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Structural Estimates of the U.S. Sacrifice Ratio Author(s): Stephen G. Cecchetti and Robert W. Rich Reviewed work(s): Source: Journal of Business & Economic Statistics, Vol. 19, No. 4 (Oct., 2001), pp. 416-427 Published by: American Statistical Association Stable URL: http://www.jstor.org/stable/1392276 . Accessed: 19/04/2012 16:57 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal of Business & Economic Statistics. http://www.jstor.org Structural Estimates of U.S. the Sacrifice Ratio StephenG. CECCHETTI Departmentof Economics, The Ohio State University,Columbus, OH 43210-1172 ([email protected]) Robert W. RICH Domestic Research Function,Federal Reserve Bank of New York,New York,NY 10045-0001 ([email protected]) This article investigatesthe statisticalpropertiesof the U.S. sacrifice ratio--the cumulativeoutput loss arising from a permanentreduction in inflation. We derive estimates of the sacrifice ratio from three structuralvector autoregressionmodels and then conducta series of simulationexercises to analyze their sampling distribution.We obtain point estimates of the sacrifice ratio that are consistent with results reportedin earlier studies. However, the estimates are very imprecise, which we suggest reflects the poor quality of instrumentsused in estimation.We conclude that the estimatesprovide a very unreliable guide for assessing the output cost of disinflationpolicy. KEY WORDS: Disinflation;Identification;Structuralshocks; Vector autoregression. The successful conduct of monetary policy requires policy makers to both specify a set of objectives for the performance of the economy and understandthe effects of policies designed to attain these goals. Stabilizing prices, one of the dual goals of U.S. monetarypolicy, is no different.It is generally agreed, and documentedby Barro (1996) and the papers in Feldstein (1999), that permanentlylow levels of inflation create long-run benefits for society, increasing the level and possibly the trendgrowthrate of output.Thereis also a strong belief that engineeringinflationreductionsinvolves short-term costs associated with losses in output. Policy makers' decisions on the timing and extent of inflation reduction depend on a balancing of the benefits and costs of moving to a new, lower level of inflation,which in the end requiresestimates of the size of each. This study focuses on the size of one of these aspects of disinflationpolicy for the U.S. over the postwarperiod. Specifically, we investigate the output cost of disinflation policy, usually referredto as the sacrifice ratio. The sacrifice ratio is the cumulative loss in output, measured as a percent of one year's gross domestic product (GDP), resulting from a onepercentage-pointpermanentreductionin inflation. Although the sacrifice ratio is a key considerationfor policy makers, estimating its size is a difficult exercise because it requiresthe identificationof changes in the stance of monetary policy and an evaluationof their impact on the path of output and inflation. Neither one of these determinationsis straightforwardbecause it is difficult to gauge the timing and extent of shifts in policy as well as to separatethe movements in output and inflation into those that were caused by policy and those thatwere not. Finally,even if an estimateof the sacrifice ratio can be constructed,it is critical for policy makers to know something about the precision of the estimate. To investigatethe U.S. sacrificeratio and its statisticalproperties, we employ structuralvector autoregression(VAR) estimation methods. Within this framework,we study models of increasingcomplexity,beginningwith Cecchetti's (1994) twovariablesystem, then consideringShapiroand Watson's(1988) three-variablesystem, and finally examining Gali's (1992) four-variablesystem. In each of these models, we derive estimates of the sacrifice ratio under a differentset of identifying restrictionsfor the structuralshocks. To assess the reliability of the sacrifice-ratioestimates, we undertakea series of simulation exercises, based on parametricbootstraps,that allow us to constructconfidence intervals. Our modeling strategyfollows the conventionalpractice in the structuralVAR literatureof decomposing monetary policy into a systematic and a random component to identify changes in the stance of policy. The systematic component can be thoughtof as a reactionfunction and describesthe historical response of the monetaryauthorityto movements in a set of key economic variables, while the random component is labeled as "monetary-policyshocks" and signifies actions of the monetary authority that cannot be explained by the policy rule. Using the estimated structuralVAR system, we can trace the dynamic responses of variables to a monetarypolicy shock and thereby assess the quantitativeimpact of a shift in policy on outputand inflation.Thus, our evaluationof the sacrifice ratio focuses on unanticipatedpolicy shocks, in which the effects of a contractionarypolicy correspondto a "pure"(exogenous) monetarytighteningratherthan a systematic (endogenous) response to other shocks. As an alternativeto the use of the structuralVAR framework and its focus on the dynamic responses to innovations in the system, one might consider studying the effects of the systematiccomponentof monetarypolicy. Attemptingto measure the sacrificeratio, and its precision, by looking at periods in which the monetary authorityadopted a stronger (or easier) anti-inflationarypolicy would requirethat experimentsin which there are statisticallymeasurableshifts in policy regime exist in the data. It is our belief that such episodes are not available. This is confirmedby a series of tests for parameter stability, in which we find that the models display little evidence of structuralbreaks over the sample period we examine. But beyond the practical issue of whether we have an appropriatehistory in the sample period, there is the problem of finding an econometric technique that would permit 416 ? 2001 American Statistical Association Journal of Business & Economic Statistics October 2001, Vol. 19, No. 4 Cecchettiand Rich: StructuralEstimatesof the U.S. SacrificeRatio a straightforwardevaluation of the effect of changes in the policy rule on the economy. Because of their reduced-form nature, structuralVAR techniques do not allow us to distinguish between the direct effects of shocks and the indirect effects that arise from their eliciting systematic responses by the monetaryauthority. We examine quarterlyU.S. data over the period 1959-1997 and consider the behavior of the sacrifice ratio over a one- to five-yearhorizon. The analysis generatesestimates of the sacrifice ratio that vary substantiallyacross the three structural VAR models. Specifically,the estimatesrange from about 1 to nearly 10, suggesting that somewherebetween 1%and 10%of one year's GDP must be sacrificedfor inflationto fall one percentage point. Although the point estimates are broadly consistent with the results of previous studies, the analysis also suggests that the sacrifice ratio is very imprecisely estimated. For example, a 90% confidence intervalcovers 0 for the point estimates in each model. Further,the sacrifice-ratioestimates display even greaterimprecisionas the models allow for additional structuralshocks. Although the simplest two-variable system indicates that the true value of the sacrifice ratio may lie somewherebetween -0.5 and +3.8, the four-variablesystem suggests a possible range that extends from -43 to +68. We do not take these latter estimates too literally because the values seem extremely implausible given our reading of the recent history. Nevertheless, they serve as a caution to policy makers in that the point estimates do not seem to provide a very reliable guide for gauging the output cost of disinflation policy. After presenting the basic results, we look into the possible sources of the imprecision in the estimates. Our findings do not suggest that the imprecisionarises from structural instability of the reduced-formVAR's or from a deterioration in the forecasting performance of higher-dimensionalmodels. Rather,the evidence points to weak instrumentsimplicitly used to estimate the models. This latter finding is consistent with the conclusions of Pagan and Robertson (1998), who reported substantialrandomnessin structuralVAR estimates of the liquidity effect and documentedthat poor instrument quality is an important source of the imprecision. Because the models examined in this study and by Pagan and Robertsonshare similaridentificationschemes, futureresearch employing the structuralVAR methodology may need to be particularlyconscious of the issue of instrumentquality and its implications for the reliability of the estimates. 1. THE SACRIFICERATIO Most people believe that attemptson the partof a monetary authorityto lower the inflation rate will lead to a period of increasedunemploymentand reduced output.The reason disinflationaryepisodes have this effect on real economic activity is that inflation displays a great deal of persistence or inertia. That is, price inflation [measuredby indexes such as the consumerprice index (CPI)] tends to move slowly over time, exhibiting very different behavior from things like stock or commodity prices. Thus, the adjustmentprocess during a disinflation requires the monetary authority to slow aggregate 417 demand growth, creating a period of temporaryslack in the economy that will lower the inflation rate only eventually. There are a number of explanations for inflation's slow adjustment and the absence of costless disinflation. Fuhrer (1995) provided a review of recent discussions and focused on three possibilities. First, inflation persistence may arise from the overlap and nonsynchronizationof wage and price contracts in the economy. Because wages and prices adjust at different times, as well as to each other, slowing the process takes time. Second, people's inflationexpectations may adjustslowly over time, being based on a sort of adaptive mechanism.Because decisions aboutwages and prices depend on expectations of future changes, slow adaptationis selffulfilling, creating inertia. And third, if people do not believe that the monetary authority is truly committed to reducing inflation, then inflation will not fall as rapidly. That is, the credibilityof the policy makeris importantin determiningthe dynamics of inflation, with less credibility leading to more persistence. The view that reductions in inflation are accompaniedby a period of decreased output (relative to trend) has generated considerabledebate among economists on how to lessen the costs of disinflation. Some discussions, including those of Okun (1978), Gordon and King (1982), Taylor (1983), Sargent (1983), Schelde-Andersen (1992), and Ball (1994), have focused on the speed of disinflation and whether the monetaryauthorityshould adopt a gradualistapproachor subject the economy to a "cold turkey"remedy.Otherdiscussions have focused on identifying the sources of inflation persistence and analyzing the implications of the level of inflation (Ball, Mankiw,and Romer 1988), the degree of nominal wage rigidity (Grubb, Jackman, and Layard 1983), and the extent of centralbank independence(Jordan1997) for the pursuitof cost-reducingstrategies. Although the question of how to lessen the costs of disinflation raises a numberof importantissues, the present analysis does not focus on this debate. Rather, we contend that discussions about the design and implementationof disinflation policy cannot proceed without some understandingof the quantitativeimpact of monetary policy on output and inflation. Thus, it is our view that the measurementof the sacrifice ratio is a prerequisiteto any study attemptingto evaluate its key determinantsor the impact of alternativepolicies on the costs of disinflation. With this practicalissue in mind, a numberof authorshave estimated sacrifice ratios for the United States using a variety of techniques. Okun (1978) examined a family of Phillips curve models and derived estimates that range from 6% to 18% of a year's gross nationalproduct,with a mean of 10%. Gordon and King (1982) used traditionaland VAR models to obtain estimates of the sacrifice ratio that range from 0 to 8. Mankiw (1991) examined the 1982-1985 Volckerdisinflation estiand used Okun'slaw to arriveat a "back-of-the-envelope" movements Ball examined mate of 2.8. More recently, (1994) in trend output and trend inflation over various disinflation episodes and obtainedestimates that vary from 1.8 to 3.3. Although the estimates calculated by Ball (1994) and Mankiw (1991) are of roughly the same order of magnitude, suggesting that a consensus may exist about the size of the 418 Journalof Business & Economic Statistics,October2001 4 3 2- l 01 -1 _ -2-358 68 78 88 98 Figure 1. QuarterlyGrowthRate of Output:Real GDP: 1959:Q11997:Q4. sacrifice ratio, there are several issues remaining.First, prior studies do not, in our view, adequatelycontrol for the impact of nonmonetaryfactors on the behavior of output and inflation. Consider the plots of the quarterlygrowth rate of real GDP and the CPI displayed in Figures 1 and 2. Although some of the movements in output and inflation over the postWorldWarII period are surely attributableto monetary-policy actions, it is unreasonable on either theoretical grounds or from visual inspectionto believe they all are. Thus, computing a meaningfulestimate of the sacrificeratio requiresmore than simply calculating a measure of the association between output and inflationduringarbitrarilyselected episodes. Rather,it S depends critically on isolating which movements result from monetaryinfluences. Second, previous studies such as that of Gordon and King (1982) have failed to account for the policy process. Some of the actions undertakenby a monetary authorityare intended to accommodate or offset shocks to the economy. However, the analysis of Gordonand King did not allow the movements in a policy variableto be separatedinto those associated with a shift in policy and those reflecting a systematic response to the state of the economy. This type of decomposition, which is necessary to assess the effects of monetary policy on the economy, requiresthe specificationand estimationof a structural economic model. Another importantissue concerns the periods selected for the empiricalanalysis. Studies such as thatof Ball (1994) have focused solely on specific disinflationaryepisodes-periods when contractionarymonetary policies are thought to have resultedin the reductionin both inflationand output.However, it is not obvious that estimates of the sacrifice ratio should exclude a priori episodes in which inflation and output are both increasing. Such an approachwould only be justified if there were an accepted asymmetryin the impact of monetary policy on output and prices. In the absence of such evidence, 4.0 3.5 3.0 C 2.5 2.0 . 1.5s 1.0 0.5 0.0 economic expansionswould contain episodes in which output and inflation increased as a result of expansionarymonetary policy. Such episodes would then be as informativeabout the sacrifice ratio as a disinflation. Filardo (1998) was an exception because he presentedevidence thatthe sacrificeratiofor the United States varies across threeregimes correspondingto periodsof weak, moderate,and strong output growth. Using a measure of the sacrifice ratio similar to that used in this study, Filardo derived sacrificeratio estimates of 5.0 in the weak-growthregime and 2.1 in the strong-growthregime. Although the findings of Filardo offer new and interesting insights into the outputcost of disinflation,his study, as well as previous work, was silent on the accuracyof the estimates. Specifically, there has been no serious attemptto characterize the statisticalprecision of sacrifice-ratiomeasures.Estimation of economic relationshipsand magnitudesinherentlyinvolves some uncertainty,and it is extremely importantto quantify their reliability.For example, policy makersmay be reluctant to undertakecertain policy actions unless they can attach a high degree of confidence to the predictedoutcomes. Characterizing the precision of sacrifice-ratioestimates is a primary goal of our analysis. We now turnto a discussion of the structuralVAR methodology and a descriptionof the models that we use to construct sacrifice-ratioestimates. 2. STRUCTURAL VECTORAUTOREGRESSIONS 2.1 IdentifyingMonetary-PolicyShocks and DerivingEstimates of the Sacrifice Ratio The structuralVAR methodology remains a popular technique for analyzing the effects of monetary policy on the economy. A structuralVAR can be viewed as a dynamic simultaneous-equationsmodel with identifying restrictions based on economic theory. In particular,the structuralVAR relates the observed movementsin a variableto a set of structural shocks-innovations that are fundamentalin the sense that they have an economic interpretation.In the formulation of their identificationassumptions,the models we employ appealto economic theories that then allow us to interpretone of the structuralinnovations as a monetary-policyshock. For this reason, we find the structuralVAR methodology attractive in evaluating the impact of monetary policy on output and inflation and giving us a measure of the sacrifice ratio. Admittedly, the structuralVAR methodology is not without its limitations. The estimated effects of shocks can vary considerablyas a result of slight modificationsto the identifying restrictions.Thus, it may be quite importantto examine a set of models when drawing inferences based on this approach. Our strategy for deriving an estimate of the sacrifice ratio can be illustratedwithin a relatively simple system that only includes output and inflation.Following Cecchetti (1994), we consider the following structuralVAR model: , Ayt = 58 Figure 2. 1997:Q4. 68 78 88 98 Quarterly Growth Rate of Prices: CPI Inflation: 1959:Q1- b lAy,_i + b 2A t+ i=1 A1rr,= b , Ay, + -b2 ti+ i=l i=1 bi , Ayti +" i=1 b227 ti Et (1) Cecchettiand Rich: StructuralEstimatesof the U.S. SacrificeRatio 419 that can be rewrittenmore conveniently as Because the structuralshocks are not observable,we obtain estimates of the structuralimpulse responses by using the B(L) A•, (2) reduced-formVAR representationof (1) in conjunction with et identifying restrictions.Formally,the unrestrictedVAR reprewhere yt is the log of output at time t, 7rtis the inflation rate sentation is given by between time t - 1 and t, A denotes the difference operator = = DX - D2Xt-2 (5) DkXt-k ..D(L)Xt -t, Xt -D(1 - L), B(L) - [Bij(L)] for i, j = 1, 2 is a (2 x 2) matrix of polynomial lags, and E, = [E45, Et]' is a vector innovation where Xt is an (n x 1) vector of endogenous variables,D(L) to that contains the shocks aggregatesupply (Ey) and is a kth-orderlag polynomial matrix, and t, is the (n x 1) process It assumed that Et has zero mean vector of innovationsto the system. It is assumed that has demand is aggregate (E7). ,t and is serially uncorrelatedwith covariance matrix E[Et,,] = zero mean and is serially uncorrelatedwith covariancematrix fl for all t. The model includes the change in the inflation E[/xjt/] = I for all t. rate to allow shocks to have a permanenteffect on its level. Equation(5) can be estimatedand invertedto yield its unreOurprimaryinterestis in the impactof the structuralshocks strictedVMA representation: on output and inflation over time. To evaluate these magnitudes, we can look at the vector moving average (VMA) rep(6) X, = L,t + ClL-1 + C2.t-2 +'-' = C(L)jt. of the of which resentation (1), impulse responses provides From (2) and (3), the VAR and VMA representationsof the the system to the structuralshocks. This is written as structuralsystem can be written,respectively,as aIylEt-i+ i=o Ay,] Lata I -i=O i Y _+ a2et-i BoXt = B X_1 +B2Xt-2 a2rt-i i=O X, = AoEt+ AI •E,_ -' i=O [A11(L)A12(L) rY (7) t+ and a 22 t-i SA21(L) A22(L) LEt +- - +BkXt-k (3) If we initially use aggregatedemand shocks to identify shifts in monetarypolicy, then (3) providesa particularlyconvenient representationto assess the dynamicimpactof a monetarypolicy shock on outputand inflation.An estimate of the sacrifice ratio can then be computed based on the structuralimpulse response functions from (3). For inflation, the sum of the first r coefficients in A22(L) measures the effect of a monetary-policyshock on its level 7 periods forward.In the case of output, however, the sacrifice ratio requiresus to consider the cumulativeeffect on its level resulting from a monetarypolicy shock. This quantitycan be expressed as a function of the coefficients in A12(L). Taken together,the relative impact of monetarypolicy on outputand inflation, and hence the sacrifice ratio, over the time horizon 7 is just the ratio of these effects and can be calculated as = A(L)Et. (8) Equations(6) and (8) imply that E[(AoEt)(Ao0t)'] = AoIAo= = E[(,t)(, tt)] (9) and A(L) = C(L)A0. (10) The ability to link the unrestrictedVAR to a structuralVAR model hinges crucially on the estimation of the matrix A0. Because A0 has (n x n) unique elements, complete identification requiresa total of n2 restrictions.This is a necessary but not sufficient condition for identificationbecause sufficiency requiresthe matrix A0 to be invertible. One set of identifying restrictionsis based on the assumption that the structuralshocks are uncorrelatedand have unit variance. That is, fl = I, where I, is the (n x n) identity matrix. In addition to assuming that fl is the identity matrix, there are three other sets of identifying restrictions that are employed in the estimation of A0. Two of these sets focus on the effects of structural shocks on particular variables df lfl S(7)= (4) and involve short-runrestrictions(A0 restrictions)or long-run Lj=oa) restrictions[A(1) restrictions].The third involves restrictions on the coefficients of contemporaneousvariablesin the struct re u For aimt2 -aioaa12 diinf2m oi=n /io=0 ii=0 tural equations(B0 restrictions). Following Blanchardand Quah (1989), our additionalidentifying restriction for the Cecchetti model is that aggregate i=O demand shocks have no permanent effect on the level of ai output. This is equivalent to the condition that A12(1) = L ai=a2 = 0. (4) ( The Cecchetti model only identifies two shocks and associates shifts in monetary policy with the aggregate demand For a disinflationarymonetary strategy undertakenat time t, disturbance.Although this identificationscheme is useful for the numeratormeasuresthe cumulativeoutputloss throughthe illustrating the structuralVAR methodology and may profirst 7 periods (ignoring discounting), while the denominator vide a good approximationfor analyzing the relative importance of nominal and real shocks, the frameworkcould yield is the difference in the level of inflation 7 periods later. ( Journalof Business & Economic Statistics,October2001 420 misleading estimates of the sacrifice ratio. Specifically, the restrictions in the Cecchetti model fail to identify separate components of the aggregate demand disturbance.Thus, the estimated monetary-policyshock would not only encompass policy shifts but also other shocks related to government spending or shifts in consumptionor investmentfunctions. To provide a more detailed analysis, we also derive estimates of the sacrificeratio from models developedby Shapiro and Watson (1988) and Gali (1992). These models allow us to decompose the aggregatedemanddisturbanceinto separate componentsand thereforecan be used to judge the sensitivity of the results to alternativemeasures of the monetary-policy shock. Drawing on the work of Shapiro and Watson (1988), we consider a three-variablesystem given by E SAt [ E Ar"tj= A(L) (it - 7Tt) (11) L L where it is a short-termnominal interestrate, (it - rt) is an ex post real interestrate, and A(L) is a (3 x 3) matrixof polynomial lags. Although we refer to this as the "Shapiro-Watson model," our specification actually differs from theirs in two small ways. First, Shapiroand Watsondecomposedthe aggregate supply disturbanceinto a technology shock and a laborsupply shock. Second, they also included oil prices as an exogenous regressorin their structuralVAR system. Based on this three-variablesystem, we are able to identify three structuralshocks, where eY continues to denote an aggregatesupply disturbanceand where the aggregatedemand disturbanceis now decomposed into an LM shock and an IS and E1s. We identify the shock, denoted, respectively,by "LM structuralshocks using both short-runand long-run restrictions. The Blanchard-Quahrestrictionallows us to identify the aggregatesupply shock. In addition,we discriminatebetween IS and LM shocks by assuming that monetarypolicy has no contemporaneouseffect on output (a?2= 0). We estimate the sacrifice ratio by identifying monetary-policyshifts with LM shocks. The Gali (1992) model, which allows for the identification of a fourth structuralshock, can be written as Ayt L i - 'lt) (Amt( =iEMS A(L) (12) M,,D L sI where mt is the log of the money supply, (Amt --r) is the growth of real money balances, and A(L) is a (4 x 4) matrix of polynomial lags. The structural demand shocks els, eMD, and EtS denote (a2 = a3 = 0). The second assumptionis that contemporaneous prices do not enter the money-supplyrule (b?3+ b4 = 0). Our estimate of the sacrifice ratio uses money-supply shocks to identify changes in the stance of monetarypolicy. Although the inflationrate does not appearas an individual variablein (12), we can recoverits impulse responsefunctions from those estimated for Ait and (i, - rt). Specifically, we can use the following relationship 7rt = it[= (1 - L)-'Ait] - (it - 7rt) (13) to obtainan estimateof the impactof a monetary-policyshock on the level of inflation 7 periods forward: T +'r aTrt t ) i r a22 --a32. i=O (14) 2.2 ComputingConfidenceIntervals The sacrifice-ratioestimates we compute are not the true values but are randomvariableswith distributions.It is natural to compute and reportsome measure of the precision for our estimates. Thereare severalpossible proceduresfor computing confidence intervals for S, (r). One possibility would be to note that the sacrifice ratio is a function of the parametersof the structuralVMA and use the delta method.This is computationally demandingand infeasible for the more complex models we study. Insteadwe employ simulation-basedmethods to constructexact small-sampledistributionsof the estimatorof the sacrifice ratio. The procedurecan be briefly described as follows. Let 0 denote the vector of parametersthat constitute the reduced-formVAR model, and let 6 and I denote, respectively, the estimatedparametervector and estimatedvariancecovariance matrix of the residuals of the reduced-formVAR using the observed set of data. Our simulation approachis based on specific distributionalassumptionsabout (0, 1) and follows the procedure described by Doan (1992, chap. 10). Specifically, we construct a sequence of random draws of I from an inverted Wishart distribution,each of which is then used to generate a draw for 0. For each draw of (0i, Vi), we can impose the relevant identification restrictions and estimate an artificial structuralVAR model, compute its impulse response functions, and then construct an estimate of the sacrifice ratio denoted by Si(Oi, iW). In this manner, a series of N simulations can be undertakenand used to construct N estimates of the sacrifice ratio denoted by Sl(O', E'), S2(02, 2) ..., SN(ON, CN). We can then construct confidence intervalsfor S, (7) based on the range that includes the specified percent of the values for Si(Oi,Ei) Simulationmethods can also provide insights into the presence of bias in the point estimates. Specifically, we can use informationfrom the simulations to report median biascorrected point estimates of the sacrifice ratio using the following formula: a money-supply shock, a money-demandshock, and an IS shock, respectively.The identificationof the structuralshocks are again based on short-runand long-run restrictions. We retain the Blanchard-Quahrestriction and assume that the (15) s, (7) = Sr (7) -[S;5 (r7)- SE (7)], three aggregate demand shocks have no permanent effect on the level of output. Further,we follow Gali and adopt where S•, 7) is the bias-adjustedpoint estimate of the sacritwo additional assumptions. The first is that neither money fice ratio, S,,(7) is the sacrifice-ratioestimate from the strucdemand nor money supply affect output contemporaneously tural VAR model, and S?;5(7) is the median sacrifice-ratio Cecchettiand Rich: StructuralEstimatesof the U.S. SacrificeRatio estimate from the simulations.The term in bracketsmeasures the estimated bias. A similar procedurecan be used to construct bias-adjustedestimates of the impulse response functions and confidence intervals. As a final note, one could arguethatour resultsmay actually understatethe imprecision associated with the sacrifice-ratio estimates. The analysis abstractsfrom real-time data considerations and its implicationsfor the conduct of monetarypolicy. For example, variables such as output (or unemployment rate) gaps are typically constructedand used by the monetary authorityfor forecasting purposes or in Taylor-typerules to gauge policy actions. Because these variables are unobserved and very imprecisely measured (Orphanides2000a,b), their inclusion in the models we study would only lead to greater unreliabilityin the sacrifice-ratioestimates. The analysis now turns to a presentationof the results and a discussion of the point estimates of the sacrifice ratio and their associated confidence intervals. RESULTS 3. EMPIRICAL We construct sacrifice-ratioestimates from the three structuralVAR models using quarterlydata over the sample period 1959:Q1-1997:Q4. Outputis measuredby real GDP and inflation is measuredby the CPI. The sacrifice-ratioestimates are based on a differenttransformationof the data for output and prices. Outputgrowth is measured at a quarterlyrate in percentage terms, while inflationis measuredat an annualrate in percentage terms. The short-terminterest rate representsthe yield on three-monthTreasurybills and the monetary aggregate is measuredby Ml. The appendix describes the data in furtherdetail. It is worth noting that preliminary analysis of the data supports the specification of the models. In particular,we examined the stationarityproperties of the various series to determinetheir degree of integrationand the presence of cointegrating relationships. The results from the application of Dickey-Fuller (1981) tests providedevidence that y, ir, i, and Am contain a unit root, but that (i - 7) and (Am - Ir) are stationary variables. The findings that both output and inflation contain a unit root are particularlyimportantfor the identification schemes and the concept of a sacrifice ratio. The evidence of a unit root in the outputprocess allows the long-run restriction on the effects of aggregate demand shocks to be well defined and meaningful,while the evidence of a unit root in the inflationprocess allows for permanentshifts in its level. The lag length of the reduced-formVAR was set equal to 8 for the Cecchetti model and equal to 4 for the ShapiroWatsonand Gali models. Estimationof the sacrificeratio also requiresthe selection of a horizon for the long-runrestriction on aggregatedemand shocks. Following Cecchetti (1994), we assume that aggregatedemandshocks completely die out after 20 years and truncatethe structuralVMA representationsat 80 quarters. Table 1 presents the point estimates of the sacrifice ratio at horizons of one to five years, and Figure 3 displays the (median) bias-adjusted estimated responses of output and inflation to a one-unit monetarypolicy shock across the three 421 Table1. Sacrifice-RatioEstimatesforthe UnitedStates, 1959:Q11997:Q4;CumulativeOutputLoss as a Percentage of Real GDP Model 7= 4 1.3219 Cecchetti Shapiro-Watson 0.1854 0.8016 Gall r=8 r = 12 r = 16 r = 20 1.3204 0.6211 3.7092 1.5700 1.0014 6.1791 1.5219 1.1548 8.1434 1.3763 1.2768 9.8709 models, together with 90% confidence bands. The point estimates should be interpretedas the cumulativeoutputloss correspondingto a permanentone-percentage-pointdecline in the rate of inflation measuredon an annualbasis. The Cecchetti model yields sacrifice-ratioestimates that are relatively constant as the horizon grows, while the other two models show a clear upward trend, with higher output loss at longer horizons. Interestingly,the 20-quarterhorizon estimates from the Cecchetti and Shapiro-Watsonmodels, 1.38 and 1.28, respectively,are very similarto the values of 1.8 and 1.4 calculated by Ball (1994) and Schelde-Andersen(1992) for the Volckerdisinflation.In contrast,the Gali model generates a markedlyhigher estimate of 9.87, which is much closer to the mean value obtainedby Okun (1978). An examinationof the bias-adjustedimpulse responsefunctions in Figure 3 reveals a pattern that is qualitatively similar across models and accords with the predicted effects of a monetary tightening. Output declines in response to a monetary-policyshock before eventually returningto its initial level. (The nature of the identificationrestrictionsin the Shapiro-Watsonand Gali models precludes a contemporaneous responseof outputto the monetary-policyshock.) Inflation also decreases and displays a permanentdecline in response to the monetary-policyshock. As shown, there are some differencesin the extent and size of the impact of the monetary-policyshock across models. For example, most of the response of output and inflation in the Cecchetti model occurs within the first few quartersafter the shock. Relative to the Cecchetti and Shapiro-Watsonmodels, the Gali model suggests a deeper and more protractedoutput decline along with a smaller decrease in inflation. Both of these effects lead to the higher point estimates of the sacrifice ratio that emerge from this model. Figure 3 also indicates that the impulse response functions are not estimatedvery precisely. Because of the natureof the long-run identifying restrictions,one would expect the confidence intervals for output to narrow and converge around 0 as the horizon steadily increases. However, the results indicate that the confidence intervalsat shorterhorizons typically cover 0 and can be fairly wide. In the case of inflation, the confidence intervalsfor the Shapiro-Watsonand Gali models seem particularlywide and either cover or lie very close to 0. An initial reading of this evidence reveals the imprecision associated with the quantities of interest and clearly hints at the potential difficulty of obtaining reliable estimates of the sacrifice ratio. Table 2 presents the results for the simulated distributions of the sacrifice ratio based on the proceduresoutlined in the previous section, with 10,000 replications.We report(median) 422 Journal of Business & Economic Statistics, October 2001 A: DynamicResponse to a MonetaryPolicy Shock B: DynamicResponse to a MonetaryPolicy Shock Real GDP- Cecchetti 0.75 0.4 0.50 0.2 "" "* -0.25 ... . 0.0 0 - Cecchetti Inflation 0.6 - ....... .. 0.00 -0.2- -0.25 -0.4 -0.50. .10 -1.25- -1.2 -1.50 -1.4 -1.75 0 5 10 15 20 0 C: DynamicResponse to a MonetaryPolicy Shock 5 15 20 D: DynamicResponse to a MonetaryPolicy Shock Real GDP- Shapiro-Watson - Shapiro-Watson Inflation 0.6 0.4 10 0.75 " 0.2 . ... - 0.50 0.25 ... -0.2 -e -0.25 -1.0 -1.25 -1.2 -1.50 -1.4 - -1.75 0 5 10 15 20 0 E: DynamicResponse to a MonetaryPolicy Shock 5 Real GDP- Gali 0.75 0-50-1.50 .. a ...1.25 -1.4 0 0 5 10 5 101 Figur 3. -0.75 ?.. aMonetary Bias-Adjusted -1.75F: Dynamic PolicyIpulse ShockResponseFunctions wons RepfidenceIntervals.Policy E: Dynamic Respne3.to Meia 20 - Gali Inflation 0.4 -1.0 15 F: DynamicResponse to a MonetaryPolicy Shock 0.6 -1.2 0.2 -.0.25 -0.6 10 15 20 0 5 0051 Bias-Ajste Imus 10 15 Shock 20 52 Repos bias-adjustedestimates of the sacrifice ratio and 90% confidence intervalsfor each horizon. The empirical density functions for SE (7 = 20) are plotted in Figure 4. There are several strikingresults that emerge from Table 2 and Figure 4. The first is that a 90% confidence interval for the sacrifice ratio includes 0 for all three models. That Fucioswth9%Cofdncneras is, we cannot with any reasonable degree of certainty rule out the possibility that SE(7)r)= at each horizon. Second, the Shapiro-Watsonand Gali models display a marked increase in the imprecision of the sacrifice-ratioestimates as the horizon lengthens. Last, the results are sensitive to the measure of monetary-policyshocks. In particular,expanding the Cecchettiand Rich: StructuralEstimatesof the U.S. SacrificeRatio 423 Table2. MedianBias-AdjustedEstimatesand ConfidenceIntervalsfor SacrificeRatio;SimulationExperimentBased on 10,000 Replications S,7(r) Medianbias-adjusted 90%confidence interval Medianbiasadjustedestimate Horizon (quarters) Cecchettimodel r= 4 7==8 1.3072 1.3203 1.5806 1.5482 1.3591 T=12 S=16 7 = 20 (0.0110,2.4686) (-0.3782, 2.6474) (-0.5048,3.8141) (-0.3341,3.9162) (-0.4200, 3.2170) Shapiro-Watsonmodel 7= 4 7= 8 7= 12 0.4131 2.0049 3.4605 3.9607 4.3495 T=16 r = 20 (-2.4175, 4.5281) (-11.5053, 18.6673) (-22.7007, 30.0720) (-27.6783,36.4351) (-30.2247,43.0066) Galimodel 7=4 1.0911 4.9281 8.2839 11.2572 14.0334 '"=8 S=12 7= 16 7 = 20 (-2.4804, 5.2475) (-13.4865,25.2386) (-34.1267, 48.7030) (-41.9125, 55.3507) (-43.4513, 67.9279) two-variable system to identify separate components of the aggregate demand disturbanceyields ranges for the sacrifice ratio that are highly variableand implausible.Takentogether, these findings speak directly to the unreliabilityof the point estimates and suggest that we have little understandingabout CecchettiModel 0.35 - 0.3 0.25 0.2 0.15 - 0.1 - 0.05 0 -60 -40 -20 0 20 40 60 Model Shapiro-Watson 0.1 0.09- 0.08 0.07 0.060.05- the quantitative impact of monetary policy on output and inflation. Looking at Figure 4, we see that the simulated distribution of the sacrifice ratio for the Cecchetti model is nonnormal, although it is reasonably symmetric. In addition, the bias-correctedbootstrappedestimates of the sacrificeratio are nearly identical with the original estimates, providing little evidence of bias. For the Shapiro-Watson and Gali models, however, the simulated distributionsof the sacrifice ratio reveal a markedlydifferentpicture.The distributionsare nonnormal, asymmetric,and extremely long tailed. Further,there is now increasedevidence of bias in the point estimates from both models. To explore the behavior of the sacrifice-ratioestimates in furtherdetail, we examined the simulated distributionof the estimatorsof the impact responses of outputand inflationto a monetary-policyshock. Because the featuresof the estimators for each model were broadly similar across horizons, we only display and discuss the resultsfor a horizonof r = 20 quarters. Figure 5 plots the correspondingempirical density functions. The graphs relate to the relevant quantities in the numerator and denominatorof the sacrifice ratio and therefore provide estimates of the cumulative response of the level of output and the response of the level of inflationto a monetary-policy shock. The impact response estimatorshave very nonormaldistributionsthatbecome distinctlybimodal as the numberof structural shocks increases. In addition,the simulateddistributions of (Iaii'12)do not restrictthe outputresponseto be negative and display greater variability as the complexity of the models increases. In the case of the response of inflationto a monetary-policyshock, the estimatedsampling distributionof Ca2 lies exclusively below 0 for the Cecchetti model, while the corresponding sampling distributions for the ShapiroWatsonand Gali models display a pronouncedrightwardshift and are centered near 0. This latter result would seem to account for the extremely wide confidence bands associated with the sacrifice-ratioestimatesfrom the Shapiro-Watsonand Gali models. Because the estimatedinflationresponse appears in the denominatorof the expressionfor the sacrificeratio, values of this magnitudeclose to 0 will lead, for a given output response, to larger (absolute) estimates of the sacrifice ratio. The Shapiro-Watsonand Gali models also seem capable of generatingpositive estimates of the sacrifice ratio in the presence of perverse output and inflationresponses. 0.04 0.03 0.02 4. 0.01O -60 -40 -20 -60 -40 -20 0 20 40 60 0 20 40 60 EXPLORINGTHE SOURCES OF THE IMPRECISION There are a numberof possible sources for the imprecision find in the estimates of the sacrifice ratio. In this section we GallModel we examine three. First, we look at the possibility that the 0.07 0.06 estimatedVAR's contain structuralbreaks.If they do, then the wide confidence bands we obtain could reflect specification error arising from the estimation of a time-invariantmodel. We examine this possibility with a set of structuralbreak tests based on the work of Andrews (1993) and Andrews and Ploberger(1994). Much to our surprise,we are able to conclude that all three of the models we analyze are stable over Estimates. Sacrifice-Ratio for Functions 4. EmpiricalDensity the sample period we study. Figure 0.05 0.04 0.03 0.02 0.01 424 Journal of Business & Economic Statistics, October 2001 Output:CecchettiModel CecchettiModel Inflation: 0.6 a 0.6 2.5 0.5 - 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 - 0.1 2.5 22 -2 1.5 0 0 -20 -16 -12 -8 -4 0 4 8 12 16 0 0.3 - 0.25 0.25 0 -5 20 2 0.5 0.5 Model Output:Shapiro-Watson 0.3 -1.5 -4 -3 -2 -1 1 0 2 3 4 5 Model Inflation: Shapiro-Watson 0.8 0.8 0.7.0.7 0.6 -.0.6 0.2 0.2 0.15 0.15 - 0.1 0.1 - 0.05 0.05 0 0 -20 -16 -12 -8 -4 0 4 8 12 16 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0. 0.1 0. 0.1 0 20 0 -5 -4 Output:GaliModel 0.14 0.12 - 0.12 0.1 - 0.1 0.08 -0.08 0.06 0.06 0.04 0.04 0.02 -0.02 0 -16 -12 -8 -4 0 4 8 12 -2 -1 0 1 2 3 4 5 GaliModel Inflation: 0.14 0 -20 -3 16 20 1.2 ]221.2 1 0.8 - 0.8 0.6 - 0.6 - 0.2 0.2 0 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Figure5. EmpiricalDensityFunctionsfor Responses of Outputand Inflation. Second, we ask whetherthe results reflect the fact that the higher-dimensionalmodels provide worse forecasts of output and inflation. If they do, then it may not be the addition of more structuralshocks that leads to less precise estimates of the sacrificeratio. Instead,it may be that the Shapiro-Watson and Gali models are just poorer characterizationsof the data. To address this issue, we compare the out-of-sample forecasting performanceof each of these models for output and Cecchettiand Rich: StructuralEstimatesof the U.S. SacrificeRatio inflation. The findings indicate that the more complex models provide better,not worse, forecasts, so this is not the explanation for our results. Finally, we look at the issue of instrumentquality. Estimation of structural VAR models can be viewed in an instrumental-variablessetting. The use of weak instruments could lead to imprecision in the calculation of the elements that go into the computationof the sacrifice ratio. We provide some evidence here that this is a likely source of at least part of the imprecisionin the sacrifice-ratioestimates. Before we proceed to examine each of these three possible sources of imprecision in detail, we would like to comment on one avenue we do not pursue. Specifically, we do not attempt to provide a detailed investigation into the relative contributionsof A0, the matrix of the contemporaneous effects of the structuraldisturbanceson the endogenous variables, and C(L), the reduced-formVMA representation,to the variability of the sacrifice-ratioestimates. The most natural way to attempt to parse the relative impact of these on the imprecision of the sacrifice ratio would be to hold one of these two quantitiesfixed and then use simulateddata to estimate the variabilityinduced by the other. Unfortunately,this approachis not feasible because the value of AOdepends on C(L) throughthe identifying restrictions,so it cannot be held constant while new data are generatedto reestimateC(L). 425 or implicit monetary-policyreaction function and, if the policy regime were to change, then one would expect the coefficients in these equations to change as well. In other words, we have every reason to believe that the models we estimate may contain structuralbreaks. To investigate this issue, we use tests proposed by Andrews (1993) and Andrews and Ploberger(1994) to study the possibility of a break in the specification of the models. Table 3 reports the results of the three commonly used tests for structuralchange, all of which are Lagrange multiplier (LM) tests and allow for an endogenous determination of the break date. We report the value of the LM test statistics, togetherwith their p values, for each equationin each of the three models. The results are quite striking. It is only the Andrews-Plobergeraverage LM test that provides evidence of a structuralbreak for one of the equations in the StockWatson model and two of the equations in the Gali model. Even for these cases, however,the other two tests fail to reject structuralstability. From this we conclude that instability of the VAR's is unlikely to be the source of our results. 4.2 ForecastingAccuracy If the two-variablesystem is just a better model than the three- or four-variablesystems, then the explanation for our results would be straightforward.The obvious dimension in 4.1 Structural Breaks which to comparemodels of this type is to examine their foreThe results in Section 3 are estimatedover a nearly 40-year casting ability. We do this by calculatingthe accuracyof outperiod beginning in 1959. Many things may have changed in of-sample forecasts for outputand inflationusing the relevant the United States over this time, not the least of which is Fed- reduced-formequations (or their equivalents)for each of the eral Reserve policy. Each of our models includes an explicit three models. Table3. StructuralBreakTests:1959:Q1-1997:Q4 Modell equation Andrews/Quandt supremumLMtest AndrewslPloberger exponentialLMtest AndrewslPloberger average LMtest Cecchettimodel Outputequation Inflationequation Outputequation Inflationequation Ex post real interestrate equation 23.180 8.981 (0.9199) (0.8295) 27.999 10.626 (0.6211) (0.5555) Stock-Watsonmodel 16.612 (0.5084) 18.851 (0.2631) 15.455 (0.9936) 22.166 (0.6616) 24.579 (0.4600) 6.276 (0.8999) 8.841 (0.4274) 10.233 (0.2225) 11.876 (0.6187) 15.521 (0.1790) 18.722 (0.0372) 12.206 (0.3081) 12.135 (0.3176) 12.123 (0.3192) 13.860 (0.1404) 20.649 (0.1337) 19.530 (0.2067) 22.524 (0.0586) 23.910 (0.0298) Galimodel Outputequation Nominalinterestrate equation Ex post real interestrate equation Real money balances equation 30.524 (0.4320) 29.371 (0.5166) 27.114 (0.6877) 34.415 (0.2041) NOTE: Structural stabilitytests correspondto the jointtest forconstancyof all regressionparametersin a particular equation.Values of the test statisticare reportedin columns2-4, withasymptoticp values reportedbelow in parentheses. Journalof Business & EconomicStatistics,October2001 426 Table4. RollingOne-Quarter-Ahead Out-of-SampleForecasts; 1959: Q1-1975:Q1 (1997:Q4)Root Mean SquaredError Model Cecchetti Shapiro-Watson Gali Outputgrowth (Levelof) Inflation 0.7348 0.6956 0.7072 1.3943 1.3666 1.2313 Table 4 reportsthe results of an exercise in which we estimate each model, generateone-quarter-ahead forecasts,add an additionalquarter'sworthof data,and repeatthe exercise. This expanding-sampleestimation begins with data from 1959:Q1 through 1975:Q1 and rolls forward through 1997:Q4. We reportthe root mean squarederrorof these one-step-aheadoutof-sample forecasts. The conclusion is clear. As we increase the numberof variablesin the system, the models improve at forecasting output growth. The accuracyof the inflationforecasts is roughly equivalent,with the two-variablesystem performing worst. The last result is not a surprisebecause adding information rarely improves inflation forecasts. But overall, we do not find evidence indicatingthat the forecastingperformance deterioratesas the complexity of the model increases. 4.3 Instrument Quality Pagan and Robertson(1998) recently showed that structural VAR models can be cast in a generalizedmethod of moments frameworkand that the restrictionsused to identify the structural shocks generate instrumentsfor the estimation of A0. In addition, they found that structuralVAR estimates of the liquidity effect are extremely imprecise due in large part to the poor quality of instrumentsused in estimation.The results of Pagan and Robertson suggest that instrumentquality may be a relevantconsiderationfor our results. Following the approachof Pagan and Robertson,we adopt the testing procedureoutlined by Shea (1997) to evaluate the quality of instruments used in the estimation of the three structuralVAR models. An advantageof this methodology is that it can be applied to equations with multiple endogenous explanatoryvariables.The testingprocedureyields a partialR2 measure indicatingthe extent to which the instrumentset has componentsimportantto an endogenousvariablethatare independent of those importantto the other endogenous variables. Table 5 reports the partial R2 measure for the instruments used in estimating the structuralequations. Although we are able to examine the quality of instrumentsfor each equation, the results for the output equation and inflation equation in the Cecchetti and Shapiro-Watsonmodels are of particular interest. In the case of the Gali model, we will focus our attentionon the outputequationas well as on the nominaland real interest-rateequations. Across all three models, the long-run restrictions yield instrumentsof relatively low quality for estimating the coefficients of the output (aggregate supply) equation. This is especially true in the case of the Gali model, in which the instruments seem to be particularlypoor. For the inflation equation, or the nominal and real interest-rateequations in the Gali model, there is evidence of some improvementin the quality of the instruments.However, some caution may be needed in interpretingthe high observedvalues of the partial R2 measure.As Pagan and Robertsonnoted, the short-run restrictions used for identification in these latter equations generate instrumentsthat are (structuraland reduced-form) residualsfrom previouslyestimatedequations.Although these variablesare assumedto be orthogonalto the structuralerrors, the high partial R2 measure indicates that they may not be valid instruments and suggests the presence of additional bias in the instrumental-variableestimates. Taken together, the evidence of weak or invalid instrumentsspeaks directly to the tenuous natureof the identifying assumptionsand the limited ability of current empirical methodologies to allow researchersto draw reliable inferences about structuraleconomic relationships. Table5. PartialR2Measurefor Instrumentsin the StructuralVARModels Cecchettimodel Endogenous variables Outputequation Inflationequation Ayt 0.858085 Art 0.121203 Shapiro-Watsonmodel Endogenous variables Outputequation Inflationequation Real interestrate equation Ayt - At 0.932317 0.239066 - 0.772303 0.477639 (it - t) 0.071403 0.096139 Galimodel Endogenous variables Outputequation Nominalinterestrate equation Real interestrate equation Moneygrowth equation AyAi - 0.091705 (i (it) 0.048001 (Amt - ITt) 0.130205 0.058115 0.832205 0.627003 0.215293 0.266718 0.209473 0.206824 0.184180 Cecchettiand Rich: StructuralEstimatesof the U.S. SacrificeRatio 5. CONCLUSION This study examines the output cost of disinflation policy for the United States over the postwar period. Across various models, the estimates of the sacrificeratio imply that a permanent one-percentage-pointreductionin inflation entails a loss of approximately1-10% of a year's real GDP. The confidence intervals around the point estimates, however, indicate that none of the point estimatesdiffer from 0 at conventionallevels of statisticalsignificance.Further,the high degree of imprecision associatedwith the estimates suggests that our knowledge about the actual impact of monetarypolicy on the behaviorof the economy is quite limited. The evidence from this study supports the view that the identifying restrictions used in estimation are tenuous and generate weak or invalid instruments.Alternative identification schemes are unlikely to change this outcome. Unlike standard situationsthat allow for the applicationof instrumentalvariables procedures, the instrument set available for the estimation of structuralVAR models is very restricted.This lack of valid instrumentsimposes severe restrictions on the nature of the structuralshocks that can be identified. Thus, although a better understandingof the true costs of disinflation would be of particularinterest and importanceto policy makers, we are skeptical that current data and econometric techniquescan provide a meaningful set of estimates. ACKNOWLEDGMENTS We have benefited from the suggestions of attendees of the System Committee on Macroeconomics Conference in Chicago and seminar participants at the University of Arkansas,the Universityof Connecticut,and the Universityof Californiaat Davis. We thank Paul Bennett, Jean Boivin, Tim Cogley, Jeff Fuhrer,Jordi Gali, EdwardGreen, Jim Hamilton, Jim Kahn, Ken Kuttner, Rick Mishkin, Adrian Pagan, Simon Potter,the associate editor, and an anonymousreferee for helpful comments on an earlier version of this article. We also thankJordiGali and Ken Kuttnerfor their assistancewith programs used in performing the computations. The views expressed in this article are ours and do not necessarilyreflect the position of the Federal Reserve Bank of New York or the Federal Reserve System. We are solely responsible for any remainingerrors. APPENDIX:DATA This appendix discusses the data used in the estimation. All data are quarterly.The estimates are conducted over the sample period 1959:Ql-1997:Q4. Output. The outputseries GDPH is measuredas real GDP in chain-weighted 1992 dollars. Prices. The price data are a quarterly average of the monthly consumerprice series PCU for all urbanconsumers. InterestRates. 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