A Reduced Rank Regression Approach to Coincident and Leading Indexes Building*
| Date | 01 April 2007 |
| Published date | 01 April 2007 |
| Author | Gianluca Cubadda |
| DOI | http://doi.org/10.1111/j.1468-0084.2006.00196.x |
A Reduced Rank Regression Approach to
Coincident and Leading Indexes Building*
Gianluca Cubadda
Dipartimento SEFeMEQ, Universita’ di Roma ‘ Tor Vergata’, Roma, Italy
(e-mail: gianluca.cubadda@uniroma2.it)
Abstract
This paper proposes a reduced rank regression framework for constructing a
coincident index (CI) and a leading index (LI). Based on a formal definition that
requires that the first differences of the LI are the best linear predictor of the first
differences of the CI, 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.
I. Introduction
In a large number of countries, a coincident index (CI) and a leading index (LI) are
routinely built in order to provide economic analysts with early signals of the broad
swings in macroeconomic activity that compose the business cycle. These indexes
are typically constructed in two steps. The first step aims at identifying groups of
variables that move during, before or after the recession (see, e.g. Niemira and Klein,
1994). In this paper the focus is on the first two groups of variables, which are
defined as the coincident and leading indicators respectively. The second step
consists of forming composite indicators, namely the CI and LI, in order to extract
the relevant business cycle features from the individual indicators.
*Previous drafts of this paper were presented at the 59th European Meeting of the Econometric Society in
Madrid, the Common Features Conference in Maastricht, the seminar at Ente Einaudi in Rome, and the 4th
Colloquium on Modern Tools for Business Cycle Analysis in Luxembourg. I wish to thank Bertrand
Candelon, Paolo Paruolo, the associate editor Anindya Banerjee and two anonymous referees for useful
comments, as well as Alain Hecq for kindly providing me with the data that were analysed in section 4.
Financial support from MIUR is gratefully acknowledged. The usual disclaimers apply.
JEL Classification number: C32.
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 69, 2 (2007) 0305-9049
doi: 10.1111/j.1468-0084.2006.00196.x
271
Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2006. Published by Blackwell Publishing Ltd,
9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.
Among the various statistical methods for constructing a CI and LI, the procedure
developed by Stock and Watson (1989, 1991 and 1993) for the National Bureau of
Economic Research (NBER) has rapidly become a standard reference. However, a
variety of alternative approaches are also available such as those based on principal
component analysis, 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, and
Marcellino, 2005, for a thorough survey of the literature).
In a similar spirit as Emerson and Hendry (1996), the viewpointtaken in this paper is
that the construction of CIs and LIs 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 and LI from a vector of cointegrated
economic indicators. RRR has been extensively analysed in the statistical and
macroeconometric literature (see inter alia Anderson, 1984; Velu, Reinselt and
Wichern, 1986; Ahn and Reinsel, 1988; Tiao and Tsay, 1989; Johansen, 1996) but, to
the best of the author’s knowledge, it has not yet been applied to the problem at
hand. This seems a promising route to follow as there is convincing evidence (see
inter alia Reinsel and Ahn, 1992; Camba-Mendez et al., 2003) that imposing reduced-
rank structures in vector auto-regressive (VAR) models improves performance in
prediction.
In particular, the dynamic properties of the data are investigated within the
polynomial serial correlation common feature modelling (Cubadda and Hecq, 2001).
Similarly to the composite indexes built by The Conference Board (1997, TCB
henceforth), the proposed CIs and LIs are obtained as linear combinations of
observed variables. However, the weights of the new indexes are derived such that
the changes of the LI are the best linear predictor of the changes of the CI. Hence,
the suggested CIs and LIs are constructed with a mind to document and predict
variations in overall economic activity.
1
Another relevant characteristics of the new
composite indicators is that the actual existence of such CI and LI is tested for, rather
than assumed a priori and it is possible to check if the individual indicators
significantly enter in the CI and LI. Furthermore, the multivariate Beveridge-Nelson
(1981) cycle of the LI leads that of the CI.
This paper is organized as follows. Section II proposes definitions for the new CI
and LI, and shows how to build such indexes by means of RRR. In section III the
conditions for the existence of a long leading index are examined. In section IV the
methodology is applied to US business cycle indicators. Section V concludes.
II. The statistical methodology
The aim of this section is to present an RRR framework to build the CI and LI from a
set of cointegrated time series.
1
Although the proposed CI and LI are not specifically designed for predicting business cycle turning points,
they may be used for such a purpose along the lines of Wecker (1979) and Hamilton and Perez-Quiros (1996).
272 Bulletin
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