sample-size calibration used in an actual DF-GLS test.

The first transformed response is \(y_1^\alpha=2\). For \(t\geq2\), calculate each value as \(y_t-0.3y_{t-1}\):

(y_1^\alpha,\ldots,y_{10}^\alpha) =(2,\ 2.4,\ 1.1,\ 3.4,\ 3.8,\ 2.5,\ 4.8,\ 5.2,\ 3.9,\ 6.2).

For a constant regressor, \(x_1^\alpha=1\) and \(x_t^\alpha=1-0.3=0.7\) for \(t\geq2\). The sum of the nine transformed responses from \(t=2\) to \(10\) is \(33.3\). Therefore the transformed-regression estimate of the constant is

\tilde\delta =\frac{\sum_{t=1}^{10}x_t^\alpha y_t^\alpha} {\sum_{t=1}^{10}(x_t^\alpha)^2} =\frac{2+0.7(33.3)}{1+9(0.7)^2} =\frac{25.31}{5.41} \approx 4.678.

The corresponding first two detrended observations are \(\tilde y_1=2-4.678\approx-2.678\) and \(\tilde y_2=3-4.678\approx-1.678\). This example shows why the first transformed observation and the later quasi-differences must be handled separately.

From Detrending to the DF-GLS Test

After constructing \(\tilde y_t\), estimate a Dickey–Fuller regression on that series. If serial correlation requires augmentation, include lagged first differences, as in the augmented Dickey–Fuller setup. With \(k\) augmentation lags, the test regression is

\Delta\tilde y_t =\gamma\tilde y_{t-1} +\sum_{i=1}^{k}\phi_i\Delta\tilde y_{t-i} +\varepsilon_t.

The unit-root null is \(H_0:\gamma=0\); the left-tailed alternative is \(H_a:\gamma<0\). The DF-GLS statistic is the \(t\)-statistic for \(\hat\gamma\) in this regression. Under the unit-root null its distribution is nonstandard. Use DF-GLS critical values for the chosen deterministic specification and lag procedure, not the usual Student \(t\) critical values and not automatically the critical values for an ordinary ADF regression.

Worked Example: Computing the Test-Regression Coefficient

Suppose a hypothetical DF-GLS detrending step has produced six usable lagged values \(x_t=\tilde y_{t-1}\), with corresponding changes \(d_t=\Delta\tilde y_t\), as follows:

(x_1,\ldots,x_6)=(-2,-1,1,2,0,3),\qquad (d_1,\ldots,d_6)=(-0.40,-0.05,0.05,0.30,0.10,0.20).

For illustration, take \(k=0\), so the regression contains only the lagged detrended level. The least-squares estimate is

\hat\gamma =\frac{\sum_{t=1}^{6}x_td_t}{\sum_{t=1}^{6}x_t^2} =\frac{0.80+0.05+0.05+0.60+0+0.60}{4+1+1+4+0+9} =\frac{2.10}{19} \approx0.1105.

The fitted residual sum of squares is approximately \(0.07289\). With six observations and one estimated slope, the residual variance estimate is \(0.07289/(6-1)\approx0.01458\), giving \(SE(\hat\gamma)=\sqrt{0.01458/19}\approx0.02770\). Thus the coefficient \(t\)-statistic is \(0.1105/0.02770\approx3.989\). Its positive sign is opposite the left-tailed alternative, so these hypothetical data do not provide evidence against the unit-root null. In an actual application, the decision would use DF-GLS critical values for the relevant specification.

Why the Transformation Can Improve Power

The power comparison is most clearly understood under a local-to-unity alternative. Let the autoregressive coefficient be \(\rho=1+c/T\), where \(T\) is the sample size and \(c<0\) is fixed. As \(T\) grows, \(\rho\) approaches one, so these alternatives describe very persistent but stationary processes. Conventional Dickey–Fuller tests can have difficulty distinguishing them from a unit root.

For the conventional ERS calibrations, \(\bar\alpha=1+\bar c/T\), with \(\bar c=-7\) for a constant-only specification and \(\bar c=-13.5\) for a constant-plus-trend specification. These are specification-dependent choices, not universal estimates of the process's persistence. The finite-sample transformation applies the same quasi-difference to the observed series and its deterministic regressors.

Worked Example: Comparing the Local Alternative and the Filter

Consider a hypothetical sample of \(T=1000\) observations under \(\rho=1-8/1000=0.992\), so \(c=-8\). For the constant-only ERS calibration, \(\bar c=-7\), hence \(\bar\alpha=1-7/1000=0.993\). If \(y_t=\rho y_{t-1}+u_t\), then for \(t\geq2\),

y_t^\alpha=y_t-\bar\alpha y_{t-1} =(\rho-\bar\alpha)y_{t-1}+u_t =\frac{-8-(-7)}{1000}y_{t-1}+u_t =-0.001y_{t-1}+u_t.

The filtered series is therefore close to a difference, while the remaining persistence term reflects the small gap between the local alternative and the calibration. GLS detrending uses this local-to-unity structure when estimating the deterministic component. In contrast, ordinary least-squares detrending does not use the same quasi-difference weighting. Under the local alternatives for which the procedure is designed, this can preserve more information about the near-unit-root behavior and improve asymptotic local power.

The power result is comparative, not absolute: it concerns rejection probabilities under specified local alternatives at a controlled test size. It does not mean that DF-GLS has greater power for every data-generating process, sample size, lag choice, or finite-sample realization. In particular, poor lag specification or a deterministic component that does not match the chosen specification can undermine either procedure.

Choose deterministic terms.
Decide whether the tested specification has a constant only or a constant and trend; this determines the transformation and reference distribution.

Quasi-difference both parts.
Apply \(\bar\alpha\) to the series and deterministic regressors, leaving the first observation unfiltered.

Form the detrended series.
Fit the transformed regression and subtract its fitted deterministic component from the original observations.

Run the test regression.
Regress the detrended change on its lagged level and any required lagged differences; test the coefficient on the lagged level.

Use the correct reference values.
Compare the statistic with DF-GLS critical values for the deterministic specification, rather than ordinary \(t\) critical values.

Common Pitfalls and Practical Significance

DF-GLS matters because near-unit-root processes are precisely where removing a deterministic component can strongly affect the test. Its GLS transformation is a principled way to address that issue, not a cosmetic rescaling. A valid report should identify the deterministic specification, the GLS detrending calibration, the augmentation choice, the test statistic, and the DF-GLS reference values used.

Using ordinary \(t\) critical values. The DF-GLS coefficient statistic has a nonstandard null distribution. Use the appropriate DF-GLS critical values.
Filtering only the response. GLS detrending transforms the deterministic regressors as well as the series; filtering just \(y_t\) does not implement the stated procedure.
Filtering the first observation as if it had a lag. The definition uses \(y_1^\alpha=y_1\) and \(x_1^\alpha=x_1\); there is no \(y_0\) term in this construction.
Treating the recommended \(\bar\alpha\) as an estimated root. It is a calibration for quasi-differencing, not the estimated autoregressive coefficient of the observed series.
Claiming universal power dominance. The improvement is motivated by local alternatives and does not guarantee a better result in every finite sample or under every specification.

Key takeaway: DF-GLS applies GLS quasi-differencing to estimate deterministic components, then tests the detrended series with a Dickey–Fuller-type regression. Its power advantage is tied to local-to-unity alternatives, and valid inference requires DF-GLS rather than standard \(t\) critical values.

Check Your Understanding

Answer each question by referring to the transformation, regression, or local-power argument.

For a constant-only specification, what are \(x_1^\alpha\) and \(x_t^\alpha\) for \(t\geq2\)?
Why must the deterministic regressors be quasi-differenced along with the observed series?
State the null and left-tailed alternative for the coefficient on \(\tilde y_{t-1}\) in the DF-GLS regression.
Why are ordinary Student \(t\) critical values inappropriate for the DF-GLS statistic under the unit-root null?
In what sense can GLS detrending improve power, and why is that not a guarantee of better finite-sample performance in every application?