A Method for Working with Sign Restrictions in Structural Equation Modelling

AuthorSam Ouliaris,Adrian Pagan
Date01 October 2016
DOIhttp://doi.org/10.1111/obes.12137
Published date01 October 2016
605
©2016 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 78, 5 (2016) 0305–9049
doi: 10.1111/obes.12137
A Method for Working with Sign Restrictions in
Structural Equation Modelling*
Sam Ouliaris† and Adrian Pagan‡,§
Institute for Capacity Development, International Monetary Fund, Washington, DC, USA
(e-mail: sam.ouliaris@gmail.com)
Melbourne Institute for Applied Economic and Social Modelling, University of Melbourne,
Melbourne, Australia (e-mail: adrian.pagan@sydney.edu.au)
§School of Economics, University of Sydney, Sydney, Australia
Abstract
The paper sets out a method for handling sign restrictions in systems of simultaneous
equations which are only partially identified. These sign restrictions might apply to either
structural equation parameters or functions of them such as impulse responses. Initially
a range of values for the unidentified parameters are generated and then the role of sign
restrictions is to narrow the range. It is simple to apply and can be handled in packages
such as EViews and Stata. Examples are givenof how to implement it in a number of cases
where there are both parametric and sign restrictions.
I. Introduction
Consider a system of nstructural equations in an n×1 vector of variables ytwith plags.
This will be written as
A0yt=A1yt1+···+Apytp+B0t,(1)
where tis a n×1 vector of shocks taken to be N(0,In), while "t=B0twill contain the
structural equation shocks. Exogenous variables can be subsumed into ytby appropriate
definitions of A0and B0. In the traditional simultaneous equations format estimation of
the unknown parameters was done by imposing zero restrictions on the Aj. In the exactly
identified Structural Vector Autoregression (SVAR) case A1,…, Apwere left unconstrained
and restrictions were imposed on A0and B0. Defining A=A0and B=B0one gets the form
of SVARs pioneered byAmisano and Giannini (1997) and used in programs such as EViews
and Stata. We refer to this as the (A,B) technology and will use it extensively in this paper.
In most structural model applications A0has been taken to have unity on its diago-
nal (reflecting a normalization) and B0to be a diagonal matrix containing the standard
deviations of the shocks. In such a form, there are n2unknown elements in A0and B0.
*We are grateful to Francesco Zanetti and three referees for their comments on a previous version of the paper.
The second author’swork was supported by the ARC Grant DP160102654.
JEL Classification numbers: C32, C36, C51
606 Bulletin
Accordingly, to estimate these parameters requires some moment conditions. n(n+1)
2of
these are available using the fact that the shocks "t=B0tare uncorrelated. This leaves
n2n(n+1)
2=n(n1)
2‘free’ parameters whose values need to be found by some method. Ar-
ranging these parameters in a n(n1)
2×1 vector , this vector is estimated in SVAR work by
imposing n(n1)
2restrictions on A0, e.g. Sims (1980) made A0triangular.
There have of course been other suggestions for setting these n(n1)
2restrictions. One of
these involving A0and the Aj(j>0)is the long-r un restriction employed by Blanchard and
Quah (1989). However, in many instances, one might only have Kplausible parametric
restrictions, where Kn(n1)
2. In those situations, it is only possible to estimate Kof the
unknown parameters, leaving n(n1)
2Kthat are unidentified. Consequently, might be
divided into a K×1 vector 1of estimable parameters leaving2to capture the remainder,
namely those that are unidentified.
In this paper, we suggest that values for the parameters 2be generated by some
procedure, following which 1can be estimated. This will be done conditional upon the
generated values of 2and by using the Krestrictions on Aj. Clearly 1will not be unique
and there will be a range of values for it.To discriminate between these, one might use some
extra information. A popular example of this – often described as ‘agnostic’– has been the
use of sign information. This might come directlyfrom the 1and 2parameters themselves
but mostly it has been about some functions of them, such as impulse responses. This will
narrow the range, although it will rarely make it possible to get a unique set of valuesfor 1.
Nevertheless, sign restrictions on impulse responses in SVARs have become a very popular
way of doing empirical work.They do not provide a single set of impulse responses but a
range of possible outcomes. There are manyapplications of this methodology, e.g. Canova
and De Nicol´o (2002), Elekdag and Han (2015), Faust (1998), J¨askel¨a and Jennings
(2011), Uhlig (2005), Mumtaz and Zanetti (2012), and Stˆang˘a (2014).
Section II sets out our method for finding a range of values for 1and 2(and hence
statistics that might be constructed from them) in the context of a simple demand and
supply system that we term the market model. In section III, the ideas are applied to larger
systems, along with a number of types of parametric restrictions, i.e. K=0. Our strategy
will be to formulate the system so as to be able to use the (A,B) technology. Some of
the parameters in Aj,B0can be estimated using parametric restrictions, namely 1,but
there will be others that cannot (2), and we therefore propose that they be generated by
some mechanism. The use of the (A,B) technology means that our method can mostly be
implemented with programs such as EViews and Stata. Our method will be designated as
SRC – sign restrictions with generated coefficients.
Section IV considers how our method relates to the standard (conventional) sign
restriction approach used in the SVAR literature. This was begun by Faust (1998), Canova
and De Nicol´o (2002) and Uhlig (2005), but has recently been given a general treatment in
Arias, Rubio-Ram´
irez and Waggoner (2014). Because the examples of section III mostly
involved signs of impulse responses, we can ask what the advantage of our method would
be over the method proposed inArias et al. (2014). It is argued that our method is relatively
easy to implement because of its use of existing software such as EViews and Stata; it
can be applied in a wider context than SVARs; and is possibly more transparent, which
©2016 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd

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