The Less-Volatile U.S. Economy: A Bayesian...

The Less-Volatile US EconomyA Bayesian Investigation of TimingBreadth and Potential Explanations

Chang-Jin KIMDepartment of Economics Korea University Anam-Dong Seongbuk-ku Seoul 136-701 Korea(cjkimkoreaackr)Department of Economics University of Washington Seattle WA 98195(changjinuwashingtonedu)

Charles R NELSONDepartment of Economics University of Washington Seattle WA 98195(cnelsonuwashingtonedu)

Jeremy PIGERResearch Department Federal Reserve Bank of St Louis St Louis MO 63102(pigerstlsfrborg)

Using a Bayesian model comparison strategy we search for a volatility reduction in US real gross do-mestic product (GDP) growth within the postwar sample We nd that aggregate real GDP growth hasbeen less volatile since the early 1980s and that this volatility reduction is concentrated in the cyclicalcomponent of real GDP Sales and production growth in many of the components of real GDP display sim-ilar reductions in volatility suggesting the aggregate volatility reduction does not have a narrow sourceWe also document structural breaks in in ation dynamics that occurred over a similar time frame as theGDP volatility reduction

KEY WORDS Bayes factor Business cycle Stabilization Structural break

1 INTRODUCTION

The US economy appears to have stabilized consider-ably since the early 1980s as compared with the rest of thepostwar era For example the standard deviation of quar-terly growth rates of real gross domestic product (GDP) for1950ndash1983 was more than twice as large as that for 1984ndash1999 This observation has sparked a growing literature rig-orously testing the statistical signi cance of the volatilityreduction and documenting various stylized facts about thenature of the stabilization (see eg Niemira and Klein 1994Kim and Nelson 1999 McConnell and Perez-Quiros 2000Kahn McConnell and Perez-Quiros 2000 Ahmed Levin andWilson 2002 Warnock and Warnock 2000 Blanchard andSimon 2001 Chauvet and Potter 2001 Stock and Watson2002)

A primary goal of this research agenda is to determine thecause of the observed volatility reduction Many explanationshave been proposed including improved policy (speci callymonetary policy) structural changes (eg a shift to servicesemployment better inventory management) and good luck(a reduction in real or external shocks) Determining the rela-tive importance of these competing explanations is importantin that they have very different implications for the sustain-ability of the volatility reduction and the evaluation of pol-icy effectiveness In assessing the viability of explanations formacroeconomicevents it is of course useful to have a clear pic-ture of the nature of the event In the present case this includescompiling a list of stylized facts describing the volatility reduc-tion One can then ask whether these stylized facts invalidate

any potential explanations for the reduction In this article werevisit an existing list of stylized facts that document the per-vasiveness of the volatility reduction within broad productionsectors of aggregate real GDP We also investigate structuralchanges in the dynamics of in ation around the same time asthat found in real GDP

This article has four main ndings

1 Aggregate real GDP underwent a volatility reduction inthe early 1980s that is shared by its cyclical componentbut not by its trend component

2 A volatility reduction similar to that found in aggregatereal GDP is present in many of the broad production sec-tors of real GDP Thus the volatilityreduction in aggregateGDP is not con ned to any one sector

3 The volatility reduction is apparent in nal sales as wellas in production

4 The dynamics of in ation display structural breaks inpersistence and conditional volatility over a similar timeframe as the volatility reduction observed in real GDP

Our results can be compared with those obtained byMcConnell and Perez-Quiros (2000) (hereinafter MPQ) andWarnock and Warnock (2000) These authors concluded that avolatility reduction in broad measures of activity (real GDP forMPQ aggregate employment for Warnock and Warnock) is re- ected in within-sector stabilizationfor only one sector durable

copy 2004 American Statistical AssociationJournal of Business amp Economic Statistics

January 2004 Vol 22 No 1DOI 101198073500103288619412

80

Kim Nelson and Piger Volatility Reduction in US GDP 81

goods MPQ went on to show that measures of nal sales havenot become more stable suggesting that the volatility reductionis focused in the behavior of inventories The results of MPQcan lead to striking conclusions regarding the viability of com-peting explanations for the source of the volatility reductionFor example MPQ argued that explanationsbased on improvedmonetary policy are hard to reconcile with the limited breadthof the volatility reduction particularly the failure of measuresof nal sales to show greater stability They argued that expla-nations based on improved inventory management are muchmore consistent with the stylized facts In contrast the resultspresented here are consistent with a broad range of potentialexplanations Indeed we argue that the evidence from broadproduction sectors of real GDP is not suf ciently sharp to helpinvalidate potential candidates for the source of the volatilityreduction

Whereas MPQ investigated structural breaks in volatility us-ing classical tests based on work of Andrews (1993) and An-drews and Ploberger (1994) here we investigate this issue us-ing a Bayesian model comparison framework This methodcompares the marginal likelihood from models with and with-out structural breaks using the Bayes factor As discussed byKoop and Potter (1999) Bayes factors have an advantage overclassical tests for evaluating structural change in the way inwhich information regarding the unknown break date is incor-porated in the test The unknown break date which is a nui-sance parameter present only under the alternative hypothesisleads to nonstandard asymptotic distributions for classical teststatistics Solutions to this problem in the classical frameworkfail to incorporate sample information regarding the unknownbreak pointOn the contrarymodel comparison based on Bayesfactors incorporatessample information regarding the unknownbreak date and as a byproductof the procedure yield the poste-rior distribution of the unknown break date a very useful pieceof information for determining when the break occurred Giventhe substantial differences in methodology the evidence in fa-vor of structural change providedby the Bayes factor may differsubstantiallyfrom that yieldedby classical tests Thus a primarygoal of this article is to evaluate the robustness of the MPQ re-sults based on classical techniques to a Bayesian model com-parison methodology

Section 2 discusses the basic model speci cation andBayesian methodology that we use to investigate structuralchange in the various series considered in this article Section 3presents the results for aggregate and disaggregate real GDPand contrasts these results with the existing literature based onclassical hypothesis tests It also discusses evidence of struc-tural change in in ation dynamics Section 4 discusses possibleexplanations for the different results produced by the classicaland Bayesian methodologies Section 5 concludes

2 MODEL SPECIFICATION BAYESIAN INFERENCEAND MODEL COMPARISON TECHNIQUES

To investigate a possible volatility reduction in growth ratesof aggregate and disaggregate real GDP we use the following

empirical model

yt DkX

jD1

Aacutejytiexclj C et

et raquo N0 frac34 2Dt

frac34 2Dt

D frac34 20 1 iexcl Dt C frac34 2

1 Dt

Dt Draquo

0 1 middot t middot iquest

1 iquest lt t lt T iexcl 1

(1)

where yt is the demeaned growth rate of the output series underconsiderationand Dt is a discrete latent variable that determinesthe date of the structural break iquest To allow for the possibilityofa permanent but endogenous structural break in the conditionalvariance we follow Chib (1998) in treating Dt as a discretelatent variable with the following transition probabilities

PrDtC1 D 0 j Dt D 0 D q

PrDtC1 D 1 j Dt D 1 D 1

0 lt q lt 1

(2)

That is before a structural break occurs or conditional onDt D 0 there always exists nonzero probability 1 iexcl q that astructural break will occur or DtC1 D 1 Thus the expected du-ration of Dt D 0 or the expected duration of a regime beforea structural break occurs is given by Eiquest D 1

1iexclq Howeveronce a structural break occurs at t D iquest we have Diquest Cj D 1 for allj gt 0 We estimate two versions of the model given in eqs (1)and (2) The rst version which we call model 1 freely esti-mates the model parameters The second version which we callmodel 2 constrains the model parameters to have no structuralbreak that is frac34 2

0 D frac34 21

Bayesian inference requires speci cation of prior distribu-tions for the model parameters In order to evaluate the sensi-tivity of the results to the choice of priors we use the followingthree sets of prior speci cations for model 1

Prior 1A [Aacute1 Aacutek]0 raquo N0k IkI1

frac34 20

raquo gamma12I

1

frac34 21

raquo gamma11I q raquo beta8 1

Prior 2A [Aacute1 Aacutek]0 raquo N0k 2IkI1

frac34 20

raquo gamma14I

1

frac34 21



frac34 20

raquo gamma11I

1

frac34 21

raquo gamma1 5I q raquo beta8 05

Priors used for model 2 were based on those for [Aacute1 Aacutek]0 andfrac34 2

0 in priors 1Andash3A and yielded results very close to the maxi-mum likelihood estimates In the interest of brevity we presentonly results for prior 1A in the following sections however allresults were quite robust to choice of prior

To evaluate the evidence for a structural break at an unknownbreak date based on the above models we compare the modelallowing for structural change and the model with no structuralbreak using Bayes factors

BF12 DmeYT jmodel 1

meYT jmodel 2

82 Journal of Business amp Economic Statistics January 2004

where eYT D [y1 yT ]0 and meYT j cent is the marginal likelihoodconditional on the model chosen Among the various methodsof computing the Bayes factor introduced in the literature wefollow Chibrsquos (1995) procedure in which the Bayes factor isevaluated througha direct calculationof the marginal likelihoodbased on the output from the Gibbs sampling procedure Kassand Raftery (1995) provided a general discussion on the use ofBayes factors for model comparison Details of the Gibbs sam-pling procedure for the speci c models considered here havebeen described by Kim and Nelson (1999)

To aid interpretationof the Bayes factor we refer to the well-known scale of Jeffreys (1961) throughout

lnBF lt 0 Evidence supporting the nullhypothesis

0 lt lnBF middot 115 Very slight evidence against thenull hypothesis

115 lt lnBF middot 23 Slight evidence against the nullhypothesis

23 lt lnBF middot 46 Strong to very strong evidenceagainst the null hypothesis

lnBF gt 46 Decisive evidence against the nullhypothesis

It should be emphasized that this scale is not a statistical cal-ibration of the Bayes factor but is instead a rough descriptivestatement often cited in the Bayesian statistics literature

3 EVIDENCE OF A VOLATILITY REDUCTION INAGGREGATE AND DISAGGREGATE REAL GDP

In this section we evaluate the evidence of a volatility re-duction in the growth rates of aggregate and disaggregate USreal GDP data All data were obtained from Haver Analyt-ics are seasonally adjusted are expressed in demeaned andstandardized quarterly growth rates and cover the sample pe-riod 19532ndash19982 The lag order k was chosen based onthe Akaike Information Criterion (AIC) for the model with no

Table 1 Bayesian Evidence of a Volatility Reduction in Aggregate andDisaggregate Real GDP

PosteriorVariable lnBF12 mean of iquest frac34 2

1 =frac34 20

GDP 1813 19841 20Trend Consumption-based iexcl38Trend HP lter iexcl8285Cycle Consumption-based 1394 19834 23Cycle HP lter 1741 19831 20Goods production 1294 19841 25Services 3450 19582 09Structures production 683 19841 37Durable goods production 1620 19841 21Nondurable goods production 365 19864 47Final sales of domestic product 664 19833 40Final sales of durable goods 441 19913 36Final sales of nondurable goods 796 19861 34

structural break model 2 with k D 6 the largest lag length con-sidered The AIC chose two lags for all series except consump-tion of nondurable goods and services for which it chose onelag All inferences are based on 10000 Gibbs simulations af-ter the initial 2000 simulations were discarded to mitigate theeffects of initial conditions

Table 1 summarizes the results of the Bayesian estimationand model comparisons for all of the aggregate and disaggre-gate real GDP series considered The second column presentsthe results of the Bayesian model comparison summarizedby the log of the Bayes factor in favor of a structural breaklnBF12 The third column shows the mean of the posteriordistribution of the break date iquest The fourth column presentsthe ratio of posteriormeans of the variance of et before and afterthe structural break frac34 2

1 =frac34 20 Finally Figures 1ndash13 plot the esti-

mated probabilityof a structural break at each point in the sam-ple PrDt D 1jeYT and the posterior distribution of the breakdate for each series considered

31 Aggregate Real GDP Trend and Cycle

The rst row of Table 1 contains results for the growth rateof aggregate real GDP The posterior mean of frac34 2

1 is 20 of frac34 20

(a) (b)

Figure 1 Growth Rate of GDP (a) Probability of Structural Break and (b) Posterior Distribution of Break Date


(a) (b)

Figure 2 Growth Rate of Consumption of Nondurable Goods and Services (a) Probability of Structural Break and (b) Posterior Distribution ofBreak Date

consistent with a sizable reduction in conditionalvolatilityThelog of the Bayes factor is 181 decisive evidence against themodel with no volatility reduction The posterior mean of theunknown break date is 19841 the same date reported by Kimand Nelson (1999) and MPQ Figure 1(b) shows that the poste-rior distribution of the unknown break date is tightly clusteredaround its mean

We next turn to the question of whether both the trend andthe cyclical component of real GDP have shared in this ag-gregate volatility reduction It seems reasonable that at least aportion of the volatility reduction is due to a stabilization ofcyclical volatility Many plausible explanations for the volatil-ity reduction (eg improved monetary policy and better inven-tory management) would mute cyclical uctuations Howeversome explanations (eg a lessening of oil price shocks) mightalso affect the variability of trend growth rates Thus investi-gating a stabilization in the trend and cyclical components may

help shed light on the viability of competing explanations forthe aggregate volatility reduction

There are of course numerous ways of decomposing realGDP into a trend and cyclical component Unfortunately thechoice of decomposition can have nontrivial implications forimplied business cycle facts (see eg Canova 1998) Thisseems particularly relevant for investigating a structural breakin volatility because assumptions made to identify the trendmay affect how much of a volatility reduction in the aggregateseries is re ected in the ltered trend or cycle To abstract fromthis problem in this article we consider a theory-based mea-sure of trend de ned as the logarithm of personal consump-tion of nondurables and services (LCNDS) Such a de nitionof trend is both theoretically and empirically plausible Neo-classical growth theory (see eg King Plosser and Rebelo1988) suggests that log real GDP and LCNDS share a commonstochastic trend driven by technological change This analy-sis suggests that log real GDP and LCNDS are cointegrated

(a) (b)

Figure 3 Growth Rate of HP Trend (a) Probability of Structural Break and (b) Posterior Distribution of Break Date


(a) (b)

Figure 4 Consumption-Based Cycle (a) Probability of Structural Break and (b) Posterior Distribution of Break Date

with cointegrating vector 1iexcl1 Recent investigations of thecointegration properties of these series con rm this result (seeeg King Plosser Stock and Watson 1991 Bai Lumsdaineand Stock 1998) If one also assumes a simple version of thepermanent income hypothesis which suggests that LCNDS isa random walk then LCNDS is the common stochastic trendshared by log real GDP and LCNDS Although it is now wellknown that LCNDS is not a random walk in the sense that onecan nd statistically signi cant predictors of future changes inLCNDS Fama (1992) and Cochrane (1994) have argued thatthese deviations are so small as to be economically insignif-icant Based on our measure of trend we de ne the cyclicalcomponent of log real GDP as the residuals from a regressionof log real GDP on a constant and LCNDS Given its popularitywe also show results for a measure of trend and cycle obtainedfrom the HodrickndashPrescott (HndashP) lter with smoothingparame-ter set equal to 1600

The second row of Table 1 shows that there is no evidence infavor of a break in the volatility of the growth rate of LCNDS

and thus in the growth rate of our theory-based measure of thetrend of log real GDP The log of the Bayes factor is iexcl4 pro-viding slight evidence in favor of model 2 the model with nochange in variance The third column of Table 1 shows that theevidence for a break in the growth rate of the HndashP measureof trend is even weaker with model 2 strongly preferred overmodel 1 This suggests that the volatility reduction observed inaggregate real GDP is focused in its cyclical component In-deed the log of the Bayes factor for the level of the cyclicalcomponent derived from the theory-based measure of trend re-ported in the fourth row of Table 1 is 139 providing decisiveevidence against model 1 The posterior mean of frac34 2

1 is 23of frac34 2

0 close to the percentage for aggregate GDP The posteriormean of the unknownbreak date is 19834 with [from Fig 3(b)]the posterior distribution of the break point clustered tightlyaround this mean Even stronger results are obtained for thecyclical component based on the HndashP lter shown in the fthrow of Table 1 Here the posterior mean of frac34 2

1 is 20 of frac34 20 and

(a) (b)

Figure 5 HP Cycle (a) Probability of Structural Break and (b) Posterior Distribution of Break Date


(a) (b)

Figure 6 Growth Rate of Goods Production (a) Probability of Structural Break and (b) Posterior Distribution of Break Date

the log Bayes factor is 174 These results which suggest thatonly the cyclical component of real GDP has undergone a largevolatility reductionmakes explanationsfor the aggregate stabi-lization based on a reduction of shocks to the trend componentless compelling

32 How Broad Is the Volatility ReductionEvidence From Disaggregate Data

In this section we attempt to evaluate the pervasivenessof thevolatility reduction observed in aggregate real GDP across sec-tors of the economy To this end we apply the Bayesian testingmethodology to the set of disaggregated real GDP data inves-tigated by MPQ This dataset comprises the broad productionsectors of real GDP goodsproductionservices productionandstructures production We investigate goods production furtherby separating durable goods and nondurable goods productionFinally we investigate the evidence for a volatility reductionin measures of nal sales in the goods sector MPQ found a

volatility reductiononly in the productionof durable goods anddetermined that the volatility reduction was not visible in mea-sures of nal sales Thus we are particularly interested in theevidence for a volatility reduction outside of the durable goodssector and in measures of nal sales

Rows 6ndash8 of Table 1 contain the results for the productionof goods services and structures Consistent with MPQ we nd strong evidence of a volatility reduction in the goods sec-tor The log of the Bayes factor for goods production is 129providing decisive evidence against the model with no changein volatility The volatility reduction is quite large with theposterior mean of frac34 2

1 25 of frac34 20 The posterior mean of the

break date is 19841 close to the date estimated by MPQ usingclassical techniques Also consistent with MPQ we nd littleevidence of a volatility reduction in services productionthat oc-curs around the time of the break in aggregate real GDP Fromrow 7 of Table 1 the log Bayes factor for services productionis 345 extremely strong evidence in favor of a volatility reduc-tion However the estimated mean of the break date is 19582

(a) (b)

Figure 7 Growth Rate of Services (a) Probability of Structural Break and (b) Posterior Distribution of Break Date


(a) (b)

Figure 8 Growth Rate of Structures Production (a) Probability of Structural Break and (b) Posterior Distribution of Break Date

not in the early 1980s Figure 7(b) con rms this showing nomass in the posterior distribution of the break date in the 1980sHowever because the model used here allows for only a singlestructural break there may be evidence of a second structuralbreak in the early 1980s that is not captured To investigate thispossibility we re-estimated the model for services productionusing data beginning in 1959 The log Bayes factor fell to 34still strong evidence for a structural break However the esti-mated break date was again well before the early 1980s withthe mean of the posterior distribution in the second quarter of1966

We turn now to structures production where we obtain the rst discrepancy with the results obtained by MPQ Againusing classical tests MPQ were unable to reject the null hy-pothesis of no structural break in the conditional volatility ofstructures production However the Bayesian tests nd deci-sive evidence of a volatility reduction From row 8 of Table 1the log Bayes factor is 68 with the posterior mean of the breakdate equal to 19841 the same quarter as for aggregate GDP

Figure 8(b) shows that the posterior distribution of the breakdate is clustered tightly around the posterior mean The volatil-ity reduction in structures is quantitatively large as well theposterior mean of frac34 2

1 is 37 of frac34 20 Thus the evidence from

the Bayesian model comparison suggests that the volatility re-duction observed in real GDP not only is re ected in the goodssector as suggested by MPQ but also is found in the productionof structures

Following MPQ we next delve deeper into an evaluation ofthe goods sector by separately evaluating the evidence for avolatility reduction in the nondurable and durable goods sec-tors Row 9 of Table 1 contains the results for durable goodsThe log Bayes factor is 162 decisive evidence in favor ofstructural change The posterior mean of the unknown breakdate is the same as for aggregate GDP 19841 From row 10the log Bayes factor for nondurable goods production is 37smaller than for durable goods but still strong evidence againstthe model with no structural break The volatility reduction in

(a) (b)

Figure 9 Growth Rate of Durable Goods Production (a) Probability of Structural Break and (b) Posterior Distribution of Break Date


(a) (b)

Figure 10 Growth Rate of Nondurable Goods Production (a) Probability of Structural Break and (b) Posterior Distribution of Break Date

nondurablegoods is quantitativelylarge with frac34 21 47 of frac34 2

0 al-though smaller than that found for durable goods The posteriormean of the break date is also somewhat later than for durablegoods falling in 19864 Finally from Figures 9(b) and 10(b)the posterior distributionof the break date for nondurablegoodsproductionis more diffuse than those for durable goodsproduc-tion

We now turn to an investigation of measures of nal salesAgain a striking result found by MPQ is that classical tests donot reject the null hypothesisof no volatility reduction for eitheraggregatemeasures of nal sales or nal sales of durable goodsThis result is strongly suggestive of a primary role for the be-havior of inventories in explaining the aggregate volatility re-duction MPQ also argued that the failure of nal sales to showany evidence of a volatility reduction casts doubt on explana-tions for the aggregate volatility reduction based on monetarypolicy To investigate the possibility of a volatility reduction in

nal sales we investigate three series The rst series is an ag-gregate measure of nal salesmdash nal sales of domestic productWe then turn to measures of nal sales in the goods sector Inparticular we investigate nal sales of nondurable goods and nal sales of durable goods

From row 11 of Table 1 the log of the Bayes factor for -nal sales of domestic product is 66 decisive evidence againstthe model with no volatility reduction The volatility reductionin nal sales is quantitatively large with the posterior mean offrac34 2

1 40 of frac34 20 The posterior mean of the break date is 19833

consistent with that observed for aggregate real GDP From Fig-ure 11(b) the posterior distribution is clustered fairly tightlyaround this mean although less so than for aggregate real GDPThat we nd evidence for a volatility reduction in aggregate -nal sales is perhaps not surprising given that we have alreadyshown that the Bayesian model comparison nds evidence ofa structural break in structures production a component of ag-gregate nal sales Thus we are perhaps more interested in the

(a) (b)

Figure 11 Growth Rate of Final Sales of Domestic Product (a) Probability of Structural Break and (b) Posterior Distribution of Break Date


(a) (b)

Figure 12 Growth Rate of Durable Goods Final Sales (a) Probability of Structural Break and (b) Posterior Distribution of Break Date

results with regard to nal sales in the goods sector the onlysector for which the distinction between sales and productionismeaningful We now turn to these results

From row 12 of Table 1 the log of the Bayes factor for -nal sales of durable goods is 44 strong evidence against themodel with no structural break However the log Bayes factoris weaker than when inventories are included suggesting thatinventory behavior may be an important part of the volatilityreduction in durable goods production in the early 1980s Alsothe posterior mean of the break date for nal sales of durablegoods is in 19913This break is 7 years later than that recordedfor aggregate GDP and durable goods production Thus it ap-pears that the MPQ nding of no volatility reduction in nalsales of durable goods is not con rmed by the Bayesian tech-niques used here However there does appear to be signi canttiming differences in volatility reduction in the production and nal sales of durable goods

From row 13 of Table 1 we see that there is much ev-idence of a volatility reduction in nal sales of nondurablegoods The log Bayes factor is 80 decisive evidence againstthe model with no structural break Interestingly this is muchstronger evidence than that obtained for the production of non-durable goods Also the volatility reduction for nal sales ofnondurable goods appears to be quantitatively more importantthan that for production of nondurableswith frac34 2

1 34 of frac34 20 for

nondurable nal sales and 47 of frac34 20 for nondurable produc-

tion The posterior mean of the break date is in 19861 closeto the break date for nondurable goods production Howeveras seen in Figure 13(b) the posterior distribution of the breakdate is much more tightly clustered around this mean than thatobtained for nondurable goods productionThus the results for nal sales of nondurable goods suggests the opposite conclu-sion than was obtained for durable goodsThe evidence in favorof a break in volatility in the nondurable goods sector is morecompelling when only nal sales are considered This could be

(a) (b)

Figure 13 Growth Rate of Nondurable Goods Final Sales (a) Probability of Structural Break and (b) Posterior Distribution of Break Date


used as evidence against an inventory management explanationfor the volatility reduction in this sector

In summary the evidence presented here demonstrates thatthe results of MPQ are not robust to a Bayesian model compar-ison strategy Speci cally the volatility reduction appears to bepervasive across many of the production components of aggre-gate real GDP and is also present in measures of nal sales inaddition to productionThis evidence provides less ammunitionto invalidate any potential explanationfor the aggregate volatil-ity reduction than that obtained by MPQ Indeed the evidenceseems consistent with a broad range of explanations includ-ing improved inventory management and improved monetarypolicy and thus is not very helpful in narrowing the eld ofpotential explanations

One could still argue that our nding of reduced volatility ofdurable goods production in 1984 with no reduction in durable nal sales volatilityuntil 1991 makes improved inventoryman-agement the leading candidate explanation for the volatility re-duction within the durable goods sector However even thismay not be a useful conclusion to draw from the stylized factsThe results presented here suggest a difference in timing onlynot the absence of a volatility reduction in the nal sales ofdurables goods as in MPQ Such a timing difference could beexplained by many factors including idiosyncratic shocks to nal sales over the second half of the 1980s Another explana-tion is that the hypothesis of a single sharp break date is mis-speci ed and in fact volatility reduction has been an ongoingprocess Stock and Watson (2002) and Blanchard and Simon(2001) have presented evidence that is not inconsistent withthis hypothesis If this is the true nature of the volatility reduc-tion then the discrepancy in the estimated (sharp) break datefor durable goods production and nal sales of durable goodsbecomes less interesting

33 Robustness to Alternative Prior Speci cations

We obtained the results presented in Section 32 using theprior parameter distributions listed as ldquoprior 1Ardquo in Section 2We also estimated the model with priors 1B and 1C and ob-tained results very similar to those given in Table 1 In this sec-tion we further evaluate the sensitivity of the results to priorspeci cation by investigating alternative priors on the Markovtransition probability q This is a key parameter of the modelbecause it controls the placement of the unknown break dateThus the econometrician can specify beliefs regarding the tim-ing of the structural break by modifying the prior distributionfor q Classical tests such as those of Andrews (1993) andAndrews and Ploberger (1994) assume no prior knowledge ofthe break date Thus we are interested in the extent to whichthe differences between the results that we nd here and thoseobtained by MPQ can be explained by the prior distributionsplaced on q

We investigate this by re-evaluating one of the primary se-ries considered earlier for which we draw differing conclusionsfrom MPQmdashstructures production Using classical tests MPQwere unable to reject the null hypothesis of no structural breakin the volatility of structures production Here we found a logBayes factor of model 1 versus model 2 of 68 strong evi-dence in favor of a structural break We obtained this factor

using the prior on q listed in prior 1A q raquo beta8 1 the his-togram of which based on 10000 simulated draws from thedistribution is plotted in Figure 14(a) This distribution hasmean Eq frac14 988 and standard deviation 037 How restric-tive is this prior on q for the placement of the break date iquestFrom the 10000 draws of q plotted in Figure 14(a) roughly80 corresponded to Eiquest D 1=1 iexcl q gt 90 quarters which isthe midpoint of the sample size considered in this article Cor-respondingly roughly 20 of the draws from the distributionfor q yielded expected break dates less than 90 quarters into thesample

Table 2 considers the robustness of the result for structuresproduction to alternative prior speci cations on q keeping theprior distributions for the other parameters of the model thesame as in prior 1A Figure 14(b) holds the histogram for the rst alternativeprior that we considerq raquo beta11 This prioris relatively at with mean equal to 5 and roughly equal prob-ability placed on all values of q However this yields a some-what more restrictive prior distribution for the placement ofthe break date iquest than that used in prior 1A From the 10000draws of q plotted in Figure 14(b) only 1 corresponded toEiquest D 1=1iexclq gt 90 quarters Furthermore 98 of the drawsof q yielded expected break dates in the rst 50 quarters of thesample From the second row of Table 2 the log Bayes fac-tor for this prior speci cation is 37 less than that obtained us-ing prior 1A but still strong evidence in favor of a structuralbreak We next investigate a prior that places the mean valueof q at a very low value thus placing most probability masson a break date early in the sample We specify this prior asq raquo beta1 1 the histogram of which is contained in Fig-ure 14(c) This prior could be considered quite unreasonableFrom the 10000 draws of q plotted in Figure 14(c) less than1 corresponded to Eiquest D 1=1 iexcl q gt 90 quarters More-over more than 99 of the draws of q yielded expected breakdates in the rst 12 quarters of the sample However despitethis unreasonable prior the log Bayes factor shown in the thirdrow of Table 2 is still above 3 strong evidence in favor of astructural break To erase the evidence in favor of a structuralbreak one must move to the prior in the nal row of Table 2q raquo beta1 2 This prior distributiongraphed in Figure 14(d)places almost all probability mass on a break very early in thesample Of the 10000 draws of q plotted in Figure 14(d) nonecorresponded to Eiquest D 1= 1 iexcl q gt 50 quarters Under thisprior the model with no structural break is preferred slightlyover the model with a structural break

Thus it appears that the evidence for a structural break instructures production is robust to a wide variety of prior distri-butions This suggests that the sample evidence for a structuralbreak is quite strong making the choice of prior distribution onthe break date of limited importance for conclusions based onBayes factors

Table 2 Evidence of Structural Breakfor Alternative Priors on q

Structures Production

Prior distribution Log Bayes factor

q raquo beta8 1 68q raquo beta11 37q raquo beta11 31q raquo beta12 iexcl5


(a) (b)

(c) (d)

Figure 14 Histogram for (a) 10000 Draws From q raquo beta(8 1) (b) 10000 Draws From q raquo beta(11) (c) 10000 Draws From q raquo beta(11)and (d) 10000 Draws From q raquo beta(12)

34 Structural Change in In ation Dynamics

In this section we evaluate whether the reduction in thevolatility of real GDP is also present in the dynamics of ination Unlike the case of real GDP changes in conditionalmean and persistence also appear to be important for in ationThus to investigate structural change in the dynamics of in a-tion we use the following expanded version of the model inSection 2

yt D sup1Dcurrent

C frac12Dcurrentytiexcl1 C

piexcl1X

jD1

macrjDcurrent1ytiexclj C et

et raquo Niexcl0 frac34 2

Dt

cent

(3)

where yt is the level of the in ation rate The speci cationof the error term and its variance is the same as that inSection 2 However we also allow for a structural break inthe persistence parameter and conditional mean For exam-ple whereas frac34 2

Dtcaptures a shift in conditional volatility the

parameter frac12Dcurrent which is the sum of the autoregressive coef-

cients in the DickeyndashFuller style autoregression captures a

one-time shift in the persistence We allow for the possibil-ity that a shift in the persistence parameter and conditionalmean can occur at a different time than the shift in condi-tional variance Thus we assume that Dcurren

t is independent of Dt

and that it is governed by the following transitional dynam-ics

Dcurrent D

raquo0 1 middot t middot iquest curren

1 iquestcurren lt t lt T iexcl 1

PrDcurrentC1 D 0 j Dcurren

t D 0 D qcurren


t D 1 D 1

(4)

For in ation we consider a shorter sample period than forthe real variables beginning in the rst quarter of 1960 Pre-liminary analysis suggested that including data from the 1950syielded very unstable results that were generally indicativeof astructural break in the dynamics of the series in the late 1950sThat the 1950s might represent a separate regime in the be-havior of in ation is perhaps not surprising Romer and Romer(2002) argued that monetary policy in the 1950s had more in


Table 3 Posterior Moments CPI Ination

With structural break Without structural break

Parameter Mean Std Dev Mean Std Dev

sup10 302 145 203 129sup11 396 156frac120 941 040 910 042frac121 722 092macr10 iexcl235 131 iexcl277 078macr11 iexcl292 107macr20 iexcl158 138 iexcl314 075macr21 iexcl398 098frac34 2

0 883 118 808 094frac34 2

1 171 050qcurren 987 008q 992 013

common with that in the 1980s and 1990s than that in the 1960sand 1970s Because here we are interested in a structural breakthat corresponds to the break found in real variables in the early1980s we restrict our attention to the post-1960s sample Thisis a common sample choice in studies investigating structuralchange in US nominal variables (see eg Watson 1999Stockand Watson 2002)

Table 3 contains the mean and standard deviation of the pos-terior distributionsfor the parameters of the model in (3) and (4)applied to the consumer price index in ation rate The log ofthe Bayes factor comparing this model with one with no struc-tural breaks is 78 decisive evidence against the model with nostructural change The structural change appears to be comingfrom two places a reduction in persistence and a reduction inconditional variance The posterior mean of the persistence pa-rameter falls from frac120 D 94 to frac121 D 72 The posterior mean ofthe conditional volatility falls from frac34 2

0 D 88 to frac34 21 D 17 The

conditionalmean given by sup10 and sup11 is essentially unchangedover the sample Figure 15(b) shows the posterior distributionsof the two break dates The break date for the change in per-sistence iquest curren and the change in volatility iquest both have poste-

rior distributionsclustered tightly around their posterior means19792 and 19912 Given that the reduction in persistence im-plies a reduction in unconditional volatility these results sug-gest two reductions in volatility one reduction at the beginningof the 1980s and another at the beginning of the 1990s

4 WHAT EXPLAINS THE DISCREPANCY BETWEENTHE CLASSICAL AND BAYESIAN RESULTS

The results of MPQ and this article demonstrate that for sev-eral of the disaggregate real GDP series classical tests fail toreject the null hypothesis of no volatility reduction whereas aBayesian model comparison strongly favors a volatility reduc-tion One simple explanation for this discrepancy is that clas-sical hypothesis tests and Bayesian model comparisons are in-herently very different concepts Whereas classical hypothesistests evaluate the likelihoodof observing the data assuming thatthe null hypotheses is true a Bayesian model comparison eval-uates which of the null or alternative hypotheses is more likelyGiven this difference in philosophy the approach preferred bya researcher will likely depend on that researcherrsquos preferencesfor classical versus Bayesian techniques

This explanation is somewhat unsatisfying because it sug-gests that ldquofrequentistrdquo researchers will prefer the results ofMPQ whereas ldquoBayesianrdquo researchers will nd the results pre-sented in this article more convincing Thus in the remainderof this section we report on a series of Monte Carlo experi-ments performed to explore the discrepancy between the twoprocedures from a purely classical perspective Based on theresults of these experiments we argue that for this applicationthe conclusionsfrom the Bayesian model comparison shouldbepreferred even when viewed from the viewpoint of a classicaleconometrician

We rst consider the case in which a volatility reductionhas occurred in the series for which the classical and Bayesian

(a) (b)

Figure 15 CPI In ation Rate (a) Probability of Structural Break and (b) Posterior Distribution of Break Date


methodologies provide con icting results In this case theBayesian model comparison correctly identi es the true modelwhereas the classical hypothesis test fails to reject the false nullhypothesis a type II error Given that the alternative hypothesisis true how likely would it be to observe this type II error Toanswer this question we perform a Monte Carlo experiment tocalculate the power of a classical test used by MPQ against tworelevant alternative hypotheses In the rst of these hypotheses(DGP1) data are generated from an autoregression in whichthere is an abrupt change in the residual variance Formally

yt D sup1 C Aacute1ytiexcl1 C Aacute2ytiexcl2 C et

et raquo N0 frac34 2t

frac34 2t D

(frac34 2

0 t D 1 2 iquest

frac34 21 t D iquest C 1 T

frac34 20 gt frac34 2

1

(5)

In the second hypothesis (DGP2) data are generated from anautoregression in which there is a gradual change in the residualvariance



frac34 2t D frac34 2

0 1 iexcl PDt D 1 C frac34 21 PDt D 1


1

PrDt D 1 D

8lt

0 1 middot t middot iquest1

0 lt PDt D 1 lt 1 iquest1 lt t middot iquest2

1 t gt iquest2

PrDt D 1 middot PDs D 1 for t lt s

(6)

DGP2 is motivated by the posterior distributions of iquest for thedisaggregate real GDP data series As is apparent from Fig-ures 6ndash13 the posterior distribution of iquest is fairly diffuse inseveral cases a result suggestive of gradual rather than sharpstructural breaks Notably this tends to be the case for thoseseries for which the results of MPQ differ from the Bayesianmodel comparison structures production nondurable goodsproduction and all of the nal sales series

We base the calibration of DGP1 and DGP2 on a series forwhich the classical and Bayesian methodologies are sharply atodds nal sales of domestic product For this series the logBayes factor is 664 decisive evidence against the null of nostructural change However MPQ were unable to reject the nullhypothesis of no structural change in this series using classicaltests We replicate this result here using the Quandt (1960) sup-Wald test statistic the critical values of which were derived byAndrews (1993) Following MPQ we perform this test basedon the following equation for the residuals of (5)

jetj D reg01 iexcl Dt C reg1Dt C t

Dt Draquo

0 for t D 1 2 iquest

1 for t D iquest C 1 T

(7)

We proxy et with the least squares residuals from (5) Oet Wethen estimate (7) by least squares and the Wald statistic forthe test of the null hypothesis that reg0 D reg1 calculated foreach potential value of iquest Again following MPQ we use 15trimmingmdashthe break date is not allowed to occur in the rst or

last 15 of the sample The largest test statistic obtained is thesup-Wald statistic whereas the value of iquest that yields the sup-Wald statistic is the least squares estimate of the break date Oiquest Consistent with MPQ the sup-Wald test statistic for nal salesof domestic product is 705 below the 5 critical value of 885

To calibrate DGP1 we set the parameter iquest D Oiquest for nal salesof domestic product The parameters sup1 Aacute1 and Aacute2 are then setequal to their ordinary least squares estimates from (5) wherethe estimation is performed conditional on iquest D Oiquest Finally frac34 2

0and frac34 2

1 are set equal to

1iquest

iquestX

tD1

Oe2t and

1T iexcl iquest

TX

tDiquestC1

Oe2t

For DGP2 we set PDt D 1 D PDt j eYT obtained from theestimation of model 1 de ned in Section 21 for nal sales ofdomestic productThis probability is displayed in Figure 11(a)We set the other parameters equal to the means of the respec-tive posterior distributions from estimation of model 1 for nalsales of domestic productWe generate 10000sets of data fromDGP1 and DGP2 each set with a length equal to the samplesize of our series for nal sales of domestic product For eachdataset generated we perform a 5 sup-Wald test and recordwhether the null hypothesis is rejected

For DGP1 the sup-Wald statistic rejects the null hypothesis75 of the time suggesting that the power of this test againstDGP1 is only fair Under DGP2 arguably the more relevantdata-generating process for the disputed series the power ofthe sup-Wald test falls considerablywith the test rejecting only43 of the time Thus these results suggest that if the alternativehypothesis were true then it would not be unlikely for the sup-Wald test to fail to reject the null hypothesis This provides apossible reconciliationof the discrepancy between the classicaland Bayesian results

We next turn to the case in which the null hypothesis is trueIn this case the classical tests used by MPQ correctly fail toreject the null hypothesis whereas the Bayesian model com-parison calibrated to Jeffreysrsquos scale incorrectly prefers the al-ternative hypothesis From a classical perspective the Bayesianmodel comparison commits a type I error Thus another possi-ble reconciliationof the con icting results given by the classicaland Bayesian methodologies is that the Bayesian model com-parison tends to commit a larger number of type I errors thanthe classical test The question then is if the Bayes factor as in-terpreted in this article were treated as a classical test statisticwhat would its size be

To investigate this we perform a Monte Carlo experimentwhere data are generated under the null hypothesis of no struc-tural change Speci cally we generate data from model 2 de-scribed in Section 2 We calibrated the parameters to the meansof the posterior distributions for the parameters of model 2 tto nal sales of domestic product For each Monte Carlo simu-lation we computed the log Bayes factor using the same priorsas de ned in Section 2 Due to the lengthy computation timerequired for each simulation we reduced the number of MonteCarlo simulations to 1000

In this experiment 5 of the log Bayes factors exceeded34 Thus if one were to treat the log Bayes factor as a clas-sical test statistic then the 5 critical value would be 34 In


the results reported in Table 1 the log Bayes factor exceedsthis value in all cases except the two measures of trend GDPThis suggests that the discrepancy in the results of MPQ andthe Bayesian model comparison are not likely to be explainedby the Bayesian model comparison having an excessive sizewhen viewed as a classical test Indeed given that the values ofthe Bayes factors in Table 1 are often far above the 5 criticalvalue of 34 this reconciliation seems very unlikely

In summary these Monte Carlo experiments suggest thatwhen viewed from a classical perspective it would not be un-usual for classical tests to fail to reject the null hypothesiswhenthe alternative was the correct model However it would behighly unlikely to observe values for the log Bayes factor aslarge as those observed in Table 1 if the null hypothesis of nostructural change were true

5 CONCLUSION

In this article we used Bayesian tests for a structural break invariance to document some stylized facts regarding the volatil-ity reduction in real GDP observed since the early 1980s Firstwe found a reduction in the volatility of aggregate real GDPthat is shared by its cyclical component but not by its trendcomponentNext we investigated the pervasiveness of this ag-gregate volatility reduction across broad production sectors ofreal GDP Evidence from the existing literature based on clas-sical testing procedures has shown that the aggregate volatilityreduction has a narrow sourcemdashthe durable goods sectormdashandthat measures of nal sales fail to show any volatility reduc-tion This evidence has been used to cast doubt on explanationsfor the volatility reduction based on improved monetary policyIn contrast we have found that the volatility reduction in ag-gregate output is visible in more sectors of output than simplydurable goods production Speci cally we found evidence ofa volatility reduction in the production of structures and non-durable goods We also found evidence of a reduced volatilityof aggregate measures of nal sales similar to that in aggregateoutput Finally we found evidence of reduced volatility in nalsales of both durable goods and nondurable goods Based onthese results we argue that the evidence that one obtains frominvestigating the pattern of volatility reductions across broadproduction sectors of real GDP is not suf ciently sharp to castdoubt on any potential explanations for the volatility reductionWe also document that alongside the reduction in real GDPvolatility the persistence and conditional volatility of in ationhave also fallen

ACKNOWLEDGMENTS

Chang-Jin Kim and Charles R Nelson acknowledge supportfrom National Science Foundation grant SES-9818789 Kimacknowledges support from a Korea University special grantand Jeremy Piger acknowledges support from the Grover andCreta Ensley Fellowship in Economic Policy at the Universityof Washington The authors thank without implicating MarkGertler Andrew Levin James Morley the editor associate edi-tor an anonymousreferee and seminar participantsat the Johns

Hopkins UniversityUniversity of WashingtonFederal ReserveBoard and the 2001 AEA annual meeting for helpful com-ments The views expressed in this article do not necessarilyrepresent of cial positions of the Federal Reserve Bank of StLouis or the Federal Reserve System

[Received August 2000 Revised March 2003]

REFERENCES

Ahmed S Levin A and Wilson B A (2002) ldquoRecent US MacroeconomicStability Good Policy Good Practices or Good Luckrdquo International Fi-nance Discussion Paper no 730 Board of Governors of the Federal ReserveSystem

Andrews D W K (1993) ldquoTests for Parameter Instability and StructuralChange With Unknown Change Pointrdquo Econometrica 61 821ndash856

Andrews D W K and Ploberger W (1994) ldquoOptimal Tests When a NuisanceParameter Is Present Only Under the Alternativerdquo Journal of Econometrics70 9ndash38

Bai J Lumsdaine R L and Stock J H (1998) ldquoTesting for and DatingCommon Breaks in Multivariate Time Seriesrdquo Review of Economic Studies65 395ndash432

Blanchard O and Simon J (2001) ldquoThe Long and Large Decline in USOutput Volatilityrdquo Brookings Papers on Economic Activity 1 135ndash174

Canova F (1998) ldquoDetrending and Business Cycle Factsrdquo Journal of Mone-tary Economics 41 475ndash512

Chauvet M and Potter S (2001) ldquoRecent Changes in US Business CyclerdquoThe Manchester School 69 481ndash508

Chib S (1995) ldquoMarginal Likelihood From the Gibbs Outputrdquo Journal of theAmerican Statistical Association 90 1313ndash1321

(1998) ldquoEstimation and Comparison of Multiple Change-Point Mod-elsrdquo Journal of Econometrics 86 221ndash241

Cochrane J H (1994) ldquoPermanent and Transitory Components of GNP andStock Pricesrdquo Quarterly Journal of Economics 109 241ndash263

Fama E F (1992) ldquoTransitory Variation in Investment and Outputrdquo Journalof Monetary Economics 30 467ndash480

Jeffreys H (1961) Theory of Probability (3rd ed) Oxford UK ClarendonPress

Kahn J McConnell M M and Perez-Quiros G P (2000) ldquoThe ReducedVolatility of the US Economy Policy or Progressrdquo mimeo Federal ReserveBank of New York

Kass R E and Raftery A E (1995) ldquoBayes Factorsrdquo Journal of the AmericanStatistical Association 90 773ndash795

Kim C-J and Nelson C R (1999) ldquoHas the US Economy Become MoreStable A Bayesian Approach Based on a Markov-Switching Model of theBusiness Cyclerdquo Review of Economics and Statistics 81 608ndash616

King R G Plosser C I and Rebelo S T (1988) ldquoProduction Growth andBusiness Cycles II New Directionsrdquo Journal of Monetary Economics 21309ndash341

King R G Plosser C I Stock J H and Watson M W (1991) ldquoSto-chastic Trends and Economic Fluctuationsrdquo American Economic Review 81819ndash840

Koop G and Potter S M (1999) ldquoBayes Factors and Nonlinearity EvidenceFrom Economic Time Seriesrdquo Journal of Econometrics 88 251ndash281

McConnell M M and Perez-Quiros G P (2000) ldquoOutput Fluctuations inthe United States What Has Changed Since the Early 1980srdquo AmericanEconomic Review 90 1464ndash1476

Niemira M P and Klein P A (1994) Forecasting Financial and EconomicCycles New York John Wiley amp Sons Inc

Quandt R (1960) ldquoTests of the Hypothesis That a Linear Regression ObeysTwo Separate Regimesrdquo Journal of the American Statistical Association 55324ndash330

Romer C D and Romer D H (2002) ldquoA Rehabilitation of Monetary Pol-icy in the 1950rsquosrdquo American Economic Review Papers and Proceedings 92121ndash127

Stock J H and Watson M W (2002) ldquoHas the Business Cycle Changed andWhyrdquo NBER Macroeconomics Annual 17 159ndash218

Warnock M V and Warnock F E (2000) ldquoThe Declining Volatility of USEmployment Was Arthur Burns Rightrdquo International Finance DiscussionPaper no 677 Board of Governors of Federal Reserve System

Watson M W (1999) ldquoExplaining the Increased Variability of Long-Term In-terest Ratesrdquo Federal Reserve Bank of Richmond Economic Quarterly Fall1999


goods MPQ went on to show that measures of nal sales havenot become more stable suggesting that the volatility reductionis focused in the behavior of inventories The results of MPQcan lead to striking conclusions regarding the viability of com-peting explanations for the source of the volatility reductionFor example MPQ argued that explanationsbased on improvedmonetary policy are hard to reconcile with the limited breadthof the volatility reduction particularly the failure of measuresof nal sales to show greater stability They argued that expla-nations based on improved inventory management are muchmore consistent with the stylized facts In contrast the resultspresented here are consistent with a broad range of potentialexplanations Indeed we argue that the evidence from broadproduction sectors of real GDP is not suf ciently sharp to helpinvalidate potential candidates for the source of the volatilityreduction

Whereas MPQ investigated structural breaks in volatility us-ing classical tests based on work of Andrews (1993) and An-drews and Ploberger (1994) here we investigate this issue us-ing a Bayesian model comparison framework This methodcompares the marginal likelihood from models with and with-out structural breaks using the Bayes factor As discussed byKoop and Potter (1999) Bayes factors have an advantage overclassical tests for evaluating structural change in the way inwhich information regarding the unknown break date is incor-porated in the test The unknown break date which is a nui-sance parameter present only under the alternative hypothesisleads to nonstandard asymptotic distributions for classical teststatistics Solutions to this problem in the classical frameworkfail to incorporate sample information regarding the unknownbreak pointOn the contrarymodel comparison based on Bayesfactors incorporatessample information regarding the unknownbreak date and as a byproductof the procedure yield the poste-rior distribution of the unknown break date a very useful pieceof information for determining when the break occurred Giventhe substantial differences in methodology the evidence in fa-vor of structural change providedby the Bayes factor may differsubstantiallyfrom that yieldedby classical tests Thus a primarygoal of this article is to evaluate the robustness of the MPQ re-sults based on classical techniques to a Bayesian model com-parison methodology

Section 2 discusses the basic model speci cation andBayesian methodology that we use to investigate structuralchange in the various series considered in this article Section 3presents the results for aggregate and disaggregate real GDPand contrasts these results with the existing literature based onclassical hypothesis tests It also discusses evidence of struc-tural change in in ation dynamics Section 4 discusses possibleexplanations for the different results produced by the classicaland Bayesian methodologies Section 5 concludes

2 MODEL SPECIFICATION BAYESIAN INFERENCEAND MODEL COMPARISON TECHNIQUES

To investigate a possible volatility reduction in growth ratesof aggregate and disaggregate real GDP we use the following

empirical model

yt DkX

jD1

Aacutejytiexclj C et

et raquo N0 frac34 2Dt

frac34 2Dt

D frac34 20 1 iexcl Dt C frac34 2

1 Dt

Dt Draquo

0 1 middot t middot iquest

1 iquest lt t lt T iexcl 1

(1)

where yt is the demeaned growth rate of the output series underconsiderationand Dt is a discrete latent variable that determinesthe date of the structural break iquest To allow for the possibilityofa permanent but endogenous structural break in the conditionalvariance we follow Chib (1998) in treating Dt as a discretelatent variable with the following transition probabilities

PrDtC1 D 0 j Dt D 0 D q

PrDtC1 D 1 j Dt D 1 D 1

0 lt q lt 1

(2)

That is before a structural break occurs or conditional onDt D 0 there always exists nonzero probability 1 iexcl q that astructural break will occur or DtC1 D 1 Thus the expected du-ration of Dt D 0 or the expected duration of a regime beforea structural break occurs is given by Eiquest D 1

1iexclq Howeveronce a structural break occurs at t D iquest we have Diquest Cj D 1 for allj gt 0 We estimate two versions of the model given in eqs (1)and (2) The rst version which we call model 1 freely esti-mates the model parameters The second version which we callmodel 2 constrains the model parameters to have no structuralbreak that is frac34 2

0 D frac34 21

Bayesian inference requires speci cation of prior distribu-tions for the model parameters In order to evaluate the sensi-tivity of the results to the choice of priors we use the followingthree sets of prior speci cations for model 1

Prior 1A [Aacute1 Aacutek]0 raquo N0k IkI1

frac34 20

raquo gamma12I

1

frac34 21



frac34 20

raquo gamma14I

1

frac34 21



frac34 20

raquo gamma11I

1

frac34 21

raquo gamma1 5I q raquo beta8 05

Priors used for model 2 were based on those for [Aacute1 Aacutek]0 andfrac34 2

0 in priors 1Andash3A and yielded results very close to the maxi-mum likelihood estimates In the interest of brevity we presentonly results for prior 1A in the following sections however allresults were quite robust to choice of prior

To evaluate the evidence for a structural break at an unknownbreak date based on the above models we compare the modelallowing for structural change and the model with no structuralbreak using Bayes factors

BF12 DmeYT jmodel 1

meYT jmodel 2














1 =frac34 20









(a) (b)



(a) (b)






(a) (b)



(a) (b)





1 is 23of frac34 2



(a) (b)



(a) (b)









(a) (b)



(a) (b)








(a) (b)



(a) (b)









(a) (b)



(a) (b)








(a) (b)

















(a) (b)

(c) (d)




yt D sup1Dcurrent


piexcl1X

jD1



Dt

cent

(3)








Dcurrent D




t D 0 D qcurren


t D 1 D 1

(4)







0 883 118 808 094frac34 2

1 171 050qcurren 987 008q 992 013



0 D 88 to frac34 21 D 17 The







(a) (b)






frac34 2t D

(frac34 2

0 t D 1 2 iquest



1

(5)







1

PrDt D 1 D

8lt



1 t gt iquest2


(6)




Dt Draquo



(7)




0and frac34 2

1 are set equal to

1iquest

iquestX

tD1

Oe2t and

1T iexcl iquest

TX

tDiquestC1

Oe2t









5 CONCLUSION


ACKNOWLEDGMENTS




REFERENCES







































1 =frac34 20









(a) (b)



(a) (b)






(a) (b)



(a) (b)





1 is 23of frac34 2



(a) (b)



(a) (b)









(a) (b)



(a) (b)








(a) (b)



(a) (b)









(a) (b)



(a) (b)








(a) (b)

















(a) (b)

(c) (d)




yt D sup1Dcurrent


piexcl1X

jD1



Dt

cent

(3)








Dcurrent D




t D 0 D qcurren


t D 1 D 1

(4)







0 883 118 808 094frac34 2

1 171 050qcurren 987 008q 992 013



0 D 88 to frac34 21 D 17 The







(a) (b)






frac34 2t D

(frac34 2

0 t D 1 2 iquest



1

(5)







1

PrDt D 1 D

8lt



1 t gt iquest2


(6)




Dt Draquo



(7)




0and frac34 2

1 are set equal to

1iquest

iquestX

tD1

Oe2t and

1T iexcl iquest

TX

tDiquestC1

Oe2t









5 CONCLUSION


ACKNOWLEDGMENTS




REFERENCES



























(a) (b)






(a) (b)



(a) (b)





1 is 23of frac34 2



(a) (b)



(a) (b)









(a) (b)



(a) (b)








(a) (b)



(a) (b)









(a) (b)



(a) (b)








(a) (b)

















(a) (b)

(c) (d)




yt D sup1Dcurrent


piexcl1X

jD1



Dt

cent

(3)








Dcurrent D




t D 0 D qcurren


t D 1 D 1

(4)







0 883 118 808 094frac34 2

1 171 050qcurren 987 008q 992 013



0 D 88 to frac34 21 D 17 The







(a) (b)






frac34 2t D

(frac34 2

0 t D 1 2 iquest



1

(5)







1

PrDt D 1 D

8lt



1 t gt iquest2


(6)




Dt Draquo



(7)




0and frac34 2

1 are set equal to

1iquest

iquestX

tD1

Oe2t and

1T iexcl iquest

TX

tDiquestC1

Oe2t









5 CONCLUSION


ACKNOWLEDGMENTS




REFERENCES



























(a) (b)





1 is 23of frac34 2



(a) (b)



(a) (b)









(a) (b)



(a) (b)








(a) (b)



(a) (b)









(a) (b)



(a) (b)








(a) (b)

















(a) (b)

(c) (d)




yt D sup1Dcurrent


piexcl1X

jD1



Dt

cent

(3)








Dcurrent D




t D 0 D qcurren


t D 1 D 1

(4)







0 883 118 808 094frac34 2

1 171 050qcurren 987 008q 992 013



0 D 88 to frac34 21 D 17 The







(a) (b)






frac34 2t D

(frac34 2

0 t D 1 2 iquest



1

(5)







1

PrDt D 1 D

8lt



1 t gt iquest2


(6)




Dt Draquo



(7)




0and frac34 2

1 are set equal to

1iquest

iquestX

tD1

Oe2t and

1T iexcl iquest

TX

tDiquestC1

Oe2t









5 CONCLUSION


ACKNOWLEDGMENTS




REFERENCES



























(a) (b)









(a) (b)



(a) (b)








(a) (b)



(a) (b)









(a) (b)



(a) (b)








(a) (b)

















(a) (b)

(c) (d)




yt D sup1Dcurrent


piexcl1X

jD1



Dt

cent

(3)








Dcurrent D




t D 0 D qcurren


t D 1 D 1

(4)







0 883 118 808 094frac34 2

1 171 050qcurren 987 008q 992 013



0 D 88 to frac34 21 D 17 The







(a) (b)






frac34 2t D

(frac34 2

0 t D 1 2 iquest



1

(5)







1

PrDt D 1 D

8lt



1 t gt iquest2


(6)




Dt Draquo



(7)




0and frac34 2

1 are set equal to

1iquest

iquestX

tD1

Oe2t and

1T iexcl iquest

TX

tDiquestC1

Oe2t









5 CONCLUSION


ACKNOWLEDGMENTS




REFERENCES



























(a) (b)








(a) (b)



(a) (b)









(a) (b)



(a) (b)








(a) (b)

















(a) (b)

(c) (d)




yt D sup1Dcurrent


piexcl1X

jD1



Dt

cent

(3)








Dcurrent D




t D 0 D qcurren


t D 1 D 1

(4)







0 883 118 808 094frac34 2

1 171 050qcurren 987 008q 992 013



0 D 88 to frac34 21 D 17 The







(a) (b)






frac34 2t D

(frac34 2

0 t D 1 2 iquest



1

(5)







1

PrDt D 1 D

8lt



1 t gt iquest2


(6)




Dt Draquo



(7)




0and frac34 2

1 are set equal to

1iquest

iquestX

tD1

Oe2t and

1T iexcl iquest

TX

tDiquestC1

Oe2t









5 CONCLUSION


ACKNOWLEDGMENTS




REFERENCES



























(a) (b)









(a) (b)



(a) (b)








(a) (b)

















(a) (b)

(c) (d)




yt D sup1Dcurrent


piexcl1X

jD1



Dt

cent

(3)








Dcurrent D




t D 0 D qcurren


t D 1 D 1

(4)







0 883 118 808 094frac34 2

1 171 050qcurren 987 008q 992 013



0 D 88 to frac34 21 D 17 The







(a) (b)






frac34 2t D

(frac34 2

0 t D 1 2 iquest



1

(5)







1

PrDt D 1 D

8lt



1 t gt iquest2


(6)




Dt Draquo



(7)




0and frac34 2

1 are set equal to

1iquest

iquestX

tD1

Oe2t and

1T iexcl iquest

TX

tDiquestC1

Oe2t









5 CONCLUSION


ACKNOWLEDGMENTS




REFERENCES



























(a) (b)








(a) (b)

















(a) (b)

(c) (d)




yt D sup1Dcurrent


piexcl1X

jD1



Dt

cent

(3)








Dcurrent D




t D 0 D qcurren


t D 1 D 1

(4)







0 883 118 808 094frac34 2

1 171 050qcurren 987 008q 992 013



0 D 88 to frac34 21 D 17 The







(a) (b)






frac34 2t D

(frac34 2

0 t D 1 2 iquest



1

(5)







1

PrDt D 1 D

8lt



1 t gt iquest2


(6)




Dt Draquo



(7)




0and frac34 2

1 are set equal to

1iquest

iquestX

tD1

Oe2t and

1T iexcl iquest

TX

tDiquestC1

Oe2t









5 CONCLUSION


ACKNOWLEDGMENTS




REFERENCES









































(a) (b)

(c) (d)




yt D sup1Dcurrent


piexcl1X

jD1



Dt

cent

(3)








Dcurrent D




t D 0 D qcurren


t D 1 D 1

(4)







0 883 118 808 094frac34 2

1 171 050qcurren 987 008q 992 013



0 D 88 to frac34 21 D 17 The







(a) (b)






frac34 2t D

(frac34 2

0 t D 1 2 iquest



1

(5)







1

PrDt D 1 D

8lt



1 t gt iquest2


(6)




Dt Draquo



(7)




0and frac34 2

1 are set equal to

1iquest

iquestX

tD1

Oe2t and

1T iexcl iquest

TX

tDiquestC1

Oe2t









5 CONCLUSION


ACKNOWLEDGMENTS




REFERENCES































0 883 118 808 094frac34 2

1 171 050qcurren 987 008q 992 013



0 D 88 to frac34 21 D 17 The







(a) (b)






frac34 2t D

(frac34 2

0 t D 1 2 iquest



1

(5)







1

PrDt D 1 D

8lt



1 t gt iquest2


(6)




Dt Draquo



(7)




0and frac34 2

1 are set equal to

1iquest

iquestX

tD1

Oe2t and

1T iexcl iquest

TX

tDiquestC1

Oe2t









5 CONCLUSION


ACKNOWLEDGMENTS




REFERENCES






























frac34 2t D

(frac34 2

0 t D 1 2 iquest



1

(5)







1

PrDt D 1 D

8lt



1 t gt iquest2


(6)




Dt Draquo



(7)




0and frac34 2

1 are set equal to

1iquest

iquestX

tD1

Oe2t and

1T iexcl iquest

TX

tDiquestC1

Oe2t









5 CONCLUSION


ACKNOWLEDGMENTS




REFERENCES





























5 CONCLUSION


ACKNOWLEDGMENTS




REFERENCES


























The Less-Volatile U.S. Economy: A Bayesian...

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