SVAR

COMPSTAT’2004Symposium c Physica-Verlag/Springer2004 FINDING DIRECTED ACYCLIC GRAPHS FOR VECTOR AUTOREGRESSIONS

Oxley L.,Reale M.and Tunnicli?e Wilson G.

Key words:Graphical models,structural VAR,causality.

COMPSTAT2004section:Time series analysis.

Abstract:In this paper graphical modelling is used to select a sparse struc-ture for a multivariate time series model of New Zealand interest rates.In particular,we consider a recursive structural vector autoregressions that can subsequently be described parsimoniously by a directed acyclic graph,which could be given a causal interpretation.A comparison between competing models is then made by considering likelihood and possibly economic theory.

1Introduction

Technology has impacted extensively on the operations of?nancial markets which are inhabited by a rich array of?xed-income securities,each bearing a particular rate of interest.The relationship between the yields on these vari-

ous securities is the province of the term structure of interest rates literature which has a long history and can be traced-back formally to Keynes.

With the popularity of cointegration and VAR/SVAR approaches to es-timation in econometrics,a separate literature using these approaches to estimate and test term structure models and implications has developed.

Here the papers are typically motivated by a concern to understand the term structure for the related monetary policy control issues and focus either upon technical estimation issues and often the validity of inferences derived including,importantly,causal inference,the e?ects of structural change or

the testing of various hypotheses.Causality is a particularly important and popular issue given the role of monetary policy intervention.

In this paper we wish to add a signi?cant extra dimension to the debate by using graphical modelling to identify causal mechanisms within multivariate time series models.

This paper considers an application to the term structure of interest rates where little consensus seems to exist on the causal nexus and direction be-tween long and short rates of interest.In particular,there are three alterna-

tive views on causality;short rates cause long rates(broadly the traditional Expectations Hypothesis view);long rates cause short rates(here rational

in?ation expectations have a role);or the market segmentation,or preferred habitat approaches,where causality is discontinuous across maturity periods. The outcome in an empirical sense will be crucial for the e?cacy of monetary policy design and implementation.In Section2of the paper the Graphical Modelling(GM)approach will be outlined as it is a relatively new statis-tical approach.Section3will identify the relationship between the acyclic

2Oxley L.,Reale M.and Tunnicli?e Wilson G. graph and more traditional multivariate structural VAR models.Section4 will present the empirical example which relates to New Zealand interest rate data.The results from the research presented here can be used to assess(ex-post)empirical support for the choices made and as an example of how the GM technique can be used in practice.

2Graphical modelling

Graphical modelling(GM)is a relatively new statistical approach,whose initial ideas were proposed by Dempster[5]and later devoleped by Darroch et al.[4].The major attraction of the approach in empirical research is its ability to provide a convenient way to present pairwise relationships between random variables taken from a multivariate context.

The initial step in the approach is the computation of the partial cor-relations between the variables in the particular multivariate system under study.Once the numerical values are known we can test their signi?cance by using an opportune statistic.Finally the results are presented as a graph, where the random variables are represented by nodes and a signi?cant par-tial correlation between two random variables is denoted by a line that links them named edge.If the variables in the graph are jointly distributed as a multivariate Gaussian distribution,a signi?cant partial correlation implies the presence of conditional dependence.For this reason the graph is called a conditional independence graph or(CIG).

A more informative object in GM is the directed acyclic graph(DAG). This is a directed graph where there are arrows linking the nodes and where the joint distribution of the variables can be expressed as a sequence of marginal conditional distributions.

Although the DAG and the CIG represent a di?erent de?nition of the joint probability,there is a correspondence between the two which is embodied by the moralization rule(Lauritzen and Spiegelhalter[10]):because of this result we can obtain the CIG from the DAG by transforming the arrows into lines and linking unlinked parents with moral edges.

While the CIG represents the associations among the variables either in terms of conditional dependence or simply in terms of partial correlation,the DAG has a natural interpretation in terms of causality.As it is not the aim of this paper to enter into a philosophical,we just refer to some of the main contributions on the causality implied by directed acyclic graphs:Lauritzen [8],Pearl[15],Spirtes et al.[16],Lauritzen and Richardson[9].

The DAG is a very attractive because of its causal interpretation but in practice all we can observe is the CIG implied by the sample partial corre-lations.In order to obtain the DAG from the CIG we have to apply the inverse operation of the moralization,we name it demoralization.Unfortu-nately while the transformation of a DAG into a CIG is unique,the inverse operation of identi?cation and removal of moral edges is not.To this end we need to use all the information we have about the relationships among the

Finding DAG’s for VAR’s3 random variables in the system.

In this paper we apply this process within the context of multivariate structural VAR models considering?rst its saturated speci?cation,where there are links between every pair of variables(including the contemporane-ous variables),with the aim of?nding a parsimonious form.As an illustration we consider the application to the transmission mechanism between interest rates in New Zealand.

3The multivariate time series context

The relationship between several autoregressions can be modelled via the vector autoregression

x t=c+Φ1x t?1+Φ2x t?2+...+Φk x t?k+e t(1) of order k,VAR(k),where x t,x t?1,...,x t?k are n-dimensional vectors with the corresponding coe?cient vectorsΦ1,Φ2,...,Φk,c is the constant and e t is the error vector,which is assumed IID.If the covariance matrix,H,of e t is not diagonal,the set of linear equations(1)corresponds to a system of seemingly unrelated regressions(Zellner[18])where the relations among the components of x t are hidden in H.To highlight such relations we can represent the canonical VAR(k)in(1)in its structural form(SVAR):

Θ0x t=d+Θ1x t?1+Θ2x t?2+...+Θk x t?k+u t(2) whereΘi=Θ0Φi for i=0,...,k,d=Θ0c and u t=Θ0e t with covariance matrixΘ0HΘ 0=D,which is diagonal.

If there are no zeros in the coe?cient vectors,the SVAR is saturated,but in many cases some lagged variables on the RHS in(2)do not play any role in explaining the current variables,x t.In this case the value of the corre-sponding coe?cient is zero and hence the SVAR is sparse.An examination of the covariance matrix of the variables involved,both current and lagged,can assist in identifying the sparse structure by the computation of the partial correlations.Their signi?cance can be tested using the appropriate sampling properties(Reale and Tunnicli?e Wilson[13][14]).The model(2)may be represented by a directed acyclic graph(DAG)in which the components of x t, x t?1,...,x t?p form the nodes,and causal dependence is indicated by arrows linking nodes.The nature of the model is that all arrows end in nodes rep-resenting the contemporaneous variables on the left hand side of(2).Some arrows will start from past values,and some from other contemporaneous variables.

The coe?cients can be estimated by single equation ordinary least squares (OLS)regression which is fully e?cient under the assumption that the vector series is Gaussian but is also applicable and the properties of the estimates reliable,under wider conditions,such as e t being I.I.D.

Next consider the exploratory tools used to identify the model.The?rst step is to identify the overall order p of a VAR model for the series.The

4Oxley L.,Reale M.and Tunnicli?e Wilson G.second and central step is to construct a sample conditional independence graph (CIG)for the variables x t ,x t ?1,...,x t ?p which form the nodes of the graph.At this stage the only causality we can assume is the one indicated by the arrow of time.Nevertheless,it may serve well to suggest the direc-tion of dependence between contemporaneous variables.The corresponding structural VAR models are then ?tted and re?ned by regression and a model selection criterion such as AIC,Akaike [1],used to select the best in terms of likelihood.

The statistical procedures are based on a data matrix X which in the general case consists of m (P +1)vectors of length n =N ?P ,composed of elements x i,t ?u ,t =P +1?u,...N ?u ,for each series i =1,2,...,m ,and each lag u =0,1,...,P ,for some chosen maximum lag P .In the ?rst stage of overall order selection,for each order p we ?t,by OLS,the saturated structural VAR regressions of the m contemporaneous (lag 0)vectors on all the vectors up to lag p .Using the sums of squares S i from these regressions we form the AIC as n log S i +2k ,where k =pm 2+m (m ?1)/2is the total number of regression coe?cients estimated in the regressions.For the saturated model the causal order of the contemporaneous variables does not a?ect the result,each one is included only as a regression variable for a subsequent variable in the chosen ordering.Then select the order p which minimizes the AIC.

The next step is to construct the sample CIG for the chosen model order p .In general a CIG is an undirected graph,de?ned by the absence of a link between two nodes if they are independent,conditional upon all the remaining variables.Otherwise the nodes are linked.In a Gaussian context this conditional independence is indicated by a zero partial autocorrelation:

ρ(x i,t ?u ,x j,t ?v |{x k,t ?w })=0,(3)

where the set of conditioning variables on the right is the whole set up to lag p ,excluding the variables on the left.

The set of all such partial correlations required to construct the CIG is conveniently calculated from the inverse W ,of the covariance matrix V of the whole set of variables,as

ρ(x i,t ?u ,x j,t ?v |{x k,t ?w })=?W rs / W rr W ss )(4)

where r and s respectively index the lagged variables x i,t ?u and x j,t ?v in the matrices V and W .

In the wider linear least squares context,de?ning linear partial autocor-relations as the same function of linear unconditional correlations as in the Gaussian context,the absence of a link still usefully indicates a lack of linear predictability of one variable by the other given the inclusion of all remaining variables.

To estimate the CIG we replace V with the sample covariance matrix ?V formed from the data matrix X ,but including only lags up to p .From

Finding DAG’s for VAR’s 5here we need a statistical test to decide which links are absent in the graph.We are only concerned with links between contemporaneous variables and between contemporaneous and lagged variables,because these are the only ones that appear in the structural model DAG.The test we use is to retain a link when |ρ|>z/ (z 2+ν))≈z/√n ?p ,where z is an appropriate critical value of the standard normal distribution.This derives from two results.The ?rst is the standard,algebraic,relationship between a sample partial correlation ?ρand a regression t value given by ?ρ=t/ (t 2+ν)(see Greene

[6]p 180).The second is the asymptotic normal distribution of the t value for time series regression coe?cients,given for example by Anderson [2](p211).Generally,we might wish to apply multiple testing procedures when applying the test simultaneously to all sample partial autocorrelations,but that is not a practical option.Here we follow the arguments of Box and Jenkins

[3]in the identi?cation of autoregressive models using time series partial autocorrelations.The application of GM to VAR systems has been extended by demonstrating that the sampling properties of GM’s for stationary VAR’s are still valid for for I(1)VAR processes (Tunnicli?e Wilson and Reale [17]).We then specify the DAG’s as recursive equation systems which can be estimated by ordinary least squares.

The next stage in the process is to establish which DAG representations are consistent with the CIG or are nearly so,allowing for statistical uncer-tainty,considering demoralization .

As we mentioned above by this term we mean the inverse operation of moralization which allows to construct a CIG from a given DAG by inserting an undirected link between any two nodes a and b when there is another node c with incoming directed edges a →c and b →c .In this case c is known as a common child of a and b ,and the insertion of a new,moral,link will marry the parents.After this operation for the whole graph,the directions are removed from the original links.

Of course we attach the arrow of time to links from the past to the present,so the challenge is to clarify the directions of the recursive ordering of con-temporaneous variables.Normally there are alternative competitive models and eventually we compare them by using likelihood based methods.

4Identifying an interest rate transmission for New Zealand We apply the methodology explained in the previous sections to the interest rate mechanism in New Zealand after the implementation of the Reserve Bank Act in February 1990.To this aim we consider the model proposed by Oxley

[11]who identi?ed a structural VAR using standard procedures.The paper by Oxley also provides a thorough discussion of the economic background for the interested reader.

The application is of interest to the economists as the issues involved for this New Zealand case are multi-faceted and involves the presence of inderect e?ects.

6Oxley L.,Reale M.and Tunnicli?e Wilson G.

The data used are monthly,seasonally unadjusted interest rates taken from the Reserve Bank of New Zealand Financial Statistics database for the period February1990-April2002.The individual series considered are the rates on money at call(denoted A);90day bank bills(B);the yield on1,3 and5year Government stock(C,D and E respectively);base lending rate (F)?rst mortgage housing rate(G)and the uncovered interest parity with the US(H).

We identi?ed a VAR(2)and hence cansidered all the variables up to the second lag.Once the sample partial correlation matrix was computed we tested with the appropriate procedures explained above the signi?cance of its elements and constructed the the CIG in?gure1.

We then considered all the models consistant with the CIG and used subset regression to eliminate the moral links.

The?nal step was to used likelihood based measures to compare the di?er-ent models.In particular we considered the Akaike information criterion,the Schwarz information criterion[15](SIC)and the Hannan-Quinn information criterion[7](HIC).

A0B0C0D0E0F0G0H0 A1B1C1D1E1F1G1H1

A2B2C2D2E2F2G2H2

Figure1:Conditional independence graph.

Here we present the two best models together with a table providing their values in terms of parameters,deviance and the di?erent information criteria. They are represented in?gure2and3.

For both of them we can observe some common features as the lack of rel-evance of the uncovered interest parity and the central role of the90day bank bills interest rate.This application although interesting for the economist is meant as an example and so we will not make here more economic consider-ations.

We just conclude by saying that this methodology can be very useful for the applied scientist as it allows for any prior information when considering possible alternative DAG’s.

Finding DAG’s for VAR’s7 A2B2C2D2E2F2G2H2 A1B1C1D1E1F1G1H1

A0B0C0D0E0F0G0H0

Figure2:Best model.

A2B2C2D2E2F2G2H2 A1B1C1D1E1F1G1H1

A0B0C0D0E0F0G0H0

Figure3:Alternative model.

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Model k Dev AIC HIC SIC

Best42130.15-97.85-230.68-424.75

Alternative37141.13-96.87-235.53-438.11

Table1:Information criteria.

8Oxley L.,Reale M.and Tunnicli?e Wilson G.

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Address:Economics Department,University of Canterbury,Private Bag 4800,Christchurch,New Zealand;Mathematics and Statistics Department, University of Canterbury,Private Bag4800,Christchurch,New Zealand; Mathematics and Statistics Department,Lancaster University,Lancaster LA14YF,UK.

E-mail:les.oxley@https://www.wendangku.net/doc/9c2182724.html,;marco.reale@https://www.wendangku.net/doc/9c2182724.html,;

g.tunnicliffe-wilson@https://www.wendangku.net/doc/9c2182724.html,