Munich Personal RePEc Archive
Testing for the cointegration rank between Periodically Integrated processes
del Barrio Castro, Tomás
University of the Balearic Islands
2021
Online at https://mpra.ub.uni-muenchen.de/106603/
MPRA Paper No. 106603, posted 22 Mar 2021 09:59 UTC
Testing for the cointegration rank between Periodically Integrated processes
Tomás del Barrio Castro University of the Balearic Islands
March 11, 2021
Abstract
Cointegration between Periodically Integrated (PI) processes has been analyzed among other by Birchen- hall, Bladen-Hovell, Chui, Osborn, and Smith (1989), Boswijk and Franses (1995), Franses and Paap (2004), Kleibergen and Franses (1999) and del Barrio Castro and Osborn (2008). However, so far there is not a method, published in an academic journal, that allows us to determine the cointegration rank between PI processes. This paper …lls the gap, a method to determine the cointegration rank between a set PI Processes based on the idea of pseudo-demodulation is proposed in the context of Seasonal Cointegration by del Barrio Castro, Cubadda and Osborn (2020). Once a pseudo-demodulation time series is obtained the Johansen (1995) procedure could be applied to determine the cointegration rank.
A Monte Carlo experiment shows that the proposed approach works satisfactorily for small samples.
Keywords: Reduced Rank Regression,Periodic Cointegration, Periodically Integrated Processes.
JEL codes: C32.
1 Introduction
The two main ways of modeling non-stationary Integration in the seasonal time series are Seasonal Integration (SI) and Periodic Integration (PI), (see Ghysels and Osborn (2001) for details about the main characteristics and di¤erences between PI and SI). PI could be more attractive than SI as the nonstationary behavior is ruled by a common stochastic trend shared by the seasons on the time series, contrary in the case of a SI process, where each season of the time series hs his own stochastic trend (see Osborn (1993) and Ghysels and Osborn (2001) for details). Furthermore, PI shows up as a suitable data generating process for seasonal time series when preferences of economic agents vary along the seasons of the year (see Hansen and Sargent (1993), Gersovitz and McKinnon (1978) and Osborn (1988)).
In terms of long-run relationships (Cointegration) that could be established between seasonal integrated processes, we could also …nd Seasonal and Periodic Cointegration. For SI processes it is possible to de…ne both Seasonal and Periodic Cointegration, but in the case of PI processes only full Periodic Cointegration could be established (see del Barrio Castro and Osborn (2008) and Ghysels and Osborn (2001) for details).
In the case of Seasonal Cointegration, both methods for single equation and reduced rank regression have been proposed to test for the presence of Cointegration and determining the Cointegration rank, (see for example Hylleberg, Engle, Granger and Yoo (1990), Engle, Granger, Hylleberg and Lee (1993), Johansen and Schaumburg (1998), Cubbada (2000) and Ahn and Reinsel (1994)). Periodic Cointegration was proposed by Birchenhall, Bladen-Hovell, Chui, Osborn, and Smith (1989). A single equation method to test for the presence of Periodic Cointegration was proposed by Boswijk and Franses (1995). The authors of the previous paper claim that their method could be applied to both SI and PI processes, but del Barrio Castro and Osborn (2008) show that the asymptotic distribution of the error-correction test for periodic cointegration derived by Boswijk and Franses (1995) does not apply for PI processes. Also del Barrio Castro and Osborn (2008) proposed a residual based cointegration test for Periodic Cointegration between PI processes. But to the best of our knowledge, only the working paper by Kleibergen and Franses (1999) has tried to propose a method to determine the cointegration rank between a set of PI processes, (see also Franses and Paap (2004) for details) the method proposed by Kleibergen and Franses (1999) relies on Periodic Vector Autoregressive (VAR) Models and implies the use of GMM and reduced rank regression techniques. Finally a full dynamic system
be applied (Ghysels and Osborn (2001) pp 171–176) as it was done in the application of Haldrup, Hylleberg, Pons, and Sansó (2007), but the VAR becomes quite over-parameterized, hence this approach is feasible in practice only where data of a relatively high frequency are available.
In this paper, we proposed a simple method to determine the cointegration rank between PI processes inspired in the demodulation method proposed by del Barrio Castro, Osborn and Cubada (2020), that only needs the use of the procedure proposed by Johansen (1995) once the PI processes or time series are "…ltered"
or "demodulated".
The paper is organized as follows, in the next section, we describe and summarize the main characteristics of PI processes and their consequences in terms of cointegration between PI processes. After that, we present our reduced rank approach to determine the cointegration rank between PI processes, followed by a Monte Carlo Section where it is shown that our approach works well in small samples. Finally, the last section concludes.
It is useful to introduce some notation at this stage. Our analysis is concerned with seasonal processes which have S observations per year; for example, S = 4 for quarterly seasonal data. In this paper the vector of seasons representation that indicates a speci…c observation within the year it is used, and also the double subscript notation and it is important to appreciate that, in this vector notation,xs indicates thesth observation within the thyear; for example with quarterly dataxs is thesthquarter of year within the available sample. Assuming thatt= 1represents the …rst period within a cycle, the identityt=S( 1) +s provides the link between the usual time index and the vector notation.
2 Periodic Integration and Cointegration between Periodically In- tegrated Processes
In …rst place, we will focus on the main characteristics of Periodic Integrated processes. One of these characteristics is going to be very important and crucial for the approach proposed in this paper to determine the cointegration rank between PI processes. In second place, we will pay attention to the Cointegration possibilities between PI processes.
2.1 Periodic Integration (PI)
A Periodic autoregressive (PAR) process of order p is a generalization of an autoregressive process where the parameters are allowed to vary with the season of the year, hence we have:
ys = 1sys 1; + 2sys 2; + + psys p; +"s (1)
s= 1;2; ; S = 1;2; ; N
where "s is the innovation of the process and we assume that"s iid 0; 2" . In order to understand the concept of Periodic Integration, let focus on the PAR process of order one:
ys = sys 1; +us : (2)
In (2) we assume that us is a stationary innovation, this assumption will help us later on to connect (2) with (1)1. The condition of Periodic Integration in (2) is
YS
s=1
s= 1and implies that between the seasons of the time series we have S 1 cointegration relationships or equivalently the seasons of the process share a common stochastic trend. This situation is clearly shown in the so called Vector of Seasons representation of a PAR process, where the S seasons of the time series are stacked in aS 1vectorY = [y1 ; y2 ; : : : ; yS ]0 and have:
A0Y =A1Y 1+U (3)
1Note that (1) is connected with (2) if we write (1) as:
1 1sL 2sL2 psLp ys ="s
and factorize the polynomial 1 1sL 2sL2 psLp as:
1 1sL 2sL2 psLp = (1 sL) 1 1sL p 1;sLp 1 hence (1) is connected with (2) as ifus in (2) is de…ned as follows:
1 1sL p 1;sLp 1 us ="s :
where, U = [u1 ; u2 ; :::; uS ]0 andA0 andA1areS S matrices with generic elements
A0(h;j)= 8<
:
1 h=j; j= 1; :::; S
h h=j+ 1; j= 1; :::; S 1
0 otherwise
(4)
A1(h;j)= 1 h= 1; j=S
0 otherwise
in which the subscript(h; j)indicates the(h; j)thelement of the respective matrix. As in the quarterly case studied by Paap and Franses (1999), successively substituting in (3) yields
Y = [A01A1] Y0+A01U + X1
j=1
[A01A1]jA01U j
=A01A1Y0+A01U +A01A1A01 X1
j=1
U j: (5)
Note, that this result follows because matrix A 1
0 A1 is idempotent. First, note that the matrix A0 (see chapter 2 pp 45-48 of Pollock (1999)) is anS S lower-triangular Toeplitz matrix associated with the poly- nomial(1 sL). Hence the matrixA 1
0 collects the coe¢cients of the expansion of the inverse polynomial associated with(1 sL)2. Based on the form of the matricesA 1
0 andA1, it is clear that the resulting ma- trixA 1
0 A1is anS Smatrix with the …rstS 1columns having elements equal to zero and the last column equal to the column vectorv=
"
1 1 2 1 2 3
YS
s=1 s
#0
. Finally note that the last element of
v, that is, YS
s=1
s;is equal to1, as we have Periodic Integration. Also, as the …rstS 1columns ofA01A1are equal to zero and the lower left element of this matrix is equal to one, implies that[A 1
0 A1]j=A 1
0 A1 for j = 2;3; :::. Clearly, (5) provides a representation of (2), where the matrixA 1
0 A1A 1
0 gives the e¤ect of the accumulated vector of shocks P 1
j=1U j (see for example Boswijk and Franses (1996), Paap and Franses (1999) and del Barrio Castro and Osborn (2008)). The matrixA 1
0 A1A 1
0 has rank one and hence can be written as
A 1
0 A1A 1
0 =ab0 (6)
where, for (6),
a=
"
1 2 2 3
YS
s=2 s
#0
b=
"
1 1
YS
s=2
s 1
YS
s=3
s 1
#0
: (7)
Therefore, the scalar partial sumb0P 1
j=1U j in (5) is the common stochastic trend shared by the seasons ofY . As we have a common stochastic trend shared by the Sseasons of the PI process, we will haveS 1 cointegration relationships between the seasons of (3), Re-write (3) as:
Y =A 1
0 A1Y 1+A 1
0 U
Y Y 1= A01A1 I Y 1+A01U (8)
2That is:
A01= 2 66 66 66 66 66 64
1 0 0 0 0
2 1 0 0 0
2 3 3 1 0 0
2 3 4 3 4 4 1 0
... ... ... ... . .. ... YS
j
YS j
YS j
YS
j 1
3 77 77 77 77 77 75 :
and note that matrix A01A1 I has rank S 1 and hence he have S 1 cointegration relationships between theS seasons of the PI process. Clearly A 1
0 A1 I = 0 where both and have dimension S (S 1), one possible choice for the columns of are the lastS 1rows ofA03. Finally, It is clear that we have cointegration between the seasons ofY , if we multiply by the left (5) by 0 we have:
0Y = 0A 1
0 A1Y0+ 0A 1
0 U + 0A 1
0 A1A 1
0
X1
j=1
U j
= 0A 1
0 A1Y0+ 0A 1
0 U + 0ab0 X1
j=1
U j:
As 0a= 0 we clearly show that 0Y I(0)and that we have S 1 cointegration relationships between theS seasons ofys (orY ). In the following lemma we summarize the stochastic behavior ofY in (3).
Lemma 1 ForY = [y1 ; y2 ; y3 ; : : : ; yS ]0withys s= 1;2; ; Sde…ned in (2-3) and with(1 1sL
p 1;sLp 1 us ="s and"s iid 0; 2 then p1
TYbT rc ) A01A1A01 (1) 1W(r) = ab0 (1) 1W(r) (9)
= aw(r)
where a and b are de…ned in (7), W(r) is a S 1 multivariant Brownian Vector and w(r) is a scalar Brownian motion de…ned in the appendix. Finally the de…nition of matrix (1) could be also found in the appendix.
In the following subsection we pay attention the cointegration possibilities betweenP I processes.
2.2 Cointegration between P I processes
Ghysels and Osborn (2001) and del Barrio Castro and Osborn (2008) analyze the cointegration possibilities between P I processes, and show that between P I processes the only cointegration possibilities are full Periodic Cointegration or full Non-Periodic Cointegration.
Periodic Cointegration was introduced by Birchenhall, Bladen-Hovell, Chui, Osborn and Smith (1989) and implies that the long-run relationships are considered season by season, hence we have di¤erent cointegration vectors for each season. Periodic Cointegration could be established for both Seasonal Integrated (SI) processes and Periodically Integrated (P I) processes. Boswijk and Franses (1995) distinguished between full and partial Periodic Cointegration, the latter (partial) applies when stationary linear combinations between seasonal non-stationary time series could be established only for some seasons s = 1;2; : : : ; S: And full Periodic Integration implies that the stationary linear combinations exit for all the seasons. Finally, full Non-Periodic Cointegration implies that the same cointegration vectors are shared for all the seasons.
In this paper, we follow the de…nition of Periodic Cointegration proposed in del Barrio Castro and Osborn (2008) (see de…nition 1 in section 2.2). Let consider them 1vector processYs(m)= ys1 y2s : : : ys2 0 where each of the elementsysk areP I processes, such that:
ysk = ksyks 1; +uks YS
s=1
ks = 1; s= 1;2; : : : ; S; k= 1;2; : : : ; S: (10)
3Note that we haveS 1cointegration relationships between the seasons of (2) of the formys sys 1; ,that are clearly identi…ed with the lastS 1rows of matrixA0. that is:
0=
2 66 64
2 1 0 0 0
0 3 1 0 0
... ... ... . .. ... ...
0 0 0 S 1
3 77 75:
Note also, that equivalently we could also use its normalized version
0=
2 66 66 66 64
1 0 0 0 1
0 1 0 0 1 2
..
. ... ... . .. ... ...
0 0 0 1
S 1
Y
i=1 i
3 77 77 77 75 :
where eachu(k)s is a stationary periodic autoregressive process:
1 1skL pk 1;sLp 1 uks ="ks :
Finally the vector Es(m) = "1s "2s : : : "ms 0 is a white noise vector with positive de…nite variance- covariance matrix Eh
Es(m)Es(m)0
i
= . De…nition 1 in del Barrio Castro and Osborn (2008), established periodic cointegration for a m 1 vector Ys(m) of periodic processes satisfying (10) if there exist m r matrices s of rank r such that the linear combinations s0Ys(m) are (periodically) stationary for each season s = 1;2; : : : ; S. Boswijk and Franses (1995) de…ne Partial Periodic Cointegration when stationary linear combinations s0Ys(m) exists for only some seasons, and Full Periodic Cointegration when the linear combinations exist for all the seasons. Full Non-Periodic Cointegration is a particular case of Full Periodic Cointegration where the same m rmatrix allow us to obtain stationary linear combinations for all the seasons.
In Lemma 1 in del Barrio Castro and Osborn (2008) it is shown that between P I processes such as the ones collected in vector Ys(m) above with elements de…ned in (10), the only cointegration possibilities are Full Periodic Cointegration and Full Non-Periodic Cointegration. The intuition behind this result is that as shown in Lemma 1 of the previous subsection the S seasons of a P I process are driven by the same stochastic common trend, hence if we have cointegration between one of the seasons of P I processes recursive substitution implies that cointegration hold for the rest of seasons, with cointegration vector that will change for each season unless all the P I processes share the same coe¢cients associated with the P I condition
YS
s=1
ks= 1that is ks= sfork= 1;2; : : : ; mands= 1;2; : : : ; S. And precisely in this latter case when all the P I processes share the same coe¢cients ks = s associated to theP I condition we have Full Non-Periodic Cointegration.
For simplicity and to allow us to pay attention to the main facts on the problem we will focus from now onwards on the case of 3P I processes, that ism= 34. Between3P I processes we could have the following situations: (a) no cointegration, (b) one common stochastic trend shared by the3 P I processes and (c) two common stochastic trends shared by the3P I processes.
2.3 The 3 P I processes case.
Let focus on vector Ys(3) = y1s ys2 ys3 0 with the elements ysk k= 1;2 and 3, been Periodically In- tegrated (P I). In order to understand the cointegration possibilities between these Three-variantP Isystems, we use the following3-variant vector of seasonsY(3) = y11 ; y12 ; : : : ; y1S y12 ; y22 ; : : : ; y2S y13 ; y32 ; : : : ; y3S 0. For the scenarios (a) no cointegration, (b) one common stochastic trend shared by the 3 P I processes and (c) two common stochastic trends shared by the3 P I processes, hence we have the following VAR(1):
A(3)0 Y(3)=A(3)1 Y(3)1+U(3) (11) Y(3)= y11 ; y21 ; : : : ; yS1 y21 ; y22 ; : : : ; yS2 y31 ; y23 ; : : : ; yS3 0
= 2 4
Y1 Y2 Y3
3 5
U(3)= u11 ; u12 ; : : : ; u1S u21 ; u22 ; : : : ; u2S u31 ; u32 ; : : : ; u3S 0
= 2 4
U1 U2 U3
3
5: (12)
The matricesA(3)
0 and A(3)
1 are square matrices of dimension(3S) (3S)and will have a di¤erent form in the 3 scenarios.
2.3.1 No Cointegration
In scenario (a) of no cointegration between the 3 P I processes. The matricesA(3)
0 andA(3)
1 in (11) will be as follows:
A(3)
0 =diag A1
0;A2
0;A4
0 (13)
A(3)
1 =diag A1
1;A2
1;A3
1 ;
henceA(3)0 andA(3)1 are block diagonal matrices with the followingS S submatrices:
Aj0= 2 66 66 66 64
1 0 0 0 0
j
2 1 0 0 0
0 j3 1 0 0
0 0 j4 1 0
... ... ... ... . .. ...
0 0 0 jS 1
3 77 77 77 75
j= 1;2;3 (14)
Aj1= 2 66 66 66 64
0 0 0 0 j1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
... ... ... . .. ... ...
0 0 0 0 0
3 77 77 77 75
j= 1;2;3: (15)
In model (11) we could have by recursive substitution:
Y(3)= A(3)
0 1
A(3)
1 Y0(3)+ A(3)
0 1
U(3)+ X1
j=1
A(3)
0 1
A(3)
1 j
A(3)
0 1
U(3)j; (16)
…rst note that the inverse matrix A(3)
0 1
will be also block diagonal, such that:
A(3)0
1
=diagh
A10 1; A20 1; A30 1 i
with:
Aj
0 1
= 2 66 66 66 66 66 64
1 0 0 0 0
j
2 1 0 0 0
j 2
j 3
j
3 1 0 0
j 2
j 3
j 4
j 3
j 4
j
4 1 0
... ... ... ... . .. ...
YS
k=2 j k
YS
k=3 j k
YS
k=4 j k
YS
k=5 j
k 1
3 77 77 77 77 77 75
j= 1;2;3: (17)
Also, note that product A(3)
0 1
A(3)
1 is also block diagonal, with the following form:
A(3)0
1
A(3)1 =diagh
A10 1A11; A20 1A21; A30 1A31 i
with:
Aj
0 1
Aj
1= 2 66 66 66 66 66 64
0 0 0 0 j1
0 0 0 0 j1 j2 0 0 0 0 j1 j2 j3 0 0 0 0 j1 j2 j3 j4
... ... ... . .. ... ...
0 0 0 0
YS
k=1 j k
3 77 77 77 77 77 75
j= 1;2;3:
Clearly, as we have PI processes the lower right element of the sub-matrices Aj
0 1
Aj
1are equal to YS
k=1 j k = 1. Hence, it is easy to check that matrix A(3)
0 1
A(3)
1 is idempotent. Then it is possible to write for (16):
Y(3) = A(3)
0 1
A(3)
1 Y0(3)+ A(3)
0 1
U(3)+ A(3)
0 1
A(3)
1 A(3)
0
1X1 j=1
U(3)j (18)
A(3)
0 1
A(3)
1 A(3)
0 1
= 2 4
a1b01 0S S 0S S
0S S a2b02 0S S
0S S 0S S a3b03 3 5
aj =
"
1 j2 j2 j3
YS
s=2 js
#0
bj =
"
1 j1 YS
s=2 js
j 1
YS
s=3 js
j 1
#0
:
Note that, from (18), each of the 3 P I processes collected in the vectorY(3) has his own stochastic trend, that is b0
j
X1
k=1
Uj k for k = 1;2 and 3. And also we have cointegration between the seasons of each P I process. The stochastic behavior is summarized in the following lemma.
Lemma 2 ForY(3)=h
Y10; Y20; Y3i0
de…ned in (11-18) and with 1 1sj L jp 1;sLp 1 ujs ="js for j = 1;2 and 3 and Es(3) = "1s "2s "3s 0 is a white noise vector with positive de…nite variance- covariance matrix Eh
Es(3)Es(3)0
i
= then p1
TYb(3)T rc) A(3)
0 1
A(3)
1 A(3)
0
1 (3)(1) 1[P IS]W(3)(r) (19)
= 2 4
a1b01 0S S 0S S
0S S a2b02 0S S
0S S 0S S a3b0 3
3
5 (3)(1) 1[P IS]W(3)(r)
= 2 4
!1a1w1(r) 0S S 0S S
0S S !2a2w2(r) 0S S
0S S 0S S !3a3w3(r) 3 5
whereaj andbj forj= 1;2and3 are de…ned in (18),W(3)(r)is a (3S) 1 multivariant Brownian Vector and wj(r)for j = 1;2 and 3 are scalar Brownian motions de…ned in the appendix. Finally, the de…nition of matrix (3)(1)and the scalar terms !j forj= 1;2and3 could also be found in the appendix and Pis a 3 3 matrix such that =PP0.
Lemma 2 above is a particular case of Lemma 3 in del Barrio Castro and Osborn (2008) as in Lemma 2, we only have threeP I processes but on the other hand Lemma 2 is de…ned for a general number of seasons and the results in del Barrio Castro and Osborn (2008) are for quarterly data. Clearly, in Lemma 2 we show that between the S seasons of each P I process we have S 1 cointegration relationships and a common stochastic trend for eachP I process, that is, re‡ected in each scalar Brownian motion wj(r)with j= 1;2 and 3.
2.3.2 One Common stochastic trend shared between the three P I processes
In the case of cointegration betweenP I processes we know from Lemma 1 in del Barrio Castro and Osborn (2008) that we should have Full Periodic Cointegration or Full Non-Periodic Cointegration, the latter situa- tion is restricted to the case where all theP I processes share the same vaule for the coe¢cients associated to the periodic integration restriction. In the threeP I system, a common stochastic trend implies the existence
of two periodic cointegration relationships. Let’s consider the following situation5: ys1 = sy3s +u1s
ys2 = sys3 +u2s (20)
ys3 = 3sys3 1; +u3s s= 1;2; : : : ; S:
with sand ssuch that:
S = =
SY1
i=0 3 S i SY1
i=0 1S i
S = =
SY1
i=0 3 S i SY1
i=0 2S i
S 1=
3S 1S
S 1=
3S 2S
(21)
S 2=
3S 3 S 1 1 S 1
S 1
S 2=
3S 3 S 1 2 S 2
S 1 S 3=
3 S 3
S 1 3 S 2 1S 1
S 1 1 S 2
S 2=
3 S 3
S 1 3 S 2 2S 2
S 1 2 S 2
...
1=
SY2
i=0 3 S i SY2
i=0 1 S i
1=
SY2
i=0 3 S i SY2
i=0 2 S i
:
The system (20)-(21) admits a Vector of Seasons representation like (11) with matrices A(3)0 and A(3)1 in (11) that will be as follows:
A(3)
0 =
2 64
IS 0S S A(y01) 0S S IS A(y02) 0S S 0S S A(y03)
3
75 (22)
A(3)
1 =
2 4
0S S 0S S 0S S
0S S 0S S 0S S
0S S 0S S A(y13) 3 5;
with theS S submatricesA(y3)
0 andA(y3)
1 , de…ned equivalently as (14) and (15) that is:
A(y3)
0 =
2 66 66 66 64
1 0 0 0 0
32 1 0 0 0
0 33 1 0 0
0 0 34 1 0
... ... ... ... . .. ...
0 0 0 3S 1
3 77 77 77 75
A(y3)
1 =
2 66 66 66 64
0 0 0 0 31
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
... ... ... . .. ... ...
0 0 0 0 0
3 77 77 77 75
: (23)
5Trought the paper, we are going to use the normalization collected for example Lütkepohl (2005) pp 250.
Finally both theS S submatricesA(y1)
0 andA(y2)
0 are diagonal matrices of the form:
A(y1)
0 =diag[ 1; 2; 3; : : : ; S]
=diag 2 66 66 4
SY2
i=0 3 S i SY2
i=0 1 S i
;
SY3
i=0 3 S i SY3
i=0 1 S i
;
SY4
i=0 3 S i SY4
i=0 1 S i
; : : : ; 3 77 77 5
(24)
A(y2)
0 =diag[ 1; 2; 3; : : : ; S]
=diag 2 66 66 4
SY2
i=0 3 S i SY2
i=0 2 S i
;
SY3
i=0 3 S i SY3
i=0 2 S i
;
SY4
i=0 3 S i SY4
i=0 2 S i
; : : : ; 3 77 77 5 :
We could have also recursive substitution as in (16), note that, it is possible to check that the inverse of matrixA(3)
0 in (22) will be as follows:
A(3)
0 1
= 2 66 66 4
IS 0S S A(y01) A(y03)
1
0S S IS A(y02) A(y03)
1
0S S 0S S A(y03)
1
3 77 77 5
; (25)
note the inverse of sub-matrixA(y3)
0 , that is, A(y3)
0 1
is a lower triangular matrix as in (17), that is:
A(y3)
0 1
= 2 66 66 66 66 66 4
1 0 0 0 0
32 1 0 0 0
32 3
3 3
3 1 0 0
32 3 3 3
4 3
3 3
4 3
4 1 0
... ... ... ... . .. ...
YS
k=2 3 k
YS
k=3 3 k
YS
k=4 3 k
YS
k=5 3
k 1
3 77 77 77 77 77 5
: (26)
Based on the form of A(3)
0 1
andA(3)
1 in (25-22-26-24-??) it is possible to see that the product A(3)
0 1
A(3)
1 has the following expression:
A(3)
0 1
A(3)
1 =
2 4
0S S 0S S 0S (S 1)v1 0S S 0S S 0S (S 1)v2 0S S 0S S 0S (S 1)v3
3
5; (27)
hence all the elements of the(3S) (3S)matrix A(3)
0 1
A(3)
1 are equal to zero, except for its last column.
This last column is the concatenation of theS 1 vectorsvj j= 1;2and 3. Where the vectors are de…ned
as follows:
v1=
"
1 3
1 2 3
1 3
2 3 3
1 3 2 3
3 S
YS
s=1 3 s
#0
= 1 3
1 2 3
1 3
2 3 3
1 3 2 3
3 S 0
v2=
"
1 3
1 2 3
1 3
2 3 3
1 3 2 3
3 S
YS
s=1 3s
#0
(28)
= 1 3
1 2 3
1 3
2 3 3
1 3 2 3
3 S 0
v3=
"
3
1 3
1 3
2 3
1 3 2 3
3
YS
s=1 3 s
#0
= 31 31 32 31 32 33 1 0: Note that the lower left element of matrix is A(3)0
1
A(3)1 is equal to one. And due to its form, it is clear that A(3)
0 1
A(3)
1 is idempotent. Hence in this case we also have:
Y(3) = A(3)
0 1
A(3)
1 Y0(3)+ A(3)
0 1
U(3)+ A(3)
0 1
A(3)
1 A(3)
0
1X1 j=1
U(3)j (29)
A(3)0
1
A(3)1 A(3)0
1
= 2 4
0S S 0S S v1u03 0S S 0S S v2u03 0S S 0S S v3u03
3 5
u03=
" S Y
k=2 3k
YS
k=3 3k
YS
k=4 3k
YS
k=5
3k 1
#
Note that u0
3 is the last row of matrix A(y3)
0 1
(26). And it is also possible to write:
A(3)0
1
A(3)1 A(3)0
1
= 2 4
0S S 0S S (a1b03) 0S S 0S S (a2b03) 0S S 0S S a3b03
3
5 (30)
aj =
"
1 j2 j2 j3
YS
s=2 js
#0
j= 1;2;3
b0
3=
"
1 31 YS
k=3 3k 3
1
YS
k=4 3k 3
1
YS
k=5
3k 3
1
# :
Hence clearly the three PI processes share the same stochastic trend b03 X1
k=1
U3 k. As in the previous subsection, the following lemma summarizes the stochastic behavior of the vector of seasons.
Lemma 3 ForY(3)=h
Y10; Y20; Y3i0
de…ned in (11-29-30) and with 1 j1sL jp 1;sLp 1 ujs =
"js forj= 1;2 and 3and E(3)s = "1s "2s "3s 0 is a white noise vector with positive de…nite variance- covariance matrix Eh
Es(3)Es(3)0
i
= then p1
TYb(3)T rc) A(3)0
1
A(3)1 A(3)0
1 (3)(1) 1[P IS]W(3)(r) (31)
= 2 4
0S S 0S S (a1b0
3) 0S S 0S S (a2b03) 0S S 0S S a3b03
3
5 (3)(1) 1[P IS]W(3)(r)
= 2 4
0S S 0S S !3a1w3(r) 0S S 0S S !3a2w3(r) 0S S 0S S !3a3w3(r)
3 5
whereaj forj= 1;2and3 andb3 are de…ned in (30),W(3)(r)is a (3S) 1 multivariant Brownian Vector and w3(r)is a scalar Brownian motions de…ned in the appendix. Finally, the de…nition of matrix (3)(1) could be also found in the appendix and Pis a3 3 matrix as in the previous lemma.
Clearly, Lemma 3 above shows that the common stochastic trend shared by the Seasons of the three PI processes is identi…ed with the scalar Brownian Motion w3(r), hence we have cointegration within the Seasons of each PI process and also between the Seasons of all the PI processes.
2.3.3 Two Common stochastic trend shared between the three P I processes.
In the three P I system, two common stochastic trends imply the existence of one periodic cointegration relationship. Let consider the following situation:
y1s = 1;sys2 + 2;sys3 +u1s
y2s = 3sy2s 1; +u2s (32)
y3s = 3sy3s 1; +u3s s= 1;2; : : : ; S:
with 1;s and 2;s are such that:
1;S = 1= 1 SY1
i=0 2S i
SY1
i=0 1S i
2;S= 2= 2 SY1
i=0 3S i
SY1
i=0 1S i
1;S 1= 1 2S 1 S
2;S 1= 2 3S 1 S
(33)
1;S 2= 1 2 S 2
S 1 1 S 1
S 1
2;S 2= 2 3 S 3
S 1 1 S 1
S 1 1;S 3= 1
2S 2 S 1 2
S 2 1
S 1 S 1 1
S 2
2;S 2= 2 3S 3
S 1 3 S 2 1
S 1 S 1 1
S 2
...
1;1= 1 SY2
i=0 2S i
SY2
i=0 1S i
2;1= 2 SY2
i=0 3S i
SY2
i=0 1S i
:
The system (32)-(33) admits a Vector of Seasons representation like (11) where the matricesA(3)0 andA(3)1 in (11) will be as follows:
A(3)
0 =
2 64
IS A(y1y2)
0 A(y1y3)
0
0S S A(y2)
0 0S S
0S S 0S S A(y3)
0
3
75 (34)
A(3)1 = 2 64
0S S 0S S 0S S
0S S A(y12) 0S S
0S S 0S S A(y13) 3 75;
where the S S submatrices A(y03) and A(y13) are de…ned as in (23) and A(y02) and A(y12) are de…ned equivalently, that is:
A(y02)= 2 66 66 66 64
1 0 0 0 0
22 1 0 0 0
0 23 1 0 0
0 0 24 1 0
... ... ... ... . .. ...
0 0 0 2S 1
3 77 77 77 75
A(y12)= 2 66 66 66 64
0 0 0 0 21
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
... ... ... . .. ... ...
0 0 0 0 0
3 77 77 77 75
: (35)
Finally theS S sub-matrices A(y01y2) andA(y01y3)are diagonal matrix, de…ned as follows:
A(y01y2)=diag[ 11; 12; 13; : : : ; 1S]
=diag 2 66 66 4
1 SY2
i=0 2S i
SY2
i=0 1 S i
; 1
SY3
i=0 2S i
SY3
i=0 1 S i
; 1
SY4
i=0 2S i
SY4
i=0 1 S i
; : : : ; 1
3 77 77 5
(36)
A(y01y3)=diag[ 21; 22; 23; : : : ; 2S]
=diag 2 66 66 4
2 SY2
i=0 3S i
SY2
i=0 1S i
; 2
SY3
i=0 3S i
SY3
i=0 1S i
; 2
SY4
i=0 3S i
SY4
i=0 1S i
; : : : ; 2
3 77 77 5 :
In this case, it is also possible to use recursive substitution as in (16), note that, it is possible to check that the inverse of matrixA(3)0 in (34) will be as follows:
A(3)0
1
= 2 66 66 4
IS A(y1y2)
0 A(y2)
0 1
A(y1y3)
0 A(y3)
0 1
0S S A(y2)
0 1
0S S
0S S 0S S A(y3)
0 1
3 77 77 5
: (37)
Also as mentioned previously for A(y0i)
1
withi= 2and3we have:
A(y0i)
1
= 2 66 66 66 66 66 4
1 0 0 0 0
i
2 1 0 0 0
i2 i
3 i
3 1 0 0
i2 i 3 i
4 i
3 i
4 i
4 1 0
... ... ... ... . .. ...
YS
k=2 i k
YS
k=3 i k
YS
k=4 i k
YS
k=5 i
k 1
3 77 77 77 77 77 5
i= 2;3: (38)
The resulting matrix A(3)0
1
A(3)1 :
A(3)0
1
A(3)1 = 2 4
0S S 0S (S 1)w12 0S (S 1)w13 0S S 0S (S 1)w2 0S S
0S S 0S S 0S (S 1)w3 3
5; (39)
