Bridging Economic Theory Models and the Cointegrated Vector Autoregressive Model

Examples of simple economic theory models are analyzed as restrictions on the Cointegrated VAR (CVAR). This establishes a correspondence between basic economic concepts and the econometric concepts of the CVAR: The economic relations correspond to cointegrating vectors and exogeneity in the economic model is related to econometric concepts of exogeneity. The economic equilibrium corresponds to the so-called long-run value (Johansen 2005), the long-run impact matrix, C; captures the comparative statics and the exogenous variables are the common trends. The adjustment parameters of the CVAR are related to expectations formation, market clearing, nominal rigidities, etc. Finally, the general-partial equilibrium distinction is analyzed. --

are found in Section 5, while Section 6 concludes.

Basic Concepts of Economic Theory Models
The purpose of most economic models is to explain a set of variables, the endogenous variables, as a function of exogenous variables 1 . How the latter are generated is, by construction of the model, not explained. In the present paper exogeneity is referred to as "economic exogeneity", to distinguish it from the econometric concepts of weakand strong exogeneity (Engle, Hendry, and Richard 1983): A variable is economically exogenous if it is not in ‡uenced at any point in time by any other variable in the system under study, including other exogenous variables.
The economic model contains behavioral relations for the endogenous variables. These may be plans contingent on either observed outcomes, or expectations, and may often be regarded as solutions to optimization problems.
An equilibrium condition is imposed to secure a solution with consistency between the plans of di¤erent agents, and hence with no inherent tendency for the system to change. If existent, this solution de…nes the economic equilibrium. I shall consider two types of economic equilibrium conditions: One for the static models stating that demand equals supply, the other for the dynamic models, stating that expectations are correct (see below) 2 . The usual motivation for both types is the hypothetical state in which there have been no shocks to any of the variables for a long time, and all previous shocks have propagated fully. In such a state it seems natural that demand and supply will be equal, since otherwise incentives to change variables (e.g. prices) would remain. Likewise, it would also seem natural that expectations will become correct (expected -and actual values coincide), since sustained expectational errors seem implausible in this static state.
To derive positive statements from theory models, the comparative static analysis is conducted: the study of the e¤ects on the endogenous variables in economic equilibrium from hypothetical changes in the exogenous variables (Samuelson 1941).
A basic example, building on the above concepts is the static supply-and demand model, Q d = a 0 a 1 P + a 2 W; (1) where Q d and Q s are, respectively, demanded and supplied quantity, P; the price level, W; wage income and Z, the price of an input used in the production of Q: All parameters are positive, and all variables are in logarithms. The endogenous variables are Q d , Q s and P; while W and Z are economically exogenous. The equations (1) and (2), de…ne the two behavioral relations, and (3) is the equilibrium condition. The economic equilibrium is, with comparative static e¤ects, @Q @W = b 1 a 2 a 1 + b 1 ; @Q @Z = a 1 b 2 a 1 + b 1 ; @P @W = a 2 a 1 + b 1 ; @P @Z = b 2 a 1 + b 1 : The model is illustrated as the famous economic cross in Figure 1, where D(P; W ) and S(P; Z) denote the demand-and supply curves respectively. P is on the vertical axis and Q on the horizontal, following the convention in economics.
As is well-known, in order for the static model and its comparative statics to have any empirical relevance, equilibria must be attainable (stable), and the adjustment towards them cannot be too slow (see e.g. Chiang and Wainwright 2005). As these are dynamic properties, the static model can only implicitly assume that this is the case rather than explain it. This major limitation of static models in general, motivates the introduction of dynamic theory models, and I consider two examples in Section 4.3.
The purpose is now to suggest a set of restrictions on a VAR, consistent with simple theory models like the above, when data are persistent. However, …rst, a precise notion of persistence and the econometric tools of the CVAR analysis are needed.

The Persistence of Macroeconomic Data and the CVAR
The type of economic theory models under study can be written as sub-models of the general linear p -dimensional model, Ax t = B 1 x t 1 + ::::B k x t k + B 0 D t + u t ; where D t is a d 1 term of d deterministic components, the initial values, x 1 k ; ::::x 0 ; are …xed, A has full rank and ones on the diagonal, u t i:i:N (0; ); with > 0 and diagonal, and B i are unrestricted 3 . The corresponding reduced form VAR(k) model is, x t = 1 x t 1 + :::: k x t k + D t + " t ; with " t A 1 u t ; A 1 B 0 ; and i = A 1 B i for i = 1; ::; k: This can be reparameterized in the Error-(or Equilibrium-) Correction-Mechanism form (ECM) as, The dynamic properties are summarized in the roots, z; of the characteristic equation corresponding to (8), det(A(z)) = 0; where, In practice, we typically have a relatively short sample of time series which gives a set of estimated roots, b z, all with jb zj > 1, but some close to 1, and where j j denotes the modulus. I refer to such time series as being persistent. To conduct inference, assumptions about the underlying Data Generation Process (DGP) are needed, so that asymptotic distributions can be used as approximations of the unknown …nite sample distributions of estimators and statistics. In this case, the choice is between assuming that all roots have jzj > 1; or, that some are at 1 while the rest have jzj > 1. Under the …rst assumption asymptotics are standard Gaussian based. However, when some roots are close to 1, as suggested by the estimates, the asymptotic distributions will be poor approximations for typical sample lengths, implying unreliable inference (See e.g. Johansen 2006). From a statistical inferential point of view, it is then probably more useful to impose z = 1 for some roots, as an approxi-mation, and use the corresponding asymptotic inference theory for cointegrated I(1) processes described in Johansen (1996), cf. the second assumption.
However, though a useful statistical approximation, this unit root restriction may, or may not, contradict the economic model. We can distinguish between three cases: First, if the economic theory predicts unit roots, we of course impose them and continue the analysis, to …nd out whether these are generated in the manner according to the theory.
Second, the theory model may instead involve a steady state, implying a stationary VAR model. Given the persistence and the sample at hand, it is however, not possible to conduct inference on the steady state relations and multipliers to a satisfactory extent. As a result, the price of valid inference is that we are forced to give up the stationarity assumption of the model, hopefully in order to learn about other assumptions of the model (Møller and Sharp 2008). In this case, one would not necessarily claim that the evidence is inconsistent with the stationary "theory-VAR", but simply that inference under stationarity is unreliable.
Third, it may also be the case that the assumption of stationarity or nonstationarity -here the order of integration -is not implied by the theory model. This is the case for the static model and the simple examples of dynamic models considered here (see the discussion however): The level of the economically exogenous variables cause the level of the endogenous variables, that adjust passively. As a result, persistence in the system variables must originate from the generation of the former. As mentioned in Section 2, this is outside the theory model, implying that imposing z = 1; i.e. estimating a CVAR, is not contradicting the theory model, and since it delivers better inference, it may be the obvious thing to do (see Section 4.1).
Whichever of the three cases, it follows from (9) and (10), that imposing a root at 1, means det(A(1)) = det( ) = 0; and therefore imposing reduced rank on , which can be parameterized as, where the matrices and are p r; of rank r; being the adjustment coe¢ cients, and ; the r cointegrating vectors. The model (8), under the restriction on in (11), but otherwise unrestricted parameters, including and , and r < p; is thus a sub-model of the VAR, and is called a cointegrated I(1) model, denoted by H(r). The theory models below are viewed as sub-models of H(r).
As alluded to above, a relevant assumption about the DGP is that, The roots of (9) have jzj > 1 or z = 1: Under (12) and, where ? and ? are the orthogonal complements, the I(1) model can be represented www.economics-ejournal.org in Moving Average (MA) form, where C ? ( 0 ? ? ) 1 0 ? is the long-run impact matrix, C(L); a convergent lag polynomial, and C 0 depends on initial values, with 0 C 0 = 0 (Theorem 4.2, Johansen 1996).
The long-run movement of the series is described by the p r dimensional vector of common (stochastic) trends, (CT t ), given by, Usually, the stochastic trend, C t i=1 " i ; is decomposed into CT t ; and the so-called loadings matrix, given by, describing how each of the p r common trends a¤ect the individual variables. Below, the focus is primarily on the VAR with one lag. This keeps the analysis simple while still illustrating the main points clearly. More importantly, the VAR(1) has a particular status since any VAR(k) can be rewritten as a VAR(1), using the companion form.
I shall also assume that the deterministic term, D t ; is a constant term, which is restricted, so that it does not produce a trend in the series. Again, this is to keep it simple and generalizing deterministics (trends, indicator variables etc.), does not a¤ect the conclusions, but merely blurs the illustrations. Hence, I assume that, in (8), where s is r 1: The resulting CVAR(1), used repeatedly below, can be written as, In a VAR(1), = I; and the condition in (13) reduces to, and the MA representation becomes, with C = ? ( 0 ? ? ) 1 0 ? ; C i = ( 0 ) 1 (I r + 0 ) i 0 . Under the assumption that r( ) = r < p, the assumptions, (12), and (19) together, are equivalent to A(z) having exactly p r roots at z = 1; while the rest have jzj > 1: Either of these equivalent conditions imply that the eigenvalues of the www.economics-ejournal.org matrix I r + 0 ; all have modulus less than 1, or equivalently that, where ( ) is the spectral radius, which in turn implies that 0 has full rank, r: From these assumptions one can then establish the identity, which can be used to derive (20).
Given (20), the Impulse Response Function (IRF) is, where C h ! 0 follows from (21). This coincides with the structural IRF, when A = I in (6), as in Section 4.1 .The structural IRF is the one of economic interest, as it captures the economically interpretable dynamic response to a given isolated (i.e., statically and dynamically uncorrelated) shock. For A 6 = I; it is based on (20) with e C = CA 1 instead of C; e C i = C i A 1 instead of C i ; and u t instead of " t , and is therefore given by e C + e C h : The so-called attractor set, for the VAR(1), is usually de…ned as, Finally, two concepts of econometric exogeneity are needed, weak-and strong exogeneity, (See Engle, Hendry, and Richard 1983). They are both de…ned with respect to the parameters of interest, which is in this context. These concepts are usually discussed in connection with e¢ cient estimation and forecasting from partial models respectively (Ericsson, Hendry, and Mizon 1998). Here, the focus is on their relation to (and distinction from) the above concept of economic exogeneity. A variable is said to be weakly exogenous for ; if it has a zero row in (Johansen 1992). This in turn implies that the cumulation of shocks to this variable is a common trend. If, in addition, this variable is not Granger Caused by the endogenous variables, the variable is said to be strongly exogenous for (Johansen 1992). Partitioning x t as (x 0 1t ; x 0 2t ) 0 ; and correspondingly the matrices in (8), under the restriction, (11), as, weak exogeneity of x 2t for ; is the restriction that 2 = 0; while strong exogeneity requires 21;i = 0; in addition. For the VAR(1) the two concepts coincide.

A Static Theory Model
Suppose that a VAR(1) describes the variation in the time series (Q t ; P t ; W t ; Z t ); corresponding to the variables in Section 2. Assume that these series are persistent, cf. Section 3. Under this assumption, a set of restrictions on the VAR, consistent with the simple demand-and supply model (1) -(3) is now suggested. In general, the equations de…ning an economic model involve latent constructs, such as expectations and plans. Hence, they are not directly empirically implementable. Here, the relations, (1) and (2), are the plans involving the latent variables, Q s and Q d . In particular, introducing a time index, I shall assume that these relations are contingent plans, conditional on observed outcomes (see e.g. Hendry 1995): For example, for the demand relation, (1), Q d t = a 0 a 1 P t + a 2 W t ; denotes demand at time t; where the plan, Q d t ; is unobserved while P t and W t are realized values. This is a point on the demand curve at time t, which is denoted by D(P; W t ) a 0 a 1 P + a 2 W t .
In contrast, the VAR model is formulated in the observables. As a consequence, a mapping relating the latent variables and plans to the observables is needed. Usually, such mappings come in the form of an observation equation, for Q t outside equilibrium, and an adjustment equation for P t outside equilibrium 4 .
Consider the price adjustment mapping. I assume that it has the general form, where g() is a continuous and locally di¤erentiable function. It seems reasonable that the adjustment in prices from period t 1 to t depends on what is learned or observed in period t 1: Say, at the end of period t 1, …rms realize the reduction in inventories and the increased willingness to buy. As a consequence, they probably charge higher prices the next period. Suppose, that (26) has the speci…c, though still general, form, where prices adjust as a result of the discrepancy between plans and realizations for both consumers and producers. Compared to the equations often used in the literature (see e.g. Laroque and Salanie 1995), the mapping in (27) allows for di¤erent adjustment processes for demand-and supply deviations respectively, which seems empirically relevant, as these processes may involve di¤erent sets of agents.
As an equilibrium represents a state with no change, it follows that, so that a Taylor expansion of g() around the equilibrium delivers the mapping, where g 0 is a partial derivative. A special case of (29), in which, excess demand causes prices to rise, follows from assuming g 0 1 (0; 0) < 0 and g 0 2 (0; 0) = g 0 1 (0; 0); since this implies P t ' g 0 Hendry (1995).
The Taylor approximation is only useful provided that Q t Q d t and Q t Q s t are stationary. However, as shown below, this is exactly what cointegration means in this case.
As the form of the observation equation is similar to (29), this is given by, The partial derivatives, h 0 1 ; h 0 2 and g 0 1 ; g 0 2 ; are evaluated in the equilibrium. They are thus constants and henceforth they are denoted as, 11 ; 12 and 21 ; 22 ; respectively.
The price-and quantity adjustment in equations (29) and (30), represent the systematic, or anticipated part of the change from one period to the next. It seems reasonable to add the error terms, " P t and " Qt ; respectively in these equations, representing unanticipated and unmodelled in ‡uences. Their stochastic properties are given below.
As argued in Section 3 the economically exogenous variables, W and Z; are the source of persistence, and since the theory model is not concerned with how these are generated, it seems uncontroversial to empirically model them as I(1) processes. Assume therefore that, Hence, this is where the persistence is approximated by imposing the unit roots (see below). Finally, it is assumed, as is usual, that, where " t = (" Qt ; " P t ; " W t ; " Zt ; ) 0 and > 0 is diagonal. The shocks, " W and " Z ; are referred to as demand-and supply shocks respectively, not to be confused with " Q and " P . Collecting all this, the system, (29) -(32) can be written as the following CVAR(1), or, in the compact notation from Section 3, with x 0 t = (Q t ; P t ; W t ; Z t ), and matrices given by, and the corresponding orthogonal complements, Under the assumption (12), the model generates I(1) variables only, since det( 0 ? ? ) = 1: The MA representation thus implies the following components, using equations (15) and (16). Note that, whereas det( 0 ? ? ) = 1 is implied by the theory model, (12) is not.
The model in (35) and (36) is an identi…ed sub model of the I(1) model, H(2): It can be tested in H(2) by jointly imposing the zero restrictions on ; the normalization, and the corresponding two generically identifying zero restrictions on (See Johansen 1996). This can be done using the software CATS in RATS (Dennis, Hansen, and Juselius 2006).
From (36), it is seen that the consequence of the I(1) approximation is that the theoretical parameters of interest, a i and b i , should be modeled as cointegrating parameters, and that the assumption of economic exogeneity implies strong exogeneity in this case.
As alluded to above, the I(1) -, or unit root approximation corresponds to the empirical modelling of the exogenous variables, in that (31) and (32) can be interpreted as approximations of the processes, W t = w W t 1 +" W t and Z t = z Z t 1 +" Zt ; with j < 1 but close to 1, respectively. In such a case, instead of the unit roots, corresponding to (35) and (36), the true underlying process has two roots 1 w and 1 z , both close to, but above 1, while the rest are unaltered also with jzj > 1; provided that (12) applies. The true process is therefore a stationary "near unit root" process generating persistent series. Approximating j by 1, is thus equivalent to approximating the two borderline unit roots, 1 w and 1 z ; by 1. Note that, as opposed to the parameters a i and b i ; w and z are not theoretical parameters of interest, and hence, w = z = 1 is not inconsistent with the theory model, as argued in Section Figure 2: The Supply and Demand Curves, S(P; Z t ) and D(P; W t ); Demanded -and Supplied Quantities, Q d t and Q s t ; and the Actual Outcome, (Q t ; P t ); -all at time t: 3.
The model is illustrated in Figure 2. The diagram looks like that in Figure 1, but there is a fundamental di¤erence: In the equations, (1) -(3), underlying Figure  1, the time subscript is t for all variables, and is therefore suppressed. Since the system, (1) -(3), is in equilibrium, t must be interpreted as an "end period" in this case. That is, a period for which there have been no shocks for a long time and all previous shocks have fully propagated (see e.g. Sørensen and Whitta-Jacobsen 2005, p. 17). As a result, Figure 1 illustrates the economic equilibrium. In contrast, the time subscript in Figure 2 denotes any arbitrary period. As drawn the actual point, (Q t ; P t ), di¤ers from the equilibrium at time t, (Q t ; P t ); determined by the intersection of the demand curve at time t, D(P; W t ); and the supply curve at time t, S(P; Z t ).
The equilibrium at time t acts as a pulling force on the observed point, in the sense that, in the hypothetical absence of any other shocks from period t + 1 and onwards, the (Q; P )-point would converge towards (Q t ; P t ); starting in (Q t ; P t ): To see this, we simply set " = 0 for t + 1, so that the values of W and Z are …xed at W t and Z t : Using the identity in (22) it can be shown that, for h 0: Given (21), (I r + 0 ) h ! 0; for h ! 1, so that the limit of (39) is, This expression implies that x = (Q t ; P t ; W t ; Z t ) 0 ; and since W t+h = W t and Z t+h = Z t this shows the convergence of (Q t+h ; P t+h ) towards the "economic equilibrium at this economic equilibrium thus corresponds to the so-called long-run value, de…ned in Johansen (2005). From (40) it follows that, @x @x 0 t = C; describing the long-run impact of unit changes in the variables. Thus, the C matrix in (38) captures the comparative static e¤ects given in (5).
Starting from the point (Q t ; P t ) the expression for x t+h in (39), tells us exactly where the (Q; P )-allocation is located in the diagram after h periods, in the absence of shocks. Pre-multiplying with 0 in (39), we get an expression for the equilibrium error at time t + h, and by writing 0 k 0 = (1; 1): This shows that the assumption, has the economic interpretation of market clearing. For a given deviation from equilibrium; (42) shows how (and how fast) the market clears. It may involve oscillations or smooth convergence, fast or slow, depending on the eigenvalues of (I r + 0 ): Thus, (42) o¤ers a framework for formulating interesting hypotheses about the market clearing process, which could be formulated in terms of restrictions given ; provided that that (I r + 0 ) < 1. It should be possible to formulate Keynesian type of hypotheses about nominal -, or real rigidities, in this manner. For example, loosely illustrated in the right context of a simple AS-AD, with the same form as (1) -(2), a small value of 21 ; combined with a large value of 11 ; would describe little adjustment in prices while more adjustment in quantities, in the wake of a demand shock, i.e. "nominal rigidities".
The above "long run in the hypothetical absence of shocks" essentially resembles the theory model in pure form, but clearly, in each period unanticipated shocks hit all variables in the system: The demand-and supply curves are shifted by " W and " Z respectively, and in addition to the anticipated changes in Q and P; the shocks " Q and " P occur. An unanticipated realized position and an unanticipated equilibrium position have thus resulted, and adjustment in Q and P towards this equilibrium will take place in the next period, in which new shocks occur etc.. The economic equilibrium thus moves and corresponds to an attractor. This is also captured by the fact that existence of the economic equilibrium (4), requires a 1 + b 1 6 = 0; which is the requirement for the attractor set, A = sp( ? ); to exist, as seen from (37).
This moving equilibrium will induce lagged error correction, or, in other words, x t+1 will depend on the equilibrium error at time t, as is seen from, resembling (41) for h = 1, when shocks occur. Given (21) this is (asymptotically) stationary, which supports the use of the Taylor expansion in (29) and (30), as Figure 3: Illustration of the Long-run Impact of a Unit Rise in " W (Positive Demand Shock) on the Endogenous Variables, Q and P: As the demand-and supply shocks, " W and " Z ; change the locations of respectively the demand-and the supply curves permanently, it is seen from the C matrix in (38), that they have a long-run impact on the endogenous variables. In contrast, the shocks, " Q and " P ; have no long-run impact. This is essentially because they do not a¤ect the positions of the curves: Starting from an equilibrium, an unanticipated price shock say, " P < 0; will introduce excess demand inducing upward price adjustment until the initial equilibrium is restored. It is therefore the cumulation of " W and " Z ; and not " Q and " P that determines the long-run position of the endogenous Q and P; which is what CT t ; in (38) shows.
The loadings matrix, L; in (38) shows how these common trends a¤ect the endogenous variables. The interpretation of the elements in L is facilitated by use of the demand-and supply diagram: Consider a unit rise in " W t ; which according to L in (38) will have a long-run impact of a 2 a 1 b 1 +1 units on Q; and of a 2 b 1 +a 1 units on P . In Figure 3, this corresponds to a unit shock to W t which shifts the demand curve upwards by a 2 a 1 units, eventually resulting in a rise in the equilibrium value of Q and P of the same magnitudes, a 2 1+ a 1 b 1 and a 2 b 1 +a 1 respectively. Similarly, from L; we can see that the unit shock in W will have the full impact, a 2 a 1 ; on P; while no e¤ect on Q; for b 1 ! 0. That is, when the supply curve is vertical, demand shocks will have no e¤ect on quantity, while full e¤ect on prices, e.g. as in a simple classical AS-AD model. In contrast, when b 1 ! 1; the supply curve is horizontal and there is no impact on the price level from the demand shock, while full e¤ect on Q; equal to a 2 , since the horizontal shift in the demand curve is the vertical shift, a 2 a 1 ; times the absolute inverse slope, a 1 . Thus, the impact on the endogenous variables from demand and supply shocks is completely determined by the slopes (partial derivatives) of the curves, an this is exactly what the loadings matrix captures (or C; as C = L 0 ? = L(0; I p r ) = (0; L); when there are p r weakly exogenous variables).

Some Generalizations of the Empirical Model
The framework of the previous section is now generalized in three directions: First, since the mappings g() and h() are not part of the theory model, they are, to some extent, arbitrary. These are therefore generalized with respect to the lag of response, which was 1 above (see e.g. 27). Second, as the data often suggest more than one lag, I consider this case as well. Third, the endogenous variables may also be allowed to respond to current changes in the exogenous variables.
The theory model is still the same, and hence, , ; ? and ? are unaltered, and it is investigated whether the interpretations of the CVAR parameters, C and L, as describing comparative statics, can be retained.
Suppose that the response time were longer than one period, so that instead of (27), the mapping were, for u > 1 (u = 0 is considered below). Hence, the reaction to market disequilibrium takes place u periods later. I assume the same for quantity, i.e. for h(): This implies, that instead of (35) we have, where as before and are given by (36). As this can always be rewritten as, with the restrictions that, i = 0 ; for i = 1; :::u 1; we …nd that = I + (u 1) 0 ; which implies that C and L are unaltered as, C = ? ( 0 ? ? ) 1 0 ? ; and 0 ? = 0; 0 ? = 0: Hence, the result that C and L can be interpreted as above, is invariant with respect to reaction time u: This is intuitively expected, as the longrun e¤ect is the same as before. This holds also if P and Q react to equilibrium errors with di¤erent lags. Now, suppose that the CVAR needs more than one lag. Since ? and ? are the same, we see from, C = ? ( 0 ? ? ) 1 0 ? = L 0 ? ; that needs to ful…ll certain requirements, for the interpretations to be unaltered. It su¢ ces to have two lags i.e., Introduce the block notation, which enters the expression for C: Now, economic exogeneity of W and Z (cf. Section 2) implies that 21 = 0; that is, that x 1t does not Granger cause x 2t : As is unaltered, x 2t is thus strongly exogenous for (See Section 3). Moreover, economic exogeneity requires that the exogenous variables are mutually unrelated, which amounts to 22 being diagonal. The requirement of economic exogeneity is thus stronger than strong exogeneity. Denoting the diagonal elements of 22 by ii ; provided that ii 6 = 1 (the condition in (13)). This implies that, So, the C matrix has in fact changed, but this is simply because the W and Z are now modelled as AR(2) I(1) variables, implying that the long-run impact of a unit rise in " W on W; say, which is what C shows, is no longer 1, but (1 11 ) 1 : As comparative statics concern the e¤ect of a unit change in W; we simply need to normalize the shock, " W ; to obtain a long-run e¤ect of 1 on W: So, if we change " W t by 1 11 we obtain a unit change in W in the long run, resembling the comparative static experiment. As described in Johansen (2005) we can add to the variables at time t, i.e. to " t , which gives the long-run impact C : So, if we add = (0; 0; (1 11 ); 0) 0 to the variables we essentially normalize the column in C showing the impact of " W shocks and get the same as before.
Hence, under economic exogeneity nothing substantial has changed, and provided that we change the current values of the exogenous variables in order to produce a long-run unit change in them, C still captures the comparative statics. Now, consider current/simultaneous e¤ects. The econometric model in Section 4.1 is (6), with k = 1; and A = I, i.e. no current/static correlations between variables. Consider a general normalization A 6 = I. It is su¢ cient to consider the CVAR(1). The corresponding so-called "structural" CVAR is, where a A ; u t A" t distributed as i:i:N (0; ); diagonal. From the MA representation the modi…ed long-run impact matrix is (see Section 3), e C = CA 1 : Hence, the C matrix will in general change. However, under economic exogeneity, A www.economics-ejournal.org has a certain structure, which implies that we can still maintain the interpretations from before. To see this, partition matrices again, i.e., Under economic exogeneity we have that, which in turn implies that, Inserting this into (53) along with C from (54), we get, e C = C: Hence, under economic exogeneity we can also generalize with respect to (identi…ed) A 11 and A 12 parameters, without a¤ecting the conclusion about C and L. So far Q t and P t depend only on the deviation from equilibrium in the previous period cf. equations (30) and (29). When discussing simultaneous e¤ects in the present context it would seem natural to replace Q t 1 Q d t 1 and Q t 1 Q s t 1 ; by the current deviations from equilibrium, Q t Q d t and Q t Q s t . As a result quantity and price would react to disequilibrium already within the period. Instead of (35), the model would be, de…ned in terms of the theoretical parameters of interest ( e ; e s), the corresponding adjustment parameters, e ; and a diagonal e > 0. This corresponds to the structural VAR, (6), with A = (I p e e 0 ) and B k B 1 = I p ; B 0 = e e s; D t = 1 and u t = e " t The model in (58) can be viewed as a reparameterization of the usual model with lagged responses, i.e. x t = ( 0 x t 1 + s) + " t : Since the maximum likelihood estimator (MLE) is invariant, the MLEs of the parameters in (58) can be found from the MLEs of ; ; s and ; to be computed as described in Johansen (1996). Inference on the "new" parameters can then be conducted following the usual approach (see e.g. Hendry 1995, Section 10.11).
So, given the model, x t = ( 0 x t 1 + s) + " t ; consider the mapping, e = ( ); where ( ; ; s; ) and e (e ; e ; e s; e ); given by, e = (I r + 0 ) 1 ; e = ; e s = s and e = V V 0 ; where V (I p (I r + 0 ) 1 0 ); and where det(I r + 0 ) 6 = 0 has been assumed. A reparameterization implies that is a one-to-one mapping: By construction, for a given choice of ; (59) implies a unique value of e , but a one-to-one mapping requires the opposite as well -i.e. for a given value of e there is a unique solution for : To see that this is the case, we …rst note from (59) that = e ; s = e s: From the de…nition of e it follows that e = (I p e e 0 ) ; and hence, a unique value of exists for given e and e if (I p e e 0 ) is invertible. As shown in Appendix A, under the usual (testable) assumptions about ; given in Section 3, this is the case. The unique value of is = (I p e e 0 ) 1 e : To …nd , we note that V = (I p e e 0 ) which implies that = (I p e e 0 ) 1 e (I p e e 0 ) 1 : Thus, is a reparameterization and the MLE of e can be obtained as ( b )i.e. from the usual estimation. The question remains whether C in the estimated model still captures the comparative statics (given by (5) with " e " notation). To see that this is the case, use that = e ; so that ? = e ? ; which can be written as, e ? = ( e B 0 1? ; I 2 ) 0 ; as ? in (48). Note that, the 2 2 block, e B 0 1? ; gives the comparative static e¤ects. Next, the usual partitionings, e = (e 0 1 ; 0 0 ) 0 and e 0 = ( e B 0 1 ; e B 0 2 ); imply that ; which equals (I p e e 0 ) 1 e ; can be written as, = (e 0 1 K 0 ; 0 0 ) 0 ; with K (I 2 e 1 e B 0 1 ) 1 : This implies that 0 ? = (0; I 2 ); i.e. has the same form as before. It follows that, showing that C can still be interpreted as the comparative static e¤ects.

Two Simple Dynamic Theory Models based on Expectations
Embedding a static theory model in the dynamic VAR as suggested above clearly provides a means of empirically assessing, not only the model itself -i.e. its explicit assumptions, but also its implicit albeit crucial assumptions of stability and "reasonable" speed of equilibrium adjustment (see Section 2). Now, say, that a given data sample suggests that both the explicit and the implicit assumptions are reasonable. This certainly brings more credibility to the static theory model, and if we are primarily interested in the (long-run) equilibrium e¤ects our analysis may be su¢ cient -at least as a …rst approximation. However, knowing that the demand and supply equilibrium is stable, and that disequilibria are not too persistent, we still lack a theory of the adjustment process, which may be essential in understanding, for example, e¢ ciency-and distributional properties outside equilibrium. Such a theory is part of a dynamic theory model. Several assumptions make theory models dynamic. For example, related to the present context, ECMs may arise from dynamic optimization (Nickell 1985)  (68) where = c 1 (1 2 ); and the MA components, There is one characteristic root at 1, and det( 0 ? ? ) = 1: Hence, in this particular case, (12) is implied by the theory model.
The interpretations of C (and L) as capturing the comparative statics still apply: The impact from the economically exogenous variables on the endogenous variables computed in the hypothetical absence of shocks, so that expectations are realized (i.e. in economic equilibrium), is still given by C.
The equilibrium error, or the expectational error is, as I r + 0 = 0: As seen from (70), this has an anticipated component, c 1 (1 2 ) r t 1 . Hence, as long as 2 < 1; systematic expectational errors will take place. In the special case when 2 = 1; r e t coincides with the conditional mathematical expectation, E t 1 (r t ). This can be referred to as the case of "Rational Expectations", and it implies that the expectational error is, 0 " t ; i.e. unsystematic.
The model is illustrated in Figure 4. Each period, the downward sloping investment demand curve is shifted up or down by the random walk Y; while the interest rate shifts the vertical line (also as a random walk), which can be thought of as the supply curve. The adjustment in the wake of an interest shock is illustrated: Suppose that the economy has been in the equilibrium, E 0 ; up to and including period t 1: In period t there is a unit shock to the real interest rate, and no other shocks occur. Under simple backward-looking expectations, the movement is from E 0 to O 1 ; in period t; then to O 2 ; in period t + 1; and from t + 2 and onwards to the new equilibrium E 1 : Once the shock has occurred it is known and the "optimal" level of investment in period t + 1; is I t c 1 : When 2 < 1, the actual investment is higher. When expectations are rational, the movement is from E 0 to O 1 ; in period t; and then directly to E 1 in period t + 1: All information is used once it is available.

The Hybrid New Keynesian Phillips Curve -Forward-Looking Expectations of Endogenous Variables
The model is, @ @ Figure 4: The Investment Demand Schedule at time t, together with a shift in the interest rate. In the case of Simple Backward-Looking Expectations, the movement is from E 0 to O 1 ; in period t; to O 2 ; in period t + 1; and to E 1 in period t + 2. When expectations are Rational, the movement is from E 0 to O 1 ; in period t; and then directly to E 1 ; in period t + 1: where p t is in ‡ation, ( ) e denotes the subjective expectation, z is the exogenous forcing variable stated as a steady state deviation, e.g. output gap, unemployment, or wage share (see Gali and Gertler (1999) and Bårdsen et al. 2005, Chapter 7). The shocks are i:i:N (0; ); > 0 and diagonal. Compared to the previous example, there are at least three important di¤erences: First, as seen from (71), the expectation now concerns an endogenous variable. Second, Expectations are forward-looking (they concern p t+1 ), and thirdly, di¤erent types of New Keynesian models underlying (71) typically have rigorous microeconomic foundations (Roberts 1995). The presence of the forward-looking term together with the backward-looking term p t 1 motivates the adjective "Hybrid". The case, 2 = 0; is the pure New Keynesian Phillips curve, whereas , 1 = 0; resembles a more traditional Expectation Augmented Phillips curve (see e.g. Sørensen and Whitta-Jacobsen 2005, Chapter 18).
In Gali and Gertler (1999) -and commonly elsewhere -the assumption of Rational Expectations is invoked, i.e., where E t ( ) is the conditional mathematical expectation. The coe¢ cient 4 is usually restricted as j 4 j < 1 (see Bårdsen et al. 2005). For illustration, suppose that all i obey, 0 < i < 1; and insert (72) in (71). Imposing the transversality condition that 1 < j 2 j ; where 2 = 1+ where 1 = 1 p 1 4 1 2 2 1 ; 3 1 ( 2 4 ) ; " pt 1 1 2 " pt and j 1 j < 1 is assumed. Under these assumptions, (73) implies the following VAR in ECM-form, with, where x t = ( p t ; z t ) 0 and e t = ( " pt + " zt ; " zt ) 0 : This is (asymptotically) stationary as det( ) = ( 1 1)( 4 1) 6 = 0 and the characteristic roots are z 1 = 1 1 and z 2 = 1 4 which both have modulus greater than 1. Now, as argued previously, given the persistence it may be better to use the approximation, 4 = 1; to obtain more reliable inference. As a result we get (in the usual notation), and as det( 0 ? ? ) = 1; we have a cointegrated I(1) model. The long-run impact matrix is, Analogous to the previous example, this gives the comparative static e¤ect, (1 1 ) ; from z on p: To see this, …rst note that in the hypothetical long-run equilibrium with no shocks, z t is …xed, expectations are realized, ( p t+1 ) e = p t+1 and p t+1 = p t = p t 1 ; so that (71) implies that p t = 3 1 1 2 z t : Hence, the comparative static e¤ect from z t on p t is 3 1 1 2 : It can be shown that this is equal to the element in C; (1 1 ) ; using that 1 + 2 = 1 1 and 1 2 = 2 1 : Compared to static models, these two examples show how dynamic theory models completely specify, not only ; but also the adjustment parameters, and i : As a result, in the absence of shocks, the theory model determines the exact position of, say, (Q t ; P t ), at any t; in a diagram corresponding to Figure 2. In other words, the matrix (I r + 0 ); and hence, the process of market clearing, is fully speci…ed by the economic model.

General Equilibrium
The framework established in Section 4.1 can readily be generalized to consider the important distinction in economics, between partial-and general equilibrium. It is well-known that general equilibrium comparative static e¤ects may be radically di¤erent from the corresponding e¤ects based on partial equilibrium -quantitatively but also qualitatively. As a result, even though we are only interested in the supplyand demand elasticities in one market, we might have to model the markets for related goods as well.
The basic ideas can be illustrated with a model with two markets, and as the example we could extend the partial equilibrium model (1) -(3), by including the labour market thereby endogenizing the wage, W: Instead, another equally simple theory model is considered, which illustrates exactly the same point.
Consider the markets for two related goods, chicken and beef, say, with quantities and prices denoted Q 1 ; Q 2 ; and P 1 ; P 2 ; respectively. The demand for Q 1 ; is related negatively to P 1 ; and positively to P 2 (Q 1 and Q 2 are substitutes). Supply depends positively on P 1 , and negatively on some input price, denoted P I : The partial equilibrium model for market 1 assumes that P 2 and P I are exogenous (resembling the model in Section 4.1), and is given by, Q s 1 = e 0 e 1 + 1 e 1 P 1 e 2 e 1 P I ; where, as before, all coe¢ cients are positive. The chosen normalization on prices (divisions with d 1 and e 1 ) is purely notational, implying that the inverse demand expressions, which are the ones we draw, enter the cointegrating relations.
The assumption that the price, P 2 ; is exogenous, in the partial equilibrium model, implies that when P 2 changes there is no feed back on it from P 1 ; which seems unrealistic: An increase in P 2 shifts demand, Q d 1 ; which will ignite an increase in P 1 ; which, in turn, will raise demand for good 2, causing a higher price P 2 : This will further feed back positively on demand for good 1, so that the increase in P 1 would be reinforced, and so on. Hence, the partial equilibrium results are invalidated, and we need to include the market for good 2, i.e. impose general equilibrium.
To keep the exposition as simple as possible, I shall assume that the supply of good 2, Q s 2 ; and P I are economically exogenous. Hence, I retain the endogenousexogenous dichotomy, but P 2 has become endogenous. The demand for good two is, and the equilibrium condition is, www.economics-ejournal.org The general equilibrium model is described by (78) -(82), with P I and Q s 2 as exogenous. Solving the model yields the general equilibrium, where D d 1 e 1 (d 2 f 2 1) is the determinant of the coe¢ cient matrix to the system. Thus, the equilibrium exists, if and only if, D 6 = 0; which is assumed. Again, the comparative static e¤ects are readily computed as the partial derivatives with respect to P I and Q 2 in (83).
The embedding of this theory model in the VAR can be done exactly as in Section 4.1, introducing the mappings from latent plans to the observable variables. For simplicity, the observation mapping for Q 2t is, and only one new adjustment coe¢ cient, 33 ; is introduced. Price-and quantity adjustment for the good 1 market are as before, so that, altogether, the CVAR (1) is, Q 1t = 11 (P 1 (d 0 d 1 Q 1 + d 2 P 2 )) t 1 + 12 (P 1 (e 0 + e 1 Q 1 + e 2 P I )) t 1 + " Q 1 t ; P 1t = 21 (P 1 (d 0 d 1 Q 1 + d 2 P 2 )) t 1 + 22 (P 1 (e 0 + e 1 Q 1 + e 2 P I )) t 1 + " P 1 t ; P 2t = 33 (P 2 (f 0 f 1 Q 2 + f 2 P 1 )) t 1 + " P 2 t ; P It = " P I t ; Q 2t = " Q 2 t : This corresponds to the matrices, in equation (18), with orthogonal complements, and common trends and loadings (as det( 0 ? ? ) = 1), e 0 + e 2 P It ? r r Figure 5: The Economy in General Equilibrium. Note how the demand curves are drawn for the equilibrium values of the price on the related market.
resulting in the long-run matrix, The general equilibrium at time t, for given values of the exogenous variables, P I and Q 2 is illustrated in Figure 5. Equations (86) - (89) show that the interpretations from Section 4.1 generalize straightforwardly: The economically exogenous variables become the common trends, and L and C capture the comparative static e¤ects.
In terms of Figure 5, the transition from one equilibrium to the next, in the comparative static experiment, now involves a sequence of shifts in the curves, since the markets interact, as opposed to the model in Section 2. As before, this (static) theory model abstracts from this interaction altogether. It can be shown, that for the theory model to have a stable equilibrium, D > 0 is necessary, and gives reasonable comparative static e¤ects, cf. Samuelson's Correspondence Principle (Samuelson 1941). This was also assumed in Section 2, as a 1 + b 1 > 0: This principle is discussed further in Section 5.
As opposed to the simple model in Section 2, this model involves two notions of equilibrium: The short-run equilibrium, i.e. the equilibrium at time t; which is involved in the sequential interaction between the markets, and the long-run equilibrium, or "steady state". In Section 2 these equilibrium concepts coincide. Furthermore, as before, this "steady state" is a moving (stochastic) equilibrium, when the shocks, " P I t and " Q 2 t ; are introduced.
The C matrix in (89) demonstrates the central point in economics, that general equilibrium comparative statics might be qualitatively di¤erent from those in partial equilibrium: First, note that the partial equilibrium model corresponds to the restriction, f 2 = 0; in which case there is an in ‡uence from P 2 on P 1 but not vice versa. Now, consider the e¤ect of, say, a supply shock, " P I ; on Q 1 . The partial equilibrium e¤ect, is e 2 d 1 +e 1 ; setting f 2 = 0 in the …rst row, fourth column in (89). This is unambiguously negative. In contrast, in general equilibrium, the e¤ect is , which is negative only if d 2 f 2 < 1. In terms of the graphs, the partial equilibrium model shows the initial upward shift in the supply curve for good 1, and then the story ends. In general equilibrium, the resulting rise in P 1 ; spills over to the market for good 2, and shifts the demand curve on this market upwards, which feeds back and shifts demand for good 1 upwards etc.. Hence, in the wake of the shift in the supply curve in market 1, there is a sequence of upward shifts in the demand curve as well. If d 2 f 2 > 1; the sum of these demand shifts is greater than the initial supply shift, and Q 1 will therefore rise.

Discussion and further Generalizations
So far, many practical econometric and theoretical issues have been disregarded, in order to provide an accessible and explicit exposition. A thorough discussion of all these issues is beyond the present scope, and instead a few remarks about the presented framework are given.
First, the approximation of persistence by I(1) can be generalized to I(d), d > 1: As an example, consider the simple static model in Section 4.2, in which more lags were added. For example, it could happen that W t and Z t were even more persistent, and hence, better approximated as I(2) than I(1). As a result, ii = 1; and the MA representation in (50) would be invalid. Instead, the MA representation for I(2) processes would apply (Theorem 4.6, Johansen 1996). As before, the I(2) property would merely be an assumption about the statistical-, and not the theoretical parameters of interest.
Second, one should note that in the case when the endogenous-exogenous dichotomy is rejected the common trends no longer have the simple form, i.e. 0 ? = (0; I p r ); as in (36) say. The common trends in CT t ; will now involve linear combinations of di¤erent cumulated shocks.
Third, as discussed in Section 4.1, the Taylor approximation (29) and (30), rests on stationarity of the equilibrium error, (43). However, given stationarity, the approximations may work better in some case than others. In general, it depends on the degree of non-linearity of the mappings, which probably depends on whether variables are in logarithms or not, whether transaction costs are negligible or not, etc. Moreover, the continuity assumption of g() and h(); should also be viewed as a rough approximation, as transaction costs are likely to introduce discontinuous adjustment to disequilibria.
Fourth, Samuelson's Correspondence Principle may be related to the CVAR analysis above (Samuelson 1941). This simple but useful principle states that stability of the equilibrium implies comparative statics with "reasonable" signs. For example, above the equilibrium price, it is often assumed that supply exceeds demand, so that the price level falls, implying stability. This assumption, thus involves a restriction on the slopes of the demand-, and supply curves, which make comparative statics have reasonable signs (See Samuelson 1941). This correspondence is not needed for the CVAR models in sections 4.1 and 4.4, as stability depends on the in addition to ; of which it is the latter that the principle concerns. As a result we can have stable equilibria with "strange" comparative statics.
Finally, in Section 3 it was argued that since the economically exogenous variables cause the endogenous variables and not vice versa, persistence in the system variables must originate exclusively from the former. There is however a special case of particular interest for which this is not the case. Consider the Hybrid New Keynesian Phillips Curve again (Section 4.3.2). Now, suppose instead that 0 < 4 < 1; so that the exogenous variable is I(0). As seen from (75), it is still possible to have det( ) = 0; i.e. a unit root, namely when 1 = 1. This condition, often invoked in the literature, requires that 2 = 1 1 ; and is referred to as dynamic homogeneity (Bårdsen et al. 2005). Substituting 1 = 1 into (73) we see that the hybrid Phillips curve relation becomes, 2 p t = z t + " pt : This means that the rate of change in in ‡ation, 2 p t ; rather than the level of in ‡ation, now depends on the level of the I(0) variable, z t ; and thus becomes I(0), implying that the level, p t ; itself is I(1), i.e. persistent.

Summary and Conclusion
In an attempt to bridge the gap between economic theory models and the CVAR, I have focused on facilitating the formulation and understanding of economic theory models as restrictions on the CVAR. As most economic models build on the same fundamental concepts, simple static-and dynamic theory models were considered to keep the exposition clear.
The hypothetical point of departure was a well-speci…ed VAR as the statistical model, with some of the estimated roots close to unity, corresponding to persistence of the series. Under the endogenous-exogenous dichotomy of the theory model, this persistence originates from the generation of the exogenous variables which is outside the theory model. Hence, roots at unity, do not contradict the theory, and should be imposed, as an approximative assumption about the DGP to obtain reliable inference from short samples of persistent series.
Approximating the exogenous variables as I(1) unit root processes, static -and simple dynamic models were thus analyzed as restrictions on a CVAR. This established an explicit correspondence between the basic concepts of theory models and the econometric concepts of the CVAR.
This correspondence shows that: The theoretical relations, i.e. demand-, and supply relations, correspond to the cointegrating vectors. The concept of exogeneity in economics is stronger than the econometric concept of strong (and thus weak-) exogeneity for . The existence of the economic equilibrium implies the existence of the attractor set, and the economic equilibrium correspond to the so-called long-run value. The comparative statics are captured by the long-run impact matrix, C: The common trends, which determine the long-run movement of the system variables, correspond to the exogenous variables in the economic model. The loadings matrix can be interpreted to describe how the slopes of the demand-, and supply curves determine the impact on the endogenous variables, from shifts in the curves (i.e. in the exogenous variables). The matrix, I r + 0 ; and, in particular, its largest eigenvalue relates to the concept of market clearing, and interesting adjustment hypotheses (e.g. nominal rigidities etc.) can be related to this matrix. The examples of the dynamic theory models, also demonstrate how hypotheses about expectations are related to the adjustment parameters of the CVAR, and i .
In a generalization of the basic framework the distinction between general-, and partial equilibrium was also related to the CVAR: It was shown how to investigate whether comparative statics in general equilibrium di¤er from those in partial equilibrium, and how the empirical validity of the partial equilibrium model can be tested in the general equilibrium model.
As alluded to, given explicit hypotheses derived from detailed microeconomic assumptions about optimization, information, expectations etc., this paper should, to some extent, facilitate the formulation of such hypotheses as restrictions on the CVAR.