STAT 497
LECTURE NOTES 5
UNIT ROOT TESTS
1
UNIT ROOTS IN TIME SERIES MODELS
• Shock is usually used to describe an
unexpected change in a variable or in the
value of the error terms at a particular time
period.
• When we have a stationary system, effect of a
shock will die out gradually.
• When we have a non-stationary system, effect
of a shock is permanent.
2
UNIT ROOTS IN TIME SERIES MODELS
• Two types of non-stationarity:
– Unit root i.e.,|i| = 1: homogeneous non-stationarity
– |i| > 1: explosive non-stationarity
• Shock to the system become more influential as time goes
on.
• Can never be seen in real life
3
UNIT ROOTS IN TIME SERIES MODELS
e.g. AR(1)
Yt  Yt 1  at
Yt   Yt T  at  at 1   at 2     at T
T
2
T
  1   T  0 as T  .
  1    1 as T  .
T
  1     as T  .
T
4
UNIT ROOTS IN TIME SERIES MODELS
• A root near 1 of the AR polynomial
differencing
• A root near 1 of the MA polynomial
over-differencing
5
UNIT ROOTS IN AUTOREGRESSION
1. DICKEY-FULLER (DF) TEST: The simplest
approach to test for a unit root begins with
AR(1) model
Yt   0  Yt 1  at
where at ~ Normal _ WN 0, .
• DF test actually does not consider 0 in the
model, but actually model with 0 and not 0
gives different results.
2
a
6
DF TEST
• Consider the hypothesis
H 0 :   1  Yt ~ I 1
H1 :   1  Yt ~ I 0 
• The hypothesis is the reverse of KPSS test.
7
DF TEST
• To simplify the computation, subtract Yt-1 from
both sides of the AR(1) model;
Yt  Yt 1  0  1   Yt 1  at
Yt  0  Yt 1  at
• If =0, system has a unit root.
H0 :   0
H1 :   0
8
DF TEST
• DF (1979)
Yt  Yt 1  at  Pure RW
Yt   0  Yt 1  at  RW with drift
Yt   0  1t  Yt 1  at  adds drift and a linear time trend
9
DF TEST
• Applying OLS method and finding the
estimator for , the test statistic is given by
ˆ  1
t 1 
s.e.ˆ 
• The test is a one-sided left tail test. If {Yt} is
stationary (i.e.,|φ| < 1) then it can be shown
2
ˆ
n     N 0, 1   .
d
• This means that under H0, the limiting
distribution of t=1 is N(0,1).
10
DF TEST
• Under the null hypothesis,
A
ˆ ~ N 1,0 
which does not make any sense. Under the unit
root null, {Yt} is not stationary and ergodic, and
the usual sample moments do not converge to
fixed constants. Instead, Phillips (1987) showed
that the sample moments of {Yt} converge to
random functions of Brownian motion
11
DF TEST
n
n
3 / 2
n
d
1
 Yt 1   W r dr
t 1
2
0
d
2
 Yt 1 
t 1
n
n
1
  W r  dW r 
d
2
2
0
1
n 1  Yt 1at   W r dr
t 1
0
where W(r) denotes a standard Brownian
motion (Wiener process) defined on the unit
interval.
12
DF TEST
• Using the above results Phillips showed that
under the unit root null H0 : φ = 1
1
 W r dW r 
n  1ˆ  1 0 1
d
 W r  dr
2
0
1
d
t 1 
 W r dW r 
0
1
1/ 2
 W r 2 dr 


0

13
DF TEST
SOME RESULTS p
• ˆ is super-consistent; that is, ˆ  at rate n
instead of the usual rate.
• ˆ is not asymptotically normally distributed and
tφ=1 is not asymptotically standard normal.
• The limiting distribution of tφ=1 is called the
Dickey-Fuller (DF) distribution and does not have
a closed form representation. Consequently,
quantiles of the distribution must be computed
by numerical approximation or by simulation.
14
DF TEST
• Since the normalized bias (n1)( ˆ − 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 : φ = 1.
15
DF TEST
• EXAMPLE:
Yˆt  0.9481Yt 1
 0.0509 
n  35  n  1ˆ  1  340.9481  1  1.7646
=0.05
Critical value: 7.3 for n=25
7.7 for n=50
Not reject H0. There exist a unit root. We need
to take a difference to be able to estimate a
model for the series
16
DF TEST
• With a constant term:
The test regression is
Yt  0  Yt 1  at
and includes a constant to capture the
nonzero mean under the alternative. The
hypotheses to be tested
H 0 :   1, 0  0  Yt ~ I 1 withoutdrift
H 1 :   1  Yt ~ I 0 with mean
This formulation is appropriate for non-trending
economic and financial series like interest
rates, exchange rates and spreads.
17
DF TEST
• The test statistics tφ=1 and (n − 1)( ˆ − 1) are
computed from the above regression. Under
H0 : φ = 1, c = 0 the asymptotic distributions of
these test statistics are influenced by the
presence, but not the coefficient value, of the
constant in the test regression:
18
DF TEST
1

r dW r 
W

n  1ˆ  1 0 1
d

r  dr
W

2
0
1
d
t 1 
where W


r dW r 
W

0
1
1/ 2
 W  r 2 dr 


0

1
r   W r    W r dr is a de - meaned Wiener
0
1
process, i.e.,  W  r dr  0
0
Inclusion of a constant pushes the distributions
of tφ=1 and (n1) (ˆ − 1) to the left.
19
DF TEST
• With constant and trend term
The test regression is
Yt  0  t  Yt 1  at
and includes a constant and deterministic
time trend to capture the deterministic trend
under the alternative. The hypotheses to be
tested
H 0 :   1,   0  Yt ~ I 1 with drift
H 1 :   1  Yt ~ I 0 with deterministic time trend
20
DF TEST
• This formulation is appropriate for trending time
series like asset prices or the levels of
macroeconomic aggregates like real GDP. The test
statistics tφ=1 and (n − 1) (ˆ − 1) are computed
from the above regression.
• Under H0 : φ = 1, δ = 0 the asymptotic
distributions of these test statistics are influenced
by the presence but not the coefficient values of
the constant and time trend in the test regression.
21
DF TEST
1

W
 r dW r 
n  1ˆ  1 0 1
d

W
 r  dr
2
0
1
d
t 1 

W
 r dW r 
0
1
1/ 2
 W  r 2 dr 


0

1 1  1 

where W r   W r   12 r     s  W r dr
 2 0  2 
is a de - meaned and de - trended Wiener process.

22
DF TEST
• The inclusion of a constant and trend in the
test regression further shifts the distributions
of tφ=1 and (n − 1)(ˆ − 1) to the left.
23
DF TEST
• What do we conclude if H0 is not rejected? The
series contains a unit root, but is that it? No!
What if Yt∼I(2)? We would still not have rejected.
So we now need to test
H0: Yt∼I(2) vs. H1: Yt∼I(1)
• We would continue to test for a further unit root
until we rejected H0.
24
DF TEST
• This test is valid only if at is WN. If there is a serial
correlation, the test should be augmented. So, check
for possible autoregression in at.
• Many economic and financial time series have a
more complicated dynamic structure than is
captured by a simple AR(1) model.
• Said and Dickey (1984) augment the basic
autoregressive unit root test to accommodate
general ARMA(p, q) models with unknown orders
and their test is referred to as the augmented
Dickey- Fuller (ADF) test.
25
AUGMENTED DICKEY-FULLER (ADF)
TEST
• If serial correlation exists in the DF test
equation (i.e., if the true model is not AR(1)),
then use AR(p) to get rid of the serial
correlation.
 p B Yt   0  at
 
 
whereat  ~ W N 0, a2 with E at4  
and  p B   1  1B     p B may contain a unit root.
p
26
ADF TEST
• To test for a unit root, we assume that
 p  B   1  B  p 1  B 
where  p 1  B   1  1B     p 1B p 1 has roots
lying outsidethe unit circle.
 p 1 B 1  B Yt  0  at
 p 1 B Yt  0  at
p 1
Yt    j Yt  j  0  at
j 1
27
ADF TEST
• Hence, testing for a unit root is equivalent to
testing =1 in the following model
p 1
ADF test equation: Yt  Yt 1   j Yt  j  0  at
j 1
or
p 1
Yt    1Yt 1    j Yt  j  0  at

j 1

p 1
ADF test equation: Yt  Yt 1   j Yt  j  0  at
j 1
28
ADF TEST
• Hypothesis
H0 :   1
H1 :   1
Reject H0 if t=1<CV
H0 :   0
H1 :   0
Reject H0 if t=0<CV
• We can also use the following test statistics:
n  p ˆ  11
where 1  1  1  2   obtained from RSF of the model.
29
ADF TEST
• The limiting distribution of the test statistic
n  p ˆ  11
is non-standard distribution (function of
Brownian motion _ or Wiener process).
30
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.
31
Choosing the Lag Length for the ADF
Test
• Ng and Perron (1995) suggest the following data
dependent lag length selection procedure that
results in stable size of the test and minimal
power loss:
• First, set an upper bound pmax for p. Next,
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.
32
Choosing the Lag Length for the ADF
Test
• A useful rule of thumb for determining pmax,
suggested by Schwert (1989), is
pmax
  n 1/ 4 
 12
 
  100  
where [x] denotes the integer part of x. This
choice allows pmax to grow with the sample so
that the ADF test regressions are valid if the
errors follow an ARMA process with unknown
order.
33
ADF TEST
• EXAMPLE: n=54
Examine the original model and the
differenced one to determine the order of AR
parameters. For this example, p=3.
Yt  0  Yt 1  1Yt 1  2Yt 2  at
Fit the model with t = 4, 5,…, 54.
OLS equation:
Yt  139  0.856 Yt 1  0.141 Yt 1  0.326 Yt  2
87.97  0.08699
0.1353
0.1369
34
ADF TEST
• EXAMPLE (contd.) Under H0,

1  B Yt    B at

where   0 / 1  1B  2 B 2 and   B  
1
 p 1 B 

1
.
2
1  1B  2 B
1
1
  1 

 1.23
1  0.141  0.326 0.815
n  p ˆ  1 1  9.033  13.3
n=50, CV=-13.3
• H0 cannot be rejected. There is a unit root.
The series should be differenced.
35
ADF TEST
• If the test statistics is positive, you can
automatically decide to not reject the null
hypothesis of unit root.
• Augmented model can be extended to allow MA
terms in at. It is generally believed that MA terms
are present in many macroeconomic time series
after differencing. Said and Dickey (1984)
developed an approach in which the orders of the
AR and MA components in the error terms are
unknown, but can be approximated by an AR(k)
process where k is large enough to allow good
approximation to the unknown ARMA(p,q) process.
36
ADF TEST
• Ensuring that at is approximately WN
 p  B Yt   q  B at 
 p B 
q B 
Yt  at

 k  B  where k 3 n
37
PHILLIPS-PERRON (PP) UNIT ROOT TEST
• Phillips and Perron (1988) have developed a more
comprehensive theory of unit root nonstationarity. The
tests are similar to ADF tests. The Phillips-Perron (PP)
unit root tests differ from the ADF tests mainly in how
they deal with serial correlation and heteroskedasticity
in the errors. In particular, where the ADF tests use a
parametric autoregression to approximate the ARMA
structure of the errors in the test regression, the PP
tests ignore any serial correlation in the test regression.
• The tests usually give the same conclusions as the ADF
tests, and the calculation of the test statistics is
complex.
38
PP TEST
• Consider a model
Yt  0  Yt 1  at
DF: at ~ iid
PP: at ~ serially correlated
PP test equation: Yt  0  Yt 1  at
• Add a correction factor to the DF test
statistic. (ADF is to add lagged ΔYt to ‘whiten’
the serially correlated residuals)
39
PP TEST
• The hypothesis to be tested:
H0 :   0
H1 :   0
40
PP TEST
• The PP tests correct for any serial correlation and
heteroskedasticity in the errors at of the test
regression by directly modifying the test statistics t=0
and nˆ . These modified statistics, denoted Zt and Z,
are given by
ˆ 2
1  ˆ2  ˆ 2  ns.e.ˆ 

Zt 
tˆ  

2
2
2

2  ˆ
ˆ

 ˆ
2
ˆ  2


1
n
s
.
e
.

ˆ  ˆ 2
Z  nˆ 

2 ˆ 2


The terms ˆ 2 and ˆ are consistent estimates of the
variance parameters
1
n
  lim n  E
2
n 
t 1
 
at2
1 n 2
  lim  E   at 
n  t 1  n t 1

2
n
41
PP TEST
• Under the null hypothesis that  = 0, the PP Zt
and Z 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 at. Another
advantage is that the user does not have to
specify a lag length for the test regression.
42
PROBLEM OF PP TEST
• On the other hand, the PP tests tend to be more
powerful but, also subject to more severe size
distortions
– Size problem: actual size is larger than the nominal
one when autocorrelations of at are negative.
– more sensitive to model misspecification (the order
of autoregressive and moving average components).
• Plotting ACFs help us to detect the potential size
problem
– Economic time series sometimes have negative
autocorrelations especially at lag one, we can use a
Monte Carlo analysis to simulate the appropriate
critical values, which may not be attractive to do.
43
Criticism of Dickey-Fuller and
Phillips-Perron Type Tests
• Main criticism is that the power of the tests is low
if the process is stationary but with a root close
to the non-stationary boundary.
• e.g. the tests are poor at deciding if φ=1 or
φ=0.95, especially with small sample sizes.
• If the true data generating process (dgp) is
Yt= 0.95Yt-1+ at
then the null hypothesis of a unit root should be
rejected.
• One way to get around this is to use a stationarity
test (like KPSS test) as well as the unit root tests
we have looked at (like ADF or PP).
44
Criticism of Dickey-Fuller and
Phillips-Perron Type Tests
• The ADF and PP unit root tests are known (from MC
simulations) to suffer potentially severe finite sample
power and size problems.
1. Power – The ADF and PP tests are known to have low
power against the alternative hypothesis that the series is
stationary (or TS) with a large autoregressive root. (See,
e.g., DeJong, et al, J. of Econometrics, 1992.)
2. Size – The ADF and PP tests are known to have severe
size distortion (in the direction of over-rejecting the null)
when the series has a large negative moving average root.
(See, e.g., Schwert. JBES, 1989: MA = -0.8, size = 100%!)
45
Criticism of Dickey-Fuller and
Phillips-Perron Type Tests
• A variety of alternative procedures have been
proposed that try to resolve these problems,
particularly, the power problem, but the ADF
and PP tests continue to be the most widely
used unit root tests. That may be changing!
46
STRUCTURAL BREAKS
• A stationary time-series may look like
nonstationary when there are structural breaks in
the intercept or trend
• The unit root tests lead to false non-rejection of
the null when we don’t consider the structural
breaks  low power
• A single breakpoint is introduced in Perron (1989)
into the regression model; Perron (1997)
extended it to a case of unknown breakpoint
• Perron, P., (1989), “The Great Crash, the Oil Price Shock and the
Unit Root Hypothesis,” Econometrica, 57, 1361–1401.
47
STRUCTURAL BREAKS
Consider the null and alternative hypotheses
H0: Yt = a0 + Yt-1 + µ1DP + at
H1: Yt = a0 + a2t + µ2DL + at
 Pulse break: DP = 1 if t = TB + 1 and zero otherwise,
 Level break: DL = 0 for t = 1, . . . , TB and one otherwise.
 Null: Yt contains a unit root with a one–time jump in
the level of the series at time t = TB + 1 .
 Alternative: Yt is trend stationary with a one–time
jump in the intercept at time t = TB + 1 .
48
Simulated unit root and trend stationary
processes with structural break.
n= 100
at ~ i.i.d. N(0,1)
y0=0
H0: -----• a0 = 0.5,
• DP = 1 for n = 51
zero otherwise,
• µ1 = 10.
H1:
• a2 = 0.5,
• DL = 1 for n > 50.
• µ2 = 10
49
Power of ADF tests: Rejection
frequencies of ADF–tests
• ADF tests are biased toward non-rejection of the null.
• Rejection frequency is inversely related to the magnitude of
the shift.
• Perron: estimated values of the autoregressive parameter in
the Dickey–Fuller regression was biased toward unity and
that this bias increased as the magnitude of the break
increased.
50
Testing for unit roots when there are
structural changes
 Perron suggests running the following OLS
regression:
p
yt  a0  a1 yt 1  a2t  2 DL   i yt i   t
i 1
 H0: a1 = 1; t–ratio, DF unit root test.
 Perron shows that the asymptotic distribution of
the t-statistic depends on the location of the
structural break, = TB/n
critical values are supplied in Perron (1989) for
different assumptions about l, see Table IV.B.
2015/4/8
51
EXAMPLE
• Consider following time series plots
52
KPSS RESULTS
> kpss.test(y,c("Level"))
KPSS Test for Level Stationarity
data: y
KPSS Level = 3.4581, Truncation lag parameter = 2, p-value = 0.01
Warning message:
In kpss.test(y, c("Level")) : p-value smaller than printed p-value
> kpss.test(y,c("Trend"))
KPSS Test for Trend Stationarity
data: y
KPSS Trend = 0.5894, Truncation lag parameter = 2, p-value = 0.01
Warning message:
In kpss.test(y, c("Trend")) : p-value smaller than printed p-value
There is a stochastic trend. We need differencing to observe stationary
series.
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PP TEST RESULTS
> pp.test(y)
Phillips-Perron Unit Root Test
data: y
Dickey-Fuller Z(alpha) = -11.1817, Truncation lag parameter = 4,
p-value = 0.4673
alternative hypothesis: stationary
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ADF TEST RESULT
• To decide the lag order
55
ADF TEST RESULTS
• Load ‘fUnitRoot’ package
> adfTest(y, lags = 5, type = c("nc"))
Unit root test with no drift
Title:
Augmented Dickey-Fuller Test
Test Results:
PARAMETER:
Lag Order: 5
STATISTIC:
Dickey-Fuller: -0.3663
P VALUE:
0.4964
> adfTest(y, lags = 5, type = c("c"))
Unit root test with drift
Title:
Augmented Dickey-Fuller Test
Test Results:
PARAMETER:
Lag Order: 5
STATISTIC:
Dickey-Fuller: 0.6517
P VALUE:
0.99
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ADF TEST RESULTS
> adfTest(y, lags = 5, type = c("ct"))
Unit root test with drift and trend
Title:
Augmented Dickey-Fuller Test
Test Results:
PARAMETER:
Lag Order: 5
STATISTIC:
Dickey-Fuller: -2.4843
P VALUE:
0.3759
Combining results of ADF and KPSS tests, we can say that there is a
stochastic trend. Differencing is needed.
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REPEAT TESTS ON DIFFERECED
SERIES
> kpss.test(ydif,c("Level"))
KPSS Test for Level Stationarity
data: ydif
KPSS Level = 0.1041, Truncation lag parameter = 2, p-value = 0.1
Warning message:
In kpss.test(ydif, c("Level")) : p-value greater than printed p-value
> kpss.test(ydif,c("Trend"))
No need after getting the above result.
KPSS Test for Trend Stationarity
data: ydif
KPSS Trend = 0.0763, Truncation lag parameter = 2, p-value = 0.1
Warning message:
In kpss.test(ydif, c("Trend")) : p-value greater than printed p-value
58
ADF TEST ON DIFFERENCED SERIES
> adfTest(ydif, lags = 5, type = c("nc"))
So, use this test result
Title:
Augmented Dickey-Fuller Test
Test Results:
PARAMETER:
Lag Order: 5
STATISTIC:
Dickey-Fuller: -0.1808
P VALUE:
0.5555
When you apply ADF test on a
differenced series, use unit root test with
no drift term. Differencing makes the
constant part zero.
> adfTest(ydif, lags = 5, type = c("c"))
Title:
Augmented Dickey-Fuller Test
Test Results:
PARAMETER:
Lag Order: 5
STATISTIC:
Dickey-Fuller: -5.1038
P VALUE:
0.01
59
ADF TEST ON DIFFERENCED SERIES
> adfTest(ydif, lags = 5, type = c("ct"))
Title:
Augmented Dickey-Fuller Test
Test Results:
PARAMETER:
Lag Order: 5
STATISTIC:
Dickey-Fuller: -5.0923
P VALUE:
0.01
60
PP TEST ON DIFFERENCED SERIES
> pp.test(ydif)
Phillips-Perron Unit Root Test
data: ydif
Dickey-Fuller Z(alpha) = -97.7996, Truncation lag parameter = 3,
p-value = 0.01
alternative hypothesis: stationary
Warning message:
In pp.test(ydif) : p-value smaller than printed p-value
• After the first order difference, the series became stationary. We don’t need
the second difference. Model identification and estimation can be done on
the first order differenced series.
• You don’t need to use ADF and PP test at the same time. Both of them are
unit root tests. If you don’t want to determine the order of lags during testing,
use just PP test.
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