Econ. 513, Time Series Econometrics Fall 2002 Chris SimsStructural VAR’s∗1. STRUCTURAL VS. BEHAVIORAL MODELSThe original meaning of a “structural” model in econometrics is explained in anarticle by Hurwicz (1962). A model is structural if it allows us to predict the effectof “interventions” — deliberate policy actions, or changes in the economy or in na-ture of known types. To make such a prediction, the model must tell us how theintervention corresponds to changes in some elements of the model (parameters,equations, observable or unobservable random variables), and it must be true thatthe changed model is an accurate characterization of the behavior being modeledafter the intervention.In the traditional simultaneous equations models that Hurwicz had in mind, theintervention was ordinarily taken to correspond to changing the parameters in anequation or block of equations in the model. The simplest conceptual example, cor-responding to the monetary VAR literature, is where one block of equations de-scribes policy behavior and another describes private sector behavior. The model isclaimed to be structural because one set of policy equations can be replaced by an-other, while leaving the private sector equations unchanged, to obtain a predictionabout the behavior of the economy with the new monetary policy.However, there is no need for the intervention to correspond to changing an equa-tion. In a model derived from a general equilibrium, for example, the natural pa-rameters of the model (from utility functions, production functions, policy makers’objective functions) are likely to appear in many equations of the model. Such amodel will claim to be structural relative to changes in at least some of these nat-ural parameters — policy makers’ objective functions, for example. One way todescribe the Lucas critique of econometric policy advice is to say that he pointedout that parameters characterizing monetary policy behavior are likely to appear,via expectations, in many equations of the model, not just in the “policy equations”.Thus an attempt to predict the effects of a policy change by changing only the pol-icy equation, holding other equations in the model fixed, will fail, because the otherequations will in fact change when the policy changes.There is no sharp distinction among interventions that change equations, changeparameters in equations, change disturbance terms in equations, or change the valueof variables in the system. For example, in a monetary policy model there may be areaction function describing monetary policy behavior, sayrt= α + Xtβ + εt, (1)∗Copyright 2002by Christopher A. Sims. This document may be reproduced for educational andresearch purposes, so long as the copies contain this notice and are retained for personal use ordistributed free.12where Xtis a vector of explanatory variables. We might claim that the model isstructural relative to changes in monetary policy, with these changes represented aschanges in the monetary policy equation. Then we might ask what would be the dis-tribution of the variables in the VAR system over the period t0to t0+ s, conditionalon policy setting rtequal to some non-random fixed path¯rtover this period. To doso, we could replace the policy equation (1) with the equation rt=¯rt, or we could re-place the time path of the disturbance to the policy equation with¯εt=¯rt− α − Xtβ,or we could replace the fixed α and β in the equation with a sequence αt, βtsatisfying¯rt= αt+ Xtβtand set εtto zero. These would all deliver exactly the same implica-tions for the behavior of the economy, because all retain the non-policy equations ofthe model unchanged, while fixing rtat the¯rtpath.Nowadays a model is often called “structural” when its parameters have behav-ioral interpretations, regardless of whether the old definition of the term applies,and on the other hand models that are in fact structural in the old sense are thoughtof as “reduced form” because they contain parameters or equations that do not haveunique behavioral interpretations. Monetary policy VAR’s, which single out a policyblock and a non-policy block of equations, are certainly structural in the old sense(or at least claim to be), but because the separate equations in the non-policy blockare often left uninterpreted, they are thought of as non-structural. Some real busi-ness cycle models, in contrast, are specified without explicit variables or equationsrepresenting monetary and fiscal policy, but are nonetheless calibrated to matchsome aspects of the behavior of macroeconomic data. There is no apparent inter-esting intervention with respect to which such models are structural in the originalsense, but because all the parameters in the models have explicit behavioral inter-pretations, they are often referred to as structural.My own preference is to reserve “structural” for its original meaning, and to use“behavioral” to characterize models with complete behavioral interpretations.2. STRUCTURAL VAR’S AND SIMULTANEOUS EQUATIONS MODELS (SEM’S)Both these classes of models can be thought of as versions of the general linearstochastic difference equation modelΓ(L)n×nyt= c + εt, (2)where Γ is a matrix-valued polynomial in positive powers of the lag operator Land Γ0is full rank. The usual structural VAR framework specializes this setup byrequiring that the elements of the εtvector be independent (in the Gaussian casethat Σ = Var(εt) be diagonal). In most of the structural VAR literature it is assumedalso that εtspans the space of the y(t) innovation vector, i.e. that if we multiply thesystem through by Γ−10to arrive atΓ−10Γ(L)yt= B(L)yt= Γ−10c + Γ−10εt= γ + νt, (3)3the result is the autoregressive representation of y, with Γ−10εtthe innovation in yt.The usual SEM framework has two standard forms. In one, the system satisfies aGranger causal ordering, i.e. it can be written as·Γ11(L) Γ12(L)0 Γ22(L)¸·xtzt¸=·εxtεzt¸, (4)with Σ block diagonal conformably with the x, z partition of the y vector. In thiscase the z’s are called exogenous, or strictly exogenous. And in this case there is norequirement that εxtbe an innovation. What is required is only that the full sample’s{εxt}vector be unrelated to (uncorrelated with, or independent of, depending on thecontext) the full sample’s {zt} vector.In the other standard form, only Γ110is block triangular, Σ is block diagonal