1 Choosing the Lag Length for the
ADF Test
An important practical issue for the implementation of
the ADF test is the specification of the lag length p.
• If p is too small then the remaining serial correlation
in the errors will bias the test.
• If p is too large then the power of the test will suffer.
• Monte Carlo experiments suggest it is better to error
on the side of including too many lags.
Ng and Perron “Unit Root Tests in ARMA Models with
Data-Dependent Methods for the Selection of the Trun-
cation Lag,” JASA, 1995.
• Set an upper bound pmax for p.
• Estimate the ADF test regression with p = pmax.
• If the absolute value of the t-statistic for testing the
significance of the last lagged difference is greater
than 1.6 then set p = pmax and perform the unit
root test. Otherwise, reduce the lag length by one
and repeat the process.
• A common rule of thumb for determining pmax, sug-
gested by Schwert (1989), is
" µ ¶1/4#
T
pmax = 12 ·
100
where [x] denotes the integer part of x. However,
this choice is ad hoc!
Ng and Perron “Lag Length Selection and the Construc-
tion of Unit Root Tests with Good Size and Power,”
ECTA, 2001.
• Select p as pmic = arg minp≤pmax MAIC(p) where
2(τ T (p) + p)
MAIC(p) = ln(σ̂ 2p) +
T − pmax
P
π̂ 2 T
t=pmax+1 yt−1
τ T (p) =
σ̂ 2p
1 T
X
σ̂ 2p = ε̂2t
T − pmax t=pmax+1
• Procedure is implemented in Eviews and S+FinMetrics
2.0
2 Phillips-Perron Unit Root Tests
The test regression for the PP tests is
∆yt = β0Dt + πyt−1 + ut
ut ∼ I(0)
The PP tests correct for any serial correlation and het-
eroskedasticity in the errors ut of the test regression by
directly modifying the test statistics tπ=0 and T π̂. These
modified statistics, denoted Zt and Zπ , are given by
à ! à !
2 1/2 2 2
σ̂ 1 λ̂ − σ̂ T · SE(π̂)
Zt = 2 · tπ=0 − 2 ·
λ̂ 2 λ̂ σ̂ 2
1 T 2 · SE(π̂) 2 2)
Zπ = T π̂ − (λ̂ − σ̂
2 σ̂ 2
2
The terms σ̂ 2 and λ̂ are consistent estimates of the
variance parameters
T
X
σ2 = lim T −1 E[u2t ]
T →∞
t=1
T
X h i
λ2 = lim E T −1ST2 = lrv
T →∞
t=1
T
X
ST = ut
t=1
The sample variance of the least squares residual ût is a
consistent estimate of σ 2, and the Newey-West long-run
variance estimate of ut using ût is a consistent estimate
of λ2.
Result: Under the null hypothesis that π = 0, the PP Zt
and Zπ statistics have the same asymptotic distributions
as the ADF t-statistic and normalized bias statistics.
3 Some Problems with Unit Root
Tests
The ADF and PP tests are asymptotically equivalent but
may differ substantially in finite samples due to the dif-
ferent ways in which they correct for serial correlation in
the test regression.
• Schwert “Test for Unit Roots: A Monte Carlo Inves-
tigation,” JBES, 1989, finds that if ∆yt ∼ARMA
with a large and negative MA component, then the
ADF and PP tests are severely size distorted (reject
I(1) null much too often when it is true) and that
the PP tests are more size distorted than the ADF
tests.
• Perron and Ng “Useful Modifications to Some Unit
Root Tests with Dependent Errors and their Local
Asymptotic Properties,” RESTUD, (1996), suggest
useful modifications to the PP tests to mitigate size
distortion of PP tests.
• ADF and PP tests have very low power against I(0)
alternatives that are close to being I(1).
• The power of unit root tests diminish as deterministic
terms are added to the test regressions.
• For maximum power against very persistent alterna-
tives the so-called efficient unit root tests should be
used.
— Elliot, Rothenberg, and Stock “Efficient Tests for
an Autoregressive Unit Root,” ECTA, 1996.
— Ng and Perron “Lag Length Selection and the
Construction of Unit Root Tests with Good Size
and Power,” ECTA, 2001.
— Tests are implemented in Eviews and S+FinMetrics
2.0.
4 Stationarity Tests
Kwiatkowski, Phillips, Schmidt and Shin (1992) derive
their test by starting with the model
yt = β0Dt + µt + ut, ut ∼ I(0)
µt = µt−1 + εt, εt ∼ W N (0, σ 2ε )
Dt = deterministic components
The hypotheses to be tested are
H0 : σ 2ε = 0 ⇒ yt ∼ I(0)
H1 : σ 2ε > 0 ⇒ yt ∼ I(1)
The KPSS test statistic is the Lagrange multiplier (LM)
or score statistic for testing σ 2ε = 0:
T
X 2
KP SS = T −2 Ŝt2 /λ̂
t=1
t
X
Ŝt = ûj
j=1
where
• ût is the residual of a regression of yt on Dt
2
• λ̂ is a consistent estimate of the long-run variance
of ut using ût.
Asymptotic results: Assume H0: yt ∼ I(0) is true.
• If Dt = 1 then
Z 1
KP SS ⇒ V1(r)dr
0
V1(r) = W (r) − rW (1)
• If Dt = (1, t)0 then
Z 1
d
KP SS → V2(r)dr
0
V2(r) = W (r) + r(2 − 3r)W (1)
Z 1
+6r(r2 − 1) W (s)ds
0
Note: V1(r) = W (r) − rW (1) is called a standard
Brownian bridge. It satisfies V1(0) = V1(1) = 0.
Right tail quantiles
Distribution 0.90 0.925 0.950 0.975 0.99
R1
V (r)dr 0.349 0.396 0.446 0.592 0.762
R01 1
0 V2(r)dr 0.120 0.133 0.149 0.184 0.229
Table 1: Quantiles of the distribution of the KPSS sta-
tistic
• Critical values from the asymptotic distributions must
be obtained by simulation methods
• The stationary test is a one-sided right-tailed test
so that one rejects the null of stationarity at the
100 · α% level if the KPSS test statistic is greater
than the 100·(1−α)% quantile from the appropriate
asymptotic distribution.
