Causality in Economics∗

Stephen F. LeRoyUniversity of California, Santa Barbara

March 28, 2005

Abstract

Formal analysis of causal relations using graphical methods has becomeincreasingly popular in the natural and social sciences, but has been used muchless in economics. The reason may be that heretofore graphical methods havebeen based on models that are structural in the sense that the equality symbol istaken to denote causation directly. We argue that current economic models arenot structural in this sense. This paper proposes a formal analysis of causalitythat applies to economic models and develops its properties. Among otherresults, it is shown that Granger causality can be connected to causality onlyunder very strong assumptions. Also, it is shown that graphical methods arenot likely to Þnd much application in determining causal orderings as deÞnedin this paper in economic models.

Formal analysis of causal relations using graphical methods has become increas-ingly popular in the natural and social sciences. Pearl [22] is a good sample of thiswork. Graphical methods are a development of analytical techniques that originatedin economic theory�speciÞcally, they take as their basis structural economic mod-els as deÞned by the Cowles group half a century ago. The standard reference oncausation is Simon [26].The term �structural� was was given several distinct, though related, meanings

by the Cowles economists, as has frequently been observed. At a minimum, the termrefers to the distinction between the structural form and the reduced form of a model.In the structural form each internal variable is expressed as a function of some otherinternal variables and some external variables, whereas the reduced form referred tothe solution of a model, in which each internal variable is expressed as a function ofthe external variables alone. As the term implies, the structural form was viewed asmore fundamental than the reduced form. It was seen as containing information not

∗First draft August 25, 1999. I am indebted to Nancy Cartwright for many discussions andmuch correspondence. I have also received comments from Daniel Hausman, Damien Fennell andJulian Reiss.

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present in the reduced form, such as exclusion restrictions. For example, a structuralsupply-demand system might specify that one or several of the external variablesaffect demand but not supply, or vice-versa. These restrictions were used to identifystructural coefficients.In many analyses �structural� has a further meaning: a structural equation is

one in which the right-hand side variables are causes of the left-hand side variable.In structural equations so deÞned the equals sign has the meaning of the assignmentoperator in computer languages, as Pearl [22] observed. This is distinguished from itsmeaning in mathematics, which by deÞnition treats the right-hand side and left-handside of an equation symmetrically. Graphical analyses of causation have heretoforebeen based on this causal interpretation of structural equations: the graphical rep-resentation of the equals sign is an arrow from the right-hand side variables to theleft-hand side variable.1

With a few exceptions, economists have been conspicuous in their absence fromthese developments in recent years. It is true that the topic of causality has been ofpassing interest to economists�witness Granger causality�but there is essentiallyno connection between the lines of inquiry that economists have pursued and thegraphical methods used in the other disciplines. It is clear why economists have notadopted the new graphical methods: economic models use the equality symbol withits usual mathematical meaning, not with the meaning of the assignment operator.2

Therefore economic models are not expressible as graphs, at least insofar as graphsare based on the causal interpretation of the equality symbol. That being so, econo-mists� lack of interest in graphical analyses of causation is not surprising. Further,economic models in the tradition of general equilibrium theory (including modernmacroeconomics) make no use of the structural form/reduced form distinction.These issues require more careful examination. We begin with Simon, whose

analysis of causation is in fact completely different from that implicit in the graphical

1A third meaning of �structural�, more basic than those discussed in the text but less directlyrelevant to the current discussion, has to do with invariance under intervention. The Cowleseconomists implicitly distinguished two classes of hypothetical experiments: (1) determining theeffects of routine realizations of external variables, and (2) analyzing changes in model structure.There is no formal justiÞcation for the distinction since hypothetical model shifts can always be

parametrized, allowing the analyst to represent an intervention on the structure of a model via ahypothetical shift in an external variable. Thus in a model that is structural in this sense coefficientscan be treated as if they were external variables: an intervention on one or several model coefficientsby deÞnition leaves other coefficients unchanged.This conception of structural models has left its tracks in current practice in macroeconomics:

the Lucas critique, with its assertion that policy change is properly modeled as an interventionon parameters rather than variables, is an example. The distinction between deep and shallowparameters is another.

2Some contemporary studies of causation in economic models carry over the interpretation ofright-hand side variables as causes of the left-hand side variable. For example, in a supply-demandmodel Heckman [12] justiÞed putting quantity on the left-hand side and price on the right-hand sideon the grounds that in competitive models individuals are modeled as price-takers. It is not clearto what extent Heckman�s formal analysis depends on this interpretation.

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treatments. Contrary to the implication in the preceding paragraph and the pre-sumption in many expositions of graphical analysis of causation, Simon�s deÞnitionof causal orderings does not require a model that is structural in the sense that theequals sign is interpreted as causal.

1 Causality

1.1 Notation

One problem that the reader of the literature on causality encounters is that termi-nology is sometimes not clearly deÞned and, further, many of the terms are used withdifferent meanings by different authors. To forestall confusion we begin by deÞningthe terminology used here. It seems to me that these deÞnitions are close to stan-dard usage, insofar as standard usage exists, but some analysts appear to disagree(for example, Hoover [15], p. 171, referred to the paper preceding this one, which hassubstantially the same terminology, as offering a �complex and difficult terminologicallandscape�).One essential distinction, stressed in logic and mathematics but often blurred in

philosophical discourse and economic analysis, is that between variables and con-stants.3 A variable is the argument or value of a function, while a constant is astandin for a number. By specifying that θ is a constant, the analyst stipulatesthat it makes no sense to consider alternative values of θ�doing so makes no moresense than asking what would happen in mathematics if π took on a value other3.14159. Variables, in turn, may be classiÞed into external and internal variables.Internal variables are determined by the model; external variables are taken as modelinputs. Therefore a model consists of a map from the space of external variablesto that of internal variables. Of course, in some exercises involving formal models,such as data-description exercises, variables are not classiÞed as between external andinternal variables, making it impossible to discuss causation.4 Since causation is thesubject here, we will assume that the analyst is willing to make this speciÞcation. Itis assumed that there are no functional relations linking external variables, but thesupport of each external variable may depend on the value taken on by other externalvariables.The terms �exogenous variable� and �endogenous variable� are often used with

the same meaning as �external variable� and �internal variable�, and the etymologyof the former pair of terms supports this usage. However, econometricians sometimesuse the term �exogenous variable� with a different meaning: roughly, an exogenousvariable is an observable variable of which an unobservable equation error is inde-

3As observed in note 1, the Cowles economists were sometimes unclear as to whether modelcoefficients were to be interpreted as constants or external parameters.

4See Shafer [25] for an extended formal discussion of causation that dispenses almost completelywith the distinction between external and internal variables.

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pendent or mean-independent. Therefore we follow Leamer�s [17] recommendationthat the terms �external variable� and �internal variable� be substituted to avoidconfusion.In multidate models we distinguish between parameters and processes. A para-

meter is a variable, while a process is a collection of variables, one for each elementof some index set representing time. Equivalently, one could deÞne a parameter asa process each element of which is assumed to take on the same value. In manydiscussions the terms �constant� and �variable� are used where we use �parameter�and �process�, but this usage would invite confusion in the present paper because ofthe different deÞnitions of constant and variable presented above. Parameters andprocesses, like variables, may be external or internal.In some other discussions the term �parameter� is used with a different meaning.

For example, Hoover [15] deÞned a parameter as a variable that is subject to directcontrol (p. 61). This deÞnition appears to coincide with our deÞnition of an externalvariable. In some of Hoover�s applications parameters are assumed to take on differ-ent values at different dates, although Hoover did not time-subscript parameters oridentify them as processes, as would seem to be appropriate at least according to thenotation adopted here. The same practice is followed in many other sources (Engle,Hendry and Richard [5] is an example). The merits of Hoover�s terminology are notclear; however, our point here is to set out the notation of the present paper, not toargue against other possible choices.Variables may be either observable to the analyst, or unobservable. External un-

observable variables will be assigned probability distributions, and these will induceprobability distributions on internal variables, both observable and unobservable (as-suming that models have unique solutions, as we do throughout this paper). We willassume that constants are unobservable. The following notation is adopted:

external observable variables or processes xinternal observable variables or processes y

unobservable constants a, b, A,B, Cunobservable external variables or processes ε

external parameters θinternal parameters ψ

Note that this classiÞcation is incomplete; other possibilities, such as internal unob-servable processes (latent variables, in some characterizations) are deleted becausethey are not considered in this paper.

1.2 Simon�s DeÞnition

We begin with a (somewhat unconventional) review of Simon�s deÞnition of causation.Suppose that a model is representable as a linear operator fromRm, a space of external

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variables, onto Rn, a space of internal variables, where m ≥ n. This operator isassumed to be representable by a rank-n n×m matrix C of constants:

y = Cx. (1)

Here (1) is the reduced form.Assume that (1) A is an n×nmatrix of constants with zeros on the main diagonal,

(2) B is an n×m matrix of constants, and (3) A and B satisfy

(I − A)−1B = C. (2)

Under these assumptions the model (1) can be written in the form

y = Ay +Bx, (3)

as is readily veriÞed by substituting the left-hand side of (2) for C in (1).Simon deÞned causal orderings from (3): for two internal variables y1 and y2, y1

causes y2�denoted y1 → y2�if y1 appears in the block of equations that determiney2, and also in a block of equations of lower order (see Simon [26] for deÞnitions ofthese terms). For example, in the model

y1 = b11x1 + b12x2 (4)

y2 = a21y1 + b23x3, (5)

we have y1 → y2 because y1 appears in equation (5), which determines y2, but also inequation (4), which by itself constitutes a lower-order block.5 Formally, the causalordering on Y , the set of internal variables, associated with a given structural modelis a subset of Y × Y ; y1 → y2 means that (y1, y2) is in the ordering.There is a well-known difficulty with this: algebraic operations on the equations

of (3) can apparently alter causal orderings. For example, substituting (4) in (5)results in

y2 = a21b11x1 + a21b12x2 + b23x3. (6)

In the model (4), (6) neither y1 nor y2 causes the other according to Simon�s deÞnition.Different algebraic operations result in models in which y1 and y2 are simultaneouslydetermined, or obey y2 → y1, even though each of these models represents the samelinear operator C. It appears as if apparently innocuous mathematical operationsalter causal orderings.This problem reßects the fact that the matrices A and B associated with a given

C by (2) are not unique. Therefore on Simon�s deÞnition the causal ordering of theinternal variables depends on which of an inÞnite number of pairs of matrices A andB satisfying (2) is chosen.

5This statement, of course, presumes that a21 6= 0. Hereafter this qualiÞcation will not berepeated.

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To avoid the problem posed by the apparent dependence of causal orderings onalgebraic operations, we impose a further restriction: we require that each equationcontain at least one external variable not found in any other equation. Hereafterwe refer to this condition as the exclusion condition.6 The exclusion condition rulesout algebraic operations that involve more than one equation (because if the originalmodel satisÞes the exclusion condition, the modiÞed model will not). For example,the model (4), (6) does not satisfy the exclusion condition: the external variablesthat appear in (4)�x1 and x2�also appear in (6).If the exclusion condition is satisÞed and if in addition the model has the same

number of internal as external variables, then causal orderings are unique. To seethis, note that (1) can be solved to yield

x = C−1y. (7)

If F is deÞned by F = I − C−1, where I is the identity matrix, this becomes

y = Fy + x. (8)

Since F is unique in (8), it is clear that the causal ordering is unique.If m ≥ n, with any C there is always associated at least one pair A and B that

satisÞes the exclusion condition: the fact that C is of rank n implies that one canalways Þnd a square nonsingular matrix C1 and a matrix C2 such that the externalvariables x can be partitioned (perhaps after reordering) into (x1, x2) and (1) can bewritten in the form

y = C1x1 + C2x2. (9)

Premultiplying by C−11 results in

C−11 y = x1 + C−11 C2x2. (10)

As before, if F is deÞned by F as I − C−1, (10) can be written as

y = Fy + x1 + C−11 C2x2 (11)

Here each x1 enters one and only one of the equations. The variables in x2 can enterin any or all of the equations.Damien J. Fennell [6] pointed out that if m > n causal orderings under Simon�s

deÞnition are not unique even if the exclusion condition is imposed. This is sobecause with m > n different subsets of the external variables can be selected tosatisfy the exclusion condition, and each choice implies a different causal ordering.To see this, consider the system

6The subset condition is essentially the same as Hausman�s independence condition ([10], p. 64).See also Hoover [14], p. 103 ff.

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y1 = b11x1 + b12x2 (12)

y2 = b22x2 + b23x3, (13)

in which y1 and y2 are not causally ordered. The exclusion condition is satisÞed bythe presence of x1 in (12) and x3 in (13). However, if (13) is solved for x2 and theresult is substituted in (12), there results the system

y1 = b11x1 + a12y2 − (b12b23/b22)x3. (14)

y2 = b22x2 + b23x3, (15)

where

a12 = b12/b22. (16)

In the system (14)-(15) the exclusion condition is again satisÞed because of the pres-ence of x1 in (14) and x2 in (15). In (14)-(15) we have y2 → y1. Since (12)-(13)is mathematically equivalent to (14)-(15), it follows that causal orderings are notunique.Despite this, causal orderings are unique generically: in (14)-(15) we have y2 → y1

for almost all values of a12; it is only in the special case (16) that (14)-(15) is equivalentto a system in which y2 does not cause y1. Since we have already ruled out nongenericspecial cases (see note 2), it is seen that Fennell�s observation about nonuniquenessof causal orderings when m > n does not involve anything new.What is noteworthy is that assuming that a model satisÞes the exclusion condition

is weaker than assuming that it is structural in the sense that that equality symbolis asymmetric: imposition of the exclusion condition allows renormalization of in-dividual equations (i.e., expressing them so that a different variable appears on theleft-hand side), so it does not matter which variable is located on the left-hand side.It follows that causal orderings as Simon deÞned them are not altered by such renor-malizations. In contrast, under the causality deÞnition based on the asymmetricinterpretation of the equality symbol, renormalizations of individual equations resultin a different model with a different causal ordering.The foregoing discussion is very close to Simon�s development, at least on a gen-

erous reading of Simon. On a superÞcial comparison of the above discussion withSimon�s paper, it appears that the exclusion condition has nothing to do with Simon�sSection 6 discussion of when causal structures are �operationally meaningful�. Infact, however, Simon�s discussion is entirely consistent with the discussion here; theapparent differences are terminological.

Simon�s Section 6 marks a change from the discussion that preceded it in hispaper. Prior to that section Simon did not explicitly incorporate external variablesin his discussion (except in Example 4.2), as that term is used here. His examples

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contained only variables x and constants a (or α). Simon�s x corresponds to our y;his a (or α) corresponds to our x and a. Simon used the terms �exogenous variable�and �endogenous variable�, but he assigned them a meaning that is derived fromhis deÞnition of causal orderings: on Simon�s usage if we have y1 → y2, then y1 isexogenous in the set of equations that determine y2, and y2 is endogenous in thatset.However, in Section 6 in dealing with the fact that algebraic operations can appar-

ently alter causal orderings, Simon considered �interventions� in the a terms, implyingthat in that section he was viewing the a terms as variables that are external in thesense of this paper, as opposed to constants as in the earlier sections.Simon did not distinguish between the coefficients and the intercept terms, im-

plying that he was allowing for interventions in either. Here, in contrast, we are sim-plifying relative to Simon by maintaining the assumption that the coefficient termsare constants (i.e., are not subject to intervention), so that only the intercept termsare treated as external variables. Treating the coefficients as variables would convertwhat is a linear model into a bilinear model. Following Simon here would complicatethe discussion unnecessarily.For Simon, causal orderings are operationally meaningful only if the equations of

a structural model have �individual identities�. The equations of a structural modelhave �individual identities� insofar as interventions can be associated with particularequations or subsets of equations. In the terminology of the present paper, theseinterventions are associated with external variables. Therefore, translating into theterminology of the present paper, Simon�s criterion for operational meaningfulnessis that particular external variables be associated with particular equations. Thiscorresponds exactly to our exclusion condition.Simon stated this explicitly: �The causal relationships have operational meaning,

then, to the extent that particular alterations or �interventions� in the structure canbe associated with speciÞc complete subsets of equations� (p. 65). Continuing,�[w]e found that we could provide [a causal] ordering with an operational basis if wecould associate with each equation of a structure a speciÞc power of intervention, or�direct control.� ... Hence, ... structural equations are equations that correspond tospeciÞed possibilities of intervention� (p. 66).Simon�s discussion would have been clearer if he had explicitly incorporated this

idea in his deÞnition of causal orderings, as we have, rather than implicitly attachingthe relevant condition later as a condition for causal orderings to be operationallymeaningful. This is, of course, a criticism of exposition, not substance.We simpliÞed Simon�s discussion by distinguishing intercepts from coefficients:

we treated intercepts as external variables and coefficients as constants. In con-trast, Simon treated both intercepts and coefficients as external parameters. OursimpliÞcation does not alter the substance of Simon�s argument: it is easy to seefrom examination of examples that if y1 → y2 when the coefficients are treated as

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constants, the same is true when the coefficients are treated as external parameters.7

However, a critical assumption that Simon adopted without discussion, that thecoefficients are all external, greatly limits the applicability of his treatment of cau-sation. Macroeconomists in particular frequently deal with linearized versions ofmodels. In such settings the coefficients�the �shallow parameters� are functions ofa few deep parameters representing preferences, production possibilities and the like.With fewer external than internal variables, there is no possibility that the exclusioncondition is satisÞed.

1.3 Causality as Sufficiency8

Part of the reason Simon�s characterization of causality is not much used currently isthat Simon did not provide a clear explanation of what follows if one variable causesanother. What interventions are admissible if y1 causes y2, but not otherwise? Whatis the interpretation of these interventions? What does the fact that the cause variableis determined in a lower-order subsystem relative to the effect variable have to do withcausation? What is the content of �operationally meaningful� in this context, andwhat is the connection between this concept and the exclusion condition?The best way to supply intuitive content to causality is to consider simple exam-

ples. We will see that in some such cases it is clear that causal statements are notappropriate, while in other cases it is equally clear that they are. Examination of thedifference between these examples will suggest the general principle. This principleis stated more formally in the next subsection.Consider the supply-demand model:

qs = αs + αspp+ αsww (17)

qd = αd + αdpp+ αdii (18)

qs = qd = q, (19)

where qs is quantity supplied, qd is quantity demanded, q is equilibrium quantity, i isincome, p is price and w is weather. Here weather and income are external and theother variables are internal.In the system (17)-(19) the question �What is the effect of weather on the equilib-

rium quantity?� is unambiguous: the effect can be directly calculated from the model.This is so because weather is external. However, if one were to ask �What is theeffect of price on equilibrium quantity?� the appropriate response would be that thequestion is misposed. Price and quantity are both internal; they are simultaneouslydetermined, and neither is causally prior to the other.

7However, the converse is not true: changing an external parameter to a constant reduces theset of possible interventions, implying that the causal ordering may become smaller.

8The material presented in this and the following subsections is drawn from LeRoy [18].

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The reasoning here is worth elaborating. The assumed intervention results inthe price changing from, say, p to p +∆p. The problem is to infer the effect of thisintervention on q. The reason the question is ambiguous is that any of an inÞnitenumber of pairs of shifts in the external variables �weather� and �income� couldhave caused the assumed change in price, and these interventions map onto differentvalues of q. Thus the reason the question is misposed is that it does not give enoughinformation about the intervention being considered to allow a unique answer.The suggestion is that causal statements involving internal variables as causes are

ambiguous except when all the interventions consistent with a given change in thecause variable map onto the same change in the effect variable. One is led to deÞnetwo internal variables as causally ordered when the indicated condition is satisÞed,and not otherwise.Now consider the model

qs = αs + αsww + αsff (20)

qd = αd + αdpp+ αdii (21)

qs = qd = q, (22)

where f is fertilizer. Weather, fertilizer and income are the external variables. Hereeven though q is internal there is no problem with the assertion that q causes p.This is so because all the interventions in weather and fertilizer consistent with agiven change in q map onto the same value of p, as the structure of the model makesobvious.

1.4 A Formal Statement

Let the external set Xj for a particular internal variable yj be the minimal set ofexternal variables such that yj can be written as a function of Xj. Then the model(1) can be written in the form

yj = βjXj j = 1, ..., n, (23)

where Xj is a vector of which the elements are the members of Xj , and βj is aconformable vector. Of course, βj coincides with the j-th row of C with the zeroelements deleted.Suppose that Xi ⊂⊂ Xj, where ⊂⊂ means �is a proper subset of�. Hereafter we

will call this the subset condition. Further, deÞne Xj,i as Xj − Xi (i.e., as the setconsisting of the elements of Xj that are not in Xi). DeÞne Xj,i as a vector of whichthe elements are the members of Xj,i. Suppose in addition that there exists a scalarγj,i and vector δj,i such that (23) can be written in the form

yj = βjXj = γj,iyi + δj,iXj,i. (24)

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Existence of γj,i and δj,i with this property implies that all the interventions in Xiconsistent with a given change in yi have the same effect on yj. Thus all the infor-mation relevant for yj contained in Xi is summarized in yi, so that even though theintervention in Xi associated with a given intervention on yi is ambiguous, there is noambiguity in the effect of the intervention on yj. When γj,i and δj,i exist that satisfythe above property we will say that yi is a simple cause of yj, and will write yi ⇒ yj.Thus yi ⇒ yj means that yi is sufficient for Xi in the determination of yj. We willcall the condition that there exist γj,i and δj,i with the properties just described thesufficiency condition. Thus we have yi ⇒ yj if and only if both the subset conditionand the sufficiency condition are satisÞed.In general, Xi ⊂⊂ Xj does not imply existence of γj,i and δj,i satisfying (24).

Therefore it will not generally be that case thatXi ⊂⊂ Xj implies yi ⇒ yj. However,if Xi ⊂⊂ Xj, there may exist some other internal variable yk such that all theinterventions in Xi consistent with a given change in yi and a given value of ykmap onto the same value of yj. Then we have conditional causation, indicated byyi ⇒ yj|yk. Still more generally, the conditioning set may include several internalvariables rather than just one. The set of possible yk such that yi ⇒ yj|yk is generallynot unique.Two conditions are required for conditional causation. First, yi must be variation-

free: if, in contrast, the variables held constant completely determine yi, it makes nosense to talk about the effect of variations in yi on yj, ceteris paribus. For example,if the conditioning set includes all the external variables, no variation in yi is possible.Second, the conditioning set must be such as to ensure that all values of the externalvariables e(yi) consistent with given yi and yk produce the same value of yj . Thisrequirement, which corresponds to the sufficiency requirement for simple causation,is needed to avoid ambiguity in the effect of variations in the external set for yi onyj .As long as Xi ⊂⊂ Xj, there will always exist some subset (possibly the null set,

if yi ⇒ yj) of the external and internal variables such that yi causes yj conditionalon that set of variables. We will write yi → yj if either yi ⇒ yj or yi ⇒ yj|zk forsome scalar or vector zk. Thus yi → yj under the deÞnition just given is equivalentto Xi ⊂⊂ Xj .Conditional causation raises problems of interpretation. The indicated interven-

tion requires a nonzero change in the variables in Xi, with the changes required tosatisfy a linear relation so as to hold yk constant. Existence of such functional re-lations among external variables appears to conßict with their exogeneity. If thereexists a functional relation among the variables in Xi, then assuming that these vari-ables are external is a misspeciÞcation. Thus the intervention is inappropriate to theassumed model. An example will be discussed below.It is easily veriÞed that the above deÞnition of yi → yj coincides with Simon�s

deÞnition: assuming that the exclusion condition is satisÞed, yi appears in the blockof equations that determines yj and also in a lower-order block if and only if Xi ⊂⊂

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Xj. Thus yi → yj can refer to both conditional causation as deÞned here and Simon�sdeÞnition of causation.

1.5 Causality and Parameter Interventions

Up to now we have taken coefficients to be constants rather than parameters, implyingthat external variables enter additively rather than multiplicatively. This assumptionwas for convenience only. Many problems involving causation do not satisfy thisrestriction. For example, as soon as coefficients are treated as parameters (that is, asexternal variables) rather than constants models become nonlinear. In this section wemake some observations about the consequences of treating coefficients as parameters.Neoclassical macroeconomists stress that analysis of macroeconomic policy changes

requires identifying and estimating �deep parameters� (labeled here external para-meters). This is correct if one is considering changes in policy regimes, and if oneis modeling regime change by parameter interventions, as recommended by the Lu-cas critique (Lucas [20]), at least on some readings. I have argued elsewhere (LeRoy[19]) that policy changes in dynamic models are appropriately modeled either throughinterventions on external parameters or external processes, depending on the ques-tion being asked. An important point is that if regime changes are modeled usingprocess rather than parameter interventions, there is no need to model the depen-dence of internal parameters on external parameters. This is so because if y1t causesy2t when coefficients are treated as constants, the same is true when the coefficientsare treated as parameters, and this is so regardless of the causal ordering amongparameters. The only difference is that in the latter case the external parameters,or some subset of them, are included in the exogenous sets, but this change will notcause failure of the subset and sufficiency conditions, assuming that these are satisÞedwhen coefficients are treated as constants. This point underscores the importanceof Marschak�s [21] observation that analysis of causation does not always require acomplete characterization of a model�s causal ordering.Hoover [13] pointed out that a potentially testable implication of causality is that

interventions on the (external parameters that determine the) probability distributionof the cause variable should not affect the probability distribution of the effect variableconditional on the cause variable. Cartwright [1], p. 57, took issue with this assertion:

If x causes y, then in a two-variable model D(y|x) [the distribution ofy conditional on x] measures the strength of x�s effect on y [Cartwright�snotation has been changed]. Clearly the question of the invariance ofthe strength of this inßuence across envisaged interventions in x is oneof considerable interest in itself. But Þnding out the answer is not atest for causality, neither in the original situation nor in any of the newsituations that might be created by intervention. Even if x causes y in theoriginal situation and continues to do so across all the changes envisaged,there is in general no reason to think that interventions that change the

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distribution of x will not also affect the mechanism by which x bringsabout y, and hence also change the strength of x�s inßuence on y.

This observation appears to be incorrect, at least under the implementation ofcausality analyzed in this paper. Suppose that we write the model as

y1 = θ1 + θ2y2 + θ3ε (25)

y2 = θ4η (26)

where θ1, ..., θ4 are external parameters (or variables; in a static model there is nobasis for distinguishing parameters from variables), and ε and η are independentlydistributed unobserved external variables. For concreteness we will take ε and η tohave zero mean and unit variance. Taking θ4 as a parameter rather than a constantallows the analyst to consider interventions on the standard deviation of y2.In the nonlinear model (25)-(26) y2 causes y1. To see this, note that we have that

the external sets for y2 and y1 are

e(y2) = (θ4, η) (27)

e(y1) = (θ1, θ2, θ3, θ4, ε, η), (28)

which satisfy the subset condition e(y2) ⊂⊂ e(y1). The sufficiency condition for y2 ⇒y1 is also satisÞed. We have that y1 conditional on y2 is distributed with meanθ1 + θ2y2 and standard deviation θ3. Therefore an intervention on θ4 will affect themarginal distribution of y1, but will not affect D(y1|y2).As Cartwright observed in the passage just quoted, it is easy to imagine models

in which interventions on θ4 do affect the parameters of the conditional distributionof y1. However, in such models by deÞnition the conditions for y2 ⇒ y1 will fail,contrary to the assumption. It appears, then, that Hoover�s assertion is correct.

1.6 Causality and Counterfactuals

Analysis of causality inherently involves hypotheticals (or, as they are sometimescalled, counterfactuals). Clear identiÞcation of the assumptions that are needed todetermine the effect of hypothetical interventions is facilitated by formal analysis.An example, adapted from Dawid [4], makes this point.A Þrm has to decide whether to assign employees to the jungle J or the city C.

Employees assigned to the jungle will die D with probability 3/4 (and therefore liveL with probability 1/4), while those assigned to the city will die with probability 1/4.A given employee is assigned to the jungle, and he dies. The manager asks �What isthe probability that he would have died if assigned to the city?" Such hypotheticalquestions are entirely legitimate, and it is worthwhile determining conditions that

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J

C

D

D

L

L

3/4

3/4

1/4

1/4

Figure 1:

are required to arrive at speciÞc answers. In the accompanying Þgure the settingjust described is formalized as a game. The manager decides between assigning theemployee to the jungle or the city, and then a chance player determines whether theemployee dies. The probabilities assigned to the chance player are as shown, and theheavy line denotes the assumed equilibrium path.If the manager plays C instead of J, the outcome depends on the draw in the lower

chance node, not the upper chance node. Further, the relevant probability is thatconditional on D being drawn at the upper chance node. Since we are interested in aconditional probability, we need to know whether the draws at the two chance nodesof Figure 1 are independent. If so, the conditional probability that the employeewill die if assigned to the city is 1/4, equal to the corresponding marginal probability.If the analyst is not willing to assume independence, or to make some alternativeassumption, then the question cannot be answered.In most applications of game theory the question of dependence between the draws

on chance nodes on different paths of play does not come up because these pathsrepresent alternatives (in contrast, game theorists routinely assume independencebetween draws at chance nodes along given paths of play). However, it is obviouslyrelevant here.Specifying whether the draws at the two chance nodes are independent involves

a substantive speciÞcation by the analyst. To see this, consider an alternative.Suppose that there are two types of workers, strong S and weak W. Strong workerswill die with probability 5/8 if assigned to the jungle, and die with probability 1/8

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S

W

J

C

C

J

D

L

D

L

D

L

D

L

5/8

3/8

1/8

7/87/8

1/8

3/8

5/8

1/2

1/2

Figure 2:

if assigned to the city. Weak workers die with probability 7/8 if assigned to thejungle and 3/8 if assigned to the city. The probability that any worker is strong is1/2. The manager cannot distinguish between strong and weak workers in settingthe work assignment. Since 3/4 is the average of 7/8 and 5/8, and 1/4 is the averageof 3/8 and 1/8, this speciÞcation is identical to that given above to any analyst whocannot distinguish strong from weak workers. The game is as follows:The root node is a chance node; then the manager chooses J or C (he cannot tell

which node he is at, and therefore must choose J at both nodes or C at both nodes).The rightmost nodes are chance nodes.Given that the manager chose J and the worker died, an application of Bayes�

theorem shows that the posterior probability that the worker is strong is 5/12 ratherthan 1/2 as in the simpler speciÞcation. Consequently, the probability that theworker would have died if assigned to the city is 13/48 rather than 1/4. In thisversion the analyst no longer treats the chance nodes of Figure 1 as representingindependent events; instead, the event D if J is positively associated with the eventD if C, since Figure 1 does not condition on the worker�s type. However, as Figure 2shows, conditioning on the worker�s type (S or W), these events are independent. Atone level or another the analyst must make an independence assumption if the effectof a counterfactual is to be determined.

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2 Causality and IdentiÞcation

2.1 IdentiÞcation and Exclusion Restrictions

Exclusion restrictions play a central role in determining causal orderings. As is wellknown, they also play a central role in determining whether the parameters in a modelare identiÞed (Fisher [7]). Despite the common role of exclusion restrictions in theirdetermination, causality and identiÞcation are very different notions: causality isan ordering on internal variables, whereas identiÞcation has to do with whether theeconometrician can make inferences about parameter values from the (population)distribution of observable variables. Causality can be deÞned and analyzed withouteven specifying which variables are observable, and in fact this is exactly what wehave done up to now. However, estimation of causal parameters requires assumptionsthat assure identiÞcation, and this goes beyond causality. Whether parameters areidentiÞed depends not just on which variables are excludable from which equations,but also on which variables are observable and what distributional assumptions areimposed on those (external) variables that are not observable.In bringing empirical evidence to bear on assessing the causal relation (or lack

thereof) between variables yi and yj in a given model, then, two questions must bedistinguished. First, are the two variables causally ordered? If not, the question ofidentiÞcation of causal parameters does not come up since the relevant parameter isundeÞned. If the two variables are causally ordered, then the associated parameteris well deÞned conceptually, but it is not necessarily identiÞed.An assumption that plays a central role in assuring identiÞability of causal pa-

rameters is that all unobserved external variables are (statistically) independent ofobserved external variables and of each other. Henceforth we will call this the inde-pendence assumption.9 The role of this requirement is to force the model-builder tostate explicitly what is assumed about causation, rather than burying causation inuninterpreted correlations among external variables.To understand the role of the independence assumption, consider two observed

variables y1 and y2. Knowledge of their joint probability distribution obviously doesnot allow any inference about which variable, if either, causes the other. However,the analyst can certainly deÞne an unobserved variable ε and a parameter β suchthat y1 and y2 satisfy

9The independence assumption is essentially the same as the principle of the common cause, whichsays that if two variables are correlated, then either one causes the other or they have a commoncause (Reichenbach [24]). Here, however, independence is interpreted as a formal restriction onmodels rather than as a philosophical proposition. As such, the suitability of the assumption isevaluated according to whether it is analytically fruitful, rather than according to whether it isconsistent with, for example, the correlation between British bread prices and the sea level in Venice(see Hoover [16] for discussion).

16

y2 = βy1 + ε, (29)

where the speciÞed distribution of ε and the chosen value of β generate the observedjoint probability distribution of y1 and y2. If the model-builder is willing to interprety1 and ε as external variables, then we have y1 ⇒ y2. Of course, an inÞnite numberof pairs of parameters β and random variables ε are available which satisfy thisconstruction, implying that β is unidentiÞed.Correpondingly, the model-builder can simply project y2 onto y1:

y2 = δy1 + η. (30)

Here by construction η is uncorrelated with y1, and the random variable η and parame-ter δ are unique. However, in the absence of further assumptions this decompositionof y2 into y1 and δ has nothing to do with causation; causal variables can be projectedonto effect variables as well as vice-versa.As is obvious, these two decompositions of y2 into y1 and an unobserved random

variable are very different operations: the construction of ε and the assumption thatit is external amount to assuming that y1 ⇒ y2, but the associated parameter isunidentiÞed. In contrast, the construction of η guarantees its uncorrelatedness withy1, but has nothing to do with causation. If we are willing to assume further thatε = η�or, equivalently, that y1 and ε are uncorrelated, or that η is external�we caninterpret the coefficient δ of the projection of y2 on y1 as a causal parameter.One implication of the independence assumption is that if one variable causes

another, the two variables are necessarily correlated.10 In some settings this mayseem counterintuitive. Suppose that yt is generated according to

yt = ayt−1 + bxt + εt, (31)

where xt is a regulator, the behavior of which is generated by

xt = cyt−1. (32)

Then the behavior of yt follows

yt = (a+ bc)yt−1 + εt, (33)

where the external variables are ε1, ε2, .... and y0. The deÞnitions of causation implythat we have xt ⇒ yt and yt−1 ⇒ yt under the (heretofore unstated) assumption thata+ bc 6= 0.10Of course, the converse is not true: if the external sets for two internal variables have a nonempty

intersection, the independence assumption implies that two variables will be correlated. However,if neither external set is a subset of the other, then the two variables are not causally ordered, eitherunconditionally or conditionally.

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However, suppose that the regulator is operated by choosing c so as to minimizethe unconditional variance of yt. This results in a+bc = 0 or, equivalently, c = −a/b,implying that (33) becomes

yt = εt. (34)

In this special case the regulator xt is no longer correlated with yt, and we no longerhave causation: xt ; yt and yt−1 ; yt. This result may seem to run counter to theordinary-language usage of the term �causality�, but it is an unavoidable consequenceof the formal deÞnitions.Under any particular parametrization of a model, imposing the independence

requirement is obviously very restrictive. However, imposing independence on somerelated parametrization of the model amounts only to requiring the modeler to stateexplicitly what he is or is not willing to assume about causation. This is not anunreasonable requirement, if the goal is to arrive at causal conclusions. For example,the modeler who is willing to assume y2 ⇒ y1 instead of y1 ⇒ y2 would generate y1from y1 = γy2 + ξ, with y2 and ξ assumed external and independent. Finally, if themodel-builder were unwilling to assume that neither y1 nor y2 causes the other, hecould specify the parametrization

y1 = βy2 + ε (35)

y2 = δy1 + η. (36)

Here ε and η are independent by the independence assumption, but this impliesneither y1 ⇒ y2 nor y2 ⇒ y1.

2.2 Causality and the Variation-Free Condition

Assume, as in Subsection 1.2, that external vector-valued variables x and internaly are observed, and deÞne u as a vector of unobserved external variables. It isassumed that the covariance matrix of u is nonsingular, so that the realizations of uare variation-free. Then we can write a linear model as

y = Ay +Bx+ u. (37)

Barring other restrictions, coefficient aij is identiÞed�its value can be inferreduniquely from x and the distribution of u�if the number of variables excluded fromequation i at least equals the number of equations in the system less one (see, forexample, Fisher [7]). Barring other restrictions, coefficient aij is underidentiÞed ifthere are fewer than this number of exclusion restrictions.Surprisingly, in such models there turns out to be a close link between the causal

interpretability of coefficients and their identiÞability. The basic result is that coeffi-cient aij represents the effect of internal variable j on internal variable i, conditional

18

on the other variables in equation j being held constant, if and only if aij is identi-Þed. If, in contrast, aij is underidentiÞed then the condition for a conditional causalordering that yj be variation-free fails.Since identiÞability guarantees only that coefficients are associated with condi-

tional causation, not simple causation, the envisioned interventions generally involvemore than one external variable, and are subject to linear restrictions. We arguedabove that such interventions are sometimes difficult to reconcile with exogeneity.However, in special cases this problem does not occur. For example, consider anequation containing exactly two internal variables, but also all but one of the ex-ternal variables. This equation is just identiÞed, implying that the variation-freecondition is satisÞed. Further, the implicit interventions involve only the excludedexternal variable. The supply-demand model presented in the next paragraph Þtsthis description.The supply-demand model analyzed above can be written as

q = αs + αspp+ αsww (38)

q = αd + αdpp+ αdii, (39)

in which weather w affects supply but is excluded from demand, and vice-versa withincome i. Both equations of this model are just identiÞed, so αsp measures the effectof price on quantity holding constant weather, while αdp measures the effect of priceon quantity holding constant income. These, of course, are the slopes of the supplyand demand functions, respectively.If i had not been excluded from the supply function, αsp would not measure the

effect of variations in p on q, cet. par., since if w and i are held constant no variationin p is possible. Correspondingly, αsp is underidentiÞed.

2.3 Causality and Regression

In LeRoy [18] it was pointed out that, under the condition that all external variablesare independently distributed, coefficients associated with causation can be estimatedconsistently by ordinary least squares. This is so because under the stated restrictionvariables representing causes are statistically independent of error terms. It followsthat econometric theory can sometimes be used to determine causality: knowledgethat ordinary least squares results in inconsistency implies absence of causation. Theexample given to illustrate this point was the model

xt = λxt−1 + ut (40)

ut = ρut−1 + et, (41)

where the external variables, et, et−1, ..., e1, u0 and x0, are assumed to be independent.When ρ 6= 0 we have that xt−1 is correlated with ut, so a least-squares regression of xt

19

on xt−1 will not produce a consistent estimate of λ. From the stated result it followsthat xt−1 ; xt.In response to this, Hoover [14] observed that applying the Koyck transformation

results in

xt = (ρ+ λ)xt−1 − ρλxt−2 + et, (42)

in which the parameters ρ+ λ and ρλ are in fact estimated consistently by ordinaryleast squares. Further, separate consistent estimates of ρ and λ are easily calculatedfrom the estimated regression coefficients. Hoover appeared to view this resultas raising questions about the validity of the inference that failure of ordinary leastsquares implies nonexistence of causation, although he did not spell out the argument.In fact, the multiple regression of xt on xt−1 and xt−2 is different from the univariateregression of xt on xt−1. The fact that ordinary least squares is valid in (42) suggests11

that even though we do not have the simple causation xt−1 ⇒ xt, we might have theconditional causation xt−1 ⇒ xt|xt−2, and it can be directly veriÞed from the deÞnitionof conditional causation that this is the case.

2.4 Exogeneity, Causality and Granger-Causality

In many analyses the ideas of exogeneity and causality are closely linked. However, insuch discussions the assumed concept of exogeneity sometimes bears no close relationto the idea of external variables, which is the basis of the deÞnition here of causality.Further, causality usually means Granger-causality. It is necessary to determine therelation between exogeneity and Granger-causality in such discussions and causalityas treated here, and to determine when exogeneity is testable. Sims [27], p. 24, gavea direct answer to the question whether exogeneity is testable:

When, as econometricians estimating models ordinarily do, someoneasserts that a particular variable or group of variables is strictly exogenousin a certain regression, that assertion is, in time series models, testable.�Exogeneity� here is given its standard econometrics textbook deÞnition.Exogeneity tests are thus an easily applied test for speciÞcation error, pow-erful against the alternative that simultaneous-equations bias is present.The usefulness of these speciÞcation tests ought not to be controversial....

Sims� argument is direct. In the model

y = Xb+ u (43)

X is strictly exogenous if E(u|X) = 0. �If exogeneity holds, ... we can add to theright-hand side of (43) the variable Z ... to get

11We use �suggests� rather than �implies� because the converse of the above result�that consis-tency of ordinary least squares implies causation�is not generally true.

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y = Xb+ Zc+ u. (44)

On the null hypothesis that (43) satisÞes the assumptions of the Gauss Markov the-orem, (44) does also, with c = 0. Testing c = 0 by standard methods thus tests thenull hypothesis of strict exogeneity of X in (43).�This argument is difficult to follow. It is not clear whether b and u are intended

to refer to the same constant and random variable in (43) and (44). If so, there isnothing to test: subtracting (43) from (44) gives Zc = 0 directly, implying c = 0. Ifnot, (44) should be written as

y = Xc+ Zd+ v. (45)

In (45) we have d = 0 if Z is uncorrelated with y conditional on X. It is not clearwhat this has to do with whether u is mean-independent of X in (43).A process y2 Granger-causes another process y1 if lagged values of y2 predict y1

conditional on lagged values of y1. If y2 fails to Granger-cause y1 then, accordingto Granger [9], correlations between the two processes can be taken to represent thecausal inßuence of y1 on y2. It shortly was made clear by a number of critics thatthis conception of causality bore no obvious relation to causality as deÞned eitherin ordinary language or in formal analysis.12 However, the point was not made assharply as it might have been because of the lack of a suitable formal deÞnition ofcausality to compare to Granger causality, or so it appears with hindsight. ThedeÞnition of causality developed in this paper makes possible a precise comparisonwith Granger causality.Suppose that processes y1 and y2 are generated by

y1t = a12y2t + b11y1,t−1 + b12y2,t−1 + ε1t (46)

y2t = a21y2t + b21y1,t−1 + b22y2,t−1 + ε2t, (47)

where the independence assumption is satisÞed (so that the external variables ε1tand ε2t are uncorrelated with each other contemporaneously, with their own andeach other�s lagged values, and with the initial values y10 and y20. This model isunderidentiÞed. By inspection the lagged internal variables are causally prior totheir contemporaneous counterparts conditional on each other: y1,t−1 ⇒ y1t|y2,t−1 andy2,t−1 ⇒ y1t|y1,t−1, but y1t is not causally prior to y2t, either simply or conditionally(because the external sets of these variables are the same).13

12Recently the point was made in passing by Heckman [11] in reviewing the Cowles contributionsto econometric theory.

13Of course, there are similar results with the roles of the variables reversed. Hereafter we do notnote these separately.

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Under the restriction a12 = 0 the model generates y1t ⇒ y2t|y1,t−1, y2,t−1, whichgives a precise sense in which the process y1 is causally prior to the process y2. Therestriction a12 = 0, being just-identifying, by itself has no observable implications,and therefore is not testable in the absence of other restrictions. Under the restrictiona12 = b12 = 0, so that y1 is strictly exogenous, we have y1t ⇒ y2,t+j , j = 1, 2, ....Granger-causality is deÞned from the reduced form for the model (46)-(47), writ-

ten as

y1t = c11y1,t−1 + c12y2,t−1 + η1t (48)

y2t = c21y1,t−1 + c22y2,t−1 + η2t, (49)

where c12 = (b12 + a12b22)/(1 − a12a21). Here y2 fails to Granger-cause y1 if c12 = 0.Plainly c12 = 0 is neither necessary nor sufficient for a12 = 0, so Granger-noncausalityis neither necessary nor sufficient for y1t ⇒ y2t|y1,t−1, y2,t−1. Granger-noncausality isnecessary for strict exogeneity, so c12 6= 0 is evidence against y1t ⇒ y2,t+j, j = 1, 2, ....,subject to the usual caveats about sampling error and the like, under the maintainedassumptions of the model. However, Granger non-causality is not sufficient for strictexogeneity, and it is easy to construct theoretical examples with spurious exogeneity(Granger-noncausality without strict exogeneity). Sims [27] expressed the view thatspurious exogeneity is a problem only under exceptional circumstances.

3 Comparing the Causality DeÞnitionsAssume that we have in hand a model that we are willing to treat as structural (inthe sense that it is representable using a causal graph with arrows representing theequals sign). Assume also that the exclusion condition is satisÞed (otherwise theSimon deÞnition is inapplicable). Then we have three distinct deÞnitions of causalorderings: (1) that encoded in the graph, (2) that of Simon (which coincides with thedeÞnition here of conditional causation, as noted), and (3) the deÞnition of simplecausation proposed here. We now show that in general these orderings are different,although they may coincide in special cases.First we produce an example in which the causal ordering under the deÞnition of

Simon is the same as that under the deÞnition offered here, but different from thatimplied by the associated causal graph. This is the birth-control pills/thrombosisexample, which is familiar from the philosophy literature. Taking birth control pillsincreases the risk of thrombosis, but so does pregnancy. In turn, suppose that theincidence of pregnancy is determined by whether or not a woman takes birth controlpills, and also by her rate of sexual activity. In this model pregnancy and thrombosisare internal, while birth control pills and sexual activity are external.Write the model as

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T = btbB + atpP (50)

P = apbB + bpsS. (51)

The question is whether pregnancy, an internal variable, causes thrombosis undereach of the three deÞnitions. According to the causal graph suggested by the accountjust given, with the equality symbol directly representing causation, the answer is yes:the structural equation which has the incidence of thrombosis on the left-hand sidehas pregnancy and birth control pills on the right-hand side, and this is the criterionfor causation.According to Simon�s deÞnition, however, the answer is no, and the same is true

under the deÞnition of simple causal orderings proposed here (the latter conclusionis immediate because y1 9 y2 implies y1 ; y2). In terms of Simon�s deÞnition,pregnancy is determined in the same subsystem as thrombosis, not a lower-ordersubsystem. This is so because the exclusion condition fails: each external variable inthe exogenous set for P also appears in the exogenous set for T . Therefore pregnancyis not causally prior to thrombosis. In terms of our deÞnition of simple causation, thequestion �What is the effect of pregnancy on thrombosis?� is inherently ambiguous.The change in pregnancy might be due to a change in either pill use or sexual activity,and these affect the incidence of thrombosis differently. Until we are told why thepregnancy rate changed, there is no way to answer the question.As this example makes clear, under both Simon�s deÞnition of causation and that

proposed here, adopting language like �the incidence of pregnancy is a determinant ofthrombosis� to describe a model informally is not sufficient to justify a formal state-ment that pregnancy causes thrombosis. One might view this disconnect betweenan informal usage that is virtually unavoidable and the formal deÞnition as a disad-vantage. Perhaps so, but the alternative of using causal language to describe theeffect of one variable on another when the envisioned intervention is not adequatelycharacterized does not seem like an improvement.In other models Simon�s deÞnition gives a different causal ordering from the deÞn-

ition of simple causation proposed here. The deÞnition proposed here gives a sparsercharacterization of causal orderings: whenever y1 causes y2 according to our deÞ-nition, the same is true under Simon�s deÞnition (this follows immediately becauseX1 ⊂⊂ X2 immediately implies that y1 is determined in a lower-order system thany2). However, the converse is not true. Consider the uninterpreted model

y1 = a12y2 + b11x1 (52)

y2 = a21y1 + b22x2 (53)

y3 = a31y1 + a32y2 + b33x3, (54)

which contains the recursive model

23

y1 = b11x1 (55)

y2 = a21y1 + b22x2 (56)

y3 = a31y1 + a32y2 + b33x3 (57)

as a special case if a12 = 0. In both these models we have y2 → y3 according toSimon�s deÞnition, but not y2 ⇒ y3 according to our deÞnition of simple causation.The fact that y2 → y3 follows from the fact that the subset condition for our deÞnitionis satisÞed. This, however, does not imply that the sufficiency condition is satisÞed,and it is not satisÞed in the present case. The variable y2 is not a sufficient statisticfor X2 for the determination of y3, since x1 and x2 also affect y3 through y1.In this model we have y2 ⇒ y3|y1. In the recursive model (55)-(57) there is

no difficulty determining the intervention associated with this conditional causation:it involves an intervention in x2. However, in the model (52)-(54) the indicatedintervention involves simultaneous changes in x1 and x2, with the changes restrictedso that y1 remains constant. If, as this experiment suggests, there exists a functionalrelation between x1 and x2, the model should be altered to incorporate this relation.As a digression, it is appropriate to return to our discussion of the role of the

exclusion condition in Simon�s deÞnition of causal orderings. It will be recalledthat if m > n satisfaction of the exclusion condition does not imply uniqueness ofthe causal ordering. However, we show now that it does imply generic uniqueness.To see this, note that if y1 appears in the equation for y2 and also in a lower-orderblock (and if the exclusion condition is satisÞed) then the subset condition is satisÞedexcept under special parameter values (such as those of the example discussed above).Satisfaction of the subset condition is not altered by algebraic operations, so we havegeneric uniqueness.

4 Comparing with PearlAs noted in the introduction, Pearl [22] presented an alternative formalization ofcausality. Like our development, his is based on the Cowles analysis of the 1950s,particularly that of Simon [26]. Pearl�s view is that after a promising beginning dur-ing the Cowles years, social scientists lost touch with the idea of structural modelingand failed to develop the original formal analysis of causation. He criticized sharplythe tendency of economists�as exempliÞed in this paper�to interpret the equalssign in (supposedly) structural models as having its usual mathematical meaning,rather than as directly representing causation. He also would reject the assertion inSection 1.2 that Simon�s analysis of causation is relevant to current practice preciselybecause it does not depend on the interpretation of structural equations as directlyincorporating causation.

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Under Pearl�s interpretation of a structural model, each structural equation rep-resents a distinct causal law for one of the internal variables. Pearl�s method ofanalyzing interventions is to delete the equation determining a particular internalvariable and setting the value of that internal variable at a preassigned level. Inthis paper, in contrast, we have followed the current economics literature in model-ing interventions by the straightforward device of simply specifying values for causalvariables. This is a distinction without a difference when the causal variable is ex-ternal. When the cause variable is internal, however, Pearl�s algorithm can lead todifficulties.The assumption that it makes sense to delete one or more of the structural equa-

tions and replacing the value of the internal variable so determined by a constantwithout altering the other equations has been termed �modularity�.14 In specialcases Pearl�s assumption of modularity is satisÞed, implying that his algorithm isvalid even when the causal variable is internal. For example, modularity for allpossible interventions is satisÞed if the external sets for the internal variables are dis-joint. This property, however, is virtually never satisÞed in economic models sinceeach external variable typically affects equilibrium values of more than one internalvariable. In fact, it is difficult to think of nontrivial models in any area of researchin which the modularity assumption is satisÞed (Cartwright [2]). In any case, whenmodularity is satisÞed the resulting causal ordering on internal variables is empty, socausal analysis is rendered trivial.When modularity fails, Pearl�s method of analyzing interventions is valid if the

variable Pearl treats as a cause is in fact causally prior to the effect variable in thesense deÞned in this paper. If not, however, replacing the equation determining thepurported cause variable with direct determination of the equilibrium value of thatvariable amounts to changing the model so as to render an inherently ambiguousquestion�what is the effect of one variable on another?�unambiguous. There is noreason to expect that causal analysis based on the altered model has any relevancefor the original model.To get a clearer idea of the problems Pearl�s algorithm entails when the requisite

causal ordering fails, we consider Pearl�s application in a supply-demand model likethose analyzed above. Pearl (p. 215) wrote the model as follows:

q = aqpp+ bqii (58)

p = apqq + bpww, (59)

where we have deleted the error terms since they play no role in the analysis. Here(58) is a structural demand equation and (59) is a structural supply equation. Asbefore, i is income and w is weather. These external variables enter the demand

14This term was used by Cartwright and Reiss [3], whose criticism of Pearl is similar to thatpresented here.

25

equation and the supply equation, respectively. This model conforms to Pearl�sinterpretation of the equations of structural models as representing distinct causallaws, one for each internal variable; here price causes quantity in the demand equation,whereas quantity causes price in the supply equation. Economists will be puzzled bythis asymmetric modeling of supply and demand; however, some such speciÞcation isrequired under Pearl�s characterization of structural models.Pearl noted that three queries can be distinguished:

1. What is the expected value of �the demand q� [quotation marks supplied] if theprice is controlled at p = p0?

2. What is the expected value of �the demand q� if the price is reported to bep = p0?

3. Given that the current price is p = p0, what would be the expected value of�the demand q� if we were to control the price at p = p1?

Observe the syntax here: despite Pearl�s terminology, the symbol q refers to (equi-librium) quantity, which equals quantity demanded and quantity supplied equiva-lently, as above. Pearl�s use of the phrase �the demand q� reßects his speciÞcationthat quantity is determined by price in the demand equation (58). In contrast, pricedetermines quantity in the supply equation (59). One wonders whether Pearl wouldaccept the question �What is the expected value of �the supply q� if the price is con-trolled at p = p0?� as being equivalent to question 1, on the grounds that quantitydemand equals quantity supplied in equilibrium, or whether instead he would regardthat question as inapplicable in the system (58)-(59), in which supply quantity is acause of price, not an effect.In a footnote Pearl reported that he has presented this model and these questions

to well over one hundred econometrics students and faculty. He found that therespondents had no trouble answering 2, but only one person could solve 1, and nonecould solve 3. With the exception of the one respondent who could answer question1 to Pearl�s satisfaction, this is exactly the response pattern that one would hope forbased on the analysis of this paper: if the unsatisfactory phrase �the demand q� isreplaced by simply �q�, the correct response to questions 1 and 3 is that they areambiguous because price does not cause quantity in the system (58)-(59).Under Pearl�s algorithm, however, questions 1 and 3 are not ambiguous. They

are answered by deleting (59) and replacing it with the equation p = p0 or p = p1,respectively. Thus the relevant causal parameter is aqp; the supply elasticity apq byassumption plays no role.These difficulties arise because, as we have seen, Pearl�s representation of an

economic model differs in key respects from the representation of a model whichmost economists would feel comfortable working with. In contrast to Pearl�s view,we have argued that deÞning �structural� models as models that directly encodecausal ideas is a dead end.

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5 Graphical MethodsWe noted in the introduction that graphical analysis of causation has seen few ap-plications in economics. One suspects that the reason is that most economists havedifficulty understanding the logic underlying graphical analyses of causation, at leastas set out in most expositions (Pearl [22], Pearl and Verma [23] and Geiger, Vermaand Pearl [8] are good examples). In these developments no attempt is made toderive causal orderings from deeper properties of models relating to exogeneity, asSimon did and as we have done here.15 Even though the notion of causality presumedin graphical analysis is not reduced to any deeper idea, we are still presented withmethods for deriving graphs from patterns of conditional independence among inter-nal variables. For example, if one observes cov(x1, x3|x2) = 0, we are told that onecan infer either y1 ⇒ y2 ⇒ y3 or y3 ⇒ y2 ⇒ y1. The validity of this inference is ap-parently taken to be self-evident: when x1 inßuences x3 through the mediation of x2,if we condition on x2, should not x1 and x3 be uncorrelated? However appealing thisintuitive argument is, one wonders whether the proposition cannot be demonstratedformally. But taking graphs as primitive seems to rule out this possibility.In this paper we have derived causal orderings from underlying assumptions about

exogeneity. Therefore rather than attempting to construct graphs from statisticalindependence relations as in the papers just cited, we would derive them from assumedmodel speciÞcations, and then inquire whether causal orderings in fact imply theconditional independence relations assumed in graph theory.

We construct a graph from a model as follows: Þrst, connect by an arrow eachinternal variable with each external variable in that variable�s external set (exceptwhen there exists another internal variable that causes the internal variable underconsideration). Second, connect each pair of internal variables in the causal orderingwith an arrow (except when there exists another internal variable that is an effect ofthe cause variable and a cause of the effect variable). To keep the discussion brief weignore conditional causation here; the treatment of conditional causation is similar.As observed above, without adopting the independence assumption there is obvi-

ously no way to make inferences about statistical independence, either unconditionalor conditional, from causal orderings, or vice-versa. However, if we adopt the inde-pendence assumption, it turns out that the statistical independences of graph theoryare derivable consequences of causal orderings as deÞned here. For example, supposewe have a 3-variable system parametrized as

y = Ay + ε, (60)

where ε is a triple of external�and therefore uncorrelated, by the independence

15Despite this, it is clear that the notion of causality underlying graphical methods is not that ofSimon: as we have seen, Simon�s deÞnition of causality precludes cyclicality in causation, because ofthe role of the subset condition. However, cyclicality is not ruled out, at least as a logical possibility,in graphical analyses, even if attention usually centers on acyclic graphs.

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assumption�random variables and A is a 3 × 3 matrix with zeros on the maindiagonal. It is easily veriÞed that the internal variables are causally ordered asy1 ⇒ y2 ⇒ y3 if A is nonzero only in the 2,1 and 3,2 places. Assume this isthe case. Then it is easy to compute the covariance matrix of y as a functionof the variances of the εi and the terms of A. It is immediately veriÞed thatcov(x1, x3|x2) = σ12−σ223/σ22 = 0. Thus we have a testable implication of an assumedcausal ordering. A similar derivation justiÞes the other conditional independences ofgraph theory as consequences of causal orderings.As an example, return to the price-quantity model (17)-(19)

q = αs + αsww + αsff (61)

q = αd + αdpp + αdii (62)

the graph so constructed is shown as Figure 3. In graph-theoretic terms (Pearl [22],for example), the chain w ⇒ q ⇒ p (or, equivalently, f ⇒ q ⇒ p) is blocked by q.This means that all the causal inßuence of weather on price passes through quantity.If the external variables weather, fertilizer and income are uncorrelated, then it maybe directly veriÞed that weather and price are uncorrelated, conditional on quantity(even though they are necessarily correlated unconditionally). This is a testableimplication of the assumed model.Causal orderings produce testable implications only when they are overidentifying.

The causal ordering y1 ⇒ y2 ⇒ y3 is overidentifying, as we just saw. However,other causal orderings may not be overidentifying, and these will not have testableimplications. For example, in 3-variable systems the Wold ordering y1 ⇒ y2 andy2 ⇒ y3|y1 is produced by a triangular coefficient matrix with the variables orderedas y1, y2, y3. This causal order is just-identifying, as is evident from the fact that anycoefficient matrix can be reparametrized in triangular form by applying the Choleskidecomposition. Hence the Wold causal ordering cannot be tested.From the foregoing development, there is no doubt that graph-theoretic methods

have a role to play in testing overidentifying restrictions implied by causal models.However, proponents of graph-theoretical methods go farther than this: they assertthat computer-based methods can be used to identify causal structures (for example,see Pearl [22] or Spirtes, Glymour and Scheines [28]. The essential assumptionthat is held to underly such exercises is that of �faithfulness� (Spirtes et al.) or,in Pearl�s terminology, �stability�. The idea is to rule out parameter values thatimply extraneous zero conditional covariances�that is, zero conditional covariancesthat are not produced by causal orderings�so as to be able to reverse the chain ofreasoning and treat zero conditional covariances as evidence in favor of the causalorderings that produce them.Proponents of graphical methods view the assumption of faithfulness as innocu-

ous. Spirtes et al. assert (p. 41) that all they are doing is ruling out a set of Lebesguemeasure zero. To see the basis for this argument, consider the three-variable example(60) analyzed above. The parameter space is 9-dimensional (A has 6 off-diagonal

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weather fertilizer

quantity income

price

Figure 3:

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elements, and there are 3 error variances). Generically, all the covariances, bothunconditional and conditional, of the variables in this space are nonzero. However,an 8-dimensional manifold of the parameter space will satisfy the single restrictioncov(y1, y3|y2) = 0; call this region B. Further, a 5-dimensional manifold of the para-meter space will satisfy y1 ⇒ y2 ⇒ y3; call this region C. As noted above, C is theset of parameter values such that A is zero except in the (2, 1) and (3, 2) elements.Since y1 ⇒ y2 ⇒ y3 implies cov(y1, y3|y2) = 0, we have that C is a subset of B (seeFigure 2).The assumption of faithfulness amounts to assuming that C−B is the empty set,

so that the above reasoning can be reversed, and a Þnding of cov(y1, y3|y2) = 0 canbe interpreted as implying y1 ⇒ y2 ⇒ y3. The fact that the 8-dimensional manifoldB has zero Lebesgue measure relative to the 9-dimensional parameter space is thebasis for Spirtes et al.�s representation of the faithfulness assumption as innocuous.However, there is another way to view the faithfulness assumption. Essentially,

the role of faithfulness is to generate the conclusion that if a parameter set is in the 8-dimensional manifold B, it is necessarily in the 5-dimensional manifold C. Thereforean empirical Þnding that cov(y1, y3|y2) = 0 allows us to conclude that y1 ⇒ y2 ⇒ y3(or, of course, y1 ⇐ y2 ⇐ y3; it is not asserted that graph theory can determine thedirection of causation in this case). Since C is of zero Lebesgue measure relativeto B, from the present vantage the assumption of faithfulness amounts to restrictingattention to a set of zero Lebesgue measure, not ruling out a set of zero measure.Doing so is anything but innocuous.Geiger, Verma and Pearl [8] defended the faithfulness assumption on grounds of

Occam�s razor: the simplest model that explains the empirical facts is to be chosen.However, it is not clear that models with nonempty causal orderings are preferable ongrounds of simplicity to models with empty causal orderings. Thus we are left withthe conclusion that algorithms that purport to determine causal orderings depend onthe validity of strong and probably unrealistic assumptions. Attributions of causalityrequire strong prior theoretical inputs; whether these are sought from preexistingtheory, as the Cowles economists envisioned, or from an assumption of faithfulness,as proponents of graphical methods prefer, may be a matter of taste.

6 ConclusionWe have pointed out that formal analyses of causation fall into two groups: those inwhich the idea of causation is derivable from exogeneity (that of Simon and that pro-posed here) and those in which structural equations are taken to connote causationin and of themselves. We have expressed the view that the former characterizationof causation is better suited to current economic models. The deÞnition of simplecausation proposed here has a particularly straightforward interpretation: it is ap-plicable whenever one wishes to know whether an intervention characterized in termsof a hypothetical variation in an internal variable has an unambiguous effect on some

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other variable.Other meanings for causation are less easy to interpret. We noted that when

one variable causes another according to Simon�s deÞnition but not according to thedeÞnition proposed here, the intervention usually involves joint restrictions on theexternal variables. This, we suggested, appears to conßict with their exogeneity.Graphical analyses along the lines developed by Pearl and others involve still moreserious problems: they require a model of a fundamentally different sort than thoseeconomists use. However, we showed in the preceding section that graphical methodsare applicable under the characterization of causation proposed here, and appear tooffer promise in identifying testable implications of causality speciÞcations.

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