Topics in Structural VAR Econometrics

144 Pages·1992·5.29 MB·English

by  Prof. Carlo Giannini (auth.)

Lecture N otes in Economics
and Mathematical Systems 381
Editorial Board:
H. Albach, M. Beckmann (Managing Editor)
P. Dhrymes, G. Fandel, G. Feichtinger, W. Hildenbrand
W. Krelle (Managing Editor)
H. P. Künzi, K. Ritter, U. Schittko, P. Schönfeld, R. Selten, W. Trockel
Managing Editors:
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. W. Krelle
Institut für Gesellschafts-und Wirtschaftswissenschaften
der Universität Bonn
Adenauerallee 24-42, W-5300 Bonn, FRG
Carlo Giannini
Topics in Structural
VAR Econometrics
Springer-Verlag Berlin Heidelberg GmbH
Author
Prof. Carlo Giannini
Department of Economics
University of Ancona
Via Pizzecolli 68
I-60121 Ancona, Italy
ISBN 978-3-540-55262-8 ISBN 978-3-662-02757-8 (eBook)
DOI 10.1007/978-3-662-02757-8
This work is subject to copyright. All rights are reserved, whether the whole or part of
the material is concemed, specifically the rights of translation, reprinting, re-use of
illustrations, recitation, broadcasting, reproduction on microfilms or in any other way,
and storage in data banks. Duplication of this publication or parts thereof is permitted
only under the provisions oftheGerman Copyright Law of September 9, 1965, in its
current version, and permission for use must always be obtained from Springer-Verlag.
Violations are liable for prosecution under the German Copyright Law.
©Springer-Verlag Berlin Heidelberg 1992
Originally published by Springer-Verlag Berlin Heidelberg in 1992.
42/3140-543210-Printedon acid-free paper
To Vittoria and Andrea
CONfENTS
Foreword IX
1. Introduction 1
2. Identification Analysis and F.I.M.L. Estimation for the K-Mode1 10
3. Identification Analysis and F.I.ML. Estimation for the C-Model 23
4. Identification Analysis and F.I.M.L. Estimation for the AB-Model 32
5. Impulse Response Analysis and Forecast Error Variance
Decomposition in SVAR Modeling 44
5 .a Impulse Response Analysis 44
5.b Variance Decomposition (by Antonio Lanzarotti) 51
6. Long-run A-priori Information. Deterministic Components.
Cointegration 58
6.a Long-run A-priori Information 58
6.b Deterministic Components 62
6.c Cointegration 65
7. The Working of an AB-Model 71
Annex 1: The Notions ofReduced Form and Structure in Structural VAR
Modeling 83
Annex 2: Some Considerations on the Semantics, Choice and
Management of the K, C and AB-Models 87
Appendix A 93
Appendix B 96
Appendix C (by Antonio Lanzarotti and Mario Seghelini) 99
Appendix D (by Antonio Lanzarotti and Mario Seghelini) 109
References 128
Foreword
In recent years a growing interest in the structural VAR approach (SVAR) has followed
the path-breaking works by Blanchard and Watson (1986), Bemanke (1986) and Sims
(1986), especially in U.S. applied macroeconometric literature.
The approach can be used in two different, partially overlapping directions: the
interpretation ofbusiness cycle fluctuations of a small number of significantmacroeconomic
variables and the identification of the effects of different policies.
SVAR Iiterature shows a common feature: the attempt at "organizing"-in a "structural"
theoretical sense -instantaneous correlations between relevant variables. In non-structural
VA R modelling, instead, correlations are normally hidden in the variance-covariance matrix
of the innovations of VA R models.
Structural VA R analysistri es to isolate ("identify") a set of independent shocks by means
of a number of meaningful theoretical restrictions. The shocks can be regarded as the
ultimate source of stochastic variations ofthe vector ofvariables which, moreover, could be
seen as potentially all endogenous.
Looking at the development of SVA R Iiterature I felt that it stilllacked a formal general
framework which could embrace the several types ofmodel so farproposed for identification
and estimation.
Following Rotheuberg (1971, 1973) the present monograph tries to develop a
methodological framework for three types of model which encompass a11 the different
models used in applied literature. I have also tried to generalise the identification and
estimation set-up using the most general type of linear constraints available for the
representation of ideas about the organisation of instantaneous co-movements of variables
in response to "exogenous" independent shocks.
Trying to adapt recent work by Lütkepohl, section 5 contains calculations of the
asymptotic distributions of impulse response functions and forecast error variance
decompositions. This allowed me to avoid using bootstrapping or Monte Carlo integration
X
techniques in the three types of model. Paragraph S.b of this section was written by Antonio
Lanzarotti.
Section 6, introductory in nature, includes some suggestions and warnings that may be
useful in the treatment of detenninistic components, typically long run constraints in a
stationary context and a way to match a cointegrating set-up.
Section 7 tries to offer deeper insights into Structural VAR modelling. It contains very
few technicalities and qualitatively discusses the results of an exercise carried out on Italian
data using an AB-model. The exercise draws upon the economic framework put forward by
O.J. Blanchard in its 1989 paper.
Annex 1 deals with the notion of structure in SVA R modelling, while Annex 2 contains
my point ofview on the significance ofthe three types of model discussed in this monograph.
I also try to suggest some criteria on which model to choose in different applications tagether
with some generat considerations on their overall working.
Appendix A briefly summarizes rules and conventions of matrix differential calculus
adopted in this monograph.
Appendix B contains the calculation of the first order conditions for the maximization
of the lik.elihood of the "Key" model and the corresponding Hessian matrix.
Appendices C and D have been written jointly by Antonio Lanzarotti and Mario
Seghelini: the fmmer contains some examples of symbolic identification analysis for the K,
C and AB models; the latter contains two RATS programs that implement the ideas put
forward in this monograph.
Looking at a selected choice of recent SVA R applied papers one can see the following
correspondence with regard to the categorization put forward in this monograph: O.J.
Blanchard, M.W. Watson, 1986 (K-model); O.J. Blanchard, D. Quah, 1989 and M. Shapiro,
M.W. Watson, 1988 (C-model); B. Bernanke 1986 and O.J. Blanchard 1989 (AB-model).
When I started to write the first draft of this monograph, no available work covered the
fu1l range ofmethods and issues used in VA R econometric literature. Since then, Lütkepohl 's
1991 book entitled "lntroduction to Multiple Time Series Analysis" has filled the gap. It now
enables me, to my great relief, to avoid discussing the problems of vector autoregressive
modelling and usual structuralization through Cholesky decompositions.
XI
This monograph surely overlooks a number of important topics in SVA R modelling, the
most important of which is probably how to choose between alternative structuralizations
of the same unstructured VA R model. Although the issue could be treated as a problern of
testing non-nested hypotheses, I believe that a recent paper by Pollack and Wales (1991) on
the likelihood dominance criterion offers the most Straightforward solution.
A first version of this monograph has already had a limited circulation as "Topics in
Structural VAR Econometrics", Quaderni di ricerca of the Department of Economics of
Ancona University, July 1991. I have since added Section 7, Annex 2 and made some minor
changes in other sections. The present version is stilltobe regarded as something in between
a first draft and a final version; comments and suggestions are therefore wiumly welcomed.
In preparing this monograph I have been supported by a M. U. R.S.T. 40% research grant
at Ancona University labelled "Modelli macroeconomici e analisi econometrica dinamica".
I wish to thank G. Amisano, S. Yadav and R. Mosconi for helpful discussions and M.
Faliva for providing useful algebraic references while I was working on the first version. I
am also indebted toS. Calliari, J.D. Hamilton, M. Lippi, J.R. Magnus, H. Neudecker, R.
Orsi, P.C.B. Phillips, D.S.G. Pollock, H.-E. Reimersand to the unknown Springer referee
for their suggestions and/or encouragement after reading the first version. Thanks are also
due to Ubaldo Stecconi of Cooperativa Logos, Ancona, who revised the English manuscript
and managed the typesetting.
Special thanks are due to my students at Pavia University Antonio Lanzarotti and Mario
Seghelini both for their contribution and suggestions. They have accompanied me through
a joumey which had started in a fog of confused ideas.
The usual claims obviously apply.
Ancona, January 1992
Carlo Giannini
Dipartimento di Economia
Universita di Ancona
Via Pizzecolli 68
60121 Ancona, Italy
1. lntroduction
In order to introduce the basic elements of Structural VA R Analysis, let us suppose that
we can represent a set of n economic variables using a vector (a column vector) Yt of
stochastic processes, jointly covariance stationary without any detenninistic patt and
possessing a finite order (p) autoregressive representation.
A(L)yt= Et
A(L)=l-AIL- ... -Aplf
The roots ofthe equation det(A(L)) = 0 areoutside the unit circle in the complex domain
and Et has an independent multivariate nonnal distribution with [0] mean.
Et ~I MN([O],:E)
=
E(er) [0]
E(ere'r) =r . det (:E) ::f:. 0
E(ete's) = [0] S::f:.t
(in other words Et is a nonnally distributed vector white noise)
The Yt process has a dual Vector Moving Average representation (Wold representation)
Yt= C(L)et
C(L) =A(L)-l
C(L)=l +C1L+C2L2+ ...
where C(L) is a matrix polynomial which can be of infinite order and for which we assume
that the multivariate invertibility conditions hold, i.e. det(C(L)) = 0 has all roots outside the
unit circle, so
C(Lf1 =A(L)