Lowering the Anchor: How the Bank of England's Inflation-Targeting Policies have Shaped Inflation Expectations and Perceptions of Inflation Risk

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Lowering the Anchor: How the Bank of England’s Inflation-Targeting Policies have Shaped Inflation Expectations and Perceptions of Inflation Risk Meredith J. Beechey 2008-44 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Lowering the Anchor: How the Bank of England(cid:146)s In(cid:135)ation-Targeting Policies have Shaped In(cid:135)ation Expectations and Perceptions of In(cid:135)ation Risk Meredith Beechey (cid:3) August 13, 2008 Abstract In(cid:135)ation targeting as practiced by the Bank of England has undergone several changes since its adoption in 1992, including rede(cid:133)nition of the goal, measures to increase transparency and the granting of independence to the central bank. These changes are likely to have a⁄ected long-run in(cid:135)ation expectations and perceptions of future in(cid:135)ation risk. To that end, this paper estimates a no-arbitrage, a¢ ne, factor model of the term structure of in(cid:135)ation compensation in the United Kingdom. The model yields time series of expected in(cid:135)ation and in(cid:135)ation risk premia at short and long horizons estimated in a theoretically consistent manner. The results reveal that long-run in(cid:135)ation expectations drifted down slowly during the (cid:133)rst (cid:133)ve years of in(cid:135)ation targeting, but in(cid:135)ation risk premia moved down abruptly only once the Bank of Englandwasgrantedindependence. Thisevent,whicharguablysignalledmorecredible commitmentbythecentralbanktoitsin(cid:135)ationanchor,appearstohavebeenmoreimportantinshapingin(cid:135)ationexpectationsandperceptionsofin(cid:135)ationriskthanchanges in the de(cid:133)nition of the target or measures to increase transparency. (cid:3)Division of Monetary A⁄airs, Board of Governors of the Federal Reserve System, Washington DC 20551. Email: meredith.j.beechey@frb.gov. I am grateful to Francesco Bianchi, Spencer Dale, Benson Durham, Jordi Gali, Mike Joyce, Don Kim, Andy Levin, Peter Lildholdt, Paul Tucker, Jonathan Wright and P(cid:228)r (cid:214)sterholm for helpful conversations and to participants at the conference on In(cid:135)ation Targeting, Central Bank Independence and Transparency, June 2007, at Cambridge University for valuable feedback. Theviewsinthispaperaresolelytheresponsibilityoftheauthorandshouldnotbeinterpretedasre(cid:135)ecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. 1

1 Introduction In(cid:135)ation targeting conducted by the Bank of England has undergone several changes since its adoption 15 years ago. A quantitative goal for retail price in(cid:135)ation was (cid:133)rst announced in October 1992 and was subsequently rede(cid:133)ned several times, from a range to a point target, (cid:133)rst asymmetric then symmetric. More recently the goal was lowered in tandem with the shift of the target measure to consumer price in(cid:135)ation. Amidst these changes, the Bank of England was granted independence to pursue its in(cid:135)ation goal and measures were taken to increase the Bank(cid:146)s transparency. These adjustments were aimed at fostering greater public understanding and credibility of the central bank(cid:146)s goals. How, then, have they in(cid:135)uenced in(cid:135)ation expectations and perceptions of in(cid:135)ation risk? This paper turns to (cid:133)nancial market data to answer that question, taking advantage of the long history of in(cid:135)ation-indexed debt issued by the government in the United Kingdom toestimateano-arbitrage, a¢ nefactormodelofthein(cid:135)ationcompensationtermstructure. The gap between the nominal and real term structures represents the compensation that investors receive to hold nominal rather than real government debt and is comprised of expected in(cid:135)ation and in(cid:135)ation risk premia. The term structure model returns time series ofexpectedin(cid:135)ationandin(cid:135)ationriskpremiaatshortandlonghorizons.Long-runin(cid:135)ation expectations inferred from the model line up closely with those from surveys of professional forecasters, butunlikesurveys, themodelcanbe(cid:133)ttedtohigh-frequencydata. Riskpremia areestimatedinatheoreticallyconsistentmannerwithinthemodelratherthanasaresidual after observing expectations. Disentangling in(cid:135)ation expectations from in(cid:135)ation risk premia matters to monetarypolicy makers. In(cid:135)ation expectations are an important input to the policy decision process, and(cid:133)nancialmarketsprovidereal-timeinformationabouttheexpectationsofabroadrange ofagents. Asabarometerofin(cid:135)ationuncertaintyinthebroadereconomy,(cid:133)nancialmarkets are able to quickly convey information that the central bank may need to factor into its deliberations, perhaps requiring the central bank to clarify its policy actions or reiterate its commitment to its long-run goal. Understanding when rising in(cid:135)ation risk premia re(cid:135)ect upon the credibility of the central bank or re(cid:135)ect broader macroeconomic sources of 2

in(cid:135)ation risk matters to the conduct of policy. The empirical results indicate that market participants(cid:146)long-run expectations of retail pricein(cid:135)ationtrendeddownfrom1992,butnotuntilaftertheBankofEnglandwasgranted independence did they approach and then remain close to 2.5 percent. In(cid:135)ation risk premia showed little sign of declining until 1997, when they dropped abruptly in the months following independence. This event, which arguably signalled more credible commitment of the central bank to its in(cid:135)ation objective, appears to have been more important in shaping in(cid:135)ation expectations and in(cid:135)ation risk than changes to the de(cid:133)nition of the target or measures to increase transparency, such as publication of the minutes of the Monetary Policy Committee (MPC) or releasing details of internal forecasting models. Following independence, expected future in(cid:135)ation and in(cid:135)ation risk premia remained low and fairly stable, until the in(cid:135)ation target measure was rede(cid:133)ned in December 2003 from retail prices excluding mortgage interest payments (RPIX) to consumer prices (CPI). SincetheannouncementofthenewCPIin(cid:135)ationtarget,riskpremiaonretailpricein(cid:135)ation outcomesfarintothefuturehaverisen30to40basispointsabovetherangeofthepreceding decade. Moreover, much of the increase in in(cid:135)ation risk premia was concentrated in the weeks immediately following the switch to the CPI target. Whydidrede(cid:133)ningthetargetraisein(cid:135)ationriskpremia?Thiscanbebetterunderstood by noting that in(cid:135)ation-indexed government debt in the United Kingdom is linked to the retailpriceindex(RPI).Inthelongrun,beyondthetypicalmonetarypolicycycle,forecasts of RPI and RPIX tend to coincide and their trend rates of growth are similar. But RPI and CPI outcomes are less correlated and the gap between them has been persistent at times. Disbanding the retail price in(cid:135)ation anchor arguably increased the compensation thatinvestorsrequiredtoholdlong-runretailpricein(cid:135)ationrisk. Thisisnottosaythatthe Bank of England became less credible in its commitment to controlling in(cid:135)ation; rather, it illustrates the role that an in(cid:135)ation anchor plays in moulding perceptions of future in(cid:135)ation risk. The paper is structured as follows. Section 2 outlines the no-arbitrage, a¢ ne, factor model used in the paper, deriving the general asset-pricing condition and outlining the model assumptions. An overview of the data is also given. Section 3 discusses issues 3

regarding estimation and presents the results for the sample at hand. Section 4 uses the output of the model to investigate how markets have responded to changes in the Bank of England(cid:146)s in(cid:135)ation targeting policies and Section 5 concludes. 2 A no-arbitrage factor model of in(cid:135)ation compensation This paper adapts a no-arbitrage, a¢ ne, factor model to the term structure of in(cid:135)ation compensation, a novel application of a standard (cid:133)nance model. The model is based on three key assumptions: i) assets are priced by a single discount factor such that no arbitrage opportunities remain in trading in(cid:135)ation compensation at di⁄erent maturities, ii) the dynamicsofthein(cid:135)ationcompensationtermstructurecanbewelldescribedbythreelatent factors, and iii) the discount factor used to discount returns is log-normal. In other words, yields are essentially a¢ ne in the three factors. Related research has modelled the nominal and real term structures jointly, imposing the restriction that the same three factors describe both term structures and the in(cid:135)ation process. (See Risa, 2001, for an application to the United Kingdom and D(cid:146)Amico, Kim and Wei,2006,andKimandWright,2005forapplicationstotheUnitedStates.) Oneadvantage ofmodellingthetermstructureofin(cid:135)ationcompensationdirectly,asisdonehere,istoavoid such restrictions on the factors and instead estimate those factors and parameters that best (cid:133)t the data. This does not imply that the factors underlying the in(cid:135)ation compensation term structure are uncorrelated with those underlying the nominal or real term structures. On the contrary, the factors may indeed covary but I do not impose restrictions about the natureofthatcovariance. Anotherreasontomodelin(cid:135)ationcompensationratherthanreal and nominal rates (jointly or separately) is to bypass concerns about structural change in the equilibrium real interest rate. Finally, the purely latent-factor approach to modelling the yield curve used here and elsewhere is more (cid:135)exible than models that employ observable macroeconomic factors, such as Diebold, Rudebusch, and Aruoba (2006) and Rudebusch and Wu (2004). While the inclusion of observable factors lends greater interpretability to the results, it also leads to a deterioration in overall (cid:133)t and greater concerns about misspeci(cid:133)cation. 4

2.1 Pricing of in(cid:135)ation-contingent assets The model builds upon a fundamental asset-pricing condition. Begin with the familiar condition for pricing nominal bonds, MN E t+nRN = 1 (1) t MN n;t (cid:20) t (cid:21) where MN is the continuous-time stochastic discount factor (SDF) used to price the gross t nominal return on a bond from t to t+n, RN . The nominal stochastic discount factor, n;t MN; and the real stochastic discount factor, MR (which prices real bonds) are linked via t t the price level, Q , t MN = MR=Q : (2) t t t We can express equation (1) in terms of the price of a zero-coupon nominal bond that pays $1 at time t+n and thus has a gross return of RN = 1=PN n;t n;t MN MR Q MR Q PN = E t+n = E t+n E t +cov t+n t : (3) n;t t MN t MR t Q t MR Q (cid:20) t (cid:21) (cid:20) t (cid:21) (cid:20) t+n (cid:21) (cid:20) t t+n (cid:21) Rewriting equation (3) in terms of the nominal yield, 1 yN = ln PN = yR +(cid:25)e +(cid:30) (4) n;t (cid:0)n n;t n;t n;t n;t (cid:0) (cid:1) where 1 Q (cid:25)e = ln E t : n;t (cid:0)n t Q t+n (cid:18) (cid:20) (cid:21)(cid:19) Intuitively this states that in(cid:135)ation compensation, the spread between nominal and real yields, is compromised of an in(cid:135)ation expectation, (cid:25)e , and an in(cid:135)ation risk premium, (cid:30) n;t n;t (including a Jensen(cid:146)s inequality term). Thus we can write down a general asset pricing condition for in(cid:135)ation compensation. The in(cid:135)ation compensation yield spread, yIC = yN yR , can be viewed as the yield on a n;t n;t n;t (cid:0) synthetic in(cid:135)ation bond that pays $1 at time t+n. This bond will have price PIC, where n;t PIC = exp n:yIC = exp n((cid:25)e +(cid:30) ) : (5) n;t n;t n;t n;t (cid:0) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) 5

This can in turn be rewritten as a more familiar general asset pricing condition, MIC PIC = E t+n (6) n;t t MIC (cid:20) t (cid:21) where MIC is the stochastic discount factor that prices in(cid:135)ation-contingent securities.1 t The model described below assumes that the pricing relationship (6) holds for synthetic in(cid:135)ationsecuritiesatallmaturities, n. Thatis, therearenoarbitrageopportunitiestrading combinations of securities at di⁄erent horizons. It is straightforward to show that if no arbitrage prevails in the nominal and real bond markets, no arbitrage opportunities exist for in(cid:135)ation compensation. 2.2 An a¢ ne factor model The model of the term structure employed here is based upon a model proposed by Du¢ e (2002) with three latent factors, using the factor normalisation implemented by Kim and Orphanides (2005). As stated above, it assumes that equation (6) prices all in(cid:135)ation compensation returns and that the time-series behaviour of the in(cid:135)ation compensation yield curve can be described by three underlying latent factors. The presentation of the model follows closely that found in Kim and Wright (2005). Assume that the stochastic discount factor, MIC, is speci(cid:133)ed as an a¢ ne function of a t 3 1 vector of latent factors, X , and evolves in a manner that mirrors equations (5) and t (cid:2) (6): dMIC t = r (cid:21) d" (7) MIC (cid:0) t (cid:0) 0t t t where r is the instantaneous rate of in(cid:135)ation, (cid:21) is a 3 1 vector, and r and (cid:21) are a¢ ne t t t t (cid:2) functions of the factors, r = (cid:26) +(cid:26)X (8) t 0 0 t (cid:21) = (cid:30) +(cid:8)X : (9) t 0 0 t 1That is, E Mt I + C nRIC =1, where the return RIC =1=PIC. t (cid:20) Mt IC t+n (cid:21) t+n n;t 6

The coe¢ cient (cid:26) is a scalar, (cid:26) and (cid:30) are 3 1 vectors, (cid:8) is a 3 3 matrix and the vector 0 0 (cid:2) (cid:2) of latent factors evolves as a continuous-time analogue of a vector autoregression dX = KX dt+(cid:6)d" (10) t t t where K and (cid:6) are 3 3 coe¢ cient matrices and the shock " is a three element vector of t (cid:2) Brownian motions. To achieve identi(cid:133)cation, K is transformed to be lower triangular and (cid:6) to a diagonal matrix, (cid:6) = diag((cid:27) ;(cid:27) ;(cid:27) ). 1 2 3 The interpretation of equation (7) is that the growth rate of the stochastic discount factor re(cid:135)ects a risk-free in(cid:135)ation expectation plus an adjustment for the sources of risk in the model: the shocks, " , and elements of (cid:21) , which are the time-varying market prices t t of those risks. In contrast to the Fisher hypothesis, which interprets the spread between nominal and real interest rates as an implied in(cid:135)ation expectation, this model explicitly allows for a time-varying in(cid:135)ation risk premium. The in(cid:135)ation risk premium is comprised of term risk (the risk associated with uncertainty about future in(cid:135)ation rates), default risk, and liquidity risk.2 As the markets for nominal and indexed government debt in the UnitedKingdomaredefault-freeandsu¢ cientlyliquid,riskduetouncertaintyaboutfuture in(cid:135)ation rates is likely to be the primary driver of the in(cid:135)ation risk premium. Other factors unrelated to in(cid:135)ation expectations may at times a⁄ect in(cid:135)ation compensation, such as changesindemandforindexeddebtinducedbyinstitutionalreform,taxissuesorconvexity. These are not identi(cid:133)ed separately and to the extent that they do not conform to the estimated dynamics of the instantaneous in(cid:135)ation rate are largely attributed to the risk premium.3 The model permits tractable and nonnegative bond prices. Substituting equations (7), 2Liquidity in theindexed giltmarketishigh by internationalstandards. Sinceitsinception in the early 1980s, the share of gilt issuance that is indexed-linked has risen gradually from 17 percent in (cid:133)scal year 1992-93 to 26 percentin 2005-06 (United Kingdom DebtManagementO¢ ce). Turnoverin the indexed-gilt marketisaboutone(cid:133)fththatinthenominalgiltmarket(reportedinMcGrathandWindle,2006)butgiven thematurityandsizeoftheU.K.market,liquiditypremiaarelikelytobeverysmall. Bywayofcomparison, in the less mature and lower volume U.S. TIPS market, the liquidity premium is estimated to be at most 5 to 10 basis points (D(cid:146)Amico, Kim and Wei, 2006). There is also an indexation-lag risk premium implicit intheyieldonindex-linkedgiltsbutpreviousresearch,namelyRisa(2001),hasfoundthatthispremium is small and varies within a narrow range of just a few basis points. 3The convexity e⁄ect in this type of model is constant over time and varies only by maturity (see Joyce andLildholdt,2006). Whetheritisimputedtoexpectedin(cid:135)ationorriskpremiawillnota⁄ecttimevariation of either of these components. 7

(8), (9) and (10) into the asset-pricing equation (6) yields PIC = exp a +b X ; n;t n 0n t (cid:0) (cid:1) where a and b are functions of the structural parameters of the model: (cid:14) ; (cid:14) ;v ;v ;(cid:8) n n 0 1 0 1 3 and (cid:27)2 . Yields are thus a¢ ne functions of the three latent factors, j j=1 n o 1 1 yIC = ln(PIC) = a +b X ; n;t (cid:0)n n;t (cid:0)n n 0n t (cid:0) (cid:1) as are forward rates, expected future short rates and term premia. The model can be written in state-space form, with measurement equation y = a+B X +(cid:17) t 0 t t wherey t isaq (cid:2) 1vectorofzero-couponyieldsofmaturitiesn 1 ;n 2 :::n q ;a = a n1 ;a n2 :::a nq 0; B is a 3 q matrix, the ith column of which is b and (cid:17) is a vector of mea(cid:2)surement erro(cid:3)rs, (cid:2) ni t assumed to be Gaussian. The transition equation is X = expKX +" t t 1 t (cid:0) where " t (cid:24) N(0; 0 1 eKs(cid:6)(cid:6) 0 eK 0 sds), a discrete time version of equation (10). The model is estimated by maxRimum likelihood and the structural parameters recovered as functions of the reduced-form parameters a and b . It is then straightforward to construct expected n n in(cid:135)ation at any horizon by forecasting the factors and substituting into equation (8). The forward in(cid:135)ation risk premium at a given horizon is then constructed as the (cid:133)tted forward rate less expected in(cid:135)ation. The model can be (cid:133)tted to monthly, weekly or daily data. 8

3 Estimation and Results 3.1 Data Iestimatethemodelusingzero-coupon in(cid:135)ation compensationyieldsprovided bytheBank of England. In(cid:135)ation compensation is calculated as the spread between the nominal and realzero-coupon yield curves, smoothed usingthe Bankof England(cid:146)spreferred spline-based method and, in the case of the real yield curve, adjusted for the indexation lag on indexlinked gilts; see Anderson and Sleath (2001) for details.4 The estimation sample consists of monthly data from October 1992 to May 2007, the sample start date coinciding with the announcement of the Bank of England(cid:146)s (cid:133)rst in(cid:135)ation target. The sample range deliberately captures the period during which the Bank of England actively pursued an in(cid:135)ation-targeting policy with an in(cid:135)ation objective at or close to 2.5 percent RPIX in(cid:135)ation(cid:151)or its expected equivalent for CPI in(cid:135)ation(cid:151)lessening concern about structural change in long-run policy objectives during the estimation period. The shortest-maturity yield available over the whole sample is 4 years, so the model is estimated with yields of maturity 4, 5, 6, 10 and 15 years (data shown in Figure 1). The lack of short-maturity data makes the model better suited to address questions about long-run in(cid:135)ation expectations and risk premia. While one can construct a (cid:133)tted term structure of short-maturity yields and forward rates using the parameters estimated from longer-maturity yields, the (cid:133)t is likely to be worse at short horizons. Once estimated, the parameters describing the term structure can be used to (cid:133)t the model to higher-frequency data, such as daily or weekly, which will prove valuable in analysing the market reaction to changes in Bank of England policy. Principal components analysis of the (cid:133)ve yields in the data set suggests that the data are well described by three factors (accounting for 99.99 percent of the variance) but could be modelled with two (the (cid:133)rst two principal components account for 99.73 percent of the variance). As is commonly found in the literature, the (cid:133)rst principal component seems to proxy for level of the term structure of interest rates, in this case the level of in(cid:135)ation 4The Bank of England(cid:146)s smoothed yield curves are publicly available at http://www.bankofengland.co.uk/statistics/yieldcurve/. 9

Figure 1: U.K. In(cid:135)ation Compensation Yields, 1992 to 2007 Percent 5 year 7 10 year 15 year 6 5 4 3 2 1 1992 1994 1996 1998 2000 2002 2004 2006 Source: Bank of England, www.bankofengland.co.uk/statistics compensation, as the coe¢ cient loadings are fairly constant across maturity and the factor is very persistent. The second and third principal components appear to capture slope and curvature respectively. The three principal components are highly correlated with the respectivemodel-estimatedfactors,oneofwhichissimilarlypersistentandseemstocapture information about the level of in(cid:135)ation. Results of a two-factor model estimated over the same sample period are shown in Figure 8 at the end of the paper, but the speci(cid:133)cation is not able to meet the stylised facts of in(cid:135)ation expectations in the United Kingdom. Speci(cid:133)cally, two factors are insu¢ cient to capture the long-run dynamics of in(cid:135)ation compensation, with a deterioration in root mean square error particularly for long-maturity yields (see column 3 of Table 2 at the end of the paper) and overly-fast mean reversion of the most persistent factor. 3.2 Estimation The three-factor state-space model outlined in Section 2.2 is estimated with maximum likelihood; parameter estimates and (cid:133)t diagnostics are shown in Table 2 at the end of the paper. The model assumes that the latent factors are stationary processes and accordingly, theinstantaneousexpectedfuturein(cid:135)ationraterevertstoitslong-runmean,(cid:26) ,asmaturity 0 10

increases. Risk premia also revert to a stationary mean, as they are functions of the same three latent factors, but the estimated factor loadings result in a much slower rate of mean reversion. When estimated, the most persistent factor has annual persistence of 0.853, equivalent to a half-life of about four and half years. Mean reversion of this factor is su¢ ciently gradual that the model can accommodate much of the reduction in the level of in(cid:135)ation compensation over the sample.5 And because risk premia and expected in(cid:135)ation both place some loading on this factor, they too can be far from their means for long periods. 3.2.1 A question of persistence Estimates of persistence in time-series models are known to be biased downwards in small samples and this bias becomes more pronounced as the stationary series approaches a random walk (see Shaman and Stine, 1988, among others). Given that we have only 15 years of data with which to estimate the dynamics of the factors, it is likely that persistence, especially that of the most persistent factor, is underestimated. Such downward bias imparts too much mean reversion tolong-run in(cid:135)ation expectations and, as a result, too much movement to far-horizon risk premia. This small-sample estimation problem is not unique to the term-structure estimation in this paper. Kim and Orphanides (2005) address the problem by including survey data in their estimation to help pin down the dynamics of the interest-rate process of US nominal interest rates. For the United Kingdom, however, available survey forecasts pertain to RPIX, not RPI in(cid:135)ation, which complicates their use in estimation. An alternative is to seek an econometric solution to the estimation problem and use the survey data ex post as a cross check of the model(cid:146)s results. Inlightofthedownwardbiasproblem,Ire-estimatethemodelrestrictingthepersistence of the most persistent factor to its median unbiased estimate based upon Monte Carlo simulations of a (cid:133)rst-order autoregressive process.6 Given the original estimate of annual 5Indeed it is not possible to reject the null hypothesis that the factor contains a unit root using an augmented Dickey-Fuller test. 6Speci(cid:133)cally, ten thousand simulations are performed of a (cid:133)rst-order, univariate autoregressive process with the sample length equivalent to monthly data from October 1992 to May 2007. The factor being restricted is the (cid:133)rst, which, given the lower triangular nature of the coe¢ cient matrix (cid:20), is orthogonal to 11

Figure 2: Survey and Model-Based Long-Run In(cid:135)ation Expectations Unrestricted Estimation Percent 5 Consensus Economics RPIX inflation expectations 4 3 Model based inflation expectations 2 out of sample 1 1990 1992 1994 1996 1998 2000 2002 2004 2006 Restricted Estimation Percent 5 Consensus Economics RPIX inflation expectations 4 3 Model based inflation expectations 2 out of sample 1 1990 1992 1994 1996 1998 2000 2002 2004 2006 12

persistence, 0.853, the median unbiased estimate is 0.913, which extends the half life of the factor to about seven and a half years. The remainder of the parameters are freely estimated and their estimates and measures of (cid:133)t are shown in column 2 of Table 2 at the end of the paper. Based on the likelihood values of the restricted and unrestricted models, alikelihoodratiotestwithonedegreeoffreedomcannotrejecttheimposedrestriction. Nor is there any noticeable deterioration in (cid:133)t according to the root mean square error statistics in Table 2. Moreover, the restricted speci(cid:133)cation accords better with other macroeconomic evidence. Speci(cid:133)cally, in(cid:135)ation expectations from the restricted model more closely resemble the in(cid:135)ation expectations of professional forecasters. To the extent that professional forecasters share similar expectations to (cid:133)nancial market participants, survey responses are a useful cross check of the model(cid:146)s results. Figure 2 plots average six- to ten-year ahead RPIX in(cid:135)ation expectations reported by Consensus Economics each April and October against the six- to ten-year ahead instantaneous expected RPI in(cid:135)ation rate implied by the unrestricted and restricted speci(cid:133)cations of the model for the same months. The (cid:133)gure also shows the (cid:133)tted values to two out-ofsample years, 1990 and 1991. Although the model pertains to RPI in(cid:135)ation and survey responses to RPIX in(cid:135)ation, at horizons beyond the usual monetary-policy cycle, these forecasts should reasonably converge. Indeed, outcomes of both measures of in(cid:135)ation have averaged 3.3 percent since 1990 (see Figure 3). Both speci(cid:133)cations capture the disin(cid:135)ation of the mid 1990s, but the persistencerestrictedspeci(cid:133)cationdoesabetterjobofmatchingthehighexpectedin(cid:135)ationearlyinthe 1990s. Slower mean reversion also enables the restricted speci(cid:133)cation to track the continued decline in in(cid:135)ation expectations through to 2003 before rising modestly in recent years. The gap between model- and survey-based expectations in 1992 and 1993 likely re(cid:135)ects the prolonged period during which RPI in(cid:135)ation outcomes fell short of RPIX in(cid:135)ation, perhaps creating expectations that such a gap would persist well into the future. Ultimately, the gap between the two measures of in(cid:135)ation was closed by mid-1994 and model and survey expectations converge at about the same time. the other two factors. 13

Figure 3: Retail Price In(cid:135)ation, twelve-month ended change Percent 12 11 10 9 8 RPIX 7 6 5 4 3 RPI 2 1 0 1990 1992 1994 1996 1998 2000 2002 2004 2006 Source: United Kingdom Office of National Statistics. Because of its coherence with survey evidence, the restricted model is preferred and will be the basis of discussion from here on. However, the two speci(cid:133)cations tell qualitatively and, to a large extent, quantitatively similar stories about the evolution of in(cid:135)ation expectations and risk premia, so the choice of model does not detract from the bigger picture. Fitted decompositions from the unrestricted model are shown in Figure 6 at the end of the paper. 3.3 Expected in(cid:135)ation and in(cid:135)ation risk premia during the in(cid:135)ationtargeting years Most of the steep decline in far-forward in(cid:135)ation compensation between late 1992 and 1998 is attributable to a reduction in in(cid:135)ation risk premia. The upper panel of Figure 4 plots (cid:133)tted instantaneous forward in(cid:135)ation compensation eight years ahead, alongside the model(cid:146)s decomposition into expected in(cid:135)ation and risk premia at that horizon. Eight-year ahead in(cid:135)ation expectations match closely the six-to-ten year average shown in Figure 2; forward rates at more maturities are shown in Figure 7 at the end of the paper. During the early years of in(cid:135)ation targeting, far forward rates of in(cid:135)ation compensation declined about 3 percentage points, of which two percentage points are estimated to have 14

Figure 4: Forward rates of in(cid:135)ation compensation, expected in(cid:135)ation and in(cid:135)ation risk premia Decomposition of 8 year instantaneous inflation compensation 6 Forward Rate Expected Inflation Inflation Risk Premium 5 4 3 tn e c re p 2 1 0 1 Jan 93 Jan 95 Jan 97 Jan 99 Jan 01 Jan 03 Jan 05 Jan 07 Decomposition of 2 year instantaneous inflation compensation 6 5 4 tn e c 3 re P 2 1 0 Jan 93 Jan 95 Jan 97 Jan 99 Jan 01 Jan 03 Jan 05 Jan 07 15

been lower risk premia. The compression of risk premia during 1997 stands out, with the sharp one-day move in that year coinciding with the announcement of the Bank of England(cid:146)s operational independence. Through to 1998, expected in(cid:135)ation trended down from 3.5 percent to around 2.5 percent, which as we have already seen, matches closely the movements in private forecasters(cid:146)long-run in(cid:135)ation expectations. After a long period of low and relatively stable far-forward rates, in(cid:135)ation compensation began to trend up from early 2004, a movement largely attributable to rising in(cid:135)ation risk premia. Indeed, by late 2006, risk premia had reached a decade high. Section 4 discusses some of the potential reasons underlying this movement. Turning to the lower panel of Figure 4, which plots in(cid:135)ation compensation and its componentsatthetwo-yearhorizon,twofeaturesstandout. First,moreofthemovementin in(cid:135)ationcompensationisattributedtoexpectedin(cid:135)ation. Thisisintuitivegiventheshorter forecast horizon, where business cycle (cid:135)uctuations are likely to be played out. From late 1992 to early 1998, two-year ahead in(cid:135)ation expectations declined by about 2 percentage points (from just under 5 percent to just under 3 percent), which accords well with the decline in near-term survey forecasts of RPIX in(cid:135)ation over the same period.7 The second notable feature is that short-run in(cid:135)ation risk premia have shown no systematic trend over the 15 years since the introduction of in(cid:135)ation targeting, unlike long-horizon in(cid:135)ation risk premia which trended down noticeably. In terms of their magnitude, long-maturity risk premia fell but it appears that as the Bank of England earned credibility in its commitment to in(cid:135)ation targeting, the greatest reductions in perceptions of risk concerned long-run in(cid:135)ation rather than the near-term outlook. Short-maturitytermpremiahaveattimesbeenquitevariable,withtheepisodebetween January 1999 and January 2001 likely owing to the Minimum Funding Requirement for pensionfundswhichcameintoe⁄ectduringthelate1990s,andthethreatoffurtherreforms. Pension funds, which hold either directly or indirectly about one half of index-linked gilts, sharplyincreasedtheirdemandforindexedgovernmentdebtatdesirablematurities,raising 7While being cautious about comparisons of RPI and RPIX in(cid:135)ation expectations at short forecast horizons, Consensus Economics reports that average two-year ahead RPIX in(cid:135)ation expectations fell from 4.7percentinOctober1992to2.9percentinOctober1997,broadlyinlinewiththemodel(cid:146)s(cid:133)tforexpected RPI in(cid:135)ation. 16

forwardin(cid:135)ationcompensationdramaticallyatsomehorizons(andnarrowingforwardrates at very long horizons). Reassuringly, the model attributes very little of this movement to expected in(cid:135)ation, which accords well with the interpretation of developments at the time (Clews, 2002). 4 Changes in the practice of in(cid:135)ation targeting Numerous researchers have argued that clarity, transparency and commitment by a central banktoitsin(cid:135)ationobjectivesshouldhelptoanchorin(cid:135)ationexpectations. Adjustmentsof the Bank of England(cid:146)s in(cid:135)ation-targeting policies o⁄er natural experiments to assess these claims. The term-structure model described above(cid:151)(cid:133)tted to high-frequency data(cid:151)makes it possible to observe how (cid:133)nancial market participants(cid:146)long-run in(cid:135)ation expectations and perceptions of risk responded to those adjustments. A short appendix at the end of the paper gives the chronology of re(cid:133)nements to the Bank of England(cid:146)s policies, and the following section addresses three major changes that have occurred. 1. Announcement of the in(cid:135)ation target and its early years The Bank of England announced its (cid:133)rst in(cid:135)ation target on 8th October, 1992, a range target between 1 and 4 percent as measured by annual RPIX in(cid:135)ation. Forward rates of in(cid:135)ation compensation (cid:133)ve to ten years ahead declined about 30 basis points over the followingtwodays, 20basispointsofwhichappeartohavebeenareductioninthein(cid:135)ation riskpremium. However,bothin(cid:135)ationexpectationsandriskpremiawerefairlyvolatileearly in the sample and these moves were quickly eroded. Over the following (cid:133)ve years, long-run in(cid:135)ation expectations did drift down gradually from about 3.5 percent to 3 percent, while the in(cid:135)ation risk premium moved in a 1 to 2 percent band (Table 1 reports end-of-year observations and selected averages). On 14th June, 1995, the rede(cid:133)nition of the target to 2.5 percent or less annual RPIX in(cid:135)ation elicited little market reaction on the day, although near-term in(cid:135)ation expectations and risk premia appeared to decline somewhat in subsequent months. 17

Table 1: Far-horizon forward in(cid:135)ation compensation and its components Forward Rate Expected In(cid:135)ation Risk Date In(cid:135)ation Premium End of year: 1992 5.25 3.43 1.82 1993 3.62 2.90 0.72 1994 4.45 3.36 1.09 1995 4.59 3.12 1.48 1996 4.24 3.08 1.16 1997 3.02 2.69 0.33 1998 2.39 2.36 0.03 1999 2.48 2.71 -0.23 2000 2.44 2.43 0.01 2001 2.50 2.43 0.07 2002 2.38 2.33 0.05 2003 2.70 2.58 0.12 2004 2.83 2.55 0.28 2005 2.87 2.54 0.34 2006 3.15 2.65 0.49 31 May 2007 3.24 2.66 0.58 Averages: 1 Oct 1992 to 5 May 1997 4.66 3.27 1.39 6 May 1997 to 9 Dec 2003 2.63 2.52 0.11 10 Dec 2003 to 31 May 2007 2.96 2.60 0.36 Notes: Fittedmodelestimatesofeight-yearaheadinstantaneousin(cid:135)ationcompensationfrom thepersistence-restricted speci(cid:133)cation. Numbersin thetablecorrespond tothoseplotted in the upper panel of Figure 4. 18

2. Independence The announcement on 6th May, 1997 of the granting of operational independence of the BankofEnglandandthecreationoftheMPCwasmetbyasharpdeclineinforwardratesof in(cid:135)ation compensation at both short and long horizons, an event also noted by G(cid:252)rkaynak, Levin, and Swanson (2006) and Geraats, Eij¢ nger, and van der Cruijsen (2006) among others. Five-, eight and ten-year ahead forward rates of in(cid:135)ation compensation declined about35basis pointson thedayof theannouncement, cumulatingtonearly60basis points of decline within another two days. The model allocates about one third to a decline in long-run expected in(cid:135)ation and about 40 basis points to lower in(cid:135)ation risk premia. The sharp decline was never retraced and over the following months risk premia declined even further. Long-run expected in(cid:135)ation stabilised around 2.5 percent by late in the year. The reduction in in(cid:135)ation risk premia during 1997 identi(cid:133)ed by the model appears consistentwithdeclinesinothermeasuresofforecasteruncertainty. Speci(cid:133)cally,thedispersion of point forecasts of in(cid:135)ation across respondents to the Bank of England(cid:146)s Survey of External Forecasters also started to decline from the second quarter of 1997 and leveled out from early 1999 (shown in Boero, Smith and Wallis, 2007).8 The survey question pertains to a much shorter horizon than the model(cid:150)the fourth quarter of the following year(cid:150)but is suggestive that perceptions of in(cid:135)ation risk moderated following the Bank of England(cid:146)s independence. The reaction to the granting of independence has broader implications for monetary policy. Actions that enhance the perceived commitment of a central bank to its in(cid:135)ation goal not only align the public(cid:146)s in(cid:135)ation expectations with the central bank(cid:146)s target, but reduce uncertainty about future in(cid:135)ation outcomes. The U.K. experience also suggests that in(cid:135)ation targeting helps lower long-term interest rates by reducing the in(cid:135)ation risk premia embedded in market rates, and not just by reducing expected in(cid:135)ation, as argued by Geraats et al (2006). The results described above also correlate well with the (cid:133)ndings of Gurkaynak et al (2006) that post-1997, long-horizon U.K. in(cid:135)ation compensation ceased to 8Dispersioncapturestheextentofdisagreementbetweenforecastersandnottheuncertaintycontainedin anindividual(cid:146)soraggregateprobabilitydensityfunctionin(cid:135)ationforecast. However,severalmethodological changesintheBankofEngland(cid:146)sSurveyofExternalForecastersduringthelate1990scomplicateinference about forecaster uncertainty, but do less harm to the calculation of dispersion statistics. 19

Figure 5: Far-horizon in(cid:135)ation risk premia and the switch to the CPI in(cid:135)ation target Percent Percent 0.6 0.6 Switch to CPI target Dec 10 2003 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0.1 0.1 2001 2002 2003 2004 2005 2006 2007 be sensitive to the surprise components of macroeconomic data releases: the reduction in in(cid:135)ation uncertainty seems to have gone hand-in-hand with decreased sensitivity to news. The target was rede(cid:133)ned a month later, on 12th June, 1997, to a symmetric target of 2.5 percent. Much like the earlier rede(cid:133)nition, the change elicited little reaction in the term structure of in(cid:135)ation compensation in the days following the announcement, most likely dwarfed by the changes a month earlier. 3. Rede(cid:133)ning the target from RPIX to CPI in(cid:135)ation Announced in June 2003 and implemented on 10th December 2003, the Bank of England(cid:146)s in(cid:135)ation target was rede(cid:133)ned from 2.5 percent measured by RPIX in(cid:135)ation to 2 percent measured by annual CPI in(cid:135)ation. The wedge between the two re(cid:135)ected the expected long-run di⁄erence between the two measures arising from di⁄erences between their formulae (geometric versus arithmetic weighting), although the likely long-run di⁄erence was understood to be closer to three quarters of a percentage point.9 To the extent that market participants expect CPI and RPIX in(cid:135)ation to covary closely in the future (around their respective means), a CPI in(cid:135)ation target of 2 percent is an 9This was addressed in a speech given by MPC member Steve Nickell on 16 September 2003, as well as in the Bank of England(cid:146)s February 2004 In(cid:135)ation Report. 20

e⁄ective anchor for RPIX in(cid:135)ation and a good long-run guide to RPI in(cid:135)ation.10 However, as pointed out in o¢ cial communications at the time, the formulae are not the only di⁄erence between the two in(cid:135)ation measures, with the inclusion of several house-price related components likely to make the gap between CPI and RPIX in(cid:135)ation volatile. Thus, in the switch to the CPI in(cid:135)ation target, two additional sources of risks to RPI in(cid:135)ation emerged. First, even were both RPIX and CPI in(cid:135)ation expected to return to known, respective means, the central bank had ceased to promise to address persistent deviations of RPIX in(cid:135)ation from its mean. Second, the expected trend rates of growth of CPI and RPIX in(cid:135)ation could drift apart permanently. Compositional di⁄erences could accentuate the formula e⁄ect and long-lived trends in housing-related components could drive a wedge between the two measures, greater than had previously been the case between RPI and RPIX. In other words, the new target for CPI in(cid:135)ation translates into an unclearly-de(cid:133)nedgoalforRPIin(cid:135)ationanduncertaintyaboutthecentralbank(cid:146)swillingness to stabilise retail prices. It is not altogether surprising then that the switch to the new in(cid:135)ation target was accompanied by an increase in long-run in(cid:135)ation risk premia in (cid:133)nancial markets. Indexed debt in the United Kingdom continued to be linked to RPI after the rede(cid:133)nition of the target and the additional risk inherent in that measure of in(cid:135)ation appears to have been embeddedintomarket-basedin(cid:135)ationcompensation. Figure5illustrateshowabruptlythat occurred. Fitted values from the model indicate that in(cid:135)ation risk premia rose sharply in the weeks following the adoption of the CPI in(cid:135)ation target to a level about 20 basis points above the average of earlier years. The model attributes no such sharp increase to in(cid:135)ation expectations around those weeks. In(cid:135)ation risk premia continued to rise through to mid-2004, with concerns about rapid house-priceincreasesspillingintoRPIin(cid:135)ationlikelycontributingtotherise. However,the bulk of the increase in risk premia was not retraced even after concerns about house-price appreciation had alleviated. Beginning late 2005, in(cid:135)ation risk premia rose further, with rising energy prices and high in(cid:135)ation readings during 2005 and 2006 appearing to heighten 10However, such close covariance is unlikely to be the case. The historical correlation (Oct 1992 to Dec 2003)betweenRPIandCPItwelve-monthendedin(cid:135)ationwas0.17. BetweenRPIandRPIXthecorrelation was somewhat higher, 0.37. 21

uncertainty. At short horizons, Consensus Economics survey responses indicate that in mid-2006, average RPIX in(cid:135)ation expectations for the coming year rose above 2.5 percent for the (cid:133)rst time since 1998, and that CPI in(cid:135)ation expectations rose above 2 percent. For longer-horizon questions, survey-based RPIX and CPI in(cid:135)ation expectations remain close to 2.5 and 2 percent respectively. This is consistent with the model evidence that most of the recent rise in RPI-based in(cid:135)ation compensation owes to heightened perceptions of risk concerning long-run in(cid:135)ation outcomes and an increase in the compensation demanded for in(cid:135)ation risk. 5 Conclusion Applying a no-arbitrage, a¢ ne factor model to the term structure of in(cid:135)ation compensation in the United Kingdom provides insights into how the Bank of England(cid:146)s policies have in(cid:135)uenced in(cid:135)ation expectations. To the extent that in(cid:135)ation risk premia mirror perceptions of in(cid:135)ation risk, the model provides a way of assessing not only the level of long-run in(cid:135)ation expectations but also uncertainty about future in(cid:135)ation outcomes. It also has the advantagethatitcanbe(cid:133)ttedtohigher-frequencydatathantraditionalsourcesofin(cid:135)ation expectations, namely surveys, but yields consistent results. The model(cid:146)s (cid:133)tted results provide insights into the e⁄ects of the Bank of England(cid:146)s in(cid:135)ation-targeting policies. Market participants(cid:146)long-run expectations of retail price in- (cid:135)ation trended down from 1992, but not until after the Bank of England was granted independence did expectations approach and remain close to 2.5 percent. In contrast, in(cid:135)ation risk premia showed little sign of declining until 1997, when they dropped abruptly in the months following independence. The abrupt drop suggests that independence signalled more credible commitment by the central bank to its in(cid:135)ation goal, bringing about a reduction in perceptions of long-horizon in(cid:135)ation risk. By comparison, various rede(cid:133)nitions of the target and measures to increase transparency, such as earlier publication of the MPC(cid:146)s minutes or publication of details of the Bank(cid:146)s forecasting model, did not have the same impact or lasting e⁄ect on expectations. The (cid:133)ndings of this paper(cid:150)that term premia were substantially lower following independence(cid:150)are consistent with a long-lived reduction 22

in long nominal term premia post independence found by Bianchi, Mumtaz, and Surico (2008). Indeed, the results presented here suggest that much of the moderation in nominal term premia owed to a moderation in in(cid:135)ation risk. As a (cid:133)nal observation, long-run in(cid:135)ation expectations and in(cid:135)ation risk premia were low and fairly stable until the in(cid:135)ation target was rede(cid:133)ned in December 2003 from RPIX in(cid:135)ationtoCPIin(cid:135)ation. Sincethen,long-runRPIin(cid:135)ationriskpremiahaverisenabout40 basis points, above the range established since 1997. Disbanding the RPIX in(cid:135)ation target has arguably increased the compensation that investors require to hold long-run retail price in(cid:135)ationrisk,asthenewtargetforCPIin(cid:135)ationtranslatesintoanunclearly-de(cid:133)nedgoalfor RPI in(cid:135)ation and uncertainty about the central bank(cid:146)s willingness to stabilise retail prices. This need not re(cid:135)ect upon the Bank of England(cid:146)s credibility as an in(cid:135)ation targeter. To the contrary, survey-based in(cid:135)ation expectations do appear to be well anchored on the new CPI in(cid:135)ation measure and model-based RPI in(cid:135)ation expectations have moved relatively little. However, the experience highlights a more general point. Even in the presence of a general objective for price stability, unclear de(cid:133)nition of and uncertain commitment to an in(cid:135)ation goal is likely to foster heightened perceptions of long-run in(cid:135)ation risk. 23

References Anderson, N., and J. Sleath (2001): (cid:147)New Estimates of the UK Real and Nominal Yield Curves,(cid:148)Bank of England Working Paper, No. 126. Bianchi, F., H. Mumtaz, and P. Surico (2008): (cid:147)The Great Moderation of the Term Structure of U.K. Interest Rates,(cid:148)Unpublished manuscript, Bank of England. Boero, G., J. Smith, and K. Wallis(2007): (cid:147)Uncertaintyanddisagreementineconomic prediction: the Bank of England Survey of External Forecasters,(cid:148)Forthcoming in the Economic Journal. Clews, R. (2002): (cid:147)Asset Prices and In(cid:135)ation,(cid:148) Bank of England Quarterly Bulletin, 42(2), 178(cid:150)185. D(cid:146)Amico, S., D. Kim, and M. Wei (2007): (cid:147)Tips from TIPS: The Informational Content of Treasury In(cid:135)ation-Protected Security Prices,(cid:148) Forthcoming in the FEDS Working Paper Series, Board of Governors of the Federal Reserve. Diebold, F., G. Rudebusch, and S. B. Aruoba (2006): (cid:147)The Macroeconomy and the Yield Curve: A Dynamic Latent Factor Approach,(cid:148)Journal of Econometrics, 131(1-2), 309(cid:150)338. Duffie, G. (2002): (cid:147)Term Premia and Interest Rate Forecasts in A¢ ne Models,(cid:148)Journal of Finance, 57, 405(cid:150)443. Geraats, P., S. Eijffinger, and C. van der Cruijsen (2006): (cid:147)Does Central Bank Transparency Reduce Interest Rates?,(cid:148)CEPR Discussion Paper 5526. G(cid:252)rkaynak, R., A. Levin, and E. Swanson (2006): (cid:147)Does In(cid:135)ation Targeting Anchor Long-Run In(cid:135)ation Expectations? Evidence from Long-Term Bond Yields in the U.S., U.K., and Sweden,(cid:148)Federal Reserve Bank of San Francisco Working Paper 2006-09. Joyce, M., and P. Lildholdt (2006): (cid:147)Understanding the real rate conundrum: an application of no-arbitrage (cid:133)nance models to the UK real yield curve,(cid:148) Unpublished manuscript, Bank of England. Kim, D., and A. Orphanides (2005): (cid:147)Term Structure estimation with survey data on interest rate forecasts,(cid:148)FEDS Working Paper Series No. 2005-48, Board of Governors of the Federal Reserve. McGrath, G., and R. Windle (2006): (cid:147)Recent Developments in Sterling In(cid:135)ation- Linked Markets,(cid:148)Bank of England Quarterly Bulletin, 46(4), 386(cid:150)396. Risa, S. (2001): (cid:147)Nominal and In(cid:135)ation Indexed Yields: Separating Expected In(cid:135)ation and In(cid:135)ation Risk Premia,(cid:148)Unpublished manuscript, Columbia University. Rudebusch, G., and T. Wu (2004): (cid:147)A Macro-Finance Model of the Term Structure, Monetary Policy and the Economy,(cid:148)Federal Reserve Bank of San Francisco Working Paper 2003-17. Shaman, P., and R. Stine (1988): (cid:147)The Bias of Autoregressive Coe¢ cient Estimators,(cid:148) Journal of the American Statistical Association, 83, 842(cid:150)848. 24

Appendix: Key Dates during In(cid:135)ation Targeting at the Bank of England 1992, 8 October First in(cid:135)ation target, de(cid:133)ned as a rate of 1 to 4 percent annual RPIX in(cid:135)ation. 1994, 13 April Minutes of the Chancellor of the Exchequer(cid:146)s meeting with the Governor Bank of England (cid:133)rst published. 1995, 14 June In(cid:135)ation target rede(cid:133)ned as 2:5 percent or less annual RPIX in(cid:135)ation. 1997, 6 May Bank of England granted operational independence. 1997, 12 June In(cid:135)ation target rede(cid:133)ned as a symmetric target around 2.5 percent annual RPIX in(cid:135)ation 1998, 21 October Publication timetable for minutes of MPC meetings shortened to two weeks. 2003, 9 April Intention to switch to an in(cid:135)ation target measured by CPI in(cid:135)ation announced by the Chancellor of the Exchequer. 2003, 10 December Switch to an in(cid:135)ation target of 2 percent annual CPI in(cid:135)ation announced by the Chancellor of the Exchequer. 2005, 31 January Publication of details of the Bank of England Quarterly Model, including equations, data sources and parameter values. Source: Bank of England, http://www.bankofengland.co.uk/ 25

Table 2: Parameter Values and Fit Diagnostics for Model Speci(cid:133)cations Three-factor model Two-factor model End of year Unrestricted Restricted Unrestricted (cid:20) 2.0695 0.0913* 0.2313 1;1 (cid:20) -0.4172 -0.7290 -2.8732 2;1 (cid:20) -0.3318 -0.1026 - 3;1 (cid:20) 0.5990 1.0481 2.3693 2;2 (cid:20) 0.3479 -0.1393 - 3;2 (cid:20) 0.1587 1.3183 - 3;3 (cid:27) 0.0126 0.0026 0.0009 1 (cid:27) 0.0039 0.0138 0.0154 2 (cid:27) 0.0051 0.0010 - 3 (cid:26) 0.0290 0.0299 0.0267 0 (cid:21) -0.3895 -0.1988 -0.1320 1 (cid:21) 0.1171 -0.3538 -0.4484 2 (cid:21) -0.1494 0.1780 - 3 Largest annual persistence 0.853 0.913 0.793 Log-likelihood -37.1730 -37.1710 -32.3345 RMSE for zero-coupon yields basis points 5 year 1.9889 1.9888 1.9144 10 year 1.1477 1.1471 14.1819 15 year 3.6798 3.6848 32.1835 * This parameter restricted to the median unbiassed estimate of the most persistent factor from the unrestricted speci(cid:133)cation. 26

Figure 6: Fitted forward rates and components from unrestricted estimation of the threefactor model. 7 7 0 0 4 4 0 0 4 4 0 0 0 0 2 2 e ta r s u o e n a tn a ts n i ra e y 5 7 7 0 0 8 6 0 0 0 0 0 2 e ta r s u o e n a tn a ts n i ra e y 5 1 7 7 0 0 8 6 0 0 0 0 0 2 6 6 9 9 9 9 1 1 7 7 0 0 0 0 1 1 6 5 4 3 2 1 0 1 2 9 6 5 4 3 2 1 0 1 2 2 9 9 9 tnecrep 1 tnecrep 1 7 7 0 0 4 4 0 0 4 4 0 0 0 0 2 2 e ta r s u o e n a tn a ts n i ra e y 2 7 7 0 0 8 6 0 0 0 0 0 2 e ta r s u o e n a tn a ts n i ra e y 0 1 7 7 0 0 8 6 0 0 0 0 0 2 6 6 9 9 9 9 1 1 7 7 0 0 0 0 1 1 6 5 4 3 2 1 0 1 2 9 6 5 4 3 2 1 0 1 2 2 9 9 9 tnecrep 1 tnecrep 1 27

Figure 7: Fitted forward rates and components from persistence-restricted speci(cid:133)cation of the three-factor model. 7 7 0 0 4 4 0 0 4 4 0 0 0 0 e ta r s u o e n a tn a ts n i r a e y 5 7 7 0 0 8 6 0 0 0 0 0 2 2 e ta r s u o e n a tn a ts n i r a e y 5 1 7 7 0 0 8 6 0 0 0 0 0 2 2 6 6 9 9 9 9 1 1 7 7 0 0 0 0 1 1 2 2 6 5 4 3 2 1 0 19 6 5 4 3 2 1 0 1 29 9 9 tnecrep 1 tnecrep 1 7 7 0 0 4 4 0 0 4 4 0 0 0 0 e ta r s u o e n a tn a ts n i r a e y 2 7 7 0 0 8 6 0 0 0 0 0 2 2 e ta r s u o e n a tn a ts n i r a e y 0 1 7 7 0 0 8 6 0 0 0 0 0 2 2 6 6 9 9 9 9 1 1 7 7 0 0 0 0 1 1 2 2 6 5 4 3 2 1 0 19 6 5 4 3 2 1 0 1 29 9 9 tnecrep 1 tnecrep 1 28

Figure8: Fittedforwardratesandcomponentsfromunrestrictedestimationofatwo-factor model. 7 7 0 0 4 4 0 0 4 4 0 0 0 0 e ta r s u o e n a tn a ts n i r a e y 5 7 7 0 0 8 6 0 0 0 0 0 2 2 e ta r s u o e n a tn a ts n i r a e y 5 1 7 7 0 0 8 6 0 0 0 0 0 2 2 6 6 9 9 9 9 1 1 7 7 0 0 0 0 1 1 2 2 6 5 4 3 2 1 0 19 6 5 4 3 2 1 0 1 29 9 9 tnecrep 1 tnecrep 1 7 7 0 0 4 4 0 0 4 4 0 0 0 0 e ta r s u o e n a tn a ts n i r a e y 2 7 7 0 0 8 6 0 0 0 0 0 2 2 e ta r s u o e n a tn a ts n i r a e y 0 1 7 7 0 0 8 6 0 0 0 0 0 2 2 6 6 9 9 9 9 1 1 7 7 0 0 0 0 1 1 2 2 6 5 4 3 2 1 0 19 6 5 4 3 2 1 0 1 29 9 9 tnecrep 1 tnecrep 1 29