Phillips-Perron (PP) Unit Root Tests

Phillips-Perron (PP) Unit Root Tests
The Dickey–Fuller test involves fitting the regression model
Δy t = ρy t−1 + (constant, time trend) + u t (1)
by ordinary least squares (OLS), but serial correlation will present a problem. To account for this, the
augmented Dickey–Fuller test’s regression includes lags of the first differences of yt. The Phillips–
Perron test involves fitting (1), and the results are used to calculate the test statistics. They estimate
not (1) but:
y t = πy t−1 + (constant, time trend) + u t (2)
In (1) u t is I(0) and may be heteroskedastic. The PP tests correct for any serial correlation and
heteroskedasticity in the errors u t non-parametrically by modifying the Dickey Fuller test statistics.
Phillips and Perron’s test statistics can be viewed as Dickey–Fuller statistics that have been made
robust to serial correlation by using the Newey–West (1987) heteroskedasticity- and
autocorrelation-consistent covariance matrix estimator.
Under the null hypothesis that ρ = 0, the PP Zt and Zπ statistics have the same asymptotic
distributions as the ADF t-statistic and normalized bias statistics. One advantage of the PP tests over
the ADF tests is that the PP tests are robust to general forms of heteroskedasticity in the error term
ut. Another advantage is that the user does not have to specify a lag length for the test regression.
We have not dealt with it, but the Dickey Fuller test produces two test statistics. The normalized bias
T (π− 1) has a well defined limiting distribution that does not depend on nuisance parameters it can
also be used as a test statistic for the null hypothesis H0 : π = 1. This is the second test from DF and
relats to Z π in Phillips and Perron.
Continued...............

EXTRACT FROM STATA MANUAL
Note the regression is y on lagged y, not differenced y on lagged y.
Z T is the adjusted t statistic as in Dickey Fuller.
is just the equivalent in tthe t stat in the DF test. S 2 n is an unbiased estimator (OLS) of the variance of the
error terms.
when j=0 this is a (maximum likelihood) estimate of the variance of the error terms, when j>0 its an
estimator of the covariance between two error terms j periods apart.
q is the number of lagged covariances looked at. Now when the covariances are zero – i.e. no
autocrrelation between error terms is zero for j>0. Hence the second terms disappears and
. In this case we can replace with in:

In this case = 0 and the second term disappears. = 1 thus reduces to
and
= . This is just the t statistic in the standard Dickey Fuller equation. Hence.when there is no
autocorrelation between error terms this part of the Phillips-Perron test is equal to the Dickey Fuller albeit
one estimated on (2) rather than (1). This perspective helps us understand that the PP test corrects the DF
one for autocrrelation amongst error terms non-parametrically (i.e. outside of a regression framework). The
critical values, have the same distribution as the Dickey–Fuller statistic
Although we have not done it when there is no autocorrelation between error terms, when the covariances
are equal then again the second term in the other PP statistic collapses to zero because
In this case
which again is the same as the Dickey Fuller test