The Conquest of U.S. Inflation: Its Implications for the FisherHypothesis and the Term Structure of Nominal Interest Rates

Shingo Goto†and Walter Torous‡

November 2003

ABSTRACT

A regime shift in the inflation process is the collective outcome of shifts in individual agents’rational expectations resulting from an important policy initiative. Taking this view, we finddecisive evidence of a shift in the inflation process during the Volcker experiment reflectingthe Federal Reserve’s enforcement of an anti-inflationary interest rate policy consistent withthe Taylor (1993) rule. Subsequent to the regime shift, inflation and real interest rates movetogether, especially in the long run, contradicting both the Fisher hypothesis and the presence ofMundell-Tobin e ects. Contrary to conventional wisdom, today’s yield curve is not informativeabout expected inflation and yields now command a significant inflation risk premium suggestingyet another violation of the Fisher hypothesis.

Keywords: Fisher Hypothesis; term structure; inflation risk premium; monetary policy; regime shifts

JEL classification: E43; E44; E47; G10

This is a significant revision of our earlier manuscript, “Evolving Inflation Dynamics, Monetary Policy, and TheFisher Hypothesis.” We would like to thank Cli Ball, Anton Braun, Chip Chappell, Wayne Ferson, Scott Harrington,Steve Mann, Ted Moore, Lee Ohanian, Richard Roll and Rossen Valkanov for helpful comments and suggestions onthe earlier draft. All errors are our own.

†The Moore School of Business, University of South Carolina, 1705 College Street, Columbia, SC 29208. Phone:803-777-4927. Email: shingo.goto@moore.sc.edu.

‡The Anderson School at UCLA. 110 Westwood Plaza, Los Angeles, CA 90095-1481. Phone: 310-825-4059. Email:walter.n.torous@anderson.ucla.edu.

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1 Introduction

Since the Federal Reserve Board’s “Volcker experiment” from October 1979 to October 1982, infla-tion in the U.S. has been stabilized. The “conquest” of U.S. inflation (Sargent (1999)) in no smallpart reflects the fact that “the changing economic theories and opinions about inflation are theultimate cause of the changes in actual inflation” (Taylor (1997)). Under Chairman Volcker, theFed signaled a new direction for its interest rate policy consistent with the so-called Taylor (1993)rule in which real rates are raised in an attempt to suppress any increase in inflation expectations.

Research investigating U.S. inflation dynamics and causes of the recent stabilization of inflationrelies on the Lucas (1976) critique to recognize that inflation is an aggregate outcome of the in-dividual decisions of forward looking rational agents. In other words, to the extent that the Fedpolicy shift was unanticipated and credible, it had important consequences for the decision rulesof optimizing agents and hence collectively for the dynamic behavior of inflation. Cogley and Sar-gent (2002, 2003), for example, explicitly assume that the inflation process evolves over time withgradual, albeit unexpected, shifts in monetary policy. An alternative view, adapted in this paper,is that the policy shift was abrupt as opposed to gradual. This is a viable alternative in light ofthe overwhelming empirical evidence consistent with a significant structural shift in the inflationprocess in the early 1980s1 which, as argued by Judd and Rudebusch (1998), Taylor (1999), andClarida, Gali, and Gertler (2000), among others, reflects the Fed’s credible enforcement of an anti-inflationary interest rate policy that e ectively shifted the behavior of optimizing agents and hencecollectively inflation dynamics.

The aggressive anti-inflationary interest rate policy embodied in the Taylor rule implies that nom-inal short term interest rates will move more than one-for-one with expected inflation resulting ina positive relation between expected inflation and real rates. Prima facie, this violates the Fisherhypothesis which posits independence between expected inflation and real rates. It is also incon-sistent with the presence of so called Mundell-Tobin e ects, that is, the negative e ects of inflationexpectations on real rates (e.g., Mundell (1963), Tobin (1965), and Fama and Gibbons (1982)).This positive relation between expected inflation and real rates implies, at least in the long run,that less than one dollar needs to be invested in short term Treasury bills to protect the dollar’spurchasing power, making bills e ective hedges against inflation, contrary to the conclusions ofFama and Gibbons (1982) and others. By contrast, an aggressive interest rate rule increases theuncertainty surrounding real rates and may make long term investors view bills as riskier invest-ments than bonds (Campbell and Viceira (2001) and Cochrane (2001)). The interaction of thesee ects can potentially have important implications for the relative demand for long term bondsversus short term bills and hence on the term structure of interest rates.2

1See, for example, Evans (1991), Kim (1993), Evans and Wachtel (1993), and Garcia and Perron (1996)).2 In this paper we focus on nominal Treasury bonds, as opposed to inflation linked bonds. For a related study

making use of U.K. inflation linked bonds, see Evans (1998). U.S. inflation linked bonds (Treasury Inflation ProtectedSecurities or TIPs) have been issued since January 1997. Roll (2003) o ers a comprehensive analysis of TIPs.

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Assuming that bills e ciently incorporate inflation expectations, in this paper we ask whetherregime shifts in expected inflation were incorporated in bill yields and, furthermore, if such shiftshad important e ects on the term structure of interest rates. Establishing the potential significanceof regime shifts in rational forecasts of inflation, however, necessitates us to explicitly account for thee ects of stochastic volatility. In particular, as Sims (2002) and Stock (2002) argue in their critiquesof Cogley and Sargent (2002), time variation in estimated vector autoregression (VAR) coe cientsmay simply reflect the time varying nature of the volatility of underlying shocks.3 Therefore, weextend the inflation forecasting model of Fama (1975) by positing stochastic volatility for inflationshocks and interest rate shocks, while also explicitly accommodating the possibility of regime shiftsin its parameters. While this inflation forecasting model will be relied upon to approximate thepossibly regime dependent rational expectations and conditional variances of inflation and interestrates, estimation risk is potentially large because underlying regimes, stochastic volatilities, andagents’ underlying expectations are unobservable. Nevertheless, we are able to account for the ef-fects of any resultant parameter uncertainty by implementing a likelihood function based estimationmethodology that is not conditioned on particular parameter estimates.

After accounting for the e ects of stochastic volatility, we indeed find decisive evidence of a regimeshift in the dynamic relation between inflation and interest rates (the “Fisher relation”) during theVolcker experiment. Subsequent to this regime shift, a significant positive relation prevails betweenexpected inflation and real rates, contradicting both the Fisher hypothesis as well as the presenceof Mundell-Tobin e ects. This positive correlation subsequent to the regime shift is supported outof sample by Roll’s (2003) analysis of TIP securities over the 1997 to 2002 sample period. Also,consistent with the results of Clarida, Gali, and Gertler (2000) and Cogley and Sargent (2002,2003), the persistence of the inflation process has decreased after the regime shift, resulting in asubstantial reduction in long term inflation uncertainty and making long term bonds less riskyfor long term investors. While nominal interest rates closely track expected inflation prior to theregime shift, for example, Campbell and Ammer (1993) argue that changes in inflation expectationsare the main drivers of long term interest rates through the early 1980s, this is no longer the caseafter the regime shift. In particular, it is real rate expectations that now play an important rolein determining long term interest rates. In other words, the conventional wisdom that long terminterest rates primarily reflect inflation expectations (e.g., Fama (1990), Mishkin (1990a,b), andCampbell and Ammer (1993)) is no longer supported by the data subsequent to the regime shift.

We also find that yield risk premia have increased subsequent to the regime shift and are nowsignificantly positive. While the increase in the yield risk premia during the early 1980s has beendocumented previously by Mankiw (1986), little is known about its origin. We present evidence

3To see this, consider a stationary VAR in companion form:

+1 = + ˜ +1 ˜ +1 (0 1)

Time varying heteroscedasticity is captured by time variation of Assuming the stability of the VAR, we canrewrite it as

+1 = ( ) 1 ˜ +1

where is the lag operator. If is assumed constant when is in fact time varying, we would find time variationin the VAR coe cients even if is constant. This makes distinguishing time varying VAR coe cients ( ) fromtime varying heteroscedasticity ( ) di cult.

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that the increase in yield risk premia reflects a substantial increase in the term structure’s inflationrisk premium. Our estimates of the inflation risk premium di er from the corresponding estimatesof Campbell and Shiller (1996), Ang and Bekaert (2003) and Buraschi and Jiltsov (2003). Thisprevious research did not uncover the shift in the Fisher relation and, consequently, their estimatesof the inflation risk premium are larger than ours before the regime shift but lower subsequently.

To better understand this result, recall that under the Fisher hypothesis, the purchasing power ofmoney is independent of technology as well as investors’ preferences and, as a result, nominal interestrates do not command an inflation risk premium. However, our evidence suggests that inflationnow covaries with investors’ marginal utility, and hence nominal bonds must yield an inflation riskpremium to induce substitution away from consumption. This conclusion is independently verifiedby evidence of a statistically reliable increase in the mean excess returns to inflation trackingportfolios. That is, portfolios of stocks and bonds designed to track changes in long term inflationexpectations earn positive mean excess returns subsequent but not prior to the regime shift.

The “conquest” of U.S. inflation has therefore been achieved at the expense of increasing theinflation risk premium and so increasing the costs of issuing nominal debt. The increase in theinflation risk premium after the regime shift can be intuitively linked to the Fed’s aggressive anti-inflationary interest rate policy in a number of ways. For example, when the Fed is crediblyseen as controlling inflation, investors believe that any increase in inflation expectations will beassociated with an increase in real yields. As the Fed’s systematic response to an increase in inflationexpectations exerts contractionary real e ects and so increases the marginal rate of substitution,nominal bonds must yield a higher premium to induce demand. Alternatively, since the regime shiftresulting from the Fed’s systematic anti-inflationary policy, inflation has been driven primarily bymoney demand shocks, rather than money supply shocks. These shocks increase not only the levelof expected inflation but also its conditional volatility because of the Friedman (1977) e ect, that isthe positive relation between the level and the volatility of inflation. Given a positive market priceof inflation risk, the resultant increase in volatility, in turn, leads to an increase in the inflationrisk premium resulting in a higher real cost of nominal debt. Of course, these two interpretationscomplement each other as in both cases the increase in the inflation risk premium can be ultimatelytraced to the Fed’s enforcement of a credible anti-inflationary policy.

The plan of this paper is as follows. We detail our data in Section 2 while Section 3 puts forwardour empirical specification and methodology. Section 4 provides evidence of the regime shift in theFisher relation in the early 1980s and corresponding shifts in the dynamic properties of inflationand interest rates. Section 5 investigates the implications of this shift in Fed policy on the termstructure of interest rates, with particular attention being paid to its e ects on the inflation riskpremium in nominal interest rates. Section 6 concludes.

2 Data

Let and denote the monthly inflation rate and the nominal short rate, respectively. We definethe inflation rate by (ln ln 1) where denotes the level of the consumer price

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index for urban consumers, seasonally unadjusted and retrospectively corrected for owner-occupiedhousing costs using the rental-equivalence methodology.4 Our sample period is from January 1967to December 2002. The nominal short rate is given by the one month Treasury bill spot rate(continuously compounded) as of the last business day of each month. Interest rate data, includingthe one month spot rate and discount yields on Treasury bonds of longer maturities, are taken fromMcCulloch and Kwon (1993) for the period January 1967 to February 1991, and are provided byRobert Bliss for the period March 1991 to December 2002.5

Table 1 provides summary statistics for the and series. The average annualized inflation rateover the sample period is 4 50 percent while the average annualized short rate is 6 06 percent.Both series are skewed and leptokurtic relative to the normal distribution and are highly persistentas evidenced by their twelfth-order autocorrelation coe cients, 0 50 for inflation and 0 66 for theshort rate. Augmented Dickey Fuller (ADF) and Phillips-Perron (PP) tests do not reject thenull hypothesis of a unit root for the nominal short rate series but yield mixed evidence for theinflation series. These results should be interpreted cautiously since these tests tend not to rejectthe null hypothesis in the presence of heteroscedasticity and possible regime shifts. Our subsequentempirical methodology, however, does not depend on whether these underlying data are stationary.

3 The Dynamic Relation between Inflation and Interest Rates

3.1 Reduced-form Dynamics

We let [ ]0 and express the joint dynamics of the monthly inflation rate and the nominalshort rate using the following reduced-form vector autoregression (VAR):

+1 = + ( ) + ˜ +1 (1)

where ˜ +1 = [˜ +1 ˜ +1]0 is a vector of innovations, and ( ) =

P=1

1 is a poly-nomial in the lag operator with maximal lag length .6 We allow for possible regime shiftsin the model parameters. Regime dependence is indicated by the subscript in expression (1).

4Before the Bureau of Labor Statistics (BLS) adopted the rental-equivalence measure in January 1983, mortgagepayments were used to measure owner-occupied housing costs. Because of this, the conventional CPI inflation ratebefore 1983 was overly sensitive to interest rates, and generally overstated the true cost of living. The BLS recomputedthe CPI on a post-1983 basis from January 1967 to December 1982 and we use this CPI-X series from January 1967to December 1982 and the conventional CPI series from January 1983 onwards. We obtain the BLS’s CPI-X seriesfrom Bidarkota and McCulloch (1998). We also conducted our analysis using alternative measures of inflation, suchas the chained price index of the personal consumption expenditure and the CPI less food and energy (the “core”CPI). These alternative measures did not change our conclusions.

5We thank Robert Bliss for providing his programs and data. Bliss’s data improve upon the cubic-spline procedureof McCulloch and Kwon (1993). See Bliss (1997) for details. The McCulloch-Kwon data are adjusted for the di erencebetween the ordinary income tax rate and capital gains tax rate while Bliss’s data are not. This, however, does notpresent a significant problem since the tax adjustments are extremely small or even zero after the 1986 Tax ReformAct (Green and Odegaard (1997)).

6A variety of other economic variables have been suggested as being important in measuring the market’s inflationexpectations. Fama (1990) and Mishkin (1990a,b), for example, argue that the slope of the term structure contains

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The regime indicator sequence =1 is assumed to be characterized as an independent Bernoullisequence following a two state hidden Markov chain.

Markov regime switching models provide a convenient and tractable means of characterizing manydata generating processes in finance and economics. We appeal to the Lucas (1976) critique inmotivating our use of this specification. As argued by Lucas, inflation is an outcome of manyforward looking agents’ decisions that depend on their expectations of events a ecting their budgetsets.7 Consequently, a shift in the inflation process, if any, must result from an unanticipated event,like a change in monetary policy, that alters agents’ expected future budget sets and hence shiftscurrent decision making. Only unexpected regime shifts in the inflation process can materialize andthe probability of such a regime shift cannot have been incorporated into forward looking agents’rational expectations.8

Notice that there is a timing di erence between and since the nominal short rate is observedat the end of month while the inflation rate then reflects consumer price data collected overmonth . Any factors a ecting will consequently be known to bill market participants and soincorporates all the information a ecting . Therefore, interest rate innovations ˜ +1 contain

both a response to inflation shocks as well as to shocks to the nominal short rate itself, enabling usto express the forecast error vector ˜ +1 as:

˜ +1 =

·1 0

1

¸˜ +1 ˜ +1

·˜ +1

˜ +1

¸(2)

where ˜ +1 and ˜ +1 represent the assumed mutually independent inflation and interest rateshocks, respectively. The recursive identification explicit in (2) reflects the fact that the bill marketincorporates all relevant information about inflation.9 We interpret ˜ +1 as a combination of

useful information about future inflation. As noted by Stock and Watson (1999), however, term structure forecastsof inflation are not stable while Phillips curve related variables forecast inflation well. Hence real economic variablessuch as the unemployment rate (Cogley and Sargent (2002, 2003)), capital utilization rates (Kozicki and Tinsley(2001)) and help-wanted advertising indices (Ang and Piazzesi (2003)) could be incorporated to better measure themarket’s inflation expectations. But the usefulness of the Phillips curve as a forecaster of future inflation is notwithout criticism. For example, Stock (2002) argues that the reduced form Phillips curve linking inflation withlagged inflation and a measure of real economic activity has been highly unstable while Atkeson and Ohanian (2001)show that the Phillips curve relation does not forecast inflation out of sample. While our bivariate VAR specificationadmittedly provides a simplistic characterization of the market’s expectations of inflation and interest rates, we rely onexpression (1) because it o ers a direct interpretation of the Fisher relation. Furthermore, our econometric analysis,detailed below, does not depend on particular parameter estimates and hence is not susceptible to potential omittedvariable problems.

7See, for example, Gali and Gertler (1999) for a recent example of a forward looking model of inflation. Existingmonetary asset pricing models (e.g., Bakshi and Chen (1996) and Buraschi and Jiltsov (2003)) do not incorporatethe price setting behavior of agents nor do they account for the feedback of the inflation process to shifts in monetarypolicy.

8By contrast, Evans and Lewis (1995) characterize agents’ rational expectations by a regime switching processto illustrate a possible “peso problem” in which regime switching probabilities are explicitly accounted for (see alsoAng and Bekaert (2003)). Nevertheless, both Evans and Lewis’s and our regime switching models are observationallyequivalent reduced form models and neither o ers a structural interpretation of inflation dynamics.

9Koustas and Serletis (1999) use quarterly business day averages of interest rate in their VAR. In this case,disentangling the e ects of interest rate and inflation shocks from the observed innovations is not straightforward.Koustas and Serletis demonstrate that the choice of a particular identifying assumption has an important e ect onthe interpretation of their results.

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various exogenous shocks to which investors and the monetary authority respond to.10

3.2 Stochastic Volatility of Shocks

It is also important to accurately characterize the second moments of and , especially givenour subsequent investigation of the inflation risk premium (Boudoukh (1993)). The importance ofstochastic volatility in short rate dynamics has been well documented, both theoretically as well asempirically. Furthermore, Sims (2002) and Stock (2002) argue that most of the important changesin inflation dynamics are associated with changes in the variance of inflation shocks, and that timevarying variances are di cult to distinguish from changes in the persistence of inflation reportedby Cogley and Sargent (2002). Therefore, the potential significance of regime shifts in the inflationprocess can only be established after accounting for the e ects of stochastic volatility.

Consequently, our empirical analysis of the Fisher relation explicitly incorporates the stochasticvolatility of inflation and interest rate shocks:

˜ +1 ¯ exp

µ1

2+1

¶˜ +1; ˜ +1 i.i.d. (0 1) (3)

˜ +1 exp

µ1

2+1

¶˜ +1; ˜ +1 i.i.d. (0 1) (4)

where =1 and =1 are assumed characterized by latent, regime-dependent autoregressiveprocesses:

+1 = + ( ) + ˜ +1; ˜ +1 (0 1) (5)

+1 = + ( ) + ˜ +1; ˜ +1 (0 1) (6)

We assume that the stochastic volatility processes are stationary with and representingtheir respective long run, possibly regime dependent, means as in So, Lam, and Li (1998). Assuggested by Evans and Wachtel (1992), a policy shift can potentially alter the susceptibility ofinflation and interest rates to structural disturbances through its e ects on individual agents’ pricingdecisions. This can result in a long run shift in the level of these volatilities as captured by theregime dependence of and . Also measures the elasticity of inflation volatility to thelevel of inflation, ¯.11 The elasticity of interest rate volatility with respect to the level of thenominal short rate is measured by .

10For example, Evans and Marshall (1998, 2002) argue that a majority of interest rate shocks are attributable tosystematic responses of the Fed to underlying economic shocks. We do not associate ˜ +1 with “monetary policyshocks”. Doing so would require a more careful discussion of identification, as in Cochrane and Piazzesi (2002).11Since inflation often takes negative values, we use the nominal short rate to proxy for the level of inflation, i.e.,

¯ = . We have also used moving averages and exponentially smoothed measures of inflation. Our results, however,are insensitive to di erent proxies of ¯.

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3.3 The Fisher Relation

For expositional convenience, we write the elements of ( = 1 2 ) as

=

" | || |

#

With this notation, expression (1) implies the following equation that generalizes Fama’s (1975)inflation forecasting equation (suppressing the constant term):12

[ +1] =

P=1

|

1P

=1| (7)

Fama argues that the e ciency of the bill market together with the constancy of the real interestrate imply that [ +1] = ; that is, |

1 = 1, |= 0 for = 2 3 and |

= 0 for= 1 2 in expression (7).13 Because of measurement errors and high frequency variation in

the real rate,14 however, we do not expect an exact one-for-one correspondence between expectedinflation and the nominal short rate on a period by period basis. Alternatively, the Fisher hypothesiscan be formulated as a one-for-one correspondence between observed inflation and nominal interestrates in the long run. In fact, Mishkin (1992) argues that the Fisher hypothesis is only a long runphenomenon and may very well be time period dependent.15

Building upon Mishkin’s insight, we define and measure the Fisher relation by the possibly regimedependent long run sensitivity of changes in inflation expectations to changes in nominal short ratesat frequency zero:

=

P=1

|

1P

=1| (8)

The Fisher hypothesis implies a long run one-for-one correspondence between inflation and interestrates, and hence = 1 for each in the absence of income taxes.16 Mundell-Tobin e ects suggestthat nominal short rates move less than one-for-one with expected inflation and so 1. We canalso interpret as the amount to be invested in short term bills and rolled over to hedge againstthe depreciation of the purchasing power of a unit of currency (dollar). Intuitively, 1 impliesthat more than a dollar must be invested in short term bills to protect the dollar’s purchasing power12Since our regime switching specification is motivated by the Lucas critique, agents’ conditional expectations

and conditional variances depend on the particular policy regime at the time of the forecast. However, for ease ofexposition, we suppress the regime dependence of the conditional moments unless necessary.13 In fact, Fama’s conditions are too restrictive for testing the Fisher hypothesis. First, the constancy of the real

rate is more stringent than assuming the independence between the real rate and inflation. Secondly, the CPI andother measures of inflation are noisy proxies for the true cost of living.14For example, Evans and Wachtel (1992) observe that high frequency variation in nominal short rates is mostly

due to real rate variation and only “sustained variation” in the nominal short rate is indicative of expected inflation.15Barsky (1987) also argues that the estimated Fisher relation depends on the time series property of inflation

which may be time period dependent.16The tax adjusted Fisher hypothesis implies = 1 where is the marginal income tax rate (e.g., Crowder

and Ho man (1996)).

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over the long run. Consequently, in this case short term bills cannot provide complete protectionagainst inflation even in the long run.

Our measure of the Fisher relation, is more general than the cointegration based measure thathas been widely used in recent studies (e.g., Mishkin (1992), Evans and Lewis (1995), and Crowderand Ho man (1996)). While specifies the Fisher relation at frequency zero, that is, the relationbetween changes in inflation and changes in nominal short rates following a “permanent” shock,it also provides a meaningful interpretation even when the system (1) is stable and all shocks tothe system are transitory. To see this, notice that the stability of the system implies the followingrelation between the two shocks and changes in long run expectations of inflation and the shortrate:

( +1 )

ÃX=1

+

!= ˜ +1 (9)

where

=1 1

P=1

|+

P=1

| P=1

|P=1

|+

³1

P=1

| ´1

P=1

|

with =³1

P=1

| ´³1

P=1

| ´ ³P=1

| ´³P=1

| ´6= 0 Defining ( )

(1 2) as the ( )-th element of we see that =(1 2) (2 2) where (1 2) and (2 2)

are the long run responses of inflation and interest rate expectations, respectively, to interest rateshocks, ˜ +1. For example, suppose that a certain economic factor increases the nominal short

rate by one percent. Expression (9) implies that it shifts expected inflation by (1 2) percent andshifts expected interest rates by (2 2) percent.17

When and are cointegrated, the rank deficiency of (1) i.e., = 0 implies=

(1 2) (2 2)=

(1 1) (2 1), where gives the cointegrating parameter. In this case,can be used directly to test the Fisher hypothesis, as has been done by, among others, Mishkin(1992), Evans and Lewis (1995), and Crowder and Ho man (1996). To see this, consider the errorcorrection representation of our VAR, expression (1):

4 +1 = + +

1X=1

4 +1 + ˜ +1 (10)

where =P

= +1 = 1 2 1 +P

=1 and 4 (1 ) denotesthe first di erence operator. Given this representation, the error correction term is given by

=

" P=1

|1P

=|

#( )

17Koustas and Serletis (1999) define the Fisher relation from the long run responses of inflation and interest ratefollowing an inflation shock, assuming that and are di erence stationary and not cointegrated. This approachcorresponds to using =

(1 1) (2 1) instead of in our context. However, focusing on the e ects of interest rateshocks is more desirable than focusing on the e ects of inflation shocks, because inflation shocks are likely to containmeasurement errors. Rational investors react only to “true shocks” to their cost of living, not to measurement errors.For example, to the extent that measurement errors are large relative to the true shocks, (2 1) tends to underestimatethe true e ects of inflation shock on interest rate expectations.

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The stationarity of 4 implies the stationarity of and hence a long run equilibriumrelation between inflation and the nominal short rate.

3.4 Estimation Methodology and Prior Elicitation

We use our VAR model to approximate agents’ rational expectations and conditional variancesas well as to capture the possibility of regime shifts therein. Unfortunately, the econometriciandoes not observe these conditional moments and latent regimes. That being the case, we treat theparameters and latent variables as random variables that must be inferred after observing the data.To the extent that our model may be a poor approximation, these parameters will be impreciselyestimated. But the e ects of this parameter uncertainty on our conclusions regarding the Fisherhypothesis and other asset pricing implications can be explicitly accounted for.

To do so, we implement a Bayesian estimation methodology based on the likelihood function thatis not conditioned on particular parameter estimates. In particular, we treat the observed data asfixed and derive joint densities of the model parameters and latent state variables conditional onthe data. In this framework, we probabilistically describe parameters of interest by appropriatelyintegrating the likelihood function. For example, confidence regions for functions of parameters,impulse response functions, conditional moments, and the inflation risk premium, all depend onlatent stochastic volatilities and hidden Markov regimes as well as other nuisance parameters.However, as demonstrated by Sims and Zha (1999), a wide range of values of these latent variablesand nuisance parameters can yield similar high likelihood values while producing vastly di erentconfidence regions. In this situation, asymptotic confidence regions tend to lead to poor inferencesin finite samples since they rely on a particular set of estimated values of nuisance parameters andhence fail to account for the shape of the likelihood function. So to provide accurate representationsof the data, we report posterior probability intervals (“posterior bands”) by integrating over thenuisance parameters. These integrations are carried out by Monte Carlo simulation using the Gibbssampler.18

To implement this Bayesian approach, however, requires us to specify our prior beliefs. While wewill be more explicit about our priors concerning the stationarity of the inflation and short rateprocesses, that is, the elements of , the priors for the other model parameters are standardand are summarized in Table 2.19

Regarding the order of integration of the inflation and short rate processes, we incorporate thefollowing prior beliefs.20 The main prior belief that we impose is that there exists a long runequilibrium relation between inflation and interest rates within each regime, and hence at least

18Our Gibbs sampler is built upon Bayesian non-Gaussian filtering algorithms developed by, among others, Kim,Shephard, and Chib (1998), Mahieu and Schotman (1998), and So, Lam, and Li (1998). An appendix, which detailsour estimation procedures, is available upon request.19Our subsequent analyses use a lag length of = 3 months ( = 1 2 3) for which the model’s marginal likelihood

is highest.20This prior is motivated by the analysis of De Jong (1992).

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one stationary linear combination of and exists within each regime. This implies that oursystem (1) contains at most one unit root in each regime and so a stable relation prevails betweeninflation and the nominal short rate within a regime.21 While a flat or non-informative prior hasfrequently been used, perhaps to facilitate communication between researchers, Phillips (1991)argues that a flat prior does not necessarily reflect ignorance regarding integration and that in factit actually favors stationarity. Therefore, rather than pretending to be ignorant about the timeseries properties of inflation and short rates, we base our priors on previous research.

Specifically, after orthogonalizing the error terms of our VAR using (2) we use Bayesian regressiontechniques to separately estimate our inflation and short rate specifications relying on a truncatedmultivariate normal prior for the corresponding regression coe cients.22 The resultant estimatedregression coe cients are used to compute ( = 1 2 3) which determine the stability of ourVAR system in each regime. At the mean of the normal density prior to truncation, we assumethat and are cointegrated with common stochastic trend given by the nominal short rateprocess itself. This implies that the nominal short rate follows an independent unit root processand hence ( +1) = . This assumption is consistent with the known behavior of nominal shortrates since unit root tests do not reject the null hypothesis of a unit root in nominal short ratesand since previous studies often characterize the nominal short rate as being exogenous. It alsoimplies a long run one-for-one correspondence between ( +1) and at the mean of the normaldensity. With this choice, the prior probability of having a stationary VAR in inflation and shortrates is approximately six percent, implying a prior probability of a cointegration between andof approximately ninety-four percent (see Panel B of Table 2). Given the widespread reliance on

cointegration to characterize the Fisher relation, our prior is broadly consistent with the beliefs ofmost economists regarding the time series properties of inflation and short rates.23

4 Empirical Evidence

4.1 Regime Shifts in Inflation and the Fisher Relation

We begin our empirical analysis by investigating whether we have evidence of a permanent shiftin the inflation forecasting relation (7) after formally acknowledging stochastic inflation volatility.To investigate the significance of regime shifts, we conduct formal Bayesian hypothesis testingrelying on marginal likelihoods and corresponding Bayes factors24 for two models that explicitly21This also implies that, when and are di erence stationary and 6= 1, ( +1) is also di erence

stationary. Nelson and Schwert (1977) and Garbade and Wachtel (1978) suggest the possibility of being di erencestationary.22We truncate the normal density to preclude explosive roots in our system.23With about four hundred and thirty observations, a particular choice of the prior has only negligible influence

on the posterior distributions of our parameters including latent state variables and possibly complicated functionsof these parameters such as impulse response functions, conditional means and variances, and other variables ofinterest reported below.24A Bayes factor is the ratio of the marginal likelihoods of two competing models and is the posterior odds ratio

when the two models are equally probable a priori. See Kass and Raftery (1995) for a review. Bayes factors areespecially useful in evaluating nonlinear time-series models like ours.

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incorporate stochastic volatility. In the null model, the Fisher relation parameter does notshift over the sample period while, by contrast, the alternative model specifies that parameters canshift between the two regimes. Panel A of Table 3 reports the log marginal likelihoods of the twocompeting specifications, (with regime shifts) versus (without regime shifts). Assumingthat the two models are equally probable a priori, the posterior odds of choosing the RSSV modelover the SV model after observing the data is exp

¡59 82

¢1013. This clearly suggests that a time

series model of inflation is misspecified when it does not incorporate the possibility of discrete shiftsin parameters. Our decisive evidence in favor of a regime shift in the inflation process even afteraccounting for stochastic volatility verifies the robustness of the previous findings of Evans (1991),Kim (1993), Evans and Wachtel (1993), Evans and Lewis (1995), and Garcia and Perron (1996) tothe criticisms of Sims (2002) and Stock (2002).

Panel B of Table 3 indicates that regime shifts in the inflation process have taken place onlyoccasionally with the posterior mean or median probability of a regime shift being approximatelyone percent per month. From Figure 1 we see that the period until the end of 1980 is generallycharacterized by the regime = 0. A substantial regime shift, however, took place between late1980 and 1981 when the posterior probability of being in regime = 1 increased sharply fromless than five percent (1980:11) to more than ninety-five percent (1981:12). By contrast, the periodsince 1982 is generally characterized by the regime = 1.

The posterior simulation results in Panel C of Table 3 indicate that the regime = 0 is broadlyconsistent with the Fisher hypothesis. In particular, the posterior mean of =0 is 0 95 with valuesof =0 ranging from 0 61 to 1 23 in its ninety percent posterior bands. The result of =0 1 isconsistent with Fama and Schwert (1977), who show that T-bills provide a “complete hedge againstexpected inflation” during the 1953-1971 period. By contrast, =1 is significantly less than 1 or,with tax e ects, less than 1 for reasonable tax rate assumptions. Its posterior mean or medianis approximately 0 40 with values of =1 ranging from 0 28 to 0 51 in its ninety percent posteriorbands.25 Thus neither the Fisher hypothesis nor Mundell-Tobin e ects are consistent with the datadrawn from the post-1981 sample period.

The results in Panel C also suggest that the regime shift in the inflation process closely correspondsto a shift in the Fisher relation itself. For example, from the joint posterior of ( =0 =1),the posterior probability of =0 =1 is less than one percent, suggesting an observationalequivalence between the shift in the inflation process and a shift in the Fisher relation.26 This ob-servational equivalence reinforces our interpretation that the regime shift occurred when monetarypolicy shifted and is consistent with the argument of Clarida, Gali, and Gertler (2000) that theenforcement of an anti-inflationary interest rate policy rule over this time period stabilized inflation.

The observational equivalence between the regime shift in inflation and the regime shift in theFisher relation can be tested formally by re-estimating the inflation forecasting equation with the

25 In order for this value to be consistent with the tax-adjusted Fisher hypothesis, the marginal income tax ratemust be 60 percent with corresponding confidence bands of between 49 percent and 72 percent. These tax rates aretoo high for our sample period.26By contrast, the regime shift is not associated with a shift in the volatility process of inflation shocks (Panel D

of Table 3).

12

prior restriction that =0 =1. From Panel E of Table 3 we see that the restricted modelactually yields a higher marginal likelihood, though the statistical significance of the di erenceis not strong.27 This result suggests that the regime shift in the inflation process is statisticallyequivalent to the regime shift in the Fisher relation.

4.2 Dynamic Behavior of Inflation and Interest Rates

Figure 2 plots posterior medians and ninety percent posterior bands of the impulse responses ofinflation to independent one percent inflation and interest rate shocks within each regime. Priorto the regime shift, both inflation shocks and interest rate shocks have persistent e ects on thefuture level of inflation. These e ects, however, become substantially smaller after the regime shift.For example, inflation shocks have almost no e ects on future inflation in regime = 1. On theother hand, interest rate shocks forecast inflation in both regimes, though the magnitude is muchsmaller after the regime shift.

Impulse responses of the nominal short rate to these independent shocks are displayed in Figure 3.Before the regime shift, inflation shocks forecast subsequent increases in nominal short rates butthey do not do so subsequent to the regime shift. Interest rate shocks have persistent e ects onfuture short rates in both regimes.

In general, the results summarized in Figures 2 and 3 suggest that interest rate shocks have persis-tent e ects on future inflation and interest rates. That is, interest rate shocks are associated withlong run changes in inflation expectations as well as interest rate expectations.28

Finally, Figure 4 plots impulse response functions of the real interest rate. Prior to the regimeshift, both shocks have only transitory e ects on future real interest rates which respond negativelyto inflation shocks and positively to interest rate shocks for up to approximately three months.Subsequent to the regime shift, a positive inflation shock is followed by a short run increase in thereal rate while positive interest rate shocks forecast a persistent increase in real rates.

These findings are, in general, consistent with a policy based interpretation of the regime shift.For example, the impulse responses of inflation and real interest rates prior to the regime shiftsuggest the possibility of shocks triggering what Clarida, Gali, and Gertler (2000) call “self-fulfillingchanges in inflation expectations”. That is, in the absence of anti-inflationary policy responses, anunexpected increase in inflation leads to a short run decline in real interest rates, as argued by

27The Bayes factor is reported as the twice the di erence of the log marginal likelihood values of our null modelwith the restriction =0 =1 and the alternative unrestricted model. Since the null model is nested within thealternative, the magnitude of the Bayes factor can be interpreted analogously to the likelihood ratio test statistic.Unlike likelihood ratio tests, however, the nested model may yield a higher marginal likelihood value than thealternative if the restriction leads to lower parameter variability without sacrificing model fit.28This supports our view that interest rate shocks are more informative about changes in both inflation and

interest rate expectations, thereby making =(1 2) (2 2) a better measure of the Fisher relation than =

(1 1) (2 1)

13

Mundell (1963), Tobin (1965), and Fama and Gibbons (1982). This decline stimulates the economywhich, in turn, results in a persistent increase in actual inflation. By contrast, subsequent to theregime shift, unexpected inflation leads to a short run increase in the real interest rate. Thisis consistent with an anti-inflationary interest rate policy on the part of the Fed that raises realinterest rates in response to an increase in inflation expectations (the Taylor rule). To the extentthat this action curbs inflation, inflation shocks then do not forecast future inflation ex post inregime = 1.

A positive correlation between inflation expectations and real rates is supported by Roll’s (2003)investigation of TIP yields.29 Roll suggests taxes and the business cycle as possible sources of thispositive correlation. For example, a positive correlation between inflation and real rates followswhen the business cycle is caused by demand shocks (e.g., policy shocks) that move inflation andreal rates together.30 Evidence of the presence of aggregate demand-like shocks has been reportedby Evans and Marshall (2002) under a wide variety of structural assumptions. As suggested byEvans and Marshall, the systematic conduct of monetary policy subsequent to the regime shiftplays an important role in reinforcing and transmitting the e ects of demand shocks.

4.3 Information Content of Inflation and Interest Rate Shocks

Using our time series framework, changes in inflation expectations and real rate expectations canbe ultimately traced back to underlying inflation shocks or interest rate shocks. In this section, weexplore the relative importance of these shocks in driving changes in these expectations and askwhether the systematic change in monetary policy underlying the regime shift altered their relativeimportance.

In order to simplify the notation in the subsequent analysis, we express our VAR in its companionform (suppressing the constant term):

+1 = + ˜ +1 (11)

where [ 1 1 2 2]0 ˜ +1 [ ˜ +1 ˜ +1]

0 and

1 2 3

2×2 02×2 02×202×2 2×2 02×2

1 01

04×1 04×1

Changes in inflation expectations and real rate expectations, respectively, are driven by the under-lying shocks

( +1 )X=1

+ = 1

X=1

1 ˜ +1

29For example, Roll concludes that “At the present, we must remember simply that there is definitely a positiveand strong relation between real yields and anticipated inflation, whatever the cause. TIP bond real yields are notindependent of inflation.” (emphasis by Roll).30Supply shocks (e.g., technology shocks), on the other hand, tend to produce opposite e ects on inflation and real

output (e.g., Fama (1981) and Fama and Gibbons (1982)).

14

( +1 )X=1

+ 1 =

( h2P 1

=11

1P

=11

i˜ +1;

1 ˜ +1;

if 2if = 1

where is the selector vector picking the -th element, such that 1 = and 2 = andterms in the square bracket corresponding to the impulse response functions reported in Figures 2and 4.

Since the shocks are mutually independent, we can measure the relative importance of interest rateshocks on long term inflation expectations by the following variance decomposition:

| =

hP=1

³+

˜ +

´i2( ˜ +1)hP

=1

³+

˜ +1

´i2( ˜ +1) +

hP=1

³+

˜ +1

´i2( ˜ +1)

where³

+

˜ +

´= 1

1 02 and

³+

˜ +

´= 1

1 01 are the impulse responses of

inflation to interest rate shocks and inflation shocks in regime , respectively. The relative infor-mativeness of interest rate shocks on long term real rate expectations, | is defined analogously.

Panel A of Figure 5 plots the posterior median together with the ninety percent posterior bandsfor the relative importance of interest rate shocks on long term inflation expectations after mar-ginalizing over the latent Markov regimes.31 Prior to the regime shift, interest rate shocks explainabout fifty to sixty percent of the variance of expected inflation at the sixty month horizon whilesubsequent to the regime shift about eighty percent of the variance of expected inflation at thishorizon can be traced to interest rate shocks.

Panel B of Figure 5 plots the relative importance of interest rate shocks on long term real rateexpectations. Before the regime shift, interest rate shocks explain, on average, about fifty percentof real rate expectations at the sixty month horizon. This suggests that inflation shocks have animportant e ect on real rate expectations, though the posterior bands are too large to draw aconclusion regarding the importance of each shock. After the regime shift, however, almost all ofthe variance of real rate expectations can be traced to interest rate shocks. These results suggestthat the importance of interest rate shocks on inflation expectations and real rate expectationsincreased significantly after the regime shift.

Our variance decompositions demonstrate that both before and after the regime shift interest rateshocks signal changes in inflation expectations as well as real rate expectations. Subsequent to theregime shift, however, inflation shocks lose their predictive power for both the long term inflationrate and real interest rates. These results are again consistent with our policy based interpretationof the regime shift. Before the regime shift, agents did not anticipate an anti-inflationary responseby the Fed and so nominal interest rates increase by less than the rate of inflation, or equivalently,real interest rates fall with increasing inflation. Accordingly, prior to the regime shift, inflationshocks have predictive power for long term inflation as well as real interest rates. By contrast,

31That is, |=P1

=0 Pr ( = )| . Its complement, 1 | , gives the relative importance of inflation

shocks.

15

the Fed’s credible anti-inflationary policy after the regime shift causes agents to anticipate onlytransitory e ects arising from inflation shocks since any inflation surges are expected to be curbedby the Fed. As a result, inflation shocks no longer result in self-fulfilling increases in expectedinflation. Mundell-Tobin e ects are not found in the data after the regime shift since real ratesmove together with expected inflation. Subsequent to the regime shift, only interest rate shocksreveal changes in agents’ inflation expectations and accompanying changes in real rate expectations.

Consistent with Evans and Marshall (1998, 2002), these interest shocks subsequent to the regimeshift can be viewed as systematic responses of the Fed to inflationary shocks. When the Fedfollows a credible anti-inflationary interest rate policy, the bond market perceives that increases ininflation expectations will be countered by increases in real rates. To the extent that this policyis successful in stabilizing inflation, the expected response of realized inflation will be small andshort-lived, especially when compared to the response of real rates. As such, subsequent to theregime shift, long term interest rates are more informative about real rate expectations as opposedto inflation expectations. Furthermore, the increased informativeness of interest rate shocks forlong term interest rates also reflects the shift in the Fed’s systematic policy. In particular, the Fed’sresponse to inflationary surges, revealed as interest rate shocks, curbs the e ects of inflation shockson inflation and so curbs their e ects on long term interest rates.

4.4 Persistence of Inflation and Nominal Short Rates

The persistence and possibility of unit roots in the inflation and nominal short rate processeshas been widely investigated.32 For example, Evans and Wachtel (1993) consider a Markov regimeswitching model in which the inflation process switches between a random walk and a mean revertingprocess. Interestingly, Evans and Wachtel find the inflation process is better characterized as arandom walk between the early 1960s and the early 1980s, roughly corresponding to the periodbefore our regime shift.

This section investigates whether the persistence of the inflation and nominal short rate processeshave shifted as well. One reasonable measure of persistence is the variance ratio (e.g., Campbell,Lo, MacKinlay (1997, Chapter 2)). That is, as inflation becomes more persistent, its long runconditional volatility per period becomes increasingly larger compared to its one period conditionalvolatility. We define the -month variance ratio of inflation as

( )=

³P=1 +

´P

=1 ( + )= 1 +

P=1 6=

P=1 ( + + )P

=1 ( + )

The variance ratio for nominal short rate, ( ), is defined analogously. We consider ( ) and( ) as functions of the unknown parameters including latent stochastic volatilities, and examine

32Despite our choice of prior that favors cointegration over stationarity, the posterior probability of a stationary VARsystem exceeds fifty percent in both regimes. This indicates the di culty of distinguishing between stationary versuscointegration for inflation and nominal short rates a priori. Our analysis incorporates both cases probablistically.

16

their posterior distributions after marginalizing over the parameters as well as the latent Markovregimes. Since we explicitly account for the time varying nature of variances, our measures ofpersistence, ( ) and ( ), address the concerns raised by Stock (2002).

Figure 6 plots the natural logarithm of ( ) and ( ) for = 60 months. As we can see in PanelA, at the sixty month horizon, the log variance ratio of inflation has a posterior median of aboutfour before the regime shift. However, after the regime shift, it drops to less than two, clearlyshowing that the persistence of inflation has substantially declined. The persistence of nominalshort rates (Panel B), by contrast, is much higher than that of inflation. For example, at the sixtymonth horizon, the volatility is twenty-five times of what we would see in the absence of persistencebefore the regime shift.33 After the regime shift, consistent with Watson (1999), there is a slightincrease in the persistence of the nominal short rate process. However, unlike the inflation process,the shift in the persistence of the nominal short rate process is not statistically significant.34

As argued by Clarida, Gali, and Gertler (2000) and Cogley and Sargent (2002, 2003), the reducedpersistence of inflation process results from the Fed’s inflation stabilizing monetary policy.35 Aslong as forward looking agents view the inflation stabilizing policy as being credible, inflationarysurges do not raise inflation expectations and hence do not raise the actual inflation rate. Thisresult is particularly important for our analysis since the reduced persistence of inflation implies alower long run inflation uncertainty, which reduces the risk of nominal bonds for long term investors.

4.5 What Moves Long Term Yields?

Campbell and Ammer (1993) use a homoscedastic VAR to decompose long term bond returns intothe e ects of changes in inflation expectations, real rate expectations, and bond return expectations.Relying on ten year T-bond returns for the 1952:01-1987:02 sample period, Campbell and Ammerdocument that long term bond returns are mainly driven by news about inflation expectationswith real rates having little e ect on these returns. To complement their analysis, we investigatethe e ects of the regime shift on the variance of long term (five year in our case) T-bond returns,though we restrict our attention to assessing the importance of only inflation expectations versusreal rate expectations.

33exp¡6 52

¢25 8

34 In order to see why the persistence of nominal short rate did not shift down with inflation persistence, it is usefulto rewrite ( ) as

( )= 1 +

³P=1 +

´+ 2

³P=1 +

P=1 ˜ +

´P

=1 ( + )

where + + 1 ( + ) is the one-month ahead expected inflation at + 1. The decline in the inflation

persistence is due to a substantial downward shift in³P

=1 +

P=1 ˜ +

´, and not due to a shift in the

persistence of + . Thus the persistence of nominal short rates would not shift downwards even if real rates wereconstant.35Clarida, Gali, and Gertler (2000) suggests that the estimated pre-Volcker monetary policy rule leaves open the

possibility of explosive behavior on the part of inflation.

17

In particular, we decompose the variance of monthly changes in the sixty month spot rate acrossboth regimes as³

4 (60)+1

´=

³4¯(60)+1

´+

³4¯(60)+1

´+ 2

³4¯(60)+1 4¯(60)+1

´+ remainder

where 4 (60)+1

(60)+1

(60) 4¯(60)+1160 +1

³P60=1 +1+

´160

³P60=1 +

´4¯(60)+1

160 +1

³P60=1 +

´160

³P60=1 + 1

´The remainder term contains the variance of changes

in expected excess bond returns and their covariances with 4¯(60)+1 and 4¯(60)+1 In Table 4 we report

the variance share of each component,³4¯(60)+1

´ ³4¯(60)+1

´ ³4¯(60)+1 4¯(60)+1

´and

the remainder component, each scaled by³4 (60)

+1

´. Prior to the regime shift, changes in

inflation expectations are more important than real rate expectations in explaining long term bondyields, though the variance shares are not precisely estimated. These results for regime = 0are broadly consistent with the findings of Campbell and Ammer (1993) and we find no statisticallyreliable correlation between changes in inflation expectations and changes in real rate expectations.

By contrast, subsequent to the regime shift, changes in real rate expectations become more im-portant than changes in inflation expectations, suggesting that changes in long term bond yieldsbecome informative about changes in real rate expectations. This conclusion is consistent withCochrane (2001) (page 380) who observes that in recent years real rate news drives long term in-terest rates but contradicts the conventional wisdom that long term interest rates reflect long terminflation expectations (e.g., Fama (1990) and Mishkin (1990a,b)). More importantly, the covariancebetween changes in real rate expectations and changes in inflation expectations has become positiveand constitutes an important share of the variance of changes in long term interest rates. In fact,they are highly correlated at the sixty month horizon with a median or mean correlation coe cientof 0 99.36

4.6 Conditional Variances of Inflation and Real Rates — Have Short Term BillsBecome Riskier than Long Term Bonds?

Relying on a homoscedastic VAR, Campbell and Viceira (2001) suggest that the recent episodeof low inflation risk plus high real interest rate risk makes short term bills riskier than long termbonds for long term investors. In this section, we use our stochastic volatility framework to explicitlycompare the time-varying riskiness of bills and bonds.

By holding nominal default free bonds until maturity, a long term investor is exposed solely toinflation risk. Thus the conditional variance (per month) of the real return to holding an -month bond until maturity coincides with the conditional variance of the -month inflation rate,

36At the six month horizon, the correlation is lower at 0 38 but still significant. This contrasts with Fama’s (1990)conclusion for 1952-1988 that expected changes in inflation and real rates are significantly negatively correlated,especially at horizons less than one year.

18

1³P

=1 +

´. Alternatively, if the investor rolls over short term bills for months, the

conditional variance (per month) of the resultant real return is 1³P

=1 + 1P

=1 +

´.

Figure 7 compares the annualized conditional volatilities of the real returns to holding sixty monthbonds until their maturity versus rolling over short term bills for sixty months. Prior to the regimeshift, the prevailing inflation uncertainty results in the real return volatilities of bonds exceeding thereal return volatilities of bills. However, subsequent to the regime shift, the real return volatilitiesof long term bonds decline substantially, so much so that there are no statistically significantdi erences in the conditional volatilities of bonds and bills. While, as suggested by Campbell andViceira (2001), we observe that short term bills appear to have higher real return risk than bondsat the five year horizon (Panel B of Figure 7), the di erence in the volatilities, however, is notstatistically significant.37

This result can be traced back to the increased importance of interest rate shocks on inflation andreal interest rates documented previously. Recall that after the regime shift, interest rate shocksincrease both the long term uncertainty surrounding inflation as well as real rates. As a result,the long term volatilities of inflation and real rates are closely tied to the volatility of interest rateshocks, making it di cult to distinguish inflation uncertainty from interest rate uncertainty.

5 Implications for the Term Structure of Interest Rates

5.1 Historical Behavior of Expected Inflation, Expected Real Rates, and YieldRisk Premia

To investigate the e ects of this regime shift on the term structure, we rely on the standard discretetime log approximation to the no-arbitrage condition underlying the pricing of nominal default-freebonds (e.g., Campbell, Lo, and MacKinlay (1997), Chapter 11):

( ) 'X=1

+

X=1

+ +1

2

X=1

+

X=1

+ (12)

Here ( ) (with (0)= 0) denotes the log price at month of a nominal default-free discount bond

maturing at month + while is the log one-period pricing kernel. All expectations and variancesare conditioned on the regime .38

37 It is also possible that the increased idiosyncratic variations of Treasury bills (Du ee (1996)) can account for apart of the increased real return volatility of short-term bills.38We suppress their regime dependence unless necessary. Since our regimes are motivated by the Lucas (1976)

critique, agents in this economy do not explicitly incorporate regime switching probabilities in forming their condi-tional expectations. This treatment of regime shifts di ers from that of Evans and Lewis (1995) and Ang and Bekaert(2003). A somewhat similar approach to ours is used by Kozicki and Tinsley (2001) who use a homoscedastic timeseries model in which the market’s perception of the long term policy target of inflation is subject to regime shifts.

19

Let ( ) be the -month yield risk premium, that is, the di erence between the observed -month

yield and the corresponding interest rate expectation, ( ) ( ) 1³P

=1 + 1

´. Note

that ( ) is simply the expected excess return to holding the bond until maturity, i.e., ( )=

1³P

=1( +1)+ 1

´, where ( )

+1( 1)+1

( ) is the excess return to holding an -month

bond from month to month + 1 By decomposing interest rate expectations³P

=1 + 1

´into corresponding inflation expectations and real rate expectations, we can express the -monthspot rate as follows:

( ) ' 1 X=1

+ +1 X

=1

+ 1 +( ) (13)

Panels A and B of Figure 8 display our posterior estimates of -month expected inflation,1

³P=1 +

´, for = 12 and = 60 months, respectively.39 In both cases, prior to the regime

shift, we see that yields (dotted lines) are closely related to inflation expectations. By contrast,subsequent to the regime shift, there is only a weak link between nominal interest rates and expectedinflation.

Posterior estimates of -month real rate expectations, 1³P

=1 + 1

´, for = 12 and = 60

months are given in Panels A and B of Figure 9, respectively. Before the regime shift, expectedreal rates are extremely low and close to being constant as asserted by Fama (1975). During thistime period, expected real rates are typically less than one percent at both maturity horizons withthe corresponding ninety percent posterior bands often including zero percent. With the regimeshift, however, expected real rates increase dramatically. This increase is consistent with the viewthat the Fed increased real rates during the Volcker experiment to stabilize inflation (e.g., Walsh(1998), p. 419).

Taken together, Figures 8 and 9 clearly show the e ects of the regime shift on the components of theyield curve. Prior to the regime shift, nominal bond yields are closely related to expected inflationand not to real rate expectations. Subsequent to the regime shift, by contrast, nominal bondyields closely track real rate expectations and are not informative about expected inflation. Thisconclusion complements our previous finding that changes in bond yields are driven by inflationnews before the regime shift and by real rate news after the regime shift. It also contrasts withthe conventional wisdom (e.g., Fama (1990) and Mishkin (1990a,b)) that, while short term interestrates, typically with maturities of six months or less, reveal real rate expectations, long term interestrates are associated with inflation expectations.40

Figure 10 displays our posterior estimates of -month yield risk premia, ( ), for = 12 and = 60months. In general, yield risk premia are higher after the regime shift than before, especially at39This VAR based forecast of inflation is similar to the inflation forecasts of Michigan Household Survey, the most

accurate survey based measure of expected inflation (e.g., Thomas (1999)).40 In fact, Mishkin (1990a) reports substantial shifts in estimated coe cients in his regression of short term interest

rates on future inflation in early 1980s. However, due to a sample size constraint, Mishkin (1990b) does not conducta subsample analysis for long term interest rates.

20

longer horizons. Contrary to the Expectations hypothesis, there has been substantial time variationin the yield risk premia across the sample period.

Figure 11 reports the slopes of the term structures of the three components of bond yields. Through-out the entire sample period, the slopes of the term structures of expected inflation (Panel A) andexpected real rates (Panel B) are flat. The e ects of the regime shift, however, are most clearlyseen in the term structure of yield risk premia. In particular, the slope of the yield risk premiumshifts upward with the regime shift and remains positive thereafter. On average, after the regimeshift, the yield risk premium of five year bonds is approximately two to three percent higher thanthat of three month bills, implying that expected excess returns are higher for long term bondsthan short term bills after the regime shift.

This empirical evidence has several important implications. For example, today’s long term interestrates are indicative of real interest rates and risk premia rather than expected inflation. Thispresumably increases the informativeness of long term interest rates regarding future real economicactivity and hence improves the allocation of scarce productive capital. At the same time, theincreased risk premium in the term structure substantially increases the cost of issuing long termdebt for both the U.S. Government and corporations. Little is known, however, of the source of thisincreased risk premium. In the next section we analyze one important element of the risk premium,that is, compensation for the risk of inflation.

5.2 Inflation Risk Premium in the Term Structure

The no-arbitrage identity, expression (12), implies that the inflation risk premium can be ex-pressed as a covariance between the pricing kernel and inflation. Assuming an -month zerocoupon real bond, its yield, ¯( ), gives the real riskless yield where ¯( ) ' 1

³P=1 +

´12

³P=1 +

´. Using this theoretical real yield, we can rewrite expression (12) as

( ) 1 X=1

+ +1

2

X=1

+ ' ¯( ) + 1 X=1

+

X=1

+ (14)

The left hand side of expression (14) is the expected real return per period to holding this -monthdiscount bond until maturity, after adjusting for Jensen’s inequality, while the right hand sidedecomposes the expected real return to nominal bonds into the theoretical real riskless yield, ¯( ),plus the inflation risk premium, that is, the conditional covariance between the -month pricingkernel and inflation.

A high conditional covariance betweenP

=1 + andP

=1 + implies that agents’ desire toconsume tends to increase more when the purchasing power of money depreciates more. In thiscase, nominal bonds must yield a higher risk premium to induce demand since these bonds are

21

claims against future dollars. Under the Fisher hypothesis, however, individuals’ preferences donot depend on the purchasing power of money and so the pricing kernel is independent of inflation.Hence, the presence of an inflation risk premium violates the Fisher hypothesis. An even strongerrejection of Fisher hypothesis would result if the inflation process directly a ects the pricing kernelitself and so the real riskless yield ¯( ).

In this section we ask two empirical questions. First, is there any evidence of a regime shift inthe real riskless yield ¯( )? Real riskless yields depend on the conditional moments of the pricingkernel that, in turn, depend on real state variables (e.g., aggregate real consumption, aggregatereal wealth) as well as “deep” parameters that govern agents’ preferences as well as productiontechnology. A significant shift in ¯( ) would suggest that the Fisher relation has had importantlong run e ects on these real state variables and on the pricing kernel itself. Second, is inflationrisk significantly compensated in each regime, i.e.,

³P=1 +

P=1 +

´6= 0? It is only

natural to investigate the consequences of the documented shift in the long run relation betweeninflation and real interest rates on the inflation risk premium.

As expression (12) is merely a no-arbitrage identity, to separately identify the inflation risk premiumand the real riskless yield requires further assumptions. We first assume that the Sharpe ratio forholding an -month discount bond until maturity is constant within each regime. The Sharpe ratio,denoted by ( ), is measured by the ratio of the expected real holding return of the discount bond

in excess of the corresponding real riskless yield³¯( )´to the conditional volatility of real returns,

that is, ( ) 1³P

=1 +P

=1 +

´ r1

³P=1 +

´41 With this assumption, we

can write expression (14) as

( ) 1 X=1

+ +1

2

X=1

+ ' ¯( ) + ( )

vuuut 1 X=1

+ (15)

Expression (15) says that the expected real return to holding an -month bond until maturitydepends on both the real riskless yield and inflation uncertainty. Under the Fisher hypothesis, ( )

must be zero, since the pricing kernel is independent of inflation. On the other hand, a positive ( )

implies that the market requires compensation for bearing the inflation risk inherent in nominalbonds. For this reason, we refer to ( ) as the market price of inflation risk.42 Equation (15) alsoreflects Boudoukh’s (1993) argument that an inflation volatility factor is important in determiningthe expected excess returns on bonds.

41The Sharpe ratio is constant when, for example, the pricing kernel and inflation have a constant conditionalcorrelation and the pricing kernel follows a homoscedastic process.42We assume that the market price of inflation risk, ( ) is regime dependent but remains constant within each

regime. Recent term structure literature (e.g., Dai and Singleton (2002), Du ee (2002), Ang and Piazzesi (2003))has emphasized the importance of incorporating time varying market prices of risk to explain the observed variationof the yield risk premium. We restrict our attention to the case of regime dependent market price of inflation riskwhile incorporating the time varying nature of the inflation volatility.

22

Our second assumption is that the real riskless yield, ¯( ), follows a homoscedastic autoregressiveprocess driven by independent Gaussian innovations:

¯( ) ( ) = ( )

³¯( )1

( )´+ ˜ ;

¯( )¯

1 ˜ (0 ( ( ))2) (16)

where ( ) is the real riskless yield’s long run mean that may shift between the two regimes with( )

0 in each regime.43

If the real riskless yield is independent of³P

=1 +

´, we can estimate ( ) and ( ) as

well as ( ) and ( ) by casting expressions (15) and (16) into the following linear regression withautoregressive disturbances:44

( ) 1 X=1

+ +1

2

X=1

+ = ( ) + ( )

vuuut 1 X=1

+ + ˜( )

where ˜ = ( )˜ 1 + ˜ ; ˜ (0 (( ))2) . The inflation risk premium in the -

month yield is estimated as ( )

r1

³P=1 +

´, while the remainder, ( )

+ ˜( ), provides

our estimate of the real riskless yield ¯( ).45 A positive (negative) value of ( ) indicates that theinflation risk premium is positively (negatively) related to the conditional volatility of inflation.The market price of inflation uncertainty at a specific horizon is given by ( ) and only (1) canbe interpreted as the market price of an inflation shock. The Fisher hypothesis implies ( )

= 0for all maturity horizons.

To investigate whether the long run mean of the real riskless yield, ( ), has shifted, Panels Aand B of Table 5 report our estimates of ( )

=0 and( ) ( )

=1( )=0 for = 6 to 60 month

horizons. Although the parameters ( )=0 are not precisely estimated, similarly to Ang and Bekaert

(2003), the long run means of ¯( ) are seen to be hump shaped peaking at a one year maturity.The shifts ( ) are not reliably di erent from zero as the corresponding ninety percent posteriorbands include zero. Therefore, the evidence of a permanent shift in real riskless yields is weak,

43Evans and Wachtel (1992) and Boudoukh (1993) relate the pricing kernel to consumption growth. Evans andWachtel also include an autoregressive process similar to equation (16). In the absence of widely accepted specificationfor the pricing kernel, we rely on statistical, rather than economic, assumptions.44While

³P=1 +

´and

³P=1 +

´are not observed, conditional on each simulated trajectory of³P

=1 +

´and

³P=1 +

´we can simulate ( ) and ( ) as well as ( ) and ( ) that are functions of

estimated parameters (of our VAR with regime shifts and stochastic volatility) and the data. This enables us tosimulate the posterior distributions of ( ) and ( ) as well as ( ) and ( ) over the full parameter space without

relying on particular estimates of³P

=1 +

´and

³P=1 +

´Using flat priors to facilitate comparison

with classical regression, we simulate from the posterior distributions of ( ) ( ) ( ) and ( ) as in Chib (1993).45The real riskless yield ¯( ) and the corresponding yield obtained from TIPs are not identical since taxes cause

the latter to covary with expected inflation. See Roll (2003).

23

suggesting that the stabilization of inflation has but insignificant long run e ects on real risklessyields, or more generally, on the pricing kernel.46

Our estimates of ( ) and ( ) (Panels C and D of Table 5) are consistent with Ang and Bekaert(2003) who report that real riskless yields are more variable at shorter maturities but are moresmooth and persistent at longer maturities.

By contrast, we observe striking di erences in the inflation risk premia between the two regimes.Panel E of Table 5 shows that the market does not require positive compensation for inflationrisk before the regime shift. Point estimates (posterior means or medians) of the market priceof inflation volatility are not significantly di erent from zero as the corresponding ninety percentposterior bands include zero at all maturity horizons. However, after the regime shift, the ( )

=1

parameters become significantly positive at horizons of two years and longer (Panel F).47 Forexample, at the two year horizon, the point estimates of (24)

=1 suggests that the market requiresabout a 0 7 percent premium for a yearly inflation volatility of one percent. At the five year horizon,the market requires about a 0 4 percent premium for a five year inflation volatility of one percentper year.

Figure 12 displays the variation of the estimated inflation risk premia, ( )

r1

³P=1 +

´across our sample period. Prior to the regime shift, there is little or no inflation risk premia in theterm structure. Subsequent to the regime shift, the inflation risk premium has become significantlypositive, especially at long maturities. Throughout this time period, the inflation risk premiumconstitutes a major component of real interest rates and hence the term structure of nominal bondyields. On average, the inflation risk premium ranges from about three to four percent in the mid1980s to about two to three percent in the 1990s.

By comparison, our estimated inflation risk premia subsequent to the regime shift are much largerthan the corresponding estimates of Campbell and Shiller (1996), Ang and Bekaert (2003) andBuraschi and Jiltsov (2003). Campbell and Shiller (1996) estimate the inflation risk premium for a5-year discount yield to be between 0 5 percent and 1 percent. While Campbell and Shiller (1996)analyze the 1953-1994 period, they do not separately analyze the post-1981 sample period. In thecase of Ang and Bekaert (2003), a regime switching model is introduced as a convenient approx-imation of the inflation process that makes bond pricing tractable. Since they do not interpretregime shifts as outcomes of changes in the collective behavior of forward looking agents, theirmodel fails to uncover the important shift in the underlying Fisher relation. Consequently, theirestimated inflation risk premium does not exhibit a regime shift. Uncovering the shift in the under-lying dynamics, however, requires distinguishing process shifts from variance shifts, as argued by

46Likewise, we do not find evidence for a shift in ( ) or ( ) when we allow regime dependence in these parameters.47While point estimates (posterior means and medians) of the market price of inflation risk are quite stable across

maturities, statistical significance is established only for maturities of two years or longer subsequent to the regimeshift. Combined with the volatile and less persistent theoretical real riskless yields of short maturities, these findingssuggest large variations of short term bill yields that are unrelated to inflation risk. In fact, Du ee (1996) documentsthat the idiosyncratic variation of Treasury bill yields (with maturities of one year or less) have increased after theearly 1980s.

24

Sims (2002) and Stock (2002). When Markov regimes merely proxy for the stochastic volatility, aregime switching model may not identify important process shifts that a ect rational expectations.Buraschi and Jiltsov’s (2003) model does not incorporate the price setting behavior of agents nordoes it account for the feedback of the inflation process to shifts in monetary policy making theirrisk premia much smoother and smaller in magnitude than our estimates.

In order to facilitate the economic interpretation of the shift in ( ) , it is useful to rewrite ourassumption of a constant Sharpe ratio in each regime as

X=1

+

X=1

+ =( )r

1³P

=1 +

´ X=1

+

X=1

+ + ˜( )+1 (17)

where ˜( )+1 is independent of the inflation innovation,P

=1 +

³P=1 +

´.48 According

to expression (17) a positive market price of inflation risk³

( )0´implies that inflation is

conditionally positively correlated with the pricing kernel. Since real returns to holing a nominalbond are inversely related to realized inflation, ( )

0 implies a negative correlation betweenreal bond returns and the pricing kernel. Using an equilibrium interpretation, investors requirehigher compensation for the inflation risk inherent in holding nominal bonds when their marginalutility of consumption tends to increase with unexpected inflation

³( )

0´. In other words, real

bond returns tend to be lower when agents desire to increase their consumption (e.g., a recession).Thus nominal bonds must yield a positive inflation risk premium when ( )

0. On the otherhand, when inflation is uncorrelated with the preferences of investors, as is consistent with theFisher hypothesis, ( )

= 0 and hence nominal interest rates do not contain an inflation riskpremium. Our finding of an upward shift in the inflation risk premium suggests that inflation andthe marginal utility of investors are significantly correlated after the regime shift, while not beingcorrelated before the regime shift.

The upward shift in the inflation risk premium also has important implications for the increasein the yield risk premium, or expected excess bond returns. We can use the no arbitrage identity

48This implies that ¯( )=

³P=1 +

´12

2 12

³˜( )+1

´where ¯( ) is the theoretical -month real

riskless yield. Also note that expression (17) does not necessarily imply that inflation uncertaintyP

=1 +³P=1 +

´causes unexpected changes in the pricing kernel,

P=1 +

³P=1 +

´. Expression (17)

merely describes a possible correlation between inflation and the pricing kernel.

25

(equation (12)) to write the yield risk premium ( ) ( ) 1³P

=1 + 1

´as

( ) ' 1

2

X=1

+

X=1

+ +1

2

X=1

( + + )

=1

2

³( )

1´X

=1

( + )1

2

³( )

1´X

=1

( + )

+1 X

=1

+

X=1

+

X=1

( + + ) (18)

In expression (18) ( ) and ( ) are the variance ratios measuring the persistence of the pricingkernel and inflation. The first term of the right side of expression (18) summarizes the e ects of thepersistence of the pricing kernel on the yield risk premium. Similarly, the second term summarizesthe e ects of the persistence of inflation on the yield risk premium while the third term capturesthe e ects of the term structure of inflation risk premia.

It is not likely that the first term of this expression explains the observed increase in the yieldrisk premium since the pricing kernel itself is insensitive to a shift in inflation or monetary policyespecially in the long run.49 This is supported by the weak evidence for a shift in real risklessyields. We have previously reported evidence of a downward shift in inflation persistence whichmay potentially explain the upward shift in the yield risk premium. Panel A of Figure 13 plotsthe time variation of 1

2

³( )

1´P

=1 ( + ) This term is significantly negative beforethe regime shift and is close to zero after the regime shift. However, the magnitude of the shift in12

³( )

1´P

=1 ( + ) is much smaller than that of the yield risk premium. While there

was an upward shift in the inflation risk premium by as much as 4 percent in early 1980s (Figure10), the shift in the inflation persistence can account for only about 0 2 percent.

Thus the increase in the yield risk premium must be associated with a shift in the inflationrisk premium. In Panel B of Figure 13, we plot the last term of (18) where we approximate1P

=1 ( + + ) by ( +1 +1) =(1)

( +1). As can be seen, the magnitudeof the increase in the inflation risk premium does indeed match that of the yield risk premium.

Ang and Bekaert (2003) argue that variation in inflation compensation (expected inflation andinflation risk premia) accounts for the majority of the dynamics of the term structure (see, alsoRoll (2003)). Our analysis generally concurs. However, expected inflation and the inflation riskpremia play very di erent roles in the term structure of interest rates before versus after the regimeshift. Before the regime shift in the Fisher relation, most time variation in the nominal yieldcurve can be traced to changes in expected inflation. However, inflation risk premia were close to

49While an unexpected shift in monetary policy has insignificant long run e ects on the pricing kernel, analogousto “long run neutrality”, this does not preclude it possibly having short run e ects. For example, Jensen, Mercer,and Johnson (1996), among others, document significant short run e ects of unexpected monetary policy on expectedasset returns.

26

zero before the regime shift. After the regime shift, by contrast, expected inflation plays a muchweaker role in determining the nominal yield curve, while the inflation risk premium componentsof expected real rates have emerged as the major determinant of the term structure.

5.3 Evidence from Inflation Tracking Portfolios

To provide independent verification of this increase in the inflation risk premium, we now investi-gate whether inflation tracking portfolios, that is, portfolios of stocks and bonds whose unexpectedreturns are maximally correlated with contemporaneous changes in expected inflation (e.g., Bree-den, Gibbons, and Litzenberger (1989) and Lamont (2001)) indeed earn higher mean excess returnsafter the regime shift.50 To do so, we split our sample into the two subsamples, January 1967 toDecember 1979 versus January 1983 to December 2002, and then within each subsample periodwe construct zero-investment portfolios that track changes in -month ahead inflation expectations(the -month inflation tracking portfolio). In particular, let be a vector of excess returns of thebase assets and ( ) be the vector of portfolio weights of the -month inflation tracking portfoliosuch that

( 1)X=1

+ = ( ) 1 ( )+ ˜( )

where ˜( ) is orthogonal to the unexpected components of . From Lamont (2001), we obtain aconsistent portfolio weight estimate ˆ( ) from the following OLS regression:

X=1

+ = ( ) + ( )1 +

( ) (19)

where 1 is a vector of conditioning variables that are known to track expected returns 1 ( ).51

Then ˆ( ) provides a consistent estimate of the excess return of the -month inflation trackingportfolio.

Table 6 tabulates sample means, medians, and standard deviations of the inflation tracking port-folios for di erent forecasting horizons. Notice that before the regime shift, excess returns of theinflation tracking portfolios are not statistically di erent from zero at any forecasting horizon.However, after the regime shift, the mean excess returns of the inflation tracking portfolios increasesubstantially. In particular, since the regime shift portfolios tracking three-, four- and five-yearinflation expectations earn positive mean excess returns of about 0 3 percent per month. Thisevidence provides independent support for the significance of the inflation risk premium in today’slong term yields despite the modest inflation risk of the past twenty years. Interestingly, the lastcolumn of Table 6 reports estimated mean excess returns of the inflation tracking portfolios whenwe ignore the regime shift and fit the regression, expression (19), to the whole sample period. The

50We thank Wayne Ferson for suggesting this exercise. For simplicity, we ignore the time variation of the inflationrisk premium in this exercise.51The notes to Table 6 provide a detailed description of the variables used in expression (19).

27

results indicate that by ignoring the regime shift we would erroneously conclude that the inflationrisk premium is negative.52

5.4 Why Has the Inflation Risk Premium Increased After the Regime Shift?

One possible explanation for the increased inflation risk premium after the regime shift is thatinflation was a monetary phenomenon prior to the regime shift but subsequently has been drivenby negative output shocks that reduce the demand for money. Recall that inflation tends to increaseduring recessions when it is driven by negative output shocks. If the purchasing power of moneydeclines during recessions, nominal long term bonds become less attractive investments than otherreal assets and short term bills, and so must yield a higher inflation risk premium to induce theirdemand. This explanation is broadly consistent with the money demand based asset pricing theory(e.g., Marshall (1992), Bakshi and Chen (1996)) which argues that a high rate of inflation raisesthe liquidity risk premium on assets that substitute for money. In this explanation, the positivecovariance between inflation and the pricing kernel proxies for the negative covariance betweenoutput shocks and the pricing kernel. However, the money demand explanation fails to account forthe positive correlation observed between inflation and real interest rates after the regime shift. Asargued by Fama and Gibbons (1982), if negative output shocks cause inflation, they should lowerreal interest rates, actually resulting in a negative correlation between inflation and real interestrates (see, also Fama (1990)).

An alternative explanation is that while negative output shocks do indeed lower real riskless realyields, ¯( ) 1

³P=1 +

´12

³P=1 +

´, they may actually cause the inflation

risk premium³P

=1 +P

=1 +

´=

( )

r1

³P=1 +

´to rise, so much so that

taken together real yields increase. This explanation relies on the significance of the Friedman(1977) e ect, that is, the hypothesized positive relation between the level and the volatility ofinflation. If a positive relation between the level and volatility of inflation indeed exists, negativeoutput shocks can increase real interest rates by raising the level and volatility of inflation, andhence the inflation risk premium. Thus the significance of the Friedman e ect, coupled with thefact that inflation is mostly driven by output shocks, can provide a consistent explanation for thesignificance of the inflation risk premium as well as the positive correlation between inflation andreal rates.

Ball and Cecchetti (1990), Evans (1991), and Kim (1993) document evidence consistent with theFriedman e ect in the long run, but not in the short run. In our estimation, the short run Friedmane ect is captured by the parameter . The posterior mean or median of our estimate isapproximately 0 2, with the corresponding 90 percent posterior band ranging from 0 05 to 0 44and so we conclude that the short run Friedman e ect is indeed weak (Panel D of Table 3). Bycontrast, we clearly see in Figure 14 a tendency for expected inflation and inflation uncertainty tomove together in the long run.52Negative inflation risk premia have been reported by Chen, Roll, and Ross (1986) and Lamont (2001), among

others, using di erent sample periods and di erent horizons.

28

The presence of a long run Friedman e ect suggests that expected inflation and the conditionalvolatility of inflation move together in the long run. When inflation is caused by negative outputshocks which lower money demand, the pricing kernel and inflation are positively correlated andhence the market price of inflation risk is positive. Given this, negative output shocks can increasereal interest rates by increasing their inflation risk premium component.53 On the other hand, asargued by Clarida, Gali, and Gertler (2000) and Barsky and Killian (2002), among others, inflationprior to the regime shift was mostly due to an overly expansionary monetary policy as opposed tothe oil shocks of 1970s. Since money supply shocks have only a weak relation with the marginalutility of investors, the inflation risk premium is close to zero prior to the regime shift.

A potential weakness of this argument is that, as shown by Roll (2003), TIP yields and inflationare significantly positively correlated over the 1997 to 2002 sample period. Since TIP yields donot contain an inflation risk premium, this positive correlation suggests that recent inflation maynot be solely driven by output shocks. While Roll notes the potential significance of tax e ects inaccounting for the positive correlation, it is also possible that both inflation and real riskless yieldsmove together over the business cycle, especially if the business cycle is driven by demand shocks. Infact, Evans and Marshall (2002) provide robust evidence consistent with the presence of aggregatedemand-like shocks that generate positive responses of both inflation and real interest rates. Oneimportant source of these demand-like shocks is the systematic response of the Fed to exogenousshocks that produce positive correlation between inflation and real rates. When the Fed follows theTaylor rule, any increase in inflation expectations is countered by an increase in real interest rates.There exists ample empirical evidence that unexpected increases in short term interest rates havecontractionary e ects on real economic variables.54 Hence, to the extent that the increase in realinterest rates is seen to contract the economy, it leads to an increase in investors’ marginal rateof substitution. Putting this di erently, the positive inflation risk premium after the regime shiftproxies for the real e ects of the systematic response of monetary policy to inflationary shocks.Consequently, forward looking agents perceive that inflation and the pricing kernel are positivelycorrelated in the long run, and hence require a higher inflation risk premium to hold nominal longterm bonds.

6 Conclusions

A regime shift in the inflation process is the collective outcome of shifts in the decision making ofindividual forward looking agents reflecting shifts in their rational expectations. By recognizing thesignificance of the regime shift brought about by the new direction for monetary policy initiatedby the Fed under Chairman Volcker provides us with very di erent implications for the relationbetween inflation and interest rates and consequently the dynamics of the term structure.

53This result is related to Boudoukh’s (1993) argument in favor of an inflation volatility factor as a determinantof expected excess bond returns. Boudoukh’s estimated risk premium is, however, smaller in magnitude than ourestimate since Boudoukh does not consider the possibility of regime shifts. However, he does acknowledge in hisconclusions that possible parameter shifts may yield very di erent implications about the shape and dynamics of theterm structure than those obtained under constant parameters.54For example, see Bernanke and Blinder (1992).

29

In particular, subsequent to the regime shift, we find that after explicitly recognizing the stochasticvolatility of inflation shocks, inflation and real interest rates move together, especially in the longrun. This result is consistent with the anti-inflationary interest rate rule of Taylor (1993) andcontradicts both the Fisher hypothesis as well as the presence of Mundell-Tobin e ects. Ourevidence suggests that, consistent with the enforcement of the Fed’s anti-inflationary interest raterule, the regime shift in the inflation process is equivalent to a shift in the dynamic relation betweeninflation and interest rates. We also show that the persistence of inflation declines significantly withthe regime shift, reinforcing our policy based interpretation of the regime shift.

The regime shift has important implications for the dynamics of the term structure. Contrary tothe conventional wisdom that long term interest rates reflect long term inflation expectations, wefind that long term interest rates are no longer informative about future inflation subsequent tothe regime shift. In fact, today’s yield curve is primarily driven by real rate expectations and thepositive covariance between inflation expectations and real rate expectations. While the inflationrisk premium is insignificant before the regime shift, it is significantly positive afterwards. In partic-ular, we estimate that subsequent to the regime shift, bond yields typically require approximately a0 4 to 0 7 percent inflation risk premium for bearing one percent conditional volatility of inflation.These estimates are larger than reported in previous research that did not account for the regimeshift in agents’ rational expectations. The significance of the regime shift on the inflation riskpremium is independently verified by corresponding shifts in the mean excess returns of inflationtracking portfolios. We trace the increase in the inflation risk premium to the Fed’s enforcementof a credible anti-inflationary policy.

While we observe time variation in the yield risk premia and in its relation with the inflation riskpremia, our bivariate system does not allow us to further investigate the sources of the expectationspuzzle of the term structure. To do so would require a larger and more complicated estimationframework. How much we can learn about the expectations puzzle and more generally about thetime variation of the risk premium of bonds when accounting for possible regime shifts in agents’rational expectations is left for future research.

30

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Tobin, J., 1965, “Money and Economic Growth,” Econometrica, 33-4, 671-684.

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34

Table 1: Summary statistics

(January 1967 — December 2002, n = 432 observations)

Mean S.D. Max. Min. Skew. Kurt. Jq-Bera AC(1) AC(12) ADF PPπ 4.50 3.39 19.27 −5.50 0.70 4.34 67.41∗∗ 0.60 0.50 −2.01 −12.3∗∗i 6.06 2.64 16.21 1.17 1.16 4.83 157.9∗∗ 0.96 0.66 −2.27 −2.26

Notes to Table 1:

Here π denotes the CPI inflation rate that retrospectively incorporates the rental equivalence methodologyfor the period 1967:01-1982:12 and i denotes the one-month Treasury spot rate. Both π and i are expressedin annual percentage points. Jq-Bera denotes the Jarque-Bera statistic for testing the null hypothesis ofnormality. AC(n) denotes the autocorrelation coefficient of order n. ADF and PP are the Augmented DickeyFuller and Phillips-Perron statistics, respectively, for testing the null hypothesis of a unit root, corrected forserial correlation of order 12. Critical values for the two unit root tests are −3.45 (1 percent) and −2.87 (5percent). ∗∗ denotes significance at the 1 percent level.

Table 2: Priors

Panel A: Prior Distributions

Parameters Description of the Prior DistributionsΦk,st ; k = 1, 2,3 See the text for detailed description.νπ,st , νi,st, γπ, γi, ρst Independent N (0, 100) for each parameter.κπ, κi Independent N (0, 100) I|κ|<1 for each parameter where I|κ|<1 is a truncation

to ensure stationarity of zt and ht processes.µπ,st ,µi,st , z0, h0 Independent N (ln (sse) , 100) for each parameter where ln (sse) is the log of

the squared standard errors from a VAR of π and i.ξ2π, ξ

2i Independent IG

¡0.0012 , 0.0012

¢for each parameter where IG is the Inverse

Gamma distribution.Pr (st+1 = st) , st ∈ (0,1) Independent Beta(4, 1) for each parameter. (e.g., Albert and Chib (1993)).

Panel B: VAR Dominant Roots

Mean S.D. 0.025 0.05 Median 0.95 0.975 Pr (|zj| > 1) j=1,2|z1| 1.002 0.018 1.000 1.000 1.000 1.000 1.022 5.7%|z2| 1.118 0.120 1.003 1.006 1.082 1.356 1.441 100%

Notes to Table 2:

Panel A summarizes our assumed prior distributions. Panel B reports distributions of the VAR dominantroots based on our choice of the random walk prior. Specifically, letting z ≡ (z1, z2, ...) denote roots of¯I −Pp

k=1 Φkzk¯= 0, such that |z1| ≤ |z2| ≤ ...., Panel B reports our prior distributions of the first and

second dominant roots of the VAR system, based on 100, 000 Monte Carlo simulations.

35

Table 3: Regime Shifts in the Inflation Forecasting Equation

Panel A: Bayesian Hypothesis Testingfor the Presence of Regime ShiftsModels: Log marginal likelihood

Stochastic volatility only (SV) −985.7Regime shifts and stochastic volatility (RSSV) −955.8Bayes factor: 2 (lnMLRSSV − lnMLSV ) +59.8

Panel B: Markov Regime Switching Probabilities

Mean S.D. 0.025 0.05 Median 0.95 0.975Pr (st+1 = 0|st = 0) 0.986 0.010 0.959 0.965 0.988 0.997 0.998Pr (st+1 = 1|st = 1) 0.993 0.007 0.974 0.979 0.996 1.000 1.000

Panel C: Fisher RelationBst =

³Ppk=1 φ

π|ik,st

´/³1−Pp

k=1 φπ|πk,st

´.

Mean S.D. 0.025 0.05 Median 0.95 0.975Bst=0 0.939 0.191 0.531 0.607 0.949 1.234 1.294Bst=1 0.399 0.070 0.256 0.281 0.399 0.513 0.536Pr (Bst=0 > Bst=1) = 99.3 percent.

Panel D: Stochastic Volatility of Inflation Shocks

Mean S.D. 0.025 0.05 Median 0.95 0.975γπ 0.196 0.149 −0.099 −0.049 0.195 0.438 0.489κπ 0.793 0.091 0.572 0.623 0.807 0.912 0.928ξπ 0.471 0.109 0.283 0.306 0.462 0.665 0.706µπ,st=0 0.761 0.654 −0.449 −0.250 0.755 1.776 1.993µπ,st=1 0.927 0.560 −0.099 0.080 0.928 1.787 1.962∇µπ 0.165 0.372 −0.515 −0.384 0.173 0.692 0.795∇µπ ≡ µπ,st=1 − µπ,st=0

Panel E: Bayesian Hypothesis Testingfor Observational Equivalence

Models: Log marginal likelihoodRestricted RSSV model: Bst=0 > Bst=1 −953.8Unrestricted RSSV model −955.8Bayes factor: 2 (lnMLUnrest. − lnMLRestr .) −4.0

36

Notes to Table 3:

Panel A reports log marginal likelihood values of the inflation forecasting equation with stochastic volatility(SV) and of the inflation forecasting equation with Markov regime switching and stochastic volatility (RSSV).The reported Bayes factor is computed as twice the difference of the log marginal likelihood values of thetwo competing models (RSSV minus SV) and is on the same scale as the familiar likelihood ratio teststatistic. A positive Bayes factor provides evidence for the RSSV model against the SV model. See Kass andReftery (1995) for a review. Log marginal likelihoods are computed using the Newton and Raftery’s (1994)estimator. The following criteria, due to Kass and Raftery (1995), are useful in interpreting the results ofBayesian hypothesis testing:

Criteria for Bayesian Hypothesis Testingdue to Kass and Raftery (1995, p.777)

Bayes factor Evidence against the null model2 lnBF = Evidence for the alternative model0 to 2 Not worth more than a bare mention2 to 6 Positive6 to 10 Strong> 10 Very Strong

Panel B reports the estimated transition probabilities of the hidden Markov chain describing discrete shiftsin the Fisher relation. Panel C reports a measure of the Fisher relation in each regime, given by

Bst =

Ppk=1 φ

π|ik,st

1−Ppk=1 φ

π|πk,st

.

We report posterior mean, posterior standard deviation, and posterior median as well as 2.5th, 5th, 95th,and 97.5th percentiles. Panel D reports hyperparameters for the stochastic volatility process of inflationshocks.

Panel E reports log marginal likelihood values of the regime-switching stochastic volatility (RSSV) inflationforecasting equation, with and without the restriction Bst=0 > Bst=1. A negative Bayes factor indicates thatthe restricted model is favored. Log marginal likelihoods are computed using Newton and Raftery’s (1994)estimator.

All results are based on 50, 000 Gibbs sampling iterations after discarding the first 5, 000 draws.

37

Table 4: Variance Decomposition for Changes in 5-year Zero-Coupon Yields

Panel A: Regime st = 0

Shares Mean S.D. 0.025 0.05 Median 0.95 0.975

V ar³∆π

(60)t+1

´1.780 1.454 0.184 0.238 1.352 4.539 5.472

V ar³∆r

(60)t+1

´0.070 0.120 0.006 0.008 0.032 0.256 0.351

2Cov³∆π(60)t+1 ,∆r

(60)t+1

´−0.055 0.529 −1.890 −1.357 0.022 0.976 1.290

Remainder −0.795 1.263 −3.454 −2.966 −0.497 0.708 0.776

V ar³∆y(60)t+1

´0.195 0.012 0.174 0.178 0.193 0.225 0.231

Cor³∆π

(60)t+1 ,∆r

(60)t+1

´0.023 0.691 −0.980 −0.962 0.103 0.935 0.959

Panel B: Regime st = 1

Shares Mean S.D. 0.025 0.05 Median 0.95 0.975

V ar³∆π(60)t+1

´0.250 0.120 0.069 0.086 0.234 0.465 0.523

V ar³∆r(60)t+1

´0.523 0.193 0.201 0.240 0.507 0.872 0.962

2Cov³∆π(60)t+1 ,∆r

(60)t+1

´0.684 0.152 0.263 0.321 0.690 1.027 1.106

Remainder −0.456 0.437 −1.313 −1.167 −0.476 0.300 0.418

V ar³∆y

(60)t+1

´0.159 0.008 0.133 0.146 0.160 0.168 0.172

Cor³∆π(60)t+1 ,∆r

(60)t+1

´0.985 0.013 0.953 0.965 0.989 0.995 0.996

Notes to Table 4:

Table 4 reports the components of the sample variance of changes in the 60-month zero coupon bond yield,

∆y(60)t+1 ≡ y

(60)t+1 − y

(60)t , for both regimes. Here ∆π(60)t+1 ≡ 1

60Et+1

³P60j=1 πt+1+j

´− 1

60Et

³P60j=1 πt+j

´and

∆r(60)t+1 ≡ 1

60Et+1

³P60j=1 rt+j

´− 1

60Et

³P60j=1 rt+j−1

´are changes in 60-month inflation expectations and

real rate expectations, respectively. Note that V ar³∆π

(60)t+1

´, V ar

³∆r

(60)t+1

´, 2Cov

³∆π

(60)t+1 ,∆r

(60)t+1

´, and

Remainder are shown in shares scaled by V ar³∆y(60)t+1

´while V ar

³∆y(60)t+1

´and Cor

³∆π(60)t+1 ,∆r

(60)t+1

´are

shown without scaling. The last row of each table reports the correlation coefficient between ∆π(60)t+1 and

∆r(60)t+1 in each regime. The results are based on our VAR with regime shifts and stochastic volatility,implemented with 50, 000 Gibbs sampling simulations after discarding the first 5, 000 draws.

38

Table 5: Real Riskless Yields and Inflation Risk Premia

Panel A: Long-Run Means of the Real Riskless Yields in st = 0

Horizon Mean S.D. 0.025 0.05 Median 0.95 0.975a(6)st=0

1.569 1.054 0.124 0.222 1.371 3.597 4.153

a(12)st=0 2.295 1.291 0.248 0.441 2.137 4.664 5.260

a(24)st=0 2.166 1.282 0.174 0.342 2.000 4.502 5.126

a(36)st=0 2.095 1.365 0.135 0.263 1.870 4.609 5.248

a(48)st=0 2.076 1.425 0.115 0.222 1.824 4.756 5.397

a(60)st=0 2.053 1.482 0.097 0.185 1.766 4.859 5.523

Panel B: Changes in the Long-Run Mean of the Real Riskless Yield

∇a(n) ≡ a(n)st=1 − a(n)st=0

Horizon Mean S.D. 0.025 0.05 Median 0.95 0.975∇a(6) 0.760 1.630 −2.090 −1.657 0.607 3.642 4.265∇a(12) −0.383 1.776 −3.528 −2.987 −0.556 2.809 3.634∇a(24) −1.055 1.402 −3.954 −3.448 −0.962 1.021 1.436∇a(36) −0.737 1.538 −4.044 −3.431 −0.588 1.545 1.938∇a(48) −0.366 1.658 −3.929 −3.282 −0.223 2.106 2.482∇a(60) −0.034 1.747 −3.731 −3.077 0.096 2.575 2.967

Panel C: Persistence of Real Riskless Yields

Horizon Mean S.D. 0.025 0.05 Median 0.95 0.975b(6) 0.294 0.455 −0.321 −0.268 0.098 0.921 0.934b(12) 0.229 0.310 −0.225 −0.173 0.160 0.885 0.917b(24) 0.356 0.180 0.008 0.064 0.354 0.660 0.728b(36) 0.407 0.163 0.084 0.139 0.406 0.679 0.729b(48) 0.435 0.164 0.115 0.170 0.431 0.705 0.751b(60) 0.455 0.165 0.131 0.186 0.452 0.728 0.771

Panel D: Volatility of Real Riskless Yields

Horizon Mean S.D. 0.025 0.05 Median 0.95 0.975

σ(6)r 1.225 0.404 0.704 0.722 1.252 1.884 2.000

σ(12)r 1.110 0.302 0.670 0.691 1.072 1.657 1.782

σ(24)r 0.840 0.151 0.638 0.661 0.814 1.118 1.237

σ(36)r 0.774 0.115 0.602 0.621 0.759 0.977 1.046

σ(48)r 0.750 0.113 0.569 0.590 0.741 0.942 0.999

σ(60)r 0.738 0.118 0.547 0.567 0.730 0.941 1.000

39

Panel E: Market Price of Inflation Risk in st = 0

Horizon Mean S.D. 0.025 0.05 Median 0.95 0.975λ(6)st=0

−0.203 0.264 −0.823 −0.694 −0.165 0.159 0.210

λ(12)st=0 −0.246 0.232 −0.754 −0.659 −0.227 0.102 0.155

λ(24)st=0 −0.137 0.145 −0.425 −0.374 −0.134 0.098 0.149

λ(36)st=0 −0.103 0.111 −0.318 −0.280 −0.104 0.084 0.128

λ(48)st=0 −0.086 0.092 −0.263 −0.232 −0.089 0.070 0.108

λ(60)st=0 −0.074 0.080 −0.224 −0.196 −0.077 0.061 0.100

Panel F: Market Price of Inflation Risk in st = 1

Horizon Mean S.D. 0.025 0.05 Median 0.95 0.975

λ(6)st=1 0.347 0.412 −0.436 −0.316 0.336 0.991 1.057

λ(12)st=1 0.527 0.413 −0.358 −0.215 0.591 1.091 1.163

λ(24)st=1 0.682 0.240 0.147 0.269 0.704 1.027 1.090

λ(36)st=10.572 0.210 0.175 0.237 0.571 0.920 0.993

λ(48)st=10.473 0.207 0.119 0.169 0.455 0.840 0.924

λ(60)st=10.393 0.197 0.072 0.115 0.369 0.746 0.830

Notes to Table 5:

Table 5 reports simulated posterior distributions of parameters in the following regression:

y(n)t − 1nEt

nXj=1

πt+j

+ 1

2nV art

nXj=1

πt+j

= a(n)st + λ(n)st

vuuut 1

nV art

nXj=1

πt+j

+ v(n)t ; a(n)st > 0

v(n)t = b(n)v(n)t + εr,t; |b| < 1, εr,t ∼ N

µ0,³σ(n)r

´2¶,

for n = 6, 12, 24, 36, 48, and 60 month horizons. Flat priors are used to facilitate comparison with classical

regression. From the 50, 000 simulated trajectories of Et

³Pnj=1 πt+j

´and V art

³Pnj=1 πt+j

´, every 10th

trajectory is drawn for a total of 5, 000. Then for each of the 5, 000 trajectories of Et

³Pnj=1 πt+j

´and

V art³Pn

j=1 πt+j´, posterior samples of parameter values are simulated from 50 Gibbs sampling simulations

(Chib (1993)) after discarding the first 25. Panels A-F report the posterior distributions based on the 250, 000posterior samples thus obtained.

40

Table 6: Monthly Excess Returns of Inflation Tracking Portfolios

Before the shift After the shift Equality Whole (no shift)1967:01—1979:12 1983:01—2002:12 (p-values) 1967:01—2002:12

n = 12 Portfolios tracking changes in 12-month inflation expectations

Mean 0.027 (51.0%) −0.011 (68.2%) 41.6% −0.056 (3.4%)Median 0.067 (42.9%) −0.061 (41.0%) 27.5% −0.042 (1.3%)St.Dev. 0.397 0.492 0.6% 0.543Obs. 144 228 420

n = 24 Portfolios tracking changes in 24-month inflation expectations

Mean 0.038 (40.6%) 0.076 (8.3%) 56.3% −0.097 (2.7%)Median 0.016 (57.3%) 0.034 (32.4%) 79.0% −0.076 (1.4%)St.Dev. 0.518 0.639 0.6% 0.885Obs. 132 216 408

n = 36 Portfolios tracking changes in 36-month inflation expectations

Mean −0.090 (22.5%) 0.295 (0.0%) 0.0% −0.139 (2.1%)Median −0.180 (7.6%) 0.250 (0.0%) 0.0% −0.137 (0.8%)St.Dev. 0.807 0.896 19.4% 1.196Obs. 120 204 396

n = 48 Portfolios tracking changes in 48-month inflation expectations

Mean −0.151 (9.3%) 0.309 (0.0%) 0.0% −0.255 (0.3%)Median −0.183 (5.6%) 0.333 (0.0%) 0.0% −0.252 (0.1%)St.Dev. 0.926 1.013 28.3% 1.654Obs. 108 192 384

n = 60 Portfolios tracking changes in 60-month inflation expectations

Mean 0.097 (42.0%) 0.324 (0.0%) 10.3% −0.318 (0.6%)Median 0.304 (28.0%) 0.315 (0.0%) 21.4% −0.247 (0.3%)St.Dev. 1.169 1.060 29.2% 2.210Obs. 96 180 372

41

Notes to Table 6:

Table 6 reports descriptive statistics for monthly excess returns (in percentage points) of inflation trackingportfolios for the given sample periods and for n = 12-, 24-, 36-, 48-, and 60-month horizons. Numbers inparentheses next to the mean and median estimates are p-values for the null hypothesis of zero mean excessreturns based on the t-test (mean) and the Wilcoxon signed rank test (median). Numbers in bold indicatesignificance at the 5 percent level. The “Equality” column reports p-values for the null hypothesis of equalitybetween the two subsamples based on the t-test (mean), the Kruskal-Wallis median test (median), and thevariance ratio F -test (standard deviation).

A zero-investment tracking portfolio of n-month inflation expectations is constructed by first fitting an OLSregression:

nXj=1

πt+j = α(n)XRt + β(n)Zt−1 +˜(n)t

where XRt is a vector of excess returns of base assets and Zt−1 is a vector of conditioning variables thatdescribeEt−1 (XRt). Then, the excess return on the n-period inflation tracking portfolio is given by α

(n)XRt.See Lamont (2001) for details. The following variables are included in XRt and Zt−1.

Base Assets (XRt)

Fama-French 3 factors Market, HML, SMB factors (source: Kenneth French’s webpage)

Excess returns of 10 industry portfolios Consumer nondurables; Consumer durables; Oil, gas, and coalextraction and products; Chemicals and allied products;Manufacturing; Telephones and television; Utilities;Wholesale, retail, and some services; Finance; Others(source: Kenneth French’s webpage)

Excess returns of bond portfolios CRSP 90-day, 1-year, 2-year, and 5-year Treasury indexes;Ibbotson’s long-term government bond portfolio index;Ibbotson’s long-term corporate bond portfolio index;

Note: Excess returns are total returns minus Ibbotson’s monthly Treasury bill return.

Conditioning Variables (Zt−1)

Short-term interest rate Ibbotson’s 1-month Treasury bill yield.

Term Spread 10-year Treasury constant maturity yield (last business day of each month;source: FRED) less Ibbotson’s 1-month Treasury bill yield.

Default spread Moody’s Baa-rated average corporate bond yield less the Aaa-rated averagecorporate bond yield (last business day of each month; source: FRED).

Dividend-price ratio log dividend price ratio of the S&P 500 index (source: DRI).

Dividend payout ratio log dividend payout ratio of the S&P 500 index (source: DRI).

Note: A constant is also included in Zt−1.

42

Figure 1: Posterior Probability of Being in Regime st = 1

We plot the posterior probability of being in the regime st = 1 in a particular month using our two-state Markov regime-switching specification. Posterior probabilities are based on 50, 000 Gibbs samplingsimulations after discarding the first 5,000 draws.

43

Figure 2: Responses of Inflation to Inflation Shocks and Interest Rate Shocks

Panel A: Response of Inflation to a One Percent Inflation Shock

Panel B: Response of Inflation to a One Percent Interest Rate Shock

We plot the impulse responses of inflation (πt) to a one percent shocks to the monthly inflation rate andto the monthly nominal short rate over a five-year horizon. In each Panel, impulse response functions arecompared between regime st = 0 (before the regime shift; left) and regime st = 1 (after the regime shift;right). The results are based on 50, 000 Gibbs sampling simulations after discarding the first 5, 000 draws.Each graph shows posterior medians (solid) as well as ninety percent posterior bands (dashed), estimatedpoint by point.

44

Figure 3: Responses of the Nominal Short Rate to Inflation Shocks and Interest Rate Shocks

Panel A: Response of Nominal Short Rates to a One Percent Inflation Shock

Panel B: Response of Nominal Short Rates to a One Percent Interest Rate Shock

We plot the impulse responses of the nominal short rate (it) to a one percent shocks to the monthly inflationrate and to the monthly nominal short rate over a five-year horizon. In each Panel, impulse response functionsare compared between regime st = 0 (before the regime shift; left) and regime st = 1 (after the regimeshift; right). The results are based on 50, 000 Gibbs sampling simulations after discarding the first 5, 000draws. Each graph shows posterior medians (solid) as well as ninety percent posterior bands (dashed),estimated point by point.

45

Figure 4: Responses of the Real Short Rate to Inflation Shocks and Interest Rate Shocks

Panel A: Response of Real Short Rates to a One Percent Inflation Shock

Panel B: Response of Real Short Rates to a One Percent Interest Rate Shock

We plot impulse responses of the real short rate (rt ≡ it − Et (πt+1)) to a one percent shocks to the monthlyinflation rate and to the monthly nominal short rate over a five-year horizon. In each Panel, impulse responsefunctions are compared between regime st = 0 (before the regime shift; left) and regime st = 1 (afterthe regime shift; right). The results are based on 50, 000 Gibbs sampling simulations after discarding thefirst 5, 000 draws. Each graph shows posterior medians (solid) as well as ninety percent posterior bands(dashed), estimated point by point.

46

Figure 5: Variance Decompositions of Inflation and Interest Rates

Panel A: Variance of 60-month Inflation Expectations due to Interest Rate Shocks

Panel B: Variance of 60-month Real Rate Expectations due to Interest Rate Shocks

47

Panel A plots the components of the variance of 60-month inflation expectations traced to interest rateshocks. Specifically, we plot the posterior distributions of Dπ|wi

t =P1

i=0Pr (st = i)Dπ|wist where

Dπ|wist =

hPnj=1Est

³∂πt+j∂wi,t+j

´i2V ar (wi,t+1)hPn

j=1Est

³∂πt+j∂wπ,t+1

´i2V ar (wπ,t+1) +

hPnj=1Est

³∂πt+j∂wi,t+1

´i2V ar (wi,t+1)

,

for n = 60. HerePn

j=1Est

³∂πt+j∂wi,t+j

´= ι1

Pnj=1 F

j−1st Gstι

02 andPn

j=1Est

³∂πt+j∂wπ,t+j

´= ι1

Pnj=1 F

j−1st Gst ι

01 are the cumulative sums of impulse responses of inflation to in-

terest rate shocks and inflation shocks, respectively.

Panel B plots the components of the variance of 60-month real rate expectations traced to interest rateshocks. Specifically, we plot posterior distributions of Dr|wi

t =P1

i=0Pr (st = i)Dr|wist where

Dr|wist =

hPnj=1Est

³∂rt+j−1∂wi,t+j

´i2V ar (wi,t+1)hPn

j=1 Est

³∂rt+j−1∂wπ,t+j

´i2V ar (wπ,t+1) +

hPnj=1Est

³∂rt+j−1∂wi,t+j

´i2V ar (wi,t+1)

,

for n = 60. HerePn

j=1Est

³∂rt+j−1∂wπ,t+j

´= ι2

Pn−1j=1 F

j−1st Gstι

01−ι1

Pnj=1 F

j−1st Gst ι

01−ι1Gstι

01, and

Pnj=1 Est

³∂rt+j−1∂wi,t+j

´= ι2

Pn−1j=1 F

j−1st Gstι

02 − ι1

Pnj=1 F

j−1st Gst ι

02 − ι1Gst ι

02, are the cumulative sums of impulse responses of the

real rate to interest rate shocks and inflation shocks, respectively.

Each graph shows posterior medians (solid) and ninety percent posterior bands (dashed), estimated pointby point. We fix V ar (wi,t+1) and V ar (wπ,t+1) at their posterior medians. The results are based on 50, 000Gibbs sampling simulations after discarding the first 5,000 draws.

48

Figure 6: Variance Ratios for Inflation and Nominal Short Rates

Panel A: Log Variance Ratio for Inflation: 60-month Horizon

Panel B: Log Variance Ratio for Nominal Short Rates: 60-month Horizon

We plot the posterior distributions of the log variance ratios at a 60-month horizon (n = 60)

ln³V R(n)π,t

´= ln

³V art

³Pnj=1 πt+j

´/Pn

j=1 V art (πt+j)´, and

ln³V R(n)i,t

´= ln

³V art

³Pnj=1 it+j

´/Pn

j=1 V art (it+j )´, after marginalizing over parameters and the latent

Markov regimes. The results are based on 50, 000 Gibbs sampling simulations after discarding the first 5, 000draws.

49

Figure 7: Conditional Volatilities of Real Returns on Bonds and Bills

Panel A: Posterior Medians - 60-month Horizon

Panel B: Differences in Conditional Volatilities - 60-month Horizon

We plot the differences between the conditional volatilities of real returns to holding a 60-month discountbond until maturity versus rolling over one-month bills for a 60-month horizon. Panel A compares posteriormedians of conditional volatilities of real returns between holding discount bonds until maturity (solid)and rolling over short-term bills (dashed), in annual percentage points. The conditional volatility of realbond returns (solid) is the same as the conditional volatility of inflation. Panel B plots posterior medians(solid) and 90 percent posterior bands (dashed), estimated point by point, for the differences in conditionalvolatilities (bonds less bills). The results are based on 50, 000 Gibbs sampling simulations after discardingthe first 5, 000 draws.

50

Figure 8: Expected Inflation

Panel A: 12-month Expected Inflation

Panel B: 60-month Expected Inflation

Panel C: Term Structure of Expected Inflation

1967:04

1968:10

1970:04

1971:10

1973:04

1974:10

1976:04

1977:10

1979:04

1980:10

1982:04

1983:10

1985:04

1986:10

1988:04

1989:10

1991:04

1992:10

1994:04

1995:10

1997:04

1998:10

2000:04

2001:10

0

2 75 4

- 2

0

2

4

6

8

1 0

1 2

1 4

% p e r a n n u m

M a t u r it y ( m o n t h )

T e r m S t r u c tu r e o f E x p e c t e d I n f l a t i o n

51

Figure 9: Expected Real Rates

Panel A: 12-month Expected Real Rates

Panel B: 60-month Expected Real Rates

Panel C: Term Structure of Expected Real Rates

1967:04

1968:10

1970:04

1971:10

1973:04

1974:10

1976:04

1977:10

1979:04

1980:10

1982:04

1983:10

1985:04

1986:10

1988:04

1989:10

1991:04

1992:10

1994:04

1995:10

1997:04

1998:10

2000:04

2001:10

0

2 75 4

- 4

- 2

0

2

4

6

8

1 0

% p e r a n n u m

M a t u r it y ( m o n t h )

T e r m S t r u c t u r e o f E x p e c t e d R e a l R a t e s

52

Figure 10: Yield Risk Premia

Panel A: 12-month Yield Risk Premium

Panel B: 60-month Yield Risk Premium

Panel C: Term Structure of Yield Risk Premium

1967:04

1968:10

1970:04

1971:10

1973:04

1974:10

1976:04

1977:10

1979:04

1980:10

1982:04

1983:10

1985:04

1986:10

1988:04

1989:10

1991:04

1992:10

1994:04

1995:10

1997:04

1998:10

2000:04

2001:10

01 22 43 64 86 0

- 6

- 4

- 2

0

2

4

6

% p e r a n n u m

M a t u r it y ( m o n t h )

T e r m S t r u c tu r e o f Y i e ld R i s k P r e m i u m

53

Figure 11: Term Structure Slopes of Expected Inflation, Real Rates,and Yield Risk Premia

Panel A: 5-year Expected Inflation minus 3-month Expected Inflation (% per annum)

Panel B: 5-year Expected Real Rates minus 3-month Expected Real Rates (% per annum)

Panel C: 5-year Yield Risk Premium minus 3-month Yield Risk Premium (% per annum)

54

Figure 12: Inflation Risk Premia

Panel A: 12-month Horizon

Panel B: 60-month Horizon

55

Figure 8 plots inflation expectations for different maturity horizons, 1nEt

³Pnj=1 πt+j

´, based on our VAR

model with regime shifts and stochastic volatility. Panels A and B plot posterior medians as well as ninetypercent posterior bands, estimated point by point, for n = 12 and 60 months, respectively. These values areexpressed in annual percentage points and are displayed with the corresponding spot rates (dotted). PanelC plots posterior means of inflation expectations for n = 1, 2, ...60 months. The results are based on 50, 000Gibbs sampling simulations after discarding the first 5,000 draws.

Figure 9 plots real rate expectations for different maturity horizons, 1nEt

³Pnj=1 rt+j

´, based on our VAR

model with regime shifts and stochastic volatility. Panels A and B plot posterior medians as well as 90percent posterior bands, estimated point by point, for n = 12 and 60 months, respectively. These values areexpressed in annual percentage points and are displayed with the corresponding spot rates (dotted). PanelC plots posterior means of real rate expectations for n = 1, 2, ...60 months. The results are based on 50, 000Gibbs sampling simulations after discarding the first 5,000 draws.

Figure 10 plots the yield risk premium, q(n)t , based on our VAR model with regime shifts and stochasticvolatility. Panels A and B plot posterior medians as well as 90 percent posterior bands, estimated point bypoint, for n = 12 and 60 months, respectively. These values are expressed in annual percentage points andare displayed with the corresponding spot rates (dotted) to facilitate comparison with the yield curve. PanelC plots posterior means of yield risk premia for n = 1,2, ...60 months. The results are based on 50,000 Gibbssampling simulations after discarding the first 5, 000 draws.

Panel A of Figure 11 plots the difference between 5-year expected inflation and 3-month expected inflation.Panel B of Figure 11 plots the difference between 5-year expected real rates and 3-month expected real rates.Panel C of Figure 11 plots the difference between the 5-year yield risk premium and the 3-month yield riskpremium. All values are in annual percentage points. Posterior medians of the differences (solid) are shownwith ninety percent posterior bands (dashed) based on our VAR model with regime shifts and stochasticvolatility. The results are based on 50, 000 Gibbs sampling simulations after discarding the first 5, 000 draws.

Figure 12 plots the estimated inflation risk premium, λ(n)st

r1nV art

³Pnj=1 πt+j

´, corresponding to the results

of Table 5. Panels A and B plot posterior medians (solid) as well as ninety percent posterior bands (dashed),estimated point by point, for n = 12 and 60 months, respectively, in annual percentage points. See the Notesto Table 5 for details on the estimation.

56

Figure 13: Effects of the Inflation Risk Premium on the Yield Risk Premium

Panel A: Effects of Reduced Inflation Persistence on the Yield Risk Premium: 60-month Horizon

Panel B: Effects of Inflation Risk Premium on the Yield Risk Premium: 60-month Horizon

Panel A and Panel B plot 12n

³V R(n)π,t − 1

´Pnj=1 V art (πt+j) and λ

(n)st

r1nV art

³Pnj=1 πt+j

´−Covt (mt+1, πt+1),

respectively, for n = 60 months. Both panels display posterior medians (solid) as well as ninety percent pos-terior bands (dashed), estimated point by point, in annual percentage points. See the Notes to Table 5 fordetails on the estimation.

57

Figure 14: Friedman Effects

Panel A: Conditional Mean and Conditional Variance of Inflation - 12-month Horizon

Panel B: Conditional Mean and Conditional Variance of Inflation - 60-month Horizon

Panels A and B plot posterior median estimates of conditional means (solid) and conditional volatilities(dotted) of inflation for 12- and 60-month horizons, respectively. The results are based on 50, 000 Gibbssampling simulations after discarding the first 5, 000 draws.

58

Appendix to

“The Conquest of U.S. Inflation: Its Implications for the FisherHypothesis and the Term Structure of Nominal Interest Rates”

A Estimation Methodology

A.1 Setup

Our dynamic model of inflation (π) and the nominal short rate (i) is estimated equation by equationeven in the presence of stochastic volatility since the residuals are orthogonalized. Each equationcan be written in the following form:

yt = X 0tβst + ut (1)

ut = yγt exp

µ1

2αt

¶εt, εt

i.i.d.∼ N (0, 1) ,

αt = µst + κ¡αt−1 − µst

¢+ ξηt, ηt

i.i.d.∼ N (0, 1) , |κ| < 1where yt is the response variable (πt+1 or it+1), Xt ≡ [1, πt, it, πt−1, it−1, πt−2, it−2]0 for the inflationequation and Xt ≡ [1, πt, it, πt−1, it−1, πt−2, it−2, πt+1]0 for the interest rate equation. Here ytrepresents the level of the response variable that may covary with its volatility. We set yt ≡ it inboth equations. The latent log variance sequence (ztTt=1 or htTt=1) is denoted by αtTt=1, whileεtTt=1 and ηtTt=1 are unit variance Gaussian white noise sequences. We assume that the regimeindicator stTt=1 is a stationary Bernoulli sequence following a two-state hidden Markov chain withtransition probability matrix:

st = 0 st = 1

st−1 = 0 p 1− pst−1 = 1 1− q q

where the probability parameters p, q ∈ (0, 1) are time invariant and unknown.

A Gibbs sampler is used to estimate each equation. Our procedure builds upon the efficient“conditionally-Gaussian” filtering algorithm developed by Carter and Kohn (1994), DeJong andShephard (1995), Kim, Shephard, and Chib (1998), Mahieu and Schotman (1998), So, Lam, andLi (1998), and Chib, Nardari, and Shephard (2002). By casting a nonlinear non-Gaussian filteringproblem into a “conditionally-Gaussian” state space model within each Gibbs sampling iteration,we are able to draw entire trajectories of αtTt=1 and stTt=1 in a single “sweep” from the respective

1

conditional joint distributions. This algorithm is far more efficient than the generic “latent variableby latent variable” non-Gaussian filtering algorithm employed by Cogley and Sargent (2003). Werun 50, 000 posterior simulations, after discarding the first 5, 000 draws.

In the following, we first describe our sampling strategies for the latent stochastic volatility processand the hidden Markov regime. We do not discuss the sampling of model parameters for thestochastic volatility and hidden Markov regime processes since they have standard full conditionaldensities. Priors for these model parameters are standard and are summarized in Table 2 ofthe paper. On the other hand, we will be more explicit about our priors for the autoregressivecoefficients βst , since they affect the stationarity of π and i. After discussing our priors for βst , wediscuss our strategy for constructing the VAR and for computing conditional second moments of πand i.

A.2 SamplingMethodology for Stochastic Volatility andHiddenMarkov Regimes

A.2.1 Stochastic Volatility

Conditional on the regime indicator stTt=1 and the regime-dependent regression coefficients βst in(1), the stochastic volatility model can be written as a non-Gaussian state space model:

ln u2t = αt + 2γ ln yt + t (2)

αt = µst + κ¡αt−1 − µst

¢+ ξηt (3)

where t ≡ ln ε2t . The distribution of t is lnχ2 with one degree of freedom with density given by1

plnχ21 ( t) =1√2πexp

µt − exp ( t)

2

¶.

Following the method proposed by Kim, Shephard, and Chib (1998), we approximate the lnχ21density by a seven component mixture of normal densities:

t|kt ∼ N (mkt , Vkt)

where kt ∈ (1, 2, 3, 4, 5, 6, 7) is the unobserved component indicator with probability mass functionPr (kt).2 Conditional3 on the indicator variables ktTt=1, our non-Gaussian state space model canbe treated as a linear Gaussian model and the Kalman filtering algorithm and the simulationsmoothing algorithm can be applied.

1Harvey, Ruiz, and Shephard (1994) rely on Quasi-Maximum Likelihood (QML) by approximating the distributionof t by a normal distribution with mean −1.2704 and variance π2/2. However, the QML estimator has very poorsmall sample properties because t is far from being Gaussian.

2We adopt Pr (kt) ,mkt , Vkt from Table 1 of Chib, Nardari, and Shephard (2002). This seven component mixtureof normal densities gives a very accurate approximation of lnχ21 even in its tails.

3The full conditional density of the unobserved component indicator kt is given by p¡kt| ln u2t , αt

¢∝

p (kt) fN¡ln u2t |αt + 2γ ln yt +mkt , Vkt

¢for each t = 1, . . . , T (e.g., Kim, Shephard, and Chib (1998), Mahieu and

Schotman (1998)).

2

A well known problem associated with the sampling of expressions (2)-(3) is the high correlationbetween the parameters γ and µst , (st = 0, 1). High dependency among parameters leads to highserial correlation in simulated samples and hence deteriorates the efficiency of the Gibbs sampler.As such, these highly correlated parameters should be simulated as one single block to reduceserial correlation in simulated samples. Accordingly, we augment the latent state vector αt withparameters γ and µst , (st = 0, 1) and introduce an augmented state vector:

α∗t =£γ µst+1=0 ∇µ αt

¤0where ∇µ ≡ µst+1=1−µst+1=0. Given the outcome of st, ktTt=1 , we cast our augmented state spacemodel corresponding to expressions (2)-(3) into a conditionally Gaussian state-space model:

ln u2t = Ztα∗t +Gtet,

α∗t+1 = Ttα∗t +Htet,

where et ≡ [ t, ηt]0 ∼ N (0, I)4 and

Zt =£2 ln yt 0 0 1

¤, Gt =

£ pVkt 0

¤

Tt =

1 0 0 00 1 0 00 0 1 00 1− κ (1− κ) st κ

, Ht =

0 00 00 00 ξ

.Each element of α∗0 is assumed independently normally distributed with variance of 100.

The forward recursion is then given by the well known Kalman filtering algorithm:

vt = ln u2t − Ztα∗t , Ft = ZtPtZ

0t +GtG

0t, Kt = TtPtZ

0tF−1t

α∗t+1 = Ttα∗t +Ktvt, Pt+1 = TtPtT

0t +HtH

0t −KtFtK

0t.

Using the output of the Kalman filter, we run the backward recursion of the simulation smootherof DeJong and Shephard (1995). Specifically, setting rT = 0 and NT = 0 for t = T, T −1, . . . , 1, andwritingDt = F−1t +K 0

tNtK0t, nt = F−1t vt−K 0

trt, Σt = GtG0t, we simulate Z1α

∗1, . . . , ZTα

∗T |©ln u2t

ªTt=1

recursively backwards in the following manner:

Ct = Σt − ΣtDtΣt, κt ∼ N (0, Ct) , Vt = Σt¡DtZt −K 0

tNtT0t

¢, Lt = Tt −KtZt,

rt−1 = Z 0tF−1t vt + L0trt − V 0tC

−1t κt, Nt−1 = Z 0tF

−1t Zt + L0tNtLt + V 0tC

−1t Vt.

We first draw ZTα∗T and obtain γ, µ (st = 0), µ (st = 1), and αT from the Gaussian posterior density

of α∗T . Then we simulate zt −Σtnt − κt − 2γ ln ytTt=1 in a single sweep from αt|©ln u2t

ªTt=1

fort = 1, 2, ..., T.

4The possible contemporaneous correlation between εt and ηt does not play a role in our estimation since t andηt are uncorrelated regardless of the correlation between εt and ηt. See Harvey, Ruiz, and Shephard (1994).

3

A.2.2 Hidden Markov Regimes

Given the simulated stochastic volatility sequence αtTt=1 and the conditionally Gaussian statespace framework for the inflation process, we are able to sample the hidden Markov Chain stTt=1as a single block using the discrete filtering algorithm of Carter and Kohn (1994).5 Let Θ andfN denote the unknown model parameters and the normal density function, respectively. Then thefollowing algorithm enables us to draw stTt=1 as a single sweep in each Gibbs sampling iteration.

For t = 1, . . . , T

1. Compute

p³st| yτ , ατt−1τ=1 ,Θ

´=Xi=0,1

p (st|st−1 = i) p³st−1 = i| yτ , ατt−1τ=1 ,Θ

´;

2. Filter:

p∗¡st| yτ , ατtτ=1 ,Θ

¢∝ p

³st| yτ , ατt−1τ=1 ,Θ

´×fN

³yt|x0tβst , y2γt exp (αt)

´×fN

³αt|µst + κ

³αjt−1 − µst

´, ξ2´;

3. Obtain p¡st| yτ , ατtτ=1

¢using

p¡st| yτ , ατtτ=1 ,Θ

¢=

p∗¡st| yτ , ατtτ=1 ,Θ

¢Pi=0,1 p

∗ ¡st = i| yτ , ατtτ=1 ,Θ¢ .

Then we are able to draw stTt=1 from p¡st| yτ , ατtτ=1

¢for t = 1, 2, ..., T, in a single sweep.

A.3 Constructing the VAR

A.3.1 Prior Specification

In order to be explicit about our priors concerning stationarity, let us use βπ,st and βi,st to denotethe regression parameter vector

¡βst¢in expression (1) for inflation and the nominal short rate,

respectively. We define βst ≡£β0π,st , β

0i,st

¤0 ∈ A, where A ⊆ R15×1 denotes the parameter space.By rearranging the two individual equations for inflation and nominal short rates, we obtain abivariate VAR for xt ≡ [πt, it]0:

xt+1 = ν¡βst¢+

3Xk=1

Φk¡βst¢xt+1−k + Ut+1. (4)

5So, Lam, and Li (1998) use this algorithm for their stochastic volatility model of stock returns.

4

In expression (4) , ν¡βst¢and Φk

¡βst¢(k = 1, 2, 3) are functions of βst .

The main prior belief that we impose is that there exists a long run equilibrium relation betweeninflation and interest rates within each regime, and hence at least one stationary linear combinationof πt and it exists within each regime. This implies that our bivariate system, expression (4),contains at most one unit root in each regime and so a stable relation prevails between inflationand the nominal short rate within a regime. This assumption is formally incorporated into our priorbeliefs in the following manner. Let zst ≡ (zst,1, zst,2, ...) denote roots of

¯I −Pp

k=1Φk¡βst¢zk−1st

¯=

0 for each st, such that |zst,1| ≤ |zst,2| ≤ . . .. Given the regime sequence st , we write the priordensity for βst as

P¡βst¢=

0P ∗¡βst¢(

1−Rβst∩A|z1|>1 P∗(βst)dβstRβst

∩A|z1|=1P∗(βst)dβst

)P ∗¡βst¢

0

if |zst,2| ≤ 1,if |zst,2| > 1 and |zst,1| > 1,

if |zst,2| > 1 and |zst,1| = 1,if |zst,2| > 1 and |zst,1| < 1,

(5)

where P ∗ is a probability density function from which βst is drawn. Notice that A can be groupedinto three non-overlapping subsets, A|z1|>1, A|z1|=1, and A|z1|<1 where A|z1|>1 corresponds to asubspace yielding stable VARs for xt while A|z1|=1 corresponds to a subspace implying that ∆xtfollows a stable system. We preclude explosive roots (|z1| < 1) in inflation and interest rateseries, that is, Φst ∩ A|z1|<1 = ∅. Expression (5) embodies our prior belief that when |zst,1| > 1then xt follows a stationary VAR in that regime while |zst,1| ≤ 1 implies that πt and it share aunit root process (common stochastic trend) but a linear combination of πt and it is stationary.This stationary combination represents the long run equilibrium relation between inflation and thenominal short rate.

At the mean of P ∗¡βst¢, we assume that πt and it are cointegrated and the common stochastic

trend is set by the nominal short rate process itself. This implies that the nominal short ratefollows an independent unit root process and hence Et (it+1) = it. Since unit root tests do notreject the null hypothesis of a unit root in nominal short rates and since previous studies oftencharacterize the nominal short rate as exogenous, our assumption is consistent with the knownbehavior of nominal short rates. It also implies that at the mean of P ∗

¡βst¢, expected inflation

equals the nominal short rate minus the mean of the ex post real short rate, as is consistent withthe long run one-for-one correspondence between Et (πt+1) and it. As a result, the mean of P ∗

¡βst¢

is characterized by the Fisher hypothesis.

Specifically, at the mean of P ∗¡βst¢, we set

Φ1¡βst¢=

·0 10 1

¸,

and Φ2¡βst¢, Φ3

¡βst¢are zero matrices. All elements of βst are assumed independently normally

distributed with standard deviations of one. With this choice of P ∗¡βst¢, the prior probability

of having a stationary VAR (i.e., Pr (|zst,1| > 1)) is approximately six percent, implying a prior

5

probability of a cointegration between πt and it of approximately ninety-four percent. Thus ourprior favors cointegration over stationarity.

A.3.2 Sampling Methodology for the VAR

By combining each simulation of the inflation equation and interest rate equation, we can constructa reduced form VAR, expression (4). The VAR can be expressed in companion form as (suppressingthe constant term):

x¯t+1

= F¡βst¢x¯t+G

¡βst¢ ewt+1,

where x¯t≡ (πt, it, πt−1, it−1, πt−2, it−2)0 , and ewt+1 is a vector of orthogonal shocks. Our prior belief,

described above, probabilistically imposes that the VAR system is stationary or cointegrated.

Let Ψ(i) ≡µn

α(i)t

oTt=1

,ns(i)t

oTt=1

,Θ(i)¶be the i-th posterior sample of parameters including the

latent stochastic volatilities and Markov regimes, where Θ(i) is a set of all model parameters includ-ing β

(i)st . We also let

¯e(i)st,1

¯,¯e(i)st,2

¯, ... be the absolute values of eigenvalues of F

³β(i)st

´such that¯

e(i)st,1

¯≥¯e(i)st,2

¯≥ .... In order to incorporate our prior belief, we implement the following sampling

procedure. First, we compute the VAR system matrix F³β(i)st

´and corresponding eigenvalues for

each Ψ(i) (i = 1, 2, ...).6 If¯e(i)st,1

¯< 1 we accept and store Ψ(i). If

¯e(i)st,1

¯≥ 1, we reestimate our

VAR in an error correction form and the Ψ∗(i) thus obtained satisfies¯e(i)st,1

¯= 1. With our prior

specification, we accept and store Ψ∗(i) with probability one. This ensures that our VAR satisfies¯e(i)st,1

¯< 1 (stationary) or

¯e(i)st,1

¯= 1 (difference stationary). We then proceed to the next posterior

simulation to obtain Ψ(i+1) (or Ψ∗(i+1)) until we obtain 50, 000 posterior simulated samples, afterdiscarding the first 5, 000 simulations.

A.3.3 Conditional Second Moments

The n-period innovation in xt ≡ [πt, it]0 , given the regime at the time of the forecast st, is givenby7

nXh=1

xt+h −Et

ÃnX

h=1

xt+h

!=

nXh=1

n−hXl=0

F¡βst¢lG¡βst¢wt+h.

Consequently, we can express the conditional covariance matrix of xt, given st, as

nXh=1

Ãn−hXl=0

F¡βst¢l!

G¡βst¢Ωt+h|stG

¡βst¢0Ãn−hX

l=0

F¡βst¢l!0 ,

6 If¯e(i)st,2

¯≥ 1, we discard Ψ(i), though this never happened in our simulation.

7Note that F−1st does not exist when the system is cointegrated.

6

where Ωt+h|st ≡ Et

£wt+hw

0t+h

¤is the conditional forecast of the future covariance matrix made in

regime st. Letting Ω(i,j)t+h|st be the (i, j)-th element of Ωt+h|st (i, j ∈ (1, 2)) ,

Ω(1,1)t+h|st = Et

hi2γπt+h−1

iexp

(³I − κhπ

´µπ,st + κhπzt +

1

2ξ2π

µ1− κhπ1− κπ

¶2),

Ω(1,2)t+h|st = Ω

(2,1)t+h|st = ρstΩ

(1,1)t+h|st ,

Ω(1,1)t+h|st = ρ2stΩ

(1,1)t+h|t +Et

hi2γit+h−1

iexp

(³I − κhi

´µi,st + κhi ht +

1

2ξ2i

µ1− κhi1− κi

¶2),

where¡γπ, µπ,st , κπ, ξπ

¢and

¡γi, µi,st , κi, ξi

¢are the model parameters for the stochastic volatility

of inflation shocks and interest rate shocks, respectively. We evaluate Et

hi2γπt+h−1

iand Et

hi2γit+h−1

iby a first order Taylor approximation around i

2γπt and i

2γit .

References

[1] Carter, C.K. and R. Kohn, 1994, “On Gibbs sampling for State Space Models,” Biometrika, 81,541-553.

[2] Chib,S., F. Nardari, and N. Shephard, 2002, “Markov Chain Monte Carlo Methods for Stochas-tic Volatility Models,” Journal of Econometrics, 108, 281-316.

[3] Cogley, T. and T. Sargent, 2003, “Drifts and Volatilities: Monetary Policies and Outcomes inthe Post WWII U.S.,” working paper.

[4] DeJong, P., and N. Shephard, 1995, “The Simulation Smoother for Time Series Models,” Bio-metrika, 82, 339-350.

[5] Harvey, A., E. Ruiz, and N. Shephard, 1996, “Multivariate Stochastic Variance Models,” Reviewof Economic Studies, 61, 247-264.

[6] Kim, S., N. Shephard and S. Chib, 1998, “Stochastic Volatility: Likelihood Inference andComparison with ARCH Models,” Review of Economic Studies 65: 361-394.

[7] Mahieu, R.J. and P. Schotman, 1998, “An Empirical Application of Stochastic Volatility Mod-els,” Journal of Applied Econometrics 13, 333-360.

[8] Phillips, P.C.B., 1991, “To Criticize the Critics,” Journal of Applied Econometrics, 6(4), 333-364.

[9] So, M.K.P., K. Lam, and W.K. Li, 1998, “A Stochastic Volatility Model with Markov Switch-ing,” Journal of Business and Economic Statistics, 16-2, 244-253.

7