on cointegration for processes integrated at di⁄erent...
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On cointegration for processes integrated at di¤erent frequencies(Periodic Polynomial Cointegration)
Tomás del Barrio CastroUniversity of the Balearic Islands
Gianluca CubaddaUniversità degli Studi di Roma "Tor Vergata"
Denise R. OsbornUniversity of Manchester
Febrary 2020
Abstract
This paper explores the possibility of having cointegration relationships between processes integrated atdi¤erent frequencies. We found that it is possible to establish this kind of cointegration and that the onlypossible cointegration relationships between processes integrated at di¤erent frequencies is full periodicpolynomial cointegration. We explore the connection of this kind of cointegration with the demodulatoroperator and �nally propose a simple way to test for the presence of cointegration between processesintegrated at di¤erent frequencies based on the use of the demodulator operator.Keywords: Periodic Cointegration, Polynomial cointegration, Demodulator Operator.
JEL codes: C32.
1 Introduction
To date, the vast literature on cointegration has focused primarily on the long-run characteristics of economictime series through the analysis of zero frequency unit roots. Nevertheless, economic and �nancial time seriesmay exhibit unit roots at other frequencies; in particular, Engle, Granger, Hylleberg and Lee (1993), Johansenand Schaumburg (1999), Ahn and Reinsel (1994) and Bauer and Wagner (2012) analyze the seasonal case,while Bierens (2001) and Caporale, Cuñado and Gil-Alana (2013) consider unit roots associated with thebusiness cycle. However, such analyses typically examine a speci�c frequency, without allowing the possibilitythat the responses of economic agents may vary over the seasonal or business cycle. In contrast, this paperstudies long-run linkages between time series with unit roots at di¤erent frequencies and, speci�cally, thenature of any cointegration between a conventional zero frequency I(1) series, denoted I0(1), and a serieswith a unit root at a business cycle or seasonal frequency. To our knowledge, no previous study has examinedthe nature of any such cointegration. Succinctly stating our main result, we show that cointegration can existbetween time series that are integrated at di¤erent frequencies, with this being a speci�c type of time-varyingpolynomial cointegration. More speci�cally, the cointegrating relationship is dynamic with coe¢ cients thatexhibit cyclical variation, so that (for example) a long-run relationship can vary over the business cycle.Polynomial cointegration is discussed in the literature in the contexts of (so-called) seasonal cointegra-
tion and multicointegration (see Hylleberg, Engle, Granger and Yoo, 1990, and Granger and Lee, 1989,respectively), while Gregoir (1999a, 1999b) undertakes a general analysis of these cases. Cubadda (2001)provides an alternative representation of the polynomial cointegration arising in the seasonal case in termsof complex-valued cointegration, which is developed further by Gregoir (2006, 2010). Although we take asimilar approach to these latter authors, we relax the restrictions that cointegration applies only at a singlefrequency and that cointegrating vectors are time invariant.As examined by Park and Hahn (1999) and Bierens and Martin (2010), time-varying cointegration allows
the relevant coe¢ cients to change smoothly over time in any direction. Such a general speci�cation is,however, problematic in that it raises the question of what underlying mechanism drives these changes andhence it is not surprising that other authors place some economic structure on the nature of the temporalvariation exhibited by the long-run relationship. For example, Hall, Psaradakis and Sola (1997) allow thecointegrating relationship to change with the economic environment through the use of a Markov-switching
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speci�cation, while Birchenhall, Bladen-Hovell, Chui, Osborn and Smith (1989) apply periodic cointegration,in which the long-run coe¢ cients vary with the time of the year.The present paper generalizes periodic cointegration to show that temporal variation in the coe¢ cients
of a long-run relationship, with this variation being of a cyclic nature, can deliver cointegration betweenvariables that are individually integrated at di¤erent frequencies. This approach encompasses not onlyvariation associated with the seasons, but also over a cycle at a business cycle frequency, and the approachprovides a speci�c form of regime-switching cointegration. Some of our results are implicit in analyses ofperiodic cointegration (see, in particular, Ghysels and Osborn, 2001, and Franses and Paap, 2004), but thecross-frequency cointegration implications have not previously been drawn out.In our analysis, a central role is played by the complex demodulation operator, which transforms a real
valued process integrated at a frequency di¤erent from zero to a complex valued process that is integratedat frequency zero. The idea of complex demodulation has long history in time series analysis (see, e.g.,chapter 6 of Bloom�eld (1976) and the references therein) but, to the best of our knowledge, it has not beenpreviously used to investigate the presence of cointegration among series that are integrated at di¤erentfrequencies.This paper is organized as follows. Section 2 reviews the notions of integration at a given frequency.
Section 3 presents our theoretical results. First, we show that two complex valued processes integratedat di¤erent frequencies can cointegrated and the connection of this fact with the demodulator operator.Second, we examine in details the various forms of cointegration that may exist between real valued timeseries integrated at di¤erent frequencies. Third, we tackle inferential issues. In Section 4, a Monte Carlosimulation exercise documents the small sample properties of the tests that we suggest. Section 5 presentsan empirical application to illustrate concepts and methods. Finally, Section 6 concludes.
2 Integration at a frequency
For what follows, it will be useful to have a notation for the operator that removes a single unit root at aspectral frequency ! 2 [0; �]. To this end, and following Gregoir (1999a) and Cubadda (1999), we adopt thenotation
�! =
�1� e�i!L; ! = 0; �1� 2 cos!L+ L2 = (1� e�i!L)(1� ei!L); ! 2 (0; �) (1)
where L is the conventional lag operator. Special cases of this operator include:
�0 = 1� L�� = 1 + L
��=2 = 1 + L2
��=3 = 1� L+ L2
so that �0 is the conventional �rst di¤erence operator, �� and ��=2 are the operators that remove unitroots at the semi-annual and annual frequencies, respectively, for a seasonally integrated quarterly process(Hylleberg et al., 1990), while ��=3 will remove a unit root corresponding to a cycle of six years duration inannual data.To pin down the concept of integration at some frequency !, we adopt the following de�nition, used by
Gregoir (1999a):
De�nition 1. A purely nondeterministic real-valued random process xt is integrated of order d, for non-negative integer d, at frequency ! 2 [0; �] if �d!xt is a covariance stationary process such that, for zero meanwhite noise "t, its Wold representation
�d!xt = c(L)"t =1Xi=0
ci"t�i
satis�esP1
i=0 c2i <1 and c(ei!) 6= 0.
Following Hylleberg et al. (1990) and Gregoir (1999a), a process xt satisfying De�nition 1 is denoted
xt � I!(d):
Although Gil-Alana (2001) and Gray, Zhang and Woodward (1989) allow fractional d, which is particularlyrelevant for �nancial time series, we are interested in cointegration for unit root economic time series which
2
are typically I!(1) after taking account of deterministic e¤ects. Obviously, xt � I0(1) corresponds to aconventional (single) unit root process integrated at the zero frequency.Although the di¤erencing operator �! of (1) is de�ned for a real valued series, it is useful for the analysis
that follows to consider complex valued processes. Speci�cally, when xt � I!(1) we consider individuallyeach of the two factors
�1� e�i!L
�of �! for ! 2 (0; �). Then, for
xt = (2 cos!)xt�1 � xt�2 + �t; (2)
with �t � I!(0) real valued, de�ne the complex valued process
x�t = xt � ei!xt�1: (3)
It is then straightforward to see thatx�t = e
�i!x�t�1 + �t: (4)
Successive substitution from (4) yields
x�t = e�i!tx�0 +t�1Xj=0
e�i!jvt�j
= e�i!tx�0 +tX
k=1
e�i!(t�k)vk
= e�i!t
"x�0 +
tXk=1
ei!kvk
#: (5)
As noted by Gregoir (1999a, 2006) and del Barrio Castro, Rodrigues and Taylor (2017a,2017b), (5) isequivalent to:
x�t = e�i!t
"x�0 +
tXk=1
ei!kvk
#= e�i!tx
(0)�t : (6)
the complex valued process x(0)�t = x�0 +Pt
k=1 ei!k�k is integrated of order one at the zero frequency
(I0 (1))1 , while e�i!t is the demodulation operator that shifts the zero frequency peak to frequency !,leading to a complex valued I!(1) process. The demodulation operator provides the key to cointegrationbetween processes integrated at di¤erent frequencies, examined in subsequent sections2 .Using an analogous line of argument, it can be seen that this real valued process is also implied by the
complex valued process
x+t = ei!x+t�1 + vt; (7)
x+t = ei!tx(0)+t
x(0)+t = x�0 +
tXk=1
e�i!k�k
which forms a complex conjugate pair with x�t .In the appendix it could be found Lemma 1 where the stochastic characteristics of 4) are summarized.
To do that, we use the vector of seasons representation of (4) and also we use the double subscript notation(see the appendix for a detailed explanation of the use of double subscript notation xn� ) Also in Lemma1 we present the results for !j = 2�j=N with j = 1; : : : ; (N � 1) =2, that completes a full cycle every N=jperiods.Finally note that for (5) x�t = e
�i!jtx(0)�t and (7) x+t = e
i!jtx(0)+t it is possible to write x(0)�t = ei!jtx�t
and x(0)+t = e�i!jtx+t respectively.
1Note that as �t � I!(0) acts as the innovation for the real valued I!(1) process (2), ei!kvk is the complex valued innovationfor x(0)�t :
2The demodulator operator it is also used in Theorem 4 of Johansen and Schaumburg (1999) in a multivariant setting.
3
3 Cointegration between processes integrated at di¤erent frequen-cies
In this section we initially focus on the long run relationships between complex valued process associated todi¤erent frequencies and show that it is possible to establish long-run relationships between complex valuedintegrated processes associated to di¤erent frequencies and that this cointegration is periodic. In the secondsubsection we show that between real valued processes the long run relationship between processes integratedat di¤erent frequencies should be periodic and polynomial. As we will work with two processes associatedto di¤erent frequencies, hence we will de�ne !j = 2�j=N and !k = 2�k=N with j 6= k.
3.1 Cointegration between complex I!j(1) and I!k(1) processes
Based on the results of the previous section, it is possible to de�ne the following triangular system of a longrun relationship (cointegration) between two complex valued process associated to the zero frequency:
y(0)�t = �x
(0)�t + ut (8)
x(0)�t = x
(0)�t�1 + e
ti!j�n�
where both the parameter of the long run relationship � and the I (0) innovation ut are complex valued.Note that if we multiply by e�i!jt both sides y(0)�t = �x
(0)�t +ut we obtain the triangular system of Gregoir
(2010) (pp 1500):
e�i!jty(0)�t = �e�i!jtx
(0)�t + e�i!jtut (9)
e�i!jtx(0)�t = x�t
x�t = e�i!jx�t�1 + �t:
Note also that it is possible to write x(0)�t = ei!jtx�t , hence if we replace it in (8) we get:
y(0)�t = �ei!jtx�t + ut (10)
x�t = e�i!jx�t�1 + �t
Hence now we have a long-run relationship between y(0)�t a complex valued integrated process at the zerofrequency I0(1) and x
�t a complex valued integrated process at the frequency !j , I!j (0). It is also important
to note that between y(0)�t and x�t we have a periodic cointegration, as the cointegration vector varieswith the seasons3 �ei!jt = �ei!j(N(��1)+n) = �ei!jn. Lemma 5 in the appendix summarize the stochasticbehaviour. of the triangular system (10) using bi-variant vector of seasons for the triangular system de�nedin (10). It is clear from (34) that y(0)�t and x�t share a common stochastic trend and hence that we haveN�1 cointegration relationships between the seasons of the variables included in bi-variant vector of seasons.Note also that if in (10) we premultiply by the demodulator operator e�i!kt that will shift the complex val-
ued zero frequency process y(0)�t from the zero frequency to frequency !k = 2�k=N with k = 0; 1; : : : ; bN=2cand with j 6= k4 we have:
e�i!kty(0)�t = �ei[!j�!k]tx�t + e
�i!ktut:
Where we could de�ne y�t = e�i!kty
(0)�t , clearly y�t is a complex valued process integrated at frequency !k
sharing a common stochastic trend with x�t , that is a complex valued process integrated at frequency !j .Hence we have the following system with one I!j (1) and one I!k(1) with a periodic cointegration relationship�1;��ei[!j�!k]t
�0, as we have di¤erent coe¢ cient per each season:
y�t = �e�i[!k�!j ]tx�t + e�i!ktut (11)
x�t = e�i!jx�t�1 + �t:
In Lemma 6 in the appendix we summarize the stochastic behaviour. of the triangular system (11) using abi-variant vector of seasons and double subscript notation.
3As !j = 2�j=N it is evident that !j (N(� � 1) + n) is periodic and hene the identity �ei!j(N(��1)+n) = �ei!jn holds.4 If j = k it is clear that we are in the case of (9).
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Note also that if in (11) e�i!kt is such that !N=2 = � we �nally have a long relationship between a processintegrated at an harmonic frequency !j = 2�j=N and a process integrated at the Nyquist frequency (�).To summarize with (10), (34), (11) and (43) we have shown that it is possible to de�ne long-run relation-
ship between complex valued integrated processes associated to di¤erent frequencies. In the next section weshow that it is possible extend the situation to the case of real valued processes.
3.2 Cointegration between real valued I!j(1) and I!k(1) processes
In �rst place we are going to show that it is possible to establish a long-run relationship between a real valuedprocess integrated at an harmonic frequency and a real valued process associated to the zero frequency. Basedon the complex valued triangular representation (10) and in its complex conjugate system we have:
y(0)�t = �ei!jtx�t + ut (12)
x�t = e�i!jx�t�1 + �t
y(0)+t = ��e�i!jtx+t + �ut (13)
x+t = ei!jx+t�1 + �t
where �ut and �� = [�R � i�I ] are the complex conjugates of ut and � = [�R � i�I ] respectively. Note that x�tand x+t are connected with a real valued process integrated at frequency !j say
�1� 2 cos (!j)L+ L2
�xt = �t
by x�t =�1� ei!jL
�xt and by x
+t =
�1� e�i!jL
�xt respectively. Using this fact we get:
y(0)�t = [�R + i�I ] e
i!jt�1� ei!jL
�xt + ut (14)
y(0)+t = [�R � i�I ] e�i!jt
�1� e�i!jL
�xt + �ut (15)�
1� 2 cos (!j)L+ L2�xt = �t:
If we add both cointegration relationships (14) and (15) we obtain:�y(0)�t + y
(0)+t
�=2 = [�R cos (!jt)� �I sin (!jt)]xt (16)
� [�R cos (!j [t+ 1])� �I sin (!j [t+ 1])]xt�1+(ut + �ut)�
1� 2 cos (!j)L+ L2�xt = �t:
Where�y(0)�t + y
(0)+t
�=2 is a real valued process associated to zero frequency I0(1), that shares the sto-
chastic non-stationary behaviour of�1� 2 cos (!j)L+ L2
�xt = �t a real valued process I!j (1) associated to
frequency !j . Clearly in (16) we have periodic cointegration but also polynomial cointegration, note that ast = N (� � 1) + n we could write:
[�R cos (!jt)� �I sin (!jt)] = [�R cos (!jn)� �I sin (!jn)] = �1n (17)
[�R cos (!j [t+ 1])� �I sin (!j [t+ 1])] = [�R cos (!j [n+ 1])� �I sin (!j [n+ 1])] = �2n:
In Lemma 8 in the appendix we summarize the stochastic behaviour of the triangular system (16).In the case of long-run relationship between two real valued process integrated at di¤erent harmonic
frequencies, based on the complex valued triangular representation (11) and its complex conjugate system5 :
y�n� = �e�i[!k�!j ]nx�n� + e�i!knun� (18)
x�n� = e�i!jx�n�1;� + �n�
y+n� = ��ei[!k�!j ]nx+n� + ei!kn�un� (19)
x+n� = ei!jx+n�1;� + �n�
5Using the double subscript notation described in the appendix , also note that as t = N (� � 1) + n, we have for examplee�i[!k�!j ]t = e�i[!k�!j ](N(��1)+n) = e�i[!k�!j ]n.
5
where as in the previous case � and �� are complex conjugates and also un� and �un� . Using the connectionbetween the complex valued processes y�n� , y
+n� , x
�n� and x
+n� and the real valued processes yn� and xn� it is
possible to write:
�1� ei!kL
�yn� = [�R + i�I ] e
�i[!k�!j ]n�1� ei!jL
�xn� + e
�i!knun� (20)�1� e�i!kL
�yn� = [�R � i�I ] ei[!k�!j ]n
�1� e�i!jL
�xn� + e
i!kn�un� (21)�1� 2 cos (!j)L+ L2
�xn� = �n� :
by adding and subtracting both cointegration relationships we obtain:
yn� = cos (!k) yn�1;� + [�R cos ([!k � !j ]n) + �I sin [!k � !j ]n]xn�� [�R cos (!kn� !j [n+ 1]) + �I sin (!kn� !j [n+ 1])]xn�1:� (22)
+�e�i!knun� + e
i!kn�un��=2
sin (!k) yn�1;� = [�R sin ([!k � !j ]n)� �I cos ([!k � !j ]n)]xn�� [�R sin (!kn� !j [n+ 1])� (23)
�I cos (!kn� !j [n+ 1])]xn�1;�+�e�i!knun� � ei!kn�un�
�= (�2i) :
Hence we have two cointegration relationships as in Gregoir (2010). In the case of !k = !j (22)-(23)reduce to the result reported in Gregoir (2010). In our case we also have polynomial cointegration but alsoperiodic cointegration.For our convenience we summarize (22)-(23), in one expression using (22)+cos(!k)= sin(!k)(23), and
obtain the following triangular system:
yn� =
��R
�cos ([!k � !j ]n) +
cos (!k)
sin (!k)sin ([!k � !j ]n)
�� �I
�sin ([!k � !j ]n)�
cos (!k)
sin (!k)cos ([!k � !j ]n)
��xn�
���R
�cos (!kn� !j [n+ 1]) +
cos (!k)
sin (!k)sin (!kn� !j [n+ 1])
�(24)
��I�sin (!kn� !j [n+ 1])�
cos (!k)
sin (!k)cos (!kn� !j [n+ 1])
��xn�1;� +$n�
�1� 2 cos (!j)L+ L2
�xt = �t:
In Lemma 9 of the appendix we summarized the stochastic behaviour of the triangular system (24).
3.3 Econometric strategy
Once we have shown that it is possible to �nd long-run relationship between processes integrated at di¤erentfrequencies. We need an Econometric strategy to detect this situation. To do that, we propose two di¤erentways based on the Johansen (1996) procedure to determine the cointegration rank in a VAR model. Our�rst and simpler approach consist in testing the cointegration rank in a VAR model applied to a system of2N variables given by vector of seasons representation of each of the two individual time series6 , which treatsthe intra-cycle n = 1; :::; N as distinct time series. This approach is discussed by, for example, Ghysels andOsborn (2001, Chapter 6) in the context of contemporaneous periodic cointegration. However, it has theobvious disadvantage of typically requiring the use of high dimensional systems. For example if we have a twointegrated time series one associated to the zero frequency and another associated to an harmonic frequency,if we do not have cointegration we should �nd 3 common trends or 2N�3 cointegration relationships betweenthe elements of the 2N � 1 vector formed with the vector of seasons of the two time series. But if we havecointegration, that is, Periodic Polynomial Cointegration, we should �nd evidence in favour of the existenceof 2 common trends or 2N � 2 cointegration relationships between the elements of the 2N � 1 system. Thebasic problem of this �rst approach is that is going to be quite overparametrizised and it is also well knownthat the Johansen procedure do not works well when we have VAR models with a high dimension.
6 In the case of k variables the system will be of kN where each of the time series is treated as N � 1 vector of seasons.
6
The second approach to test for the presence of cyclic cointegration between yt � I0(1) and xt � I!(1)is to apply the transformation e�i!t(1 � e�i!L)xt and ei!t(1 � ei!L)xtto the latter. Then using softwarethat allows computations with complex variables, a test for cointegration is conducted in the bivariatesystem consisting of yt and either e�i!t(1 � e�i!L)xt or ei!t(1 � ei!L)xt. In the case of two real valueprocesses associated to harmonic frequencies yt � I!j (1) and xt � I!k(1) is to apply the transformationse�i!jt(1 � e�i!jL)yt and e�i!kt(1 � e�i!kL)xt and test for cointegration between then and proceed in aequivalent way with ei!jt(1� ei!jL)yt and ei!kt(1� ei!kL)xt. We apply the method proposed by Cubadda(2001) to test for cointegration in harmonic frequencies. Hence in this case we use the critical values collectedin the paper by Cubadda (2001) as in our case the distribution of the test should be the same as the onereported by Cubadda (2001) in theorem 2.1. Note that in the Lemmas in the appendix we always havecomplex Brownian Motions as in theorem 2.1.Finally the e¤ect of the demodulator operator e�i!jt on the deterministic part could be easily understand
following the lines of chapter 7 in Bloom�eld (2000). Note that for example seasonal dummies have aone to one correspondence with their trigonometric representation in terms of cos (!jt) and sin (!jt) notethat cos (!jt) = e�i!jt + ei!jt, and hence (1 � e�i!kL) cos (!jt) = ei!jt � ei!j(t�2) and �nally e�i!kt(1 �e�i!kL) cos (!jt) = 1� e�i!j2 hence becomes a constant once demodulated.
4 Monte Carlo
In this section we undertake a Monte Carlo experiment to illustrate the nature of long-run relationshipsbetween processes that are nonstationary at di¤erent frequencies and also show how to test for cointegrationin such situations. In particular consider examples with N = 6, and for two sample size of 100 and 200complete cycles, and hence the total number of observations is 6T = 600 and 6T = 1200, respectively, forthe two cases. The Monte Carlo results are based on 5; 000 replications.For the cointegration situation we work with two cases
Case I: Where we reproduce situation collected in (16) with !j = 2�=3 and �=3. Hence we are in a situationof cointegration between an I0 (1) process and an I!j (1) process. Table 1 collects the results (16) with!j = �=3, �R = cos (��=3), �I = cos (��=3) and T = 200. Clearly the results of table 1 show that it ispossible to establish a long-run relationship between a process integrated in the frequency �=3 and a processintegrated in the zero. For the sample size of T = 200 both method proposed in the previous section performwell. In table 2 we also collects the results from (16) but in this case for !j = 2�=3, �R = cos (�2�=3),�I = cos (�2�=3) and T = 200 and 100. The results of table 2, also show that it is possible to establishcointegration between processes integrated at the zero and 2�=3, but it also clearly show a poor performanceof the vector of seasons approach with the sample size of T = 100 but the one based on the demodulatoroperator performs equally well for both sample sizes.Case II: Where we reproduce situation collected in (24) with !k = 2�=3 and !j = �=3, �R = cos (��=3),�I = cos (��=3), T = 200 and 100 As in case I we could appreciate that with T = 200 both method performwell but with T = 100 only the demodulator approach perform properly.Finally, in order to complete the picture, we also show the performance of the previous tests in situations
when we do not have cointegration between the processes integrated at di¤erent frequencies. In particularwe pay attention to the following cases:
xn� = 2 cos�2�
3
�xn�1;� � xn�2;� + "xn� "xn�~Niid(0; 1) (25)
yn� = yn�1;� + "yn� "yn�~Niid(0; 1)
xn� = 2 cos�2�
3
�xn�1;� � xn�2;� + "xn� "xn�~Niid(0; 1) (26)
yn� = 2 cos��3
�yn�1;� � yn�2;� + "yn� "yn�~Niid(0; 1)
With T = 200, the results are collected in table 4, and show that the two methods proposed work properlyfor a sample size of T = 200.Despite that both methods proposed in the previous section to detect the presence of cointegration
between processes integrated at di¤erent frequencies when the sample size is big, but the vector of seasonsapproach do not work so well as the sample size decreases, hence we advice to use the approach based onthe demodulator operator.
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5 Empirical application
In this section we explore the presence of cointegration relationship between processes integrated at di¤erentfrequencies using data quarterly data from the Balearic Islands, in particular we analyze the connectionbetween total employment (empn� ) and tourist arrivals (arrn� )form the �rst quarter of 1979 to the fourthquarter of 2015. Picture 1 and 2 show the time series in logs. Using the HEGY test we �nd clear evidenceof the presence of a unit root at the zero frequency for ln (empn� ). In the case of ln (arrn� ) we �nd clearevidence of a unit root at the zero frequency and a pair of complex unit roots at frequency �=2 and mildevidence of the presence of a unit root at the Nyquist frequency. We have explore the possibility of a longrun relationship between ln (empn� ) and (1� L) ln (arrn� ) and also ln (empn� ) and (1� L) (1 + L) ln (arrn� ).That is, we are going to check if the non-stationary behaviour associated to frequency �=2 present in thetourism arrivals could a¤ect the non-stationary behaviour observed in the employment. The following tablecollect the results obtained using the demodulator approach, the results consider the inclusion of quarterlyseasonal dummies:
ln (empn� ) (1� L) ln (arrn� )VAR order ei
�2 t e�i
�2 t
q0 = 2 q0 = 1 q0 = 2 q0 = 12 4,8658 123,9097��� 4,8658 123,9097���
3 4,4071 95,9816��� 4,4071 95,9816���
4 0,2953 104,1451��� 0,2953 104,1451���
5 0,1190 107,1158��� 0,1190 107,1158���
6 0,1641 113,9965��� 0,1641 113,9965���
7 0,4588 117,1246��� 0,4588 117,1246���
8 0,0590 127,2409��� 0,0590 127,2409���
ln (empn� ) (1� L) (1 + L) ln (arrn� )VAR order ei
�2 t e�i
�2 t
q0 = 2 q0 = 1 q0 = 2 q0 = 12 4.7300 130.1296��� 4.7300 130.1296���
3 4.3086 101.9734��� 4.3086 101.9734���
4 0.2436 111.3259��� 0.2436 111.3259���
5 0.0447 118.5896��� 0.0447 118.5896���
6 0.1865 124.5702��� 0.1865 124.5702���
7 0.4064 125.7330��� 0.4064 125.7330���
8 0.0253 127.1980��� 0.0253 127.1980���
6 References
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Bauer D. and Wagner M. (2012) A state space canonical form for unit root processes, Econometric Theory,28, 1313-1349.
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del Barrio Castro, T. (2007). Using the HEGY procedure when not all unit roots are present, Journal ofTime Series Analysis, 28, 910-922.
del Barrio Castro, T. and Osborn, D.R. (2008). Testing for seasonal unit roots in periodic integratedautoregressive processes, Econometric Theory, 24, 1093-1129.
del Barrio Castro, T., P.M.M. Rodrigues & A.M.R. Taylor (2015) Semi-Parametric Seasonal Unit RootTests, DEA Working Papers 72.
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9
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10
7 Appendix
It will be useful to introduce a double subscript notation xn� where the subscripts n� indicate the nth
observation within the � th cycle and the total number of observations per cycle N . As in the later sectionswe are going to work with processes integrated at di¤erent spectral frequencies it will we convenient todenote the frequency associated to process xn�as !j = 2�j=N with j = 0; 1; : : : ; bN=2c and b:c denotes deinteger part. Hence for example !1 = 2�=N completes a full cycle every N observations. A double subscriptnotation when considering an I!j (1), with (2) then written as
xn� = (2 cos!j)xn�1;� � xn�2;� + vn� ; n = 1; 2; � � � ; N (27)
where the subscripts n� indicate the nth observation within the � th cycle at frequency !j . When using thedouble subscript notation, it is understood that yn�k;� = yN�n+k;��1 for n�k � 0. Adopting the conventionthat t = 1 corresponds to n = � = 1, then t = N(� � 1) + n provides the one-to-one mapping between thetwo notations.De�ne the N �1 vector of seasons X�
� =�x�1� ; x
�2� ; x
�3� ; : : : ; x
�N�
�0to (4) and present the following lemma
where we summarize the stochastic characteristics of (4):
Lemma 1 For X�� =
�x�1� ; x
�2� ; x
�3� ; : : : ; x
�N�
�0with x�n� n = 1; 2; � � � ; N de�ned in (4) and with vn� �
iid�0; �2
�it is possible to write:
1pTX�bTrc ) �C�j W (r) = �v�j v
+0
j Wv (r) (28)
= � (N=2)1=2v�j (wR (r) + iwI (r))
where C�j is circulant matrix of rank one C�j = Ci rc
�1; e�i!j ; e�i2!j ; � � � ; e�i(N�1)!j
�, and the vectors v�j
and v+j are de�ned as v�j =�e�i!j e�i2!j e�i3!j � � � e�iN!j
�0and as v+j =
�ei!j ei2!j ei3!j
� � � eiN!j�. W v (r) =
�W v1 (r) W v
2 (r) � � � W vN (r)
�0is a N � 1 vector Brownian motion and
wvR (r) and wvI (r) are two scalar Brownian motions de�ned as w
vR (r) = (S=2)
�1=2PNk=1 cos (k!j)W
vk (r)
and wvI (r) = (S=2)�1=2PN
k=1 sin (k!j)Wvk (r) respectively.
Remark 2 Note that the result in Lemma 1 above also applies to x+t (7) and it is straightforward to seethat for X+
� =�x+1� ; x
+2� ; x
+3� ; : : : ; x
+N�
�0it is possible to write T 1=2X+
bTrc ) �C+j Wv (r) = �v+j v
�0
j Wv (r) =
� (N=2)1=2v�j (w
vR (r)� iwvI (r)) with C+j = Ci rc
�1; ei!j ; ei2!j ; � � � ; ei(N�1)!j
�. Hence we have a pair of
complex conjugate complex valued scalar Brownian motions wvR (r) � iwvI (r) as in Gregoir (2010) (seepp.1494). Note also that in del Barrio Castro, Rodrigues and Taylor (2017) it is shown a similar resultbut considering complex valued Near-Integrated processes and also allowing for serial correlation in the in-novations (see pp 9 (3.12) and (3.13)).
Remark 3 From (28) and (5) it is clear that (wvR (r) + iwvI (r)) collects the behaviour. of the stochastic
trendhx�0 +
Ptk=1 e
i!jkvk
iand the vector v�j =
�e�i!j e�i2!j e�i3!j � � � e�iN!j
�0the e¤ect of the
demodulator operator e�it!j . Another interesting point of (28) is that it clearly show that the seasons associ-ated to X�
� share a common stochastic trend, or equivalently that between the seasons of X�� we have N � 1
cointegration relationships.
Remark 4 In the case of process xn� (27) as shown in Smith, Taylor and del Barrio Castro (2009) Lemma 1and remark pp 540 we have the circulant matrix of rank 2 Cj = Ci rc
hsin(!j)sin(!j)
;sin(N!j)sin(!j)
;sin([N�1]!j)
sin(!j); : : : ;
sin(2!j)sin(!j)
iand note that we have the following relationship between Cj, C
�j and C
+j Cj =
e�i!j
e�i!j�ei!j C�j +
ei!j
ei!j�e�i!j C+j .
Proof of Lemma 1: First note that the process (4) admits the following vector of seasons representationX�� =
�x�1� ; x
�2� ; :::; x
�N�
�0, then
��0 X�� = �
�1 X
���1 + V� (29)
where, V� =�v�1� ; v
�2� ; :::; v
�N�
�0and ��0 and �
�1 are N �N matrices with the generic element of
��0(h;j) =
8<: 1 h = j; j = 1; :::; N�e�i! h = j + 1; j = 1; :::; N � 10 otherwise
��1(h;j) =
�e�i! h = 1; j = N0 otherwise
11
in which the subscript (h; j) indicates the (h; j)th element of the respective matrix. As in the quarterly PI(1)model studied by Paap and Franses (1999), successively substituting in (29) yields
X�� = [(��0 )
�1��1 ]�X�
0 + (��0 )
�1V� +��1Xj=1
[(��0 )�1��1 ]
j(��0 )�1V��j
= [(��0 )�1��1 ]
�X�0 + (�
�0 )
�1V� + [(��0 )
�1��1 ](��0 )
�1��1Xj=1
V��j : (30)
Note that this result follows because (��0 )�1��1 is idempotent, which can be seen by generalizing the form
of (��0 )�1��1 presented by Paap and Franses (1999) and noting that e�i!N = 1 for x�n� � I!(1); and
hence7 [(��0 )�1��1 ]
j = [(��0 )�1��1 ] for j = 2; 3; :::.Clearly, (30) provides an alternative representation of
(5), but now expressed in terms of the vector of (complex-valued) observations over an entire cycle, where[(��0 )
�1��1 ](��0 )
�1 gives the e¤ect of the accumulated vector of shocksP��1
j=1 V��j (see for example Boswijkand Franses (1996), Paap and Franses (1999), del Barrio Castro and Osborn (2008), and for complex-valuedprocesses in the context of seasonally integrated processes by del Barrio Castro, Rodrigues and Taylor(2017)). The matrix [(��0 )
�1��1 ](��0 )
�1 has rank one and hence we can write it as
C�j = [(��0 )
�1��1 ](��0 )
�1 = (a�j )(b�j )
0 (31)
where, for (31),
a�j =�1 e�i!j e�i2!j � � � e�i(N�1)!j
�0b�j =
�1 e�i(N�1)!j e�i(N�2)!j � � � e�i!j
�0: (32)
Therefore, in (30), the scalar partial sum (b�j )0P��1
j=1 V��j is integrated at the zero frequency, while a�j
allocates this across the N observations of the cycle at frequency !j , yielding an I!j (1) process. This is, ofcourse, the same as in (5)-(6), but the cyclicality of the resulting process is now made clear by representingthe demodulation operator through the vector a�j . Note also that the N � N matrix C�j is a Circulant
matrix C�j = Ci rc�1; e�i!j ; e�i2!j ; � � � ; e�i(N�1)!j
�. Note also that we have 1p
T
PbTrcj=1 Vj ) �W (r) where
W (r) is a N � 1 vector Brownian motion. Also for the circulant matrix C�j it is possible to write:
C�j = (a�j )(b�j )
0 = (a�j )e�i!jei!j (b�j )
0 = v�j v+0j (33)
v�j =�e�i!j e�i2!j e�i3!j � � � e�iN!j
�0v+j =
�ei!j ei2!j ei3!j � � � eiN!j
�0:
Finally note that v+0j W (r) =PN
k=1 eik!jWk (r) =
PNk=1 cos (k!j)Wk (r)+i
PNk=1 sin (k!j)Wk (r), and that
wR (r) = (S=2)�1=2PN
k=1 cos (k!j)Wk (r) and wI (r) = (S=2)�1=2PN
k=1 sin (k!j)Wk (r).�
Lemma 5 For Z(0;!j)�� =hy(0)�1� ; y
(0)�2� ; : : : ; y
(0)�N� ; x
�1� ; x
�2� ; : : : ; x
�N�
i0with x�n� and y
(0)�n� n = 1; 2; � � � ; N
de�ned in (10) and with vn� � iid�0; �2
�and un� � iid
�0; �2u
�it is possible to write:
1pTZ(0;!j)�bTrc ) � (N=2)
1=2
��1v�j
�(wvR (r) + iw
vI (r)) (34)
with (wvR (r) + iwvI (r)) and v
�j as in Lemma 1 and �nally 1 is a N � 1 vector of ones.
Proof of Lemma 5: In the case of triangular system (10) we then de�ne the vectors Z(0;!j)�� =hy(0)�1� ; y
(0)�2� ; � � � ; y(0)�N� ; x
�1� ; x
�2� ; � � � ; x�N�
i0and V Z� = [u1� ; u2� ; � � � ; uN� ; v1� ; v2� ; � � � ; vN� ]0 = [V u0� V 0� ]
0 with
7 Note that matrix ��0 (see chapter 2 pp 45-48 of Pollock (1999))is a N �N lower-triangular Toeplitz matrix associated to
the polynomial�1� e�i!jL
�. Hence in matrix
���0
��1it will be collected the coe¢ cientes on the expansion of the inverse
polynomial associated to�1� e�i!jL
�. Based on the form of matrices (��0 )
�1 and ��1 it is clear that he resulting matrix(��0 )
�1��1 will be a N �N matrix with the �rsts N � 1 columns with elements equal to zero and the last column equal to thecolumn vector v�j =
�e�i!j e�i2!j e�i3!j � � � e�iN!j
�0. �nally note that the last element of v�j , that is, e�iN!j isequal to 1 as !j = 2�j=N and hence the idempotency of (��0 )
�1��1 could be deduced easily.
12
V� = [v1� ; v2� ; � � � ; vN� ]0 as in Lemma 1 and V u� = [u1� ; u2� ; � � � ; uN� ]0, using the same line of argument as
in the proof of Lemma 1 we can write
��0 Z(0;!j)�� = ��1 Z
(0;!j)���1 + V Z� (35)
where
��0 =
�IN�N ��120
0N�N ��220
�; ��1 =
�0N�N 0N�N0N�N ��221
�in which all sub-matrix are N �N , with
��120 = Diag [��1;��2; � � � ;��N ]= Diag
��ei!j�;�ei!j2�; :::;�ei!jN�
�Diag
h�e�i!j [N�1]�;�e�i!j [N�2]�; :::; �
i��220(h;k) =
8<: 1 h = k; k = 1; :::; N�e�i!j h = k + 1; k = 1; :::; N
0 otherwise
��221(h;k) =
�e�i!j h = 1; k = N0 otherwise
;
� is the zero frequency cointegration coe¢ cient of (8) and the subscripts (h; j) refer to the (h; j)th elementof the respective sub-matrix, ��120 or ��221 . Note that the inverse of matrix ��0 using results for the inverseof partitioned matrices will be as follows:
���0��1
=
"IN�N ���120
���220
��10N�N
���220
��1#
Note also that sub matrix���220
��1as previously stated is the inverse of a N �N lower-triangular Toeplitz
matrix associated to the polynomial�1� e�i!jL
�. Hence matrix
���0��1
will collect the coe¢ cients on the
expansion of the inverse polynomial associated to�1� e�i!jL
�. Once we know the form of
���0��1
it is
easy to check that the resulting matrix���0��1
��1 will be a 2N � 2N matrix with its �rst columns with allthe elements equal to zero and the last column collecting in �rst place the elements of the �rst column of���120
���220
��1multiplied by8 e�i!j , followed by the �rst column of matrix
���220
��1multiplied by e�i!j
as well9 , hence as the last element of the last column of���0��1
��1 is equal to e�iN!j = 1. it is clear that
matrix���0��1
��1 is idempotent.
� 1. Recursive substitution from (35), yields
Z(0;!j)�� =���0��1
��1 Z(0;!j)�0 +
���0��1
V Z� +���0��1
��1���0��1 ��1X
j=1
V Z��j (36)
We can write ���0��1
��1���0��1
=
�0N�N C
(0)�y
0N�N C�x
�(37)
where the N �N sub-matrices C(0)�y and C�x each have rank one, so that
C�x = a�x b�0x (38)
C(0)�y = a(0)�y b�0x
with
a�x =�1 e�i!j e�i2!j � � � e�i(N�2)!j e�i(N�1)!j
�0a(0)�y = ei!j�
�1 1 1 � � � 1 1
�0(39)
b�x = b�y =�1 e�i(N�1)!j e�i(N�2)!j � � � e�i2!j e�i!j
�0:
8That is e�i!jh�1; e
�i!j�2; � � � ; e�i!j [N�1]�Ni0= e�i!j
he�i!j [N�1]�; e�i!j [N�2]e�i!j�; � � � ; e�i!j [N�1]�
i0=
� [1; 1; � � � ; 1] :9That is e�i!j
h1; e�i!j ; � � � ; e�i!j [N�1]
i0=�e�i!j ; e�i2!j ; � � � ; e�iN!j
�013
It is clear from (36) to (39) that���0��1
��1���0��1
, C(0)�y and C�x have rank 1, and hence
there is one common stochastic trend shared by the seasons of both time series y(0)�n� and x�n� onequivalently we have 2N � 1 cointegration relationships between the seasons of the vector Z�� .Note also that C�x in (38) has the same expression that (33) them it is possible to write:
C�x = (a�x )(b�x )
0 = (a�x )e�i!jei!j (b�x )
0 = v�x v+0x (40)
v�x =�e�i!j e�i2!j e�i3!j � � � e�iN!j
�0v+x =
�ei!j ei2!j ei3!j � � � eiN!j
�0:
And in the case of C(0)�y note that:
C(0)�y = a(0)�y b�0x = �1ei!j b�0x = �1v+0x (41)
with 1 a N � 1 vector of ones. Based of the fact that:
1pT
bTrcXj=1
V Z� =
"1pT
PbTrcj=1 V
u�
1pT
PbTrcj=1 V
u�
#)��uW
u (r)�W v (r)
�where Wu (r) and W v (r) are two N � 1 vector Brownian motions. It is possible to see that forZ�� in (36) we have:
1pTZ(0;!j)�bTrc )
�0N�N C
(0)�y
0N�N C�x
� ��uW
u (r)�W v (r)
�(42)
and the result in Lemma 5 came from taking together (42),(40) and (41).�
Lemma 6 For Z(!k;!j)�� =�y�1� ; y
�2� ; : : : ; y
�N� ; x
�1� ; x
�2� ; : : : ; x
�N�
�0with x�n� and y
�n�n = 1; 2; � � � ; N de�ned
in (11) and with vn� � iid�0; �2
�and un� � iid
�0; �2u
�it is possible to write:
1pTZ(!k;!j)�bTrc ) � (N=2)
1=2
��v�kv�j
�(wvR (r) + iw
vI (r)) (43)
with (wvR (r) + iwvI (r)) and v
�j as in Lemma 1 and �nally v
�k =
�e�i!k e�i2!k e�i3!k � � � e�iN!k
�0.
Remark 7 As particular case of (11) and (43) we could de�ne a triangular system between a couple ofcomplex valued integrated processes one associated to the Nyquist frequency (�) and another associated to anharmonic frequency !j, if we multiply (10) by e�i�(N(��1)+n).
Proof of Lemma 6: In the case of triangular system (11) we then de�ne the vectors Z(!k;!j)�� =�y�1� ; y
�2� ; � � � ; y�N� ; x
�1� ; x
�2� ; � � � ; x�N�
�0and V Z� = [u1� ; u2� ; � � � ; uN� ; v1� ; v2� ; � � � ; vN� ]0 = [V u0� V 0� ]
0 with V� =
[v1� ; v2� ; � � � ; vN� ]0 as in Lemma 1 and V u� =��ei!ku1� ;�eiw!ku2� ; � � � ;�eiN!kuN�
�0, using the same line
of argument as in the proof of Lemma 5 we can write
��0 Z(!k;!j)�� = ��1 Z
(!k;!j)���1 + V Z� (44)
where
��0 =
�IN�N ��120
0N�N ��220
�; ��1 =
�0N�N 0N�N0N�N ��221
�in which all sub-matrix are N �N , with
��120 = Diag [��1;��2; � � � ;��N ]
= Diagh�ei(!j�!k)�;�ei(!j�!k)2�; :::;�ei(!j�!k)N�
iDiag
h�e�i(!j�!k)[N�1]�;�e�i(!j�!k)[N�1]�; :::;�e�i(!j�!k)�
i��220(h;k) =
8<: 1 h = k; k = 1; :::; N�e�i!j h = k + 1; k = 1; :::; N
0 otherwise
��221(h;k) =
�e�i!j h = 1; k = N0 otherwise
;
14
Recursive substitution from (44), and recognizing that���0��1
��1 is idempotent10 , yields
Z(!k;!j)�� =���0��1
��1 Z(!k;!j)�0 +
���0��1
V Z� +���0��1
��1���0��1 ��1X
j=1
V Z��j (45)
We can write ���0��1
��1���0��1
=
�0N�N C�y0N�N C�x
�(46)
where the N �N sub-matrices Cy and C�x each have rank one, so that
C�x = a�x b�0x (47)
C�y = a�y b�0x
with
a�x =�1 e�i!j e�i2!j � � � e�i(N�2)!j e�i(N�1)!j
�0a�y = e�i!kei!j�
�1 e�i!k e�i2!k � � � e�i(N�2)!k e�i(N�1)!k
�0(48)
b�x = b�y =�1 e�i(N�1)!j e�i(N�2)!j � � � e�i2!j e�i!j
�0:
It is clear from (46) to (47) that���0��1
��1���0��1
, C�y and C�x have rank 1, and hence there is onecommon stochastic trend shared by the seasons of both time series y�n�and x
�n� on equivalently we have
2N � 1 cointegration relationships between the seasons of the vector Z(!k;!j)�� . Note also that C�x in (47)has the same expression that (33) them it is possible to write:
C�x = v�x v+0x (49)
v�x =�e�i!j e�i2!j e�i3!j � � � e�iN!j
�0v�+x =
�ei!j ei2!j ei3!j � � � eiN!j
�0:
And in the case of C�y note that:
C�y = a�y b�0x = �v�y v
+0x (50)
v�y =�e�i!k e�i2!k e�i3!k � � � e�iN!k
�0:
Based of the fact that:1pT
bTrcXj=1
V Z� =
"1pT
PbTrcj=1 V
u�
1pT
PbTrcj=1 V
u�
#)��uW
u (r)�W v (r)
�where W v (r) is a N � 1 vector Brownian motion as in Lemma 2 and Wu (r) is a N � 1 vector complexvalued Brownian motion. Hence it is possible to see that for Z(!k;!j)�� in (45) we have:
1pTZ(0;!j)�bTrc )
�0N�N C�y0N�N C�x
� ��uW
u (r)�W v (r)
�(51)
and the result in Lemma 3 came from taking together (51),(49) and (50).�10First note that the for the inverse of ��0 we have:
���0
��1=
24 IN�N ���120
���220
��10N�N
���220
��135 :
Note also that the resulting matrix���0
��1��1 is a 2N � 2N matrix with its �rst colunms with all the elements equal to zero
and the last column collecting in �rst place the elements of the �rst column of ���120
���220
��1multiplied by e�i!j , followed
by the �rst column of matrix���220
��1multiplied by e�i!j as well, in this case also the last element of the last column of�
��0
��1��1 is equal to e�iN!j = 1. it is clear that matrix
���0
��1��1 is idempotent.
15
Lemma 8 For Z(0;!j)� =h�y(0)�1� + y
(0)+1�
�=2;�y(0)�2� + y
(0)+2�
�=2; : : : ;
�y(0)�N� + y
(0)+N�
�=2; x1� ; x2� ; : : : ; xN�
i0with xn� and
�y(0)�n� + y
(0)+n�
�=2 n = 1; 2; � � � ; N de�ned in (16) and with vn� � iid
�0; �2
�,un� � iid
�0; �2u
�and un� � iid
�0; �2u
�it is possible to write:
1pTZ(0;!j)
bTrc ) � (N=2)1=2
"12 [(�R + i�I)1 (w
vR (r) + iw
vI (r)) + (�R � i�I)1 (wvR (r)� iwvI (r))]
e�i!j
�2i sin(!j)v�j (w
vR (r) + iw
vI (r)) +
e�i!j
2i sin(!j)v�j (w
vR (r)� iwvI (r))
#(52)
with (wvR (r) + iwvI (r)), v
�j and v
+j as in Lemma 1, (w
vR (r)� iwvI (r)) the complex conjugate of (wvR (r) + iwvI (r))
and 1 and N � 1 vector of ones.
Proof of Lemma 8: In the case of triangular system (16) we then de�ne the vectors Z(0;!j)� =h�y(0)�1� + y
(0)+1�
�=2;
�y(0)�2� + y
(0)+2�
�=2; : : : ;
�y(0)�N� + y
(0)+N�
�=2; x1� ; x2� ; : : : ; x
0N�
iand V Z� = [(u1� + �u1� ) =2; (u2� + �u2� ) =2; � � � ;
(uN� + �uN� ) =2; v1� ; v2� ; � � � ; v0N� ] = [V u0� V 0� ]0 with V� = [v1� ; v2� ; � � � ; vN� ]0 and V u� = [(u1� + �u1� ) =2; (u2� + �u2� ) =2;
� � � ; (uN� + �uN� ) =2]0, using the same line of argument as in the proof of previous lemmas we can write
�0Z(0;!j)� = �1Z
(0;!j)� + V Z� (53)
where
�0 =
�IN�N �1200N�N �220
�; �1 =
�0N�N �1210N�N �221
�in which all sub-matrix are N �N , with
�120 =
266666664
��11 0 0 0 � � � 0��22 ��12 0 0 � � � 00 ��23 ��13 0 � � � 00 0 ��24 ��14 � � � 0...
......
.... . .
...0 0 0 � � � ��2N ��1N
377777775with �1n and �2n de�ned in (17).
�220 =
26666666664
1 0 0 0 � � � 0�2 cos (!j) 1 0 0 � � � 0
1 �2 cos (!j) 1 0 � � � 00 1 �2 cos (!j) 1 � � � 00 0 1 �2 cos (!j) � � � 0...
......
.... . .
...0 0 � � � 1 �2 cos (!j) 1
37777777775
�121 =
266666664
0 0 0 � � � 0 �120 0 0 � � � 0 00 0 0 � � � 0 00 0 0 � � � 0 0.........
. . ....
...0 0 0 � � � 0 �0
377777775
�221 =
266666664
0 0 � � � 0 �1 2 cos (!j)0 0 � � � 0 0 �10 0 � � � 0 0 00 0 � � � 0 0 0......
. . ....
......
0 0 � � � 0 0 �0
377777775First note that, using results form the inverse of a partitioned matrix, we have:
(�0)�1=
"IN�N ��120
��220
��10N�N
��220
��1#:
16
Matrix �220 is a lower-triangular Toeplitz matrix associated to the polynomial�1� 2 cos (!j)L+ L2
�, hence�
�220��1
will collect the coe¢ cients of the expansion of the inverse of�1� 2 cos (!j)L+ L2
�, that is:
��220
��1=
1
sin (!j)
266666664
sin (!j) 0 0 0 � � � 0sin (2!j) sin (!j) 0 0 � � � 0sin (3!j) sin (2!j) sin (!j) 0 � � � 0sin (4!j) sin (3!j) sin (2!j) sin (!j) � � � 0
......
......
. . ....
sin (N!j) sin ([N � 1]!j) sin ([N � 2]!j) sin ([N � 3]!j) � � � sin (!j)
377777775In the case of the case of ��120
��220
��1it is easy to check that:
��120��220
��1=
1
sin (!j)�26664
�11 sin (!j) 0 � � � 0�12 sin (2!j) + �22 sin (!j) �12 sin (!j) � � � 0
......
. . ....
�1N sin (N!j) + �2N sin ([N � 1]!j) �1N sin ([N � 1]!j) + �2N sin ([N � 2]!j) � � � �1N sin (!j)
37775Matrix (�0)
�1�1 is a 2N � 2N matrix, but as matrix �1 only has 4 elements di¤erent from zero (see de
de�nition �121 and �221 above) it is possible to check that for (�0)�1�1 it is possible to write:
(�0)�1�1 =
�02N�2(N�1) Vy
02N�2(N�1) Vx
�(54)
where Vy and Vx are S � 2 matrices with the following form11 :
Vy =1
sin (!j)
266666664
� [�11 sin (!j)] �11 sin (2!j) + �12 sin (!j)� [�12 sin (2!j) + �22 sin (!j)] �12 sin (3!j) + �22 sin (2!j)� [�13 sin (3!j) + �23 sin (2!j)] �13 sin (4!j) + �23 sin (3!j)� [�14 sin (4!j) + �24 sin (3!j)] �14 sin (5!j) + �24 sin (4!j)
......
� [�1N sin (N!j) + �2N sin ([N � 1]!j)] �1N sin ([N + 1]!j) + �2N sin (N!j)
377777775(55)
Vx =1
sin (!j)
266666664
� sin (!j) sin (2!j)� sin (2!j) sin (3!j)� sin (3!j) sin (4!j)� sin (4!j) sin (5!j)
......
� sin (N!j) sin ([N + 1]!j)
377777775: (56)
It is easy to check that the matrix ��10 �1 is idempotent12 . Recursive substitution from (53) yields:
Z(0;!j)� = ��10 ��1 Z(0;!j)o +��10 V Z� +��10 �1�
�10
��1Xj=1
V Z��j
11Based on the form of �1 it is possible to check that the �st colunm of Vy and Vx is going to be equal to minus th �rstcolumn of
��220
��1 and ��120 ��220
��1 respectively. Finally in the case of the second column of Vy and Vx is the weigthed
sum of the �rst and second column of��220
��1 and ��120 ��220
��1 with weigths 2 cos (!k) and �1 respectively. Finally noteto obtain the second column for (55) and (56) we use the following recurrent expression for multiple angle:
sin (j!k) = 2 cos (!k) sin ([j � 1]!k)� sin ([j � 2]!k)cos (j!k) = 2 cos (!k) cos ([j � 1]!k)� cos ([j � 2]!k)
12The lower 2� 2 submatrix of Vx is equal to:264 � sin([N�1]!j)sin(!j)
sin(N!j)sin(!j)
� sin(N!j)sin(!j)
sin([N+1]!k)
sin(!j)
375and it is equal to I2�2. Hence
h(�0)
�1 �1ih= (�0)
�1 �1 for h = 2; 3; : : : :
17
and matrix ��10 �1��10 displays the impact of the accumulation shocks
P��1j=1 V
Z��j and has the form:
��10 �1��10 =
�0N�N VyU
0x
0N�N VxU0x
�U0x =
1
sin (!j)
�sin ([N � 1]!j) sin ([N � 2]!j) sin ([N � 3]!j) sin ([N � 4]!j) � � � 0sin (N!j) sin ([N � 1]!j) sin ([N � 2]!j) sin ([N � 3]!j) � � � sin (!j)
�:
First we are going to pay attention to VxU0x, in this case it is check that:
VxU0x = Cj = Ci rc
�sin (!j)
sin (!j);sin (N!j)
sin (!j); � � � ; sin (2!j)
sin (!j)
�=
e�i!j
�2i sin (!j)C�j +
ei!j
2i sin (!j)C+j : (57)
Where Cj is de�ned in remark 2 and C�j and C
+j in Lemma 1. In the case of Vy it is possible to show that
(55) could be simpli�ed as follows:
Vy =
266666664
� [�R cos (!j)� �I sin (!j)] �R� [�R cos (!j)� �I sin (!j)] �R� [�R cos (!j)� �I sin (!j)] �R� [�R cos (!j)� �I sin (!j)] �R
......
� [�R cos (!j)� �I sin (!j)] �R
377777775(58)
In order to obtain (58) form (55), we use the expression of �1n and �2n in (17) and the formula connectingproducts of sin and cos to sums13 jointly with the expressions for the double angle sin and cos14 . Clearlythe two columns of Vy are linearly dependent. Note also that it is possible to check that VyU
0x is a matrix
with generic row element �R cos (!jh)��I sin (!jh) with h = 1; 2; : : : ; N , that is, all the element of the �rstcolumn are equal to �R cos (!j)��I sin (!j), the elements of the second column to �R cos (!j2)��I sin (!j2),and so on a so forth. Finally it is possible to check that for VyU
0x it is possible to write:
VyU0x =
1
2
�(�R + i�I)1v
+0j + (�R � i�I)1v�0j
�:
Where 1 is aN�1 vector of ones and v�j =�e�i!j e�i2!j e�i3!j � � � e�iN!j
�0v+j =
�ei!j ei2!j ei3!j
� � � eiN!j�0. Hence the result in (52) it is easily obtained, following the lines of the proofs of the previous
lemmas.�
Lemma 9 For Z(!k;!j)� = [y1� ; y2� ; : : : ; yN� ; x1� ; x2� ; : : : ; xN� ]0 with xn� and y1� n = 1; 2; � � � ; N de�ned in
(24) and with vn� � iid�0; �2
�and $n� � iid
�0; �2$
�it is possible to write:
1pTZ(!k;!j)
bTrc ) � (N=2)1=2
26664n(�R+i�I)
2 v�k (wvR (r) + iw
vI (r)) +
(�R+i�I)2 v+k (w
vR (r)� iwvI (r))+
+ cos(!k)sin(!k)
h(�R+i�I)
�2i v�k (wvR (r) + iw
vI (r)) +
(�R+i�I)2i v+k (w
vR (r)� iwvI (r))
ione�i!j
�2i sin(!j)v�j (w
vR (r) + iw
vI (r)) +
e�i!j
2i sin(!j)v�j (w
vR (r)� iwvI (r))
o37775
(59)with (wvR (r)� iwvI (r)), v�j and v+j as in Lemma 8, and v�k =
�e�i!k e�i2!k e�i3!k � � � e�iN!k
�0and v+k =
�ei!k ei2!k ei3!k � � � eiN!k
�0.
Proof of Lemma 9: In the case of triangular system (24) we then de�ne the vectors Z(!k;!j)� =[y1� ; y2� ; : : : ; yN� ; x1� ; x2� ; : : : ; xN� ]
0 and V Z� = [$1� ; $2� ; � � � ; $N� ; v1� ; v2� ; � � � ; vN� ]0 = [V $0� V 0� ]0 with
V� = [v1� ; v2� ; � � � ; vN� ]0 and V $� = [$1� ; $2� ; � � � ; $N� ]0, using the same line of argument as in the proof
of previous lemmas we can write�0Z
(!k;!j)� = �1Z
(!k;!j)� + V Z� (60)
13 cos (!) sin (') =sin(!+')�sin(!�')
2and sin (!) sin (') = cos(!�')�cos(!+')
2
14 sin (2!) = 2 cos (!) sin (!) and 1�cos(2!)2 sin(!)
= sin (!)
18
where �0 and �1 in (60) have the same expression as in Lemma 8 but in this case �1n and �2n are de�nedas:
�1n =
��R
�cos ([!k � !j ]n) +
cos (!k)
sin (!k)sin ([!k � !j ]n)
�(61)
��I�sin ([!k � !j ]n)�
cos (!k)
sin (!k)cos ([!k � !j ]n)
���2n = �
��R
�cos (!kn� !j [n+ 1]) +
cos (!k)
sin (!k)sin (!kn� !j [n+ 1])
���I
�sin (!kn� !j [n+ 1])�
cos (!k)
sin (!k)cos (!kn� !j [n+ 1])
��Hence �120 and �121 are de�ned as in Lemma 8 but with �1n and �2n following the expression collected in
(61), and �220 and �221 with exactly the same expression as in Lemma 8. So also (�0)�1 and (�0)
�1�1 have
equivalent expression has in Lemma 8. Clearly (�0)�1�1 is idempotent and it has the form reported in (54)
with Vx de�ned in (56) and Vy as in (55) but with �1n and �2n de�ned in (61) by recursive substitutionfrom (60) yields:
Z(!k;!j)� = ��10 ��1 Z(!k;!j)o +��10 V Z� +��10 �1�
�10
��1Xj=1
V Z��j
matrix ��10 �1��10 is also equal to:
��10 �1��10 =
�0N�N VyU
0x
0N�N Cj
�Ux and Cj are exactly as in Lemma 8 (57) and Vy is equal to (55) with �1n and �2n de�ned in (61).Replacing �1n and �2n de�ned in (61) in (55) and using the expressions cos (!) sin (') =
sin(!+')�sin(!�')2 ,
sin (!) sin (') = cos(!�')�cos(!+')2 , sin (2!) = 2 cos (!) sin (!) and 1�cos(2!)
2 sin(!) = sin (!) we obtain that thegeneric element on the �rst column of Vy will be:
��R�cos (!kh� !j) +
cos (!k)
sin (!k)sin (!kh� !j)
�� (62)
��I�cos (!kh� !j)�
cos (!k)
sin (!k)sin (!kh� !j)
�and the generic element for the second column of Vy will be:
�R
�cos (!kh) +
cos (!k)
sin (!k)sin (!kh)
�+ (63)
+�I
�cos (!kh)�
cos (!k)
sin (!k)sin (!kh)
�:
With h in (62) an (63) been the row position of in each column, that is h = 1; : : : ; N .It is possible to check (using trigonometric identities) that the generic element of VyU
0x is going to have
the following expression:
�R
�cos (!kh� !jf) +
cos (!k)
sin (!k)sin (!kh� !jf)
�+ (64)
+�I
�sin (!kh� !jf)�
cos (!k)
sin (!k)cos (!kh� !jf)
�h = 1; : : : ; N
j = 1; : : : ; N
Where h refers to the rows position an f to the column position in VyU0x. Finally it is possible to write:
VyU0x =
�1
2
�(�R + i�I)v
�k v
+0j + (�R � i�I)v+k v
�0j
�+ (65)
+cos (!k)
sin (!k)
�(�R + i�I)
�2i v�k v+0j +
(�R + i�I)
2iv+k v
�0j
��:
19
With (65) and (57) and following the lines of the proof of the previous lemmas de result reported in (59) iseasily obtained.
20
Table 1: Results for case (16) with !j = �=3.
Johansen Vector of SeasonsT q0 = 12 q0 = 11 q0 = 10 q0 = 9 q0 = 8 q0 = 7 q0 = 6 q0 = 5 q0 = 4 q0 = 3 q0 = 2 q0 = 1200 1,0000 1,0000 1,0000 1,0000 1,0000 1,0000 1,0000 1,0000 1,0000 1,0000 0.0592 0.0050
Johansen Demodulatedei
�3 t e�i
�3 t
T q0 = 2 q0 = 1 q0 = 2 q0 = 1200 0.9942 0.0398 0.9942 0.0398
21
Table 2: Results for case (16) with !j = 2�=3.
Johansen Vector of SeasonsT q0 = 12 q0 = 11 q0 = 10 q0 = 9 q0 = 8 q0 = 7 q0 = 6 q0 = 5 q0 = 4 q0 = 3 q0 = 2 q0 = 1200 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0586 0.0050100 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9970 0.7448 0.0528 0.0052
Johansen Demodulatedei2
�3 t e�i2
�3 t
T q0 = 2 q0 = 1 q0 = 2 q0 = 1200 0.9950 0.0462 0.9950 0.0462100 1.0000 0.0354 1.0000 0.0354
22
Table 3: Results for case (??) with !k = 2�=3 and !j = �=3
Johansen Vector of SeasonsT q0 = 12 q0 = 11 q0 = 10 q0 = 9 q0 = 8 q0 = 7 q0 = 6 q0 = 5 q0 = 4 q0 = 3 q0 = 2 q0 = 1200 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0606 0.0028100 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9970 0.7448 0.0528 0.0052
T Johansen Demodulatedei
�3 t and ei2
�3 t e�i
�3 tand e�i2
�3 t
q0 = 2 q0 = 1 q0 = 2 q0 = 1200 1.0000 0.0324 1.0000 0.0324100 1.0000 0.0354 1.0000 0.0354
23
Table 4: Results for cases (25) and (26)
Johansen Vector of Seasonsq0 = 12 q0 = 11 q0 = 10 q0 = 9 q0 = 8 q0 = 7 q0 = 6 q0 = 5 q0 = 4 q0 = 3 q0 = 2 q0 = 1
(25) 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0666 0.0036 0.0000(26) 1.0000 1.0000 1.0000 1.0000 1.0000 0.9988 0.9478 0.6140 0.0834 0.0078 0.0008 0.0000
Demodulated ei2�3 t Demodulated e�i2
�3 t
q0 = 2 q0 = 1 q0 = 2 q0 = 1(25) 0.0656 0.0048 0.0656 0.0048Demod. ei2
�3 t and ei
�3 t Demod. e�i2
�3 t and e�i
�3 t
q0 = 2 q0 = 1 q0 = 2 q0 = 1(26) 0.028 0.0032 0.028 0.0032
24
25