Conditional and Structural Error Correction Models

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 487 October 1994

CONDITIONAL AND STRUCTURAL ERROR CORRECTION MODELS Neil R. Ericsson

NOTE: International Finance Discussion Papers are preliminary materials circulated to stiraulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.

ABSTRACT

A “structural” error correction model (in Boswijk’s sense) is a representation of a conditional error correction model that satisfies certain restrictions. This paper examines the conditions under which such a structural error correction model exists and when the associated representation is of interest. To clarify the nature of such models, several analytical and empirical examples are considered, which violate those conditions. Structural error correction models are economically appealing, but their liraitations imply that some care must be taken when applying them in practice.

Key words and phrases: Boswijk, cointegration, conditional models, dynamic specification, encompassing, error correction, exogeneity, general-to-specific modeling, sequential reduction, structural models, vector autoregression.

Conditional and Structural Error Correction Models

Neil R. Ericsson*

1 Introduction

In his innovative paper “Efficient inference on cointegration parameters in structural error correction models,” Boswijk (1994) proposes the notion of a “structural error correction model,” which is a representation of a conditional error correction model (ECM) that satisfies certain restrictions. His Section 2 discusses the relations between vector autoregression, conditional ECM, structural ECM, and static regression approaches to cointegration; and it formulates an identification procedure for cointegrating vectors. Boswijk’s Section 3 derives various estimators and associated test statistics for the structural ECM, and states their asymptotic properties. Section 4 presents Monte Carlo evidence comparing the estimators’ properties.

Boswijk (1994) carefully develops the analytical structure, historical perspective, and statistical framework for this new model class; and Boswijk (1992) additionally demonstrates how to implement structural ECMs, using data on money demand in the United Kingdom. Given such thorough coverage, I will focus on the nature of the structural ECM itself, employing its relation to the conditional ECM. That leads to examining the conditions under which the structural ECM exists and when such a representation is of interest.

Conditional ECMs are very popular empirically; cf. Davidson, Hendry, Srba, and Yeo (1978), Hendry, Pagan, and Sargan (1984), Ericsson and Hendry (1985), Hendry and Ericsson (1991), and the papers in Ericsson (1992) and

Ericsson and Irons (1994) inter alia. If weak exogeneity is valid, neglect of

*Forthcoming in the Journal of Econometrics as invited discussion of Boswijk (1994). The author is a staff economist in the Division of International Finance, Federal Reserve Board, Washington, D.C. The views expressed in this paper are solely the responsibility of the author and should not be interpreted as reflecting those of the Board of Governors of the Federal Reserve System or other members of its staff. Iam grateful to Peter Boswijk, Julia Campos, Jon Faust, David Hendry, John Irons, and Michel Lubrano for helpful comments

and discussions.

the marginal process is without loss of information; and conditional ECMs are almost invariably much simpler to model than the whole system. A structural ECM enforces a certain economic interpretation on a conditional ECM, wi:h corresponding restrictions. Thus structural models are very appealing in that they offer the empirical, statistical, and economic advantages of a conditional ECM within a given economic framework.

The restrictions associated with a structural ECM lend it strength by providing structure; they are also a weakness in that they need not be satisfied in practice. To clarify the nature of structural ECMs, several analytical and ernpirical examples are considered, which violate those conditions. Most examples are under the condition of weak exogeneity, in which case a valid conditional ECM exists but a structural ECM does not. While structural ECMs offer an important and economically appealing subclass of conditional ECMs, their limitations imply that care must be taken when applying the former in practice. Section 2 derives the structural ECM from a vector autoregression via the conditional ECM and comments on some essential elements of that derivaticn. Section 3 provides the examples.

Before continuing, a semantic point is in order. The adjective “structure” has many meanings in econometrics. Boswijk has added his own definition, so “structural” is used in Boswijk’s sense in Sections 2 and 3 below. Caution is required, however. The examples in Section 3 are not structural in Boswijk’s sense. Yet, they are structural in their authors’ sense, which corresponds to Hendry’s (1993, p. 1) definition of structural, i.e., being “invarian[t] over extensions of the information set in time, interventions or variables.” The examp.es

also are all ECMs, so they are structural ECMs, albeit not in Boswijk’s sense.

2 Cointegrated Vector Autoregressions

The nature of the structural ECM is best understood by considering the standard cointegration analysis of a vector autoregression (VAR). Transformations of and restrictions on the VAR lead to the structural ECM (Sections 2.1 and 2.2) and suggest a sequential modeling strategy for obtaining it (Section 2.3).

2.1 Transformations of the VAR

For an n x 1 vector of variables z, at time t, the €"-order Gaussian VAR can be parameterized as:

e-1 Az, =mti1+ > AjAnjte, en NO.) t=1,...,7, (1)

j=1 where A is the difference operator, and the A; are n X n matrices of coefficients, and €; is the n x 1 vector of independently and normally distributed disturbances. The rank of m (denoted r, 0 <r < n) determines the number of

cointegrating vectors, so (1) may be written as: é-1

j=l fo: an n x r matrix of r cointegrating vectors 6 and an n x r weighting matrix a. Partitioning x; into subvectors of g and k variables as (y; : 21)’, (2) may be

rewritten as a conditional model for Ye given 2;:

e-1 Ay: = Bo Az, + a.’ r41 + >> AT; Ar; +u; uf ~ NIQ(0, %3) (3) j=l and a marginal model for z: e-1 Az, = a2 8'r4-1 + > Ag; Ary; + Et Eat ~ NI(O, Xo) (4) j=l

without loss of generality, where Bg = S125}, ae = y — Bjoae, Ai; = Aj; -

Bj Aj, vf = eu — Boex, UX = Vy — Yi29 Dat, matrices are partitioned

conformably with (yj: z/), and subscripts indicate the relevant submatrix. Premultiplying (3) by a nonsingular g x g matrix [9 obtains another con-

ditional model, isomorphic to (3):

PoAy, = BoAzt A(Tyz-a + Bu-1)

é-1

j=l where Bo = ToB, A = Toa, (I: B) = B’, (Tj : Bj) =ToAt;, ve =Tovy, and Ly = [oL*T% for suitable partitionings of matrices. Equations (1), (2), (3), (4), and (5) are equations (3), (5), (10), (12), and (17) in Boswijk (1994), with

the notation modified slightly for convenience of the discussion below. Under

the assumptions detailed in Section 2.2, (5) above is a structural ECM. It is the focus of Boswijk (1994), with (4) as the accompanying marginal model.

2.2 Restrictions on the VAR Equation (1) is the basis for Johansen’s (1988, 1991) and Johansen and

Juselius’s (1990) complete system analysis of cointegration. Because 7 may be of full rank, neither (2) nor (3)-(4) nor (4)-(5) as such entails any loss of generality relative to (1). Under the assumptions below, (2), (3)-(4), and (4)-(5) are with loss of generality with respect to the unrestricted VAR. Boswijk (1994) analyzes the class of structural ECMs. A structural ECM is a certain representation of a VAR such as (1) that satisfies restrictions involving cointegration [Assumption (i)], weak exogeneity [Assumptions (ii) and (iii)], and structurality [Assumptions (iv)-(viii)]. For ease of discussion in Section 3, these assumptions are presented below in greater detail than they appear in Boswijk (1994). i. The number of cointegrating vectors is r and satisfies 0 < r < n with a; being integrated of order one; ii. the parameters of interest are (3; ili, ag = 0; lv. g=T; v. rank(AD) =r; vi. @ is identified by a set of restrictions {R;6; = 0,7 =1,...,r}, where the matrices R; are known and satisfy certain rank conditions; vii. the matrix A is diagonal with nonzero elements on the diagonal; and viii. the matrix [9 is normalized to have unit elements on the diagonal. Assumptions (i)—(viii) map into Boswijk’s (1994) assumptions as follows. Assumption (i) is Boswijk’s Assumption 1; (ii) is implicit for the most part; (iii) is explicit in the text (e.g., Section 1), when used; (iv) and (v) are Boswijk’s Assumptions 3(i) and 3(ii); (vi) is Boswijk’s restriction for the generic identification of 6 (his Assumption 2); and (vii) and (viii) are Boswijk’s restrictions for identifying Io, A, and &, conditional on the identification of B. Assumptions (i), (ii), and (iii) together imply weak exogeneity of z, for 3; see Engle, Hendry, and Richard (1983). That is, the parameters of interest,

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which are all the cointegrating vectors, can be obtained from the conditional mcdel (3) alone without loss of information. Assumptions (i) and (iii) are both testable, as discussed in Johansen (1988, 1991, 1992a, 1992b) and Boswijk (1992), whereas Assumption (ii) is (supposedly) guided by economic theory and the purpose of the model. Even if the parameters of interest depend on Bj, a, &*, and the Az ; as well, the conditions for weak exogeneity remain the sarae, provided that (a, 8, Bj, {Aj;},=%) and ({A2;}, 22) are variation free.

Assumption (ii) appears in Johansen (1992a, 1992b) as well as Boswijk (1994), albeit with both authors generally including a in the parameters of interest. However, if the parameters of interest include only some of the cointegrating vectors, (ili) is an overly strong condition for the weak exogeneity of z. Any individual empirical investigation might reasonably restrict its focus to only a subset of the cointegrating vectors in the economy, so the examples below consider this situation in greater detail.

Assumption (iv) states that the number of equations (g) in the conditional model equals the number of cointegrating vectors (r). Weak exogeneity and Assumption (iv) jointly imply that | a. | 4 0, and so all the cointegrating vectors enter the conditional model. Assumption (iv) is testable in that g need not equal r. For instance, for a given r and a given partitioning of z;, (iv) implies (iii), which is testable.

Assumption (v) prohibits the conditional equations from including cointegration relations that involve the z;, only, noting that AT =Toa,. Both a, anc. [ must be of full rank.

In one common version of Assumption (vi), the set. of cointegrating vectors is identified such that each of the g variables in y enters one and only one cointegrating vector: that is, 6’ = (J, : B). For instance, Phillips (1991, 1994), Phillips and Loretan (1991), and Stock and Watson (1993) assume a given number of cointegrating vectors with I of full rank and identify 8 by (I, : 0)6 = I,. While the “problem” of identifying multiple cointegrating vecors is often associated with Johansen’s procedure, it arises from dealing with more than one cointegrating vector and is not related to estimation per Se.

Assumption (vii) is without loss of generality, provided Assumptions (iv)

and (v) are satisfied. Assumption (vii) restricts each cointegrating vector to

enter one and only one conditional equation, with the normalized variabl:2 in the cointegrating vector matching the normalized endogenous variable of the conditional equation.

Assumption (viii) is without loss of generality because Ip is nonsingular. Even so, the interpretability of the conditional model may be affected by imposing (vi), (vii), and (viii). While these three assumptions may uniquely identify the parameters of the structural model, they do not guarantee a model

interpretable in light of economic theory; cf. Hendry (1993, pp. 25ff).

2.3 Modeling the VAR A structural ECM is equation (5) under Assumptions (i)—(viii), and it may

be motivated as follows. Suppose the practitioner approaches modeling in three steps. First, r is determined by Johansen’s procedure from the rank of m in (1), the unrestricted VAR. Second, g is determined, e.g., in light of exogeneity and conditioning arguments from economic theory. Third, given those values of r and g and the partitioning of z,, the hypothesis az = 0 is tested by Johansen’s (1992a) test of weak exogeneity. In practice, the number of cointegrating vectors r, the economically interesting conditioning set (which determines g), and the presence or lack of cointegrating vectors in various equations (thereby determining a2) need bear no relation whatsoever to each other. However, if g =r > 0, a2 = 0, rank(AT') =r, and the parameters of interest are , then analysis of (5) as a statistically valid (conditional) structural ECM proceeds. The economic interpretability of (5) under Assumptions

(i)-(viii) remains an issue specific to the application at hand.

3 Examples

As Boswijk (1994) correctly argues, structural ECMs are an appea.ing framework for analyzing systems that satisfy Assumptions (i)-(vili). If one or more of these assumptions is violated, a different approach is required. Analysis of the system as a whole is valid in any case, and conditional modeling is valid under Assumptions (i)-(iii) alone. If Assumptions (i)-(viii) are invalid but are imposed, inferences about the supposed structural ECM may: be misleading. Equally, it is feasible to discover (rather than impose) that these

assumptions hold, to the extent that they are with loss of generality.

To clarify the nature of the assumptions, each example below specifies a given data generation process or empirical model, explains the interest in it, and shows how it violates one or more of Assumptions (i)~(viii). The first two examples are analytical. The remaining three examples are empirical and are taken from Hendry and Mizon (1993), Juselius (1992), and Kamin and Ericsson (1993).

3.1 Separate Subsystems Consider the system (1) with block-diagonal 7, {A;}, and U. That is, y;

and z, are generated by two completely unrelated subsystems:

é-1 Ay: = a By yi + > AyjAm-jtenu en ~ NI(O, 441) (6) j=1 and é-1 Ax = 0223 592t-1 + > Aga; Az_; +E En ~ NI(O, d22) ; (7)

j=1 with E(€,,€5,) = 0 for all t and s. Because Bg = 0, (6) is also the conditional

inodel for y; given z; and their lags. The long-run impact matrix 7 is:

r= af _ ay 0 Buy 0 _ anf}, 0 (8) 0 are 0 By 0 22359 |

A system such as (6)-(7) might arise when analyzing two possibly related markets or countries that have no interactions in fact.

Under Assumption (ii), z; is not weakly exogenous unless a2 = 0 because the parameters of interest are 8, which include Ay, and Bo2. While 81, can be retrieved from (6) alone without loss of information, 322 can not even be identified from (6). However, Assumption (ii) can not be taken for granted. In analyzing a conditional model, the parameters of interest might be only those cointegrating vectors that enter the conditional model itself. For the system (6)-(7), z; is weakly exogenous for those parameters of interest, which are (4,;. The presence or lack of weak exogeneity depends upon what the parameters of interest are, and they need not include all the cointegrating

vectors of the system. The conditions for weak exogeneity also depend upon

what the parameters of interest are. For instance, weak exogeneity of z; for Bi, does not require Assumption (iii), noting that a2: may be nonzero in (7).

In general, Assumption (iv) is not satisfied by (6)-(7), noting that the number of variables in (6) need not equal the number of cointegrating vectcrs in the two subsystems combined. Unless y; is stationary, rank( (41) is less than the number of variables in y,; and rank((22), as a part of (7), is unrelated to the determination of y,. Even if g = r, Assumption (v) can not be satisfied because some of the cointegrating vectors involve z; alone.

For (6)-(7), conditional modeling of y; given z; is feasible, provided the parameters of interest are 3,,;. Structural ECMs as defined by Assumptions (i)-(viii) do not exist for (6)-(7). Assumptions (ii) and (iii) could be modified so that the parameters of interest are 6,,. Even then, if rank($11) < g, (iv) and (vii) can not be satisfied because of the dimensions of y, and A; and if

rank(@1,) = g, then y; is stationary, violating (i).

3.2 Multiple Cointegrating Relations

Consider the system (1) with two endogenous variables (g = 2), two weakly exogenous variables, a block-diagonal ¥, and two cointegrating vectors (f}, 0)’ and (0 (3,)’, with the cointegrating vectors entering the conditional equations only. For ease of exposition, let A; = 0 except for the 2 x 2 submatrix Aoi, which contains the coefficients on Ay;_, in the equation for Az;. This system

is: Ayt = 01184, ye-1 + 0128 5221-1 + €1t (9)

Az: = AmAyt-1 + E22 ’ (10)

where @41, Q12, (11, and {22 are all 2 x 1 matrices. The vector 2x; is integrated of order one, provided the coefficients in A211 satisfy certain mild conditions; see Johansen [(1992b), equation (5)]. Here, as in (6)-(7), Bj = 0; so (9) is the conditional model for y; given z; and their lags. A system such as (9)-(10) might arise when analyzing two related markets where disequilibria in both markets affect one of the markets directly and the other market indirectly. Equations (9)-(10) satisfy Assumptions (i)-(iii) with r = 2, so z, is weakly exogenous for 3, making (9) a valid conditional ECM. Equations (9)-(10) a.so

satisfy Assumption (iv) that g = r, contrasting with (6)-(7). Even so, (9) is

not a structural ECM: Assumption (v) is violated because:

rank) = cak (| / =1<2=r. (11)

If @ were incorrectly identified by Assumption (vi) imposing full rank on TI, statistical inference would be adversely affected. Relatedly, regression-based cointegration techniques due to (e.g.) Phillips and Loretan (1991) and Stock and Watson (1993) are inappropriate here because one of the cointegrating

vectors involves z; alone.

3.3 U.K. Narrow Money Demand

Hendry and Mizon (1993) model narrow money demand in the United Kingdom. The data are quarterly over 1963(1)-1984(4) for nominal M, (M ); real total final expenditure at 1985 prices (TFE), the corresponding deflator (P), and the three-month local authority interest rate (R3). The variables in Hendry and Mizon’s cointegration analysis are m — p, Ap, tfe, r3, a constant, and a trend, where lower case denotes variables in logarithms. Hendry and Mizon [(1993), Table 18.7] derive a congruent system with the trend entering the cointegration space, with:

-—0.095 0 _

170 -1 0.7.0 (mp) ,| 2 0 —0.235 Ap

aB = 0 1 —0.28 0 0.0014

t 0 +0.338 te, 0 0 r3, t

a, 0 By Lt .

= 10 a] 16} ] ¢ 1, (12)

0 0

where the notation is modified to incorporate the trend. The first cointegrating vector in (12) is interpretable as a standard money demand function. The second cointegrating vector relates inflation to the output gap, noting that to-

tal final expenditure grows at approximately 0.5% per quarter over the sample.

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The nonzero feedback coefficients in a are also easily interpretable. Current “excess money” lowers next period’s demand for money. If inflation is alsove its equilibrium rate (for a given output level relative to trend), inflation falls absolutely and output grows at more than trend in the next period. Hendry and Doornik (1994) obtain similar results on an updated data set.

If the parameters of interest are both cointegrating vectors [(ii)], only r3; is weakly exogenous. That implies that g = 3 > 2 = r, so Assumption (iv) is not satisfied.

Even if g were equal to two, Assumption (vi) identifying 8 may reduce the interpretability of the resulting cointegrating vectors. Specifically, suppose B' is identified as B’ = (J, : B), following Phillips (1991) and Stock and Watson (1993). While this identification appears innocuous, choosing any feasible pair of variables as y; (corresponding to J, in #) implies that at ‘east one of the cointegration vectors in (12) is confounded with the other, and possibly both are confounded. Because r3; is weakly exogenous and the trend is deterministic, neither variable can be included in y;. Yet, these two variables and (m— p); are the only ones for which such an identification of 8 woulc. not confound the economically interpretable cointegrating vectors.

If money demand (and so /;) is the only relation of interest, then Ap,, tfe:, and r3,; are weakly exogenous. Assumptions (iv)-(viii) are satisfied because g1 = 71 = 1 (where the subscripts on g and r match the subscript on the cointegrating vector), so the conditional money demand equation is a st:ructural ECM. If the output-inflation relation (and so (2) is the only relation of interest, then (m — p); and r3; are weakly exogenous, but g. = 2 > 1 == ra,

violating Assumption (iv) and so Assumption (vii) as well.

3.4. Danish Inflation

Juselius (1992) models the determinants of Danish inflation, positing disequilibrium effects from the internal labor market, from domestic money demand, and from the external sector via foreign prices and foreign interest rates. Because of the large number of variables involved, Juselius analyzes the three sectors separately. She extracts four cointegrating vectors, which correspond to equilibrium conditions for wages and prices, money demand, purchasing

power parity, and uncovered interest rate parity. Short-run dynamics and all

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four cointegrating relations are included in a single-equation model of inflation, with the feedback coefficients on the cointegrating relations reflecting the importance of the respective disequilibria in determining inflation. All error correction terms are statistically significant, with foreign disequilibria numerically and statistically dominating domestic disequilibria in the determination of inflation.

Because the number of variables (n = 12) and the number of cointegrating vectors (r = 4) are large relative to the sample (T = 57 quarterly observations), estimation of the complete system (1) or even the structural ECM (5) is infeasible. Juselius’s two-stage approach is feasible and delivers data-coherent, economically interpretable results. Assumptions (iii)-(v) may or may not be satisfied; and here as in the previous example, some applications of Assumption (vi) identifying @ may not deliver the most economically interpretable set of cointegrating vectors.

Assumption (vii) precludes more than one cointegrating vector from entering the inflation equation. However, multiple cointegrating vectors in a single equation are sensible in this model, given multiple equilibrium relations for the price level. By restricting A to be diagonal, Assumption (vii) is mathematically convenient for (e.g.) evaluating the stability of (5) and may arise naturally from certain sorts of optimization by economic agents, but the assumption also can be economically unappealing (as here).’ In general, economic theory may suggest that more than one disequilibrium might enter a given equation, so the economic interpretability of a diagonal A must be specific to the problem at hand and is not generic. Also, if agents’ decisions are sequential, economic theory may suggest that [9 is upper triangular, ©, is diagonal, and A is unrestricted, which provides a different (and historically common) set of

identification restrictions; see Juselius (1993).

3.5 Argentine Broad Money Demand

Kamin and Ericsson (1993) model broad money demand in Argentina.

The data are monthly over January 1977-January 1993 for nominal M3 (M)

’

‘As another example violating Assumption (vii), Hendry and von Ungern-Sternberg (1681) find two error correction terms in their conditional consumption function: the

corsumption-income ratio and the liquidity-income ratio.

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the domestic consumer price index (P), the interest rate on domestic pescdenominated fixed-term bank deposits (R), and the free-market exchange rate (E). The variables in their cointegration analysis are m—p, Ap, R, Ap™°*, and Ae, where Ap™®* is the maximum inflation rate to date. For = 7, Kamin and

Ericsson [(1993), Table 3] find one cointegrating vector, with af’z; estimated

as: —0.042 _ [1 6.57 —6.72 1.19 6.14] (m ~ p)s 0.027 An, apr, = 0.051 R, . (18) 0.015 Apr*

—0.076 Aer

The cointegrating vector in (13) is interpretable as a Cagan (1956) money demand relation, with money demand increasing in response to a higher own rate and decreasing in response to higher returns on alternative assets (domestic goods and dollars). The coefficient on the ratchet Ap™** is significant and implies hysteresis of money demand with respect to the inflation rate. The inflation rate is weakly exogenous at the 95% level: Johansen’s (1992a) x?(2) test statistic for Ap and Ap™* jointly is 5.56 [0.062], where the asymptotic p-value is in square brackets. Real money, the interest rate, and the exchange rate are all endogenous, with the corresponding individual x?(1) test statistics being 9.30, 8.09, and 6.05. The nonzero feedback coefficients are all economically sensible. Current excess money lowers next period’s demand for money, raises next period’s nominal interest rate, and slows next period’s depreciation of the peso.

An ECM of (m — p)t, Rr, and Ae; given inflation is a valid conditional model. However, no valid structural ECM exists because either or both of Assumptions (iii) and (iv) are rejected, depending upon the partitioning of r; into (yj: z/). For instance, if yj = [(m — p):, Ri, Ae], then (iii) is satisfied but (iv) is not (g = 3 >1=7r). If y% = [(m— p):], then (iv) is satisfied (g =r = 1) but (iii) is not (the coefficients in a for R and Ae are nonzerc). The beauty of structural ECMs comes in fair part from having weak exogeneily and just as many cointegrating vectors as endogenous variables. The specificity

of structural ECMs also arises from those conditions.

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3.6 A Synthesis

As the examples above illustrate, structural ECMs require assumptions about the rank and identification of certain matrices in addition to the assumption of weak exogeneity. When conditioning is acceptable, the additional assumptions still may be invalid or economically unappealing. Thus, it would be helpful to develop formal testing procedures for Assumptions (iv)-(viii), to the extent that they are testable. The structural ECM [equation (5)] is nested in the VAR [equation (2)] from which it is derived, so Hendry and Mizon’s (1993) VAR-encompassing test is a natural one to use. Even when the assumptions for a structural ECM are statistically valid, the economic interpretability of the identified @ and diagonal A will depend upon the particular data and economic theory being examined.

More generally, conditional ECMs may be structural in senses other than Beswijk’s. Hendry and Mizon’s ( 1993) money demand model is structural in Hendry’s (1993) sense, and Hendry and Mizon (1993, p. 272) explicitly refer to their model as a “structural econometric model.” The cointegrating vector Ai; in Sections 3.1 and 3.2 also is structural in Hendry’s sense, since it remains invariant to the addition of z; and its lags to a VAR of yz alone. Likewise, the empirical models in Juselius (1992) and Kamin and Ericsson (1993) are structural in this sense, noting that those authors find empirically constant

pa:ameters for their models over extensions of their data sets.

4 Remarks

Boswijk has masterfully developed a new and special class of conditional ECMs, called structural ECMs. They generalize the empirically successful single-equation conditional ECMs in a direction that adds more economic stricture. Suitable testing procedures are easy to implement. Many tests of weak exogeneity are readily available; and a test of structurality can be calculated as a test of over-identifying restrictions, but with a new interpretation. While the examples above have indicated some limitations of structural models in particular instances, the general empirical importance of structural ECMs remains to be seen. With the statistical foundations now in place, I thus look forward to seeing the modeling of conditional ECMs as structural

ECMs in practice.

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Kamin, S.B. and N.R. Ericsson, 1993, Dollarization in Argentina, International Finance Discussion Paper No. 460 (Board of Governors of the Federal Reserve System, Washington, D.C.) November.

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Phillips, P.C.B., 1994, Some exact distribution theory for maximum likelihood estimators of cointegrating coefficients in error correction models, Econometrica 62, 73-93.

Phillips, P.C.B. and M. Loretan, 1991, Estimating long-run economic equilibria, Review of Economic Studies 58, 407-436.

Stock, J.H. and M.W. Watson, 1993, A simple estimator of cointegrating vectors in higher order integrated systems, Econometrica 61, 783-820.

15

IFDP Number

487

486

485

484

483

482

481

480

479

478

477

476

475

International Finance Discussion Papers

Titles

1994

Conditional and Structural Error Correction Models

Bank Positions and Forecasts of Exchange Rate Movements

Technological Progress and Endogenous Capital Depreciation: Evidence from the U.S. and Japan

Are Banks Market Timers or Market Makers? Explaining Foreign Exchange Trading Profits

Constant Returns and Small Markups in U.S. Manufacturing

The Real Exchange Rate and Fiscal Policy During the Gold Standard Period: Evidence from the United States and Great Britain

The Debt Crisis: Lessons of the 1980s for the 1990s

Who Will Join EMU? Impact of the Maastricht Convergence Criteria on Economic Policy Choice and Performance

Determinants of the 1991-93 Japanese Recession: Evidence from a Structural Model of the Japanese Economy

On Risk, Rational Expectations, and Efficient Asset Markets

Finance and Growth: A Synthesis and Interpretation of the Evidence

Trade Barriers and Trade Flows Across Countries and Industries

The Constancy of Illusions or the Illusion of Constancies: Income and Price Elasticities for U.S. Imports, 1890-1992

Author(s)

Neil R. Ericsson

Michael P. Leahy Robert Dekle John Ammer

Allan D. Brunner

Susanto Basu John G. Fernald

Graciela L. Kaminsky Michael Klein Graciela L. Kaminsky

Alfredo Pereira

R. Sean Craiz

Allan D. Brunner Steven B. Kamin

Guy V.G. Stevens Dara Akbarian

Alexander Galetovic

Jong-Wha Lee Phillip Swagel

Jaime Marquez

Please address requests for copies to International Finance Discussion Papers, Division of International Finance, Stop 24, Board of Governors of the Federal Reserve System, Washington, D.C. 20551.

16

IFDP Number

474

473

472 471

470

469

468

467

466

465

464 463

462

International Finance Discussion Papers

Titles

1994

The Dollar as an Official Reserve Currency under EMU

Inflation Targeting in the 1990s: The Experiences of New Zealand, Canada, and the United Kingdom

International Capital Mobility in the 1990s

The Effect of Changes in Reserve Requirements on Investment and GNP

International Economic Implications of the End of the Soviet Union

International Dimension of European Monetary Union:

Implications For The Dollar

European Monetary Arrangements: Implications for the Dollar, Exchange Rate Variability and Credibility

Fiscal Policy Coordination and Flexibility Under European Monetary Union: Implications for Macroeconomic Stabilization

The Federal Funds Rate and the Implementation of Monetary Policy: Estimating the Federal Reserve’s Reaction Function

Understanding the Empirical Literature on Purchasing Power Parity: The Post-Bretton Woods Era

Inflation, Inflation Risk, and Stock Returns

Are Apparent Productive Spillovers a Figment of Specification Error?

When do long-run identifying restrictions give reliable results?

17

Author(s)

Michael P. Leahy John Ammer Richard T. Freeman Maurice Obstfeld

Prakash Loungani Mark Rush

William L. Helkie David H. Howard Jaime Marquez

Karen H. Johnson

Hali J. Edison Linda S. Kole

Jay H. Bryson

Allan D. Brunner

Hali J. Edison Joseph E. Gagnon William R. Melick

John Ammer

Susanto Basu John S. Fernald

Jon Faust Eric M. Leeper