Building a Structural Model: Parameterization and Structurality

Mouchart, Michel; Orsi, Renzo Working Paper Building a Structural Model: Parameterization and Structurality Quaderni - Working Paper DSE, No. 1039 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Mouchart, Michel; Orsi, Renzo (2015) : Building a Structural Model: Parameterization and Structurality, Quaderni - Working Paper DSE, No. 1039, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4406 This Version is available at: https://hdl.handle.net/10419/159877 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ ISSN 2282-6483 Building a Structural Model: Parameterization and Structurality Michel Mouchart Renzo Orsi Quaderni - Working Paper DSE N°1039 Building a Structural Model: Parameterization and Structurality∗ Michel MOUCHARTaand Renzo ORSIb aInstitut de Statistique, Biostatistique et Sciences Actuarielles (ISBA) and CORE, UCLouvain, Belgium bDepartment of Economics, University of Bologna, Italy November 27, 2015 Abstract A specific concept of structural model is used as a background for discussing the structurality of its parameterization. Conditions for a structural model to be also causal are examined. Difficulties and pitfalls arising from the parameterization are analyzed. In particular, pitfalls when considering alternative parameterizations of a same model are shown to have lead to ungrounded conclusions in the literature. Discussion of observationally equivalent models related to different economic mechanisms are used to make clear the connection between an economicall meaningful parameterization and an economically meaningful decomposition of a complex model. The design of economic policy is used for drawing some practical implications of the proposed analysis. Keywords: structural model, recursive decomposition, exogeneity, causality, model identification, observationally equivalent models, structural invariance, misleading constraints. JEL Classification: C10, C18, C50, C51, C54 Corresponding Author: Michel Mouchart, ISBA, Université catholique de Louvain, 20 Voie du Roman Pays, B-1348 Louvain-la-Neuve, Belgium. e-mail: [email protected] ∗The work underlying this paper started in 1984 under the heading “ A Fallacious Argument in Econometric modeling” and since that period took advantage of a large number of comments from many colleagues. Incorporating these comments, the present version benefited from particularly useful comments from G. Wunsch and from the participants of the Seminar of the Economic Department of the University of Bologna (I) 1 Contents 1 Introduction 3 2 Econometric Models as Statistical Models 4 3 Recursive decomposition and sub-mechanisms 5 3.1 Recursive decomposition . . . . . . . . . . . . . . . . . . . . 5 3.2 Explanation and structurality . . . . . . . . . . . . . . . . . . 7 3.3 Parsimonious modelling . . . . . . . . . . . . . . . . . . . . . 9 3.4 Causal and Structural Models . . . . . . . . . . . . . . . . . . 9 4 Structural model and parametrization 10 5 Pitfalls with alternative parameterizations 12 5.1 Reparameterizations suggesting erroneous constraints . . . . . 13 5.1.1 A simple pedagogic example . . . . . . . . . . . . . . . 13 5.1.2 A case in simultaneous equations . . . . . . . . . . . . 14 5.2 Reparameterizations involving different but observationally equivalent sub-mechanisms . . . . . . . . . . . . . . . . . . . . 17 6 Concluding remarks 20 6.1 Summarizing: The basic framework . . . . . . . . . . . . . . . 20 6.2 On the use of models for the design of economic policy . . . . 21 References 23 2 1 Introduction In this paper, we develop a concept of structural econometric model through a specific model building strategy and embed the concept of causality within the framework of a suitably constructed structural model, justifying accordingly that a structural model be also called a causal model. We show that the structurality of a model depends both on its probabilistic structure and on its parameterization. We also expose some difficulties, and pitfalls, in the treatment of identification and of reparameterization, particularly when treating observationally equivalent models through different parameterizations. Following Mouchart, Wunsch and Russo (2015), we rely on a specific approach to structural modeling that combines two main econometric traditions. On the one hand, one of these traditions starts from a “theory ” (i.e. economic theory) and develops a structural model from the statistical implications of an economic theory. This approach has been proposed by the Cowles Commission (CC), in particular Koopmans and Havelmoo. On the other hand, another tradition starts from the idea of a “Data Generating Process ” (DGP), representing how the data have been generated: a structural model then looks for a structure underlying the DGP. This approach has been launched by D. Sargan at LSE and further developed by D. Hendry and others. The order of exposition is as follows. In the next three sections, we propose a statistical approach to the concept of structural model by successively examining econometric models as a class of statistical models and the recursive decomposition of a model as a device for providing explanatory power to a model. We also show that the parameterization of a decomposed model is part of the explanatory process. The structurality of the model is related to field knowledge, in particular under some form of economic theory, and to properties of stability, or invariance, with respect to a class of interventions or of changes of the environment. Section 5 presents two major difficulties when treating the parameterization of a structural model. A first one deals with illegitimate constraints when blending two parameterizations of a same model; we show, in particular, that this error has been made repeatedly in the econometric literature. Another difficulty deals with the structural interpretation of the parameterizations of two different economic mechanisms leading nevertheless to observationally equivalent models. The last section takes an helicopter view of the achievements of this paper and points out, in particular, some implications for the design of an economic policy based on an econometric model. 3 2 Econometric Models as Statistical Models In this paper, we approach an econometric model as a particular type of statistical model. Formally, a statistical model Mmay be viewed as a set of probability distributions, explicitly: M={S, P ωω∈Ω}(1) where S, the sample space or observation space, is the range space of an observable random variable (or vector of variables) and for each ω∈Ω,Pωis a probability distribution on the sample space, i.e. the sampling distribution. In other words, ωis a characteristic, or parameter, of the corresponding distribution and Ωdescribes the set of all sampling distributions belonging to the model. The basic idea is that the data are to be analyzed as if they were a realization of one of those distributions. A statistical model can accordingly be viewed as a set of plausible hypotheses regarding the Data Generating Process (for short, DGP); for more detail see Mouchart and Russo (2011), and Wunsch, Mouchart and Russo (2014). A statistical model is based on a stochastic representation of the world. The random component of the model delineates the frontier, or the internal limitation, of the statistical explanation. More explicitly, the randomness represents what is not explained by the model, while the parameters of the distributions are the cornerstone of the statistical explanation. A structural econometric model endeavors at unfolding the structure of an underlying economic mechanism assumedly generating the observed data. In the words of Illari and Williamson (2012): “A mechanism for a phenomenon consists of entities and activities organized in such a way that they are responsible for the phenomenon”. This definition is general enough to be applicable to social contexts too. For economic phenomena, a mechanism may be viewed as a mathematical structure that models choices of economic agents or institutions through which economic activity is guided and coordinated. This mathematical structure provides a representation of a mechanism either in a deterministic form when the model is assumed to provide a complete explanation of a phenomenon or in a probabilistic form when the explanation is considered as an incomplete one. An econometric model, being a statistical one, belongs to the second alternative and takes the form of a conditional distribution where the endogenous variable is the one generated by the mechanism and the conditioning variables are those under which the mechanism is operating. 4 3 Recursive decomposition and sub-mechanisms An econometric model is not always structural. Its aim may be to provide an insightful description of the observed data without the ambition of unfolding the mechanisms and sub-mechanisms underlying the DGP. Thus, many macro-econometric models as well as models of financial econometrics are based on the literature concerning the empirical issue of interest and on some stylized facts that the author aims to interpret. These models are of a descriptive nature, as they do not aim at unfolding a structural mechanism underlying a DGP. 3.1 Recursive decomposition A structural econometric model is not only aimed at providing a stochastic representation of a global mechanism, but should also provide an explanation of that process. Once the model is dealing with a large number of variables, the usual way of explaining a complex process is given by a decomposition of the global mechanism into an ordered sequence of simpler sub-mechanisms. This is the objective of a recursive decomposition of a statistical model. More explicitly, let us consider a partition of the data Xinto pcomponents: X= (X1, X2,· · · Xp). Suppose that the components of Xhave been ordered in such a way that in the complete marginal-conditional decomposition of the joint distribution of X: pX(x|ω) = pXp|X1,X2,···Xp−1(xp|x1, x2,· · · xp−1, θp|1,···p−1)· pXp−1|X1,X2,···Xp−2(xp−1|x1, x2,· · · xp−2, θp−1|1,···p−2)· · · · pXj|X1,X2,···Xj−1(xj|x1, x2,· · · xj−1, θj|1,···j−1)· · · · pX1(x1|θ1)(2) each component of the right hand side is characterized by mutually independent parameters, i.e. variation-free in a sampling theory framework: ω= (θp|1,···p−1, θp−1|1,···p−2· · · , θ1)∈Θp|1,···p−1×Θp−1|1,···p−2· · · × Θ1(3) or a priori independent in a Bayesian framework. When each factor in (2) stands for a univariate conditional distribution, equations (2) and (3) characterize a completely recursive system. When some, or all, factors represent the conditional distribution of a vector of variables, equations (2) and (3) characterize a partially recursive, or block-recursive, system. The right-hand side of equation (2) may be interpreted as an ordered sequence of psub-mechanisms, each one characterized by a distribution generating a variable conditionally on (an increasing set of ) conditioning variables. 5 This is in line with Illari and Williamson (2012) and is compatible with an interpretation of the recursive decomposition in terms of sub-mechanisms in a structural model, as detailed in Wunsch, Mouchart, Russo (2014). Equation (3) requires that there are no restrictions binding the parameters of different factors of (2), in particular that there are no common parameters, and allows one to interpret each factors of the decomposition as independent sub-mechanisms. A reference to the Simultaneous Equations Model (SEM) may be useful. In its standard formulation, the structural form of the SEM may be written as follows. By +Cz =u u ∼ N(0,Σ) u⊥⊥z(4) The condition u⊥⊥zimplies that the equation in (4) is derived from the conditional distribution of (y|z)and the conditional expectation IE (y|z) is given by the reduced form: y= Π z+vwhere Π = −B−1C v =B−1u(5) By +Cz =u(6) y+ (B−I)y+Cz =u(7) y= (I−B)y−Cz +u(8) This model is said recursive when the matrix Bis lower triangular (along with the usual normalization rule of making the elements of the main diagonal equal to 1) and Σis diagonal. Under a normality assumption, this recursive system of equations stands for an ordered sequence of conditional expectations and the errors are mutually independent. In this case, the SEM corresponds to a completely recursive decomposition of the joint distribution of the endogenous variables yconditionally on the exogenous variables zand the global mechanism generating (y|z)is "explained" through an ordered sequence of sub-mechanisms represented by an ordered sequence of conditional univariate distributions. When the SEM is not recursive, equation (4) does not refer to conditional distributions, does not stand for conditional expectations (along with errors) and may be interpreted as referring to "notional" sub-mechanisms in a spirit similar to that of counterfactuals. For instance, consider a two-equations elementary market model, generating price and quantity under a competitive equilibrium. The equation representing the quantity demanded as a function of the price and other exogenous variables, may be interpreted as the notional 6 demand that would operate if the prices were exogenously fixed whereas the econometric model specifies that the actually observed prices, and quantities, have been jointly generated under an equilibrium mechanism. From a causality, or explanation, point of view, equation (5) reflects a situation of block-recursivity: a global mechanism “explains” that vector z“causes” vector ybut no explanation is given about the functioning of that mechanism. Only if model (4) were recursive would one obtain an explanation of the functioning of that global mechanism through a causal ordering in terms of sub-mechanisms. 3.2 Explanation and structurality The recursive decomposition provides a structure for the explanation of a complex system through an ordered sequence of sub-mechanisms that compose the global mechanism. However, conditions (2) and (3) are not sufficient for ensuring that a given recursive decomposition is the valid one among the p!possible recursive decompositions (corresponding to the number of ordered permutations of pvariables). We also require conditions for ensuring that each factor of the product in (2) represents a valid sub-mechanism. The required conditions are twofold. Firstly, each putative sub-mechanism, represented by a specific conditional distribution, should be congruent with field knowledge, i.e. with economic theory, as long as this one may be viewed as an organized synthesis of a large body of out-of-sample observations relative to the phenomenon of interest. Secondly, the putative sub-mechanism should be stable, or invariant, relatively to a large class of interventions or of modifications of the environment. Indeed, a conditional distribution that would be different, say, for each observation could not be deemed to represent an underlying structure neither would it be useful, at least for accumulating empirical information. This refers to the fundamental issue of defining a “population of reference”; indeed neither an economic theory nor an econometric model may reasonably claim to be “universal” in time and in space. The discussion of models such as (5) often concerns the distinction between the use of the model to make causal statements or simply use it to characterize a relationship between the observed data, i.e. a purely descriptive model. If model (5) correctly represents a (block-)causal relationship, it should be invariant under interventions that change any of the independent variables z. Conversely if (5) reveals not to be invariant under a wide range of interventions, it may be appropriate to describe a statistical relationship supported by the data, but it will not be explanatory. Moreover model (5) 7 Figure 1: A same converging sequence in two parameterizations. 5.1.2 A case in simultaneous equations As naive as it may seem, the previous fallacious argument has been met in less elementary situations, when considering different parameterizations of a given model. As an example, let us consider the simplest version of Haavelmo (1947)’s model: ct=δ0+δyt+utwith ut∼IN(0, σuu) yt=ct+ztut⊥⊥zt (16) with reduced form: yt=α0+αzt+vt(17) where α0=δ0(1 −δ)−1,α= (1 −δ)−1,vt= (1 −δ)−1ut. Let us also assume that zt∼IN(µz, σzz)independently of ut(or of vt). The reduced form implies σyy =α2σzz +σvv and therefore α2< σyyσ−1 zz . This (true) inequality appears to have been erroneously interpreted as a constraint on the parameter space. For instance, Genberg (1972) suggests to estimate α under the restriction α2< q, where qwould be a consistent estimator of the variance ratio σyyσ−1 zz and notices that if one were to use a ratio of sample moments to estimate σyyσ−1 zz the restriction α2< q would not be binding when estimating αby ordinary least squares. Remark that Gensberg’s proposal could also be used to assert that Haavelmo’s model implies a (wrong) restriction over σzz, namely σzz < σyyα−2. As a matter of fact, we have two parameterizations (structural form and reduced form) of a same model, namely (µz, σzz, δ0, δ, σuu) = θ, say and 14 (µz, σzz, α0, α, σvv) = λ, say. These parameterizations are in bijection and each is variation-free: there is no constraint among the parameters within a same parameterization. The variance of y,σyy, is a function of these parameters, indifferently of θor of λ, is not a new parameter and introduces no new constraint neither on θnor on λ. In particular, the (true) inequality σyy > α2σzz does not introduce a constraint within the parametrization θ1. In order to ascertain how fallacious these arguments of “constraints often overlooked” (see Zellner, 1972 and Maddala, 1976) are, we consider, in rather obvious notation, two equivalent parameterizations of the set of bivariate normal distributions on (y, z): θ∗= (µy, µz, σyy, σyz, σzz)∈Θ∗=IR2× C(2) (18) λ∗= (µz, α0, α, σvv, σzz)∈Λ∗=IR3×IR2 +(19) where α0=µy−αµz,α=σyzσ−1 zz ,σvv =σyy −σ2 yzσ−1 zz and C(2) is the cone of the (2 ×2) SP DS matrices. It should be pointed out that there is no restriction on the range of α, neither on that of σzz nor on that of σyy. As in the pedagogic example, inequalities like α2< σyyσ−1 zz involve two different parametrizations and represent no restriction on any parameter space. Before qualifying the use of such inequalities as misleading restrictions, it may be illustrative to consider a slightly more general version of the very same example. Let us consider the model: (y|z)∼Nm(Π0 az, Σvv) z∼Nk(0,Σzz)(20) where Παis a (k×m)matrix the elements of which are known functions of a vector of unknown parameters α. This model may also be written as: y z∼Nm+k 0 0,Σyy Π0 αΣzz ΣzzΠα,Σzz  (21) where Σyy = Σvv +Π0 αΣzzΠα. Thus we consider two equivalent parametrizations: θ∗∗ = (Σzz,Σvv, α)∈Θ∗∗ =C(k)× C(m)×A(22) λ∗∗ = (Σzz,Σyy, α)∈Λ∗∗ ={(Σzz,Σyy, α)∈ C(k)× C(m)×A |Σyy −Π0 αΣzzΠα∈ C(m)}(23) 1But if the estimation of σyy introduces new data, then we have a case of mixed estimation, “à-la-Theil”, or of mixed Bayesian estimation and the constraints may have a role in the procedure of blending two sources of data but not for introducing new constraints on the parameter space. 15 where Ais the set of possible values for α, the “free” parameters of the (possibly overidentified) reduced form. Here it is valid to recognize that the parameter space Λ∗∗ is restricted by the condition: Σyy −Π0 αΣzzΠα∈ C(m), i.e. Σyy −Π0 αΣzzΠα(= Σvv)should be SP DS, and thus Σyy,Σzz, α are not variation free . Note however that the relationship Σyy−Σvv ∈ C(m), although true, does not provide any effective restriction as it grounds on two different parametrizations. The danger of overlooking this fact can be illustrated in the particular case where m= 2, α = (β, γ),Πα= (βγ, β) = β(γ, 1) with β∈IRkand γ∈IR. Thus, in this particular case: Σyy = Σvv +β0Σzzβγ2γ γ1 Here it is valid that, in Λ∗∗, parameters α, Σzz and Σyy should be restricted by: Σyy −β0Σzzβγ2γ γ1∈ C(2) or, more explicitly: σy1y1≥γ2β0Σzzβ σy2y2≥β0Σzzβ σy1y1σy2y2−σ2 y1y2≥β0Σzzβ(γ2σy2y2−2γ σy1y2+σy1y1) (Note that one of the first two inequalities is redundant). A fallacious use of the (true) relationship β0Σzzβ=γ−1(σy1y2−σv1v2)would be to assert that, from the above inequalities, γis constrained to satisfy restrictions such as the following ones: σy1y2−σv1v2 σy2y2 ≶γ≶σy1y1 σy1y2−σv1v2 according to : σy1y2≶σv1v2(24) γ2σy2y2(σy1y2−σv1v2)−γ[σy1y1σy2y2−σ2 y1y2+ 2σy1y2(σy1y2−σv1v2)] +σy1y1(σy1y2−σv1v2)≤0(25) In other words, we disagree with the contention that (24) or (25) would mean that “certain parameter values are subject to bounds flowing from usual specifying assumptions”. Indeed inequalities (24) and (25) do not represent restrictions neither on Θ∗∗ nor on Λ∗∗: they only express some properties of the correspondence between Θ∗∗ and Λ∗∗. As a consequence, estimating γ, for instance, under the restriction implied by consistent estimates of the 16 bounds of inequalities (24) would either be ineffective, if a correct parameterization is employed, or would consist of using twice the same sample information if one were to step from one parameterization to another one. This later would be the case when estimating e.g. bounds in one parameterization before estimating the other parameterization without taking due account of the double use of the same sample information. When pooling two sources of information as is done with bayesian methods or in pooling time series and cross-section data or in mixed estimation, the use of two parameterizations may be justifiable. In the above example, while θ∗∗ may be the most natural parameterization for the final inference, it may be that there is available some prior information on, say, Σyy leading to consider also the λ∗∗-parameterization as a natural recipient for that prior information. If inference only concerns parameters common to θ∗∗ and λ∗∗, i.e. αand/or Σzz, one may also start by first incorporating the prior information in the λ∗∗-parameterization and thereafter reparameterize in θ∗∗ before incorporating the second information. Such a stepwise procedure seems to be the only available one in the case of inference on Σvv. It should nevertheless be noticed that even in such a case inequalities (24) or (25) will never appear as restriction on any parameter space. The reader may like to compare this analysis with the debate between Maddala (1976, a and 1976,b) and Zellner (1972, 1976, a and 1976,b) that has been inconclusive for missing the issue of the relationships between alternative parameterizations. 5.2 Reparameterizations involving different but observationally equivalent sub-mechanisms The previous section illustrates possible abuse of restrictions due to a fallacious argument involving alternative parameterizations. In this section, we consider two different structural models that are nevertheless observationally equivalent and, therefore, correspond to two different parameterizations of a same model. In such cases the two parameterizations correspond to two different (sub-)mechanisms possibly suggesting different contextually relevant parametric restrictions. As a matter of fact the price equations of the two models below represent two structurally different mechanisms, and the corresponding parameters capture quite a different economic meaning. “Model 1” (from Bowden, 1978 b ) 17 Consider a simple price adjustment model: Dt=α0 1xD t−α2Pt+u1t(26) St=β0 1xS t+β2Pt+u2t(27) Pt=µPt−1+ (1 −µ)P∗ t+u3t(28) where Dt(demand), St(supply) and Pt(price) are endogenous variables, xD t and xS tare exogenous variables and P∗ tis a “clearing” price defined as: P∗ t= (β2+α2)−1(α0 1xD t−β0 1xS t+u1t−u2t)(29) Therefore, the reduced form of the price equation is: Pt=µPt−1+ (1 −µ)[(β2+α2)−1(α0 1xD t−β0 1xS t+u1t−u2t)] + u3t =µPt−1+ (1 −µ)(β2+α2)−1α0 1xD t−(1 −µ)(β2+α2)−1β0 1xS t +(1 −µ)(β2+α2)−1[u1t−u2t)] + u3t(30) “Model 2” (from Fair and Jaffee, 1972) Consider another model given by equations (26), (27) and the alternative specification of the price adjustment mechanism Pt−Pt−1=η(Dt−St) + u∗∗ 3t(31) equivalently: Pt=Pt−1+η(Dt−St) + u∗∗ 3t(32) =Pt−1+η[α0 1xD t−α2Pt+u1t−(β0 1xS t+β2Pt+u2t)] + u∗∗ 3t = [1 + η(β2+α2)]−1[Pt−1+ηα0 1xD t−ηβ0 1xS t+η(u1t−u2t) + u∗∗ 3t] Therefore, the reduced form of the price equation is: Pt= [1 + η(β2+α2)]−1Pt−1+ [1 + η(β2+α2)]−1ηα0 1xD t −[1 + η(β2+α2)]−1ηβ0 1xS t +[1 + η(β2+α2)]−1[η(u1t−u2t) + u∗∗ 3t](33) Notice that (28) has sense only if 0≤µ≤1whereas in (31) one should require η∈IR+. Therefore the parameter spaces corresponding to Models 1 and 2 are: θ= (α1, β1, α2, β2, µ, Σ) ∈Θ = IRkD+kS×IR2 +×[0,1] × C(3) (34) λ= (α1, β1, α2, β2, η, Σ∗∗)∈Λ = IRkD+kS×IR3 +× C(3) (35) 18 where kDis the number of variables in xD,kSis the number of variables in xS(for the sake of presentation, we assume no common exogenous variable in xSand xD), Σ = V ar(u1, u2, u3)and Σ∗∗ =V ar(u1, u2, u∗∗ 3). The two price equations (28), along with (30) and (31) with (33), represent two different sub-mechanisms. Model 1 and Model 2 are nevertheless observationally equivalent because they provide two different parameterizations of a same set of distributions. Indeed, from the coefficients of Pt−1the mapping between θand λis given by: µ= [1 + η(β2+α2)]−1η=1−µ µ(β2+α2)(36) implying that in equations (34) and (35) we indeed have: µ∈[0,1] corresponding to η∈IR+ Moreover, for the coefficients of (u1t−u2t), we may check that: (1 −µ)(β2+α2)−1= [1 + η(β2+α2)]−1η and conclude that the two models are observationally equivalent under the identification relationship: u3t= [1 + η(β2+α2)]−1u∗∗ 3tu∗∗ 3t=µ−1u3t(37) As a matter of fact, equation ( 37) may be viewed as a short-hand notation for: σj3= [1 + η(β2+α2)]−1σ∗∗ j3σ∗∗ j3=µ−1σj3j= 1,2(38) σ33 = [1 + η(β2+α2)]−2σ∗∗ 33 σ∗∗ 33 =µ−2σ33 (39) Note that µand Σare variation free in Θ; so are also ηand Σ∗∗ in Λ. It may be nevertheless tempting to erroneously conclude from (37) , as in the previous section, that “ µ→0implies σ33 →0because u3t=µu∗∗ 3t”. Once Model 1 and Model 2 are recognized as observationally equivalent, does it imply that the two price sub-mechanisms are also equivalent? The answer is: no! Indeed, equations (28) and (31) explain the disequilibrium of the market by two different price sub-mechanisms : in equation (28) the price sub-mechanism partially adjusts the past price to the current equilibrium price whereas equation (31) partially adjusts the past price in function of the disequilibrium in quantity. Thus the parameters µand ηhave a clearly 19 different economic meaning. In other words, the choice between the specifications of Models 1 and 2 is a choice between two different parameterizations of a same (conditional) distribution and should be based on economic and contextual plausibility and on structural stability. This example illustrates several issues. Firstly, the two models are observationally equivalent: no data may decide in favor of one against the other one. Nevertheless, the two price equations represent two structurally different sub-mechanisms, the corresponding parameters being endowed with quite a different economic meaning: only field knowledge, structural stability (or invariance) or new information may decide which one is actually structural. Secondly, again, one should operate a clear distinction between relationships among alternative parameterizations, bringing no restrictions on the parameter space, and genuine restrictions that might be used either to improve inference, if accepted, or to be subjected to testing, if put in doubt. Thirdly, alternative parameterizations may possibly bring interesting information in an encompassing spirit. For instance, if one is willing to interpret the parameters of Model 1 at the light of Model 2, the relationship (36) would tell that a value of µclose to 0(resp. 1) in Model 1 would correspond to a great (resp. small) value of ηin model 2. But this relationship implies no restriction neither on µnor on η. More recently, An and Schorfheide (2007) produced another example of two observationally equivalent models corresponding to two different parametrizations of a same family of conditional distributions. These two models describe different sub-mechanisms characterized by parameters with a different economic interpretation. Through an implied identification problem, they propose to work out a bayesian solution. This example shows that when specifying a structural model it is not sufficient to specify a family of distributions: a particular parametrization, with a specific economic meaning, should also be specified. 6 Concluding remarks 6.1 Summarizing: The basic framework In this paper, we make explicit the link between causal model and structural model. In particular, the structural explanation of a model is based on a recursive decomposition of the model itself, together with the interpretation of each term of the structural decomposition as an economic sub-mechanism. When we can reach this structuring of the model, the explanatory variables can be interpreted as causal variables. 20 We also propose an approach for building a structural econometric model, i.e. a statistical model that provides (i) an appropriate representation of a global economic mechanism and (ii) an explanation of the working of that global mechanism i.e. each factor of the recursive decomposition should provide a suitable representation of an economically meaningful sub-mechanism. The necessary conditions for a recursive decomposition to be interpreted as a structural model, are: (i) there is congruence with the underlying economic theory (ii) there is invariance or stability of the parameters characterizing the economic sub-mechanisms as well as of the recursive decomposition itself. One limitation of this approach is that the explanatory power of the model relies on its recursive decomposition. A completely recursive decomposition provides a complete causal ordering of the variables. When a completely recursive decomposition is not possible, i.e. the case of blockrecursivity where the recursive decomposition is only partial, there is a simultaneity of the action of several sub-mechanisms within the generation of a block of endogenous variables. In this case, the model cannot claim for causal effects within the block of endogenous variables because outcomes cannot cause each other simultaneously. From a narrow statistical point of view, the parametrization of a family of distributions is arbitrary. In a structural modeling approach the issue is more subtle. Indeed, a basic requirement of structurality is the stability of the recursive decomposition, involving both the stability of the decomposition itself and the stability of the parameters of the different distributions. As a matter of fact, the stability of these characteristics is, in general, not complete. As mentioned in an example of Section 4, some factors of the decomposition may be more stable than others. Therefore the specification of the parametrization should be based on the search of the parameters that are likely to be more stable; this ensures that these parameters are endowed with a reliable economic meaning. This paper also makes explicit some difficulties and pitfalls when handling alternative parameterizations, namely the danger of introducing illegitimate constraints in case of alternative parameterizations of a same model, or selecting among two different sub-mechanisms leading to observationally equivalent models. 6.2 On the use of models for the design of economic policy An intervention, such as an economic policy, should be based on a structural, or causal, model rather than on a descriptive model. More specifically, 21 econometric models used for the design of economic policy should represent actual behavior, be it macroor micro-, rather than provide a representation based on theoretically grounded behavior. This is precisely the meaning of an econometric structural, or causal, model. Thus the model builder should carefully check the structural stability of the model, in particular its resilience to a suitable class of transformations of the environment. Indeed the parameters and the recursive structure of the model should not be thought in a universal sense, in space and in time, and it is crucial to evaluate how the “universe” should be circumscribed. Lucas’ critique may indeed be interpreted as referring to the fact that an intervention may modify the structure of the causal model, because a model is developed within a given environment and the difficulty may be to evaluate to what extent a modification of the environment might modify some properties of the causal model. Thus the strategy for building an econometric model should identify from the apparent instability of the global process the relevant sub-mechanisms and identify the stable aspects of the working of economic mechanisms. 22 References An Sungbae and F. Schorfheide (2007), Bayesian analysis of DSGE Models, Econometric Reviews,26 2-4: 113-172. DOI: 10.1080/07474930701220071 Barndorff-Nielsen O. 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