For a multivariate time series model with structural breaks, explicit representations of the Beveridge-Nelson and Granger-Gonzalo-Proietti permanent trends are derived from the Johansen maximum likelihood estimates.Multivariate Time Series, Structural Breaks, Permanent Trends

Clifford L.F. Attfield

English

Research Papers in Economics

Structural Breaks and Permanent Trends
C.L.F. Attﬁeld
January 2003
Discussion Paper No 03/545
Department of Economics
University of Bristol
8 Woodland Road
Bristol BS8 1TN
UKStructural Breaks and Permanent Trends
C.L.F. Att…eld¤
University of Bristol
January 2003
Abstract
For a multivariate time series model with structural breaks, explicit rep-
resentations of the Beveridge-Nelson and Granger-Gonzalo-Proietti perma-
nent trends are derived from the Johansen maximum likelihood estimates.
Keywords: Multivariate Time Series, Structural Breaks, Permanent Trends.
JEL Classi…cation: C32, C51, E3
1 Introduction
A good deal of recent research in econometric time series modelling is concerned
withtheimpact ofstructural breaks-breaksinmeanandtrends-incointegrating
systems (e.g., Johansen et al [10, 2000], Banerjee et al [2, 1998], Gregory and
Hansen [9, 1996], Bai and Perron [3, 1998] and Bai and Perron [4, 2001]). Models
of permanent stochastic trends which rely on the notion of cointegration and
vector error correction areobviously a¤ected by structural breaks and the object
ofthispaperisto derivepermanent trendsin thepresenceofbreaks. Weconsider
amaximum oftwo breaksandobtainexplicit expressionsforthepermanent trend
de…nitions of Beveridge and Nelson [1, 1981], hereafter B-N, as extended in a
multivariate context by King et al [11, 1991] and for the Granger and Gonzalo
[8, 1995], hereafter G-G, de…nition as interpreted by Proietti [13, 1997]. The
next section de…nes the model and derives the permanent trends for the case of
structural breaks with the main algebraic results in an appendix.
2 The Multivariate Model with Structural Breaks
Suppose xt isa (p£1) vectorofI(1) variables with cointegrating rank r: Suppose
further that there are two breaks in the sample with T1 observations in the
…rst period, and T2 ¡T1 observations in the second period, and T observations
¤I am indebted to Jonathan Temple for helpful discussions on material in this paper. The
research was undertaken with the assistance of a University of Bristol Fellowship and an award
from the ESRC. Any errors or omissions are my own responsibility.
1altogether so that there are T ¡T2 observations in the third period. Johansen et
al [10, (2000)] derive a likelihood ratio test for cointegration in the presence of
breaks in trend and mean at known points. In general the model can bewritten:
¢xt =µo¥t +
k X
j=1
µj¢xt¡j +¯(®0; °0)
µ
xt¡1
t¥t
¶
+
k+1 X
i=1
3 X
j=2
·jiDjt¡i +³t (1)
where xt = (x1t; x2t; :::;xpt)0;Djt = 1 for t = Tj¡1, with To = 0; and Djt = 0
otherwise and ¥0
t = (¥0
1t;¥0
2t; ¥0
3t) with ¥jt = 1 for Tj¡1 + k + 2 5 t 5 Tj
and zero otherwise and where To = 0. This speci…cation allows for shifts in
the intercepts of both the VECM and cointegrating equations (although they
cannot be identi…ed individually), in the term µo¥t and shifts in the trend in
the cointegrating equations only, in the term °0t¥t with ¯°0 = 0: The ¥jt’s
are dummies for the e¤ective sample period for each sub-period and the Djt¡i’s
have the e¤ect of eliminating the …rst k +1 residuals of each period from the
likelihood, thereby producing the conditional likelihood function given theinitial
values in each period.
2.1 Permanent Stochastic Trends
2.1.1 Beveridge-Nelson
The de…nition of the B-N permanent trend component in a multivariate context
is:
xBN¡P
t = xt +
i=1 X
i=1
Et(¢xt+i ¡¹¢x) (2)
see, for example, Cochrane [7, 1994]. To determine a solution for (2), write the
VECM in (1) as
¢xt = KoHt +
k X
j=1
µj¢xt¡j +¯vt¡1 +³t: (3)
where Ko = (µo; {) where { contains the ·ji vectors; and:
Ht =
·
¥t
Dt
¸
where Dt contains the Djt¡is; and vt¡1 =®0xt¡1 +°0t¥t: It follows that:
vt =®0xt +°0(t+1)¥t =®0¢xt +°0¥t +vt¡1
and then:
vt =KooHt +®0µ1¢xt¡1 +¢¢ ¢ +®0µk¢xt¡k +
¡
I +®0¯
¢
vt¡1 +®0³t (4)
where:
Koo =
¡
®0µo +°0; ®0{
¢
:
2De…ning µ(L) = Ip ¡
k X
j=1
µjLj, the model in (3) can also be written:
µ(L)¢xt =KoHt +¯vt¡1 +³t: (5)
Appending (4) to the system in (3) we have a …rst order stationary vector au-
toregression of the form:
zt =AoHt +A1zt¡1 +ª³t t =1;:: :; T (6)
where z0
t is the (1 £pk +r) vector:
z0
t = (¢x0
t;¢x0
t¡1;:::; ¢x0
t¡k+1; v0
t):
The matrices Ao and A1 are de…ned as:
Ao =
2
6
6 6
6
6
6 6
4
Ko
0
0
. . .
0
Koo
3
7
7 7
7
7
7 7
5
and:
A1 =
2
6 6
6
6 6
6
6
4
µ1 µ2 ¢¢¢ µk¡1 µk ¯
I 0 ¢¢¢ 0 0 0
0 I ¢¢¢ 0 0 0
. . .
. . . ¢¢¢
. . .
. . .
. . .
0 0 ¢¢¢ I 0 0
®0µ1 ®0µ2 ¢¢¢ ®0µk¡1 ®0µk ®0¯ +I
3
7 7
7
7 7
7
7
5
(7)
and:
ª =
2
6
6
6 6
6
4
I
0
. . .
0
®0
3
7
7
7 7
7
5
:
From (6) it follows that:
E(zt) =¹z =(I¡A1)
¡1AoHt (8)
so that:
zt ¡¹z =(I¡A1L)
¡1 ª³t: (9)
De…ne the matrix:
G=
2
6
6
6
4
Ip
0
. . .
0
3
7
7
7
5
3Then G0zt selects out ¢xt and it follows from (6) that:
¢xt ¡¹¢x = G0(zt ¡¹z) =G0 (I ¡A1L)¡1 ª³t = C(L)³t (10)
which is the moving average representation. Inverting [I ¡A1]; see Appendix,
it is straightforward to show that1:
C(1) =G0[I¡A1]¡1ª =µ(1)¡1 ¡µ(1)¡1¯(®0µ(1)¡1¯)¡1®0µ(1)¡1
where µ(1) =Ip ¡
k X
1
µi:
The expectations term in equation (2), can then be written:
i=1 X
i=1
Et(¢xt+i ¡¹¢x) = G0A1[I ¡A1]¡1(zt ¡¹z): (11)
Some algebra, see Appendix, produces:
G0A1 (I¡A1)¡1 =
·
C(1)
P
1
µi;C(1)
P
2
µi; C(1)
P
3
µi;¢ ¢¢ C(1)µk;¡Q
¸
: (12)
where Q= µ(1)¡1¯(®0µ(1)¡1¯)¡1:
Let:
µ¤(L) =
X
1
µi+
X
2
µiL +::: +µkLk¡1
then from (11) obtain:
xBN¡P
t =xt +C(1)µ¤(L)(¢xt ¡¹¢x) ¡Q(vt ¡¹v): (13)
Substituting µ¤(L)(1 ¡ L) = µ(L) ¡µ(1) and vt = ®0xt +°0(t +1)¥t into (13)
gives:
xBN¡P
t =xt ¡Q®0xt ¡Q°0(t +1)¥t +C(1)[µ(L) ¡µ(1)] xt ¡C(1)µ¤(1)¹¢x +Q¹v
=C(1)µ(1)xt ¡Q°0(t+1)¥t +C(1)µ(L)xt ¡C(1)µ(1)xt ¡C(1)µ¤(1)¹¢x +Q¹v
) xBN¡P
t =C(1)µ(L)xt ¡Q°0(t +1)¥t +±o (14)
with ±o =¡C(1)µ¤(1)¹¢x +Q¹v.
¹¢x and ¹v are the means of stationary variables and can be estimated from
sample counterparts. Explicit formulae can be deduced from (8), i.e.:
G0¹z = ¹¢x =G0 (I ¡A1)
¡1 AoHt
1Proietti [13, 1997] obtains the same result using the Kalman …lter except that instead of
£(1)
¡1 he has (£(1)¡ ¯®
0)
¡1 but it is easy to show that the two forms give exactly the same
C(1):
4and:
J0¹z = ¹v =J0 (I¡A1)
¡1 AoHt
where J0 = (0;0; :::;0;I) selects out the rows of ¹z associated with ¹v: Multi-
plying out the matrices using the results in the appendix yields:
¹¢x =G0 (I ¡A1)
¡1 AoHt
=
¡
C(1)µo ¡µ(1)¡1¯(®0µ(1)¡1¯)¡1°0¢
¥t +C(1){Dt
=C(1)KoHt ¡Q°0¥t
and:
¹v =J0 (I¡A1)¡1AoHt = ¡(®0µ(1)¡1¯)¡1 £
(®0µ(1)¡1µo +°0)¥t +®0µ(1)¡1{Dt
¤
As a …rst di¤erence:
¢xBN¡P
t =C(1)µ(L)¢xt ¡Q°0¥t (15)
and substituting the VECM in (5) into (15) we obtain:
¢xBN¡P
t = C(1)(KoHt +¯vt¡1 +³t) ¡Q°0¥t
but, since C(1)¯ =0; it follows that:
¢xBN¡P
t =C(1)KoHt +C(1)³t ¡Q°0¥t
=¹¢x +C(1)³t (16)
so that the trends have no long run impact on the permanent component. The
equivalent de…nitions of trend growth in (15) and (16) derived from the B-N
decomposition are used by King et al [11, 1991] and Cochrane [7, 1994] for the
case of no structural breaks.
The de…nition of trend from the B-N decomposition can be justi…ed on the
grounds that it is only permanent shocks which impact on the trend. That is,
suppose, without any loss of generality that the non-singular matrix ¡o decom-
poses the error term ³t into permanent and transitory shocks, i.e.:
¡o³t ="t =
µ
"P
t
"T
t
¶
: (17)
We can then decompose (16) into the model:
¢xBN¡P
t =¹¢x +C(1)¡¡1
o ¡o³t
=¹¢x +¡(1)"t
and since it is only permanent shocks that have an impact on the trend, it must
be the case that:
C(1)¡¡1
o =¡(1) =[¡1;0] (18)
where ¡1 is (p£p¡r) and 0 is (p£r), where p is the number ofvariables in the
system and r is the dimension of the cointegrating space. It follows that:
¢xBN¡P
t =¹¢x +C(1)³t =¹¢x +¡1"P
t : (19)
and so it is only permanent shocks which have a long run impact on the trend.
52.1.2 Gonzalo and Granger
The B-N de…nition of trend has been criticised by Quah [6, 1989] and Lippi
and Reichlin [12, 1994] because it does not contain any changes in permanent
and transitory shocks. Gonzalo and Granger, hereafter G-G, [8, 1995] suggest
a method of decomposing the series into permanent and transitory components
to incorporate some shock dynamics into the permanent component which, in
essence, allows CHANGES in the transitory component to have an impact on
CHANGES in the permanent component and thus a transitory impact on the
LEVEL ofthepermanent component. Their procedureis to decompose xt into a
permanent component, x
GG¡P
t ; which is a linear function of the common factors
in the model, ft, and a stationary transitory component, x
GG¡T
t , i.e.:
xt =xGG¡P
t +xGG¡T
t = A1ft +e xt:
In a model without intercepts or trends they de…ne the common factors as:
ft =¯0
?xt
where ¯? represents the orthogonal complement of ¯, such that ¯0
?¯ = 0; and
the transitory component as a linear function of the stationary error correction
terms:
xGG¡T
t = e xt =A2®0xt
which ensures that the only linear combinations of xt on which e xt has no long
run impact will be ¯0
?xt: To see this multiply through equation (1) by ¯0
?:
Since:
xt =A1¯0
?xt +A2®0xt
inverting the matrix: ·
¯0
?
®0
¸
yields:
(A1; A2) =
¡
®?(¯0
?®?)¡1;¯(®0¯)¡1¢
:
where ®? is the orthogonal complement of ®:
Adding a broken trend in the cointegrating vector yields a transitory compo-
nent of the form:
xGG¡T
t =¯(®0¯)¡1®0xt +¯(®0¯)¡1°0(t+1)¥t
so the permanent component becomes:
xGG¡P
t = ®?(¯0
?®?)¡1¯0
?xt ¡¯(®0¯)¡1°0(t +1)¥t
so that, in …rst di¤erences:
¢xGG¡P =®?(¯0
?®?)¡1¯0
?¢xt ¡¯(®0¯)¡1°0¥t: (20)
6Substitute the moving average representation in (10), i.e.:
¢xt = ¹¢x +C(L)³t
into equation (20) to obtain:
¢xGG¡P = ®?(¯0
?®?)¡1¯0
?¹¢x+®?(¯0
?®?)¡1¯0
?C(L)³t¡¯(®0¯)¡1°0¥t: (21)
Then, since:
®?(¯0
?®?)¡1¯0
?¹¢x =®?(¯0
?®?)¡1¯0
?
¡
C(1)KoHt ¡Q°0¥t
¢
and ®?(¯0
?®?)¡1¯0
?C(1)=C(1) and:
®?(¯0
?®?)¡1¯0
?Q=
¡
I ¡¯(®0¯)¡1®0¢
Q
=Q¡¯(®0¯)¡1
it follows that:
®?(¯0
?®?)¡1¯0
?¹¢x =C(1)KoHt ¡Q°0¥t +¯(®0¯)¡1°0¥t
=¹¢x +¯(®0¯)¡1°0¥t:
Substituting this result into (21) yields:
¢xGG¡P =¹¢x +®?(¯0
?®?)¡1¯0
?C(L)³t:
Then using:
C(L) =C(1) +(1 ¡L)C¤(L)
we obtain:
¢xGG¡P =¹¢x +®?(¯0
?®?)¡1¯0
? (C(1) +(1¡L)C¤(L)) ³t
=¹¢x +C(1)³t +®?(¯0
?®?)¡1¯0
?C¤(L)(1¡L)¡¡1
o "t
=¹¢x +¡1"P
t +!1(L)¢"P
t +!2(L)¢"T
t : (22)
using the results in (17) and (18).
Comparing (22) with (19) we see that ¢xGG¡P = ¢xBN¡P +!1(L)¢"P
t +
!2(L)¢"T
t so that short run dynamics in both permanent and transitory compo-
nents have a short run impact in the Granger-Gonzalo formulation.
Proietti [13, 1997] noticed that the B-N decomposition can be amended to
form a G-G decomposition by replacing the term µ(L) in equation (14) with:
µ(L) =µ(1) +(1 ¡L)µ¤(L)
to give:
xBN¡P
t =C(1)µ(1)xt +C(1)µ¤(L)¢xt ¡Q°0(t +1)¥t +±o: (23)
7Adding the term -C(1)µ¤(L)¢xt to (23) results in:
xGGP¡P
t =C(1)µ(1)xt ¡Q°0(t+1)¥t +±o (24)
which includes error dynamics. To see this, take …rst di¤erences:
¢xGGP¡P
t =C(1)µ(1)¢xt ¡Q°0¥t
and substitute the moving average representation to give:
¢xGGP¡P
t =C(1)µ(1)¹¢x +C(1)µ(1)C(L)³t ¡Q°0¥t
=C(1)µ(1)
£
C(1)KoHt ¡Q°0¥t
¤
+C(1)µ(1)C(L)³t ¡Q°0¥t
=C(1)KoHt ¡Q°0¥t +C(1)µ(1)C(L)³t
=¹¢x +C(1)µ(1)C(L)³t
since C(1)µ(1)C(1) = C(1) and C(1)µ(1)Q = 0. Then using C(L) = C(1) +
(1 ¡L)C¤(L) we obtain:
¢x
GGP¡P
t =¹¢x +C(1)µ(1)(C(1) +(1 ¡L)C¤(L))³t
=¹¢x +C(1)³t +µ(1)(1¡L)C¤(L)¡¡1
o "t
=¹¢x +¡1"P
t +!¤
1(L)¢"P
t +!¤
2(L)¢"T
t (25)
where wehave used the same decomposition of the equation error ³t into transi-
tory and permanent shocks as in (17). The expression in (25) contains changes
in both permanent and transitory shocks and is only di¤erent from (22) by the
weights on these short run dynamic components.
8Appendix
Theorem 1 If
A1 =
2
6
6
6 6
6
6
6
4
µ1 µ2 ¢¢¢ µk¡1 µk ¯
I 0 ¢¢¢ 0 0 0
0 I ¢¢¢ 0 0 0
. . .
. . . ¢¢¢
. . .
. . .
. . .
0 0 ¢¢¢ I 0 0
®0µ1 ®0µ2 ¢¢¢ ®0µk¡1 ®0µk ®0¯ +I
3
7
7
7 7
7
7
7
5
and:
G=
2
6 6
6
4
Ip
0
. . .
0
3
7 7
7
5
then:
G0 (I ¡A1)
¡1 =µ(1)¡1 ¡µ(1)¡1H
µ
P
1
µiµ(1)¡1
¶
;
¡
µ(1)¡1 ¡µ(1)¡1Hµ(1)¡1¢P
2
µi; ¢¢ ¢
¢ ¢¢ ;
¡
µ(1)¡1 ¡µ(1)¡1Hµ(1)¡1¢
µk;¡µ(1)¡1¯(®0µ(1)¡1¯)¡1
where H =¯(®0µ(1)¡1¯)¡1®0 and µ(1) = Ip ¡
k X
1
µi:
Proof
Let
£ =
2
6
6 6
6
6
6 6
4
µ1 µ2µ3 ¢¢ ¢µk¡1 µk
Ip 0 0 ¢¢ ¢ 0 0
0 Ip 0 ¢¢ ¢ 0 0
0 0 Ip ¢¢ ¢ 0 0
. . .
. . .
. . .
...
. . .
. . .
0 0 0 ¢¢ ¢ Ip 0
3
7
7 7
7
7
7 7
5
¯0
k =
£
¯0 0¢ ¢¢ 0
¤
®0
k =
£
®0 0¢¢ ¢0
¤
Then:
A1 =
·
£ ¯k
®0
k£ Ir +®0
k¯k
¸
and so:
I(pk+r) ¡A1 =
·
Ipk ¡£ ¡¯k
¡®0
k£ ¡®0
k¯k
¸
:
By direct inversion:
(I ¡A1)
¡1 =
·
B¡1 ¡B¡1¯k(®0
kB¡1¯k)¡1®0
k£B¡1 ¡B¡1¯k(®0
kB¡1¯k)¡1
¡(®0
kB¡1¯k)¡1®0
k£B¡1 ¡(®0
kB¡1¯k)¡1
¸
9where B =Ipk ¡£: We requirethe …rst p rows of(I ¡A1)
¡1, i.e. G0 (I ¡A1)
¡1 :
By direct inversion we obtain:
B¡1 =(Ipk ¡£)¡1 =E +F
where:
E =
2
6
6
6 6
6
6
6
4
µ(1)¡1 µ(1)¡1
k X
2
µi µ(1)¡1
k X
3
µi¢ ¢¢ µ(1)¡1µk
. . .
. . .
. . .
. . .
. . .
µ(1)¡1 µ(1)¡1
k X
2
µi µ(1)¡1
k X
3
µi¢ ¢¢ µ(1)¡1µk
3
7
7
7 7
7
7
7
5
where:
F =
2
6 6
6
6 6
6
6
4
0 0 0 0 ¢¢ ¢ 0
0Ip 0 0 ¢¢ ¢ 0
0Ip Ip 0 ¢¢ ¢ 0
0Ip Ip Ip ¢¢ ¢ 0
. . .
. . .
. . .
. . . ¢¢ ¢ 0
0Ip Ip Ip Ip Ip
3
7 7
7
7 7
7
7
5
It follows that:
G0 (I ¡A1)
¡1 =G0(E +F)¡G0(E +F)¯k(®0
k(E+F)¯k)¡1®0
k£(E+F);
¡G0(E +F)¯k(®0
k(E +F)¯k)¡1
(A1)
or
G0 (I ¡A1)
¡1 =µ(1)¡1 ¡µ(1)¡1H
µ
P
1
µiµ(1)¡1
¶
;
¡
µ(1)¡1 ¡µ(1)¡1Hµ(1)¡1¢P
2
µi; ¢¢ ¢
¢ ¢¢ ;
¡
µ(1)¡1 ¡µ(1)¡1Hµ(1)¡1¢
µk;¡µ(1)¡1¯(®0µ(1)¡1¯)¡1
(A2)
where H =¯(®0µ(1)¡1¯)¡1®0.
Corollary 1. It follows that:
C(1) =G0 (I ¡A1)
¡1 ª
=µ(1)¡1 ¡µ(1)¡1¯(®0µ(1)¡1¯)¡1®0
Ã
X
1
µiµ(1)¡1
!
¡µ(1)¡1¯(®0µ(1)¡1¯)¡1®0
and so:
C(1) = G0 (I ¡A1)
¡1ª =µ(1)¡1 ¡µ(1)¡1¯(®0µ(1)¡1¯)¡1®0µ(1)¡1
Corollary 2. Noting that G0A1 (I ¡A1)¡1 = G0 (I ¡A1)¡1 ¡G0 from (A2) we
obtain:
G0A1 (I ¡A1)
¡1 =C(1)
P
1
µi;C(1)
P
2
µi; ¢¢ ¢C(1)µk; ¡µ(1)¡1¯(®0µ(1)¡1¯)¡1
(A3)
10References
[1] Beveridge, Stephen and Charles R. Nelson, (1981), “A New Approach to De-
composition of Economic Time series into Permanent and Transitory Com-
ponents with Particular Attention to Measurement of the ‘Business Cycle’,
Journal of Monetary Economics, 7, pp.151-74.
[2] Bai, Jushan, Robin L. Lumsdaine and James H. Stock, (1998), “Testing for
and Dating Common Breaks in Multivariate Time Series”, The Reveiw of
Economic Studies, 65, pp.395-432.
[3] Bai, Jushan and Pierre Perron, (1996), “Estimating and Testing Linear
Models with Multiple Structural Changes,” Econometrica, 66, pp. 47-78.
[4] Bai, J. andP. Perron, (2001), “Computationand Analysis ofMultipleStruc-
tural Change Models,” manuscript, Boston University.
[5] Banerjee, Anindya, Robin L. Lumsdaine and James H. Stock, (1992), “Re-
cursiveand Sequential Tests of the Unit-Root and Trend-Break Hypotheses:
Theory and International Evidence”, Journal of Business & Economic Sta-
tistics, 10, pp.271-287.
[6] Blanchard, Oliver Jean and Danny Quah, (1989), “The Dynamic E¤ects of
Aggregate Demand and Supply Disturbances”, American Economic Review,
79, pp.655-73.
[7] Cochrane, JohnH., (1994), “Permanent andTransitoryComponentsofGNP
and Stock Prices”, Quarterly Journal of Economics, pp.241-65.
[8] Gonzalo, Jesus and Clive Granger, (1995), “Estimation of Common Long-
Memory ComponentsinCointegrated Systems”, Journal of Business & Eco-
nomic Statistics, 13, pp.27-35.
[9] Gregory, Allan W. and Bruce E. Hansen, (1996), “Residual-based tests for
cointegration in models with regime shifts”, Journal of Econometrics, 70,
pp.99-126.
[10] Johansen, Søren, Rocco Mosconi and Bent Nielsen, (2000), “Cointegration
analysis in the presence of structural breaks in the deterministic trend”,
Econometrics Journal, 3, pp.216-249.
[11] King, Robert G., Charles I. Plosser, James H. Stock and Mark W. Wat-
son, (1991),“StochasticTrends and EconomicFluctuations”, American Eco-
nomic Review,81,pp.819-840.
[12] Lippi, Marco and Lucrezia Reichlin, (1994), “Di¤usion of Technical Change
and the Decomposition of Output into Trend and Cycle”, The Review of
Economic Studies, 61, pp.19-30.
[13] Proietti, Tommaso, (1997), “Short-Run Dynamics in Cointegrated System-
s”, Oxford Bulletin of Economics and Statistics, 59, pp.405-422.
11

Structural Breaks and Permanent Trends

http://www.efm.bris.ac.uk/economics/working_papers/pdffiles/dp03545.pdf

Structural Breaks and Permanent Trends

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics