This paper provides a theoretical functional representation of the density function related to the Dickey- Fuller random variable. The approach is extended to cover the multivariate case in two special frameworks: the independence and the perfect correlation of the series.Dickey-Fuller distribution, unit root

Cerqueti, Roy

Costantini, Mauro

Lupi, Claudio

English

Research Papers in Economics

Università degli Studi del Molise
Facoltà di Economia
Dipartimento di Scienze Economiche, Gestionali e Sociali
Via De Sanctis, I-86100 Campobasso (Italy)
ECONOMICS & STATISTICS DISCUSSION PAPER
No. 55/09
A Characterization of the Dickey-Fuller Distribution, 
With Some Extensions to the Multivariate Case 
by
Roy Cerqueti
University of Macerata, Dept. of Economic and Financial Institutions
Mauro Costantini
University of Vienna, Dept. of Economics
and
Claudio Lupi
University of Molise, Dept. SEGeS
The Economics & Statistics Discussion Papers are preliminary materials circulated to stimulate discussion 
and critical comment. The views expressed in the papers are solely the responsibility of the authors.A Characterization of the Dickey-Fuller Distribution,
With Some Extensions to the Multivariate Case
Roy Cerqueti Mauro Costantiniy Claudio Lupiz
Abstract
This paper provides a theoretical functional representation of the density function related to the Dickey-
Fuller random variable. The approach is extended to cover the multivariate case in two special frameworks:
the independence and the perfect correlation of the series.
key words: Dickey-Fuller distribution, unit root
JEL codes: C12, C16, C22
1 Introduction
In this paper we deal with the theoretical case where n possibly cross-dependent time series are
generated by
yi;t = yi;t 1 + ui;t (i = 1;:::;n; t = 1;:::;T)
= yi;0 +
t X
j=1
ui;j
= yi;0 + Si;t (1)
where yi;0 = ci with probability one, or it has a given probability distribution. The ui;t’s are
assumed to satisfy some regularity conditions so that a suitably normalized transform of Si;t,
S
i;T(r) := T 1=2 1
i Si;bTrc (where bc denotes the integer part and r 2 [0;1]), is such that
S
i;T(r) ) Wi(r) as T ! 1, with Wi(r) a Wiener process (see e.g., Phillips, 1987, for a detailed
discussion of such regularity conditions and of the exact meaning of the normalization). Here
we take the simplifying assumption that ui;t  iid(0;2
i ), so that the conditions for the weak
convergence of the normalized partial sums are trivially satisﬁed.
It is well known that, under the null H0 : i = 1, the t-ratio based on the OLS estimator of i, b ti,
in
yi;t = iyi;t 1 + ei;t : (2)
(corresponding author) University of Macerata, Dept. of Economic and Financial Institutions, Via Crescimbeni,
20. I-62100 Macerata (Italy). roy.cerqueti@unimc.it
yUniversity of Vienna, Dept. of Economics, Bruenner Strasse, 72. A-1210 Vienna (Austria).
mauro.costantini@univie.ac.at
zUniversityofMolise, Dept. ofEconomics, ManagementandSocialSciences, ViaDeSanctis. I-86100Campobasso
(Italy). lupi@unimol.it
12 The Univariate Case 2
has the non-standard Dickey-Fuller limiting distribution
b ti )
Z 1
0
Wi(r)dWi(r)
Z 1
0
W2
i (r)dr
  1
2
(3)
as T ! 1 (see e.g., Phillips, 1987).
Building on Ruben (1962), in Section 2 we derive an explicit formulation of the conventional uni-
variate Dickey-Fuller distribution (3). Although not easy to manage, the proposed formulation can
be used to derive analytical results, as opposed to the conventional simulation approach. In this
respect, our work is related to Abadir (1995). In Section 3 we extend the analysis to the multi-
variate case by focusing on the special cases where the series are either independent or perfectly
correlated. Section 4 concludes.
In the derivation of the theoretical results, we assume that all the random quantities introduced
in the paper are contained in a ﬁltered probability space (
;F;fFtgt>0;P).
2 The Univariate Case
Consider the Dickey-Fuller distribution (3): by deﬁnition of stochastic integral, we take a partition
of the interval [0;1] in N intervals of length 1=N. If N is large enough, we can approximate the
asymptotic distribution of b ti under the null as follows:
Z 1
0
Wi(r)dWi(r)
Z 1
0
W2
i (r)dr
  1
2


 
N X
k=1
Wi(k=N)[Wi(k=N)   Wi ((k   1)=N)]
! 
1
N
N X
k=1
W2
i (k=N)
!  1
2
=
=
1
2
 
W2
i (1)   1

 
1
N
N X
k=1
W2
i (k=N)
!  1
2
=: Xi=Yi; (4)
where
Xi :=
1
2
 
W2
i (1)   1

(5)
Yi :=
 
1
N
N X
k=1
W2
i (k=N)
! 1
2
: (6)
Therefore, by deﬁnition of Wiener process, the distribution of the univariate Dickey-Fuller test
under the stated conditions can be approximated as follows:
b ti ) Xi=Yi 
1
2
 
2(1)   1

(CvM0(1))
1
2
; (7)
where 2(1) denotes a Chi-squared distribution with 1 degree of freedom and CvM0(1) denotes a
zero level Cram´ er-von Mises distribution with 1 degree of freedom.2 The Univariate Case 3
In order to derive an analytical expression for the density function of the Dickey-Fuller distribu-
tion, we need to introduce the ratio variable Zi := Xi=Yi. The cumulative distribution function of
Zi will be denoted as Fi. For z 2 R, we have
Fi(z) = P(Zi  z)
= P(Xi=Yi  z)
=
Z +1
0
Z yz
0
fXi;Yi(x;y)dxdy ; (8)
where fXi;Yi is the joint density function of the random variables Xi and Yi.
The density function fi of the variable Zi is
fi(z) =
@
@z
Z +1
0
Z yz
0
fXi;Yi(x;y)dxdy

=
Z +1
0
yfXi;Yi(yz;y)dy : (9)
We need to ﬁnd an explicit form of the density function fXi;Yi, in order to get an expression for
fi. Fixed x;y 2 R, we have
fXi;Yi(x;y) = fYijXi(yjXi = x)fXi(x); (10)
where fYijXi is the density function of Yi conditional on Xi, and fXi is the (marginal) density
function of the random variable Xi.
We write the cumulative distribution functions of Xi as:
FXi(x) = P(Xi  x)
= P

1
2
(W2
i (1)   1)  x

= P(W2
i (1)  2x + 1)
= KXi
Z 2x+1
 1
s 1=2e s=2ds; (11)
where KXi is the normalizing constant. The density function fXi is then
fXi(x) = KXi
@
@x
Z 2x+1
 1
s 1=2e s=2ds

= 2KXi(2x + 1) 1=2exp

2x + 1
2

: (12)2 The Univariate Case 4
The cumulative distribution function of Yi conditional on Xi = x is
FYijXi(yjXi = x) = P(Yi  yjXi = x)
= P
0
@
0
@ 1
N
N X
k=1
W2
i (k=N)
!1=2
 y
 
 
 
W2
i (1) = 2x + 1
1
A
= P
0
@
 
1
N
"
N 1 X
k=1
W2
i (k=N) + 2x + 1
#!1=2
 y
1
A
= P
 
N 1 X
k=1
W2
i (k=N) + 2x + 1  Ny2
!
= P
 
N 1 X
k=1
W2
i (k=N)  Ny2   2x   1
!
: (13)
Equation(13)suggeststhatweneedtodiscussthedistributionofasumofsquarednon-independent
zero-mean Gaussian variables.
Denote the Cram´ er-von Mises distribution as
V :=
N 1 X
k=1
W2
i (k=N); (14)
and
FV (v) = P(V  v); v 2 R+: (15)
Our approach relies on an invariant symmetry property of the random V (see Ruben, 1962). In
particular, Ruben (1962) shows that the cumulative distribution function of V can be written as
series expansions of Chi-squared cumulative distribution functions, i.e. there exists a sequence of
real numbers fjgj2N such that
FV (v) =
+1 X
j=0
jFN 1+2j(v); (16)
where FN 1+2j is the cumulative distribution function of a Chi-squared random variable with
(N   1 + 2j) degrees of freedom. By substituting the explicit expression of the F’s in (16), we
obtain
FV (v) =
+1 X
j=0
j
Z w
0
e s=2s(N 3+2j)=2ds; (17)
where we assume without loss of generality that the ’s contain also the normalizing constants
related to the Chi-squared distributions. By (17), then (13) can be written as
FYijXi(yjXi = x) = P
 
N 1 X
k=1
W2
i (k=N)  Ny2   2x   1
!
=
+1 X
j=0
j
Z Ny2 2x 1
0
e s=2s(N 3+2j)=2ds: (18)3 The Multivariate Dickey-Fuller Distribution 5
The conditional density function fYijXi is
fYijXi(yjXi = x) =
+1 X
j=0
j
@
@y
"Z Ny2 2x 1
0
e s=2s(N 3+2j)=2ds
#
= 2Ny  exp

 
Ny2   2x   1
2



+1 X
j=0
j[Ny2   2x   1](N 3+2j)=2 : (19)
By substituting (12) and (19) into (10), we have
fXi;Yi(x;y) = 4N  KXi
y
p
2x + 1
 exp

 
Ny2   2(2x   1)
2



+1 X
j=0
j[Ny2   2x   1](N 3+2j)=2 : (20)
Finally, from (9) and (20) we have
fZi(z) = 4N  KXi
Z +1
0
y2
p
2yz + 1
 exp

 
Ny2   2(2yz   1)
2



+1 X
j=0
j[Ny2   2yz   1](N 3+2j)=2dy : (21)
Although rather involved, (21) can in principle be used to derive analytical results on the Dickey-
Fuller distribution.
3 The Multivariate Dickey-Fuller Distribution
This section is devoted to the analysis of the distribution of the multivariate Dickey-Fuller t-ratio.
Let’s deﬁne the random vector (Z1;:::;Zn) of asymptotic distributions under the null, ac-
cordingly with (3) and (6). More precisely, we can write
Zi :=
1
2
 
W2
i (1)   1

 
1
N
N X
k=1
W2
i (k=N)
!  1
2
i = 1;:::;n: (22)
Our aim is to provide a closed form expression for the joint density function fZ1;:::;Zn of the
random vector (Z1;:::;Zn).
Here we study the two extreme cases where the series are either independent or perfectly
correlated.
3.1 Independent Series
Assume that the series y’s are cross sectional independent. Once that the univariate (marginal)
density has been derived, this case becomes trivial. Indeed, for each (z1;:::;zn) 2 Rn, we can3 The Multivariate Dickey-Fuller Distribution 6
write the density function fZ1;:::;Zn simply as
fZ1;:::;Zn(z1;:::;zn) =
n Y
i=1
fZi(zi); (23)
where fZi(zi) is given by (21).
3.2 Perfectly Correlated Series
Fixed i = 2;:::;n, we assume that yi and yi 1 are perfectly correlated, and there exists a constant
i such that
yi = iyi 1 : (24)
Of course, the ordering of the series is purely conventional. Any ordering would be possible just
by changing the parameter i.
The dependence among the variables is reﬂected on the dependence among the Wiener processes
Wi. In particular, by using the derivation of the Dickey-Fuller asymptotic distribution, then con-
dition (24) can be rewritten in terms of the Wiener processes Wi:
Wi = iWi 1 : (25)
By substituting (25) into (22), we obtain
Zi :=
1
2
 
(iWi 1(1))2   1


1
N
PN
k=1(2
iW2
i 1(k=N)
 1
2
= jijZi 1 +
1
2(2
i   1)

2
i
N
PN
k=1 W2
i 1(k=N)
 1
2
i = 1;:::;n: (26)
Formula (26) allows us to write explicitly the conditional cumulative distribution of Zi given Zi 1.
Consider zi;zi 1 2 R. Then (18) gives
P(Zi  zijZi 1 = zi 1) =
= P
0
B
@jijZi 1 +
1
2(2
i   1)

2
i
N
PN
k=1 W2
i 1(k=N)
 1
2
 zi

 
 


Zi 1 = zi 1
1
C
A
= P
0
@
N X
k=1
W2
i 1(k=N) 
"
2jij(zi   jijzi 1)
p
N(2
i   1)
#21
A : (27)
Now, considerthejointdensityfunctionoftherandomvariable(Z1;:::;Zn). Given(z1;:::;zn) 2
Rn, the dependence condition (25) implies
fZ1;:::;Zn(z1;:::;zn) = fZ1(z1) 
n Y
i=2
fZijZi 1(zijZi 1 = zi 1); (28)4 Conclusions 7
where
fZijZi 1(zijZi 1 = zi 1) =
@
@zi
[P(Zi  zijZi 1 = zi 1)]: (29)
Consider i = 1:::;n and zi;zi 1 2 R. Then (18) gives
P(Zi  zijZi 1 = zi 1) =
+1 X
j=0
j
2
6
4
Z

2jij(zi jijzi 1)
p
N(2
i  1)
2
0
e s=2s(N 1+2j)=2ds
3
7
5 : (30)
The density function of Zi conditional on Zi 1, fZijZi 1, is then
fZijZi 1(zijZi 1 = zi 1) =
+1 X
j=0
j
@
@zi
2
6
4
Z

2jij(zi jijzi 1)
p
N(2
i  1)
2
0
e s=2s(N 1+2j)=2ds
3
7
5
=
82
i(zi   jijzi 1)
N(2
i   1)2  exp
8
<
:
 
1
2
"
2jij(zi   jijzi 1)
p
N(2
i   1)
#29
=
;


+1 X
j=0
j
"
2jij(zi   jijzi 1)
p
N(2
i   1)
#N 1+2j
: (31)
4 Conclusions
In this paper an explicit approximation of the density function of the multivariate Dickey-Fuller
random variable is provided. We proceed by analyzing at ﬁrst the univariate case. Our result is
grounded on an invariant symmetry property of some random variables involved in the Dickey-
Fuller distribution (see Ruben, 1962).
As in Abadir (1995), the followed approach allows us to avoid the conventional simulation-
based approach. The theoretical results regarding the univariate case are then extended to the
multivariate framework under the assumptions of independent and perfectly correlated series.
Although the independent and the perfectly correlated cases are two extreme settings, they
represent the starting point for exploring models with less restrictive assumptions. In this respect,
the analysis of a general cross sectional dependence case is already in our research agenda.
Moreover, while we deal here only with the Dickey-Fuller distribution in the absence of de-
terministic terms, further extensions are under scrutiny to cope with the “constant” and “constant
plus linear trend” cases.
References
Abadir, K. M. (1995), “The Limiting Distribution of the t Ratio under a Unit Root,” Econometric
Theory, 11, 775–793.
Phillips, P. C. B. (1987), “Towards a Uniﬁed Asymptotic Theory for Autoregression,” Biometrika,
74, 535–547.
Ruben, H. (1962), “Probability Content of Regions Under Spherical Normal Distributions, IV:
TheDistributionofHomogeneousandNon-HomogeneousQuadraticFunctionsofNormalVari-
ables,” Annals of Mathematical Statistics, 33, 542–570.

Probability Content of Regions Under Spherical Normal Distributions, IV:

The Limiting Distribution of the t Ratio under a Unit Root,” Econometric Theory,

Towards a Uniﬁed Asymptotic Theory for Autoregression,”

A Characterization of the Dickey-Fuller Distribution, With Some Extensions to the Multivariate Case

http://web.unimol.it/progetti/repec/mol/ecsdps/ESDP09055.pdf

A Characterization of the Dickey-Fuller Distribution, With Some Extensions to the Multivariate Case

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics