The
ADF test was the first test developed for testing the null hypothesis of a unit
root and its most commonly used test in practice. Other test have been developed
later and many of them claim that have higher power.
The power of unit root means that the test have a
more power to reject the null hypothesis of a unit root against the alternative
when alternative is true. So, a more powerful test is better able to distinguish
between a unit AR root and a root that is large but less than 1.
For
this discussion, we would like to look another unit root test proposed by Elliot,
Rothenberg and Stock (1996) what they called it as DF-GLS (Dickey-Fuller
Generalized Least Squares).
The test
is introduced for the case that, under the null hypothesis, \({{Y}_{t}}\) has a
random walk trend, possibility with drift, and under the alternative \({{Y}_{t}}\)
is stationary around a linear time trend.
The DF-GLS
test is computed in two steps.
In
the first step, the intercept and trend are estimated by GLS. The GLS
estimation is performed by computing three new variables, \({{V}_{t}}\),\({{X}_{1t}}\)
and \({{X}_{2t}}\), where;
\({{V}_{1}}={{Y}_{1}}\) (1)
\({{V}_{t}}={{Y}_{t}}-\alpha
*{{Y}_{t-1}}\) \(t=2,...T\) (2)
\({{X}_{11}}=1\) (3)
\({{X}_{1t}}=1-\alpha
*\) \(t=2,...T\). (4)
\({{X}_{21}}=1\) (5)
\({{X}_{2t}}=t-\alpha
*\left( t-1 \right)\) (6)
where
\(\alpha *\) is computed using formula \(\alpha *=1-13.5/T\).
Then \({{V}_{1}}\) is regressed against \({{X}_{1t}}\) and \({{X}_{2t}}\) by OLS to
estimate the coefficient of the population regression equation;
\({{V}_{1}}={{\delta
}_{0}}{{X}_{1t}}+{{\delta }_{2}}{{X}_{2t}}+{{e}_{t}}\) (7)
by using
observation \(t=1,...T,\) where \({{e}_{t}}\) is error term and there is no
intercept in Eq(7).
The OLS
estimation \({{\hat{\delta }}_{0}}\) and \({{\hat{\delta }}_{1}}\) are then
used to compute a “detrended” version of \({{Y}_{t}}\);
\(Y_{t}^{d}={{Y}_{t}}-\left(
{{{\hat{\delta }}}_{0}}+{{{\hat{\delta }}}_{1}}t \right)\) (8)
In the
second step, the DF test is used to Eq(8) where the DF regression does not
include an intercept or a time trend;
\(\Delta Y_{t}^{d}=\delta
Y_{t-1}^{d}+\sum\limits_{i=1}^{k}{{{\gamma }_{i}}\Delta Y_{t-i}^{d}+{{\varepsilon
}_{t}}}\) (9)
where
the number of lags \(k\) is determined, as
usual, either by expert knowledge or by using information criterion such as AIC
and BIC.
The
GLS regression in the first step of the DF-GLS test makes this test mode complicated
than the conventional ADF test, but it improves its ability to reject the null
hypothesis of unit root.
Although
the DF-GLS looks complicated especially in the first step, but luckily for us
the Stata provide the command dfgls which is design to test our time series data with DF-GLS method.
Now,
we will perform the unit root test with DF-GLS method with the real data. The
data that we dealt is same as we used in ADF test and PP test, the Macro_Stata
data.
To
perfom the DF-GLS test for gdp variable;
dfgls gdp
The results show that the null hypothesis of a unit root is failed to reject
for lags 1-5, it is rejected at the 10% level for lags 6-11, and it is rejected
at the 5% level for lags 11.
If we compared ADF test and PP test at lags 2, the DF-GLS test shows the
series is stationary at lags 2 but not for ADF test and PP test.
The comparison means that the dfuller
and pperron results are not as
strong as those produced by dfgls and
is not surprising because the DF-GLS
test with a trend has been shown to be more powerful than the standard ADF and
PP.
If
we want to test the variable gdp in the first difference,
dfgls D.gdp
which
show that the null hypothesis for a unit root is rejected for all the lags
range.
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