A reduced rank regression approach to coincident and leading indexes building.
Date: 2004-12-31
Institutions: Eurostat
Subjects: United States, econometrics, economic cycle

N
O

I
T
I

D
E

4
0
0
2

A reduced rank 
regression approach 
to coincident and leading
indexes building

E U R O P E A N
C O M M I S S I O N

 
Europe Direct is a service to help you find answers to your questions about the European Union

New freephone number:
00 800 6 7 8 9 10 11

A great deal of additional information on the European Union is available on the Internet.
It can be accessed through the Europa server (http://europa.eu.int).

Luxembourg: Office for Official Publications of the European Communities, 2004

ISBN 92-894-7532-3
ISSN 1725-4825

© European Communities, 2004

4TH EUROSTAT AND DG ECFIN  
COLLOQUIUM ON MODERN TOOLS FOR BUSINESS CYCLE ANALYSIS 

"GROWTH AND CYCLE IN THE EURO-ZONE" 

20 TO 22 OCTOBER 2003 

Luxembourg, European Parliament 
Hémicycle, Schuman building 

A reduced rank regression approach to coincident  
and leading indexes building 

Gianluca Cubadda 

Università del Molise, Campobasso, Italy 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
A Reduced Rank Regression Approach to

Coincident and Leading Indexes Building∗

Gianluca Cubadda†

January 13, 2004

Abstract

This paper proposes a reduced rank regression framework for constructing coincident

and leading indexes. Based on a formal deﬁnition that requires that the ﬁrst diﬀerences of

the leading index are the best linear predictor of the ﬁrst diﬀerences of the coincident index,

it is shown that the notion of polynomial serial correlation common features can be used

to build these composite variables. Concepts and methods are illustrated by an empirical

investigation of the US business cycle indicators.

JEL classiﬁcation: C32

Keywords: Coincident and Leading Indexes, Polynomial Serial Correlation Common

Feature, Reduced Rank Regression.

∗Thanks are due to Alain Hecq, as well as the participants in the 4th Eurostat Colloquium on
Modern Tools for Business Cycle Analysis in Luxembourg, the seminar at Ente Einaudi in Rome, and
the Conference Common Features in Maastricht for their helpful comments and suggestions. Financial
support from MIUR is gratefully acknowledged. The usual disclaimers apply.

†Università del Molise, Dipartimento SEGeS, Via De Sanctis, 86100 Campobasso, Italy, e-mail:

gianluca.cubadda@uniroma1.it.

1

1 Introduction

During the last few decades systems of coincident and leading indexes have been built and used

in many countries. Whether the goal is to predict the turning points in the business cycle or the

movements of the overall economic activity, the common feature of the diﬀerent methodologies

consists generally of two steps. The ﬁrst step aims at identifying groups of variables that move

in, before or after the recession (see e.g. Niemera and Klein, 1994). In this paper the focus is

on the ﬁrst two groups of variables, which are respectively deﬁned as the coincident and leading

indicators. The second step consists in forming composite indicators, namely the Coincident

Index [CI] and Leading Index [LI], in order to extract the relevant business cycle features from

the individual indicators.

Among the various statistical methods for constructing such CI and LI, the procedure devel-

oped by Stock and Watson (1989, 1991, and 1993) for the NBER has rapidly become a standard

reference. But other approaches exist since a while, from the well known principal component

and classical linear time series analyses to more complex non-linear methods such as smooth

transition regressions, switching regimes and probit models, and nonparametric procedures (see

Camacho and Perez-Quiros, 2002, for a comparison of the forecasting performances of some of

these procedures).

In a similar spirit as Emerson and Hendry (1996), the viewpoint in this article is that the

construction of coincident and leading indexes should be based on a formal statistical analysis

of the multivariate time series properties of the data. Hence, a Reduced Rank Regression [RRR]

approach is proposed to build a CI&LI from a vector of cointegrated economic indicators. RRR

has been extensively analyzed in the statistical and macroeconometric literature (see inter alia

Anderson, 1984; Velu et al., 1986; Ahn and Reinsel, 1988; Tiao and Tsay 1989; Johansen,

1995) but, to the best of my knowledge, it has not yet been applied for the problem at hand.

This seems a promising route to follow since there is convincing evidence (see inter alia Reinsel

and Ahn,1992; Camba-Mendez et al., 2003) that imposing reduced-rank structure in Vector

Auto-Regressive [VAR] models improve in prediction performances.

In particular, the dynamic properties of the data are investigated within the polynomial

serial correlation common feature modeling (Cubadda and Hecq, 2001). Similarly as the com-

posite indexes built by The Conference Board, the proposed CI&LI’s are obtained as linear

combinations of observed variables. However, the weights of the novel indexes are derived such

2

that the changes of the LI are the best linear predictor of the changes of the CI. Hence, the

suggested CI&LI’s are constructed in order to satisfy the purpose of documenting and predict-
ing the variations of the overall economic activity.1 Other relevant characteristics of the new

composite indicators are that the existence of such CI&LI is tested and not assumed a priori,

it is possible to check if the individual indicators signiﬁcantly enter in the CI and LI, and the

multivariate Beveridge-Nelson (1981) cycle of the LI leads that of the CI.

This paper is organized as follows. Section 2 proposes a deﬁnition of the CI&LI, and shows

how to build such indexes by means of RRR. In Section 3 the conditions for the existence of a

long leading index are examined. In Section 4 the methodology is applied to the US business

cycles indicators. Section 5 concludes.

2 The statistical methodology

The aim of this section is to present a RRR framework to build the CI&LI from a set of

cointegrated time series.

2.1 Preliminaries

Let us start with the VAR(p) model for a n-vector of I(1) time series

yt, t = 1, . . . , T
{

,

}

A(L)yt = εt,

(1)

for ﬁxed values of y

p+1, ..., y0 and where A(L)

−

In

≡

−

p
i=1 AiLi, and εt are i.i.d. Nn(0, Σε)

errors. To simplify the notation, the deterministic terms are omitted at this stage.

P

It is further assumed that the process yt is cointegrated of order (1,1), namely that 1◦)
αβ 0 with α and β both

rank(A(1)) = r, 0 < r < n, so that A(1) can be expressed as A(1) =

(n

r) matrices of full column rank r, and 2◦) the matrix α0
⊥

×

A∗(1)β

⊥

where A∗(1) denotes the ﬁrst derivative of A(z) at z = 1. The columns of β span the space of
cointegrating vectors, and the elements of α are the corresponding adjustment coeﬃcients. In

−
has rank equal to (n

r)

−

order to rewrite the system in a VECM form we use the identity A(L)

A(1)L + Γ(L)∆ where

≡

1 Although the proposed CI&LI’s are not speciﬁcally designed for predicting the business cycle turning points,
they may also be used for such purpose along the lines of Wecker (1979), and Hamilton and Perez-Quiros (1996).

3

Γ(L) = In

−

1

p
i=1 ΓiLi, and Γi =
−

−

p
j=i+1 Aj for i = 1, . . . , p

−

1. And ﬁnally we obtain

P

P

Γ(L)∆yt = αβ0yt

1 + εt.

−

The stationary process ∆yt admits the following Wold representation

∆yt = C(L)εt,

(2)

(3)

with

∞j=1 j

<

Cj
|

|

∞

, and C0 = In.

∞i+1 Cj for
Based on the polynomial factorization C(L) = C(1) + ∆C∗(L), where C∗i =
P
0, we obtain the multivariate Beveridge and Nelson (1981, BN henceforth) representation

−

P

i

≥

of the series yt

yt = τ t + ξt,

(4)

where ξt = C∗(L)εt, and ∆τ t = C(1)εt.

The multivariate BN decomposition has a natural interpretation in forecasting terms. In-

deed, we easily get from equations (3) and (4) that

E(yt+h

lim
h
→∞

Ωt) = τ t,
|

(5)

where Ωt is the σ-ﬁeld generated by

0
}
a non-stationary time series coincides with its inﬁnite-step ahead prediction (see e.g. Harvey,

. Based on the popular view that the trend of

i; i

yt

≥

{

−

1990), the processes τ t and ξt are respectively deﬁned as the stochastic trends and cycles
of variables yt. Proietti (1997) and Hecq et al.
(2000) provided explicit expressions of the
components τ t and ξt in terms of the VECM parameters.

In order to analyze non-contemporaneous short-run comovements, Cubadda and Hecq (2001)

have introduced the notion of Polynomial Serial Correlation Common Features [PSCCF] such

that

Deﬁnition 1 Polynomial Serial Correlation Common Features of order m: series

∆yt have s PSCCF of order m, henceforth PSCCF(m), iﬀ there exists a n

s polynomial

1) such that the matrix δ0 is full column rank,

×

matrix δ(L) = δ0
δm

−

= 0, and δ(L)0∆yt = δ00εt.

P

m
i=1 δiLi with m < (p

−

Notice that the notion of serial correlation common feature (Engle and Kozicki, 1993) is

4

6
obtained as a special case of the PSCCF(m) with m = 0.

The presence of the PSCCF(m) endows series yt with several interesting properties. First,

the following restrictions on the VECM (2) parameters hold

Condition 1.

δ00α = 0

Condition 2.

δ00Γi =

δ0i
0




if i

m

≤
if i > m


Second, variables yt must share at least one common trend since Condition 1 implies that
the matrix α has rank less then n. Third, the multivariate BN cycles ξt respect the following
condition

E(δ(L)0ξt+h|

Ωt) = 0, h

m,

≥

(6)

which is equivalent to say that the process δ(L)0ξt+h is a VMA(m

1).

−

2.2 The Coincident and Leading Indexes

Let us assume that the vector of n time series may be partitioned into two subvectors such

that yt = (z0t, x0t)0. The ﬁrst n1 series zt are the relevant business cycle indicators whereas the
n1 series xt must Granger-cause the reference series zt. Hence, the following
remaining n2 = n

notion of coincident and leading indexes is proposed.

−

Deﬁnition 2 CI&LI. CIt and LIt are respectively the composite coincident and leading in-

dexes iﬀ

E(∆CIt+1

Ωt) = E(∆CIt+1
|

∆LIt),
|

(7)

where CIt is a linear combinations of the reference series zt, and LIt is a linear combinations
of series (y0t, .., y0t
m+1)0.
−

The above deﬁnition can be motivated as follows.

In view of the BN decomposition in

(4), if the reference series zt exhibit business cycle features, their ﬁrst diﬀerences ∆zt must

be autocorrelated. The suggested CI&LI are constructed such that the CI exhibits a cyclical

behavior but ∆CIt+1

E(∆CIt+1

∆LIt) is an innovation with respect to Ωt. Hence, the BN

cycle of CIt+1 is cancelled after removing the inﬂuence of LIt.

−

|

5

Suppose now that series ∆yt exhibit at least one PSCCF(m) such that δ00 = (ω00, 00n2). In

view of Deﬁnition 1, we have that

E(ω00∆zt+1

|

Ωt) = δ(L)0∆yt,

where δ(L) =

m
i δiLi

1. Consequently the coincident and leading indexes are simply given

−

P

by

and

CIt = ω00zt,

LIt = δ(L)0yt.

It is easy to see that the reverse implication holds as well, i.e. if there exists a pair of CI&LI

according to Deﬁnition 2 then series ∆yt have at least one PSCCF(m) with δ00 = (ω00, 00n2).
These results, along with equation (6), imply that the CI&LI have the following important

property:

Proposition 3 Let us deﬁne the detrended CI&LI respectively as CI ξ

(ω00, 00n)ξt and LI ξ

t ≡

t ≡

δ(L)0ξt. Then we have

E(CI ξ

Ωt) = E(LI ξ

t+h|

Ωt),

t+h

1|
−

m.

h

≥

The above proposition tells us that the cyclical movements of the LI lead those of the CI

when the forecast horizon is not less than the PSCCF order. Hence, the case of the PSCCF(1)

with δ00 = (ω00, 00n2) is particularly attractive for CI&LI building. In the rest of the paper the
focus will be on such particular case.

We can make inference on the existence of such CI&LI’s by means of the following RRR

procedure. We ﬁrst solve the following canonical correlation program

CanCor










∆zt

∆yt

−

1 
−


,













6

ˆβ0yt
∆yt

1

−

1

−

2

−

∆yt
...

∆yt

p+1

−













Dt

|

,






(8)

then the LR test statistic for the hypothesis that are at most s CI&LI couples is

LR1 =

T

−

s

i=1
X

ln(1

λi),

−

s = 1, . . . , min(n1, n

r)

−

(9)

b

where

λi is the i

th smallest squared canonical correlation coming from (8) and the esti-

mates of the parameters (ω00, δ01)0 are the eigenvectors associated with the s smallest eigenvalues
b
λs.2 Under the null hypothesis the test statistic (9) is asymptotically distributed as a
λ1, ...,
χ2
(d1) with d1 = s
b

3) + r + s + n2).3

A relevant feature of the RRR approach is that it is possible to test for linear restrictions

(n(p

×

−

b

−

on the CI&LI weights. Alike Johansen (1995) in cointegration analysis, these restrictions are

expressed as follows







,

(10)



ω0
δ01 


= H

ϕ



≡

H11
g1
n1
×
H21
|{z}
g1
n

H12
g2
n1
×
H22
|{z}
g2
n





where H is matrix of known elements, the sub-matrix H11 has rank equal to g1, g = g1 + g2,

s
×
|{z}

×
|{z}

×
|{z}

×
|{z}








g

(n1+n)

g







and ϕ is a parameter matrix to be estimated.

Let us a consider the illustrative example where the reference series zt are The Conference

Board [TCB] coincident indicators, namely the industrial production, employment, real income,

and manufacturing and trade sales, and we wish to test if the reference series do enter in the

LI. Then the matrix H takes the form

ϕ0
s
g1
×
ϕ1
|{z}
s
g2
×
|{z}







I4

04

(n

4)

×

−

H = 






0n

4

×



04

(n
×
In

4)
−
4 
−









,

(11)


which means that there are no cross restrictions between ω0 and δ1, ω0 is unrestricted, and
δ01 = (0s

4, ϕ01, ).
×

2 Since such eigenvalues and eigenvectors are invariant to non-singular linear transformation of variables
p+1)0, inference on the CI&LI’s does not depend on the identiﬁcation of the cointegration

ˆβ, ∆y0t

(y0t
1
vectors β.

−

1, ..., ∆y0t
−
−

3 Based on Cubadda and Hecq (2001), a test statistic with better small-sample properties can be obtained by

applying the scaling factor (T

n(p

r)/T to (9).

−

2)

−

−

7

We can handle such linear restrictions by means of the following procedure. We ﬁrst solve

the following canonical correlation program

CanCor

Then the LR test statistic is





H 0






∆zt

∆yt

−

1 
−


ˆβ0yt
−
∆yt

1

1

2

−

−

∆yt
...

∆yt

p+1

−

,













(12)













Dt

|

.






s

LR2 = T

ln

i=1
X

ˆλi
ηi !

,

1
1

−
−

Ã

s = 1, . . . , min(g1, n

r),

−

(13)

where

ηi is the i

−

th smallest squared canonical correlation coming from (12) and the esti-

b

mates of the parameters (ϕ00, ϕ01)0 are the eigenvectors associated with the s smallest eigenval-
ues

ηs. Under the null hypothesis the test statistic (13) follows asymptotically a χ2

b
η1, ...,

(d2)

distribution where d2 = s(n1 + n

g).

−

b
Notice that when s > 1 there is not necessarily a unique CI&LI pair.4

b

In the sequel we

consider both the case where several indexes are individually identiﬁed and the most usual case

where a unique CI&LI pair must be constructed.

2.3 Identifying the CI&LI’s

This subsection shows how to identify "structural" pairs of CI&LI’s by means of overidentifying

restrictions. Coming back the previous illustrative example, we may wish to construct a leading

index that does not include the four Conference Board coincident series. In this case, we need

to test for zero canonical correlations between (∆z0t,

1)0 and the past of yt.
∆x0t
−

−

More generally, suppose that we are willing to consider only composite indexes with weights

which obey the linear restrictions (10). Then the LR test statistic for the null hypothesis that

there exist s "restricted" CI&LI’s against the alternative that no restricted CI&LI’s exist is

4 A practically relevant case for which s

monthly gross domestic product.

≤

1 is when zt is formed by a single reference series, such, e.g., the

8

given by

LR3 =

T

−

s

i=1
X

ln(1

ηi),

−

s = 1, . . . , min(g1, n

r).

−

(14)

Under the null hypothesis the test statistic (14) is asymptotically distributed as a χ2

b

(d3) with

d3 = s

(n(p

×

−

1) + r + s

g).

−

2.4 Building the Optimal CI&LI

This subsection shows how to combine several CI&LI’s in order to extract the most relevant

pair for forecasting purposes. More precisely, the following notion of optimal coincident and

leading composite indexes is proposed.

Deﬁnition 4 Optimal CI & LI. CI ∗t ≡
composite coincident and leading indexes iﬀ

ξ∗0CIt and LI ∗t ≡

ξ∗0LIt are respectively the optimal

ξ∗ = arg min

(ξ) ½

ξ0V (et)ξ
ξ0V (∆CIt)ξ

,

¾

(15)

where ξ is a generic s-vector, et

of the process in argument.

∆CIt

−

≡

E(∆CIt

∆LIt
|

−

1) and V (
) is the covariance matrix
·

When several PSCCF vectors exist (i.e., s > 1), condition (15) requires that ∆CI ∗t and
1 that

1 are the most correlated among all the linear combinations of ∆zt and ∆yt
∆LI ∗t
−
−
satisfy equation (7). Based on a standard result from canonical correlation theory, equation
1/2ζ 1, where ζ1 is the eigenvector associated to the smallest

(15) is solved by ξ∗ = [V (∆CIt)]−
eigenvalue of the matrix

[V (∆CIt)]−

1/2V (et)[V (∆CIt)]−

1/2.

(16)

Hence, the optimal CI weights are given by ω∗0 = ω0ξ∗ and the optimal LI weights are given by
δ∗1 = δ1ξ∗. We summarize the above results in the following proposition.

Proposition 5 Construction of the optimal CI&LI. Suppose that there exist s PSCCF(1)

vectors such that δ00 = (ω00, 00s
n2) and δ1
×
given by CI ∗t = ω∗00 zt and LI ∗t = δ∗01 yt, where ω∗0 = ω0ξ∗, δ∗1 = δ1ξ∗, ξ∗ = [V (∆CIt)]−
ζ1 is the eigenvector associated to the smallest eigenvalue of the matrix (16).

= 0. In this case, the optimal CI and LI are respectively
1/2ζ1, and

9

6
The optimal CI&LI weights can be estimated as follows. Compute the RRR estimates

b

ω00,

δ0
1). Then obtain

1) of the CI&LI’s weights and ﬁx (ω00, δ01) = (
δ0

ω00,
(
(15) where V (∆CIt) and V (et) are respectively substituted with the sample covariance matrices
b
1). Finally, the point estimates of ω∗0 and δ∗1 are respectively
1∆yt
δ0
of
−
δ1
δ∗
ξ∗.
1 =
b

given by
b
Linear restrictions on ω∗0 and δ∗1 may be tested by a linear switching algorithm similar as
the one proposed by Johansen (1995) in cointegration analysis. In particular, let us consider

ω00∆zt and (
ω∗0 =

ω00∆zt
ω0
b

ξ∗ by solving equation

−
ξ∗ and

b

b

b

b

b

b

b

b

b

the following system of hypothesis:

H0 : δ∗

≡

(ω∗00 , δ∗01 )0 = H ∗ϕ∗

≡

)0ϕ∗

≡

H ∗ϕ∗ vs H1 : δ∗ is unrestricted,

( H ∗00
n
g

, H∗01
n
g

×
|{z}

×
|{z}

where H ∗ is a matrix of known elements and ϕ∗ is a g

1 parameter matrix.

Let us then write δ# = (ω00, δ01)0ξ#, where ξ# = [V (∆CIt)]−

×

1/2ζ1, and ζ1 is the matrix of
1) largest eigenvalues of the matrix (16). Thus

the (s

1) eigenvectors associated to the (s

−

the iterative procedure goes as follows

−

1. Estimate δ# unrestricted by

δ

#

= (

ω00,

δ0
1)0

ξ

#

.

2. For ﬁxed δ# =

δ

#

, obtain

ϕ as the eigenvector associated with the smallest eigenvalue

b

b

b

b

coming from the solution of

b

b

ˆβ0yt
∆yt

1

−

1

−

2

−

∆yt
...

∆yt

p+1

−

∆zt





∆yt

−

1 
−


,













CanCor

H∗0






δ#
0

|

∆zt





∆yt

−

1 
−


, Dt













.

(17)






3. For ﬁxed δ∗ = H ∗

ϕ∗, obtain

δ

φ(s

1), where

φ(s

1) are the eigenvectors associated

#

= δ∗
⊥

−

−

b

b

b

b

10

with the (s

−

1) smallest eigenvalues coming from the solution of

CanCor

∆zt

∆yt

δ∗0
⊥ 

−



1 
−







ˆβ0yt
−
∆yt

1

1

2

−

−

∆yt
...

∆yt

p+1

−

,













δ∗0

|

∆zt





∆yt

−

1 
−


, Dt













.

(18)






4. Continue with 2. and 3. until numerical convergence.

The LR test statistic is

ln(1

ˆλi)

−

ln(1

ρ1)

−

−

−

ln(1

,

υi)

#

−

s = 1, . . . min(n1, n

r),

(19)

−

1

s
−

i=1
X

ρi and

where
(17) and (18). The test statistic (19) follows asymptotically a χ2

th smallest squared canonical correlations respectively coming from
(d4) distribution where d4 =

υi are the i

−

b

b

LR4 = T

s

"

i=1
X

b
(n1 + n

b

g).

−

3 The Long Leading Indicator

We have so far focused on building CI&LI’s when the time delay is one period only. However, it

is often desirable to anticipate the state of economic activity with a larger advance. Hence, the

properties of the composite indexes must be evaluated also when the forecast horizon is larger

than one. By construction of the CI&LI we get

∆CIt = ∆LIt

1 + et,

−

where et = (ω00, 00n2)εt.

Equation (20) implies in turn that

E(∆CIt+h

Ωt) = E(∆LIt+h
|

1

−

(20)

(21)

Ωt),

|

for h

≥

2. Hence, the h-step ahead forecasts of the ﬁrst diﬀerences of CI are given by the

11

(h

1)-step ahead forecasts of the ﬁrst diﬀerences of LI.

−
Notice that the left hand side of equation (21) is generally not a function of ∆LIt only. For

instance, for h = 2 we get

E(∆LIt+1

Ωt) = δ01(αβ0yt +
|

p

2

−

i=0
X

Γi∆yt

i),

−

which is generally diﬀerent from E(∆LIt+1

∆LIt).
|

Based on equation (21), the h-step ahead leading index LI h
t

is deﬁned as follows

∆LI h

t = δ00E(∆yt+h

Ωt).
|

(22)

In order to build such h-step ahead leading index we may follow two diﬀerent approaches.

The ﬁrst approach requires to derive the h-step ahead forecasts of series yt and combine

them with the estimated CI weights

δ0. A possible way to incorporate the CI&LI’s restrictions

within the VECM is to rely on the following common factor representation
b

(In

−

Γ1L)∆yt = ΛFt

1 + εt,

−

(23)

where Λ is a full-rank n

(n

×

−

s) matrix such that (ω00, 00n2)Λ = 0,

Ft

1 =

−

αβ0yt

1 +

−

Γ0i∆yt
−

i,

p

1

−

i=2
X

e

e
×

−

−

×

s)

α,

(n

Γ01,

α is a (n

r matrix, and

s) matrix for i = 2, , ..., p

Γi is n
Γ02, ...,
mates of the parameters [
1] are provided by the canonical variates coeﬃcients
Γ0p
e
−
e
s) largest eigenvalues ˆλs+1, ..., ˆλn. Fi-
p+1)0 associated to the (n
1, ..., ∆y0t
1β, ∆y0t
of (y0t
e
−
−
−
e
nally, the remaining parameters of model (23) are easily estimated by a regression of ∆yt on
(∆y0t
1, F 0t
1)0.
−
−
The second approach consists in estimating the h-step ahead leading index by a single-

1. Eﬃcient esti-

−

−

e

e

equation method. In particular, for ﬁxed (δ00, δ01) = (
can be used to estimate the equation

0,
δ0

b

δ0
1), the General Method of Moments

b

1 + eh
∆CIt+h = γh∆LIt+h
−

t+h

(24)

12

ˆβ, ∆y0t, ..., ∆y0t

p+2)0 as instruments, where CIt and LIt are a generic CI&LI pair,
−
is then obtained by subtracting the

1) error. Clearly, ∆LI h
t

t is a MA(h

using series (y0t
γh is a scalar, and eh
eh
t+h

residuals

−
1 to the observed values of ∆CIt+h.
−

Although the second approach may be preferred for its simplicity, one should keep in mind

that statistical inference on (24) is conditional on the estimated CI&LI’s weights and hence

b

their sample variability is ignored.

An interesting question to be posed is if one can build an optimal CI&LI pair such that LI ∗t
is a valid leading indicator for any forecast horizon of CI ∗t . Such CI&LI should satisfy, along
with condition (15), the following equation

E(∆CI ∗t+h|

Ωt) = E(∆LI ∗t+h

∆LI ∗t ),

1|
−

(25)

for any h

1.

≥

In view of equation (21) and keeping in mind that ∆CI ∗t and ∆LI ∗t are stationary ARMA

processes, we see that equation (25) is satisﬁed when

∆LI ∗t = ρ∆LI ∗t
−

1 + νt,

(26)

< 1, and νt = δ∗01 εt.

where ρ

= 0,

ρ
|
|

Equation (26) implies that the optimal leading index is an ARIMA(1,1,0) process. But

we need a stronger requirement that the error term of this ARIMA process is an innovation

with respect Ωt

1. In the terminology of Granger and Yoon (2001), the optimal LI must be a

−

self-generating variable.

By comparing equation (20) with equation (26) and in view of Proposition 5, we conclude

that condition (25) holds when δ∗01 = ρ(ω∗00 , 001
n2). These non-linear restrictions on the optimal
×
CI&LI weights can be tested and possibly imposed in estimation by means of a grid search of

the likelihood function over diﬀerent values of ρ.

4 Coincident and Leading Indexes for the US Economy

This section illustrates the use of the RRR framework for constructing coincident and leading

indicators for the US economy. The aim of this empirical analysis is twofold. First, the historical

components of the new indexes are extracted and compared with those proposed by Stock and

13

6
Watson (1989, SW henceforth) and TCB.5 Second, an out-of-sample forecasting exercise is

performed to asses the predictive performances of the RRR procedure.

4.1 Variable Deﬁnitions and Description

For the empirical analysis we consider the monthly Business Cycle Indicators [BCI] that TCB
used to build their own indexes.6 The ﬁrst two columns of Table 1 report the variables of

interest along with their BCI code. With the exception of the stock prices index and consumer
expectations index, the data are seasonally adjusted7 and span the period 1959.01 to 2002.12.

The sub-sample 1959.01-1999.12 is used to build the new CI&LI and the remaining observations

are left for an out-of-sample forecasting exercise.

The fourth column of Table 1 reports the results of the ADF unit root tests on the BCI

indicators that have been transformed as indicated in the third column. Only the vendor

performance, the interest rate spread, and the building permits series appear to be I(0). In order

to build the new CI&LI, these stationary indicators are integrated, namely their cumulative

sum are taken in the analysis (see e.g. Rahbek and Mosconi, 1999). This operation allows to

include in the ﬁrst diﬀerences of the CI&LI also the I(0) variables that otherwise would have

been annihilated by Condition 1 of Deﬁnition 1. Notice that the cumulated I(0) series do not

posses an exact unit root by construction. Indeed, from Table 2 we see that the ADF tests

indicate the presence of signiﬁcant but not exact unit roots in such cumulated series. Finally,

the volatility of all the transformed indicators has been adjusted as TCB suggests, that is all

the ﬁrst diﬀerences of these series have unitary standard deviations and are now all expressed

in comparable scale.

SW (1989) impose a single dynamic common factor for summarizing the information con-

tained in the past. Since such assumption is not formulated in the RRR approach to CI&LI

building, the new procedure is prone a dimensionality problem. Consequently it is required to

rely on the following step-wise procedure based on the minimization of an information crite-

rion such as the Bayes Information Criterion [BIC]. Let us start with the four BCI coincident

5 A detailed description of the data and methodology used by The Conference Board can be found at the

webpage http://www.tcb-indicators.org/

6 The data used in this section were generously provided by Alain Hecq.
7 Although seasonal ﬁltering poses problems for common features analysis (see e.g. Cubadda, 1999), the
purpose of comparing the new CI&LI with the existing ones imposes to follow the usual practice of using
seasonally adjusted data.

14

×

variables zt = (z1t, z2t, z3t, z4t)0 and select the VAR order that minimizes the BIC from 0 up to
pmax. Then, we add separately in the right hand side the lags of each of the ten TCB leading
indicators and we estimate all the VARX models with order from 0 up to pmax. Finally, we
pmax VARX models with the BIC of the previously

compare the smallest BIC of these 10

selected VAR model.

If the value of the BIC is smallest for the VAR model, we keep only

the reference series in the analysis. Otherwise, the leading indicator associated to the VARX

model with the smallest BIC is retained as exogenous variable. In the second round, each of

the nine remaining TCB indicators is included as an additional exogenous variable and the BIC
pmax VARX models. Again, we compare the smallest BIC of these
pmax VARX models with the BIC of the previously selected VARX model. The procedure

is computed for all the 9

×

9

stops when it is not possible to ﬁnd a better VARX model according to the BIC. The outcome

×

is that the ”elected” series are the average weekly hours, vendor performances building permits,

and interest rate spread. These four series respectively comprise the leading indicators vector

xt = (x1t, x2t, x3t, x4t)0 in the subsequent analysis.

4.2 Building the RRR-based CI&LI

Table 3 reports both the asymptotic and the small-sample corrected versions of the Johansen

trace statistics in a VAR(3). We can not reject the presence of three cointegrating vectors

and then ﬁve common trends. A graphical inspection of the cointegrating vector conﬁrms the

outcome of the formal analysis. Hence, we ﬁx at three the number of cointegrating vectors and

we pursue the CI&LI analysis.

The next step is testing whether there exists a PSCCF vector such that the CI is formed by

the four TCB coincident series only. We use the test statistic (9), both in the asymptotic and

the small-sample corrected version. From Table 4 we see that one cannot reject the presence of

a single CI&LI at the 5% conﬁdence level. The weights of such CI&LI are also reported in the
same Table.8

It is also possible to evaluate additional restrictions on the individual indicator coeﬃcients.

As a result of a general to speciﬁc testing procedure, we cannot reject the null hypothesis that

sales (z4t) do not enter in both the CI and the LI. In particular, the p-value associated with the

8 Notice that such coeﬃcients are normalized such that the sum of the absolute value of the CI weights is

equal to one.

15

test statistic (13) for these joint restrictions on the CI&LI’s weights is equal to 0.312.9 Table 4

also reports the value of the test statistics (14) and the associated coeﬃcients of such restricted

CI&LI. In the sequel, we will refer to these restricted CI and LI respectively as the RRR_CI

and RRR_LI.

4.3 Comparison with Other Coincident Indices

In this sub-section the RRR_CI is compared with two other composite indicators, namely TCB

[TCB_CI] and SW [SW_CI] coincident indicators. The levels of the three series, rebased to

average 100 in 1995, are graphed in Figure 2. Visual inspection suggests that these indexes

provide a rather similar picture of the business cycle.

Table 5 reports the cross-correlation functions between the monthly growth rates of the

various CI’s. It is apparent that the three indexes are clearly synchronous and highly cross-

correlated. Moreover, Table 6 shows the average spectral coherency of the alternative CI’s

growth rates in the 3-9 year period band. It emerges that these indexes are almost perfectly

coherent at the business cycle frequencies.

Table 7 compares the recessions determined by each index with the NBER oﬃcial chronology.

To facilitate the comparison, the following set of dummy variable were created

dt =

di,t =








1,

if there was a recession at date t according to NBER;

0, otherwise.

1,

if there was a recession at date t according to index i;

0, otherwise.

for i = RRR, TCB, SW, and for each di,t its average squared deviation from dt was computed:



T

1
T Pi = T −

(di,t

t=1
X

dt)2.

−

(27)

We see that TCB_CI captures the NBER reference series best, but the new index perform

very similarly. The SW_CI exhibits the same value of the above index as the RRR_CI.

9 A possible explanation of this result is that the growth rates of sales display very little autocorrelation. This

implies that z4t has a negligible BN cyclical component.

16

4.4 Comparison with Other Leading Indices

So far, the one-month ahead leading index [RRR_LI] was obtained. However, SW (1989) and

TCB built their leading indicators, denoted respectively by SW_LLI and TCB_LLI, in order

to foresee the business cycles about six months in advance. Hence, also the six-month ahead

Long Leading Index [RRR_LLI] was constructed using equation (24). Similarly as in the case

of TCB, the growth rates of RRR_LLI were adjusted in order to have the same variability

as those of RRR_CI. Moreover, the levels of RRR_LLI were computed using the values of

RRR_CI at 1959.7-8 as starting values.

The levels of the indexes RRR_LLI, SW_LLI, TCB_LLI, rebased to average 100 in 1995,

are plotted in Figure 2. The graphical comparison indicates that RRR_LLI is smoother than

its competitors, providing so a clearer picture of the business cycle.

Table 8 reports the correlations of each CI’s monthly growth rates with the lags of the

associated LI’s growth rates. We see that RRR_LI forecasts its CI changes best for shorter

lags, namely one and two, whereas RRR_LLI performs better from three up to twelve periods

in advance. One may observe that this is an unfair way of comparing the in-sample forecasting

performances of the alternative LI’s because the RRR-based LI’s are explicitly designed for

predicting the associated CI’s growth rates. Hence, Table 9 shows the correlations of the

alternative CI’s j-month growth rates with the j-th lags of the associated LI’s j-month growth

rates for j = 1, 2, ..., 12. We see that RRR_LI again forecasts best for j = 1, 2, RRR_LLI

performs better for j = 3, ...6, whereas TCB_LLI is superior to its competitors for longer lags.

4.5 Out-of-Sample Forecasting Exercise

In this sub-section we wish to evaluate the out-of-sample performances of the new CI&LI.

Hence, the weights estimated using the sub-sample from 1959.01 to 1999.11 are kept ﬁxed in

the forecasting period 2000.1-2002.12.

Let us preliminary compare the properties of the RRR_CI with those of SW_CI and

TCB_CI, which are instead built using the full sample. In Table 10 we see the cross-correlation

functions of the alternative CI’s growth rates for the period 2000.01-2002.12. We notice that

the various CI’s clearly exhibit positive contemporaneous comovements, even if the evidence is

less strong than within the sample. Table 11 shows the recessions determined by each index

and the NBER oﬃcial chronology. The index (27) indicates that the RRR_CI accords with

17

NBER chronology quite well, since the index assume an intermediate value with respect to

those associated with TCB_CI and SW_LI.

In order to check for possible structural breaks in the forecasting period, the Chow tests
for parameter stability were applied. For the unrestricted VECM, the value of the χ2(288) test

statistic is 264.69 that corresponds to a p-value equal to 0.834. After imposing the CI&LI’s re-

strictions through the common factor representation (23), the value of the test statistic becomes

252.01 and the associated p-value increases to 0.938.
Finally, the forecasting performances of ∆LI h

t built according to equation (24) are contrasted
with those of an unrestricted h-step ahead forecasts of ∆CIt+h. The latter forecasts are obtained

by estimating with Generalized Least Squares the equation

∆CIt+h = γh

00 β0yt +

p

2

−

i=0
X

γh
0i ∆yt

i + eh
−

t+h,

h = 1, . . . 6,

(28)

where γh

0 is a r-vector, and γh

i is a n-vector for i = 1, 2, ..., p

2, and eh

t is a MA(h

1) error.

−

−

Table 12 shows the tests proposed by Diebold and Mariano (1995) and modiﬁed by Harvey

et al. (1997) for the equality of the Mean Square Forecasting Errors [MSFE] of equations (24)

and (28) for h = 1, ..., 6. The third column reports the p-values for the alternative hypothesis

that the former equation has a smaller MSFE than the latter, and the p-values for the opposite
inequality are the complements to one of the third column elements. It emerges that ∆LI h
t
forecasts signiﬁcantly better than equation (28) for h = 1 at the 5% level, and h = 2 at the 10%

level, whereas none of the two predictors has a signiﬁcantly smaller MSFE for larger forecasting

horizons at the 10% level.

5 Conclusions

This paper has presented a new method to build a CI and a LI from a set of cointegrated

time series yt. Based on the notion of PSCCF (Cubadda and Hecq, 2001), the CI and LI

are respectively obtained as linear combinations of the reference series zt and yt

1 such that

−

the changes of the LI are the best linear predictors of the changes of the CI. The proposed

methodology covers also additional aspects of composite indicators building such as testing

on the CI&LI weights and the construction of long leading indicators. Finally, concepts and

methods have been illustrated by an empirical application with the US business cycle indicators.

18

References

[1] Ahn, S.K., and G.C. Reinsel (1988), Nested Reduced-Rank Autoregressive Models for

Multiple Time Series, Journal of the American Statistical Association, 13, 352-375.

[2] Anderson, T.W. (1984), An Introduction to Multivariate Statistical Analysis, 2nd Ed.,

John Wiley & Sons.

[3] Beveridge, S. and C.R. Nelson (1981), A New Approach to Decomposition of Eco-

nomic Time Series into Permanent and Transitory Components with Particular Attention

to Measurement of the Business Cycle, Journal of Monetary Economics, 7, 151-174.

[4] Camacho, M. and G. Perez-Quiros (2002), This Is What the Leading Indicators

Lead, Journal of Applied Econometrics, 17, 61-80.

[5] Camba-Mendez, G., Kapetanios, G., Smith, R.J., and M.R. Weale (2003), Tests

of Rank in Reduced Rank Regression Models, Journal of Business and Economic Statistics,

21 (1) , 145-155

[6] Cubadda, G. (1999), Common Cycles in Seasonal Non-Stationary Time Series, Journal

of Applied Econometrics, 14, 273-291.

[7] Cubadda, G. and A. Hecq (2001), On Non-Contemporaneous Short-Run Comove-

ments, Economics Letters, 73, 389-397.

[8] Diebold, F.X. and R.S. Mariano (1995), Comparing Predictive Accuracy, Journal of

Business and Economic Statistics, 13, 253-263.

[9] Emerson, R.A. and D.F Hendry (1996), An Evaluation of Forecasting Using Leading

Indicators, Journal of Forecasting, 15, 271-291.

[10] Engle, R. F. and S. Kozicki (1993), Testing for Common Features (with comments),

Journal of Business and Economic Statistics, 11, 369-395.

[11] Granger, C.W.J and G. Yoon (2001), Self-Generating Variables in a Cointegrated

VAR Framework, UCSD Department of Economics working paper series, 2.

[12] Hamilton, J.D. and G. Perez-Quiros (1996), What Do the Leading Indicators Lead?,

Journal of Business, 69, 27-49

19

[13] Harvey, A. C. (1990), Forecasting, structural time series models, and the Kalman ﬁlter,

Cambridge University Press.

[14] Harvey, D.I, Leybourne, S.J. and P. Newbold, Testing the Equality of Prediction

Mean Square Errors, International Journal of Forecasting, 13, 281-291.

[15] Hecq, A., F.C. Palm and J.P. Urbain (2000), Permanent-Transitory Decomposition

in VAR Models with Cointegration and Common Cycles, Oxford Bulletin of Economics

and Statistics, 62, 511—532.

[16] Johansen, S. (1995), Likelihood-Based Inference in Cointegrated Vector Autoregressive

Models, Oxford University Press.

[17] Niemira, M.P. and P.A. Klein (1994), Forecasting ﬁnancial and economic cycles,

Wiley Finance Editions.

[18] Proietti, T. (1997), Short-Run Dynamics in Cointegrated Systems, Oxford Bulletin of

Economics and Statistics, 59, 405-22.

[19] Rahbek, A. and R. Mosconi (1999), Cointegration Rank Inference with Stationary

Regressors in VAR Models, Econometrics Journal, 76-91.

[20] Reinsel, G.C. and S.K. Ahn (1992), Vector Autoregressive Models with Unit Roots

and Reduced Rank Structure: Estimation, Likelihood Ratio Test, and Forecasting, Journal

of Time Series Analysis, 13, 352-375.

[21] Stock, J.H. and M.W. Watson (1989), New Indexes of Coincident and Leading Eco-

nomic Indicators, in Blanchard, O. and S. Fischer (eds.) NBER macroeconomics annual.

MIT Press, 351-94.

[22] Stock, J.H. and M.W. Watson (1991), A Probability Model of the Coincident Eco-

nomic Indicators, in Lahiri, K. and G.H. Moore (eds.) Leading economic indicators: New

approaches and forecasting records. Cambridge University Press, 63-90.

[23] Stock, J.H. and M.W. Watson (1993), A Procedure for Predicting Recessions with

Leading Indicators: Econometric Issues and Recent Experience, in Stock, J.H. and M.W.

Watson (eds.), Business cycles, indicators, and forecasting. NBER Studies in Business

Cycles, 28. University of Chicago Press, 95-153.

20

[24] Tiao, G.C., and R.S. Tsay (1989), Model Speciﬁcation in Multivariate Time Series,

Journal of the Royal Statistical Society, B, 51, 157-213 with discussions.

[25] Velu, R.P., Reinsel, G.C. and D.W. Wichern (1986), Reduced Rank Models for

Multivariate Time Series, Biometrika, 73, 105-118.

[26] Wecker, W.E. (1979), Predicting the Turning Points of a Time Series, Journal of Busi-

ness, 52, 35-50.

21

Table 1

ADF unit root tests

BCI code Variable

Transf.

t-ADF

Potential Coincident Indicators

BCI-041

Employees on non-agricultural payrolls

BCI-051

Personal income less transfer payments

BCI-047

Industrial production

BCI-057

Manufacturing and trade sales

log level

-2.83

log level§

-2.41

log level

-2.15

log level

-2.98

Potential Leading Indicators

BCI-001

Average weekly hours, manufacturing

log level

-3.34

BCI-005

Average weekly initial claims for unemployment insurance

log level

-2.30

BCI-008

Mfrs’ new orders, consumer goods and materials

log level

-3.24

BCI-032

Vendor performance, slower deliveries diﬀusion index

level

-5.63**

BCI-027

Mfrs’ new orders, nondefense capital goods

log level

-2.69

BCI-029

Building permits for new private housing units

log level

-3.61*

BCI-019

Index of stock prices, 500 common stocks

BCI-106

Money supply, M2

log level

-0.43

log level

-2.85

BCI-129

Interest rate spread, 10-year Treasury bond less fed. funds

level

BCI-083

Univ. of Michigan Index of consumer expectations

level

-4.25**

-2.65

§ Two additive outliers corresponding to 1992.12 and 1993.12 were removed.

* (**) Insigniﬁcant at the 5% (10%) conﬁdence level

Table 2

ADF unit root tests

BCI code Variable

BCI-032

Vendor performance, slower deliveries diﬀusion index

BCI-029

Building permits for new private housing units

Transf.

t-ADF

Σ level

-1.43

Σ log level

-2.87

BCI-129

Interest rate spread, 10-year Treasury bond less fed. funds Σ level

-1.49

22

Table 3

Johansen’s cointegration tests

Trace

295.0**

171.0**

119.3**

73.86*

40.20

18.13

8.069

0.029

Trace§

280.6**

162.6**

113.5**

70.23*

38.22

17.24

7.673

0.028

r = 0

r

r

r

r

r

r

r

1

2

3

4

5

6

7

≤

≤

≤

≤

≤

≤

≤

§ Small-sample corrected test statistics

* (**) Insigniﬁcant at the 5% (10%) conﬁdence level

Table 4

CI&LI’s tests

Unrestricted CI&LI’s

Restricted CI&LI’s

s

s

s

s

1

2

3

4

≤

≤

≤

≤

LR1

15.05*

36.72

100.1

232.0

LR§1
14.71*

35.90

97.80

226.8

s

s

s

1

2

3

≤

≤

≤

CI&LI’s weights

LR3

17.37*

39.87

164.1

LR§3
16.98*

38.97

160.4

Unrestricted CI&LI (s = 1)

Restricted CI&LI (s = 1)

z1t

z2t

z3t

z4t

0.268

0.132

0.486

-0.115

z1t

z2t

z3t

z4t

1

1

1

1

−

−

−

−

0.075

0.059

0.186

0.002

x1t

x2t

x3t

x4t

1

1

1

1

−

−

−

−

-0.077

0.103

0.122

0.132

z1t

z2t

z3t

z4t

0.295

0.208

0.497

0.000

z1t

z2t

z3t

z4t

1

−

1

−

1

−

1

−

0.072

0.091

0.227

0.000

x1t

x2t

x3t

x4t

1

−

1

−

1

−

1

−

-0.095

0.115

0.155

0.168

§ Small-sample corrected test statistics

* (**) Insigniﬁcant at the 5% (10%) conﬁdence level

23

Figure 1: RRR, TCB, and SW coincident indexes

Table 5

Cross-correlation functions of diﬀerent CI’s growth rates

Lag

6
−

RRR vs TCB .101

5
−
.086

4
−
.186

3
−
.248

2
−
.302

1

0

1

2

3

4

5

6

.467

.910

.457

.324

.262

.204

.010

.129

RRR vs SW .046

.020

.080

.169

.281

.479

.926

.437

.326

.232

.178

.080

.122

TCB vs SW .051

.032

.095

.170

.260

.426

.916

.376

.269

.224

.155

.078

.098

Note: 95% signiﬁcance is .089

Table 6

Average spectral coherency of diﬀerent CI’s growth rates

at the business cycle frequencies (3-9 year periods)

RRR vs TCB

RRR vs SW

TCB vs SW

0.997

0.983

0.987

Note: spectra are estimated by a rectangular spectral window with width = 45

24

Table 7

Recession periods determined by alternative indexes

NBER

RRR

TCB

SW

1960.04-1961.02

1960.04-1961.02

1960.04-1961.02

1960.02-1961.02

1969.12-1970.11

1969.10-1970.11

1969.12-1970.11

1969.10-1970.11

1973.11-1975.03

1973.12-1975.04

1973.12-1975.04

1973.11-1975.05

1980.01-1980.07

1980.02-1980.07

1980.02-1980.07

1980.01-1980.07

1981.07-1982.11

1981.08-1982.12

1981.08-1982.12

1981.07-1982.12

1990.07-1991.03

1990.07-1991.04

1990.07-1991.03

1990.08-1991.03

TP index

0.0171

0.0107

0.0171

Figure 2: RRR, TCB, and SW (long) leading indexes

25

Table 8

Correlations of CI’s growth rates with past

LI’s growth rates for alternative indexes

Lag RRR_LI RRR_LLI TCB_LLI SW_LLI

1

2

3

4

5

6

7

8

9

10

11

12

0.587

0.481

0.424

0.333

0.255

0.272

0.283

0.253

0.228

0.183

0.101

0.035

0.481

0.456

0.438

0.427

0.405

0.383

0.369

0.364

0.338

0.316

0.269

0.234

0.195

0.318

0.293

0.317

0.183

0.214

0.172

0.224

0.206

0.257

0.190

0.209

0.221

0.233

0.152

0.165

0.121

0.167

0.188

0.206

0.172

0.106

0.090

-0.028

Table 9

Correlations of CI’s j-month growth rates with j-th lags

of LI’s j-month growth rates for alternative indexes

j RRR_LI RRR_LLI TCB_LLI

SW_LLI

1

2

3

4

5

6

7

8

9

10

11

12

0.587

0.601

0.572

0.552

0.540

0.514

0.476

0.426

0.377

0.330

0.285

0.244

0.481

0.543

0.579

0.605

0.619

0.615

0.604

0.583

0.557

0.526

0.492

0.456

0.195

0.416

0.516

0.556

0.572

0.607

0.639

0.656

0.652

0.639

0.626

0.611

26

0.221

0.330

0.360

0.429

0.515

0.573

0.585

0.558

0.511

0.462

0.422

0.394

Table 10

Cross-correlation functions of diﬀerent CI’s growth rates (2000.01-2002.12)

Lag

RRR vs TCB

6
−
.020

5
−
.224

4
−
.264

3
−
.342

2
−
.431

1

0

1

2

3

4

5

6

.548

.754

.585

.531

.343

.201

.238

.154

RRR vs SW -.083

.112

.170

.275

.367

.501

.798

.637

.569

.345

.271

.308

.170

TCB vs SW

.002

.183

-.012

.334

.263

.385

.884

.308

.492

.328

.137

.410

.082

Note: 95% signiﬁcance is .3267

Table 11

Recession periods determined by alternative indexes

(2000.01-2002.12)

NBER

RRR

TCB

SW

2001.03-2001.11

2000.12-2002.01

2001.01-2001.12

1999.10-2002.01

TP index

0.1389

0.0833

0.1944

Table 12

Modiﬁed Diebold-Mariano tests

Lead

Statistic

P -value

1

2

3

4

5

6

-2.344

-1.672

-0.174

0.066

-0.036

1.119

0.0125

0.0517

0.4316

0.5261

0.4856

0.8646

27