Abstract
Despite the fact that it provides a potentially useful analytical tool, allowing for the joint
modeling of dynamic interdependencies within a group of connected areas, until lately the
VAR approach had received little attention in regional science and spatial economic analysis.
This paper aims to contribute in this field by dealing with the issues of parameter
identification and estimation and of structural impulse response analysis. In particular, there is
a discussion of the adaptation of the recursive identification scheme (which represents one of
the more common approaches in the time series VAR literature) to a space-time environment.
Parameter estimation is subsequently based on the Full Information Maximum Likelihood
(FIML) method, a standard approach in structural VAR analysis. As a convenient tool to
summarize the information conveyed by regional dynamic multipliers with a specific
emphasis on the scope of spatial spillover effects, a synthetic space-time impulse response
function (STIR) is introduced, portraying average effects as a function of displacement in
time and space. Asymptotic confidence bands for the STIR estimates are also derived from
bootstrap estimates of the standard errors. Finally, to provide a basic illustration of the
methodology, the paper presents an application of a simple bivariate fiscal model fitted to data
for Italian NUTS 2 regions.
1 Introduction1
Starting from the seminal article by Sims (1980), the vector autoregressive
(VAR) methodology has been applied to a vast range of empirical topics,
including monetary and fiscal policy analysis and short-term economic
forecasting.
Also in the fields of regional science and spatial economics the scope of
issues that could be addressed by means of properly identified structural VARs
appears to be wide and includes: the analysis of the regional propagation of
demand shocks via trade linkages; the assessment of long-run spatial spillover
effects from local public expenditure to private sector performance; and the study
of dynamic knowledge externalities linking patenting activity in the business
sector to academic research in nearby areas.
However, despite the fact that the VAR approach provides a potentially
useful analytical tool allowing for the joint modeling of dynamic
interdependencies within a group of connected areas, until lately it has received
little attention in the applied spatial economics literature.
This is mainly due to the overparameterization problem encountered when a
direct transposition of the standard VAR approach is attempted by simply setting
up a system that involves an equation for each endogenous variable and each
region in the sample.
At the same time, the identification of structural impulse responses appears to
pose specific difficulties, requiring the introduction of correct hypotheses if the
bilateral nature of most economic linkages in space is to be properly accounted
for.
In this paper, in line with the approach set forth in a number of previous
contributions, the inherent overparameterization problem denoting the multi-area
VAR model is addressed by imposing a priori restrictions on parameter values,
stemming from hypotheses on the spatial decay of interactions across economic
agents derived from the spatial econometrics literature.
The main methodological insight lies in the approach to structural parameters
identification, where a block triangular scheme is introduced and motivated as a
plausible extension to a space-time context of the recursive scheme widely
adopted in the empirical time series VAR literature. Apart from structural
parameter identification, some new results are also derived in the fields of
parameter estimation and of impulse response analysis.
The remainder of the paper is structured as follows. Section 2 briefly reviews
the related literature. Model specification and identification issues are then
discussed in Section 3. Parameter estimation is dealt with in Section 4. Under the
assumptions that the number of locations considered is fixed and a sufficient
number of observations is collected over time, estimation is based on the Full
Information Maximum Likelihood (FIML) method, a standard choice in structural
VAR analysis.
1 I wish to thank Juri Marcucci, Jesús Mur, the participants at the III Jean Paelink Seminar hosted
by Universidad Politécnica de Cartagena and three anonymous referees for their useful comments
and suggestions. The views expressed in this paper are my own and do not necessarily reflect those
of the Bank of Italy.
6
The topic of impulse response analysis is dealt with in Section 5, where a
synthetic space-time impulse response (STIR) function is introduced as a
convenient tool to summarize the information conveyed by individual regional
dynamic multipliers. Asymptotic confidence bands for the STIR estimates are also
derived from bootstrap estimates of the respective standard errors.
To provide a basic illustration of the methodology and an initial test of the
model's empirical performance, the application of a simple bivariate fiscal model
estimated on the set of Italian NUTS 2 regions is carried out in Section 6. Lastly,
Section 7 summarizes and concludes the paper.
2 Literature review
In this section a few previous contributions extending the VAR methodology
to multi-area panel data are briefly reviewed.
Ruling out cross-sectional interactions and assuming fixed coefficients across
areas yields a simple panel VAR specification that has received some attention in
the panel data literature (Holtz-Eakin et al. 1988), but it is of little interest in
regional economic analysis since cross-sectional interdependence is precluded.
In a multi-country set-up, cross-section interactions were more recently dealt
with by Pesaran et al. (2004), who introduce the Global VAR specification, where
information on trade shares across countries is utilized to specify the channels of
transmission of national disturbances across the world economy.
In an intra-national context, Carlino and DeFina (1995) provide a
straightforward implementation of the original Sims approach, by fitting a VAR
model involving a single endogenous variable (GNP) to the six BEA regions in
the US. In this case the limited number of areas (6 regions) and short lag order of
the model allows the authors to estimate an unrestricted reduced form. VAR
specification. They are also among the first to employ impulse response analysis
based on VAR estimation to measure the strength of spatial spillover effects
across regions. However the identification of structural shocks hinges on the
assumption of no contemporaneous spillover effects, a hypothesis that can be
overly restrictive in many empirical settings.
Space-time impulse response analysis is also dealt with by Di Giacinto
(2006), who implements a VAR approach based on an underlying univariate
STARMA (Space-Time ARMA) specification. In this article, a priori information
on spatial contiguity is utilized both to place reasonable restrictions on VAR
coefficients matrices and to identify structural impulse responses.
Two previous contributions by Lesage and Pan (1995) and Lesage and
Krivelyova (2002) introduced information on spatial contiguity to specify the
prior distribution of VAR coefficients in a Bayesian univariate regional VAR
analysis. However the authors do not deal with the topic of structural form.
identification, as the methodology is mainly aimed at improving the out–of-
sample forecasting precision of standard Bayesian VAR models in a spatio-
temporal context.
Remaining within a Bayesian setting, Canova and Ciccarelli (2006) have
recently proposed a multi-country panel VAR specification that allows for cross-
sectional interdependence in a general framework, solving the incidental
parameter problem by imposing standard (i.e. non spatial) prior distributional
assumptions. While this specification is potentially appealing in a regional context