Unit Roots: A Selected Survey

Gabriel Rodrguez
Ponticia Universidad Catlica del Per

c Gabriel Rodrguez, 2010

Motivation (1)
Random Walk (No Drift)
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Motivation (2)

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Outline
Basic References: Campbell and Perron (1991), Stock (1994), Phillips
and Xiao (1999), Maddala and Kim (2000), Haldrup and Jansson (2006)
Data Generating Process
Classical Unit Root Statistics
Other Unit Root Statistics
Recent Unit Root Statistics
Some Issues on Unit Roots
Structural Change and Unit Roots
The Role of the Initial Condition and Unit Roots
Covariates and Unit Roots
Additive Outliers and Unit Roots
Further Issues and/or Limitations of this Survey

The Data Generating Process (DGP)

(1)
(2)

yt = dt + ut ; t = 1; :::; T;
ut = u t 1 + vt ;

u0 = 0 (initial condition);
P1
P
vt = 1
i=0 ij i j < 1 and where f t g is a martingale
i=0 i t i with
dierence sequence;
vt has a non-normalized spectral density
zero given by
P at frequency
2
2
);
E(
= 2 (1)2 ; where 2 = limT !1 T 1 1
t
t=1
Under H0 , Functional Central Limit Theorem (FCLT) says: T
W (r); W (r) is a standard Wiener process.

dt =

H0 :

= 1;

zt ; where zt is a set of deterministic components;

HA : j j < 1;

Local-to-unity framework:

= 1 + c=T: Used after.

1=2

P[rT ]
t=1

vt )

4
4.1

Classical Unit Root Statistics

The Dickey-Fuller (DF) Statistic
References: Dickey and Fuller (1979, 1981).
The regression model is
0

yt =
Assume that vt
distributions are
T (b

i:i:d:(0;

zt + yt

+ vt :

(3)

) and zt = f;g. Then, the asymptotic

R
W (r)dW
0:5[W (1)2 1]
R
1) ) R
=
;
W (r)2 dr
W (r)2 dr
R
W (r)dW
0:5[W (1)2 1]
R
=
:
tb ) R
[ W (r)2 dr]1=2
[ W (r)2 dr]1=2

(4)
(5)

If
or zt = f1; tg, then W is replaced by W i = W (r)
R zt = f1g
W Z(ZZ 0 ) 1 Z(r), for i = ; . W i is the projection of W onto the
space orthogonal to z.
Asymptotic critical values at 5.0% are: -1.94, -2.86, -3.43 for zt = f;g,
zt = f1g and zt = f1; tg; respectively.

4.2

The Parametric ADF

Reference: Said and Dickey (1984).
Now, assume that vt is I(0), as in Section 3. In general: vt is an
ARM A(p; q) process.
Assuming, as before that zt = f;g, then
T (b

where

=(

0:5[W (1)2 1] +
R
;
W (r)2 dr
f0:5[W (1)2 1]g +
R
)
;
2
1=2
v [ W (r) dr]

1) )

(6)

tb

(7)

2
2
v )=2 :

Distributions depend of nuissance parameters.

The autocorrelation is corrected using the following autoregression
yt =

zt +

0 yt

1+

k
X

bi y t

(8)

i=1

where

1: Then, H0 :

= 0:
b 0 zt , then (8) is

If we dene a detrended time series as yet = yt

equivalently written as
yet =

et 1
0y

If k ! 1; k 3 =T ! 0; then, T (b
(4) and (5).

k
X
i=1

bi yet

1) and t b converge to the expressions

Important empirical application: Nelson and Plosser (1982).

(9)

4.3

The Semi-Parametric Z b and Zt Statistics

References: Phillips (1987, 1988), Phillips and Perron (1988).

The coe cient is estimated from equation (3). Residuals vbt are used
in constructing an estimator of 2 . Therefore, the autocorrelation is
taken into account in a non-parametric way:
b

= s =T

w( ; k) = 1

k+1

vbt2

+ 2T

k
X
=1

w( ; k)

T
X

t= +1

vbt vbt ; (10)

(11)

Using (10) and (11), we have that

Zb

Zt

0:5(s2 sb2v )
P
T 2 Tt=2 yt2 1
0:5[W (1)2 1]
R
;
)
W (r)2 dr
sbv
0:5(s2 sb2v )
= ( )t b
P
s
[T 2 Tt=2 yt2 1 ]1=2
= T (b

1)

0:5[W (1)2 1]
R
:
[ W (r)2 dr]1=2

(12)

(13)

which are the same as in (4) and (5), respectively.

Asymptotic critical values of Z b at 5.0% are -8.0, -14.1, and -21.7 for
zt = f;g, zt = f1g and zt = f1; tg; respectively.
Asymptotic critical values of Zt at 5.0% are -1.94, -2.86, -3.43 for zt =
f;g, zt = f1g and zt = f1; tg; respectively.

10

4.4

The M-Statistics
References: Stock (1999), Perron and Ng (1996).
Denitions:
MZ

T
2T

M SB = [
M Zt =

1 2
y~T s2
;
P
T
2
2
y
~
t
1
t=1
PT 2
2
~t 1 1=2
t=1 y
] ;
s2

[4s2 T

1 2
y~T
P
T
2
t=1

s2

y~t2 1 ]1=2

(14)
(15)
;

(16)

P
0
where: y~t = yt ^ zt , s2 = s2ek =[1 ^b(1)]2 , s2ek = Tt=k+1 e^2tk , ^b(1) =
Pk ^
j=1 bj , obtained from the autoregression (9).

The limiting distributions of M ZR ans M Zt are the expressions (4) and

(5), respectively. The M SB ) [ W (r)2 dr]1=2 :
Asymptotically: M Zt = (M Z )

(M SB):

Asymptotic critical values: see Stock (1999).

Simulation Monte-Carlo evidence.

11

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5
5.1

Recent Unit Root Statistics

The ADF GLS
References: Elliott, Rothenberg and Stock (ERS, 1996), Ng and Perron
(2001).
Under local-to-unity framework:
= 1 + c=T: Then, T
Wc (r); where Wc (r) is an Ornstein-Uhlenbeck process.

1=2

u[T r] )

It bridges the gap between I(0) and I(1) asymptotics. If c ! 1,

T (b 1) and t b have I(0) distributions. If c ! +1, T (b 1) and t b
have a Cauchy and Normal distributions, respectively.
Particular characteristic: use of GLS detrended data with

= 1 + c=T:

Construction of GLS detrended Data:

yt = [y1 ; (1
zt = [z1; (1

L)yt ]; t = 2; ::::; T;
L)zt ]; t = 2; :::::; T;

(17)
(18)

Let ^ be the estimator that minimizes:

0

S( ) = (yt

12

zt )0 (yt

zt ):

(19)

^0

Detrended series: yet = yt

GLS zt :

All unit root statistics may be used with yet . For the ADF, see ERS
(1996) and for the M-statistics, see Ng and Perron (2001).
When zt = f1g and zt = f1; tg, the limiting distributions are:
0:5[Wc (1)2 1]
R
;
[ Wc (r)2 dr]1=2
0:5[Vc;c (1)2 1]
) R
;
[ Vc;c (r)2 dr]1=2

DF GLS )

(20)

DF GLS

(21)

where Vc;c (r; c) = Wc (r)

(1 c)=(1 c + c2 =3).

rb, b = Wc (1) + 3(1

R
) rWc (r)dr,

Asymptotic critical values: see ERS (1996), Ng and Perron (2001).

13

5.2

A Feasible Point Optimal Test

References: Dufour and King (1991), ERS (1996).
This test is denoted by PTGLS and dened by:

PTGLS (c; c) =

S( )

S(1)
s2

(22)

where S( ) and S(1) are the sums of squared errors from GLS regressions with = and = 1, respectively.
Limiting distributions:
GLS
PT;
(c; c)
GLS
PT;
(c; c)

) c

) c

Wc (r)2 dr

cWc (1)2 ;

(23)

Vc;c (r; c)2 dr + (1

c)Vc;c (1; c)2 ;

(24)

for zt = f1g, and zt = f1; tg, respectively.

Selection of c:
Asymptotic critical values: see ERS (1996), Ng and Perron (2001).

14

Some Issues on Unit Root Tests

6.1

The Asymptotic Gaussian Power Envelope

There is no uniform most powerful (UMP) or uniform most powerful
invariant (UMPI) statistic in unit root framework.
With
lope.

= 1 + c=T , derivation of the asymptotic Gaussian power enve-

Power envelope allows to judge between dierent alternative statistics.

The asymptotic Gaussian power envelope is dened by:
GLS

(c) = Pr[H PT

(c; c) < b

PTGLS

(c)];

(25)

where bPT (c) is such that

GLS

Pr[H PT
with

(0; c) < b

PTGLS

(c)] = ;

the size of the test.

Selection of c (-7.0 for zt = f1g and -13.5 for zt = f1; tg).

15

(26)

6.2

Asymptotic Power Functions

The asymptotic power functions of the tests are dened by:
J (c; c)

= Pr[H J

GLS

(c; c) < bJ

GLS

(c)];
GLS

where J( ) = M Z , M SB, M Zt , and ADF , and the constant bJ

GLS
GLS
is such that Pr[H J (0; c) < bJ (c)] = , the size of the tests.

16

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6.3

Selection of the Lag length

Information Criteria: AIC, BIC
2k
;
fkg
T
log(T )k
= arg min log(s2ek ) +
:
fkg
T

kaic = arg min log(s2ek ) +

(27)

kbic

(28)

Recursive t-sig method

Modied Information Criteria: MAIC, MBIC (Ng and Perron, 2001):
kmic = arg min log(s2ek ) +
fkg

where
^T (k) =

(s2ek ) 1 b 20

CT [^T (k) + k]
T
T
X
t=1

yet2 1 :

(29)

(30)

The MAIC uses CT = 2 and the MBIC uses CT = log(T ):

Ng and Perron (2001), based on theoretical considerations and simulations, recommended MAIC.
The advantage of the MIC is that it takes into account the possible
dependence of b 0 on k:

17

6.4

Summary of Monte-Carlo Evidence

All asymptotic valid tests exhibit nite-sample size distortions for models close to I(0) model.
Importance of data dependent methods to select lag length.
Presence of non-normality or conditional heteroskedasticity increases
size distortions.
Including additional trend terms reduce the power of the unit root test
if the trends are unnecessary.
Span is important, not the frequency.
Power of the unit root depends of the initial condition u0 .
If trend is underspecied, unit root tests and estimators are inconsistent.

18

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7.1

Structural Change and Unit Root Tests

Introduction
References: Perron (1989), Christiano (1992), Banerjee et al. (1992),
Zivot and Andrews (1992), Perron (1997), Perron and Rodrguez (2003a).
Basic idea: misspecication of the trend function is responsible for the
nonrejection of the null hypothesis of a unit root in Nelson and Plosser
(1982).
Models (I, II, III):
zt = f1; 1(t > TB ); tg;
zt = f1; t; 1(t > TB )(t TB )g;
zt = f1; 1(t > TB ); t; 1(t > TB )(t

TB )g;

where 1(:) is the indicator function and TB is the break point. Assume
that TB = T , for some 2 (0; 1):
Perron (1989)
Christiano (1992)
Zivot and Andrews( 1992)

19

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7.2

GLS Detrended Data and Structural Change

Reference: Perron and Rodrguez (2003a).
Limiting distributions of the unit root statistics (Models II and III)
M Z GLS ( ) )

0:5K1 (c; c; )
K2 (c; c; )
(K2 (c; c; ))1=2
0:5K1 (c; c; )

M SB GLS ( ) )
M ZtGLS ( ) )
ADF GLS ( ) )

H M Z (c; c; );
H M SB (c; c; );

(31)
(32)

1=2

H M Zt (c; c; );

(33)

1=2

H ADF (c; c; );

(34)

(K2 (c; c; ))
0:5K1 (c; c; )
(K2 (c; c; ))

where:
(1)

(2)

K1 (c; c; ) = Vcc (1; )2 2Vcc (1; ) 1;

Z 1
Z 1
(2)
(1)
2
Vcc (r; )dr;
K2 (c; c; ) =
Vcc (r) dr 2
0

What does happen with the Model I with GLS Detrended data? Reference: Rodrguez (2007)

20

7.3

Selection of the Break Point

Method 1: Estimating as the break point that yields the minimal
value of the statistics; see Zivot and Andrews (1992), i.e. using
inf J GLS ( )
f g

where J( ) = M Z , M SB, M Zt , and ADF .

By the Continous Mapping Theorem (CMT), the limiting distribution
using method 1 is:
inf J GLS ( ) ) inf H J (c; c; );

2(0;1)

(35)

2(0;1)

Method 2: Choose the break point such that the absolute value of the
t-statistic on the coe cient of the change in slope is maximized; see
Perron (1997):
^ = arg max jtb ( )j;
2
2(";1 ")

Limiting distribution using method 2:

^ = arg max jtb ( )j ) arg max jb4 =(

2
2(";1 ")

2(";1 ")

1=2
3 )j

Hence, the limiting distributions of the statistics are given by

J GLS (^) ) H J (c; c;

21

):

(36)

7.4

The Feasible Optimal Point Test

When

is unknown:
GLS
PT;
(c; c) = f inf

2[";1 "]

S( ; )

inf

2[";1 "]

S(1; )g=s2 :

(37)

Limiting distribution:
GLS
PT;
(c; c) )

sup M (c; 0; )
2[";1 "]

2c

Wc (r)dW (r) + (c

PTGLS

2cc)

Wc (r)2 dr

(38)

(c; c):

Derivation of the power envelope.

Selection of c (c =

sup M (c; c; )
2[";1 "]

22:5)

Asymptotic Power Functions.

Finite-Sample Size and Power.
Empirical Evidence.

22

24
Figure 1: Gaussian Local Power Envelope and the Local Asymptotic Power Functions
of the Tests

The Role of the Initial Condition

Traditionally, theoretical works assume the starting value of time series is zero or has nite expectations. The eect of initial observation
disappears asymptotically.
Exceptions: Elliott (1999), Mller and Elliott (2003) in no structural
change models.
Hui and Rodrguez (2006) introduces both an unknown structural break
and a random initial condition under the alternative hypothesis.
The data generating process (DGP) is the same as before, except that:

Condition A (Initial condition assumption). We assume that u0 is zero

2
when = 1; so u1 = v1 ; while u1 has mean zero and variance 2 =(1
)
when < 1:
The innovations fvt g satisfy the standard asumptions.
Same statistics as in Perron and Rodrguez (2003a).
For Model I and II have the following limiting distributions:
M Z GLS ( ) )

0:5g1 (c; c; )
g2 (c; c; )

M SB GLS ( ) ) (g2 (c; c; ))1=2

0:5g1 (c; c; )
M ZtGLS ( ) )
(g2 (c; c; ))1=2
0:5g1 (c; c; )
ADF GLS ( ) )
(g2 (c; c; ))1=2
Using power envelope, we obtain c =

J MZ

GLS

J M SB

(c; c; )

GLS

GLS

J M Zt

J ADF

(c; c; )

(c; c; )

GLS

(c; c; )

24:

T = 1000 and 10,000 replications to calculate the asymptotic power

function for each statistic.
The curve of power function lies under the power envelope, but not far
from it.
Using Inmum method to choose break point sometimes gives a slightly
higher power function than supremum method.

Figure 1. Gaussian Power Envelope and Asymptotic Power Functions; Inmum Method
and Fixed and Random Initial Condition.

Figure 2. Gaussian Power Envelope and Asymptotic Power Functions; Supremum

Method and Fixed and Random Initial Condition.

Covariates and Unit Root Tests

Importance of covariates in improving the power of unit root tests.
References: Hansen (1995), Elliott and Jansson (2003). For structural
change models: Hui and Rodrguez (2006).
The data generating process (DGP):
yt = dyt + uyt ;
xt = dxt + uxt ;
A (L)

[1

L]uy;t
ux;t
= 1 + cT

A (L) ut ( ) = et
1

(39)
(40)
(41)
(42)

where xt , an m 1 vector, is an arbitrary number of stationary covariates containing extra information of yt , the variable to be tested.
is dened as the spectral density at the frequency zero (scaled by 2 )
of ut ( ). Therefore R2 = ! yy1 ! yx xx1 ! 0yx is a measure of the long-run
correlation between shocks to xt and quasi-dierences of yt at the frequency zero. The value of R2 represents the contribution of covariates
to the explanation of yt , and the value of R2 ranges from zero to unity.

27

The optimal statistic is dened by

i

P (1; ) =

inf

2(0;1)

inf

2(0;1)

where u^it (r) = zt (r)

T
X

u^it ( ; )0

1 i
u^t (

u^it (1; )0

1 i
u^t

; )

t=1

T
X

(1; )

c:

(43)

t=1

dt (r)0 ^ and r = ; 1:

The Theorem establishes that for cases i = 1 and 2 :

P i (1; ) )

1 (c; c; R

)+

i
2 (c; c;

; R2 )

(44)

The asymptotic power depends on c, which corresponds to one particular point under the alternative hypothesis.
The distribution of the P i (1; ) test also depends on the parameter R2 :
When R2 = 0; there is no covariate correlated with the quasi-dierences
of yt and consequently we retrieve the same asymptotic distribution as
that derived in Perron and Rodrguez (2003). When R2 is greater than
zero, the limiting distribution is a function of R2 , indicating that extra
information contained in the covariates may make a dierence on the
performance of the test.

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Figure 1. Power Envelopes for R2 = 0:0; 0:3; 0:5; 0:7; 0:9.

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