Data Uncertainty and the Role of Money as an Information Variable for Monetary Policy

Data Uncertainty and the Role of Money ∗ ,† as an Information Variable for Monetary Policy Gu¨nter Coenen European Central Bank Andrew Levin Board of Governors of the Federal Reserve System Volker Wieland Johann-Wolfgang-Goethe Universita¨t Frankfurt and European Central Bank JEL Classification System: E31, E52, E58, E61 Keywords: euro area, Kalman filter, macroeconomic modelling, measurement error, monetary policy rules, rational expectations ∗ Correspondence: Coenen: Directorate General Research, European Central Bank, Kaiserstrasse 29, D-60311 Frankfurt am Main, Germany, tel.: +49 69 1344-7887, e-mail: gunter.coenen@ecb.int; Wieland: Professur fu¨r Geldtheorie und -politik, Johann-Wolfgang-Goethe Universit¨at, Mertonstrasse 17, D-60325 Frankfurt am Main, Germany, tel.: +49 69 798-25288, e-mail: wieland@wiwi.uni-frankfurt.de, homepage: http://www.volkerwieland.com. †Theviewsexpressedinthispaperaresolelytheresponsibilityoftheauthorsandshouldnotbeinterpreted as reflecting the views of the European Central Bank or the Board of Governors of the Federal Reserve System or any other person associated with the European Central Bank or the Federal Reserve System. Wieland served as a consultant at the European Central Bank while preparing this paper. We are grateful for helpful comments from Patrick Minford, Gabriel Perez-Quiros, Raf Wouters, and participants in the Konstanz Seminar on Monetary Theory and Policy and in a seminar at the European Central Bank. We also appreciate the excellent research assistance of Andres Manzanares from the European Central Bank. Of course, the authors are responsible for any remaining errors.

Contents Abstract 4 Non-technical summary 5 1 Introduction 7 2 Data Uncertainty in the Euro Area 10 3 A Rational-Expectations Model with Data Uncertainty 15 3.1 The Behavioral Equations 16 3.2 The Revision Process 19 4 Evaluating the Role of Indicator Variables 21 4.1 The Optimal Filtering Problem 21 4.2 Measures of Information Content 23 5 Illustrating the Information Role of Money 26 5.1 The Economy without Money 27 5.2 The Information Role of Money 29 6 The Quantitative Significance of Money as an Indicator Variable 33 6.1 Results for the Baseline Estimated Model 33 6.1.1 The Optimal Indicator Weights 33 6.1.2 Measures ofInformation Content 34 6.2 The Role of Monetary Analysis 35 6.3 Further Sensitivity Analysis 38 7 Conclusion 39 References 42 Appendix A Kalman filtering à la Svensson and Woodford 45 A.1 The state-space representation 45 A.2 The Kalman filter 46 Appendix B Measuring the information content of indicator variables 47 B.1 Within-period estimation 47 B.2 Multi-period predictions 49 Appendix C The representation of the revision process 53 Appendix D Results of further sensitivity analysis 54 European Central Bank Working Paper Series 59

Abstract This paper shows that money can play an important role as an information variable when initial output data are measured with error and subject to revision. Using an estimated modeloftheeuroareawefindthatcurrentoutputestimatesmaybesubstantiallyimproved by including money growth in the information set. The gain in precision, however, depends on the magnitude of the output measurement error relative to the money demand shock. We find noticable but small improvements in output estimates, if the uncertainty due to money demand shocks corresponds to the estimated variance obtained from the money demand equation. Money plays a quantitatively more important role with regard to output estimationifweallowforacontributionofmonetaryanalysisinreducinguncertaintydueto money demand shocks. In this case, money also helps to reduce uncertainty about output forecasts. JEL classification system: E31, E52, E58, E61 Keywords: euro area, Kalman filter, macroeconomic modelling, measurement error, monetary policy rules, rational expectations 4

Non-technical summary In this study, we perform a quantitative assessment of the role of money as an indicator variable for monetary policy when aggregate data are initially measured with error and subject to revision. We start by analyzing the sequence of revisions to euro area-wide data andfindthatmeasuresofrealoutputhavebeensubjecttosubstantialrevisionsoveraperiod of up to nine months, whereas measures of prices and money have generally been subject to relatively minor revisions that occur within a short period of the initial data release. Given this pattern of euro area data revisions, monetary aggregates have a potentially significant role in providing information about the current level of output. Toanalyzethemacroeconomicimplicationsofuncertaintywithregardtocurrentoutput, we then utilize the euro area model developed by Coenen and Wieland (2000), augmented by the estimated money demand equation of Coenen and Vega (1999), together with a calibrated specification for the output revision process. In particular, the model incorporates rational expectations and exhibits nominal inertia due to overlapping wage contracts. Furthermore, theshort-termnominalinterestrateisassumedtobetheinstrumentofmonetary policy. The quantity of money is determined recursively by money demand, as a function of the chosen nominal interest rate and the output and price level consistent with this interest rate. Since the model does not assign a causal role to the money stock in influencing output or inflation (other than through nominal interest rates), this approach may be viewed as a means of providing a reasonable lower bound on the information content of money. Moneyhasapotentiallyusefulroleasanindicatorvariableinourmodel,becausemoney demand depends on the true level of output whereas the monetary policymaker and private agents only receive a noisy measure of output. We use the Kalman filter to determine the optimal weight on money (as well as the other relevant information variables in the model) in estimating the true level of output. We then proceed to compute the reduction in output uncertainty that is achieved by including money in the information set. 5

We find that money may play an important role as an information variable and may result in major improvements in current output estimates. However, this depends on the magnitude of the output measurement errors relative to the unobserved component of the money demand shocks. If the policymaker observes the money stock but has no contemporaneous information about money demand shocks, then monetary aggregates provide relatively little information about aggregate demand. In contrast, if the policymaker conducts monetary analysis that provides contemporaneous information about money demand shocks, then the money stock provides substantial information about current output and also improves the accuracy of short-term output forecasts. 6

1 Introduction Many macroeconomic time series are subject to substantial revisions, and hence such data only provide imperfect information about the true state of the economy at a given point in time. In light of these data limitations monetary policymakers and researchers alike have long been interested in identifying indicator variables that provide precise and timely information. Atleastsincetheearly1970sresearchontheinformationcontentofalternative indicators has highlighted the potential usefulness of monetary aggregates. Some examples of this line of research are Kareken et al. (1973), Friedman (1975, 1990), Tinsley et al. (1980) and Angeloni et al. (1994). These evaluations have typically been conducted in reduced-form models and models with adaptive expectations. More recently, research on Taylor-style interest rate rules has re-emphasized the importance of “real-time” data uncertainty for the design of monetary policy albeit without considering money’s potential role as an information variable. In particular, a number of studies with U.S. data have found that uncertainty arising from revisions of output gap and inflation measurements may lead to a significant deterioration in the performance of 1 such interest rate rules. This problem may be even more important in the euro area, for which aggregate time series have only been developed fairly recently and have been subject to ongoing refinement. In this study, we perform a quantitative assessment of the role of money as an indicator variable for monetary policy in the euro area. Thus, we investigate the same idea as the earlier literature on the information content of money in a forward-looking model of the economy. However, as the more recent literature on interest rate rules and real-time data uncertainty we aim to obtain the best possible estimates of those variables entering the policy rule and we model the process of real-time measurements empirically to match data revisions. In analyzing the sequence of revisions to euro area-wide data, we find that 1SeeforexampletheevaluationsofinterestraterulesunderdatauncertaintybyOrphanides(1998),Orphanidesetal.(2000)andRudebusch(2000). Foralarge-scaleanalysisofthedifferencesbetweenalternative vintages of U.S. macroeconomic data the reader is referred to Croushore and Stark (1999). 7

measures of real output have been subject to substantial revisions over a period of up to nine months, whereas measures of prices and money have generally been subject to relatively minor revisions that occur within a short period of the initial data release. Given this pattern of euro area data revisions, monetary aggregates have a potentially significant role in providing information about the current level of aggregate demand. To analyze the macroeconomic implications of data uncertainty, we utilize the euro area model developed by Coenen and Wieland (2000), augmented by the estimated money demand equation of Coenen and Vega (1999), together with a calibrated specification for the output revision process. In particular, the model incorporates rational expectations and exhibits nominal inertia due to overlapping wage contracts. Furthermore, the short-term nominal interest rate is assumed to be the instrument of monetary policy. The quantity of money is determined recursively by money demand, as a function of the chosen nominal interest rate and the output and price level consistent with this interest rate. Thus, implicitly we assume that any effect of a change in the nominal money stock on real output and inflation comes through the associated change in the nominal interest rate. In other words, direct effects of money on output and inflation are absent from the model. This 2 assumption is typical of the current generation of macroeconomic models and is consistent 3 with optimizing behavior if certain restrictions are satisfied. Since the model does not assign a causal role to the money stock in influencing output or inflation (other than through nominal interest rates), this approach may be viewed as a means of providing a reasonable lower bound on the information content of money. Money has a potentially useful role as an indicator variable in our model, because 2This includes most of the smaller-scale models currently used for research on monetary policy (see for example Rotemberg and Woodford (1997), Fuhrer (1997) or Orphanides and Wieland (1998)) as well as large-scale policy models such as the Federal Reserve Board’s FRB/US model (see Brayton and Tinsley (1996)),theECB’sArea-WideModel(seeFagan,HenryandMestre(2001)),orthemulti-countrymodelof Taylor(1993a). Analternativeapproach,whichallowsfordirecteffectsofmoneyoninflation,wouldbethe P∗ modelofHallmanetal. (1991)estimatedmorerecentlyforGermanybyT¨odterandReimers(1994)and for the euro area by Gerlach and Svensson (2000). 3These restrictions are discussed in more detail in Ireland (2001), McCallum (2000) and Leahy (2000). Theyinclude,forexample,theseparabilityoftheutilityfunctioninconsumption,moneyandleisureorthe absence of transaction costs of purchases. 8

money demand depends on the true level of output whereas the central bank and private agents only receive a noisy measure of output. We use the Kalman filter to determine the optimal weight on money (as well as the other relevant information variables in the model) in estimating the true state of the economy (cf. Pearlman et al. (1986), Svensson 4 and Woodford (2000)). We then proceed to compute the reduction in output uncertainty that is achieved by including money in the information set. We find that money may play an important role as an information variable and may result in major improvements in current output estimates. However, this depends on the magnitude of the output measurement errors relative to the unobserved component of the money demand shocks. If the policymaker observes the money stock but has no contemporaneous information about money demand shocks, then monetary aggregates provide relatively little information about aggregate demand. In contrast, if the central bank conducts monetary analysis that provides contemporaneous information about money demand shocks, then the money stock provides substantial information about current output and 5 also improves the accuracy of short-term output forecasts. Finally,itshouldbeemphasizedthatouranalysisfocusessolelyonuncertaintyregarding actual output, and does not address the problem of estimating potential output. While uncertainty about potential output has important consequences for the determination of monetary policy, we neglect this issue here because the money stock is related to actual 6 output and thus cannot serve as a direct source of information regarding potential output. The remainder of this paper is organized as follows. Section 2 characterizes the timing and magnitude of revisions to euro area data on aggregate output, prices, and money. 4Our paper is related to recent work by Dotsey and Hornstein (2000) on the usefuleness of money for discretionary policy. However, they focus on impulse responses in a calibrated model of the U.S. economy withsimplemeasurementerror,whilewederivemeasuresoftheinformationcontentofmoneyinanempirically estimated model of the euro area with a more general empirical specification of measurement error. Further, there are important differences in the information structure, which we discuss later on. 5Details about the ECB’s approach to monetary analysis may be found in the May 2001 issue of the ECB Monthly Bulletin, and in Masuch et al. (2001). For further discussion related to the United States, see Orphanides and Porter (2001). 6Fortheimplicationsofuncertaintyaboutoutputgapsandpotentialoutput(orunemploymentgapsand theNAIRU)formonetarypolicywereferthereadertoEhrmannandSmets(2000),Orphanides(2000)and Wieland (1998) among others. 9

Section3outlinesthebehavioralequationsofthemodel, andindicatesalternativerepresentations of the output revision process. Section 4 describes our methodology for determining the optimal filtering weights and for evaluating the information content of indicator variables. Section 5 illustrates the information role of money for the case of a highly stylized money demand equation, while Section 6 uses the complete model described above to evaluate the quantitative significance of money as an indicator variable. Section 7 summarizes our conclusions and suggests several directions for future research. Finally, the Appendix reportsfurtherdetailsofourmethodologyaswellasadditionalsensitivityanalysisregarding our results. 2 Data Uncertainty in the Euro Area Some macroeconomic data series, such as nominal interest rates, exchange rates, and raw materials prices, are readily available and not subject to revision. In contrast, indicators of aggregate quantities and prices are more difficult to construct, and are frequently subject to substantial revisions as additional information becomes available to the statistical 7 agency. For the euro area, aggregate data has only become available fairly recently (with the harmonization of statistical procedures across the individual member countries), and hence the record of initial releases and revisions is necessarily limited. Nevertheless, it is useful to characterize the properties of these revisions in order to shed some light on the degree of data uncertainty in the euro area. Thus, we proceed to analyze the timing and magnitude of revisions to euro area output, price, and money data, beginning with the advent of European Monetary Union in 1999. As measures of real output, we consider monthly data on industrial production (excluding construction) as well as quarterly data on real GDP. To measure aggregate prices, we consider monthly data on the Harmonized Index of Consumer Prices (HICP) and quarterly data on the GDP price deflator. Finally, we consider monthly data for M3; we focus on this 7Further revisions occur on a less frequent basis as the result of definitional changes, such as switching toadifferentbenchmarkyearforthenationalincomeaccounts. Suchrevisionsoftenshifttheentirelevelof a data series, but may have relatively minor implications for the determination of monetary policy. 10

Figure 1: Revisions to Industrial Production Index 109.5 109.0 108.5 108.0 Jan−99 Feb−99 May−99 Jul−99 Sep−99 Nov−00 Jan−99 Mar−00 Reporting Month Source: ECB Monthly Bulletin. measure of money because Coenen and Vega (1999) found that the demand function for M3 has been remarkably stable. In each case, we utilize real-time data series over the period October 1998 through December 2000, as published in consecutive issues of the European 8 Central Bank’s Monthly Bulletin over the period January 1999 through February 2001. The nature of the revision process is best understood with an example. Figure 1 shows monthlyrevisionsofindustrialproductionatthestartofmonetaryunion. Estimatesofeuro area industrial production in January and February 1999, for instance, were first published in the May 1999 issue of the ECB Monthly Bulletin. The estimates of the index reported in May were 108.6 and 108.2 for January and February industrial output, respectively. Over the following months the statistical authorities revised these estimates upwards. Revisions only ceased by the end of the year. The magnitude of the revisions over this period was 0.7 in both cases. Clearly, these revisions suggest a significant degree of data uncertainty, which persisted for some time. 8TheECB’smonthlybulletinisaconvenientsourceforobtainingconsistentreal-timedata. Furthermore, eachbulletinrepresentsareasonablyaccuratesummaryofthedataavailabletotheECBGoverningCouncil at its first meeting each month: the cut-off date for inclusion in the bulletin predates each meeting, and the bulletin itself is published a week later. However, in future work it would be interesting to analyze the timing of revisions as published by the statistical agency that actually compiles each data series. 11

Table 1: Monthly Euro Area Data Revisions (in Percent) Month after 1 2 3 4 5 6 7 8 9 initial publication Industrial Production(a) largest upward revision 0.93 0.81 0.54 0.54 0.46 0.27 0.27 0.27 0.18 largest downward revision -0.60 -0.46 -0.55 -0.27 -0.26 -0.27 -0.27 -0.27 -0.36 mean absolute revision 0.34 0.28 0.24 0.16 0.16 0.11 0.12 0.13 0.13 Consumer Prices(b) largest upward revision 0 0 0 0 0 0 0 0 0 largest downward revision -0.10 -0.10 0 0 0 0 0 0 0 mean absolute revision 0.03 0.02 0 0 0 0 0 0 0 M3(c) largest upward revision 0.37 0.20 0.14 0.13 0.14 0.06 0.02 0.10 0.03 largest downward revision -0.18 -0.13 -0.12 -0.03 -0.03 -0.10 -0.11 -0.04 -0.02 mean absolute revision 0.16 0.08 0.06 0.03 0.05 0.04 0.03 0.03 0.02 Source: ECB Monthly Bulletin, January 1999 to February 2001. Note: (a) Index of Industrial Production (excluding construction), seasonally adjusted. (b) Harmonized Index of Consumer Prices, Dec. 1998 = 100, not seasonally adjusted. (c) M3 Index, Dec. 1998 = 100, seasonallyadjusted;calculatedfrommonthlydifferencesinlevelsadjustedforreclassifications,otherrevaluations, exchange rate variations etc. Table 1 provides summary statistics regarding the revision process for monthly euro area data. The first column is associated with the first revision (one month after the initial publication), the next column reflects the second revision (i.e., the difference between the values published one month and two months following the first publication), and so on until the revision in the tenth month following initial publication. For each series, the first row indicates the largest upward revision at each interval (as a percent of the value published in the previous month), while the second row indicates the largest downward revision, and the third row indicates the mean absolute revision. 12

Evidently, the industrial production data are subject to substantial and frequent revisions over the first year after the initial publication. For example, the first monthly revision of this series has a mean absolute value of 0.34 percent, with a maximum upward revision of 0.93 percent and a maximum downward revision of 0.6 percent. While the magnitude of revisions gradually declines as time passes, revisions exceeding 0.1 percent are not unusual during each of the next few months after the initial publication. In contrast, the consumer price data are typically not revised at all; the only exceptions are apparently due to corrections of reporting errors. Clearly, the lack of revisions does not imply that these data provide an exact measure of aggregate inflation. However, measurement biases in the consumer price index have mainly been identified with longer-term factors (such as improving product quality, introduction of new goods and services, and changes in expenditure shares), and hence these biases may not be crucial in evaluating higher-frequency fluctuations in the inflation rate. In any case, as we will see below, the GDP price deflator (which is less susceptible to measurement bias than the HICP) also exhibits relatively small revisions. Finally, the magnitude of initial data revisions is substantially smaller for M3 than for industrial output: the maximum upward and downward revisions in the first month are less than half as large (in percentage terms). Furthermore, subsequent revisions in M3 are relativelysmallandinfrequent, sothatthemeanabsoluterevisionneverexceeds0.1percent from the second month onwards. Table 2 reports summary statistics regarding the revision process for real GDP and the GDP price deflator, which are available on a quarterly basis. These statistics indicate that 9 real GDP is subject to fairly large revisions. For example, in the first revision (one quarter after the initial publication), the maximum upward revision exceeds a full percentage point, and the mean absolute revision is about 0.8 percent of the previously published value. Even three quarters after the initial publication, the mean absolute revision of real GDP is about 0.5 percent. In contrast, revisions of the GDP deflator are much smaller: the mean 9Some of these revisions have occurred as individual member countries have moved to the ESA95 harmonization of national income accounts and are likely to become smaller as the implementation process is completed in most countries. 13

Table 2: Quarterly Euro Area Data Revisions (in Percent) Quarter after 1 2 3 4 5 initial publication (a) Real GDP largest upward revision 1.49 1.21 1.14 0.20 0.19 largest downward revision -0.91 -0.95 0 -0.02 -0.08 mean absolute revision 0.80 0.69 0.47 0.11 0.14 (b) GDP Price Deflator largest upward revision 0.28 0.38 0.09 0.09 0.09 largest downward revision -0.10 -0.09 0 -0.09 0 mean absolute revision 0.11 0.14 0.03 0.06 0.05 Source: ECB Monthly Bulletin, January 1999 to February 2001. Note:(a) Seasonally adjusted.(b) Seasonally adjusted. absolute revision is only about 0.1 percent in each of the first two quarters after the initial publication, and subsequent revisions are negligible in magnitude. Evidently, revisions to nominal GDP for the euro area are primarily due to revisions regarding real output rather than prices. Thus, the monthly and quarterly data yield remarkably similar conclusions regarding real-time data uncertainty in the euro area. Industrial production and real GDP are each subject to relatively large revisions during the first several quarters after the initial publication, indicating that data uncertainty regarding the current level of real output is a 10 non-trivial issue for the euro area. By comparison, both measures of aggregate prices (the HICP and the GDP price deflator) and the broad money stock (M3) are subject to rela- 10Revisions of the growth rate of real output tend to be smaller. As discussed in the ECB’s monthly bulletinofAugust2001(seepages26-28)theaveragesizeofrevisionsofquarter-on-quartergrowthsincethe firstquarterof1999was0.2percentagepoints. Throughoutthispaper,however,wecontinuetofocusonthe levelratherthanthegrowthrate,becauseinourview,thelevelofoutputrelativetotheeconomy’spotential is more relevant for determining the appropriate stance of monetary policy and its effect on inflation than the difference between actual and potential growth rates. 14

tively small revisions during the first quarter after the initial publication, and to negligible revisions in subsequent quarters. Theseresultsraisethepossibilitythatmoneycanserveasausefulindicatorinproviding real-time information about fluctuations in real output. An additional advantage of money as an indicator variable is that money data typically becomes available earlier than output 11 data. Inthefollowinganalysiswewillprimarilyfocusontheinformationgainfrommoney inthepresenceofmeasurementerror, butwewillreturntothegainsarisingfromtheearlier availability of money in the sensitivity analysis at the end of the paper. 3 A Rational-Expectations Model with Data Uncertainty Toquantifytheinformationcontentofmoney,weutilizetheeuroareamacroeconomicmodel of Coenen and Wieland (2000), augmented by the money demand equation estimated by Coenen and Vega (1999). Since these equations are specified at a quarterly frequency, it seemsreasonabletoassume(inlightofthe resultsofthe previoussection)thatobservations on aggregate output are subject to measurement error, while aggregate prices, money, and 12 nominal interest rates are observed without measurement error. It also seems reasonable to assume that the money demand of each individual household or firm depends on its own income and expenditures (which are known to that household or firm), while neither 13 private agents nor the central bank observe the true level of aggregate output. Under these assumptions, aggregate money demand will be related to the true level of aggregate income, and hence observations on the money stock can provide useful information about movements in aggregate output. 11This holds even for monthly data. For example, in June one learns about money growth and inflation in May but about industrial production in April. 12In the model considered here, measurement errors of the money stock would have the same effect as money demand shocks in reducing the information content of money as an indicator of aggregate output. Thus, one could always capture the effect of money measurement error by considering a slightly higher variance of the money demand shocks. 13Thus, the information structure differs from Dotsey and Hornstein (2000) who assume instead that private agents know the true level of aggregate output. To us it seems more reasonable to assume that private agents face similar uncertainty regarding aggregate data as the central bank. 15

3.1 The Behavioral Equations 14 The behavioral equations of the model are indicated in Table 3. As shown in equation (1), the aggregate price level p t is determined as a weighted average of overlapping nominal wage contracts signed over the past year. The estimated weighting scheme implies that a weight of 0.32 is placed on the current wage contract w t, while smaller weights are placed on earlier contracts (e.g., w t−3 receives a weight of about 0.18). The determination of nominal wage contracts is given in equation (2). As in Taylor (1980, 1993a), eachwagecontractisdeterminedbyexpectationsaboutaggregatepricesand perceived output gaps over the duration of the contract. 15 The expectations operator Et[.] indicates the optimal projection of each variable, conditional on all information available at period t. As noted above, this information set includes the true values of aggregate wages, prices, and interest rates, and noisy observations regarding aggregate output. Since our analysis is focused on the implications of data uncertainty regarding actual output q t, we assume for simplicity that potential output q∗ is exogenously determined and known by t 16 all private agents and by the central bank. Finally, under these assumptions, it should be noted that the aggregate supply disturbance uw is known to all agents (including the t centralbank);thisimplicationfollowsfromourassumptionsthatallagentsknowthecurrent contract wage and utilize identical information in forming expectations about the variables on the right-hand side of equation (2). As shown in equation (3), the current output gap depends on the true output gap in each of the previous two quarters and on the ex ante long-term real interest rate, rl (which t 14Wages,prices,output,andmoneyareexpressedinlogarithmicform,andinterestratesareexpressedat annualized rates. 15Coenen and Wieland (2000) also considered relative real wage contract specifications of the type originally proposed by Buiter and Jewitt (1981) and later studied by Fuhrer and Moore (1995). We have performed sensitivity analysis and found that the results reported here concerning the information role of moneyarenotsubstantiallyaffected byusingarelativerealwagecontractspecificationinsteadofequation (2). 16Infact,equations(2)and(3)wereestimatedbyCoenenandWieland(2000)using“final”dataoneuro area real GDP (that is, data available at the end of 1999), and constructing the output gap by removing a log-linear time trend. In future work, it would be interesting to reestimate these equations using real-time output gap data. However, as the authors note, constructing a real-time output gap data series for the period preceding EMU would be a “courageous undertaking”. 16

Table 3: A Rational-Expectations Model with Data Uncertainty Price Level p t = f 0 w t+f 1 w t−1+f 2 w t−2+f 3 w t−3, (1) where f i = 0.25+(1.5−i)0.0456 (cid:1) (cid:3) (cid:2) (cid:2) Contract Wage w t = Et 3 i=0 f i p t+i+0.0115 3 i=0 f i y t+i +uw t , (2) where y t = q t −q t ∗ and uw t ∼ i.i.d.(0,0.0038 2 ) Aggregate Demand y t = 1.2029y t−1 −0.2225y t−2 −0.0942r t l +ud t , (3) where ud ∼ i.i.d.(0,0.0058 2 ) t Money Demand ∆(m−p)t = 0.075∆ 2q t+0.097(∆is t +∆is t−1 ) (4) −0.359∆il t−1 −1.052(∆π t+∆π t−1) −0.136[(m−p)−1.140q +0.820(il −is )+5.848π]t−2 +um , t where π t = p t −p t−1 and um t ∼ i.i.d.(0,0.0023 2 ) Monetary Policy Rule is t = π˜t+0.5(π˜t −π∗ )+0.5Et[y t], (5) where π˜t = p t −p t−4 (cid:1) (cid:3) (cid:2) Term Structure il t = Et 1 8 7 j=0 is t+j (6) (cid:1) (cid:3) Real Interest Rate r t l = il t −Et 1 2 (p t+8 −p t) (7) Note: p: aggregate price level; w: nominal contract wage; uw: contract wage shock; y: output gap; q: output; q∗: potentialoutput; rl: long-termrealinterestrate; ud: aggregatedemandshock; is short-term nominal interest rate; π∗: inflation target; il: long-term nominal interest rate; m nominal money balances; um: money demand shock. 17

is defined by equation (7)). This specification seems reasonable under the assumption that each household or firm makes spending decisions based on its own directly observed 17 incomepath,andhenceaggregatespendingdependsonthetruepathofaggregateincome. Nevertheless, since the true level of aggregate output is not directly observed, the aggregate demand shock ud is not in the information set of private agents or the central bank. t As indicated by equation (4), real money balances (m−p)t are determined by an error correctionprocessinvolvingaggregateoutputq t,theshort-termnominalinterestrateis t ,the long-term nominal interest rate il (defined by equation (6)), and the one-quarter inflation t rate π t = p t − p t−1. 18 As noted above, this money demand specification indicates that the money stock responds to the true level of aggregate output q t, and hence money has a potentially useful role as an indicator variable. Of course, this role depends on the stability of the money demand. While estimates of money demand with U.S. data have tended to be 19 unstable, results with German and euro area data have typically been more encouraging. Two additional characteristics of equation (4) are crucial in determining the specific information content of money. First, the short-run link between money and output is relatively weak, with an instantaneous income elasticity of only 0.075; evidently, the money stock would provide somewhat more information about current output if the contemporaneous relationship between these two variables were significantly larger. Second, the money demand shock um has an estimated standard error of 0.23 percent, reflecting the extent to t which money balances move in response to factors other than aggregate output, inflation, andnominalinterestrates. Intheabsenceofanyadditionalinformation, privateagentsand the central bank will have substantial difficulty in determining whether a given movement in the money stock reflects a shift in aggregate output or a shift in money demand. Thus, 17Totheextentthatindividualspendingdecisionsactuallyreflectagents’perceptionsabouttheaggregate economy,thentheoutputgapequationwouldneedtobeaugmentedbytermssuchasEt[y t−1]andEt[y t−2]. Wehaveperformedsomepreliminaryanalysisofsuchspecifications,butleavefurtherinvestigationtofuture research. Inthiscontext,amodelwithmoreexplicitmicroeconomicfoundationsthatdistinguishesbetween macro- and micro-level uncertainty will be helpful. 18Because the inflation rate π t is not annualized, the corresponding coefficients in equation (4) appear unusually large. 19For a recent study regarding U.S. money demand and money’s usefulness for U.S. monetary policy see Dotsey et al. (2000). 18

as we will see below, monetary analysis may enable the central bank to identify some of the special factors and shocks that affect money demand, and thereby enhance the information content of money. Finally, as indicated by equation (5), we assume that the monetary authorities follow the simple interest rate rule proposed by Taylor (1993b), where π˜t = p t −p t−4 indicates the annual average inflation rate and π∗ the inflation target. 20 According to this equation, the ex post short-term real interest rate moves in response to deviations of inflation from target and to perceived movements in the output gap (i.e., Et[y t]). Of course, the central bank wouldprefertoavoidmakingpolicyadjustmentsinresponsetopersistentmismeasurements ofaggregateoutput, andhenceusesallavailableinformationinestimatingthecurrentvalue of output. 3.2 The Revision Process In the state-space literature, a typical assumption is that each data point of a given time series is observed just once (possibly subject to some measurement error). In contrast, here we wish to represent a sequence of revisions to the real output data that gradually refines 21 the quality of each individual data point. A general representation of the revision process can be expressed as follows: q t (t+j) = q t + v t (t+j), (8) where q(t) is the initial observation of output at time t; q(t+j) is the jth revision of this t t observation (often referred to as the time t+j “vintage” of the data); and v(t+j) represents t the deviation from the true level of output, q t. By abstracting from deviations between 20Wemakethisassumption,becauseTaylor’sruleroughlycapturesthesystematiccomponentofmonetary policy in a number of European countries in recent years (see for example Clarida, Gal´ı and Gertler (1998) and Gerlach and Schnabel (2000)). An alternative approach, would be to assume that the central bank implements an optimal monetary policy rule in our model. 21Sinceouranalysisisfocusedonthebehaviorofprivateagentsandthecentralbank,wedonotexplicitly model how the statistical agency determines these revisions, using new information on disaggregated variables,etc.. Sargent(1989)followsadifferentapproach,andanalysesamodelinwhichthestatisticalagency uses optimal filtering to revise its data on aggregate economic variables, and hence private agents and the central bank can utilize the statistical agency’s data without any further refinement. 19

the “final” revised data and true aggregate output, and ignoring occasional redefinitional changes in the entire time series, we may assume that the sequence of revisions for each datapoint eventually converges to the true level of output; that is, v(t+j) → 0 as j → ∞. t In the simplest case, the entire revision process occurs within a single quarter (that is, v(t+j) = 0 for all j > 0): t q t (t) = q t + v t (9) q t (t+1) = q t , where v t is serially uncorrelated with mean zero and standard deviation σ v, and is uncorrelated with the structural disturbances ud , uw , and um . Under this assumption about t t t the revision process, agents learn the true value of output one period after the initial data release. As we have seen in Section 2, however, the data on real output are subject to a sequence of substantial revisions for several quarters. Therefore, we also consider the following representation of the revision process: q t (t) = q t + v t 3 + v t 2 + v t 1 q t (t+1) = q t + v t 3 + v t 2 (10) q t (t+2) = q t + v t 3 q t (t+3) = q t , where v t = [v t 1 v t 2 v t 3 ] (cid:3) is a vector of serially uncorrelated measurement errors with mean zero and positive semi-definite covariance matrix Σvv. According to this representation of the revision process, the period t data vintage includes error-prone observations on q t, q t−1, and q t−2, as well as the true value of q t−3. 22 We calibrate the covariance matrix of v t using the data on revisions from Section 2. 23 The estimated standard deviations are 0.97 percent, 0.77 percent, and 0.47 percent for v1 , t v2 , and v3 , respectively. The sample correlation between v1 and v2 is negligible, while the t t t t sample correlations with v3 are -0.638 for v1 and -0.636 for v2 . Of course, given the short t t t 22Further details regarding this representation of the revision process are provided in Appendix C. 23In constructing the sample covariance matrix, we only used data for which revisions were available for at least three consecutive quarters. 20

history of data revisions, the sample covariance matrix is not estimated very accurately, and hence in the subsequent analysis we will also consider the case in which the elements of v t are mutually uncorrelated. 4 Evaluating the Role of Indicator Variables 4.1 The Optimal Filtering Problem We obtain optimal estimates of output by applying the Kalman filter to our linear rational expectations model of the euro area. Given our assumption that private agents and the central bank have the same information concerning aggregate variables, we can follow the 24 approach of Svensson and Woodford (2000), henceforth referred to as SW2000. In particular, the model can be expressed in the following form: (cid:4) (cid:5) (cid:4) (cid:5) (cid:4) (cid:5) (cid:4) (cid:5) X X X u t+1 = A1 t +A2 t|t + t+1 , (11) E˜x t+1|t x t x t|t 0 where X t is a vector of predetermined variables, x t is a vector of non- predetermined variables and u t is a vector of serially uncorrelated shocks with mean zero and positive semidefinite covariance matrix Σuu. The coefficient matrices A1 , A2 and E˜ are matrices of appropriate dimension. For some or many of the variables, policymakers and market participants can only observe noisy measurements. The vector of observables Z t is then given by (cid:4) (cid:5) (cid:4) (cid:5) X X Z t = D1 x t +D2 x t|t +v t , (12) t t|t wherev t isavectorofseriallyuncorrelatedmeasurementerrorswithmeanzeroandpositive semi-definite covariance matrix Σvv. The measurement errors v t are assumed to be uncorrelated with the shocks u t at all leads and lags, i.e. E[u t v τ (cid:3) ] = 0 for all t and τ. The matrices D1 and D2 are selector matrices of appropriate dimension. Here we use χ τ|t = E[χ τ |I t] to denote the rational expectation (that is, the optimal projection) of any variable χ in period 24A more detailed discussion of the Kalman filter and the weights given to indicator variables such as money is provided in Appendix A. 21

τ given information in period t. The information set in period t corresponds to (cid:6) (cid:7) I t = Z τ ,τ ≤ t;A1,A2,D1,D2,E˜,Σuu ,Σvv . SW2000 show that the non-predetermined variables fulfill the relationship x t = G1X t+G2X t|t (13) and that the system of equations (11), (12) can be cast into state-space form without non-predetermined variables, X t+1 = HX t+JX t|t+u t+1 (14) Z t = LX t+M X t|t+v t , (15) where the matrices G1 , G2 , H, J, L and M are derived in SW2000. This transformation of course simplifies the remaining problem of forming the estimate X t|t considerably. 25 AccountingforthecontemporaneouseffectoftheestimateX t|t onZ t,SW2000showthat the optimal estimate of X t can be obtained by means of a Kalman filter updating equation. Thisupdatingequationisexpressedintermsoftheinnovationsinthetransformedvariables Z¯ t = Z t −M X t|t: X t|t = X t|t−1+K(Z¯ t −Z¯ t|t−1) (16) (cid:1) (cid:3) = X t|t−1+K L(X t −X t|t−1)+v t . (17) The steady-state Kalman gain matrix K is given by K = PL(cid:3) (LPL(cid:3) +Σvv) −1, (18) where the matrix P is the steady-state covariance matrix of the innovations X t −X t|t−1 given information in period t−1 and satisfies the relation (cid:1) (cid:3) P = H P −PL(cid:3) (LPL(cid:3) +Σvv) −1LP H(cid:3) +Σuu . (19) 25Having eliminated the non- predetermined variables x t, the estimation of the predetermined variables X t still requires solving a simultaneity problem. Simultaneity arises because the observable variables Z t depend on the estimate of the predetermined variables X t|t, which in turn depend on the observables used in the estimation. 22

We are particularly interested in the weights on the observed indicator vector Z t under optimal filtering. While the Kalman filter estimate X t|t is obtained in terms of the weighted innovations in the transformed variables Z¯ t, we can recover the optimal weights on the observationsofZ t bysubstitutingZ¯ t = Z t −MX t|t andZ¯ t|t−1 = Z t|t−1 −MX t|t−1 = LX t|t−1 into (16), X t|t = (I +KM) −1 (I −KL)X t|t−1+(I +KM) −1KZ t . Here we can see that the contemporaneous effect of the estimate X t|t on Z t merely shows up in the premultiplication of the matrix (I + KM) −1 . When comparing the weights assigned to different information variables in the subsequent analysis we will refer to the elements of this modified Kalman gain matrix. 4.2 Measures of Information Content We evaluate the information content of indicator variables according to the extent that they will reduce the uncertainty surrounding the estimation of structural shocks and/or predictionofkeyendogenousvariables. Onemeasureofwithin-periodestimationuncertainty is the covariance matrix of the projection errors of the vector X t, given the information set I t available at period t (that is, information obtained from current and lagged values of the observed vector Z t). As shown in Appendix B.1, this covariance matrix can be expressed as follows: Cov[X t −X t|t |I t] = P −PL(cid:3) (LPL(cid:3) +Σvv) −1LP. (20) For example, one element of X t is the unobserved aggregate demand shock, ud t , and the root mean-squared error (RMSE) of estimating this shock is given by the square root of the corresponding diagonal element of Cov[X t −X t|t |I t]. In the subsequent analysis the RMSEservesasourbaselinemeasureoftheestimationuncertaintysurroundingtheoptimal estimate X t|t produced by the application of the Kalman filter. To evaluate prediction 23

uncertainty, we will also present results regarding RMSE of multi-period-ahead predictions, for which the derivations are given in Appendix B.2. In addition to the RMSE, it is useful to consider measuring estimation uncertainty using the concept ofentropy (or “expecteduncertainty”) taken from the information theory literature. In doing so we follow Tinsley et al. (1980) who employ entropy as a formal measure of the information content of indicator variables. To explain the basic concept and its relationship with the coefficient of determination in linear regression models, R2 , 26 we restate the relevant general results from Tinsley et al.. Consider two vectors χ and ξ with joint density f(χ,ξ). The joint entropy of χ and ξ is given by H(χ,ξ) = −E[ln(f(χ,ξ))]. The entropy or ’expected uncertainty’ of χ corresponds to H(χ) = −E[ln(f(χ))], where f(χ) is the marginal density of χ, and the entropy of χ given ξ corresponds to H(χ|ξ) = −E[ln(f(χ|ξ))] with f(χ|ξ) = f(χ,ξ)/f(ξ) denoting the conditional density of χ given ξ. Since H(χ) corresponds to the prior uncertainty associated with χ and the observation ξ may provide additional information with f(χ|ξ) describing what is known about χ after having observed ξ, H(χ|ξ) reflects the posterior uncertainty about χ given ξ. The expected information of the observation ξ with respect to χ is then defined as the difference between the prior uncertainty about χ, H(χ), and the posterior uncertainty of χ given ξ, H(χ|ξ), I(χ|ξ) = H(χ)−H(χ|ξ). Using this measure of information content one can derive the expected relative information gain associated with adding a particular indicator variable ζ to the information vector 26For early uses of the concept of entropy in the economics literature see also Theil (1967). 24

ξ as follows: G(χ,ξ,ζ) = [I(χ|ξ,ζ)−I(χ|ξ)]/I(χ|ξ). Regarding χ and ξ as jointly distributed normal with covariance matrix Σ, Tinsley et al. show that I(χ|ξ) has a particularly simple form. In this case, using the properties of multivariate normal distributions, I(χ|ξ) = 0.5 ln(|Σχχ |/|Σχχ −ΣχξΣ − ξξ 1 Σξχ |), (cid:3) where Σχχ, Σξξ and Σχξ = Σ ξχ are the submatrices of Σ with appropriate dimensions and Σχχ −ΣχξΣ − ξξ 1 Σξχ is the conditional covariance matrix of χ given ξ. Thus, under normality, the measure of information content, I(χ|ξ), corresponds to the log-distance between the determinants of the covariance matrices of the marginal and the conditional distribution of χ. The case of univariate χ can then be used to develop an intuitive interpretation of the expected information content I(χ|ξ). In this case, I(χ|ξ) = 0.5 ln(σ χ 2/(σ χ 2 −ΣχξΣ − ξξ 1 Σξχ)) = 0.5 ln(1/(1−β(cid:3) Σξξ β/σ χ 2 )) = 0.5 ln(1/(1−R2 )), χ|ξ where β is the vector of regression coefficients, and R2 is the population coefficient of χ|ξ determination in the linear regression of χ on ξ. Evidently, R2 = 1−(exp{2I(χ|ξ)}) −1. χ|ξ To adapt these measures to our euro area model with rational expectations and data uncertainty, we need to obtain the joint distribution of the innovations in the observed indicator variables and the innovations in the predetermined variables. This is done in Appendix B.1. 25

5 Illustrating the Information Role of Money We have now assembled the tools necessary to assess the contribution of money in the estimation of noisy output data and the underlying shocks. To illustrate the information role of money, we start with the case in which output revisions occur within a single period (as specified in equation (9)). Furthermore, we utilize the following highly stylized money demand function in place of the more complicated dynamic specification of Coenen and Vega (1999) that was presented in section 3: m t −p t = q t+um t , (21) where the exogenous disturbance um is assumed to be serially uncorrelated with mean zero t and standard deviation σ(um ). As discussed in Section 3, we assume that money demand evolves in response to the true level of output, and hence can serve as a useful indicator variable. In fact, under our assumption that aggregate prices are known by all agents, money would be a perfect indicatoroftrueoutputinthelimitingcasewithnomoneydemandshocks(σ(um ) = 0). On the other hand, of course, money would provide no useful additional information if output were observed without measurement error (σ(v) = 0). More generally, the role of money as an indicator variable will depend on the relative magnitude of money demand shocks compared with output measurement errors. As pre- 27 viously noted, contract wage shocks are known by all agents in our model, so that the key information problem is to determine whether a given movement in output is due to an aggregatedemandshockortomeasurementerror. Thus,inthissection,wewillevaluatethe information content of money based on its contribution in estimating the current aggregate demand shock, ud . t 27Thisimplicationfollowsfromourassumptionsthatallagentsknowthecurrentcontractwageandutilize identical information in forming expectations about the variables on the right-hand side of equation (2). 26

5.1 The Economy without Money Asabenchmarkforcomparison, webeginwiththespecialcaseinwhichmoneyisnotinthe information set, or equivalently, the variance of money demand shocks is arbitrarily large. Figure 2 summarizes the characteristics of the information problem for a range of values of the standard deviation of the output measurement error, σ(v). The upper-left panel shows the optimal filter weights on the noisy observation of current output (q t +v t), as well as the true values of inflation (π t) and lagged output (q t−1). The upper-right panel indicates the root mean-squared error (RMSE) of the estimate ud of the current aggregate demand t|t shock, and the lower panel indicates the R2 of a regression of ud on the vector of observed t variables. Evidently, when output is measured without error (σ(v) = 0), the aggregate demand shock can be determined exactly as a function of the observed variables; that is, the RMSE equals zero and the R2 equals unity. As the standard deviation of the measurement errors increases, the optimal filter places lower weight on the noisy observation of current output (similarly on lagged output and inflation), the RMSE rises while the R2 falls. Finally, as noted in Section 3.2, the first-quarter revisions of euro area output have a sample standard deviation of 0.97 percent. As σ(v) approaches this value, we see that the RMSE rises to about0.5percent(stillabitlowerthantheunconditionalstandarddeviationof0.58percent for the aggregate demand shock), while the R2 falls below 25 percent. Figure 3 shows the behavior of this economy in response to a single aggregate demand shock of one standard deviation (which occurs at time 0). The solid line indicates the responseofeachvariableunderthebaselinecalibration(σ(v) = 0.97percent),whilethedotdashed line indicates the corresponding path when output is not subject to measurement errors (σ(v) = 0). Finally, in the upper-left panel, the dotted line indicates the path of perceived output in response to the shock. Of course, the aggregate demand shock immediately raises the level of real output. In this case, the reported level of output happens to be exactly correct, but the optimal filter 27

Figure 2: Optimal Filtering of the Aggregate Demand Shock without Money Optimal Indicator Weights Root Mean−Squared Error (RMSE) 1.50 0.60 0.75 0.45 0.00 0.30 −0.75 noisy current output (q+v) t t 0.15 lagged output (q ) −1.50 current inflation ( t− π 1 ) RMSE of demand shock estimate t standard deviation of demand shock 0.00 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 σ(v) σ(v) Coefficient of Determination (R2) 1.00 0.75 0.50 0.25 0.00 0.0 0.2 0.4 0.6 0.8 1.0 σ(v) causes private agents and the central bank to downweight this observation. Thus, contract wages and aggregate inflation respond slightly less than if output were not subject to measurement errors. More importantly, the central bank does not raise short-term nominal interest rates as quickly, and hence actual output rises more sharply and takes somewhat longer to return to potential. Based on these impulse responses, it is evident why indicator variables such as money can serve a useful role when output observations are noisy. 28

Figure 3: Impulse Responses to an Aggregate Demand Shock output annual inflation 0.50 0.50 0.25 0.25 0.00 with noise 0.00 perceived value with noise without noise without noise 0 2 4 0 2 4 short−term nominal interest rate long−term real interest rate 0.50 0.50 0.25 0.25 0.00 0.00 with noise with noise without noise without noise 0 2 4 0 2 4 5.2 The Information Role of Money We now consider the extent to which money can provide a more accurate estimate of the current aggregate demand shock. Since the contribution of money depends both on the standard deviations of the output measurement error and the money demand shock we report the results in a set of three-dimensional graphs. In each panel, the two axes in the horizontal plane denote the standard deviation of the measurement error on output σ(v) and the standard deviation of the money demand shock σ(um ). The top four panels of Figure 4 indicate the weights (measured on the vertical axis) on the noisy current output observation q t (t) , as well as the true values of lagged output q t−1, inflation π t, and money growth µ t = m t −m t−1. 29

Figure 4: Optimal Filtering of the Aggregate Demand Shock with Money Weight on Current Output Observation (q+v) Weight on Lagged Output (q ) t t t−1 1.5 0.0 1.0 −0.5 0.5 −1.0 0.0 −1.5 1.0 1.0 1.0 1.0 0.5 0.5 0.5 0.5 σ(v) 0.0 0.0 σ(um) σ(v) 0.0 0.0 σ(um) Weight on Inflation (π) Weight on Money Growth (µ) t t 0.5 1.5 0.0 1.0 −0.5 0.5 −1.0 0.0 1.0 1.0 1.0 1.0 0.5 0.5 0.5 0.5 σ(v) 0.0 0.0 σ(um) σ(v) 0.0 0.0 σ(um) 2 Root Mean−Squared Error (RMSE) Coefficient of Determination (R ) 0.9 1.5 0.6 1.0 0.3 0.5 0.0 0.0 1.0 1.0 1.0 1.0 0.5 0.5 0.5 0.5 σ(v) 0.0 0.0 σ(u m ) σ(v) 0.0 0.0 σ(u m ) 30

With regard to the weights on current and lagged output and inflation as a function of the output measurement error, we confirm the findings for the case without money. As the measurement error regarding output increases the weights assigned to noisy current output, lagged output and inflation decrease (in absolute terms). This is the case for any level of the standard deviation of the money demand shock as can be seen by moving from right to left along the dimension which corresponds to the output measurement error. We find that money can play an important role in estimating the current aggregate demand shock if the relative magnitude of the money demand shock is not too large. Not surprisingly,theweightassignedtomoneyislargestintheabsenceofmoneydemandshocks (that is, σ(um ) = 0). In this case, the decision maker can infer the true value of output (andconsequently, theaggregatedemandshock)directlyfromthemoneygrowthrate, since lagged output and inflation are observed exactly. As σ(um ) increases, however, the weight on money growth in the optimal estimate of output declines. With the possibility of money demand shocks, the decision maker cannot be sure whether a money growth observation that seems inconsistent with observed output is an indication of a mismeasurement of actual output or of a money demand shock. While the weight on money declines, one can see that it declines more slowly the greater the standard deviation of the output measurement error. Finally, it is of interest to note that the weight on inflation and lagged output also decreases in absolute value with the weight on money, as the standard deviation of the money demand shock increases. The bottom two panels of Figure 4 show the degree of uncertainty associated with the contemporaneous estimate of the aggregate demand shock, as indicated by the RMSE and the R2 . Of course, along either axis in the horizontal plane, the RMSE is zero and the R2 is equal to one, because either σ(v) or σ(um ) equals zero. When both σ(v) and σ(um ) are strictly positive, the RMSE is positive and the R2 is less than unity. Figure 5indicatesthreemeasuresoftheinformationcontentofmoneyinestimatingthe aggregate demand shock: the reduction in RMSE (in percentage points), the improvement in R2 , and the expected relative gain in information G. When output measurement errors 31

Figure 5: Information Content of Money in Estimating the Aggregate Demand Shock Reduction in RMSE (in Percent) Increase in R2 0 0.8 −25 0.6 −50 0.4 −75 0.2 −100 0.0 1.0 1.0 1.0 1.0 0.5 0.5 0.5 0.5 σ(v) 0.0 0.0 σ(um) σ(v) 0.0 0.0 σ(um) Information Gain (G in Percent) 2000 1500 1000 500 0 1.0 1.0 0.5 0.5 σ(v) 0.0 0.0 σ(um) are relatively large compared with money demand shocks (that is, σ(v) > 0.5 percent and σ(um ) < 0.5 percent), the improvement in the quality of the estimate can be substantial: in such cases, money reduces the RMSE by 20 percent or more and raises the R2 by at least 0.18. The information gain from utilizing money as an information variable is also very high under these conditions. 32

6 The Quantitative Significance of Money as an Indicator Variable Having illustrated the role of money in a somewhat simplified model, we now proceed to quantify the information content of money using the full model given in Section 3. In particular, we utilize the empirical money demand equation given in equation (4) with the estimated standard deviation of the money demand shock, and we consider two variants of the3-quarteroutputrevisionprocessgiveninequation(10), aswellasthesimpler1-quarter revision process (given by equation (9)) that was used in the previous section. 6.1 Results for the Baseline Estimated Model 6.1.1 The Optimal Indicator Weights Table 4 indicates the optimal weight on each indicator variable used in estimating the current aggregate demand shock, ud . The upper panel shows these weights when money t is not included in the information set, while the lower panel indicates the weights when current money growth is utilized in constructing the optimal estimate. In each case, we consider three alternative assumptions about the output revision process. When the revision process is completed in a single period, the previous period’s output is known with certainty (that is, q t ( − t) 1 = q t−1). In this case, longer lags of output do not contain any additional information regarding the period t aggregate demand shock. Thus, as shown by the first row of the upper panel, the optimal filter places non-zero weight on the noisy current output observation q t (t) , the previous period’s output level q t−1, and the current inflation rate π t. (As noted previously, we are assuming that current inflation is known by all agents and hence can always serve as a perfect indicator variable.) In contrast, when the revision process takes three periods, the current and previous two output observations contain measurement error, while the true value of q t−3 is revealed in the latest data vintage. Hence, in this case, the optimal filter places non-zero weight on all four output observations (that is, q(t) , q(t) , q(t) , and q(t) ) as well as the current inflation t t−1 t−2 t−3 rate. The second row of the upper panel shows the optimal weights when the revisions are 33

Table 4: Optimal Indicator Weights for Estimating the Aggregate Demand Shock Revision Process q t (t) q t ( − t) 1 q t ( − t) 2 q t ( − t) 3 π t µ t Filtering without Money One-Period 0.267 -0.309 0 0 0.022 — Three-Period Uncorrelated 0.136 -0.059 -0.054 -0.021 0.011 — Three-Period Correlated 0.271 -0.127 -0.054 0.016 0.022 — Filtering with Money One-Period 0.265 -0.268 0 0 0.044 0.354 Three-Period Uncorrelated 0.138 -0.053 -0.081 -0.052 0.035 0.387 Three-Period Correlated 0.275 -0.115 -0.089 0.012 0.046 0.374 uncorrelated but their variances are set to the estimated values described in Section 3.2., while the third row indicates the weights when we use the complete estimated covariance matrix. Whencurrentmoneygrowthisincludedintheinformationset,weseethatthisindicator variable receives substantial weight in estimating the current aggregate demand shock. The exactweightvariessomewhatdependingonthespecificrepresentationoftheoutputrevision process, butthenotablepointisthatthemagnitudeofthisweightisroughlysimilartothat placed on the noisy current output observation. Of course, interpreting the specific pattern of filtering weights is rather difficult, and hence we now proceed to consider the measures of information content described in Section 4.2. 6.1.2 Measures of Information Content Table 5 characterizes the information role of money in estimating the current aggregate demand shock under each of the three alternative assumptions about the revision process. As a benchmark for comparison, the first two columns indicate the RMSE of the demand 34

Table 5: The Information Role of Money in the Estimated Model Information Content Revision Process RMSE R2 %∆RMSE ∆R2 Info. Gain One-Period 0.50 0.26 -1.29 0.02 8.51 Three-Period Uncorrelated 0.54 0.13 -1.46 0.03 20.42 Three-Period Correlated 0.50 0.27 -1.59 0.02 10.29 shock estimate (that is, the square root of Et[ud t|t −ud t ] 2 ) and the associated R2 when money growth is not included in the information set. Evidently, the precision of the demand shock estimates is not very high regardless of how the output revision process is specified. TheremainderofTable 5providesthreemeasuresoftheextenttowhichcurrentmoney growth increases the precision of the estimated demand shock. By including money as an indicator variable, the RMSE is reduced by about 1.5 percent, and the R2 rises by about 0.02. Measuredintermsoflowerentropy,theinformationgainissomewhatmoreimpressive: about 10 to 20 percent, depending on the specification of the output revision process. Based on these results, one would reasonably conclude that money has noticeable but notremarkablyhighinformationcontentinestimatingthecurrentaggregatedemandshock. However, our analysis thus far has assumed that the central bank is unable to identify any of the underlying factors that generate the contemporaneous money demand shock. As we will see below, monetary analysis can dramatically raise the usefulness of money as an indicator variable. 6.2 The Role of Monetary Analysis Centralbankstendtoexpendsignificantresourcestogainabetterunderstandingofongoing monetary developments on a very detailed level, thereby identifying factors that would not be well-explained by a standard money demand model. This function of monetary analysis 35

is highlighted in the May 2001 issue of the ECB Monthly Bulletin: “The decomposition of monetary growth into its macroeconomic determinants alsoindicatestheextenttowhichmonetarygrowthisnotexplainedbythe[money demand] model. Hence it may reveal additional information contained in monetary aggregates which is not captured by the other macroeconomic variables. ... Ideally, a detailed institutional analysis can provide some additional insight by providing information concerning special events, thus reducing the unexplained part of monetary growth.” [ECB Monthly Bulletin, May 2001, pp. 46/47.] The results reported in Table 6 indicate that monetary analysis can play an important role in enhancing the information role of money when contemporaneous output estimates are afflicted by substantial measurement error. In the first column, the standard deviation of the money demand shock σ(um ) is equal to the estimated value from Coenen and Vega (1999). Intheremainingthreecolumns,weassumethatmonetaryanalysisisabletoidentify factors accounting for a substantial fraction of the variation in the money demand shock. In such cases, the information content of current money growth increases dramatically. When monetary analysis is able to identify factors accounting for 75 percent of the variance of the money demand shock, then utilizing money as an indicator variable reduces theRMSEofthecurrentaggregatedemandshockestimatebyabout10to15percent, while the associated R2 rises by about 0.2, and the entropy measure of information increases by more than 100 percent. When the central bank succeeds in identifying the factors that accountfor7/8ofthemoneydemandshock, thentheRMSEisreducedby30to40percent; the other measures of information content also rise dramatically in this case. When monetary analysis is relatively effective, we also find that current money growth 28 providessubstantialbenefitsinpredictingfuturelevelsofoutput. Toillustratethisbenefit, we consider the case in which the central bank can identify factors that account for threefourthsofthevariationinthemoneydemandshock. Theunderlyingoutputrevisionprocess 28Table D1 in Appendix D indicates the RMSE of output and inflation forecasts for the case in which output is measured without error. 36

Table 6: Monetary Analysis and the Information Content of Money Reduction in the Std. Dev. of the Money Demand Shock Information Revision 0% 50% 75% 82.5% Measure Process %∆RMSE One-Period -1.29 -4.88 -16.13 -38.99 Three-Period Uncorrelated -1.46 -4.69 -11.91 -26.01 Three-Period Correlated -1.59 -5.41 -14.41 -28.77 ∆R2 One-Period 0.02 0.07 0.22 0.46 Three-Period Uncorrelated 0.03 0.08 0.19 0.39 Three-Period Correlated 0.02 0.08 0.20 0.36 Information One-Period 8.51 32.79 115.12 323.36 Gain Three-Period Uncorrelated 20.42 66.81 176.11 418.47 Three-Period Correlated 10.29 36.05 100.58 219.55 is assumed to take three periods with the revisions being correlated. The first column of Table 7 indicates the RMSE of the output prediction for a range of forecast horizons when money growth is not included in the information set, while the fourth column indicates the reduction in RMSE that results from utilizing money as an indicator variable. Evidently, money growth enhances the precision of the contemporaneous output estimate by about 5 percent, and improves the accuracy of the two-quarter-ahead output forecast by about 2 percent. Of course, as the forecast horizon increases, output exhibits a higher degree of unpredictable variation, and hence no indicator variable would be expected to have very much predictive power. As discussed in Appendix B.2, the MSE can be decomposed into a component which relates to the within-period estimation error of the predetermined variables (component I), and the propagation of unpredictable future disturbances which will affect the evolution of the predetermined variables as well as their within-period estimates in the future (component II). Component II increases with the prediction horizon and converges to the 37

Table 7: The Role of Money in Predicting Output RMSE %∆RMSE Forecast Horizon Overall I II Overall I II 0 0.56 0.56 0 -5.42 -5.42 0 1 0.84 0.62 0.57 -3.44 -5.97 -0.57 2 1.05 0.59 0.87 -2.25 -6.14 -0.52 4 1.33 0.50 1.24 -1.32 -6.15 -0.55 8 1.60 0.34 1.56 -0.83 -6.15 -0.59 16 1.75 0.15 1.74 -0.65 -6.15 -0.61 unconditional standard deviation of output. The contribution to the RMSE coming from component I dominates initially but has a declining contribution to the overall RMSE as the forecast horizon increases. Turning to the final two columns of Table 7, we see that using money as an indicator variable causes a 6 percent reduction in the RMSE associated with component I, regardless of the forecast horizon. It is interesting to note that having money in the information set alsochangesthedynamicbehaviorofactualoutput,andhencehasasmalleffectinreducing 29 the RMSE associated with component II. 6.3 Further Sensitivity Analysis Now we briefly summarize some additional sensitivity analysis regarding the results presented above. First, it is worthwhile to consider the implications of alternative money demand specifications with a stronger contemporaneous relationship between money and output. For the preceding analysis, we have used the demand function for M3 because this money aggregate has exhibited reasonable stability in the euro area over the past two decades (cf. Coenen 29Results regarding inflation prediction are given in Table D2 of Appendix D. With regard to inflation, component I is not very important. Even though component I decreases by including money growth in the information set, component II actually increases slightly. With regard to the overall MSE money growth does not help as a result. Of course, one needs to keep in mind that the information role of money for inflation here is rather limited. 38

and Vega (1999)). However, the estimated coefficient that determines the instantaneous income elasticity of M3 is rather small. Narrower money aggregates (such as M1) typically have a much tighter relationship with current output. As shown in Table D3 of Appendix D, we find that a higher instantaneous income elasticity substantially raises the information content of money. Thus, to the extent that monetary analysis can identify structural changes and special factors that generate shifts in the demand function for a narrow aggregate (such as M1), the central bank would be able to utilize such an aggregate in reducing the data uncertainty associated with current output. Second, we have assumed for simplicity that private agents and the central bank are able to utilize a noisy estimate of contemporaneous output at each point in time. Given the actual time delays in releasing GDP data, however, it may be more realistic to assume that no output estimate is available until the subsequent period (especially since our model is specified at a quarterly frequency). As shown in Appendix Tables D4 and D5, the importance of using money as an indicator variable increases in this case. Finally, the results reported above have been derived using a structural macroeconomic model. This approach provides a clear description of the transmission mechanism of monetary policy, and explicitly considers the evolution of market participants’ expectations. Nevertheless, we recognize that structural assumptions are always somewhat controversial. Therefore,wehavealsomeasuredtheinformationcontentofmoneyusingthenon-structural time series model of the euro area estimated by Coenen and Vega (1999). As shown in Appendix Tables D6 through D9, the implications of the time series model are remarkably similar to those of the structural macroeconometric model. 7 Conclusion To explore the information role of money in the presence of data uncertainty we have extendedtheeuroareamacroeconomicmodelofCoenenandWieland(2000)byincorporating theeuroarea-widemoneydemandmodelofCoenenandVega(1999)andanempiricallycalibrated model of the revision process of aggregate euro area output. Using this framework 39

we have found that money can play an important role as an information variable and may result in major improvements in current output estimates. However, the specific nature of this role depends on the magnitude of the output measurement error relative to the money demand shock. In particular, we have found noticable but small improvements in output estimates due to the inclusion of money growth in the information set when the standard deviation of money demand shocks equals the estimated value from Coenen and Vega (1999). Money plays a quantitatively more important role with regard to output estimation if we allow for a contribution of monetary analysis in reducing uncertainty due to money demand shocks. In this case, money also helps to reduce uncertainty about output forecasts. Of course, as the construction of euro area aggregate output data is improved over time, the magnitude of the revisions discussed in Section 2 is likely to decline over time. Nevertheless, evidence concerning U.S. data vintages collected by Croushore and Stark (1999) indicates that data uncertainty will remain an important issue even once the data collection technology has matured. Throughout the paper we consider a relatively limited role of money by focusing exclusively on the information content of money with respect to output measurement and by excluding the possibility of a direct role of money in output and inflation determination. In this sense, our quantitative results only indicate a lower bound on the usefulness of money. An alternative model that allows for significant direct effects of money on inflation and ∗ could be used in future research is the so-called P model. Also, as noted earlier, we have focused attention on a framework with symmetric information regarding aggregate output data as far as private market participants and the central bank are concerned. We have also conducted some exploratory analysis under the assumption of asymmetric information regarding aggregate data that is used by Dotsey and Hornstein (2000) and Svensson and Woodford (2001). However, in our view this assumptionis undesirable ifitimpliesthat arepresentative agent by knowing hisindividual income can also infer aggregate income and demand while the policymaker only observes a noisy 40

estimate of aggregate demand. We plan to study the asymmetric case in more detail in the future in a model that would allow us to differentiate more carefully between individual and aggregate uncertainty. Finally, another interesting avenue for future research would be to compare optimal filtering to simple filtering rules in keeping with the recent debate on optimal versus simple monetary policy rules. For example, one could investigate the performance of simple rules thatrespondonlytoobservedoutputgrowth,inflationandmoneygrowthinsteadofoptimal estimates of the output gap. A recent study that considers an example of a simple filtering rule in the context of NAIRU uncertainty is Meyer, Swanson and Wieland (2001). 41

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Appendix A Kalman filtering `a la Svensson and Woodford This appendix restates and specializes the setup of Svensson and Woodford (2000) – henceforth SW – so that the Kalman filter can be applied to the variants of the wage contracting models analysed in the main text. In particular, we now work with a generic linear rational expectations model and a model of measurement which permit us to formulate the wage contracting models in state-space form to which the Kalman filter is applicable. A.1 The state-space representation The generic linear rational expectations model takes the form (cid:4) (cid:5) (cid:4) (cid:5) (cid:4) (cid:5) (cid:4) (cid:5) X X X u t+1 = A1 t +A2 t|t + t+1 , (A.1) E˜x t+1|t x t x t|t 0 whereX t isavectorofpredeterminedvariables,x t isavectorofnon-predeterminedvariables and u t is a vector of serially uncorrelated shocks with mean zero and positive semi-definite covariance matrix Σuu. The coefficient matrices A1 , A2 and E˜ are matrices of appropriate dimension. Regarding the measurement of the predetermined and non-predetermined variables, let Z t denote a vector of observables given by (cid:4) (cid:5) (cid:4) (cid:5) X X Z t = D1 x t +D2 x t|t +v t , (A.2) t t|t where v t is a vector of serially uncorrelated measurement errors with mean zero and positive semi-definite covariance matrix Σvv. The measurement errors v t are assumed to be uncorrelated with the shocks u t at all leads and lags, i.e. E[u t v τ (cid:3) ] = 0 for all t and τ. The matrices D1 and D2 are selector matrices of appropriate dimension. Information in period t is supposed to be given by (cid:6) (cid:7) I t = Z τ ,τ ≤ t;A1,A2,D1,D2,E˜,Σuu ,Σvv and we let χ τ|t = E[χ τ |I t] denote the rational expectation of any variable χ in period τ given information in period t. SW show that the non-predetermined variables fulfill the relationship x t = G1X t+G2X t|t (A.3) and that the system of equations (A.1), (A.2) can be cast into state-space form without non-predetermined variables, X t+1 = HX t+JX t|t+u t+1 (A.4) Z t = LX t+M X t|t+v t , (A.5) where the matrices G1 , G2 , H, J, L and M are provided in SW. 45

Thistransformationturnsouttosimplifytheremainingproblemofformingtheestimate X t|t considerably. A.2 The Kalman filter After having eliminated the non-predetermined variables x t there is a simultaneity problem to be solved when estimating the predetermined variables X t because the observable variables Z t depend on the estimate of the predetermined variables X t|t, while the latter depends on the observables used in the estimation. Accounting for the contemporaneous effect of the estimate X t|t on Z t, SW show that the optimal estimate of X t can be obtained by a Kalman filter updating equation in terms of the innovations in the transformed variables Z¯ t = Z t −M X t|t, X t|t = X t|t−1+K(Z¯ t −Z¯ t|t−1) (A.6) (cid:1) (cid:3) = X t|t−1+K L(X t −X t|t−1)+v t , (A.7) where the second line uses that Z¯ t −Z¯ t|t−1 = L(X t −X t|t−1)+v t. 30 The steady-state Kalman gain matrix K is given by K = PL(cid:3) (LPL(cid:3) +Σvv) −1, (A.8) where the matrix P is the steady-state covariance matrix of the innovations X t −X t|t−1 given information in period t−1 and fulfills 31 (cid:1) (cid:3) P = H P −PL(cid:3) (LPL(cid:3) +Σvv) −1LP H(cid:3) +Σuu . (A.9) In section B.1 below it is verified that the term in square brackets is the covariance matrix of the updating errors X t −X t|t given information in period t, which may serve as a measure of the estimation uncertainty surrounding the optimal estimate X t|t produced by the application of the Kalman filter. The evolution over time of the predetermined variables X t and their estimates X t|t is simultaneously determined by the transition equation (A.4) and the Kalman filter updating equation (A.7) in combination with the prediction formula X t|t−1 = (H+J)X t−1|t−1 being derived from the former. For later reference, it is convenient to express this system of dynamic equations more compactly as (cid:4) (cid:5) (cid:4) (cid:5)(cid:4) (cid:5) (cid:4) (cid:5) (cid:4) (cid:5) X t = H J X t−1 + I 0 u t .(A.10) X t|t KLH (I −KL)H +J X t−1|t−1 KL K v t 30NotethatZ¯ t =Z t −MX t|t =LX t+v tandZ¯ t|t−1 =Z t|t−1 −MX t|t−1 =LX t|t−1,andthusZ¯ t −Z¯ t|t−1 = L(X t −X t|t−1)+v t. 31Note that (LPL(cid:5)+Σvv)−1 may be replaced by a generalized inverse if (LPL(cid:5)+Σvv) is singular. 46

Appendix B Measuring the information content of indicator variables To evaluate the information content of indicator variables, we assess to which extent indicator measurements will reduce the uncertainty surrounding the estimation and/or prediction of variables of interest such as output and inflation. To do so, we build on a measure from information theory, the entropy, to evaluate the within-period estimation uncertainty and compute the Mean Square Error (MSE) of multi-period ahead predictions to evaluate the prediction uncertainty. In section B.1 and section B.2 below, we show how to properly adapt the necessary computations to the generic linear rational expectations model with imperfectly observed indicator variables. B.1 Within-period estimation Drawing on results from information theory, Tinsley et al. (1980) employ the entropy, or ‘expecteduncertainty’,asaformalmeasureoftheinformationcontentofindicatorvariables. We have restated the relevant results in Tinsley et al. in the main text and adapt them here to the generic linear rational expectations model with imperfect indicator measurements. In order to do so we need to obtain the joint distribution of the innovations in the observed indicator variables, Z t −Z t|t−1, and the innovations in the predetermined variables, X t −X t|t−1. As an intermediate step, we first consider the case of the transformed indicator variables Z¯ t ≡ Z t −M X t|t, where the contemporaneous effect of the estimate X t|t is subtracted from the observed variables Z t. Assuming that the shocks u t and the measurement errors v t are normal and that the steady-state covariance matrix P from the application of the Kalman filter is given, it is straightforward to show that the innovations in the transformed indicators, Z¯ t −Z¯ t|t−1, and the innovations in the predetermined variables, X t −X t|t−1, are jointly normally distributed with (cid:4) (cid:5) (cid:9)(cid:4) (cid:5) (cid:4) (cid:5)(cid:10) X Z¯ t t − −X Z¯ t t | | t t − − 1 1 (cid:8) (cid:8) (cid:8)I t−1 ∼ N 0 0 , LPL P (cid:3) L + (cid:3) Σvv L P P , (B.1) where we recall that Z¯ t −Z¯ t|t−1 = L(X t −X t|t−1)+v t and P = Cov[X t −X t|t−1 |I t−1]. Then, observing that the innovations in the observed indicators are a linear transformationoftheinnovationsinthetransformedindicators, Z t −Z t|t−1 = (I+MK)(Z¯ t −Z¯ t|t−1), 32 32Notice that Z t −Z t|t−1 = Z t −MX t|t+MX t|t −Z t|t−1 −MX t|t−1+MX t|t−1 = Z¯ t −Z¯ t|t−1+M(X t|t −X t|t−1) (cid:11) (cid:12) = Z¯ t −Z¯ t|t−1+MK L(X t −X t|t−1)+v t = Z¯ t −Z¯ t|t−1+MK(Z¯ t −Z¯ t|t−1) 47

it follows immediately from (B.1) that the innovations in the observed indicators and the innovations in the predetermined variables are also jointly normally distributed with (cid:4) (cid:5) (cid:9)(cid:4) (cid:5) (cid:4) (cid:5)(cid:10) X Z t t − − Z X t t | | t t − − 1 1 (cid:8) (cid:8) (cid:8)I t−1 ∼ N 0 0 , N(LP P L L (cid:3) + (cid:3)N Σ (cid:3) vv)N(cid:3) N P LP , where N = I +MK. Hence, using the properties of multivariate normal distributions, the conditional covariance matrix of X t −X t|t−1 given information in period t is Cov[X t −X t|t−1 |I t] = Cov[X t −X t|t−1 |I t−1 ,Z t −Z t|t−1] (cid:13) (cid:14) (cid:3) = P −PL(cid:3)N(cid:3) N(LPL(cid:3) +Σvv) −1N(cid:3) NLP = P −PL(cid:3) (LPL(cid:3) +Σvv) −1LP. (B.2) Apparently, while the weights of the observed indicator variables Z t in computing the conditional mean X t|t = E[X t |I t] are affected by the contemporaneous effect of X t|t on Z t (as shown below), the computation of the conditional covariance Cov[X t −X t|t−1 |I t] is not. Indeed, the conditional covariance matrix Cov[X t − X t|t−1 |I t] is equal to the conditional covariance matrix Cov[X t −X t|t−1 |Z¯ t −Z¯ t|t−1 ,I t−1] which can be obtained from (B.1). In principle, one could also aim at measuring the information content of indicator variables starting from the covariance matrices of the innovations in the non-predetermined variables, x t −x t|t−1 = G1 (X t −X t|t−1)+G2 (X t|t −X t|t−1) = G1 (X t −X t|t−1)+G2K[L(X t −X t|t−1)+v t] = (G1 +G2KL)(X t −X t|t−1)+G2Kv t , where we have used equations (A.3) and (A.7). However, since the covariance matrix of x t −x t|t−1 given information in period t−1 will depend on the choice of the vector of indicator variables via the Kalman gain matrix K and the covariance matrix of the measurement errors Σvv, it will not be feasible to measure the information content of the indicator variables by measuring the distance between the covariance matrices of x t − x t|t−1 given information in t and t − 1, respectively, since a simultaneity problem, entering via K and Σvv, exists. Finally, since Cov[X t − X t|t |I t] = Cov[X t − X t|t−1 |I t], 33 equation (B.2) also provides a measure of the estimation uncertainty surrounding the optimal estimate X t|t produced = (I+MK)(Z¯ t −Z¯ t|t−1) where the step from the second line to the third makes use of the Kalman filter updating equation (A.7) and the step from the third to the fourth uses the relationship shown in footnote 30. 33Observing that Cov[X t −X t|t |I t] = Cov[(X t −X t|t)−(X t|t −X t|t−1)|I t] 48

by the application of the Kalman filter, i.e. the covariance matrix of the updating error X t −X t|t. B.2 Multi-period predictions Whenevaluatingtheinformationcontentofindividualindicatorvariablesitisalsoofinterest to asses to which extent the indicator variables may reduce the uncertainty surrounding the multi-period predictions of variables of interest such as output and inflation. To this end this section shows how to compute the Mean Square Error (MSE) of the h-period ahead predictions of the predetermined and non-predetermined variables in linear rational expectations models when the indicator variables are subject to measurement error. In particular, it is shown that the MSE can be decomposed in a component which relates to the within-period estimation error of the predetermined variables and the propagation of unpredictable future disturbances which will affect the evolution of the predetermined variables as well as their within-period estimates in the future. In general, the magnitude of both components depends on the system matrices describing the joint dynamics of the predetermined variables and their within-period estimates. To simplify the calculations we restate the dynamic system (A.10) describing the joint evolution of the vector of predetermined variables X t and its within-period estimate X t|t more compactly as Y t+1 = AY t+Bw t+1 (B.3) with Y t = [X t (cid:3), X t (cid:3) |t ] (cid:3) , w t = [u(cid:3) t , v t (cid:3) ] (cid:3) and appropriately defined matrices A and B. Then, iterating the dynamic system (B.3) forward, we can express the realisation in period t + h, Y t+h, in terms of the current realisation Y t and the future disturbances w t+1 ,...,w t+h, h(cid:15)−1 Y t+h = AhY t+ AiBw t+h−i . i=0 The h-period ahead prediction Y t+h|t, given the available information in period t, is Y t+h|t = AhY t|t = Cov[(X t −X t|t)−KL[(X t|t −X t|t−1)+v t]|I t] = Cov[(I−KL)(X t −X t|t−1)−Kv t |I t] = (I−KL)P(I−KL) (cid:5) +KΣvv K(cid:5) with P =Cov[X t −X t|t−1 |I t−1] and K =PL(cid:5)(LPL(cid:5)+Σvv)−1, we obtain after some algebra Cov[X t −X t|t |I t] = P −PL(cid:5) (LPL(cid:5) +Σvv) −1LP = Cov[X t −X t|t−1 |I t]. 49

and, thus, the h-period ahead prediction error amounts to h(cid:15)−1 Y t+h −Y t+h|t = Ah (Y t −Y t|t)+ AiBw t+h−i . (B.4) i=0 Observing that (cid:4) (cid:5) (cid:4) (cid:5) (cid:4) (cid:5) X −X X −X I Y t −Y t|t = X t t |t −X t t |t |t = t 0 t|t = 0 (X t −X t|t) (B.5) it follows that the h-period ahead prediction error (B.4) has two sources: first, the withinperiod prediction error of the predetermined variables, X t −X t|t; and, second, the unpredictable future disturbances w t+1 ,...,w t+h. Obviously, the impact of the within-period estimation error X t −X t|t dies out with increasing prediction horizon h since Ah converges to zero with increasing h given that all eigenvalues of A have modulus less than one. Since X t −X t|t = HX t−1+JX t−1|t−1+u t (cid:1) (cid:3) − KLHX t−1+((I −KL)H +J)X t−1|t−1+KLu t+Kv t = (I −KL)H(X t−1 −X t−1|t−1)+[(I −KL) −K]w t we can express X t −X t|t as the weighted sum of current and past disturbances by solving the above equation backwards, (cid:15)∞ X t −X t|t = [(I −KL)H] i [(I −KL) −K]w t−i . (B.6) i=0 Then, combining (B.5) and (B.6) with (B.4), we can express the h-period ahead prediction error Y t+h −Y t+h|t as a weighted sum of past, current and future disturbances, (cid:4) (cid:5) I (cid:15)∞ h(cid:15)−1 Y t+h −Y t+h|t = Ah [(I −KL)H] i [(I −KL) −K]w t−i+ AiBw t+h−i . 0 i=0 i=0 SinceY t −Y t|t is uncorrelated withw t+h−i, forh−i > 0, the MSE matrixof theh-period ahead prediction Y t+h|t is (cid:1) (cid:3) MSE[Y t+h|t] = E (Y t+h −Y t+h|t)(Y t+h −Y t+h|t) (cid:3) (cid:4) (cid:5) I (cid:15)∞ (cid:15)∞ (cid:1) (cid:3) = Ah [(I −KL)H] i [(I −KL) −K]E w t−i w t (cid:3) −j 0 i=0 j=0 (cid:4) (cid:5) (cid:13) (cid:14) (cid:3)(cid:13) (cid:14) (cid:3) I (cid:3) ×[(I −KL) −K] (cid:3) [(I −KL)H] j Ah 0 50

h(cid:15)−1 h(cid:15)−1 (cid:1) (cid:3) (cid:13) (cid:14) (cid:3) + AiBE w t+h−i w t (cid:3) +h−j B(cid:3) Aj i=0 j=0 (cid:4) (cid:5) (cid:4) (cid:5) = Ah I (cid:15)∞ [(I −KL)H] i [(I −KL) −K] Σuu 0 0 0 Σvv i=0 (cid:4) (cid:5) (cid:13) (cid:14) (cid:3)(cid:13) (cid:14) (cid:3) I (cid:3) ×[(I −KL) −K] (cid:3) [(I −KL)H] i Ah 0 (cid:4) (cid:5) h(cid:15)−1 (cid:13) (cid:14) + AiB Σuu 0 B(cid:3) Ai (cid:3) , 0 Σvv i=0 where it has been used that (cid:4) (cid:5) E (cid:11) w t w τ (cid:3) (cid:12) = Σuu 0 0 Σvv for t = τ and 0 else. The first term of the MSE matrix vanishes for h → ∞ since the impact of the withinperiod estimation error dies out with increasing prediction horizon h. 34 By contrast, the secondtermismonotonicallynon-decreasingandforh → ∞itapproachestheunconditional covariance matrix of Y t. It is interesting to note that, even if the first term of the overall MSE matrix has vanished, the second term and, ultimately, the unconditional variance of Y t will generally be affected by the choice of indicator variables via the Kalman gain matrix K and the covariance matrix of the measurement errors Σvv. In particular, this will hold true for the h-period ahead predictions of the predetermined variables X t unless J (cid:13)= 0, i.e. unless the dynamic system in X t and X t|t is decoupled. Once we have determined the MSE matrix of the h-period ahead prediction Y t+h|t, we can easily recover the MSE matrix of the h-period ahead prediction of the vectors of predetermined and non-predetermined variables, X t+h|t and x t+h|t, from the former. Obviously, the MSE matrix of the h-period ahead forecast of the vector of predetermined variables, MSE[X t+h|t], is the upper left block of MSE[Y t+h|t]. To obtain the MSE matrix of the h-period ahead forecast of the vector of nonpredetermined variables, MSE[x t+h|t], we recall that x t = G1X t + G2X t|t. Thus, the realisation of the vector of non-predetermined variables in period t+h is given by x t+h = G1X t+h+G2X t+h|t+h (cid:4) (cid:5) (cid:1) (cid:3) X = G1 G2 t+h X t+h|t+h (cid:1) (cid:3) = G1 G2 Y t+h 34TheabovecalculationscanbesimplifiedbyobservingthatCov[X t −X t|t |I t]=P−PL(cid:5)(LPL(cid:5)+Σvv)−1LP – as shown in section B.1 – though this would not give insight how the first term of the MSE depends on the current and past disturbances of the dynamic system. 51

and the h-period ahead prediction, given the information available in period t, is x t+h|t = (G1 +G2 )X t+h|t (cid:4) (cid:5) (cid:1) (cid:3) I = G1 G2 I (H +J) hX t|t (cid:1) (cid:3) = G1 G2 AhY t|t (cid:1) (cid:3) = G1 G2 Y t+h|t . Consequently, the h-period ahead prediction error of the vector of non-predetermined variables equals (cid:1) (cid:3) x t+h −x t+h|t = G1 G2 (Y t+h −Y t+h|t) and the MSE matrix of the h-period ahead prediction amounts to (cid:1) (cid:3) MSE[x t+h|t] = E (x t+h −x t+h|t)(x t+h −x t+h|t) (cid:3) (cid:4) (cid:5) (cid:1) (cid:3) (cid:1) (cid:3) G1 = G1 G2 E (Y t+h −Y t+h|t)(Y t+h −Y t+h|t) (cid:3) G2 (cid:4) (cid:5) (cid:1) (cid:3) G1 = G1 G2 MSE[Y t+h|t] G2 . Apparently, even if the MSE matrix of X t+h|t does not, the MSE matrix of the h-period ahead prediction of the non-predetermined variables will depend on the choice of the indicator variables at all horizons, because the future values of the non-predetermined variables will depend on the path of the within-period estimates of the predetermined variables. 52

Appendix C The representation of the revision process According to the model of the revision process in Section 3, information in period t will comprise error-corrupted observations on q t, q t−1, q t−2 and the true value q t−3, q t (t) = q t + v t 3 + v t 2 + v t 1 q t ( − t) 1 = q t−1 + v t 3 −1 + v t 2 −1 q t ( − t) 2 = q t−2 + v t 3 −2 q t ( − t) 3 = q t−3 and the application of the Kalman filter will simultaneously determine a preliminary estimate of q t (the component q t|t), two revised estimates of last and next to last periods’ output (the components q t−1|t and q t−2|t), and a final estimate of output three periods ago (the component q t−3|t) which will be equal to the true output value q t−3. Apparently, our model of measurement shows persistence since the initial measurement error drops out only gradually and it takes three periods to learn about the true value of output. To make this setup conformable with the measurement model (12) in Section 4, we augment the vector of measurement errors v t to include lagged values of the measurement errors themselves and incorporate the augmented vector of measurement errors in the generic model (11) in the standard way by embedding the vector of measurement errors in the vector of predetermined variables X t. As a consequence, the vector of measurement errors disappears from the obervation equation (12) and, instead, the properly redefined matrix D1 picks off the current and past values of the measurement errors affecting the observations on current and past output. 53

Appendix D Results of further sensitivity analysis Table D1: Predictions in the Absence of Measurement Error RMSE (in percent) Forecast Horizon Output Inflation 0 0 0 1 0.55 0.23 2 0.84 0.41 4 1.18 0.80 8 1.49 0.93 16 1.66 0.99 Note: This table shows the RMSE of output predictions (q t+h|t)andannualaverageinflationpredictions(π˜t+h|t)at a given forecast horizon h in the absence of measurement error. Table D2: The Role of Money in Predicting Inflation RMSE %∆RMSE Forecast Horizon Overall I II Overall I II 0 0 0 0 0 0 0 1 0.23 0.02 0.23 0.01 -8.43 0.08 2 0.41 0.06 0.41 0.02 -8.13 0.17 4 0.80 0.16 0.78 0.07 -7.54 0.32 8 0.92 0.16 0.91 0.09 -5.89 0.25 16 0.98 0.07 0.98 0.10 -6.15 0.12 Note: This table shows the implications for predicting the annual average inflation rate (π˜) at a given forecast horizon when ouput is subject to measurement error. 54

Table D3: Sensitivity to the Income Elasticity of Money Demand Scaling Factor for the Income Elasticity of Money Demand Information Revision 1 2 4 8 Measure Process %∆RMSE One-Period -1.29 -4.88 -16.13 -38.99 Three-Period Uncorrelated -1.46 -5.71 -17.58 -38.08 Three-Period Correlated -1.59 -5.84 -17.46 -37.51 ∆R2 One-Period 0.02 0.07 0.22 0.46 Three-Period Uncorrelated 0.03 0.10 0.28 0.53 Three-Period Correlated 0.02 0.08 0.23 0.45 Information One-Period 8.51 32.79 115.12 323.36 Gain Three-Period Uncorrelated 20.42 81.53 268.61 665.83 Three-Period Correlated 10.29 38.96 124.21 304.21 55

Table D4: Optimal Indicator Weights when Current Output is Unobserved Revision Process q t (t) q t ( − t) 1 q t ( − t) 2 q t ( − t) 3 π t µ t One-Period — 0.052 0 0 0.030 0.474 Three-Period Uncorrelated — 0.007 -0.029 -0.039 0.028 0.442 Three-Period Correlated — 0.022 -0.038 -0.019 0.029 0.457 Note: This table shows the optimal indicator weights when money is included in the information set. Table D5: The Information Role of Money when Current Output is Unobserved Information Content Revision Process RMSE R2 %∆RMSE ∆R2 Info. Gain One-Period 0.58 0 -1.74 0.03 n.d. Three-Period Uncorrelated 0.58 0 -1.64 0.03 n.d. Three-Period Correlated 0.58 0 -1.69 0.03 n.d. Note: n.d.: The measure of information gain is not defined. 56

Table D6: Optimal Indicator Weights for the Time Series Model Revision Process q t (t) q t ( − t) 1 q t ( − t) 2 q t ( − t) 3 π t is t il t µ t Filtering without Money One-Period 0.167 -0.227 0 0 -0.050 -0.113 0.334 — Three-Period Uncorrelated 0.084 -0.032 -0.023 -0.028 -0.055 -0.101 0.323 — Three-Period Correlated 0.178 -0.073 -0.032 -0.020 -0.051 -0.082 0.273 — Filtering with Money One-Period 0.165 -0.194 0 0 -0.028 -0.111 0.291 0.286 Three-Period Uncorrelated 0.084 -0.030 -0.044 -0.051 -0.029 -0.110 0.289 0.306 Three-Period Correlated 0.181 -0.065 -0.064 -0.016 -0.026 -0.089 0.239 0.299 Table D7: The Information Role of Money in the Time Series Model Information Content Revision Process RMSE R2 %∆RMSE ∆R2 Info. Gain One-Period 0.40 0.21 -1.38 0.02 11.81 Three-Period Uncorrelated 0.42 0.12 -1.71 0.03 22.93 Three-Period Correlated 0.40 0.21 -1.83 0.03 15.46 57

Table D8: Weights for the Time Series Model with Unobserved Current Output Revision Process q t (t) q t ( − t) 1 q t ( − t) 2 q t ( − t) 3 π t is t il t µ t Filtering without Money One-Period — -0.054 0 0 -0.059 -0.134 0.404 — Three-Period Uncorrelated — -0.009 0.004 0.007 -0.060 -0.124 0.385 — Three-Period Correlated — -0.021 0.010 0.005 -0.059 -0.127 0.390 — Filtering with Money One-Period — -0.027 0 0 -0.034 -0.134 0.348 0.343 Three-Period Uncorrelated — -0.009 -0.019 -0.023 -0.033 -0.132 0.343 0.335 Three-Period Correlated — -0.014 -0.019 -0.007 -0.033 -0.132 0.344 0.342 Table D9: Money’s Role in the Time Series Model with Unobs. Current Output Information Content Revision Process RMSE R2 %∆RMSE ∆R2 Info. Gain One-Period 0.44 0.05 -1.62 0.03 56.20 Three-Period Uncorrelated 0.44 0.05 -1.46 0.03 60.57 Three-Period Correlated 0.44 0.05 -1.48 0.03 60.64 58