Macro-finance VARs and bond risk premia: A caveat

Review of Financial Economics 18 (2009) 163–171

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Review of Financial Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / r f e

Macro-finance VARs and bond risk premia: A caveat☆ Marco Taboga ⁎ Economic Outlook and Monetary Policy Department, Bank of Italy, Via Nazionale 91 00184, Roma, Italy

a r t i c l e

i n f o

Article history: Received 21 April 2008 Revised 27 January 2009 Accepted 20 May 2009 Available online 23 June 2009 JEL classification: G12 C32

a b s t r a c t At the turn of the century, US and euro area long-term bond yields experienced a remarkable decline and remained at historically low levels despite rising short-term rates (the so called “conundrum”). Estimating macro-finance VARs and no-arbitrage term structure models, many researchers find that the decline in long-term rates was primarily driven by an unprecedented reduction in risk premia. I show that this result might be an artefact of the class of models employed to study the phenomenon. © 2009 Elsevier Inc. All rights reserved.

Keywords: Bond risk premia Term structure Bond yield conundrum

1. Introduction The sharp decline in long-term interest rates, which occurred at the turn of the century in the major industrialized countries, has attracted considerable attention from both academic researchers and policy makers. Especially striking is the fact that long-term rates remained low despite rising short-term rates, which former Federal Reserve chairman Alan Greenspan dubbed a “conundrum”.1 A commonly held opinion is that the low level of long-term rates could be largely explained by a decline in the compensation for risk. This opinion is supported by a great deal of empirical work devoted to the measurement of bond risk premia and the analysis of their dynamics (e.g.: Kim & Wright, 2005; Kremer & Rostagno, 2006; Backus & Wright, 2007). Mixed evidence is provided by Rudebusch, Swanson, and Wu (2006). In this paper, I argue that models employed to study recent developments in long-term rates might fail to capture permanent

☆ The views expressed in this article are those of the author and do not necessarily reflect the views of the Bank of Italy. I wish to thank Jonathan Moore for editorial assistance and the following people for their helpful comments: Paolo Angelini, Umberto Cherubini, Giuseppe Grande, Stefano Neri, Fabio Panetta and Marcello Pericoli. Also, I wish to thank the anonymous referee and participants at the Bank of Italy seminars and at the “New Directions in Term Structure Modelling” conference. ⁎ Tel.: +39 33 56590133. E-mail address: [email protected]. 1 “Long-term interest rates have trended lower in recent months even as the Federal Reserve has raised the level of the target federal funds rate by 150 basis points. This development contrasts with most experience, which suggests that, other things being equal, increasing short-term interest rates are normally accompanied by a rise in longer-term yields... for the moment, the broadly unanticipated behavior of world bond markets remains a conundrum” (Testimony of federal Reserve Board Chairman Alan Greenspan to the U.S. Senate, February 16th, 2005). 1058-3300/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.rfe.2009.06.002

changes in the economy that have contributed to lower bond yields; as a consequence, these models might underestimate the level of risk premia in more recent years and overemphasize the severity of the bond yield conundrum. Standard term structure theory identifies two main drivers behind movements in long-term rates: changes in the expected future path of the short-term rate and fluctuations in risk premia. I provide evidence that standard term structure models, based on stationary vector autoregressions (VARs),2 might be unable to detect long-lasting shifts in the expectations about the short-term rate. These models tend to attribute most of the variability in long-term forward interest rates to forward premia and little or almost none to changing expectations. The technical reason for this problem is that estimated VARs often display a fast speed of convergence to their unique equilibrium and produce forecasts of the short-term rate at long horizons which remain roughly constant in time and are hardly different from their historical averages. I propose to use a simple macro-econometric model in which permanent shifts in the natural real rate of interest,3 the growth rate of potential output and long-run inflation determine long-lasting changes in expectations about the short-term rate. Any permanent change in one of these three variables determines a new set of equilibrium values towards which all the other macroeconomic variables of interest tend to converge. 2 The same comments apply to VARs continuos-time counterparts, such as multifactor Vasicek (1977) and Cox, Ingersoll, and Ross (1985) models. 3 In the model I propose, which is an extension of the model by Laubach and Williams (2003), the natural rate of interest is defined as in Wicksell (1936) as the real short-term interest rate consistent with a macroeconomic equilibrium in which output equals its potential and inflation is constant.

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The estimates from my model suggest that in recent years a reduction in the real natural rate of interest and in inflation expectations might have lowered the long-term forecasts of the level of policy rates, both in the US and in the euro area. According to my model, this might have been an important cause of the fall in long-term bond yields, which, if overlooked, might lead to erroneously attribute the fall to a collapse of risk premia. The main inference to be drawn from the model is that, although risk premia did diminish, the fall was probably not dramatic and their level after the turn of the century was not unusual if considered from an historical perspective. The paper is organized as follows: Section 2 discusses the challenges related to the estimation of risk premia and the possible drawbacks of standard econometric models; Section 3 presents the model I use to measure risk premia; Section 4 discusses some details of the estimation method; Section 5 contains my comments on the empirical evidence obtained with the model; and the Appendix reports the technical details of the model. 2. The motivation In this section I discuss the potential problems of estimating bond risk premia with standard stationary macro-finance VARs (and their continuous-time counterparts; see footnote 3) to estimate bond risk premia. A related discussion can be found in Kim and Orphanides (2005). Among the many possible measures of the risk premium, I focus on the simplest and most straightforward: the forward premium, i.e. the difference between ft,t + n (the forward rate at which agents agree at time t to exchange funds at time t + n for one period) and Et[it + n] (the expected value of it + n given information available at time t, where it + n is the short-term rate at which agents will agree to exchange funds at time t + n for the next period). Thus, the n-periods ahead forward premium φt,t + n is defined as: φt;t

+ n

= ft;t

+ n

− Et ½it

+ n

ð1Þ

Focusing on the forward premium is without loss of generality, because any bond yield can be decomposed into a sequence of forward rates at which agents agree today to exchange funds in the future. Since forward rates ft,t + n are observable from market prices, the above equation makes clear that the whole challenge of measuring forward premia lies in accurately estimating unobservable expectations Et[it + n] of the short-term rates likely to prevail in the future. In order to estimate Et[it + n], a possibility would be to directly survey market participants' expectations about future levels of the interest rate. However, most existing studies rely on econometric models to estimate expectations about interest rates. A reason for doing so is that the typically available survey data have limited historical depth, so that their time-series properties and their reliability are difficult, if not impossible, to study. On the contrary, simple macro-econometric models use long time-series of readily available macroeconomic data, allowing the researcher to achieve sufficient historical perspective when studying the behavior of expectations and risk premia. Furthermore, some papers (e.g.: Friedman,1980; Froot,1989) provide evidence that the informative content of interest rate survey forecasts may be questionable. Most of the econometric models which have been employed to analyze the term structure of interest rates share a similar structure: they specify the joint dynamics of the short-term interest rate and of a small number of other variables as a vector autoregression (or as a continuous-time model which can be discretized so as to yield a vector autoregression — see footnote 3). The other variables in the VAR are usually macroeconomic variables, such as inflation and the output gap, and variables explicitly or implicitly related to the shape of the yield curve. For example, Estrella and Mishkin (1997) and Evans and Marshall (1998) estimate VARs with yields of various maturities and macroeconomic variables. Numerous studies estimate no-arbitrage latent factor models in which macroeconomic variables do not play

any role (e.g.: Duffie & Kan, 1996; Dai & Singleton, 2000). A number of recent papers, starting with the seminal work by Ang and Piazzesi (2003), jointly model both macroeconomic variables and (latent) variables related to the shape of the yield curve as VARs with full noarbitrage restrictions (e.g.: Rudebusch & Wu, 2004; Hördal, Tristani, & Vestin, 2006; Ang, Piazzesi, & Wei, 2006). All the aforementioned models can in principle be used to estimate expectations of the future path of short-term rates. However, as carefully illustrated by Kim and Orphanides (2005), estimates of long-horizon expectations provided by these models can be seriously misleading. Despite ample empirical and anecdotal evidence that considerable structural changes took place in the major industrialized economies and reduced long-run expectations of the short rate (e.g.: Campbell & Viceira, 2001; Clarida, Gali, & Gertler, 2000; Cogley & Sargent, 2001; Derby, 2004; Goto & Torous, 2003), most of these models provide estimates of longhorizon expectations which remain virtually unchanged across time. To capture structural breaks and produce enough variation in long-term expectations, estimated VARs need at least one highly persistent factor with a long half-life. However, it is well known that the very presence of such persistent factors makes estimation of VARs problematic, because of finite sample biases and inefficiencies of standard estimators. Quite often, if the true data-generating process has persistent factors, this persistence is unlikely to show up in the estimates. Kim and Orphanides (2005), for example, estimate a standard no-arbitrage term structure model that allows for one or more persistent factors. They show that, despite the presence of obvious structural breaks in their sample, their model is unable to detect them and produces estimates of long-run expectations that are roughly constant across time. Furthermore, the choice of the sample length is bound to heavily influence the results and the level of estimated expectations. Below, I provide further evidence in this sense. In Fig. 1, I plot the nine-year ahead expectations of the short rate obtained from a standard macro-VAR with 12 lags of inflation, of the output gap and of the short rate, estimated from quarterly US data covering the period 1965–2006. The same figure also displays the plot of the nine-year ahead forward rate. The difference between the two series is an estimate of the forward premium. The measure of expectations thus obtained is very stable throughout the whole period: approximately 80% of the values are between 6% and 7% and the average expectation is 6.67%, which almost coincides with the sample average of the short-term rate (6.66%). After the year 2000, model-implied expectations have on average been roughly equal to 6.5%. This is in sharp contrast with statements from policy makers, who suggest a range of 3.5% to 4.5% (e.g.: Derby, 2004; Hoenig, 2005), and with survey forecasts (Kim & Orphanides, 2005). Reducing the sample period, in order to mitigate the problems arising from structural breaks, does not seem to produce improvements: estimated expectations still remain roughly constant across time, but their average shifts considerably. These shifts point to a severe lack of robustness with respect to sample choice. If instead of a standard macro-VAR, I estimate a no-arbitrage VARs with both macroeconomic and latent variables (in the canonical form suggested by Pericoli & Taboga, 2008), which in principle should be able to identify persistent factors and capture structural change, then I get results which are slightly better, in the sense that the model produces a somewhat greater variability of expectations (see Fig. 1). Nonetheless, the estimated expectations still look very stable (if compared with forward rates), they are at odds with market participants' and policymaker's perceptions during the most recent period and they also display limited robustness to sample choice. Estimates for the Euro area of both standard and no-arbitrage VARs (Fig. 2) present drawbacks that are similar to those of the estimates carried out for the US and commented on above. As a solution to the aforementioned problems, I propose a simple model where some key variables are affected by permanent shocks which dynamically induce shifts in macroeconomic equilibria. I identify long-run inflation expectations, the real natural rate of interest and the growth rate of potential output as the sources of persistence that have

M. Taboga / Review of Financial Economics 18 (2009) 163–171

165

Fig. 1. The expected short rate nine-year ahead and the forward rate — USA (75-06).

the potential to capture structural change and generate the desired variation in long-term expectations. To impose persistence on the variables driving structural change, I model the corresponding stochastic processes as random walks, in such a way that the resulting economic dynamics are not explosive, but rather allow for time-varying equilibria. 3. The model My model is an extension of the model used by Laubach and Williams (2003) to estimate the natural rate of interest. According to the Wicksellian definition (Wicksell, 1936), the natural rate of interest is the real short-term interest rate consistent with a macroeconomic equilibrium in which output equals its potential and inflation is constant. The natural rate varies through time in response to structural changes in the economy. Most theoretical frameworks show that the natural rate is dependent on productivity growth, on population growth, on the subjective discount factor of individuals, on their elasticity of intertemporal substitution and on other characteristics of agents' preferences and production technologies that tend to be subject to permanent shocks. While Laubach and Williams do not specify an equation for the nominal short-term interest rate and estimate the real natural rate of

interest only by an IS equation and an inflation equation, I do add an equation for the nominal short-term interest rate to their model. This equation enables me to recover interest rate dynamics and their dependence on the real natural rate of interest and other macroeconomic variables. As a result, I have a device through which changes in the real natural rate of interest are transmitted to the nominal short-term rate and to expectations about its path in the long-run. Furthermore, while Laubach and Williams use proxies for inflation expectations, I use proper expectations, that are consistently derived within the model. The first building block of the model is a reduced form IS equation (see e.g. Rudebusch & Svensson, 1998, 1999): xt = α1 xt − 1 + α2 xt − 2 +

αr 2 4 ∑ ðr − rt − j Þ + εx;t 2 j=1 t −j

ð2Þ

where xt is the output gap, rt is the ex-ante real policy rate, r*t is the real natural rate of interest and εx,t is a serially uncorrelated shock. Provided αr is negative, when the real policy rate rt is above (below) the natural rate r*t , output tends to decrease (increase). Eq. (2) makes the defining property of the real natural rate of interest r*t explicit: in the absence of shocks εx,t, the economy can reach an equilibrium

Fig. 2. The expected short rate nine-year ahead and the forward rate — Euro area (80-06).

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M. Taboga / Review of Financial Economics 18 (2009) 163–171

where output grows along its potential (xt = 0) only if the real interest rate rt is equal to the natural rate r*t . Stated differently, r*t is the real rate of interest that allows for an equilibrium in which the demand for goods matches the supply and no adjustments in production are necessary. Following Laubach and Williams (2003) and Rudebusch and Svensson (1998, 1999), I include two lags of the output gap in Eq. (2) to control for short-run dynamics. The output gap is defined as: 4

xt = yt − yt

ð3Þ

where yt and y*t are the logarithms of output and potential output respectively. The ex-ante real policy rate is the difference between the policy rate it and inflation expectations: rt = it − Et ½πt

+ 1

ð4Þ

Inflation πt is determined by its own lags, lagged output gap and a serially uncorrelated shock: 8

πt = ∑ βj πt − j + βx xt − 1 + επ;t

ð5Þ

j=1

The lags of inflation in the above equation can be thought of as an adaptive or autoregressive representation of inflation expectations. In my quarterly model, it implies that inflation observed during the previous two years is employed to forecast future inflation. The 8

constraints ∑ βj = 1 and βj ≥ 0,∀j are imposed to ensure verticality j=1

of the Phillips curve in the long-run (see e.g. Rudebusch & Svensson, 1998, 1999): long-run verticality means that any level of inflation can be compatible with an economic equilibrium where output equals its potential. Eq. (5), together with (2), implies that the average level of inflation is indirectly controllable by a monetary authority that sets the real interest rate rt: when βx N 0, if the monetary authority desires to reduce the level of inflation, then it can raise the interest rate causing a contraction in output (Eq. (2)). The reduction of the output gap xt thus generated will then put downward pressure on prices (Eq. (5)). Besides being able to capture rather rich inflation dynamics, the above specification of the Phillips curve is consistent with the Wicksellian definition of natural rate. Together with Eqs. (2) and (6), it implies that, absent any stochastic shock to the economy, an equilibrium, if it exists,4 is characterized by the following features: 1) the real short-term rate equals the real natural rate; 2) output converges to its potential; and 3) inflation converges to a constant level. The specification of the Phillips curve in Eq. (5) is similar to that used in the literature to estimate the natural rate of unemployment (e.g.: Gordon, 1998; Brayton, Roberts, & Williams, 1999; Laubach, 2001). I stick to the original backward-looking specification proposed by Laubach and Williams (2003), because it allows for a closed-form solution of the model (see the Appendix A). Adjustments in the nominal short-term rate happen in accordance with the following equation: it = γit − 1 + ð1 − where n

4

int

n γÞðit − 1

+ γπ ð π ¯t − 1 −

4 πt − 1 Þ

+ γx xt − 1 Þ + εi;t ð6Þ

is the nominal natural rate of interest, defined as:

it = rt + Et ½πt

+ 1

ð7Þ

4 αrb0 and βxN0 are necessary conditions for the validity of our characterization of an equilibrium. They are however, not sufficient. For the system to converge to an equilibrium in the absence of stochastic shocks, the companion form of the model must have all its eigenvalues on or inside the unit circle. In the Empirical section of the paper I will further discuss this point and check that the condition holds in the estimated models.

π*t is long-run inflation: 4

πt = πt

− 1

+ επ4 ;t

ð8Þ

and π ̅t is a measure of realized inflation, defined as: π ¯t =

1 3 ∑π 4 j=0 t −j

ð9Þ

Eq. (6) could be interpreted as a smoothing policy rule (e.g.: Bjornland, Leitemo, & Maih, 2006), whereby the central bank gradually raises interest rates when either output is above its potential or realized inflation is above the desired long-run level πt*. Provided γπ and γx are strictly positive and γ is strictly less than 1, (6) guarantees that, in the absence of transitory shocks, the economy converges towards an equilibrium where output equals its potential, inflation is equal to its long-run level and the real short-term rate equals the real natural rate of interest. This mechanism allows for multiple equilibria that differ as to the level of inflation and the interest rate. Multiple equilibria are the key feature of the model, that makes it suitable to reproduce the structural changes and the permanent shifts in long-run expectations not captured by standard stationary VARs. Specifying long-run inflation πt* as a random walk, I allow for permanent unpredictable changes in the equilibrium level of inflation. The variance of the innovation επ*,t determines the expected magnitude of these changes. Long-run inflation πt* may be thought of as a statistical device that captures other economic variables, like money growth, that are not explicitly included in the model. A permanent increase in money growth, for example, can be mimicked by a positive shock to πt*; the shock results in a reduction of the differential between realized inflation π t̅ and long-run (or targeted) inflation πt*, which, in turn, causes a reduction of the policy interest rate. This reduction is then be transmitted to output (via the IS schedule) and to inflation (via the Phillips' curve) and causes a permanent increase in the inflation rate. The potential output grows at a rate gt 4

4

yt = yt

− 1

+ gt − 1

ð10Þ

that varies through time as a random walk: gt = gt − 1 + εg;t

ð11Þ

Finally, the natural real rate of interest is the sum of two terms: rt = cgt + zt

ð12Þ

where c is a constant related to the elasticity of intertemporal substitution and zt is a shock that captures other determinants of r*t such as the subjective discount factor. Also zt is specified as a random walk: zt = zt − 1 + εz;t

ð13Þ

Eq. (12) can be derived, for example, within a standard Solow or Ramsey growth model (e.g.: Barro & Sala-i-Martin, 1999), where it is related either to a balanced growth condition or to an optimality condition for savings. In order to be able to jointly estimate the forward premium and the macroeconomic dynamics specified by the above equations, I add to the system an exogenous process for the forward premium: φt = φt − 1 + εφ;t

ð14Þ

The model can be written explicitly in companion form as a first order vector autoregression, by substituting inflation expectations with their value derived from (5). The details are reported in the Appendix A.

M. Taboga / Review of Financial Economics 18 (2009) 163–171

4. The estimation strategy

Table 1 Data and data sources.

I estimate the model by maximum likelihood, with both US and euro area quarterly data (see Table 1 for details). I adopt the methodology proposed by Boivin and Giannoni (2005) for estimation in data-rich environments: all the variables in the model are treated as unobservable, in order to allow for measurement errors and for the possibility that many data series are available for a single economic concept.5 An observation equation is defined for any data series, so that there can be multiple observation equations for a single variable. The observation equations are used to infer the values of the variables of the model via the Kalman filter. As previously observed, my model can be written in companion form as a first order vector autoregression:

Variable

ξt = Fξt − 1 + vt

ð15Þ

where ξt is a column vector that contains all the variables of the model and some of their lags: 4

4

ξt = ½xt πt it yt zt gt πt xt − 1 gt − 1 zt − 1 πt − 1 …πt − 8 it − 1 φt 

⊤

ð16Þ

Euro Area Real GDP Potential output GDP at factor costs deflator

F is a matrix whose entries are determined by the solution of the model in the previous section (the functional form of the coefficients is reported in Appendix A) and vt is a vector of error terms. All the variables in ξt are treated as unobservable. Their values are estimated by the Kalman filter, using a set of observation equations: yt = Hξt + wt

United States US GDP (ar) cona US CBO forecast–potential GDP (real) cona US CPI–all urban: all items, sadj US CPI–all items less food & energy (core), sadj US treasury bill 2nd market 3 month-middle rate US zero coupon curve (FED estimates) Consumer prices, long-term forecasts

ð17Þ

where yt is the vector of observed data series used to measure the unobservable variables, H is a matrix of loadings (see the Appendix A for details) and wt is a vector of error terms. The procedure proposed by Boivin and Giannoni (2005) is motivated by the observation that: 1) many measures of economic variables are likely to be affected by measurement error; 2) there may be conceptual differences between model variables and the data series used to measure them; 3) there may be many different data series that correspond to a unique economic concept; 4) one might want to use data series which are proxies for unobservable or only partially observable variables. I choose to adopt Boivin and Giannoni's (2005) procedure, because I have many measures of inflation and all of them are potentially relevant to my model; furthermore, I have estimates and proxies of some unobservable variables (potential output, long-run inflation) that I want to exploit in order to achieve a better identification of unobservables. The vector yt includes: real GDP, an estimate of potential GDP, two different measures of inflation, the three-month interest rate, the nine-year-ahead forward interest rate6 and a consensus forecast of long-run inflation7 (a detailed list of the data series and their sources, both for the US and for the euro area, is reported in Table 1). 5. Empirical evidence In all tables and figures, the model described in Section 3 is referred to as “model with multiple equilibria”, to distinguish it from traditional

5 In what follows, I use the same terminology of Boivin and Giannoni (2005): an ‘economic variable’ is one of the variables in the model (whose concept is uniquely defined); a ‘data series’ is one of the many possible measures of the variable. For example, if inflation is a variable in the model, many data series could be used to measure it: CPI (consumer prices) inflation, core-CPI, GDP deflator, PCE (personal consumption expenditures) deflator, etc. 6 It is standard practice in the literature to take the forward premium on the nineyear ahead forward rate as a measure of the risk premium embedded in long-term interest rates (e.g.: Kim & Orphanides, 2005). 7 Both for the US and for the euro area, consensus forecasts of long-run inflation are available only for a subsample of the whole sample period. I nonetheless use them by augmenting the set of observation equations after they become available.

HICP Short-term interest rate Long-term interest rate Consumer prices, long-term forecasts a

167

Source

Sample period

Acronym

Datastream Datastream

1961–2006 1961–2006

USGDP...D USFCGDPPD

Datastream

1961–2006

USCONPRCE

Datastream

1961–2006

USCPCOREE

Datastream

1961–2006

FRTBS3M

BIS Databank (BIS_Macro) Consensus economics

1961–2006

HSLA,HSMA

1991–2006

N/A

Euro area-wide model databasea Euro area-wide model database Euro area-wide model database Euro area-wide model database Euro area-wide model database Euro area-wide model database Consensus economics

1970–2006

YER

1970–2006

YET

1970–2006

YFD

1970–2006

HICP

1970–2006

STN

1970–2006

LTN

1991–2006

N/A

For a complete description of the dataset see Fagan, Henry, and Mestre (2001).

VAR models with a unique equilibrium. Table 2 reports the estimates of the model. Both for the US and for the euro area, the estimated companion matrix F (Eq. (15)) has five unitary eigenvalues (corresponding to the five integrated processes yt*, zt, gt, πt* and φt), while the remaining eigenvalues are all inside the unit circle. Therefore, the system does not feature explosive dynamics: as explained in Section 3, in the absence of stochastic shocks, the economy tends to converge towards an equilibrium where output equals its potential, inflation is equal to its long-run level and the real short-term rate equals the real natural rate of interest. This mechanism allows for multiple equilibria that differ as to the level of inflation and the interest rate. Convergence to an equilibrium is guaranteed by the stabilization mechanisms jointly provided by the IS schedule, the Phillips' curve and the interest rate rule. For both the US and the euro area, αr is negative and statistically different from zero at 99% confidence,8 meaning that, on average, output tends to decrease when the real policy rate is above the real natural rate. βx is positive (and significantly so): as a consequence, inflation increases (decreases) when output is above (below) trend. Finally, both γx and γπ are positive: policy rates are raised both when output exceeds its potential and when inflation is above its long-run level. For the US, the estimates of γx and γπ are quite noisy, so that, individually, they are not statistically significant. However, a LR test of their joint significance rejects at all conventional levels of confidence the null hypothesis that they are simultaneously equal to zero. It is also interesting to note that, according to the estimates, the interest rate rule is essential for the stability of the system in the euro area: in fact, without the stabilization mechanism provided by the interest rate rule, the dynamics of the output gap generated by the

8 Unless otherwise stated, the level of confidence is 99% in all statistical tests commented in this section.

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M. Taboga / Review of Financial Economics 18 (2009) 163–171

Table 2 Parameter estimates. US α1 α2 αr β1 β2 β3 β4 β5 β6 β7 β8 βx γ γx γπ c σx σπ σi σz σg σπ* σφ τ1 τ2 τ3 τ4 τ5 τ6 τ7

Euro area

Estimate

St. Dev.

Estimate

St. Dev.

1.1096 − 0.1893 − 0.3705 0.3624 0 0.4911 0 0 0.0402 0 0.1063 0.2434 0.2035 0.2134 0.2413 1.5351 0.0072 0.0110 0.0055 0.0054 0.0002 0.0009 0.0138 0.0119 0.0103 0.0018 0 0 0.0001 0

0.0539 0.0455 0.1207 0.0732 0.0565 0.0231 0.0736 0.0365 0.1259 0.0010 0.0956 0.0006 0.2802 0.1667 0.2450 0.2363 0.0001 0.0004 0.0008 0.0004 0.0001 0.0001 0.0007 0.0007 0.0011 0.0001 0.0012 0.0028 0.0002 0.0005

1.5336 − 0.4851 − 0.1842 0.8054 0.1907 0.0038 0.0001 0 0 0 0 0.0468 0.6529 0.7798 0.3462 0 0.0034 0.0036 0.0054 0.0034 0.0002 0.0009 0.0036 0.0132 0.0160 0.0024 0.0001 0 0 0

0.0173 0.0229 0.0168 0.0593 0.0344 0.0112 0.0202 0.0047 0.0098 0.0035 0.0057 0.0004 0.0075 0.0279 0.1056 0.0488 0.0004 0.0006 0.0003 0.0004 0.0003 0.0002 0.0007 0.0009 0.0005 0.0001 0.0006 0.0002 0.0003 0.0003

IS curve would be explosive (α1 +α2 N1). In the United States, on the contrary, even in the absence of an interest rate rule that tends to stabilize output, the IS curve would be stable (α1 +α2 b1). I also obtain maximum likelihood estimates of the standard deviations of the shocks to the three variables zt, gt and π*t that drive the time-variation of the equilibria and of the long-run expectations (the standard deviations are denoted by σz, σg and σπ*, respectively).

For both countries, σg is small and statistically not different from zero, indicating that I am not able to reject the hypothesis that the growth rate of potential output is constant: this result is in accordance with the evidence provided by Laubach and Williams (2003) for the US. On the contrary, σz is large and statistically significant. Since the natural rate r*t is a linear combination of the growth rate of output gt and of shocks to preferences zt (Eq. (12)), the estimates σg and σz, together with the estimates of c (see Table 2), suggest that most of the variability of the natural rate is driven by shocks to intertemporal preferences, while the growth rate of output plays a minor role. This finding is consistent with the evidence provided by other models of the natural rate of interest estimated by Neri and Taboga (2007). Furthermore, by running a χ2-test, I obtain a rejection of the hypothesis that σ2z = 0, which also constitutes a rejection of the hypothesis that the natural rate is constant in time. Finally, also σπ* is statistically different from zero, providing support to the hypothesis that there are persistent changes in the equilibrium level of inflation. As far as the dynamics of the variables of interest in the two estimated models are concerned, similar pictures emerge for both the US and the euro area. Long-run inflation expectations and the real natural rate (Fig. 3) underwent significant changes throughout the sample period. Since the mid-eighties long-run inflation expectations π*t have steadily declined, reaching levels close to 2% in both countries. Also the real natural rate of interest has declined, although with a much more volatile pattern, consistent with the fact that σg is larger than σπ* (see Table 2). As a result, the long-run expectations of the nominal short-term interest rate exhibit a clear downward trend, while also displaying a fair degree of variability. This finding is in sharp contrast with the evidence provided by the traditional macro-VARs and no-arbitrage term structure models (Figs. 1 and 2), according to which the expectations remain remarkably stable throughout the whole sample. Over the entire sample, the standard deviation of the expected short rate nine-year ahead is 0.42% in the macro-VAR and 0.67% in the no-arbitrage VAR, while it is 3.38% in my model. According to the estimated models, between 1990 and 2006, long-run inflation decreased from 4.50% to 1.95% in the euro area and from 4.40% to 2.30% in the US; the real natural rate decreased from 5.70% to 1.65% in the euro area and from 2.90% to 1.90% in the US. Taking these developments into account, I obtain estimates of the forward premium (Figs. 4 and 5) which are materially different from those

Fig. 3. Expected long-run inflation and the real natural rate.

M. Taboga / Review of Financial Economics 18 (2009) 163–171

169

Fig. 4. The estimated forward premium — USA (75-06).

provided by standard models, especially as far as the last decade is concerned. The estimated forward premium dropped sharply after 2002 in both economies, reaching levels that are positive and not unusual from an historical perspective. At the end of 2006, the nineyear ahead forward premium was estimated to be 0.45% in the euro area and 1.90% in the US. Furthermore, in the macro-VAR and in the no-arbitrage VAR the standard deviation of the forward premium is 3.21% and 2.85%, respectively, whereas in my model it is lower (2.55%): these differences reflect the fact that a greater portion of the variability of the forward rate is explained by the variability of expectations. The above estimates suggest that, although some studies correctly identify a fall in risk premia as an important cause of the recent pronounced reduction in bond yields, they might tend to exaggerate the phenomenon, because they might overlook shifts in expectations that hardly allow for a direct comparison of the current level of bond yields with that of 10 or 20 years ago.

6. Conclusions Decomposing forward interest rates into forward premia and expectations of spot interest rates that will prevail in the future is key to understanding the dynamics of long-term bond yields. I have shown that the estimates of distant-horizon expectations provided by commonly employed VAR models display little or no variability and tend to attribute most of the variabilty in forward rates to forward premia. This is due to a lack of persistence of estimated shocks to shortterm interest rates and macroeconomic variables. I propose a model that allows for more persistence and allows to capture substantial changes in distant-horizon expectations. Employing my model to study the historical evolution of long-term rates, I conclude that the sharp decline in long-term interest rates experienced at the turn of the century in the major industrialized countries might have been due to a progressive fall in expected future rates rather than to a collapse in risk premia, as other models seem to suggest.

Fig. 5. The estimated forward premium — Euro area (80-06).

170

M. Taboga / Review of Financial Economics 18 (2009) 163–171

Appendix A In order to write the model explicitly, I substitute inflation expectations in the IS equation and in the interest rate equation with their explicit expression derived from 2(5): xt = α1 xt − 1 + α2 xt − 2 +

αr 2 4 − rt − j Þ ∑ ðr 2 j=1 t −j

All the remaining entries are zero. The vector of error terms vt is:

= α1 xt − 1 + α2 xt − 2 +

αr − Et − j ½πt − j ∑ ði 2 j=1 t −j

= α1 xt − 1 + α2 xt − 2 +

αr α α 4 α 4 + r it − 2 − r rt − 1 − r rt − 2 i 2 t −1 2 2 2

−

αr 2

ð

8

2

Þ

j=1

  α = α1 − r βx xt − 1 2

4 + 1  − rt − j Þ

ð

vt = ½εx;t επ;t εi;t 0 εz;t εg;t επ ;t 0 …0 εφ;t  *

Þ

9 αr ∑ βj − 1 πt − j + βx xt − 2 2 j=2   α α α α α + α2 − r βx xt − 2 + r it − 1 + r it − 2 − r zt − 1 − r zt − 2 2 2 2 2 2

∑ βj πt − j + βx xt − 1 −

−

αr α α − r cgt − 2 − r ðβ1 πt − 1 + ðβ1 + β2 Þπt − 2 + ðβ2 + β3 Þπt − 3 Þ cg 2 t −1 2 2

−

αr ððβ3 + β4 Þπt − 4 + ðβ4 + β5 Þπt − 5 + ðβ5 + β6 Þπt − 6 Þ 2

−

it = γit − 1 + ð1 −

+ γπ ð π ¯t − 1 −

π4t − 1 Þ

4

−ð1 − γÞγπ πt

− 1

36

36

H6;20 = ðF Þ3;20 + 1 40

+ ð1 − γÞγx xt − 1

ð

+ ð1 − γÞγπ

ð18Þ

1 4 ∑π 4 j=1 t −j

= γit − 1 + ð1 − γÞcgt − 1 + ð1 − γÞzt − 1 + ð1 − γÞ

8

∑ βj πt − j + βx xt − 1

j=1

Þ

1 4 4 −ð1 − γÞγπ πt − 1 + ð1 − γÞγx xt − 1 ∑π 4 j=1 t−j

      1 1 1 π π π β1 + γπ + β2 + γπ + β3 + γπ 4 t−1 4 t−2 4 t−3    1 πt − 4 + β5 πt − 5 + β6 πt − 6 + β7 πt − 7 + β8 πt − 8 + ð1 − γÞ β4 + γπ 4

ð

References ð19Þ

Now, define the vector ξt as follows: ⊤

The whole model can be written in companion form as a first order vector autoregression: ξt = Fξt − 1 + vt The non-zero entries of the companion matrix F are as follows (Fi,j denotes the entry at the intersection of the i-th row and j-th column): αr 2

F1;6 = −

αr 2

βx ; F1;2 = −

c; F1;8 = α2 −

F1;11 = −

αr 2

F1;14 = −

αr 2

F1;17 = −

αr 2

αr 2

αr 2

β1 ; F1;3 =

αr 2

βx ; F1;9 = −

ðβ1 + β2 Þ; F1;12 = −

αr 2

ðβ4 + β5 Þ; F1;15 = −

αr 2

ðβ7 + β8 Þ; F1;18 = −

αr 2

; F1;5 = − αr 2

αr ; 2

c; F1;10 = −

αr ; 2

ðβ2 + β3 Þ; F1;13 = −

αr ðβ3 2

+ β4 Þ;

ðβ5 + β6 Þ; F1;16 = −

αr ðβ6 2

+ β7 Þ;

β8 ; F1;19 =

αr ; 2

F2;1 = βx ; F2;2 = β1 ; F2;11 = β2 ; F2;12 = β3 ; F2;13 = β4 ; F2;14 = β5 ; F2;15 = β6 ; F2;16 = β7 ; F2;17 = β8 ;   1 F3;1 = ð1 − γÞðβx + γx Þ; F3;2 = ð1 − γÞ β1 + γπ ; F3;3 = γ; F3;5 = ð1 − γÞ; 4   1 F3;6 = ð1 − γÞc; F3;7 = −ð1 − γÞγπ ; F3;11 = ð1 − γÞ β2 + γπ ; 4     1 1 F3;12 = ð1 − γÞ β3 + γπ ; F3;13 = ð1 − γÞ β4 + γπ ; F3;14 = ð1 − γÞβ5 ; 4

4

All the remaining entries are zero. The first two equations correspond to two different inflation data series. The third and fourth equation correspond to output and potential output respectively. The fifth and sixth are the measurement equations for the short-term and the nine-year ahead forward interest rates, respectively. The seventh equation corresponds to long-run (10-year ahead) consensus inflation expectations. The variances of the measurement errors in the observation equations are denoted by:

− 1

Þ

+ ð1 − γÞ

ξt = ½xt πt it yt* zt gt πt* xt − 1 gt − 1 zt − 1 πt − 1 …πt − 8 it − 1 φt 

H7;j = ðF Þ2;j ; j = 1; :::; 20

τj = Var ðwj;t Þ; j = 1; :::; 7 4

= γit − 1 + ð1 − γÞcgt − 1 + ð1 − γÞzt − 1 + ð1 − γÞðβx + γx Þxt − 1 −ð1 − γÞγπ πt

F1;1 = α1 −

I denote the variances of the non-zero terms by σ2x ,σ2π,…,σ2φ. The non-zero entries of the matrix of loadings H, defining the observation equations are as follows:

H6;j = ðF Þ3;j ; j = 1; :::; 19

+ γ x xt − 1 Þ

= γit − 1 + ð1 − γÞðrt4 − 1 + Et − 1 ½πt Þ + ð1 − γÞγπ

⊤

H1;2 = 1; H2;2 = 1; H3;1 = 1; H3;4 = 1; H4;4 = 1; H5;3 = 1;

αr ððβ6 + β7 Þπt − 7 + ðβ7 + β8 Þπt − 8 + β8 πt − 9 Þ 2

γÞðint − 1

F3;15 = ð1 − γÞβ6 ; F3;16 = ð1 − γÞβ7 ; F3;17 = ð1 − γÞβ8 ; F4;4 = 1; F4;6 = 1; F5;5 = 1; F6;6 = 1; F7;7 = 1; F8;1 = 1; F9;6 = 1; F10;5 = 1; F11;2 = 1; F12;11 = 1; F13;12 = 1; F14;13 = 1; F15;14 = 1; F16;15 = 1; F17;16 = 1; F18;17 = 1; F19;3 = 1; F20;20 = 1:

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