We provide a new method for jointly consistently estimating common trends and cycles in unit root nonstationary multivariate systems. We concentrate on the MA representation of the differenced data and we jointly impose the reduced rank restriction for the common cycles and the common trends on the MA representation coefficients.Common cycles and trends, Tests of rank, Cointegration

George Kapetanios

English

Research Papers in Economics

Department of Economics
A Note on Joint Estimation of Common Cycles and Common
Working Paper No. 483           January 2003           ISSN 1473-0278
George Kapetanios
Trends in Nonstationary Multivariate Systems ￿￿A Note on Joint Estimation of Common
Cycles and Common Trends in Nonstationary
Multivariate Systems
George Kapetanios∗
Queen Mary, University of London
January 7, 2003
Abstract
We provide a new method for jointly consistently estimating com-
mon trends and cycles in unit root nonstationary multivariate systems.
We concentrate on the MA representation of the diﬀerenced data and
we jointly impose the reduced rank restriction for the common cycles
and the common trends on the MA representation coeﬃcients.
Keywords: Common cycles and trends, Tests of rank, Cointegration JEL
Codes: C32, C14
∗Department of Economics,Queen Mary,University of London,Mile End Rd.,London
E1 4NS. Email: G.Kapetanios@qmul.ac.uk
11 Introduction
In the past two decades attention in econometrics and statistics has focused
on the long run properties of nonstationary time series as represented by
the concept of cointegration. A major implication of cointegration is that
the number of underlying random walk components of a multivariate non-
stationary system is smaller than the dimension of the system. Economic
theory has been routinely used to motivate the choice of cointegrating vec-
tors and therefore implicitly the speciﬁcation of the trend components.
At the same time the speciﬁcation of the short run dynamics of the mul-
tivariate system has attracted less attention because economic theory can
provide fewer restrictions for the short run than for the long run. However,
these dynamics underlie the cyclical behaviour of the system and are therefore
of particular relevance for business cycle analysis. Following papers suchas
Vahid and Engle (1993) and Engle and Issler (1995) we consider the cyclical
components of a multivariate series to be serially correlated stationary pro-
cesses. It is then highly likely that the number of such components is smaller
than the dimension of the system in analogy to the trend components.
Vahid and Engle (1993) and Engle and Issler (1995) have provided a
method for estimating common cycles conditional on the long run parameter
estimates of the model. No method for the joint consistent estimation prob-
lem of common trends and cycles is currently available. We provide sucha
method by concentrating on the moving average (MA) representation of the
diﬀerenced data and jointly imposing the reduced rank restriction for the
common cycles and common trends on the MA representation coeﬃcients.
2 Theory
Let the multivariate I(1) series yt,o fd i m e n s i o nm, accept the VAR repre-
sentation given by
yt = a0 + A1yt−1 + A2yt−2 + ...+ Apyt−p +  t,t =1 ,...,T (1)
2If linear combinations of the I(1) variables are I(0), i.e. if there is cointegra-
tion in the system, then the system has a vector error correction representa-
tion of the form1.
∆yt = a0 + Πyt−1 +
p−1 
i=1
Bi∆yt−i +  t (2)
where Π = A1 + ...+ Ap − I and Bi = −Ai+1 − ...− Ap, i =1 ,...,p− 1.
Π will be of reduced rank, r. This implies that it can be written as αβ
 
where α and β are m × r full rank matrices. The columns of β will contain
the linear combinations that render the I(1) variables stationary. The above
system accepts an MA representation in diﬀerences. This is given by
∆yt = µ + C(L) t (3)
where C(L)=I + C1L + C2L2 + ...By the multivariate Beveridge Nelson
decomposition, this can be written as
∆yt = µ +[ C(1) + (1 − L)C
∗(L)] t (4)
where C
∗(L)=I + C
∗
1L + C
∗
2L2 + ... and C
∗
i =

j>i−Cj. In levels the
MA representation becomes
yt = µt + C(1)
∞ 
i=0
 t−i + C
∗(L) t + y0 (5)
where for simplicity we set y0 = 0. This is a decomposition of the series
in trends and cycles as discussed by a number of authors ( see e.g. Engle
and Issler (1995)). Under the assumption of cointegration we have that C(1)
has reduced rank, equal to m − r, giving the common trends representation
by Stock and Watson (1988). Engle and Issler (1995) extend this model by
suggesting that an equivalent restriction in terms of the number of cycles be
imposed. This is expressed in terms of restrictions on the rank of the C
∗(L)
matrix polynomial. They suggest imposing the restriction C
∗(L)=ψΨ(L) 
where ψ and Ψ(L)a r ea nm × s matrix and polynomial matrix of full rank
respectively where s<m .
1We assume that no series are integrated of order 2 or higher.
33 Joint common cycle - common trend esti-
mation
Engle and Issler (1995) do not suggest straightforward ways of estimating
either the reduced rank polynomial matrix, C
∗(L), or the common cycles
it implies apart from the special case where the number of trends and the
number of cycles add up to the dimension of the system. The problem is
clear and can be appreciated by examining the VAR in levels given in (1)
under a plausible identiﬁcation structure for the common trends common
cycle model. If we restrict all coeﬃcient matrices in the VAR representation
to be of the form Ai = A1A
 
2i, i =1 ,...,p,w he r e A1 and A2i are m×s full
rank matrices, as in Velu, Reinsel, and Wichern (1986) we obtain a short-
run reduced rank representation for the series. 2 However, we also need to
impose r2 unit root restrictions on the matrix polynomial A(L) needed for
the common trends representation. Estimation of the model under those joint
restrictions clearly requires iterative numerical techniques. Additionally, the
most appropriate way to impose suchrestrictions is not clear.
3.1 Reduced rank regression
Before presenting our method we brieﬂy outline some results on multivariate
reduced rank regressions, ( see e.g. Brillinger (1981), Velu, Reinsel, and
Wichern (1986), Reinsel and Velu (1998) or Camba-Mendez, Kapetanios,
Smith, and Weale (2000)). The case of a single set of reduced rank regressors
is well known and widely discussed. We therefore discuss the extension that
is useful for our purposes and involves two sets of reduced rank regressors.
Consider a multivariate regression model of the form
Y = XAx + ZAz +   (6)
2Note that the assumption of a uniform rank for all A2i is not necessary but is assumed
for simplicity. In the case where the ranks of A2i are not equal for all i, s is equal to the
maximum rank over i.
4where Y , X and Z are T × k1, T × k2 and T × k3 matrices respectively.
The matrices Ax and Az are of reduced ranks rx and rz respectively. As a
result they can be written as Ax = βxα 
x and Az = βzα 
z where αx βx are
full rank k2 × rx and k1 × rx matrices and αz βz are full rank k3 × rz and
k1 × rz matrices. We want to estimate αx, βx, αz and βz by minimising
tr[(Y − Xβxα 
x + Zβzα 
z)(Y − Xβxα 
x + Zβzα 
z)] A number of methods
exists for this estimation. One method that is both non-iterative and simul-
taneous is given by Reinsel and Velu (1998). The method assumes initial
estimates of βx and βz which can be obtained as follows: Estimate (6) by
least squares. Conditional on the estimate of Ax run a single regressor re-
duced rank regression to estimate βz and vice versa to estimate βx.T he s e
estimates are denoted by ˜ βx and ˜ βz. Conditional on these estimates, least
squares estimation gives estimates of αx and αz denoted, by ˆ αx and ˆ αz Then,
updated estimates of βx and βz are obtained via standard minimisation of
the objective function and given by
ˆ β
 
x = ˆ α
 
x

1/T(Y
  − ˆ αz˜ β
 
zZ
 )X

[1/TX
 X]
−1
vec(ˆ β
 
z)=[ ( 1 /TZ
 Z ⊗ I) − (R ⊗ Q)]vec(P)
where R =1 /TZ
 X(1/TX
 X)−11/TX
 Z, Q =(ˆ α
 
z ˆ αx)(ˆ α
 
xˆ αz)a n d
P = ˆ α
 
z(1/TY
 Z) − (ˆ α
 
z ˆ αx)ˆ α
 
x(1/TY
 X)(1/TX
 X)−1(1/TY
 Z). These
estimates are
√
T-consistent and asymptotically normally distributed.
3.2 Estimation of Common Trends and Cycles
In what follows we suggest a consistent method of estimating jointly a com-
mon cycle common trend model. We start by estimating the unrestricted
VAR in levels in (1). This provides an estimate for the error sequence ˆ  t.
Under the presence of cointegration,  t −ˆ  t is Op(T −1/2) for all t.I fn oc o i n -
tegration occurs we obtain as a corollary of superconsistency that  t − ˆ  t is
Op(T −1). Due care should be taken to ensure that the normalisations needed
5to achieve parameter identiﬁcation are consistent with each other for the
successive reduced rank estimators.
Given the above we demean the diﬀerenced data and run the following
regression withtwo sets of reduced rank regressors
∆yt = C(1)ˆ  t + C
∗
0∆ˆ  t + C
∗
1∆ˆ  t−1 + ...+ C
∗
q∆ˆ  t−q + ˆ ηt (7)
where q is chosen to be a function of the sample size as discussed below. C(1)
has rank m−r and each C
∗
i can be written as GH
 
i where both constituent
matrices have full rank s<m . In anticipation of what follows deﬁne: ∆y =
(∆y1,...,∆yT) , ˆ  q,t =( ˆ  
 
t,∆ˆ  
 
t−1,...,∆ˆ  
 
t−q) , ˆ  q =( ˆ  q,1,...,ˆ  q,T) ,  q,t =
(  
t,∆  
t−1,...,∆  
t−q) ,a n d q =(  q,1,..., q,T) .
We now provide an asymptotic justiﬁcation of our method in terms of pa-
rameter estimate consistency for the parameter set ˆ Cq =(ˆ C(1), ˆ C
∗
1,... ˆ C
∗
q) .
We split the argument in two parts. In the ﬁrst part we assume that the
error sequence  t is known. We get consistency of ˆ C under this assumption
and then generalise to the actual setup. q i sa s s u m e dt ot e n dt oi n ﬁ n i t ya s
T tends to inﬁnity, but at a rate less that T 1/3 (see Berk (1974)). No more
stringent conditions are needed for consistency. If one wants to investigate
parameter estimate rates of convergence and asymptotic normality then more
restrictive conditions are needed (it suﬃces for that to have q rise by at least
T 1/r for some r>3). We concentrate on consistency of OLS estimates of
the parameters to simplify the analysis. As both OLS estimates and reduced
rank estimates are continuous functions of the moment matrices of the re-
gressors and regressands, consistency of one estimator implies consistency of
the other and vice versa3.
Assuming a known error sequence4 ,a n dµ = 0 for simplicity, we have
that
ˆ C
OLS
q −Cq =(  
 
q q)
−1 
 
q∆y−Cq =(  
 
q q)
−1 
 
q qCq−Cq+( 
 
q q)
−1 
 
q −qC−q =(  
 
q q)
−1 
 
q −qC−q
3In the reduced rank case,consistency,of course,only holds for assumed ranks equal
to or larger to true ones.
4This implies ηt = 0 in (7).
6where  −q,t =( ∆   
t−q−1,...) ,  −q =(  −q,t,...,)  and C−q =( C
∗
q+1,...) .B y
the fact that the data generation process is a VAR in levels, ||C
∗
q+1|| ∼ cq
where |c| < 1a n d||.|| is the supremum matrix norm. This and the assump-
tion that q tends to inﬁnity at rate of less than T 1/3 leads to consistency of
the OLS estimator for the case where the error sequence is known. More
speciﬁcally we have that ||(  
q q)−1  
q −qC−q|| conveges to zero in probability.
To see this note the following: By Lemma 3 of Berk (1974) we have that
the norm of the diﬀerence between (  
q q)−1 and its population couterpart
converges to zero if q3/T converges to zero. Further, the norm of   
q −qC−q
converges in probability to zero if q(||C
∗
q+1|| + ||C
∗
q+2|| + ...) converges to
zero which is the case if q tends to inﬁnity and ||C
∗
q+1|| ∼ cq+1, |c| < 1, for
large q, which we assume. Combining the above two results with the fact
that the norm of a product is dominated by the product of the norms gives
the result.
The second step is to show that ˆ C
OLS
q − ˆ ˆ C
OLS
q = op(1) where ˆ ˆ C
OLS
q is
obtained by using ˆ  t rather than  t.W eha v et ha t
ˆ C
OLS
q − ˆ ˆ C
OLS
q =(  
 
q q)
−1 
 
q∆y − (ˆ  
 
qˆ  q)
−1ˆ  
 
q∆y
It is suﬃcient to show that (i) 1/T(  
q∆y−ˆ  
 
q∆y)=op(1) and (ii) 1/T((  
q q)−1−
(ˆ  
 
qˆ  q)−1)=op(1). We prove (i).
||1/T( 
 
q∆y−ˆ  
 
q∆y)|| = ||1/T
T 
t=1
( q,t∆y
 
t−ˆ  q,t∆y
 
t)|| ≤ q∗max
t || t−ˆ  t||||1/T
T 
t=1
∆y
 
t||
1/T
T
t=1 ∆y 
t converges to its non zero expectation, eachof  t − ˆ  t is at
most Op(T −1/2) and therefore the whole term is op(1). (ii) follows from
similar arguments as above and the facts that, ﬁrstly for positive deﬁnite
matrices A and B,i fA − B = op(1) then A−1 − B−1 = op(1), and secondly,
if A − ˆ A = op(1) and B − ˆ B = op(1) then AB − ˆ A ˆ B = op(1).
The above analysis has concentrated on providing consistent joint esti-
mates of common cycles and common trends and has not considered neither
7the asymptotic distribution of the estimates nor the question of their asymp-
totic eﬃciency. On the issue of the asymptotic distribution we note that if
standard errors of the parameter estimates are required then they may be
derived using the distribution of coeﬃcients of reduced rank regression mod-
els withtwo sets of reduced rank regressors given in Th eorem 3.2 of Reinsel
and Velu (1998, pp. 81). Note that all regressors involved here are station-
ary and ergodic and therefore no asymptotic distributional issue arises out
of the presence of nonstationarity in yt. Of course, in our case the number
of regressors increases withth e sample size but results in Berk (1974) can be
used to obtain conditions on the rate of increase of the number of regressors
q to give consistent estimation of the moment matrices needed to justify use
of Theorem 3.2 of Reinsel and Velu (1998)5.
If the model is a ﬁnite order VAR model then the suggested method will
not provide asymptotically eﬃcient estimates since the true form of the model
is not taken under account. However, this drawback is compensated by the
fact that the method we suggest is robust to the case where the true model
is a VAR model of inﬁnite order or indeed any model where the diﬀerenced
data accept an inﬁnite MA representation6. To see this note that the MA
representation of the diﬀerenced data still holds under an inﬁnite order VAR
model. This implies that the Beveridge Nelson decomposition, in (4), which
forms the basis of the analysis, holds. If we use data dependent methods
such as information criteria to ﬁt the initial vector autoregression then, if
that vector autoregression is of inﬁnite order, the lag order chosen will be
growing withth e sample size at rate ln( T) (see Ng and Perron (1995) and for
a complete treatment see Hannan and Deistler (1988)). Then, the estimated
error sequence ˆ  t will be consistent for the true disturbance terms. In this
context cointegration is deﬁned as the occurrence of reduced rank in the C(1)
5Essentially,the conditions needed for consistency of parameter estimates need to be
tightened so that q = O(T1/r) ,for some r>3 to get asymptotic normality. (see also Ng
and Perron (1995))
6This of course occurs,under regularity conditions,whenever the diﬀerenced data are
stationary using the Wold decomposition.
8matrix following Bierens (1997). Then, the same reduced rank analysis of
the MA representation, as above, applies.
3.3 Numbers of trends and cycles
The above analysis assumed that the ranks m−r and s are known. Clearly,
this is not the case in practical applications. We suggest two methods of
determining these ranks. The ﬁrst is the standard method for determining
ranks in reduced rank regression models using two sets of reduced rank re-
gressors. Partial canonical correlation analysis is used for this as suggested
by Reinsel and Velu (1998). However, this method essentially determines one
rank conditional on the other. Alternatively, for joint determination of the
ranks, information criteria may be used. To do that we estimate the model
for the grid of all possible ranks where both the rank of C(1) and that of
C
∗(L) are allowed to vary independently between 1 and m.Ap e n a l t yt e r m
depending on the information criterion is then added to the sum of squared
residuals to provide the objective function to be minimised. Any of the usual
information criteria may be used (e.g. Akaike’s, Bayesian information crite-
rion, Hannan-Quinn information criterion) taking into account the fact that
Akaike’s criterion may choose larger ranks asymptotically since it is not a
consistent criterion.
4 Conclusion
In this note we have suggested a new method for joint determination of
common trends and common cycles in cointegrated multivariate systems. No
other joint estimation method exists in the literature. The method uses the
inﬁnite MA representation of the diﬀerenced series and applies reduced rank
regression methods to estimate the trend and cycle parameters. A method to
determine jointly the number of trends and cycles has also been suggested.
It is worth noting that the method, based on the inﬁnite MA representation
9is applicable in more general settings and in particular it is applicable for
VAR models withan inﬁnite number of lags.
References
Berk, K. N. (1974): “Consistent Autoregressive Spectral Estimates,” An-
nals of Statistics, 2(3), 489–502.
Bierens, H. (1997): “Nonparametric Cointegration Analysis,” Journal of
Econometrics, 77, 379–404.
Brillinger, D. R. (1981): Time Series: Data Analysis and Theory.H o l d e n
Day.
Camba-Mendez, G., G. Kapetanios, R. J. Smith, and M. R. Weale
(2000): “Tests of Rank in Reduced Rank Regression Models,” Forthcoming
in Journal of Business and Economic Statistics.
Engle, R. F., and J. V. Issler (1995): “Estimating Common Sectoral
Cyclea,” Journal of Monetary Economics, 35, 83–113.
Hannan, E. J., and M. Deistler (1988): The Statistical Theory of Linear
Systems. John Wiley.
Ng, S., and P. Perron (1995): “Unti Root Tests in ARMA Models with
Data-Dependent Methods for the Selection of the Truncation Lag,” Jour-
nal of the American Statistical Association, 90, 268–281.
Reinsel, G. C., and R. P. Velu (1998): Multivariate Reduced Rank
Regression. Springer-Verlag.
Stock, J. H., and M. Watson (1988): “Testing for Common Trends,”
Journal of the American Statistical Association, 83, 1097–1107.
Vahid, F., and R. F. Engle (1993): “Common Trends and Common
Cycles,” Journal of Applied Econometrics, 8, 341–360.
10Velu, R. P., G. C. Reinsel, and D. W. Wichern (1986): “Reduced
Rank Models for Multiple Time Series,” Biometrika, 73, 105–118.
11This working paper has been produced by
the Department of Economics at
Queen Mary, University of London
Copyright © 2003 George Kapetanios
All rights reserved. 
Department of Economics 
Queen Mary, University of London
Mile End Road
London E1 4NS
Tel: +44 (0)20 7882 5096 or Fax: +44 (0)20 8983 3580
Email: j.conner@qmul.ac.uk
Website: www.econ.qmul.ac.uk/papers/wp.htm

Consistent Autoregressive Spectral Estimates,”

Multivariate Reduced Rank Regression.

Nonparametric Cointegration Analysis,”

Reduced Rank Models for Multiple Time Series,”

Testing for Common Trends,”

The Statistical Theory of Linear Systems.

Time Series: Data Analysis and Theory.H o l d e n Day.

Unti Root Tests in ARMA Models with Data-Dependent Methods for the Selection of the Truncation Lag,”

Weale (2000): “Tests of Rank in Reduced Rank Regression Models,” Forthcoming in

A Note on Joint Estimation of Common Cycles and Common Trends in Nonstationary Multivariate Systems

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A Note on Joint Estimation of Common Cycles and Common Trends in Nonstationary Multivariate Systems

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