The econometric consequences of the ceteris paribus ...directory.umm.ac.id/Data Elmu/jurnal/J-a/Journal of... · The econometric consequences of the ceteris paribus condition in economic

*Corresponding author. Tel.: #1-814-865-4921; fax: #1-814-863-4775.E-mail address: [email protected] (H.J. Bierens)

Journal of Econometrics 95 (2000) 223}253

The econometric consequences of the ceterisparibus condition in economic theory

Herman J. Bierens!,",*, Norman R. Swanson!

!Department of Economics, Pennsylvania State University, 510 Kern Graduate Building,University Park, PA 16802, USA

"Tilburg University, The Netherlands

Abstract

The ceteris paribus condition in economic theory assumes that the world outside theenvironment described by the theoretical model does not change, so that it has no impacton the economic phenomena under review. In this paper, we examine the econometricconsequences of the ceteris paribus assumption by introducing a &state of the world'variable into a well speci"ed stochastic economic theory, and we show that the di!erencebetween the conditional distribution implied by the theoretical model and the actualconditional distribution of the data is due to di!erent ways of conditioning on the state ofthe world. We allow the &state of the world' variable to be, alternatively and equivalently,an index variable representing omitted variables, or a discrete random parameterrepresenting a sequence of models. We construct a probability that can be interpreted asthe upperbound of the probability that the ceteris paribus condition is correct. Theestimated upperbound can in turn be interpreted as a measure of the information aboutthe data-generating process that is provided by a theoretical model which is constrainedby a set of ceteris paribus assumptions. In order to illustrate our "ndings from botha theoretical and an empirical perspective, we examine a linearized version of the realbusiness cycle model proposed by King Plosser, and Rebello (1988b. Journal of Monet-ary Economics 21, 309}341). ( 2000 Elsevier Science S.A. All rights reserved.

JEL classixcation: C32; C52; C52; E32

Keywords: Ceteris paribus; Missing variables; Bayesian prior; Information measure;Reality bound; Fit; Stochastic general equilibrium models; Real business cycle models

0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 3 8 - X

1. Introduction

In this paper we examine some econometric implications of the ceteris paribusassumption (other things being equal, or all else remaining the same) which isused in the construction of economic theories. Most economic theories, includ-ing general equilibrium theory, are &partial' theories in the sense that only a fewrelated economic phenomena are studied. For example, supply and demandtheories, and theories of the behavior of economic agents within the context ofstylized economies that have no direct counterpart in reality are commonplacein economics. In particular, consider the stylized Robinson Crusoe type econ-omy, where a single rational economic agent decides how much of his crop ofpotatoes he should eat, how much he should plant for next year's harvest, andhow long he should work in the "elds, in order to maximize his life-time utility.The analysis of this &partial' theory is justi"ed, explicitly or implicitly, by theceteris paribus assumption. However, when simple economic models of this typeare estimated using data which are themselves generated from a much morecomplex real economy, it is not surprising that they often "t poorly. In suchsituations, some theorists blame the messenger of the bad news, econometrics,for the lack of "t, and abandon econometrics altogether, in favor of simplecalibration techniques. (See Hansen and Heckman (1996) for a review of calib-ration, and Sims (1996) and Kydland and Prescott (1996) for opposite views oncalibration.) In this scenario, one may well ask what the role of econometrics is,and why calibration has become so popular. In the following, we attempt to shedlight on this issue by formalizing the ceteris paribus assumption in economictheory and econometrics. Along these lines, one of our main goals is to providenew evidence concerning the link between economic theory and econometrics.

We "rst relax the ceteris paribus assumption and introduce a &state of theworld' variable=. Given a vector > of dependent variables, and a vector X ofpredetermined variables, let the conditional density of >, given X"x and="w, be f (y D x, w). Now impose the ceteris paribus assumption in a theoret-ical economic model by conditioning on the event that the &state of the world'variable = is constant, say ="0, which yields the theoretical conditionaldensity f

0(yDx)"f (y D x, 0). On the other hand, the true conditional density of >,

given X"x alone is f (yDx)"E[ f (y D x,=)]. Therefore, the ceteris paribuscondition will in general cause misspeci"cation of the theoretical model f

0(y D x).

This is the very reason why theoretical models often do not "t the data.Next, we allow the &state of the world' variable= to be either: (1) a vector of

omitted variables, which may be represented, without too much loss of general-ity, by a discrete scalar random variable, or (2) a discrete random parameter,with a &prior' which is the maximum probability that the ceteris paribuscondition holds. We show that both interpretations of the &state of the world'lead to the same link between the theoretical model and the data-generatingprocess, and are therefore observationally equivalent. In particular, we

224 H.J. Bierens, N.R. Swanson / Journal of Econometrics 95 (2000) 223}253

1This approach is (somewhat) related to the statistical literature on mixtures, and contaminatedsampling. See Horowitz and Manski (1995) and the references therein for the latter. As is well known,under fairly general conditions there exist many sequences of densities f

j(y) such that for a wide class

of densities f (y) we can write f (y)"limm?=

+mj/0

pm,j

fj(y), where p

m,j'0, and +m

j/0pm,j

"1. Theproblem addressed in this literature is then to determine classes of densities f

j(y) for which f (y) can be

approximated well by the "nite mixture +mj/0

pm,j

fj(y). Kernel density estimates are examples of this

kind of mixtures. Moreover, denoting p0"lim

m?=pm,0

, and fH1(y)"lim

m?=+m

j/1pm,j

fj(y)/(1!p

0),

we have f (y)"p0

f0(y)#(1!p

0) f H

1(y). In the contaminated sampling literature, f H

1(y) may be

interpreted as the contamination of the unknown density f0(y), occurring with probability 1!p

0.

The main di!erence with the statistical literature involved is that in our case both f (y) and f0(y) are

given, possibly up to an unknown parameter vector, and p0

is constructed out of f (y) and f0(y).

2A more appropriate name for p0

would be the &maximum likelihood' of the theoretical model,but that will likely cause confusion.

3Note that p0

is related to the Kullback and Leibler (1951) Information CriterionK¸IC(b)":

f0(y@b);0f0(y D b) ln[ f

0(y D b)/f (y)]dy by the inequality infb K¸IC(b))ln[1/p

0]. More-

over, ln[1/p0] as a measure of the discrepancy of f

0(y D b) from f (y) satis"es the same information

inequality as the K¸IC: ln[1/p0]*0, and ln[1/p

0]"0 if and only if for some b

0the set

My: f0(y D b

0)Of (y)N has Lebesgue measure zero. Cf. White (1994, p. 9).

show that under some regularity conditions one can write f (y)"p0

f0(y D b

0)#

(1!p0) f

1(y D b

0), where f (y) is the (conditional) density of the data-generating

process, f0(y D b) is a parametrization of the (conditional) density f

0(y) implied by

the theoretical model, with b a parameter vector, and p0

is the maximumnumber between zero and one for which f

1(y D b

0) is a density.1 In other words,

p0"supbinf

yf (y)/f

0(y D b), and b

0"arg maxbinf

yf (y)/f

0(y D b). We may interpret

p0

either as an upperbound of the probability P(="0) that the ceteris paribuscondition on the &state of the world' holds, or as the maximal Bayesian prior thatthe model f

0(y D b) is correctly speci"ed. In either case, we can estimate p

0, and its

estimate may serve as a measure of the information contents of a theoreticalmodel f

0(y D b) about the data-generating process f (y), and hence as a measure of

how realistic the theoretical model is. We call this probability, p0, the reality

bound of the theoretical model2 involved. In this paper we consider threeversions of the reality bound: (1) the marginal reality bound, where f (y) andf0(y D b) are the marginal densities of a single observation implied by the data-

generating process and the theoretical model, respectively; (2) the averageconditional reality bound, which is the average of the probabilities p

0in the case

that f (y) and f0(y D b) are conditional densities; and (3) the joint reality bound,

where f (y) and f0(y D b) are the joint densities of the sample implied by the

data-generating process and the theoretical model.These reality bounds di!er from other goodness of "t measures, for example

Watson's (1993) measures of the "t of a real business cycle model, or the variousapproaches in Pagan (1994), in that they can be interpreted as probabilities thatthe theoretical model is correct, whereas other measures of "t are more akin toR2-type measures. Moreover, p

0is an information measure3 rather than a

H.J. Bierens, N.R. Swanson / Journal of Econometrics 95 (2000) 223}253 225

4According to Webster's dictionary, the correct pronounciation of the &c' in &ceteris paribus' is asthe &c' in &classic'. However, in European academic circles this &c' is often pronounced as either the &ch'in &church', or the "rst &c' in &cycle'. It appears that the Webster pronounciation is in accordance withclassical Latin spoken by the Romans, and the latter two with dialects of &church Latin', which wasthe academic lingua franca for more than 1000 years in Europe.

goodness of "t measure: it measures how much information about the data-generating process is contained in the theoretical model.

The rest of the paper is organized as follows. Section 2 contains a briefhistorical perspective on the use of the ceteris paribus assumption in economics.In Section 3, the ceteris paribus assumption is described within the context ofa fully stochastic theoretical economic model. As mentioned above, this is doneby introducing a &state of the world' variable which may be interpreted either asan index of omitted variables, or as a random parameter. We show how thetheoretical model is related to the data-generating process, by conditioning onthe even that the &state of the world' variable is constant. In Section 3, we alsooutline the derivation of the various reality bounds. In Section 4, the theoreticaltools introduced in the previous section are used to examine the consequences ofthe ceteris paribus assumption within the context of a linearized version of thereal business cycle (RBC) model of King et al. (1988b).

2. Some historical background on the ceteris paribus assumption

The speci"cation, estimation, and testing of econometric models is closelylinked to the construction of economic theories. While this statement holds fora number of rather obvious reasons, there is at least one link which can beinterpreted very di!erently, depending upon whether one is a theorist or anempiricist. This link is the ceteris paribus4 assumption, which is common toboth econometric (or statistical) and theoretical economic models, but which isin some respects much more crucial for econometric modelling than for theconstruction and interpretation of economic theories. In order to see the linkbetween the use of the ceteris paribus assumption in economic theory andeconometrics, it is useful to start by examining the approach taken by mosttheorists when constructing economic theories. Beliefs are "rst simpli"ed intosome parsimonious set of postulates and hypotheses. The simplest theoriespossible are then built up around these postulates and hypotheses, where bysimplest theories possible, we mean that models are presented which are su$-ciently simplistic enough to both convey and contain the essentials of the theory,as well as maintain enough generality as to be realistic. The latter of these tworequirements (that of su$cient generality) is closely linked with the notion ofceteris paribus, the Latin expression for &other things being equal'. One of the


simplest theories which makes use of a ceteris paribus assumption is the Law ofDemand (i.e. quantity demanded depends negatively on price ceteris paribus,etc.). This is a very useful and convenient theory, as long as the ceteris paribusassumption is not ignored, and it is understood that complements as well assubstitutes exist for most traded goods, for example. Indeed, the ceteris paribusassumption has been invoked throughout the history of economics, probably inlarge part because simpler theories more readily convey ideas, and also becausewithout this assumption we would need a uni"ed theory of economics, and justas a uni"ed theory of physics has not been developed, and may not be developedfor many generations, so a uni"ed theory for economics also seems far away.

Examples of the use of the ceteris paribus assumption in economic theory dateback more than one hundred years. For example, consider Beardsley (1895),where the e!ect of an eight hour workday on wages and unemployment isdiscussed. Beardsley notes that one popular theory of his day was that wagesvaried positively with the productiveness of an industry, and so shorter hourswould reduce wages if they lessened production. Beardsley goes on to point outthat such a statement is dependent critically on the ceteris paribus assumption,in particular:

Ceteris paribus, wages vary with the productiveness of industry, but onlyceteris paribus. The theory that wages depend entirely on the e$ciency oflabor, or on the product of industry, is a new form of the old doctrine of thewages-fund. The characteristic feature of the classical doctrine was the as-sumption that the wages-fund was an inelastic quantum of the total circula-ting capital.

The notion of ceteris paribus has indeed been used in many historical papersto illustrate theoretical concepts (e.g. Edgeworth, 1904; Murray et al., 1913;Pigou, 1917), and is still used in much curent research (e.g. Lewis and Sappin-gton, 1992; Eisner, 1992; Ghosh and Ostry, 1997). In fact, a simple search of TheAmerican Economic Review, Econometrica, The Journal of Economic History,The Journal of Industrial Economics, The Journal of Political Economy, TheQuarterly Journal of Economics, The Review of Economics and Statistics,Journal of Applied Econometrics, and The Journal of Economic Perspectivesresulted in 2098 papers being found over the last 100 yr which included thephrase ceteris paribus. Assuming 100 articles per year for 100 yr, for each ofthese approximately 10 journals suggests that approximately 2% of economicspapers contain mention of the ceteris paribus assumption. Although this num-ber is a very crude estimate, it highlights the importance of the assumption,particularly when it is noted that most uses of versions of the assumption do notexplicitly refer to it as the ceteris paribus assumption.

While the ceteris paribus assumption, and the related concept of parsimony,are used in theory to help convey ideas, and to retain simplicity given a largely


endogenous economy, these concepts must be viewed in a very di!erent lightwhen empirical models are being speci"ed, estimated, and tested. Put anotherway, the ceteris paribus assumption can be easily understood in theoreticaleconomic models, and in most cases is rather innocuous, as long as its potentialimportance is not overlooked. However, when the same ceteris paribus assump-tion is carried over into the "elds of empirical model speci"cation, estimation, andinference, careful note of its potential impact must be made. Indeed, the notion ofceteris paribus is so important in empirical econometrics that at least one verybroad area in econometrics attempts in many ways to deal with its potentialrami"cations. This area involves the study of exogeneity in econometric models.

According to Koopmans (1950): &A variable is exogenous if it is determinedoutside the system under analysis'. Engle et al. (1983) make this conceptoperational within the context of econometrics by formulating concepts of weakexogeneity, strong exogeneity, and super exogeneity, in which relationshipsbetween contemporaneous variables and parameters of interest are examined.Essentially, Engle et al. (1983) note that the ceteris paribus assumption (i.e.ignoring variables which may be &important') may have severe consequences onestimation and inference. For example, if the true economy is a complexendogenenous system, then estimating a subsystem of the bigger endogenoussystem may result in an estimated model in which the estimated parameters arenot consistent estimates of the true parameters of interest from the originaltheoretical model. These types of problems, which do not plague theoreticaleconomics, have been well known for many years. For example, Keuzenkamp(1995) discusses the controversy between Keynes and Tinbergen on econometrictesting of business cycle theories, and notes that multiple regression, which wassometimes thought to take care of required ceteris paribus conditions, does notactually help to counter Keynes view on Tinbergen's methods. Put in modernterminology, Keynes might have argued that Tinbergen's models su!ered froma lack of weak exogeneity with respect to the parameters of interest. Obviously,this was a di$cult criticism for Tinbergen to counter, as the need for parsimonyand tractability leads econometricians to use simple stochastic speci"cations, atleast to some degree. However, Tinbergen was certainly aware of the issue. Forexample, Tinbergen (1935) notes that:

The aim of business cycle theory is to explain certain movements of economicvariables. Therefore, the basic question to be answered is in what waysmovements of variables may be generated. In answering this question it isuseful to distinguish between exogen[ous] and endogen[ous] movements, theformer being movements during which certain data vary, while, in the latter,the data are supposed to be constant. 2 We have now to sum up whichgroups of subjects contribute to the supply of the demand in each of thegroups of markets and also how these supply and demand contributionsbehave in dependence on the variables adopted. Within certain limits, this


choice of variables is free. We can state a priori, however, that categoriesremaining constant, or nearly constant, throughout a business cycle shouldnot be taken as variables.

Thus, econometricians have grappled with the assumption of exogeneity, andthe related assumption of ceteris paribus for many decades, and the paper byEngle et al. (1983) can be interpreted as a modern answer to the old problem. Asanother example of exogeneity and the importance of the ceteris paribusassumption, note that Gini (1937) states:

It is clear that, in order to measure the contribution of the di!erent circum-stances in the determination of a certain phenomenon, it is necessary not onlyto know their e!ects when each of them acts separately, but also to know themanner in which their e!ects combine when they act simultaneously.

This statement of Gini, put within the econometric context of Engle et al., saysthat we are generally interested in the joint density of a group of economicvariables, as well as in the individual marginal densities. However, in practice,estimation is often carried out by factoring a joint density into a conditional anda marginal density, and weak exogeneity states that the parameters of interest canbe consistently estimated from the conditional density. Thus, for the purpose ofestimation of the parameters of interest, and given an appropriately de"nedconditional density, the marginal density can essentially be ignored. This is clearlya form of the ceteris paribus assumption. In order to further clarify the linkbetween ceteris paribus and exogeneity, note that Hendry et al. (1984) state that

As a slight caricature, economic-theory based models require strong ceterisparibus assumptions (which need not be applicable to the relevant datageneration process) and take the form of inclusion information such asy"f (z), where z is a vector on which y is claimed to depend. While know-ledge that z may be relevant is obviously valuable, it is usually unclearwhether z may in practice be treated as &exogenous' and whether othervariables are irrelevant or are simply assumed constant for analytical con-venience (yet these distinctions are important for empirical modelling).

The main di!erence between the ceteris paribus assumption in economictheory and econometrics is that in economic theory the ceteris paribus conditionis actually imposed, even if it is clear that the &state-of-the-world' is not "xed,whereas in econometrics it is often only used as a thought experiment whichfacilitates the interpretation of estimation results. The exogeneity issue is oneexample of the latter, where one of the questions which is addressed is: Underwhat conditions is the ceteris paribus assumption harmless for the estimation ofthe parameters of interest. Another example of a ceteris paribus thought


experiment in econometrics is the interpretation of the coe$cients of a linearregression model y"b

0#b

1x1#b

2x2#u, say, as marginal e!ects: b

1is the

e!ect on the conditional expectation E[y D x1, x

2] of a unit increase in x

1, given

the ceteris paribus condition that x2

is (held) constant. If x1

and x2

areindependent, we could even impose this condition without a!ecting the consist-ency of the OLS estimate of b

1. Of course, this is not the only possible thought

experiment in this case. One can impose the ceteris paribus assumption indi!erent ways and at di!erent conditioning stages. For example, the linearregression model involved can be rewritten as y"b

0#b

1x1#b

2E[x

2D x

1]#

b2(x

2!E[x

2D x

1])#u, and now the e!ect on the conditional expectation

E[y D x1, x

2] of a unit increase in x

1, given the ceteris paribus condition that only

the &innovation' x2!E[x

2D x

1] is (held) constant, is the same as the e!ect on

E[y D x1] of a unit increase in x

1. See also Manski (1997, Section 2.4) for similar

ceteris paribus thought experiments in the case of econometric models ofresponse to treatment and covariates. In particular, Manski (1997) considerstwo types of ceteris paribus assumptions on the covariates, namely before andafter the covariates themselves have responded to the treatment.

Finally, consider a quotation (taken from Loui (1989)) which is from theseminal book on modern decision theory by Savage (1950):

In application of the theory, the question will arise as to which [description ofthe] world to use. 2 If the person is interested in the only brown egg ina dozen, should that egg or the whole dozen be taken as the world? It will beseen 2 that in principle no harm is done by taking the larger of the twoworlds as a model of the situation.

This statement summarizes succinctly our dilemma. We would like to exam-ine a large portion of the world, and given the correct model speci"cation, weshould learn more by examining this large portion of the world rather thana smaller one. However, our models are always approximations, hence the morecomplex the model, the larger the likelihood of model misspeci"cation. More-over, we are left with the problem of determining whether our &portion' of theworld is general enough to adequately mimic the characteristics of the economyin which we are interested. In this paper we address these problems by providinga measure of the extent of misspeci"cation of theoretical economic models.

3. Relating theoretical economic models to the real world

3.1. The ceteris paribus condition on the &state of the world '

Let > be a vector of dependent variables, and let X be a possibly in"nitedimensional vector of exogenous and predetermined variables. Thus, X may


5Another class of theoretical models consists of conditional expectation models of the formE.0$

(y D X"x, b]":yf.0$

(y D x, b)dy, where only the left-hand side is speci"ed by the theoreticalmodel. The approach in this paper, however, does not apply to this class of partially speci"edstochastic models.

include lagged values of> in the case of time series models. For convenience, weassume in the discussion below that X is "nite dimensional so that we cancondition on events of the form X"x, where x is a "nite-dimensional non-random vector. Our arguments, however, extend to the case of an in"nitedimensional X.

Assume that a fully stochastic theoretical economic model is the conditionaldensity5of the &model version' of >, given X"x,

f.0$

(y D x, b), (1)

where b is a parameter vector. Of course, this conditional density may not beequal to the actual conditional density, f (y D x), of > given X"x.

In order to show how model (1) is related to the actual conditional densityf (y D x), let us introduce a random variable or vector,=, which represents the&state of the world'. There are two convenient interpretations of=which we willdiscuss. First, let = be a vector of omitted variables. If the omitted variablesinvolved are countable valued (which is always the case in practice), we can mapthese variables one-to-one onto the natural numbers (see Bierens, 1988; Bierensand Hartog, 1988), and hence we may (and will) assume that = is a discretenonnegative random variable.

Alternatively, we may assume that = represents a discrete random para-meter. This second interpretation leads to a Bayesian explanation of=, whichwe discuss in a later section.

Turning again to our discussion of how one can compare theoretical modelswith actual data-generating processes, let the true conditional density of> givenX"x and ="w be

f (y D x, w). (2)

Note also that the ceteris paribus assumption can be imposed by assuming that= is a nonrandom scalar. Thus, and without loss of generality, we may assumethat the economic theorist conditions on the event that="0. The theoreticalmodel, (1), is therefore correctly speci"ed, given the ceteris paribus conditioninvolved, if

f.0$

(y D x, b)"f (y DX"x,="0), (3)


for some parameter vector b. On the other hand, the conditional density of> given only that X"x is

f (y D x)"=+

w/0

f (y D x, w)p(w D x), (4)

where p(w D x) is the conditional probability function of=, given X"x. Sincewe are only interested in the comparison of (3) with (4), we may assume that the&state of the world' variable= is a binary variable, because we can always write(4) as

f (y D x)"p(0 D x) f0(y D x)#(1!p(0 D x)) f

1(y D x), (5)

where f0(y D x)"f (y D x, 0) and f

1(y D x)"(1!p(0 D x))~1+=

w/1f (y D x, w)p(w D x).

3.2. The &state of the world ' as in index of omitted variables

As an example of the interpretation of= as representing omitted variables,suppose that the data set is a random sample of workers, and that the theoreticalmodel is a standard Mincer-type wage equation. Let > be the log of wages,and X be a vector of characteristics of the worker involves, such as yearsof schooling, years of experience, years of experience squared, and a racedummy. Let = be a vector of omitted variables. For example, let = includehigher powers and/or products of years of experience, regional dummy vari-ables, dummy variables for various types of schooling, regional economicindicators, indicators of types of industries, and a dummy variable for maritalstatus. For the sake of the argument, let us assume that the error term of themodel including the omitted variables is distributed N(0, p2H), conditionally onX and=, and that the conditional expectation of > given X"x and="w islinear. Then, the true conditional density of> given X"x and="w takes theform

f (y D x, w)"exp(!1

2(y!a

0!a@

1x!c@w)2/p2H)

pHJ2p,

and the actual conditional density of > given X"x alone is

f (y D x)"P f (y D x, w)dP(w D x), (6)

where P(w D x) is the conditional distribution function of= given X"x. If weassume conditional normality of the error term, u, in the original Mincer-typewage equation as well, i.e. uDX&N(0, p2), the conditional density implied by this


6See for example DeJong et al. (1996) for a Bayesian analysis of calibrated models.

7We realize that in the eyes of a true Bayesian our Bayesian interpretation may be consideredblasphemy.

model becomes

f.0$

(y D x, b)"exp(!1

2(y!b

0!b@

1x)2/p2)

pJ2p, b"A

b0

b1

p B,which in general di!ers from the actual conditional density f (y D x), but is relatedto the conditional density f (y D x, w). This can be seen by setting w equal toa constant vector, say w"0, in which case

f.0$

(y D x, b)"f (y D X"x,="0),

for some value of b. Thus, the ceteris paribus condition involved is that="0,or equivalently, that= is constant.

In this example the vector = contains omitted variables that are discrete.Bierens and Hartog (1988) show that for such countable-valued random vectorsalmost any linear transformation h@= is a one-to-one mapping (i.e.="Uh(h@=) for some Borel measurable mapping Uh), with the exception of h'sin a set with zero Lebesgue measure. Thus, we may pick a suitable h, and replace= by Uh(=H), where=H"h@=. Therefore, without loss of generality we mayassume in this case that the &state of the world' variable= is a scalar randomvariable rather than a random vector. Moreover, given (5), we may furtherassume that = is a dummy variable.

3.3. The &state of the world ' as a binary random parameter

We now show that under mild conditions the true conditional density, f (y D x),can be written as (5). Thus, we may interpret= as a binary random parameterwith prior (conditional) probability function p(w D x), w"0, 1. The main di!er-ence between this setup and the usual Bayesian setup6 is that the prior densityinvolved does not represent the prior belief7 of the theorist in his model, but isconstructed from the theoretical density, (3), and the true density, f (y D x), asfollows.

Consider two continuous distribution functions on R, say F(y) and F0(y), with

corresponding densities f (y) and f0(y), respectively, where F(y) is the true distri-

bution function of some random variable >, and F0(y) is the &model version' of

F(y). For convenience, we examine univariate unconditional distributions. How-ever, our arguments also apply to multivariate (conditional) distributions. Ourapproach is to squeeze the distribution function F

0(y) under the distribution


function F(y) such that for some number, p03(0, 1), we can write

F(y)"p0F0(y)#(1!p

0)F

1(y), where F

1is a distribution function. This is

possible if we can "nd a positive p0

such that f (y)!p0

f0(y)*0 on the support

of f0(y). The maximal p

0for which this is possible is

p0" inf

f0(y);0

f (y)

f0(y)

.

Note that p0

is nonnegative, and cannot exceed 1 if the supports of f0

and f arethe same, because in this special case F(y)!p

0F

0(y) is a nonnegative monotonic

nondecreasing function with limit 1!p0*0 for yPR. If the support of f (y) is

contained in the support of f0(y), and if f (y)"0 in an area where f

0(y)'0, then

p0"0. In this case the theoretical model is able to predict impossible values for>, and such models are logically inconsistent with reality. Thus, the result thatp0"0 is appropriate in this case. However, if the support of f

0(y) is contained in

the support of f (y), and if :f0(y);0

f (y) dy'0, then there is no guarantee thatp0)1. This is the case for real business cycle models, where the support of f

0(y)

is a lower-dimensional subspace of the support of f (y). We shall deal with thiscase in the next subsection.

Given the above considerations, we assume for the remainder of this sectionthat the supports of f

0(y) and f (y) are equal. However, even in this case, it is

possible that p0"0. For example, assume that f (y) is the density of the standard

normal distribution, and f0(y) is the density of the N(0, p2) distribution with

p2'1. Then

infy

f (y)/f0(y)"p inf

y

exp[!(1/2)y2(1!p~2)]"0.

In practice, though, f0(y)"f

0(y D b) depends on parameters, and so does

p0(b)" inf

f0(y@b);0

f (y)/f0(yDb).

Letting

p0"sup

bp0(b) (7)

there will be a better chance that p0'0. For example, if f

0(y D b) is the density of

the N(0, b2) distribution, and f (y) is the density of the standard normal distribu-tion, then

p0(b)"inf

yf (y)/f

0(y D b)"DbDinf

yexp[!(1/2)y2(1!b~2)]

"G0 if DbD'1,

DbD if DbD)1,.


hence p0"1. In any case we can write F

1(y)"(F(y)!p

0F0(y))/(1!p

0), where

F1(y)"F(y) if p

0"0. This distribution function is continuous itself, with

density f1(y). Hence

f (y)"p0

f0(y)#(1!p

0) f

1(y).

In the case that f (y)"f (y D x), and f0(y)"f (y D x,="0) is parametrized as

f.0$

(y D x, b), p0

depends on x and b:

p0(x D b)" inf

f.0$(y@x,b);0

f (y D x)

f.0$

(y D x, b), (8)

so that (5) becomes

f (y D x)"p0(x D b) f

.0$(y D x, b)#(1!p

0(x D b)) f

1(y D x, b).

Similarly to (7) we could take p0(x)"supb p

0(x D b) as an upperbound of

the conditional probability that f.0$

(y D x, b) is correct, but thenb0(x)"arg maxb p

0(x D b) will depend on x. Therefore we propose the following

&average' conditional reality bound:

p0"sup

bE[p

0(X D b)].

Since both interpretations of= essentially yield the same result for the actualconditional density, namely Eq. (5), we shall also call = the &the state of theworld' variable in the case where= is a random parameter.

3.4. The case of nested supports

The above Bayesian interpretation of = as a random parameter is parti-cularly convenient in the case of stochastic general equilibrium models such asthe real business cycle models advocated by Kydland and Prescott (1982) andtheir followers, because due to the highly stylized nature of these models and thesingle representative agent assumption it is not realistic to attribute their lack of"t entirely to the ceteris paribus assumption when it is equated with the presenceof omitted variables in the theoretical model.

However, in the case of real business cycle models the support of the theoret-ical density is a lower-dimensional subspace of the support of the data-generat-ing process. Thus, the approach in the previous section is not directly applicable.In this subsection we shall show why this approach is not applicable in the caseof standard real business cycle models, and in the next subsection we shalldiscuss and alternative and related approach.


To begin with, consider a simple example. Let the true density of the randomvector >"(>

1, >

2)@ be

f (y)"f (y1

y2)"

exp[!12(y2

1/p2

1#y2

2/p2

2)]

p1p22p

,

which is modelled as

f0(y D b)"f

0(y

1, y

2D b)"

exp(!12y22/b2)

bJ2pI(y

1"0),

where I( ) ) is the indicator function. Thus, f0(y D b) is the density of the singular

bivariate normal distribution:

N2A0, p2A

0 0

0 b2BB.Then, the support of f

0(y D b) is the subspace spanned by the vector (0, 1)@.

Therefore, we have that

inff0(y1,y2@b);0

f (y1, y

2)

f0(y

1, y

2D b)

" infy2|R

f (0, y2)

f0(0, y

2D b)

" infy2|R

b

p1p2J2p

expC1

2(b~2!p~2

2)y2

2D

"G0 if b2'p2

2b

p1p2J2p

if b2*p22.,

Hence,

supb

inff0(y1,y2@b);0

f (y1, y

2)

f0(y

1, y

2D b)

"

1

p1J2p

,

which is larger than 1 if p1(1/J2p.

The problem with the approach of the previous section arises because thetheoretical model, f

0(y

1, y

2Db), imposes the ceteris paribus condition (>

1"0),

which is not integrated out from f (y1, y

2). In other words, f

0(y

1, y

2D b) is

compared with the wrong data-generating process. The model densityf0(y

1, y

2D b) in this example is actually the conditional density of >

2given the

ceteris paribus condition =">1"0. Hence, we should compare it with the


marginal density

f (y2)"P f (w, y

2)dw"

exp(!12y22/p2

2)

p2J2p

,

rather than with f (y1, y

2) itself.

3.5. Reality bounds for singular normal models

Now, consider the more general case where f (y) is the density of a k-variatenormal distribution, N

k(u, X), where X is nonsingular, and let the theoretical

model f0(y D b) be the density of the k-variate singular normal distribution,

Nk(k(b), R(b)), where rank (R(b))"m(k. Also, assume that u and X are given.

This case applies in particular to linearized versions of real business cyclemodels, which is also our area of application (in Section 4). Therefore we payhere explicit attention to the singular normal case.

It is di$cult to write down the closed form of f0(y D b) on its support.

Fortunately, there is no need for this, as the shapes of f (y) and f0(y D b) are

invariant under rotation and location shifts. Therefore, instead of working withf (y) and f

0(y D b) we may without loss of generality work with the transformed

densities:

fz(z D b)"f (P)(bz#k(b))

and

fz,0

(z D b)"f0(P(b)z#k(b) D b),

respectively, where P(b) is the orthogonal matrix of eigenvectors of R(b).Partitioning P(b)"(P

1(b), P

2(b)), where P

1(b) is the k](k!m) matrix of

eigenvectors corresponding to the zero eigenvalues of R(b), and P2(b) is the

k]m matrix of eigenvectors corresponding to the positive eigenvaluesj1(b),2, j

m(b)) of R(b), we have that

fz(z D b)"f

z(z

1, z

2D b)"

exp[!12(P(b)z!q(b))@X~1(P(b)z!q(b))]

(J2p)k JdetX,

where

q(b)"u!k(b), z"Az1

z2B3Rk~m]Rm.

Also,

fz,0

(z D b)"fz,0

(z1, z

2D b)"

exp(!12z@2K(b)~1z

2)

(J2p)mJdetK(b)I(z

1"0),


where

K(b)"diag(j1(b),2, j

m(b)).

Again, the latter density is actually the conditional density of Z2

given theceteris paribus condition (that="Z

1"0), and should therefore be compared

with the marginal density:

fz(z

2D b)"P f

z(z

1, z

2D b)dz

1

"

exp[!12

(z2!P

2(b)@q(b))@(P

2(b)@XP

2(b))~1(z

2!P

2(b)@q(b))]

(J2n)mJdet(P2(b)@XP

2(b))

.

Denoting p2(b)"P

2(b)@q(b)"P

2(b)@(u!k(b)), we now have that

p0"sup

binfz2

fz(z

2D b)

fz,0

(0, z2

D b)

"supb

Jdet[K(b)1@2(P2(b)@XP

2(b))~1K(b)1@2]

]expC!1

2p2(b)@(P

2(b)@XP

2(b))~1n

2(b)D

]infz2 GexpC

1

2z@2(K(b)~1!(P

2(b)@XP

2(b))~1)z

2D]exp[z@

2(P

2(b)@XP

2(b))~1p

2(b)]H.

If the matrix K(b)~1!(P2(b)@XP

2(b))~1 is positive de"nite, which is the case if

j.!9

[K(b)1@2(P2(b)@XP

2(b))~1K(b)1@2](1,

where j.!9

[A] is the maximum eigenvalue of A, then

infz2

Mexp[12z@2(K(b)~1!(P

2(b)@XP

2(b))~1)z

2]

]exp[z@2(P

2(b)@XP

2(b))~1p

2(b)]N

"exp[!12n2(b)@(P

2(b)@XP

2(b))~1(K(b)~1!(P

2(b)@XP

2(b))~1)~1

](P2(b)@XP

2(b))~1p

2(b)],


so that

p0" sup

b,j.!9*t(b)+:1GJdet W(b)exp C!

1

20 (b)@W(b)0 (b)D

]expC!1

20 (b)@W(b)(I!W(b))~1W(b)0 (b)D, (9)

where

W(b)"K(b)1@2(P2(b)@XP

2(b))~1K(b)1@2 (10)

and

0 (b)"K(b)~1@2P2(b)@(w!u(b)). (11)

Clearly, p0)1, because det W(b))j

.!9[W(b)] if j

.!9[W(b)](1.

In view of this argument, we may assume without loss of generality that thesupports of the densities f (y) and f

0(y D b) are (or have been made) the same. This

allows us to compute various versions of (8) for a variety of theoretical models.

3.6. The prior probability as a measure of the reality content of a theoretical model

In order to compute (8), we have to estimate the density f (y D x). One way ofdoing this is by nonparametric estimation. However, nonparametric conditionaldensity estimation requires large samples, which we usually do not have,particularly when we are examining macroeconomic data. Moreover, non-parametric density estimators su!er from the curse of dimensionality. Therefore,the only practical way to proceed is to specify a parametric functional form forf (y D x), say f (y D x, h), in general.

If we adopt our Bayesian interpretation of =, then in principle we canestimate the prior conditional probability, (8), with f (y D x) replaced by f (y D x, hK ),where hK is the maximum likelihood estimator of h based on the optimization of(8) with respect to b, so that

p(0(x)"sup

b|Binf

f.0$(y @ x,b);0

f (y D x, hK )f.0$

(y D x, b),

where B is the parameter space of the theoretical model. However, in this case

bK0(x)"arg max

b|B A inff.0$(y @ x,b);0

f (y D x, hK )f.0$

(y D x, b)Bdepends on x, which means that we estimate the parameter vector b for eachx"X

tseparately! Clearly, this would paint too rosy a picture of the theoretical


model. Therefore, in the conditional case we propose the following statistic:

p(C"sup

b

1

n

n+t/1

p0(X

tD b, hK ), (12)

where

p0(x D b, h)" inf

f.0$(y @ x, b);0

f (y D x, h)

f.0$

(y D x, b). (13)

The rationale behind this proposal is that p0(x D b, h) is a conditional probability.

Therefore, p0(b, h)"E[p

0(X

tD b, h)] is the corresponding unconditional prob-

ability, so that p(C

is an estimate of the maximal unconditional probability,p0(h)"supbp0(b, h). In other words, we may interpret p(

Cas an estimate of the

maximum average probability that the conditional model f.0$

(y D x, b) is cor-rectly speci"ed, and we therefore call p(

Cthe estimated average conditional reality

bound.We may also replace the conditional densities in (8) with either the joint

densities of the data or the marginal densities of a single observation,>t, in order

to get rid of x. In particular, let>"(>@1,2,>@

n)@ be a vector of stacked variables,

and let the joint density of > implied by the theoretical model be

f.0$

(y D b), b3B,

Also, let the functional speci"cation of the true joint density of > be

f (y D h), h3H,

so that ¸.0$

(b)"f.0$

(> D b) is the likelihood implied by the theoretical model,and ¸(h)"f (> D h) is the likelihood function of the data-generating process.Moreover, let hK be the maximum likelihood estimator of h. Then

p(J"sup

b|Binf

f.0$(y @ b);0

f (y D hK )f.0$

(y D b), (14)

is an estimate of the probability that="0, which may be interpreted as themaximum probability that the theoretical joint density, f

.0$(y D b), is correctly

speci"ed. Therefore, p(J

also serves as a measure of the reality content of thetheoretical model, and the larger is p(

J, the more realistic is the theoretical

economic model. We call p(Jthe estimated joint reality bound.

The computation of (14) in the case where f and f.0$

represent the jointdensities of the sample turns out to be a formidable numerical problem. How-ever, if f and f

.0$represent only the marginal densities of a single >

t, then the

computation of (14) is quite feasible. In this case we will denote the estimated


8Although our example assumes that momentary utility is additively separable, it should be notedthat KPR (1988a,b) also consider the case where momentary utility is multiplicatively separable.However, the restricted vector autoregression which we use to characterize the RBC model is thesame, regardless of which utility speci"cation we use. For further details, including restrictions onmomentary utility, the reader is referred to KPR (1988a,b).

probability, (14), as

p(M"sup

b|Binf

f.0$(y @ b);0

f (y D hK )f.0$

(y D b), (15)

where f and f.0$

are now marginal densities. We call p(M

the estimated marginalreality bound. In the empirical application below, we construct only p(

Cand p(

M,

as the preferred reality bound, p(J, is too di$cult to compute, due to the

singularity of the real business cycle model involved.

4. Measuring the marginal and average conditional reality bounds of a realbusiness cycle model

4.1. The model

As an illustration of our approach, consider the baseline real business cycle(RBC) model of King, Plosser and Rebelo (1988b; KPR), which is derived fromKydland and Prescott (1982), and which is used in Watson (1993):8

max E0

=+t/0

jt(ln(Ct)#h ln(1!N

t)), j(1,

subject to

Qt"C

t#I

t"K1~a

t(A

tN

t)a,

ln(At)"lnc

x#ln (A

t~1)#e

t, e

ti.i.d. (0, p2), (16)

Kt"(1!d)K

t~1#I

t,

where Ctdenotes consumption, Q

tis output, K

tis capital, N

tis employment, I

tis investment, and A

tis a technology index. Note that technology follows

a random walk with drift equal to ln cx. Expect for A

t, the variables in the above

theoretical model may all be interpreted as per capita aggregates. In order toexamine the empirical predictions of this model, KPR (1988a,b) and Stock(1993) calculate the equilibrium of the model using the log linearization of theEuler equations proposed by KPR (1987). Using this approximate solution, it


9See Watson (1993) for further discussion of this model, and Sims (1980), Swanson and Granger(1997) and the citations therein, for a discussion of VAR models.

turns out that the theoretical model can be written as a restricted VAR(1),9

>t"A(b)>

t~1#k(b)#u

t, u

ti.i.d. (0, R(b)), (17)

where

>t"A

* ln(Qt)

* ln(Ct)

* ln(It)

ln(Nt) B, (18)

with b"(j, h, a, d, cx, p2)@, and A(b) a matrix with elements depending on b.

Augmenting model (17) with the assumption that utis normally distributed,

ut&N(0, R(b)),

the joint distribution of the data, and the marginal distribution of >timplied by

the theoretical model, are completely determined. The functional form ofk(b), A(b), and R(b) will be discussed below.

Following KPR (1988a,b) and Watson (1993), we parameterize the truedata-generating process as a stationary Gaussian VAR(p):

>t"m#

p+i/1

Ni>

t~i#v

t, v

t&i.i.d. N

4(0, C). (19)

Stacking all of the parameters involved in a vector, h3H, the marginal densityf (y D h) is

f (y D h)"exp[!1

2(y!u(h))@X(h)~1(y!u(h))]

(2p)2Jdet X(h), (20)

where w(h)"m#+=i/1

tim and X(h)"#+=

i/1W

iCW@

i, with

AI!p+i/1

Ni¸iB

~1"I#

=+i/1

Wi¸i.

Note that the randomness of the theoretical model version of >tis solely due

to the random variable, et, in (16). Thus, the variance matrix of the theoretical

marginal model (R(b)) is singular.


10 In this version of the theoretical model, there are 8 parameters, including p2. Therefore, thereare 2 implicit restrictions among these parameters. However, we shall ignore these two restrictions inorder to keep our analysis tractable.

4.2. VAR representation

It follows directly from Eqs. (2.3)}(2.8) in KPR (1988b) that the theoreticalVAR model (17), augmented with the normality assumption takes the form

*ln(Qt)"ln(c

x)!

pQK

pNK

(/!1)ln(n)#pQK

pNK

(/!1)ln(Nt~1

)

#A1

nNK

!

nQK

nNKBet, (21)

*ln(Ct)"ln(c

x)!

pCK

pNK

(/!1)ln(n)#pCK

pNK

(/!1)ln(Nt~1

)

#A1

nNK

!

pCK

pNKBet,

*ln(It)"ln(c

x)!

pIK

pNK

(/!1)ln(n)#pIK

pNK

(/!1)ln(Nt~1

)

#A1

pNK

!

pIK

pNKBet,

ln(Nt)"(1!/)ln(n)#/ln(N

t~1)#e

t, e

t&i.i.d. N(0, p2),

where ln Nt!ln n is the percent deviation of N

tfrom its steady-state path, with

ln n constant (recall that the solution method which KPR (1988b) use to solvefor the equilibrium of the theoretical model amounts to linearizing the "rst-order conditions of the choice problem about the steady state, after transform-ing the economy to obtain a stationery system). The parameter / above can beinterpreted in a number of ways. For example, KPR (1988b) note that when / isclose to unity, hours will be characterized by substantial serial correlation, whilethe growth rates of Q

t, C

t, and I

twill exhibit little serial correlation. Thus, when

/ is close to unity an investigator might be tempted to di!erence the log of hoursin addition to the logs of Q

t, C

t, and I

t, which would lead to misspeci"cation.

Finally, note that pQK

, pCK

, pIK

, and nNK

should be interpreted as the elasticitiesof the #ow variables with respect to deviations of the capital stock from itsstationery value.10


4.3. The marginal reality bound

We can write this theoretical model as

>t"k#A>

t~1#ie

t, e

t&i.i.d. N(0, p2), (22)

where A"(0, a) is a 4]4 matrix with a a 4]1 vector, O is a 4]3 zero matrix,and k and i 4]1 vectors. The relationship between the vectors a, k, and i is

k"ln(cx)ι13#ln(n)(ι

4!a),

i"1

pNK

ι13#

1!2/1!/

ι4#

1

1!/a (23)

"

1

pNK

ι13#

1!2a4

1!a4

ι4#

1

1!a4

a

"

1

nNK

ι13#

1!2a4

1!a4

ι4#

a4

1!a4

b13

"b4b13#(1!b

4) ι

4#b

5ι13

, and

a"a4b13"

b4

1#b4

b13

,

where

ι13"A

1

1

1

0B, ι4"A

0

0

0

1B, b13"A

a1/a

4a2/a

4a3/a

41 B, b4

"

a4

1!a4

'0, b5"

1

pNK

,

and a4"/ is the last component of a. The corresponding MA(R) representa-

tion is

>t"

=+j/0

Ajk#=+j/0

Ajiet~j

,

where A0"I. Using the facts that A2"a4A, with a

4the last component of a,

and k4"(1!a

4)ln(n) is the last component of k, it follows that

E.0$

(>t)"k#AkA

=+j/1

aj~14 B"k#

k4

1!a4

a"k#ln(n)a

"ln(cx)ι13#ln(n)ι

4"b

6ι13#b

7ι4,

where

b6"ln(c

x), b

7"ln(n).


Moreover, since the last component of i is 1, so that Ai"a, it follows that

Var.0$

(>t)

p2"ii@#

1

1!a24

Aii@A@"ii@#1

1!a24

aa@

"ii@#a24

1!a24

b13

b@13"ii@#

b24

1#2b4

b13

b@13"C(b), (24)

say. Clearly, the rank of this matrix cannot exceed 2, and it has rank 1 only if i isproportional to b

13, which is not the case. Therefore, the rank of C(b) is 2.

It is not hard to verify that the two positive eigenvalues of the matrix C(b) are:

j1(b)"

1

2 Ai@i#b@13

b13

b24

1#2b4B

!

1

2SAi@i!b@13

b13

b24

1#b4B

2#

4(i@b13

)2b24

1#2b4

, (25)

j2(b)"

1

2 Ai@i#b@13

b13

b24

1#2b4B

#

1

2SAi@i!b@13

b13

b24

1#2b4B

2#

4(i@b13

)2b24

1#2b4

,

where

i@i"b24b@13

b13#2b

4b5ι@13

b13!b2

4#3b2

5#1,

and

i@b13"b

4b@13

b13#b

5ι@13

b13!b

4#1.

Moreover, the corresponding eigenvectors, qj(b), are

qj(b)"G

(i@b13

)i!(i@i!jj(b))b

13DD(i@b

13)i!(i@i!j

j(b))b

13DD

if i@b13O0,

iDDiDD

if i@b13"0 and j

j(b)"i@i,

b13

DDb13

DDif i@b

13"0 and

jj(b)"[b2

4/(1#2b

4)]b@

13b13

.


We now have all the ingredients needed in order to compute p(M. In (9), let

K(b)"p2Aj1(b) 0

0 j2(b)B"p2KH(b),

P2(b)"(q

1(b), q

2(b)),

W(b)"p2KH(b)1@2(P2(b)@X(hK )P

2(b))~1KH(b)1@2"p2WH(b),

and

0 (b)"p~1KH(b)~1@2P2(b)@(u(hK )!b

6ι13!b

7ι4)"p~10H(b).

Note that Jdet W(b)"p2Jdet WH(b), and that the condition j.!9

[W(b)](1now becomes

p2(1

j.!9

[WH(b)].

It follows from (9) that

p(M" sup

b@p2:1@j.!9*WH(b)+ Gp

2Jdet WH(b)expC!1

20H(b)@WH(b)0H(b)D

]expC!1

2p20H(b)@WH(b)(I!p2WH(b))~1WH(b)0H(b)D, (26)

where

WH(b)"KH(b)1@2(P2(b)@X(hK )P

2(b))~1KH(b)1@2

and

0H(b)"KH(b)~1@2P2(b)@(u(hK )!b

6ι13!b

7ι4). (27)

Given b, we can concentrate out p2 by maximizing the right-hand side of (26)(over the interval (0, 1/j

.!9[WH(b)])) with respect to p2 by simple line search, and

then optimize with respect to b. However, in the current context, it turns outthat

p(M"1. (28)

The reason for this is twofold. First, observe that (27) can be made equal to zeroby choosing b

6and b

7, given b

1,2, b

5, such that

P2(b)@u(hK )"b

6P

2(b)@ι

13#b

7P

2(b)@ι

4"P

2(b)@(ι

13, ι

4)A

b6

b7B,


which is system of two equations in two unknowns, b6

and b7. Therefore, we

may set 0H(b)"0 and p2"j.!9

[WH(b)]. Then, it follows that

p(M"sup

b

Jdet WH(b)

j.!9

[WH(b)]"sup

b

Jj.*/

[WH(b)]j.!9

[WH(b)]

j.!9

[WH(b)], (29)

where j.*/

[WH(b)] is the minimum eigenvalue of WH(b). Clearly, p(M

is equal tounity if j

.*/[WH(b)]"j

.!9[WH(b)], which is a single nonlinear equation in

5 unknowns, b1,2, b

5, and can be solved. (We veri"ed that p(

M"1, using

numerical methods.)However, rather than concluding that the baseline model is perfect, it appears

that the comparison of marginal distributions in this case is not very informa-tive. Nevertheless, the inclusion of the marginal reality bound in our discussionis useful as an illustration of the general method, and because the conclusion (28)is not obvious.

4.4. The average conditional reality bound

The theoretical conditional distribution of >tis

>tD X

t&N

4(k

t(b), p2ii@),

where

Xt"(>@

t~1,2, >@

t~p)@

and

kt(b)"k#A>

t~1"k#aι@

4>t~1

"ln(cx)ι13#ln(n)(ι

4!a)#aι@

4>t~1

"ln(cx)ι13#ln(n)ι

4#(ι@

4>t~1

"ln((n))a

"b6ι13#b

7ι4#b

4(ι@4>t~1

!b7)b

13,

using the same reparametrization and notation as above. The DGP is

>tD X

t&N

4(u

t(h), X(h)),

where ut(h)"m#+p

i/1N

i>t~i

and X(h)"C.The matrix ii@ has one positive eigenvalue (j(b)"i@i), and corresponding

normalized eigenvector, q(b)!i/DDiDD. Note, however, that we can write kt(b)

in terms of i and bj, j"4,2, 7, as k

t(b)"b

6ι13#b

7ι4#(ι@

4>t~1

!b7)

](i!ι4#b

4ι4!b

5ι13

). Therefore, without loss of generality, we may setb13"i, so that

kt(b)"b

6ι13#b

7ι4#(ι@

4>

t~1!b

7)(b

13!ι

4#b

4ι4!b

5ι13

),

j(b)"b@13

b13

and q(b)"b13

/Jj(b).


Now, P2(b) and K(b) in (9) become q(b) and p2j(b), respectively, and 0 (b)

becomes

0t(b)"

q(b)@(ut(h)!k

t(b))

pJj(b)"

b@13

(ut(h)!k

t(b))

pb@13

b13

.

Also, W(b), together with the condition j.!9

[W(b)](1 becomes

W(b)"p2(b@

13b13

)2

b@13

X(hK ) b13

(1.

Moreover, b@13

kt(b) can be reparametrized as

b@13

kt(b)"b

4#b

5ι@4>

t~1,

and W(b) can be reparametrized as b26"W(b)(1. Thus, we can now write (13)

according to (9) as

p0(X

tD b, hK )"b

6expC!

1

2A1

1!b26B

(b@13

ut(hK )!b

4!b

5ι@4>

t~1)2

b@13

X(hK )b13

D. (30)

Finally,

p(C" sup

b@0:b6:1

1

n

n+t/1

p0(X

tD b, hK ),

which can be computed by numerical methods.Note how crucial the restrictions on the conditional mean of the theoretical

model are. If it were possible to choose b such that b@13

ut(hK )"b@

13kt(b) for all t,

then we could set b6"1, so that p(

C"1.

Finally, note from Jensen's inequality that

p(6 C"expC sup

b@0:b6:1

1

n

n+t/1

ln(p0(X

tD b, hK ))D)p(

C.

This lowerbound can be written as

p(6 C"

1

2sup

b@0:b6:1Gln(b2

6)!

g(b)

1!b26H

"

1

2 Gsupb

lnA1#1

2g(b)!SA1#

1

2g(b)B

2!1B

!

g(b)

1!12g(b)#J(1#1

2g(b))2!1H, (31)


where

g(b)"(1/n)+n

t/1(b@

13u

t(hK )!b

4!b

5ι@4>

t~1)2

b@13

X(hK )b13

,

and is much easier to calculate than p(C. Thus, the &optimal' b's obtained from the

construction of p(6 C

may be used as starting values in the numerical computationof p(

C.

4.5. Empirical result

In the previous subsection we have outlined the tools with which to examinethe conditional baseline RBC model of KPR (1988b). We now use these tools toconstruct an estimate of the average conditional reality bound (ACRB). Werecall that our measure is an information measure: it measures the extent ofinformation about the data-generating process that is contained in the theoret-ical model. It can be viewed as an alternative to the use of other methods, such asEuler equation restriction tests, moment matching, impulse response analysis,duration analysis, and spectral analysis (see, e.g. all of the articles in the specialissue of the Journal of Applied Econometrics on calibration edited by Pagan(1994)). The advantage of our approach is that we obtain a single probabilityvalue which summarizes the ability of the RBC model to match the data, relativeto some more general model where the ceteris paribus assumption has not beenimposed.

Our approach is perhaps closest to the approach taken by Watson (1993),where the variables in the theoretical model are augmented with just enoughstochastic error so that the model can exactly match the second moments of theactual data. Measures of "t for the model, called relative mean square approxi-mation errors (RMSAEs), are then constructed on the basis of the variance ofthis stochastic error relative to the variance of the actual series. In at least tworespects, our approach is similar to Watson's. First, both approaches attempt toobtain &overall'measures of closeness of model distribution and data-generatingprocess. Second, and more importantly, both approaches assume that thetheoretical model is only an approximation of the true underlying model, and assuch do not rely on a null hypothesis that the theoretical model is equal to thetrue model (see, e.g. Watson, 1993, p. 1015), as do standard goodness of "tmeasures. The importance of this point is perhaps best illustrated by Sargent(1998), where it is noted that viewing econometric models as approximations ofthe truth rather than assuming that the models are correctly speci"ed (under thenull) can lead to interesting results. For example, when all models are viewed asapproximations, Sargent shows that simple &adaptive' forecasting techniquesbased on rolling regressions, where parameters are updated at each point intime, actually yield forecasts which are comparable to those based on &optimal'


11 It should be noted that when we examined individual pairs of our three I(1) variables using unitcointegrating vector restrictions, we found some evidence of cointegration, as one might expect,given the empirical "ndings of King et al. (1991), for example, These results suggest that even our truemodel should perhaps only be viewed as an approximation to the truth, as the failure of simplerpairwise cointegration results to match up with our system cointegration results may be accountedfor by model misspeci"cation. However, for the purpose of constructing our true model, we foundthat little explanatory power was added by including cointegrating restrictions of any form, and thuswe assume that our simpler speci"cation is an adequate representation of the truth.

rational expectations theories. This result is perhaps surprising, given that therolling regression approach is certainly not optimal in a standard utility opti-mizing representative agent framework.

The data which we use correspond to those used by Watson (1993), althoughour sample period is updated through 1994:4, so that we have 188 quarterly U.S.observations from 1948:1 to 1994:4. As mentioned above, the variables which wemodel are the natural logarithm of hours worked (ln(N

t)) and the growth rates of

output (*ln(Qt)), consumption (*ln(C

t)), and investment (*ln(I

t)). In order to

make operational our measure of the ACRB and the LACRB, we need toconstruct u

t(hK ) and X

t(hK ), the conditional mean and variance of >

tD X

t, where

>t"(*ln(Q

t), *ln(C

t), *ln(I

t), ln(N

t))' and X

tis as de"ned above. (It should

perhaps be noted that we do not at any stage use calibration techniques in ourcalculation of p(

C.) Before forming u

t(hK ) and X

t(hK ), however, we "rst undertook

a speci"cation search for our true VAR model. To begin with, we conducted unitroot tests, and found that ln(Q

t), ln(C

t), and ln(I

t) are all I(1), so that the

di!erences of these variables are all I(0). Also, ln(Nt) is I(0), at least at a 10%

level. (In this analysis, we used augmented Dickey}Fuller and Phillips}Perronunit root tests). We then chose the lag order for our VAR model of >

tby

examining values of the Akaike and Schwarz information criteria for lag ordersfrom 1 to 12. Based on the Schwarz criterion, one lag was preferred, while theAkaike criterion picked "ve lags. We also conducted a series of Johansen tracetests to determine the rank of the cointegrating space (for the subsystemconsisting of our three I(1) variables, say Z

t"(ln(Q

t), ln(C

t), ln(I

t))@). In particu-

lar, "ve di!erence cointegration tests, corresponding to the "ve di!erent deter-ministic trending assumptions discussed in Johansen (1995), were conducted.The results based on these tests were rather mixed. As an example of our"ndings, consider the two cointegration tests which assume a linear determin-istic trend in the data (the "rst includes an intercept in the cointegrating relation,while the second includes an intercept and a trend in the cointegrating relation).For a VAR(1) in Z

t, the cointegrating ranks found based on these two tests were

3 and 1, respectively, while for a VAR(5) in Zt, the ranks were 0 and 0. (Complete

details of all of our empirical "ndings, are available upon request fromthe authors.) Given our rather mixed empirical "ndings, we chose to specify thetrue model for >

tas a VAR(5) with no cointegrating restrictions.11


12As mentioned above, there is also one parameter restriction, namely that 0(b6(1. Thus,

optimization is carried out by optimizing (1/n)+nt/1

p0(X

tD b, hK ) with respect to b

13, b

4, and b

5, given

a grid of di!erent b6

values, where the grid is incremented by some small value.

13First, one solves the generalized eigenvalue problem DC1!C

2jD"0, where C

1"=@M=,

C2"X~1(hK ),="(u@

2b13

,2, u@nb13

)@, ui, i"2, 3,2, n, are the elements of u

t(hK ), where u(h) is

de"ned in Section 4.1, hK is the maximum likelihood estimator of h, and whereM"I

n~1!X(X@X)~1X@, and X is an (n!1)]2 matrix with "rst column equal to an (n!1)]1

vector of ones, and second column equal to (i@4>

1,2, i@

4>

n~1)@. In order to do this, set

C"¸~1A¸@~1, where C2"¸¸@, and solve the standard eigenvalue problem, Cx"jx. Then the

eigenvector, say x.*/

, associated with the minimal eigenvalue, say j.*/

, can be normalized to obtainbK13

, whose last element is unity. Once bK13

is known, g(bK )"j.*/

can be plugged into Eq. (31).

14This bound was found to be the same when the increment used in the grid search for b6

wasvaried between 0.01 and 0.0001.

Our approach was to "rst construct the lower average conditional realitybound (LACRB). The associated parameter estimates from this exercise werethen used to construct the ACRB. The reason why we adopted this approach isthat there are six di!erent parameters, b

13"(b

1, b

2, b

3, 1)@, b

4, b

5, and b

6, to

jointly solve for in the optimization of (1/n)+nt/1

p0(X

tD b, hK ).12 In our empirical

illustration, joint optimization with respect to so many parameters results ina probability surface with a number of &#ats', so that when &reasonable' startingvalues are not used, standard numerical optimization procedures (such as theBroyden, Flecher, Goldfarb, and Shanno quasi-Newton method which we use)result in parameter estimates which do not &move' from their starting values.Indeed, this was what occured when we chose starting values which were notequal to those associated with the LACRB. As noted above, construction ofstarting values based on the LACRB is quite simple.13 Given these startingvalues associated with the LACRB, we obtained convergence to the ACRB afterless than 50 iterations, and the bound14 was

p(C"0.77.

The ACRB is rather high, and suggests that the reality content of the baselineRBC model is perhaps higher than one might imagine, given that the RBCversion of the true VAR model is so severely restricted. For example, it issurprising that the use of only one source of stochastic #uctuation in the RBCmodel yields such a high probability. Indeed, it appears that the very simpleRBC model which we examine provides a rather e!ective approximation of ourtrue model. Watson (1993, Table 1) "nds RMSAEs (which can be interpreted like1 minus the R2 statistic from a regression based on the theoretical model) whichrange from as low as 0.21 to as high as 1.00, depending upon which variable isexamined. Interestingly, our "nding corresponds closely with the values (fromapproximately 0.20 to 0.30) found by Watson when the variable for which he


minimizes the stochastic error (see above) is the same as the variable for whichhe constructs the RMSAE.

Acknowledgements

This paper was presented by the "rst author at the conference of Principles ofEconometrics in Madison, Wisconsin, on May 1}2, 1998. The useful commentsof Arthur Goldberger, Charles Manski, and Neil Wallace, on earlier versions ofthis paper are gratefully acknowledged. Swanson thanks the National ScienceFoundation (grant SBR-9730102) for research support.

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