Aggregation of linear dynamic microeconomic models

Journal of Mathematical Economics 31 Ž1999. 131–158

Aggregation of linear dynamic microeconomic models Mario Forni b

a,)

, Marco Lippi

b,1

a Dipartimento di Economia Politica, Õia Berengario, 51, Modena, Italy Dipartimento di Scienze Economiche Õia Cesalpino, 14-00161 Rome, Italy

Received 20 February 1998; accepted 12 June 1998

Abstract We survey a number of important results concerning aggregation of dynamic, stochastic relations. We do not aim at a comprehensive review; instead, we focus heavily on the results collected in Forni and Lippi wForni, M., Lippi, M., 1997. Aggregation and the Microfoundations of Dynamic Macroeconomics. Oxford University Press, Oxfordx. We argue that the representative-agent assumption is misleading and the microfoundation of dynamic macroeconomics should be based on explicit modeling of heterogeneity across agents. An unpleasant aspect of this modeling strategy is that macroeconomic implications of micro theory are difficult to obtain. However, difficulties are reduced by large number results. Moreover, puzzling implications of existing theories could be reconciled with empirical evidence on macro data. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Aggregation; Heterogeneity; Representative agent; Linear stochastic process; Dynamic factor model

1. Introduction This paper reviews some important results on aggregation of linear stochastic dynamic models employed in macroeconomics. A comprehensive discussion is beyond our purposes, so that some interesting contributions will not be mentioned. ) 1

Corresponding author. E-mail: [email protected] E-mail: [email protected].

0304-4068r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 8 . 0 0 0 6 0 - 3

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Rather, we report mainly results by ourselves, illustrated in detail in Forni and Lippi Ž1997.. Rigor is not our primary concern here. The exposition is based on simple examples, with a hint to generalizations, so that a non-specialist reader may get an idea of the problems, the difficulties and extant results. We assume that the reader is acquainted with elementary theory of stationary stochastic processes. However, in the Appendix A, and partly in the text, we shortly recall basic notation, definitions and results, such as stationary vector processes, white-noise processes, ARIMA processes, Wold decomposition, cointegration, Granger causality. The framework in which our aggregation problem arises is Standard Macroeconomics of aggregate consumption, income, investment, employment. This field has been dominated in the last two decades by two, not necessarily conflicting, approaches. The first is New Classical Macroeconomics, which is characterized by the strong prescription that models linking observable variables must be derived from microeconomic first principles, and in particular that the dynamics of such models must be a consequence of intertemporal optimization, rather than ad hoc superimposition to a static maximization scheme. The second is based on VAR and Structural VAR models, in which estimation of a statistical model, not based on any particular economic theory, comes first, while economic theory enters at the second stage, when identification of an economically meaningful structure is concerned. 2 We shall focus mainly on the former approach, but the latter is also briefly discussed in Section 7. We will claim that irrespective of which approach is taken up, interpretation of dynamic macroequations encounters a very serious aggregation problem: when heterogeneity of agents is allowed, an important change of dynamic shape is likely to occur between micro and macro equations. Basic properties of the micro model do not hold in general for the macromodel: for instance, micro cointegration does not imply macro cointegration, lack of Granger causality in the micro model does not imply the same property at the macro level, static microequations may be transformed by aggregation into dynamic macroequations. As a consequence, overidentifying restrictions, i.e., the model features that are consequences of the underlying economic theory, cannot be tested directly using aggregate data. This is unpleasant, since the difficulties involved in formulating a macro model are considerably increased. Aggregate implications of micro theory can only be found by explicit modeling of heterogeneity, which is likely to require a lot of additional information with respect to the traditional strategy. On the other hand, there is also a pleasant implication: in some cases, models which are at odds 2

For a vast treatment of models based on dynamic objective functions and rational expectations, see Hansen and Sargent Ž1991.. A simple presentation of the main topics can be found in Sargent Ž1987.. On Structural VAR models, see Bernanke Ž1986., Shapiro and Watson Ž1988., Evans Ž1989. and Giannini Ž1992..

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with aggregate data under the representative-agent assumption can be reconciled with empirical evidence. Though we will not make specific mention of all of the following authors or works in the sequel, our general ideas are close in spirit to Granger Ž1980, 1987, Ž1982, 1984., Stoker Ž1982, 1984, 1986., Trivedi Ž1985., Lewbel 1990., Lutkepohl ¨ Ž1992, 1994., Hildenbrand Ž1994., Pischke Ž1995.. The paper is organized as follows. In Section 2, we present a most simple example of the micro–macro effect that we want to illustrate in this paper. We have a microequation linking an independent to a dependent variable. The microequation is static, while the independent variable is autocorrelated: if heterogeneity in the microparameters is allowed, the microequation linking the aggregate dependent variable to the aggregate independent variable is dynamic. The example in Section 2 is highly artificial. In Section 3, a more realistic model for the independent variables is discussed. The model is based on the distinction between common and idiosyncratic components. When the number of individuals is huge, common components survive aggregation whereas idiosyncratic components are washed away. Recent empirical work is reported, in which it is shown that major macroeconomic variables are driven by several common components. In Section 4, we give a definition of micro and macromodels that is general enough to accommodate most of the models employed in the literature. We show that the macroparameters are analytic functions of the deep microparameters. As a consequence, restrictions on the macroparameters either hold on the whole microparameter space or hold only on a negligible subset, i.e., a subset of zero Lebesgue measure. An application of this Alternative Principle is given in Section 5, where we show that cointegration of micromodels does not imply macro cointegration, apart from negligible subsets of the microparameters space. In Section 6, we review an important case in which aggregation has a positive effect. Permanent-income theory of consumption under rational expectations is at odds with aggregate empirical evidence when the representative agent is assumed. Two well-known inconsistencies are ‘excess sensitivity’ and ‘excess smoothness’. However, allowing for some heterogeneity and assuming limited information, aggregate data can be reconciled with the theory. In Section 7, we show some unpleasant consequences of aggregation on Granger non-causality and the interpretation of VAR models. Section 8 concludes. 2. An example: a dynamic macroequation with a static microequation We will often refer to static or dynamic equations linking micro or macrovariables. Regarding the meaning of these terms: Ža. All the variables are indexed with a discrete time parameter; Žb. All the equations are linear of the form s

zt s Ý

k

k

Ý a h j wh tyj q Ý bj z tyj ,

hs1 js0

js1

Ž 1.

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in which the value of z at time t depends on the values of the s variables wh at time t, t y 1, . . . , t y k, and on the values of z itself at times t y 1, t y 2, . . . , t y k. Eq. Ž1. is static if bj s a h j s 0 for any h and j ) 0, i.e., if z t depends only on variables indexed by t, dynamic otherwise. With respect to a given t, the values assumed by a variable z at times t y j are called lagged values for j ) 0, future values for j - 0, or also lags and leads of z t . z t and wt are called contemporaneous values of the variables z and w. It is often very convenient to employ the lag operator, indicated by L, whose definition is Lz t s z ty1 , and its powers: L j z t s z t y j . Putting a hŽ L. s Ý kjs0 a h j L j and bŽ L. s Ý kjs1 bj L jy1 , Eq. Ž1. can be rewritten as s

z t s Ý a h Ž L . wh t q b Ž L . z ty1 , hs1

or s

c Ž L . z t s Ý a h Ž L . wi t , is1

where cŽ L. s 1 y bŽ L. L. Let us begin with a short review of the theoretical steps involved in the formulation of a typical ‘microfounded’ partial-equilibrium macroeconomic model. Ži. A quadratic intertemporal optimization problem for an economic agent is set up and solved; the result is a linear equation linking the dependent variable yt to the expected future values of the independent variable x t and the lagged values of both variables x t and yt : yt s f Ž x t , x ty1 , x ty2 , . . . ; yty1 , yty2 , . . . ; Et Ž x tq1 . , Et Ž x tq2 . , . . . . ,

Ž 2.

where Et Ž x tqj . denotes expectation at time t of the value taken by x at time t q j. Žii. A stochastic linear dynamic equation for the independent variable x t is specified. The example x t s u t q a u ty1 ,

Ž 3.

with u t white noise, will be useful to fix ideas. Eq. Ž3., jointly with the usual assumption of rational expectations, is employed to eliminate expected values in Eq. Ž2.. The meaning of rational expectations in the context of dynamic macroeconomics, as specified for our simple example, is that the agent knows the true process driving the variable x, i.e., Eq. Ž3., and that the minimum square error predictions of x based on Eq. Ž3. are used in Eq. Ž2. to eliminate future values of x. In our example, from Eq. Ž3. we get EŽ x tq1 . s a u t and EŽ x tqj . s 0 for j ) 1. Notice that if a s 0, so that x t is a white noise, future values of x t are completely unpredictable; for this reason in this context a white noise is often called a shock. Elimination of expected values in Eq. Ž2. leads to an equation of the form yt s byty1 q ax t q cx ty1 q e t ,

Ž 4.

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where only one-period lags have been included for simplicity and e t is a white noise. Žiii. The agent following the micro model Ž3. – Ž4. is assumed to be ‘representative’, meaning that model Ž3. – Ž4. can be interpreted as a macro model and therefore can be employed directly for estimation and testing with macro data. In this paper, we depart thoroughly from this microfoundation paradigm. The crucial difference is that we drop the representative agent assumption in step Žiii. and assume instead that agents are heterogeneous. We do not place special emphasis on the optimization and the rational expectations steps. Rather, we start by introducing heterogeneity in the micro Eqs. Ž3. and Ž4.: yi t s bi yi ty1 q a i x i t q c i x i ty1 q e i t x i t s u i t q a i u i ty1 .

Ž 5.

Then we focus on the following questions: Can we suppose that the macroequation linking the aggregate variables Yt s Si yi t and X t s Si x i t is obtained by simply averaging over the coefficients of Eq. Ž5.? Under which conditions the dynamic properties of the micro models hold true in the macro model? In general, the answer to the first question is negative. Moreover, the dynamics of the macroequation may be dramatically different from the micro dynamics. The following example should be sufficient to give the reader an idea of the complications arising. Let us assume for simplicity bi s c i s 0 and e i t s 0 for all i, so that we are left with the static, exact micro equations yi t s a i x i t .

Ž 6.

Let us assume also that the independent variables of different agents are orthogonal at any lead and lag, i.e., covŽ u i t , u jtyk . s 0 for i / j and any integer k, and varŽ u i t . s 1 for any i. Finally, let us assume that there are only two agents Žor two groups of identical agents., i.e., Yt s y 1 t q y 2 t and X t s x 1 t q x 2 t . Now consider the static regression of Yt on X t : Yt s AX t q V t .

Ž 7.

A simple calculation shows that As

a1 Ž 1 q a 12 . q a2 Ž 1 q a 22 .

Ž 1 q a 12 . q Ž 1 q a 22 . 2

2

var Ž V t . s Ž a1 y A . Ž 1 q a 12 . q Ž a 2 y A . Ž 1 q a 22 . 2

2

cov Ž V t , V ty1 . s Ž a1 y A . a 1 q Ž a 2 y A . a 2 Hence, varŽ V t . s 0 if and only if a1 s a2 , while covŽ V t , V ty1 . s 0 if and only if either a1 s a2 , or a 1 s a 2 s 0. Thus if the behavioral coefficients are heterogeneous, and there is some autocorrelation in individual incomes, a researcher estimating Eq. Ž7. will find a non-zero residual Žwhereas no residual is present in the first equation of the micromodel Ž6.. and detect autocorrelation in V t . If our

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researcher shares the common representative-agent attitude, he will conclude that the true relationship between the micro counterparts of Yt and X t is a dynamic equation like Eq. Ž4. and therefore model Ž6. must be rejected. We can stop here the analysis of this very simple example. The message is that aggregation of dynamic multivariate models can produce non-trivial changes in the dynamic shape of each single equation. Starting with a static microequation, like the first of Eq. Ž6., we may end up with a dynamic equation provided that some heterogeneity among the agents is allowed. 3 3. Independent variables: common and idiosyncratic components The examples in Section 2 are partial equilibrium models, in which one or several variables are taken as given both from the agents and the researcher. For general equilibrium models we refer to Forni and Lippi Ž1997. ŽChapter 7.. Here, we shall deal only with partial equilibrium models, which are sufficient for our illustrative purposes. In the latter models, agents may differ for two reasons: because their independent variables are different, and because their responses to their independent variables are different. Let us begin by modeling the differences in independent variables. In the example of Section 2, we have assumed that the variables x i t , corresponding to different individuals, are orthogonal at any leads and lags. Although convenient to simplify calculations, this assumption is far from being realistic. Incomes of different agents, wages faced by different firms, although different, are correlated both simultaneously and with lags. A natural way in which both difference and correlation across individual variables can be represented is the dynamic factor model ŽGeweke, 1977; Sargent and Sims, 1977.. To fix ideas suppose that the independent variable is income and that yi t is income of agent i. A very simple dynamic factor model for yi t is yi t s bi Ut q j i t , Ž 8. where we assume that: Ža. Ut and j i t are stationary stochastic variables;Žb. j i t is orthogonal to Utyk for any integer k; Žc. j i t is orthogonal to j jtyk for any j / i and any integer k. Indeed, this model is so simple that it could be confused with a static factor model; notice however that, although the response of the variables y to Ut is static, the orthogonality conditions are dynamic, i.e., they hold at any lead and lag. The variable Ut , called the common factor, is a source of variation affecting all micro incomes, even though with different impacts as measured by 3 The effects of aggregation on the dynamic shape of economic relations have been firstly highlighted by Lippi Ž1988.. The emergence of an ‘aggregation error’ is studied in detail in Lippi and Forni Ž1990.. Similar phenomena can emerge with time aggregation, seasonal adjustment, aggregation of unobserved components, omitted variables, errors in variables; see Sims Ž1971, 1974., Tiao and Wei Ž1976., Nerlove et al. Ž1979., Lutkepohl Ž1982. and Forni Ž1990.. ¨

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the coefficient bi . The term bi Ut will be called the common component of yi t . Changes in Ut can represent for instance fiscal or monetary policy changes, as well as changes in productivity or labor supply that affect the whole economy. By contrast, each of the variables j i t , called the idiosyncratic components, represents events affecting only one individual, like health or luck. Idiosyncratic components are therefore uncorrelated across individuals at any lead and lag. Model Ž8. can be generalized by introducing more than one common factor and adding lags in the response of yi t to the common shocks, thus obtaining yi t s bi1 Ž L . U1 t q bi2 Ž L . U2 t q PPP qbi h Ž L . Uh t q j i t ,

Ž 9.

where j i t is orthogonal to Ustyk for any integer k and any s s 1, h. Notice that we are not assuming that the variables Ust , or the variables j i t , are white noises. However, if the Ust ’s are assumed to be costationary, model Ž9. can be rewritten as yi t s a i1 Ž L . u1 t q a i2 Ž L . u 2 t q PPP qa i h Ž L . u h t q j i t ,

Ž 10 .

where Ž u1 t u 2 t PPP u h t . is an orthonormal white-noise vector, i.e., varŽ u i t . s 1 for any i, covŽ u i t , u jtyk . s 0 for i / j and any integer k. Moreover, since empirical incomes are non-stationary while income changes are stationary, then either we interpret the variables yi t in Eq. Ž10. as deviations from a deterministic trend, or we modify Eq. Ž10. in such a way that the stationary variable on the right hand side drives the changes of income:

Ž 1 y L . yi t s a i1 Ž L . u1 t q a i2 Ž L . u 2 t q PPP qa i h Ž L . u h t q j i t .

Ž 11 .

Factor models like Eq. Ž10. or Eq. Ž11. have been recently employed in macroeconomic literature as parsimonious representations when the number of variables is large with respect to the number of available observations over time. 4 With a small number of factors, models Ž10. or Ž11. can provide a considerable reduction of the number of parameters to be estimated as compared to an unrestricted VAR model, in which each of the variables is regressed on itself and lagged values of the others. Despite parsimony, model Ž11. is flexible enough to allow for a substantial amount of heterogeneity. Models Ž10. and Ž11. have a very important property when the number of individuals is huge. Let us fix ideas on Eq. Ž10.. Formally, it is convenient to assume that there exists a countable infinity of agents. We assume also that varŽ j i t . is bounded, i.e., there exists a real L such that var Ž j i t . G L for any i. Finally, for simplicity, we set a i s Ž L. s bs Ž L. / 0 for any i and s s 1, h. As a consequence, the per-capita common component does not depend on n and is equal to b 1 Ž L . u1 t q PPP qbh Ž L . u h t . By contrast, considering the per-capita idiosyncratic component Ý nis1 j i trn, 4

See for instance Quah and Sargent Ž1994. and Forni and Reichlin Ž1998..

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orthogonality of j i t and j jt for i / j implies that var ŽÝ nis1 j i trn. s Ý nis1 varŽ j i t .rn2 / Lrn, so that n

lim var n™`

ž

Ý j i trn is1

/

s 0.

Thus the per-capita idiosyncratic component in per-capita income yt s Ytrn disappears as n tends to infinity. 5 For large n we have approximately yt s b 1 Ž L . u1 t q b 2 Ž L . u 2 t q PPP qbh Ž L . u h t , or, more in general, yt s a1 Ž L . u1 t q a2 Ž L . u 2 t q PPP qa h Ž L . u h t , where a s Ž L. is the cross-sectional average of the a i s Ž L.. If aggregate magnitudes, instead of per-capita, are concerned, the statement above can be rephrased by saying that the variance of the aggregate common component tends to infinity as fast as n 2 , whereas the variance of the aggregate idiosyncratic component tends to infinity as n, so that the ratio of the second to the first tends to zero as n tends to infinity. The elimination of the idiosyncratic components has crucial consequences for both theory and empirical work. From a theoretical point of view, it provides a nice way to reconcile macroeconomics with micro heterogeneity. Individual variables corresponding to different agents may be almost orthogonal to one another owing to big idiosyncratic components as compared to the common components. Therefore, individual variables can be viewed as spanning a vector space with a huge number of dimensions. Nonetheless, this is perfectly consistent with a very basic idea of macroeconomic theory that aggregate variables, rather than depending on all the corresponding microvariables, can be represented as driven by a relatively small number of macroeconomic sources of variation. Regarding empirical work, the large-numbers effect can be exploited in order to estimate the model and to study the problem of how many independent common components drive the micro- and the macrovariables. On estimation we refer to Forni et al. Ž1998. and Forni and Reichlin Ž1998.. 6 Regarding inference on the number of common components, some work can be found in Forni and Lippi Ž1977. and in both of the papers quoted above. 5

This large-numbers effect is well-known for static models, particularly in the finance literature Žsee, e.g., Chamberlain and Rothschild, 1983.. Some important implications for aggregation and macroeconomics are discussed in Granger Ž1987, 1990.. Forni and Lippi Ž1997. ŽChapter 1. provide conditions under which the result can be extended to the dynamic case. 6 Forni and Reichlin Ž1998. propose to estimate the factors and the common components by using simple averages. Forni et al. Ž1998. and Forni and Reichlin Ž1997. propose procedures based on principal components. An estimator which is a linear combination of the observable variables is developed by Stock and Watson Ž1997. for a model with time-varying coefficients.

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In Forni and Lippi Ž1997. ŽChapter 2. data on incomes and wages relative to US states are employed to show that the number of common components in a representation of the form Ž11. for individual incomes is definitely bigger or equal to two. This evidence is at odds with some recent applied macroeconomic work on consumption, in which heterogeneity of incomes is introduced but the micromodel contains only one common component Žsee Section 6 below.. This outcome, i.e., more than one common component in individual major economic series is interesting both per se, and because, as we will see, many of the effects of aggregation on dynamic micromodels depend on: Ž1. heterogeneity of different agents’ behaviors, Ž2. at least two common components driving the independent variables. Let us close the section by commenting on the assumption of mutual orthogonality for the idiosyncratic components. The following two examples show that this condition, although very convenient, may rule out interesting economic models and should be relaxed in some way. As a first example, consider an n-industry constant-returns economy, where production of industry i at time t is given by the equation yi t s d i1 y 1 ty1 q d i2 y 2 ty1 q PPP qd i n yn ty1 q c i u t q x i t ,

Ž 12 .

where u t is a demand common shock, x i t is an idiosyncratic productivity shock fulfilling the orthogonality assumption, i.e., covŽ x i t , x jt y k . s 0 for i / j and any integer k, d i j is the quantity of the ith product necessary as a means of production to produce one unit of the jth product, the one-period lag on the right hand side meaning that industry i is partly producing to replace means of productions employed by other industries in the previous period. Starting with Eq. Ž12. and inverting the input–output matrix we obtain an equation like Eq. Ž10.: yi t s a i Ž L . u t q j i t , where

j 1t x1t j2t x2 t s Ž I q DL q D 2 L2 q PPP . . ... .. xn t , jn t

0

0

D being the matrix having d i j in place i, j. However, unlike in Eq. Ž10., unless D is diagonal, the assumption j i t H j jt y k for j / i and any integer k is no longer valid. The problem stems from the fact that shocks originated in sector i, while affecting directly only yi t , propagate through the system via the autoregressive linkages. As a second example, consider the model yijt s a ij Ž L . u t q bij Ž L . Õtj q c ij Ž L . x i jt .

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Here, the income of region i in nation j is driven by a world-wide shock u t , a national shock y tj and a local shock x i jt . A model like this is employed in Forni and Reichlin Ž1997. in order to study comovements between European regions and to compare them with US counties. The national shocks could be accommodated in model Ž10. as common shocks; however, if the number of nations in the model is large, parsimony would be lost. By contrast, absorbing the national component into the idiosyncratic term would violate orthogonality, since the variance–covariance matrix would be block-diagonal rather than diagonal. Now, if the orthogonality assumption on the idiosyncratic terms is dropped in Eq. Ž10. or Eq. Ž11., the theoretical distinction between idiosyncratic and common components is lost and boundedness of varŽ x i t . is no longer sufficient to ensure that the per-capita idiosyncratic variance tends to zero. However, mutual orthogonality and bounded variance can be substituted by the following more general condition. Let S nj be the variance–covariance matrix of the vector Ž j 1 t j 2 t PPP xn t ., and let l nj be its maximum eigenvalue. If l nj is bounded then the variance of n

1

Ý jit n is1

tends to zero as n tends to infinity. This is very easy to show. Indicating by w the n-dimensional vector with all its components equal to unity, we have var

ž

1

n

Ý jit n is1

/

1 s n

2

wX S nj w F

1 n

2

< w < 2l nj s

1 n

l nj .

The bounded eigenvalue condition has been introduced in place of the traditional orthogonality assumption by Chamberlain Ž1983. and Chamberlain and Rothschild Ž1983., in a static context. The resulting model is named ‘approximate factor model’. A somewhat different assumption, based on dynamic eigenvalues ŽBrillinger, 1981., is introduced in Forni and Lippi Ž1998., where the ‘approximate dynamic factor model’ is proposed and the representation theorems in Chamberlain and Rothschild Ž1983. are generalized to the dynamic framework.

4. Micro and macromodels In Section 3, we have established and commented upon a model for the independent variables. Here we give a general representation of micromodels linking dependent to independent microvariables. 7 Let us begin by an example. 7

For more details, see Forni and Lippi Ž1997., Chapters 6 and 7.

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Assume that agent i determines the variable yi t in the following way: yi t s d i E Ž x i t < Ity1 . ,

Ž 13 .

where x i t is an independent variable, EŽP< Ity1 . is the expectation conditional on the information set available at time t y 1, d i is a deep parameter. We recall that deep parameters are parameters that belong to the structural functions, utility or production functions for example, determining agents’ behavior. In general deep parameters are not directly observable. However, they can be recovered from estimated parameters if the model is over or just identified, that is, roughly speaking, if the number of estimated parameters is greater or equal to the number of deep parameters. Lastly, assume that x i t s u t q a u ty1 q j i t where u t is a common white noise with unit variance, while j i t is an order-one idiosyncratic moving average:

j i t s hi t q bihi ty1 . For simplicity we assume hi t and hjt orthogonal at all leads and lags. The model of agent i contains four parameters: d i , a , bi and sh2i , determining both the independent and the dependent variable. Notice that the parameter a is not agent specific, i.e., is equal for all the agents. To solve for the conditional expectation appearing in Eq. Ž13. we have to make some assumption about the set Ity1. Two simple alternatives are as follows. ŽA. Agent i observes and employs separately both the common and the idiosyncratic component of x i t . If this is the case yi t s d i a u ty1 q d i bihi ty1 x i t s u t q a u ty1 q hi t q bihi ty1 .

Ž 14 .

ŽB. Agent i observes and employs only the variable x i t , with no distinction between the components of yi t . If this is the case, in order to determine EŽ x i t < Ity1 . we must resort to the univariate Wold representation of x i t x i t s u t q a u ty1 q hi t q bihi ty1 s e i t q d i e i ty1 ,

Ž 15 .

where d i and se 2i s varŽ e i t . are determined by the equations 2

var Ž x i t . s se 2i Ž 1 q d i . s 1 q a 2 q sh2i Ž 1 q bi2 . cov Ž x i t , x i ty1 . s se 2i d i s a q sh2i bi .

Ž 16 .

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This system will provide two reciprocal values for d i , the one smaller than unity in modulus being the solution. Thus we end up with the equations yi t s d i d i e i ty1 x i t s e i ty1 q d i e i ty1 , i.e., using Eq. Ž15.: yi t s

d i di L 1 q di L

Ž u t q a u ty1 q hi t q bihi ty1 .

Ž 17 .

x i t s u t q a u ty1 q hi t q bihi ty1 . We can stop here with this example. What we want to retain is that in both cases we end up with a couple of linear dynamic equations linking yi t and x i t to the individual and common shocks u t and hi t , whose coefficients are functions of the common and individual parameters d i , a , bi , sh2i . Under assumption ŽA. above such functions are elementary manipulations, whereas under assumption ŽB. they imply taking the roots of the algebraic system of Eq. Ž16.. Now, these features of our example are quite general. Even though the micro-models employed as microfoundations of dynamic macromodels can be much more complicated than in the example above, the general procedure already outlined at the beginning of Section 2 can now be described more precisely as follows. Ža. One starts with intertemporal objective functions and dynamic equations for the independent variables, both depending on deep microparameters. Žb. Then one must solve algebraic equations resulting from the intertemporal optimization and possibly from the procedure necessary to obtain the Wold representation of the independent variables Žas in the example, case ŽB... The coefficients of such algebraic equations are simple functions of the deep microparameters Žas in Eq. Ž16... Žc. The final result is a system of linear dynamic equations, like Eq. Ž14. or Eq. Ž17., linking the variables of interest to the micro shocks, whose coefficients result from algebraic combinations of the deep parameters and the roots of the algebraic equations just mentioned in Žb.. Thus, sticking for simplicity to the case of two variables yi t and x i t , a micromodel can be defined as Ži. A set G : R c = R s. This is the admissibility region for the microparameters, where c is the number of common microparameters while s is the number of individual microparameters. We can assume that G is open and connected. Žii. A function associating with any element of G a system of linear dynamic equations linking the variables yi t and x i t to the common and idiosyncratic shocks. The coefficients of such system are real functions defined on G , continuous on G and analytic on G with the exception of a negligible subset, i.e., a subset of Lebesgue measure zero.

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In the example above, we have one common parameter, namely a , so that c s 1, and three individual parameters, namely d i , bi and sh2i , so that s s 3. A possible definition for G is given by the constraints 1 ) a ) y1, 1 ) bi ) y1, sh2i ) 0. Moreover, in case ŽA. the coefficients of Eq. Ž14. are elementary functions of the deep parameters, while in case ŽB., Eq. Ž17. contain both deep parameters and d i , which is a function of deep microparameters through system Ž16.. To see why in both cases the coefficients are analytic we must recall that the roots of algebraic equations are analytic as functions of their coefficients, with the exception of those values of the coefficients corresponding to multiple roots. It is reasonable to require that multiple roots may occur only for a negligible subset of G . Lastly, notice that, according to the above definition, in our example we have two different micromodels, depending on whether we make assumption ŽA. or ŽB. on the information set Ity1. Once the micromodel has been defined we must define the macromodel. Let us firstly go back to our example. By aggregation over n agents we have n

n

Yt s a u ty1 Ý d i q Ý d i bihi ty1 is1

is1 n

X t s n Ž u t q a u ty1 . q Ý Ž hi t q bihi ty1 . is1

in case ŽA., while in case ŽB. n

Yt s Ý is1

d i di L 1 q di L

n

Ž u t q a u ty1 . q Ý is1

d i di L 1 q di L

Ž hi t q bihi ty1 .

n

X t s n Ž u t q a u ty1 . q Ý Ž hi t q bihi ty1 . is1

The natural definition of the macromodel corresponding to n agents and a given micromodel is the following couple of objects. ŽI. Assuming for simplicity that G s R c = R s, the set

which is the admissibility set of the macromodel. ŽII. The function, obtained by simply summing both sides of the micromodel, associating with any element p of Gn a system of dynamic equations linking Yt

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and X t to the common and to the idiosyncratic shocks. The coefficients of such system are real functions defined on Gn , continuous on Gn , analytic on Gn with the exception of a negligible subset. In Section 3 we have seen that the importance of the aggregate idiosyncratic component as compared to the aggregate common component, as measured by the variance ratio, vanishes when n tends to infinity. Thus, assuming a huge number of agents, we can drop the idiosyncratic component in the macromodel. In our example we find n

Yt s a u ty1 Ý d i is1

X t s n Ž u t q a u ty1 . , and n

Yt s Ý is1

d i di L 1 q di L

Ž u t q a u ty1 .

X t s n Ž u t q a u ty1 . in cases ŽA. and ŽB. respectively. Notice that dropping the idiosyncratic components in the equations just above does not mean that the idiosyncratic component does not play any role in the macromodel. This is what happens in case ŽA., but in case ŽB. the coefficients of the macromodel depend on the d i ’s and therefore, through the system Ž16., on sh2i and bi . More in general, when there are h common components we can write the macromodel as Yt s a1 Ž p, L . u1 t q a2 Ž p, L . u 2 t q PPP qa h Ž p, L . u h t Ž 18 . X t s b 1 Ž p, L . u1 t q b 2 Ž p, L . u 2 t q PPP qbh Ž p, L . u h t where a s Ž p, L. and bs Ž p, L., s s 1, h, are power expansions in L, whose coefficients are functions defined on Gn , continuous on Gn , analytic on Gn with the exception of a subset of Lagrange measure zero. The property that the coefficients of the power expansions in Eq. Ž18. are analytic as functions defined on Gn is crucial. For, as we are going to show by some examples in the following sections, basic properties like cointegration, Granger non-causality, etc., result in simple algebraic relationships between the coefficients of equations like Eq. Ž18.. And since such coefficients are analytic on Gn , we can resort to an important proposition, the AlternatiÕe Principle, stating that such relationships hold either everywhere on Gn or only for a zero Lagrangemeasure subset. Equivalently, if we may find a point p˜ in Gn such that cointegration, Granger non-causality, etc., does not hold for p, ˜ then cointegration, Granger non-causality, etc., holds only for a negligible subset of Gn . 8 8

For the Alternative Principle see Forni and Lippi Ž1997., Chapter 6.

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5. Cointegration Let us firstly recall the definition of cointegration Žsee Engle and Granger, 1987 for a general definition and basic results.. Assume that yt and x t are non stationary but that Ž1 y L. Ž yt x t . is a stationary vector. The variables yt and x t are cointegrated if there exists a real c such that yt y cx t is stationary. From a theoretical viewpoint cointegration is usually interpreted as an equilibrium relationship Žbetween yt and x t in our simple case. in a linear stochastic non-stationary context. For example, cointegration of income and consumption—the most studied example—is a consequence of some optimum intertemporal consumption models: firstly, a desired level of consumption relative to the level of income is obtained; such level is never actually obtained because of shocks affecting both consumption and income and adjustment costs preventing the agent from immediately reaching the desired level; however, the dynamic behavior of the agent is such that the difference between income and consumption Žin logarithms. has finite mean and variance. 9 It is natural to ask whether micro cointegration implies macro cointegration. If this is the case, when micro cointegration is a property of the micromodel, testing for cointegration on aggregate data would make sense as a test of micro theory. Let us consider again a very simple example. Suppose that the micromodel is:

Ž 1 y L . yi t s d i a i u 1 t q d i b i u 2 t q Ž 1 y L . a i u 3 t Ž 1 y L . x i t s a i u 1 t q b i u 2 t q Ž 1 y L . bi u 3 t . For simplicity here we have no idiosyncratic component. Two of the common shocks, namely u1 t and u 2 t , are permanent, whereas the third is transitory. We assume that the coefficients a i , bi , d i , a i and bi are the deep microparameters Žthere are no common microparameters.. The microvariables yi t and x i t are cointegrated for any i, since yi t y d i x i t is obtained by integrating Ž1 y L.Ž a i y d i bi . u 3 t and is therefore stationary. Summing over individuals both sides of the above equation it is seen that cointegration of the macrovariables requires the existence of a real c such that n

n

n

n

c Ý a i s Ý d i a i , c Ý bi s Ý d i bi . is1

is1

is1

Ž 19 .

is1

There are two important cases in which a c fulfilling Eq. Ž19. exists: Ža. If d i s d j for any i and j. In this case c s d i . Žb. If there exists a t such that bi s ta i for any i. In this case c s Si d i a irSi a i . The economic meaning of the above conditions is simple. If x t is consumption and yi t is income, condition Ža. means that all agents share the same long-run 9

The results presented in this section are drawn from Forni and Lippi Ž1977.: Chapter 9, Lippi Ž1988. and Gonzalo Ž1993..

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propensity to consume. On the other hand, when condition Žb. holds, the shocks u1 t and u 2 t are ‘redundant’, this meaning that all incomes are driven by the shock u1 t q t u 2 t . This in turn implies that all individual incomes are pairwise cointegrated. More generally, if a c fulfilling Eq. Ž19. exists then n

n

n

n

Ý d i bi Ý a i s Ý d i a i Ý bi , is1

is1

is1

Ž 20 .

is1

which has conditions Ža. and Žb. as particular cases. Since we have assumed five individual microparameters and no common microparameters, the set Gn is an open connected subset of R 5n. Since Eq. Ž20. describes an algebraic hypersurface in R 5n, the subset of Gn fulfilling Eq. Ž20. is negligible. This result is fairly obvious. We have three parameters Ž a i and bi play no role. that are locally free to vary with respect to one another Ž G is an open set.. Then we take all possible combinations of points of G , and therefore a thick subset of R 5n. As an easy consequence the points of Gn that fulfill Eq. Ž20. form a negligible subset. However, the following results, based on the Alternative Principle mentioned in Section 4, are less obvious. Let us modify the example by assuming that a i , bi , d i , a i and bi are not deep parameters themselves but functions of the deep parameters, i.e., functions defined on G , analytic on G with the exception of a negligible subset. Then: 1. Each of the conditions Ža. and Žb. either holds for the whole G or for a negligible subset of G . 2. If there exists a point of G such that neither Ža. nor Žb. holds, then Yt and X t are cointegrated only for a negligible subset of Gn . 3. If Ža. or Žb. holds for the whole G then Yt and X t are cointegrated for any point of Gn . 10 Some remarks are in order. Firstly, the aggregation effect summarized in the theorem above, statement Žii., requires that there are at least two non-redundant common permanent shocks driving the microvariables Žcondition Žb. can hold only for a negligible subset of G .. On the other hand, as we have reported in Section 3, we have strong evidence that several non-redundant permanent common shocks drive the microvariables corresponding to major macrovariables. Secondly, we have already observed that the construction of Gn consists in taking all possible combinations of agents picked up from G . We impose no restriction whatsoever on the agents of a population. Moreover, many of our results refer to negligible subsets of Gn . This means that we assume a state of profound ignorance about the distribution of the microparameters in empirical populations. If informations were available, these might lead to restrictions on the set Gn . However, such restrictions would not necessarily imply fulfillment of Eq. Ž20.. 10

For a detailed treatment of cointegration and aggregation, see Forni and Lippi Ž1997., Chapter 9.

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To conclude this section, let us go back to the example and introduce the following modification: yi t s a˜ i u1 t q b˜i u 2 t q Ž 1 y L . a i u 3 t x i t s a i u 1 t q b i u 2 t q Ž 1 y L . bi u 3 t . Here a˜ i , b˜i , a i , bi , a i and bi are the microparameters. Now the microvariables yi t and x i t are no longer cointegrated, apart from a negligible subset of G . However, this does not imply the impossibility of macro cointegration. The condition is n

n

n

n

Ý a˜ i Ý bi s Ý b˜i Ý a i , is1

is1

is1

is1

which describes a Ž6 n y 1.-dimensional subset of Gn , which is 6 n-dimensional. Thus, unless there is a good reason to assume that condition Ža. or Žb. holds, macro cointegration is not more likely when microvariables are cointegrated than when micro cointegration does not occur. 6. Aggregation is not necessarily bad: the case of consumption There is a very important case in recent literature in which aggregation effects contribute to reconciling theory and empirical evidence. Assume that labor income obeys the equation D x t s aŽ L . et , Ž 21 . 2 Ž . where a L s 1 q a1 L q a 2 L q PPP , and e t is a white noise. According to the Life-Cycle Permanent-Income theory, in its simplest version, consumption should obey D ct s aŽ b . et , where b s 1rŽ1 q r ., r being a risk-free interest rate. This is a famous result by Hall Ž1978.. There are three remarkable features in Hall’s result. Ž1. Consumption changes follow a white noise process uncorrelated with labor income changes at time t y k, for any k ) 0. Ž2. As noticed by Deaton Ž1987., if labor income is highly persistent and b is near to unity, then consumption is more volatile than income. Roughly speaking, a variable x t , driven by Eq. Ž21., is more or less persistent according to whether the long-run change caused by a unitary shock is big or small. In Deaton’s paper persistence is measured by the ratio aŽ1. 2rS a 2h , which was introduced in Cochrane Ž1988.. Ž3. Consumption and income changes are driven by the same white noise, so that the vector Ž D x t D c t . has rank one as a stochastic vector. Ž4. As shown by Campbell Ž1987., total consumption and income, defined as the sum of labor income and asset returns, are cointegrated with cointegrating vector Ž1 y 1..

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The first three features of Hall’s model are at odds with aggregate empirical evidence. Consumption changes are positively autocorrelated and correlated with past values of income: this fact is known as ‘excess sensitivity’ ŽFlavin, 1981.. The variance of consumption changes is much smaller than the variance of income changes; indeed, this is what the original permanent-income theory of Friedman Ž1957. was designed to explain. At the same time, there is evidence indicating that income is persistent. This problem is known as ‘excess smoothness’ of consumption or ‘Deaton’s paradox’. Lastly, consumption changes form a rank two stochastic vector Žsee Forni and Lippi, 1997, Chapter 13.. Regarding Eq. Ž4., evidence is not clear-cut. However, cointegration between aggregate income and consumption is accepted by several authors. 11 Let us show how a very simple heterogeneity assumption can solve the excess smoothness and sensitivity problems. Suppose that income of agent i evolves according to D x i t s Ž 1 q aL . u t q Ž 1 q bL . x i t ,

Ž 22 .

which is a simplification of the example used in Section 4. We suppose that the idiosyncratic shocks x i t are orthogonal to one another and that they share the same variance: sx2i s sx2 . Notice that there are two common parameters, a and b, no individual parameters and that different incomes differ only for the idiosyncratic term x i t . Then assume as in Section 4, Assumption ŽB., that agent i observes only D x i t , rather than its components. There exist a real d, < d < F 1, and a white noise hi t such that D x i t s Ž 1 q dL . hi t s Ž 1 q aL . u t q Ž 1 q bL . x i t Žnotice that since a, b and sx2 are common, the coefficient d does not depend on i .. Lastly, apply Hall’s theory to agent i to obtain D x i t s Ž 1 q dL . hi t s Ž 1 q aL . u t q Ž 1 q bL . x i t 1qdb D c i t s Ž 1 q d b . hi t s Ž 1 q aL . u t q Ž 1 q bL . xi t . 1 q dL Indicating by x t and c t per-capita magnitudes we get D x t s Ž 1 q aL . u t 1qdb D ct s Ž 1 q aL . u t . 1 q dL

Ž 23 .

In general d / a, so that D c t is not a white noise and is correlated with the past of D x t . Moreover, it is not difficult to see that, if d is negative and a is positive, then we have both a persistent income and a smooth consumption. 11

See for instance Davidson et al. Ž1978., Engle and Granger Ž1987., Campbell and Deaton Ž1989..

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What about cointegration? For brevity, we do not introduce explicitly assets and total income in the model. However, the discussion in Section 5 will be sufficient to convince the reader that, despite heterogeneity, cointegration is retained in the macro model, consistently with empirical evidence. This because in this model, by property Ž4., all individual long-run propensities to consume are equal to 1. The possibility that heterogeneity and incomplete information might explain the excess sensitivity andror the excess smoothness puzzles, has been noted in Lippi Ž1990., Goodfriend Ž1992. and Pischke Ž1995.. 12 In Pischke Ž1995. an important step forward has been obtained by using information coming from micro data. Pischke found that model Ž22., estimated for a USA panel of household data, provided a positive a, a negative b, and a sx2 much bigger than su2 , so that negative autocorrelation prevails in individual incomes. As a consequence d is negative, so that per-capita consumption changes, according to model Ž23., should exhibit positive autocorrelation, positive correlation with past income change and low variance, consistently with empirical evidence. Although extremely interesting, model Ž23. is a rank-one vector. For that matter, only one common shock is present in the micromodel, so that both the macrovariables are driven by the same shock. On the other hand, we know that only one common shock in individual incomes is unrealistic. Thus let us consider a micromodel for incomes with two common shocks: D x i t s Ž 1 q a1 i L . u1 t q Ž 1 q a 2 i L . u 2 t q Ž 1 q bi L . x i t .

Ž 24 .

Again, we employ the univariate representation D x i t s Ž 1 q d i L . hi t s Ž 1 q a1 i L . u1 t q Ž 1 q a 2 i L . u 2 t q Ž 1 q bi L . x i t to obtain individual consumption D c i t s Ž 1 q d i b . hi ts

1 q di b 1 q di L

Ž 1 q a1 i L . u1 t

q Ž 1 q a 2 i L . u 2 t q Ž 1 q bi L . x i t . Taking per-capita magnitudes D x t s Ž 1 q a1 L . u1 t q Ž 1 q a2 L . u 2 t 1 n 1 q di b D ct s Ý Ž Ž 1 q a1 L . u1 t q Ž 1 q a2 L . u 2 t . , n is1 1 q d i L

Ž 25 .

where a k s Si a k irn. The rank of Ž D x t D c t . is less than two if and only if 12

Similar results in an overlapping generation framework are found by Galı´ Ž1990. and Clarida Ž1991..

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1 q a1 L 1 q di b det Ý 1 q d L Ž 1 q a1 i L . i iy1



n

1 q a2 L 1 q di b s 0. Ý 1 q d L Ž 1 q a2 i L . i iy1 n

0

Ž 26 .

It is not difficult to get convinced that if the coefficients a1 i , a2 i , bi , s k2 and have sufficient freedom of variation with respect to one another, then Eq. Ž26. will hold only for a negligible subset of Gn . Formally, this is an application of the Alternative Principle: since Eq. Ž26. is an algebraic equations between the coefficients of Eq. Ž25., and since such coefficients are analytic functions defined on Gn , apart from a negligible subset, then if Eq. Ž26. does not hold for a point p˜ of Gn it holds only on a negligible subset of Gn . On the other hand, under mild heterogeneity conditions, a point p˜ g Gn such that Eq. Ž26. does not hold is very easy to find. 13 In conclusion, allowing for heterogeneity of agents in a common-idiosyncratic micromodel, more than one common shock, and limited information of individual agents, only cointegration is implied at the macro level by Hall’s model, whereas the puzzling implications are avoided. Even though several alternative solutions for these puzzles have been proposed, 14 the one outlined above has the advantage of an explicit consideration of heterogeneity and aggregation.

sx2

7. Wold and autoregressive representations of the aggregate vector Dealing with cointegration and with the consumption model we had only to consider representation Ž18. of the aggregate vector, in which on the left we have the aggregate variables and on the right hand side the common microshocks. Now, in order to outline some further results we need the Wold representation of the aggregate vector. Unlike Eq. Ž18., which is structural or semi-structural and has on the right hand side a number of shocks usually bigger than the number of variables on the left hand side, the Wold representation has the form Yt s A11 Ž p, L . Õ 1 t q A12 Ž p, L . Õ 2 t X t s A 21 Ž p, L . Õ 1 t q A 22 Ž p, L . Õ 2 t ,

Ž 27 .

where: Ž1. A i j Ž p, L. is a power series in L whose coefficients are functions of p g Gn . Moreover, A11Ž p,0. s A 22 Ž p,0. s 1, A12 Ž p,0. s A 21Ž p,0. s 0. Ž2. Ž Õ 1 t Õ 2 t . is a vector white noise. Comparison of Eq. Ž27. to Eq. Ž18. shows that Õ1 t and Õ 2 t depend on p; however we do not need to further complicate notation. Ž3. A11Ž p, L. A 22 Ž p, L. y A12 Ž p, L. A 21Ž p, L. does not vanish in the open circle z - 1. 13 14

See Lippi and Forni, 1997, Section 13.7. For a comprehensive review, see Deaton Ž1992..

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Forni and Lippi Ž1997. ŽChapter 10. show that the coefficients of A i j Ž p, L. are continuous on Gn and analytic on Gn with the exception of a negligible subset. Therefore the Alternative Principle, stated in Section 3, can be applied. Furthermore, it is easily seen that the same Principle can also be applied to the coefficients of the power series Bi j Ž p, L. in the autoregressive representation obtained by inverting Eq. Ž27., i.e., B11 Ž p, L . Yt q B12 Ž p, L . X t s Õ 1 t

Ž 28 .

B21 Ž p, L . Yt q B22 Ž p, L . X t s Õ 2 t .

This result has important consequences on several aggregation problems. Here we shall discuss two of them, namely Granger-causality and structural VAR models. Let us firstly recall the definition of Granger-causation. Start with a couple of costationary variables f t and c t and suppose that we are interested in minimum mean square prediction of f tq1 , based on the values taken by f and c at t, t y 1, etc. We say that c does not Granger-cause f if the prediction of f tq1 based on both f and c is equal to the prediction of f tq1 based on f only. In model Ž28., Granger-causality can be discussed in terms of the power series Ž Bi j L.. Since representation Ž28. has been obtained by inverting Eq. Ž27., we have that B11Ž p,0. s B22 Ž p,0. s 1, B12 Ž p,0. s B21Ž p,0. s 0, so that the Eq. Ž28. are the projections of Yt and X t respectively on past values of Yt and X t . By definition, Yt does not Granger-cause X t if B21Ž p, L. s 0, this meaning that past values of Yt do not help in predicting X t once the information contained in the past of X t has been fully exploited. The Alternative Principle entails that if there exists a p˜ g Gn such that Yt Granger-causes X t , then the subset of Gn where Yt does not Granger-cause X t is negligible. Now the question is: assuming that yi t does not Granger-cause x i t , can we conclude that Yt does not Granger-cause X t? The answer is negative and an illustration of the result can be given by using the example of Section 2: yi t s a i x i t x it s Ž 1 q ai L. uit . Like in Section 2 we assume that the u i t are mutually orthogonal at any lead and lag and that there are two agents. The aggregate equations are Yt s a1 Ž 1 q a 1 L . u1 t q a2 Ž 1 q a 2 L . u 2 t X t s Ž 1 q a 1 L . u1 t q Ž 1 q a 2 L . u 2 t . The corresponding Wold representation is obtained by normalizing: Yt a1 Ž 1 q a 1 L . sQ Xt 1 q a1 L

ž/ ž sQ

ž

a2 Ž1 q a 2 L . 1 q a2 L

a1Ž 1 q a 1 L . y a 2 Ž 1 q a 2 L .

Ž a1 y a2 . L

/ž

1 y1

ya2 a1

a1 1

a2 1

Õ1 t Õ2 t

/ž /ž /

a1 a 2 Ž a 1 y a 2 . L a1 Ž 1 q a 2 L . y a 2 Ž 1 q a 1 L .

Õ˜ 1t

/ž / Õ˜ 2 t

,

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where Q s Ž a1 y a 2 .y1 . The Ž2,1. entry of the corresponding autoregressive representation is

Ž a2 y a1 . L . Ž a1 y a2 . Ž 1 q a 1 L . Ž 1 q a 2 L . Thus if a1 / a 2 and a 1 / a 2 the macrovariable Yt Granger-causes the macrovariable X t , even though no such Granger-causation occurs at the micro level. Here again we see how aggregation effects occur when both the independent variables and the responses of the agents are heterogeneous. On the other hand, given a micromodel, if there exists a point in Gn for which such aggregation effect occurs, that same effect occurs almost everywhere in Gn . Let us now turn to VAR models. Suppose that a VAR is estimated for the macrovariables Yt and X t : B Ž L.

Yt V1 t s . Xt V2 t

ž/ ž /

Let AŽ L. s B Ž L.y1 and write Yt V1 t s AŽ L . . Xt V2 t

ž/

ž /

Suppose also that according to our theory the variables Yt and X t are driven by a supply shock W1 t and a demand shock W2 t , and that Ži. W1 t and W2 t have unit variance and are orthogonal at any lead and lag, Žii. the shock W1 t has no contemporaneous effect on X t . This is sufficient to identify W1 t and W2 t , and the matrix CŽ L. such that Yt W1 t s CŽ L. , Xt W2 t

ž/

ž /

Ž 29 .

where C21Ž0. s 0. Now assume that the microvariables are driven by h common supply shocks and k common demand shocks

°e. ¶ 1t

c11 i Ž L . yi t s xit c 21 i Ž L .

ž / ž

c12 i Ž L . c 22 i Ž L .

/

.. eht , h1t ... hk t

Ž 30 .

¢ß

where c11 i Ž L. and c 21 i Ž L. are 1 = h while c 21 i Ž L. and c 22 i Ž L. are 1 = k. Moreover, assume that c 21 i Ž0. s 0, so that our identification criterion is ‘correct’,

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i.e., not inconsistent with the underlying micromodel. Aggregating Eq. Ž30. and using Eq. Ž29. we obtain

°e. ¶ 1t

W1 t y1 S c11 i Ž L . s CŽ L. W2 t S c 21 i Ž L .

ž /

ž

S c12 i Ž L . S c 22 i Ž L .

/

.. eht , h1 t ... hk t

¢ß

where W1 t and W2 t appear as linear combination of the e st ’s and the hst ’s. The question is: can we state that W1 t is a combination of the micro supply shocks e st only, so that it is reasonable to call W1 t an aggregate supply shock, or a mixing occurs? Similarly, can we state that W2 t is a demand shock? The answer is that, if the mixing occurs for one point of Gn , then it occurs almost everywhere. Moreover, a mild heterogeneity is sufficient to generate the mixing. Thus, under heterogeneity, the aggregate supply shock is a combination of both the supply and demand microshocks. 15 8. Conclusions In this paper, we have shown a number of unpleasant aggregation effects. If agents are heterogeneous, a macroequation can be dynamic even though the corresponding microequations are static. Cointegration is destroyed by aggregation, unless either the micro cointegration coefficients are equal, or there is only one permanent common shock driving all the microvariables. When consumers have limited information, the cross-correlation and volatility properties of consumption and income changes implied by Hall’s model are lost at the macro level. In general, unidirectional Granger-causality is not robust with respect to aggregation. Lastly, the shocks appearing in a structural VAR representation result from a mixing of both corresponding and non-corresponding micro shocks. We can summarize all of these results by saying that when the representativeagent assumption is dropped and heterogeneity among individuals is introduced, we cannot expect that a macro model shares the same dynamic properties as the underlying micro model. As a consequence, given an estimated macroequation, we cannot interpret its dynamic shape or other features as revealing something about the behavior of individual agents. In the same way, if the estimated macro parameters fail to fulfill the restrictions implied by the micro theory, this is not a 15 The fact that causality relations can be destroyed by aggregation has firstly been pointed out in Lippi Ž1988.. Results on VAR models are presented in Blanchard and Quah Ž1989.. Here, we have followed Forni and Lippi Ž1997., Chapters 11 and 12, in which further references can be found.

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good reason to reject the micro theory. Thus, on one hand, aggregation is not so bad after all, since theory could be reconciled with evidence, as we have seen for the permanent-income model. But, on the other hand, it should be recognized that additional difficulties arise in macroeconomic modeling. In order to obtain testable implications at the macro level, information on the joint behavior of individual independent variables is needed. Here, we have proposed a model for the independent variables: the dynamic factor analytic model, generalized to allow for cross-correlated idiosyncratic components. This model is flexible enough to accommodate a large amount of heterogeneity, while retaining a reasonably parsimonious parameterization. A remarkable feature of the model is that, when the number of individuals is large, the idiosyncratic components die out with aggregation. This enormously simplifies things, since microvariables moving in a huge-dimensional space are made consistent with macrovariables driven by a small-dimensional vector of shocks. Unfortunately, long series of individual data are seldom available, so that information on the number of common shocks and the cross-sectional distribution of individual response functions may be very difficult to collect. Nevertheless, we do not think that such difficulties should convince us to stick to the representative-agent practice, which amounts to transforming a complete lack of information about a distribution in the assumption that such distribution is concentrated on a single point of the microparameter space.

Appendix A In this appendix, we recall some definitions and results on vector stochastic processes employed in the text. A lively, though fairly technical, presentation of the use of stochastic processes in macroeconomics is Sargent Ž1987.. Stationary Õector processes. For simplicity let us focus on 2-dimensional vectors. Start with a probability space Ž V , F , P ., and consider the space of all stochastic variables defined on V with finite mean and variance, call it L2 Ž V , F , P .. A 2-dimensional process on Ž V , F , P . is a function associating with any t g Z a vector Ž x t yt . with x t and yt belonging to L 2 Ž V , F , P .. Now consider an integer k and the matrix Gt , k s

ž

cov Ž x t , x tyk .

cov Ž x t , ytyk .

cov Ž yt , x tyk .

cov Ž yt , ytyk .

/

.

The process Ž x t yt . is covariance stationary, stationary for short, if Gt, k is independent of t, i.e., if the covariances between Ž x t yt . and Ž x t y k Yt y k . depend only on the lag k. In particular, the matrix Gt,0 , i.e., the variance covariance matrix of Ž x t yt ., must be constant. If Ž x t yt . is stationary we write Gk instead of Gt, k . The components of a stationary vector are also said costationary, this

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meaning that they are both stationary and that the cross covariances between them, at any lag or lead, are constant. Vector white-noise processes. Very important stationary processes are those for which Gk s 0 for k / 0. This means that there is no correlation between x t and x t y k , yt and yt y k , x t and yt y k , for any k / 0. Such processes are called white noises because their realizations do not exhibit any particular periodic pattern Žlike the white color or a noise.. ARMA processes. Very often in applications stationary vector processes are obtained as the superimposition of difference equations to a white-noise process, as in the following example x t s ax ty1 q byty1 q u t q a u ty1 yt s cx ty1 q dyty1 q Õt q b Õty1 ,

Ž 31 .

where Ž u t Õt . is a white noise. Usually the autoregressive part, AR, i.e., the link between the variables x t and yt and their values at t y 1, is stable, so that it would produce dying solutions if the white noise did not provide, so to speak, energy to the system. Eq. Ž31. also contain an averaging of u t and Õt with their values at t y 1. Such moving averages, MA, occur in many economic models. Models like Eq. Ž31., with the assumption that the autoregressive part is stable, are called vector ARMA models and their stationary solutions vector ARMA processes Žalso called VARMA.. ARIMA models. Almost all major macroeconomic variables are evidently non-stationary. In the context of stochastic macroeconomics non-stationarity is modeled either by superimposing a deterministic trend to a stationary process, like in x t s a q b t q c t , or x t s a q b t q g t 2 q c t , where c t is stationary, an ARMA for example; or by modeling as an ARMA the first difference of the variable, like in

Ž x t y x ty1 . s a Ž x ty1 y x ty2 . q u t , with u t white noise. In this case, if < a < - 1 the variable x t y x ty1 is stationary whereas the variable x t , resulting from ‘integration’ of x t y x ty1 , is non-stationary. These variables are called ARIMA Žautoregressive, integrated, moving average.. Wold representation. Consider the following 1-dimensional ARMA x t s a x ty1 q u t ,

Ž 32 .

with < a < - 1 and u t white noise. Quite obviously this autoregressive process has the infinite moving average representation x t s u t q a u ty1 q a 2 u ty2 q PPP qa n u tyn q PPP .

Ž 33 .

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It is possible to show that a wide class of vector processes, including ARMA and therefore almost all economically interesting processes, possess a representation which generalizes Eq. Ž33.. In our 2-dimensional case, in matrix notation, xt u u u u s t q A 1 ty1 q A 2 ty2 q PPP qA n tyn q PPP , yt Õt Õty1 Õty2 Õtyn

ž/ ž/ ž / ž /

ž /

Ž 34 .

where Ž u t Õt .X is a 2-dimensional white noise, while the matrices A n are 2 = 2. Provided that the matrices A n fulfill some standard conditions, representation Ž34. is known as the Wold representation of the stationary process Ž x t yt .X . Representation Ž34., though not necessarily endowed with an immediate economic meaning, is strictly linked to VAR Žvector autoregression. models, much in the same way as Eq. Ž33. derives from Eq. Ž32..

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