Lecture 24 – Unit Root Tests, II
The initial DF unit root tests assumed that
under the unit root null hypothesis, the first
differences in the series are serially
uncorrelated. Since first differences of most
macroeconomic time series are serially
correlated, these tests were of limited value
in emirical macroeconomics.
This problem was addressed in the
development of the Augment Dickey-Fuller
test (ADF test) and the Phillips-Perron test
(PP test).
Suppose that yt has the AR(p) form
yt = a1yt-1 + … + apyt-p + εt
where
εt ~ iid (0,σ2) .
We assume that either
i)
all of the zeroes of 1-a1z - … -apzp
are greater than one in modulus
(and, therefore, yt ~ I(0))
ii)
a(1) = 0 and all the other zeroes of
1-a1z - … -apzp are greater than one
in modulus (and, therefore, yt ~
I(1))
or
We can rewrite this model as
yt = ρyt-1 + δ1Δyt-1 + … + δp-1Δyt-p+1 + εt
where
ρ = a1+…+ap
δ1 = -(a2 +…+ap)
δ2 = -(a3 + … + ap)
…
δp-1 = -ap
Then, the unit root null hypothesis is:
H0 : ρ=0
The ADF Test –
Regress yt on yt-1, Δyt-1,…, Δyt-p+1 and
compute the t-statistic
ˆ  1
 
se( ˆ )
(Or, regress Δyt on yt-1, Δyt-1,…, Δyt-p+1 and
compute the t-statistic

ˆ
se( ˆ )
)
Use the same DF distribution for τ that you
would use for the AR(1) case.
We can allow for a non-zero mean under the
alternative by adding an intercept to the
model, regressing yt (or Δyt) on
1, yt-1, Δyt-1,…, Δyt-p+1, computing the tstatistic
ˆ  1
ˆ
 
 
se( ˆ ) (or,
se( ˆ ) )
and comparing its value to the percentiles of
the DF τμ distribution.
Similarly we can allow for a trend under the
null and alternative by adding an intercept
and trend to the model, regressing yt (or Δyt)
on 1, t, yt-1, Δyt-1,…, Δyt-p+1, computing the
t-statistic
t 
ˆ  1
ˆ


t
se( ˆ ) (or,
se( ˆ ) )
and comparing its value to the percentiles of
the DF τt distribution.
Problem – How to select p?
Solutions –
 model selection criterion (AIC, SIC)
 formal testing (e.g., sequential t-tests:
the δi-hats are asymptotically normal
under the null)
 diagnostic checking for residual
“whiteness” (L-B Q test,…)
Application – Nelson and Plosser (JME,
1982)
The ADF test relies on a parametric
transformation of the model that eliminates
the serial correlation in the error term
without affecting leaves the asymptotic
distributions of the various τ statistics.
Phillips and Perron (Biometrika, 1988)
proposed nonparametric transformations of
the τ statistics from the original DF
regressions such that under the unit root
null, the transformed statistics (the “z”
statistics) have DF distributions.
So, for example, suppose our model is
yt = ρyt-1 + εt , εt ~ I(0) with mean 0
The PP procedure:
 Regress yt on yt-1
 Compute τ
 “Modify” τ to get z
 Under H0, z’s asymptotic distribution
is the DF distribution for τ
If our model is
yt = α + ρyt-1 + εt , εt ~ I(0) with mean 0
or
yt = α +βt+ ρyt-1 + εt , εt ~ I(0) with mean 0
the PP procedure is basically the same as
above, i.e.,





Regress yt on 1, (t), yt-1
Compute τμ (τt)
Modify τμ to get zμ
(Modify τt to get zt)
Under H0, zμ’s asymptotic distribution
is the DF distribution for τμ (and zt’s
asymptotic distribution is the DF
distribution for τt).
ADF vs. PP
The PP test does not require us to specify the
form of the serial correlation of Δyt under the
null. In addition, the PP test does not require that
the ε’s are conditionally homoskedastic (an
implicit assumption in the ADF test).
If we apply the ADF test and have underspecified p, the AR order, the test will be missized. If we apply the ADF test and overspecify p, the test’s power will suffer.
These problems are avoided in the PP test, but if
we can correctly specify p, the PP test will be
less powerful than the ADF test.
Also, the PP test requires a “bandwidth”
parameter selection (as part of the construction
of the Newey-West covariance estimator) that
creates finite sample problems analogous to
those associated the lag length selection issue in
applying the ADF test.