No-arbitrage macroeconomic determinants of the yield curve

61 Pages·2017·1.02 MB·English

No-arbitrage macroeconomic determinants of the yield
curve
Ruslan Bikbov, Mikhail Chernov
To cite this version:
Ruslan Bikbov, Mikhail Chernov. No-arbitrage macroeconomic determinants of the yield curve.
Econometrics, 2010, 159 (1), pp.166. ￿10.1016/j.jeconom.2010.05.004￿. ￿hal-00732517￿
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Accepted Manuscript
No-arbitragemacroeconomicdeterminantsoftheyieldcurve
RuslanBikbov,MikhailChernov
PII: S0304-4076(10)00129-6
DOI: 10.1016/j.jeconom.2010.05.004
Reference: ECONOM3382
Toappearin: JournalofEconometrics
Receiveddate: 29October2008
Reviseddate: 8October2009
Accepteddate: 24May2010
Pleasecitethisarticleas: Bikbov,R.,Chernov,M.,No-arbitragemacroeconomicdeterminants
oftheyieldcurve. JournalofEconometrics(2010),doi:10.1016/j.jeconom.2010.05.004
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No-Arbitrage Macroeconomic Determinants of
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the Yield Curve
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∗
Ruslan Bikbova, Mikhail Chernovb, ,
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aFederal Reserve Board, Washington, DC 20551, USA
bLondon Business School, London School of Economics, and CEPR, London,
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NW1 4SA, UK
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Abstract
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No-arbitrage macro-finance models use variance decompositions to gauge the ex-
tent of association between the macro variables and yields. We show that results
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generated by this approach are sensitive to the order of variables in the recursive
identificationscheme.Inafour-factormodel,onemayobtain18differentsetsofan-
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swers out of 24 possible. We propose an alternative measure that is based on levels
ofmacrovariablesasopposPedtoshocks.Weaccountforthecorrelationbetweenthe
macro and latent factors via projection of the latter onto the former. As a result,
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the association between macro variables and yields can be computed uniquely via
an R2. Macro variables explain 80% of the variation in the short rate and 50% of
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the slope, and 54% to 68% of the term premia.
JEL Classification: C13; C22; G12
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Keywords: macro-finance models; term structure; variance decomposition; Kalman
filter A
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1 Introduction
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The important work of Ang and Piazzesi (2003) extends the traditional no-arbitrage term
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structure modelsbymodellingjointlythedynamics ofyields andnonfinancialvariables. The
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nonfinancial variables chosen by researchers are typically macro variables, so we refer to the
models as no-arbitrage macro-finance models. One of the most important issues that the
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setting allows researchers to address is whether yields are associated with macro variables.
This issue could be broken down further. First, is the associaStion between yields and macro
variables determined by the monetary policy? There is a strong intuition from the Taylor
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rule literature that suggests that such macro variables as inflation and real activity should
matter for the interest rate, which is the monetary policy instrument.1 Second, is the as-
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sociation between yields and macro variables determined by the yield risk premia? It could
be that macro variables affect the yields exclusively through the spot interest rate. Finally,
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regardless of the channel, how large is this association quantitatively? A precise answer to
these questions will provide a quantitative bMenchmark for equilibrium models. If structural
implications of these models are to be taken seriously, they should be able to generate, at
the very least, the same common variation in yields and macro variables as the benchmark.
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⋆
We are grateful to the Editor, Ron Gallant, the Associate Editor and the Referee for invaluable
comments that helped us to imprEove the paper. We thank Andrew Ang, Geert Bekaert, Alan
Bester, Jean Boivin, Larry Christiano, Pierre Collin-Dufresne, Greg Duffee, Silverio Foresi, Mike
Gallmeyer, Rene Garcia, Marc Giannoni, Francisco Gomes, Mike Johannes, Lars Lochstoer, Stijn
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VanNieuwerburgh,TarunRamadorai,AndreaRoncoroni,TanoSantos,SureshSundaresan,Andrea
TambalottiandparticipantsofworkshopsatColumbia,BankofCanada,theFederalReserveBank
ofAtlanta, andAFAinBoPston,CIREQ-CIRANOFinancialEconometricsConferenceinMontreal,
the CEPRmeetings at Gerzensee, Econometric World Congressin London,EFA in Moscow, NYU
SternFive-Star Conference, andseminarsat Chicago GSB,Duke, theEuropeanCentralBank, the
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FederalReserveBoard,theFederalReserveBankofNewYork,GoldmanSachsAssetManagement,
ImperialCollege,JPMorgan,LBS,LSE,NYU,PrincetonandRice.Chernovacknowledgessupport
of the JP Morgan Chase Academic Outreach program.
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∗
Correspondingauthor.LondonBusinessSchool, SussexPlace, London,NW1 4SA, UnitedKing-
dom. Tel.: [phone]
Email addresses: [email protected](RuslanBikbov),[email protected](Mikhail Cher-
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nov).
1 However, thedegreeofassociation isnotobvious. Infact, interest raterulesand,therefore,their
ingredienAts, do not matter at all in the absence of frictions, such as price stickiness (e.g., Bils and
Klenow, 2004 and Woodford, 2003).
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The answers to these questions that are found in the literature are mixed. On the one
hand, as is well-known from the term-structure studies using financial variables only, such
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as Litterman and Scheinkman (1991) or Dai and Singleton (2000), two or three factors are
sufficient forcapturing 95%-98%ofthevariationin theyield curve. ThisIconclusion suggests
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a small role for macro variables in a empirically realistic term structure model. On the
other hand, because the macro-finance models can be cast in a vector autoregression (VAR)
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framework, it is natural to quantify the common variation in yields and macro variables
via variance decomposition. Studies that follow this route attribute a large fraction (40% to
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80%) of the unconditional variance in yields to the joint contribution of real activity and
inflation, which suggests that macro determinants are veUry important.2
Thereasonthatthetwostrandsoftheterm-structureliteratureproduceseeminglyconflicting
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resultsisthattheVARliteratureusesshocks tomacrovariablestoexplainvariationinyields,
while the work based only on financial data focuAses on the levels of variables to explain the
same variation. It should be possible to reconcile the two approaches because levels are
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aggregates of shocks. In this paper, we propose a general procedure that allows us to obtain
a correct measurement of the association between yields and macro variables.
A simple example illustrates how sucDh a reconciliation can be achieved. Consider the Ang
andPiazzesi (2003)model, where macro variablesareindependent from thelatent variables.
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The infinite horizon variance decompositions from the VAR representation of the model
characterize how much of the unconditional yield variance can be attributed to shocks in
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each of the macro variables. Our main interest is in disentangling the contribution of the
macro and latent variablPes, rather than in disentangling every single variable. Therefore, we
can exploit theindependence ofthe two types ofvariables(macro andlatent) anddetermine
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the proportion of the yield variance that can be explained by macro variables simply by
computing the corresponding R2.
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The two approaches will yield the same result under two important conditions. First, the
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shocks have to be identified correctly. Researchers often employ the recursive identification
2 For exAample, Ang et al. (2004), Ang and Piazzesi (2003), Dai and Philippon(2004), Rudebusch
and Wu (2005), among others.
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scheme in the context of reduced-form VARs, such as macro-finance models. There is no
guarantee that a particular order of variables will produce the correct identification; hence,
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it is possible for there to be multiple sets of measures. Second, the VAR has to provide a
good fit to the data, in that the variance of yields in the data and in tIhe model should be
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similar. This is because the computation of variance decomposition does not involve actual
data. In contrast, an R2 measures how much of the actual variance is explained by a model.
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The coefficient of determination, R2, appears to be a more robust measure because it relies
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on observed macro variables as opposed to unobserved shocks and, therefore, is not subject
toassumptionsaboutidentification.WeemphasizethatR2 willnotonlybeauniquemeasure
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ofassociationbetween macrovariablesandyields; itwillalsobeacorrectmeasure.However,
this measure cannot be implemented if macro and laNtent variables are correlated with each
other. This would be the case if, forexample, there is feedback from the financial sector into
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the real economy.
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We resolve this problem by exploiting the special nature of latent variables. Latent variables
do not have any macro interpretation. As Dai and Singleton (2000) point out, these factors
can be re-parameterized, or rotated, in a number of different ways without changing the
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value of yields. Thus, latent factors serve no purpose other than to fit the data well. These
observations lead us to propose a novel procedure that allows to express the traditional
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latent factors via macro variables and new “residual” latent variables that are conditionally
orthogonal to the macro oneTs. As a result, we disentangle the contribution of the macro and
the new latent variables to the variation of yields. Now we can implement the R2 that we
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advocated above.
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Thekeytoourprocedureisaconditionalmodel-basedprojectionofthelatentvariablesonto
themacroones.TCheprojectioneffectivelybreaksthelatentfactorsdowninto(i)acomponent
thatconsistsofthecurrentandpastmacrovariables,and(ii)new“projectionresidual”latent
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variables that are conditionally orthogonalto the entire history of macroeconomic variables.
By construction, the variance of the new latent factors is minimized and the variance of the
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macro component is maximized. The spot interest rate becomes a linear function of macro
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variables, their lags, and a set of new latent factors.3
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Theempiricalimplementationofourprocedureisbasedonafour-factorGaussianmodelwith
twolatentfactorsandinflationandrealactivityactingasobservablefactors.Toestimatethe
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model, we use a panel of eight yields ranging from three months to 1R0 years, with inflation
andrealactivityobserved atamonthlyfrequencyfrom1970to2002.Weprovide diagnostics
that indicate that the model fits the data reasonably well. C
We document that the variance decomposition results are extremely sensitive to the order
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of the model factors chosen for the recursive identification. For example, the fraction of the
interest rate variance that is explained by the joint shockUs to inflation and real activity can
vary between 36% and 77% depending on the assumed order of the factors. Similarly, the
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fraction of the explained slope variance varies between 25% and 81%.4 In particular, the
standardorderofthevariables,thatis,macrovariablesfollowedbylatentvariables,produces
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52%fortheshortinterestrateand37%fortheslope.Ourprocedureproducesauniquesetof
explained variances precisely because it is bMased on observed macro variables. In particular,
80%ofthevariationintheinterestrateand52%ofthevariationintheslopecanbeexplained
byinflationandrealactivity.Asimilarsetoffigures(76%and59%,respectively) isobtained
via the variance decomposition if theDorder ofthe variables is real activity followed by latent
variables, which are in turn followed by inflation. Clearly, there is no way of knowing this
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“correct” order of shocks in the context of a reduced-from model.
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The point that variance decomposition results are sensitive to the order of variables in the
recursive identification scheme is not new in itself. Nevertheless, given the popularity of
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the identification scheme in the macro-finance literature on the term structure, it is worth
revisiting. The uncovEered magnitudes of disagreement between the variance decompositions
based on different orders of variables seem to warrant a consideration of an alternative
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approach. To be sure, our projection-based procedure is nota silver bullet. It cannot replace
3 Ourlagstructureisnotarbitrary:recursiveprojectionformulasimplyrelianceonalllagsandthe
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loadings on these lags are optimal, due to the fact that they are selected to minimize the variance
of the residuals.
4 We meAasure the slope of the yield curve as the difference between the 10-year and three-month
yields.
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variance decompositions because it can only workin the presence oflatent variables and has
no causal implications. We view our procedure as a useful complement to the toolkit of a
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researcher studying VARs that are related to the yield curve.
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We further illustrate the usefulness of our procedure by using it to adRdress the issues raised
at the beginning of the introduction.5 First, we investigate how much of the short rate
variation can be explained by the two macro variables, as opposeCd to by the latent factors.
It seems natural to start with the null hypothesis that macro variables do not contribute to
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explaining the short rate, given the excellent explanatory power of latent factor models that
we discussed earlier. As we show, this hypothesis places restrictions on various parameters
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that control the dynamics and cross-section ofthe yield curve. A number of previous studies
have simplyimposed these restrictions, therebylimitingtheabilityofyields tovarywith the
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macroeconomy. We test these restrictions in terms ofthe size ofR2 thatcan be explained by
macro variables and confirm that macro variableAs are extremely important. Our projection-
basedapproachcanexplain 80%oftheshortratevariation,usinginflation,realactivity, and
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their lags exclusively as the basis.
Second,we investigate whether macroeconomicvariablesareuseful forexplainingbondterm
premia and not just short-term intereDst rates.6 Our approach enables us to determine that
inflation and real activity risk premia are both significant factors, which jointly explain 54%
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to 68% of the variation in the term premia, depending on a bond’s maturity. In the context
of time-series analysis, the contribution of inflation is countercyclical. It is also the most
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important contributor with respect to the magnitude and volatility of term premia.
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Third, we focusonthe nature ofthe residual latent variables. In order togauge the relation-
ship between additioEnal macro variables and our latent factors, we correlate them with two
measures of liquidity (AAA credit spread and the growth rate of the money zero maturity
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5 More recent works by Chernovand Mueller (2008) andMueller (2008) provide furtherexamples
of applications of our methodology.
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6 In fact, there is currently a debate in the literature over this point. Ang et al. (2004) and
Ludvigson and Ng (2009) find that inflation and real activity measures play an important role in
explaininAg bond risk premia. In contrast, Duffee (2007) concludes that macro variables make a
minimal contribution to the term premia.
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(MZM), which is a measure of money supply) and a measure of the budget deficit (public
government debt growth). The correlations are strong, but not overwhelming. We therefore
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conclude that it is more appropriate to use latent variables to model shocks. The shock as-
sociated with liquidity measures contributes to the variation in short Iinterest rate, almost
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completelyaccountingforthepartleftunexplained bythemacrovariables.Weinterpretthis
factorasanexogenous policyshock, that is, occasional reactionsofthe short interest rateto
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information beyond that contained in real activity and inflation. The shocks correlated with
the budget deficit make the largest contribution to that portion of the slope variation that
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is left unexplained by macro variables. We interpret this factor as an exogenous non-policy
shock. U
Clearly, our paper is related to the growing literatuNre on macro-finance yield curve mod-
elling.7 Our work, while having a different focus, is particularly close to the concurrent
study by Ang et al. (2004) (ADP henceforth).AADP use a macro-finance model to show
that one and the same interest rule can be interpreted in three distinct ways. Given that it
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is based on additional assumptions such as the forecasting horizon and whether exogenous
shocks depend on current or past innovations in the state variables, the interest rule can
be seen as either simple, forward-looking, or backward-looking. Their approach applies to a
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modelwithonlyonelatentfactor.Intheirempiricalwork,ADPusevariancedecompositions
to gauge the degree of associatioEn between the macro variables and yields. Our projection-
based approach implies a backward-looking interest rate rule and so appears to be similar
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to their work. However, there are important differences in the focus and approach. First,
our procedure applies regardless of the number of latent factors in the model. Second, we
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do not focus on the interest rate rules per se, precisely because structural interpretation
requiresadditionalasEsumptions. Third, regardlessoftheassumed interest raterule,variance
decompositions will be the same and our focus is on providing a robust alternative to such
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a computation.
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7 ThisliteratureincludesAngandPiazzesi(2003),Bekaertetal.(2010),DaiandPhilippon(2004),
Duffee (2A006), Diebold et al. (2006), Gallmeyer et al. (2005), Ho¨rdahl et al. (2006), Law (2004),
Rudebusch and Wu (2005), among others.
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The remainder of this paper is organized into five sections and two appendixes. Section 2
summarizes the macro-finance framework. Section 3 motivates and describes the proposed
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methodology. Section 4 discusses the estimation strategy. Section 5 presents our findings.
The final section concludes. Technical details are presented in the appeIndices.
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2 The Model U
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2.1 State variables
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FollowingAngandPiazzesi(2003)(APhencMeforth),weassumethatthestateoftheeconomy
is captured by the vector z =(m′,x′)′. In particular, the vector of macroeconomic variables
t t t
m is equal to (g ,π )′, where g and π are monthly real activity and the inflation rate,
t t t t t
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respectively. The remaining factors x are latent. The latent factors may contain the lags
t
of m , other macro variables, or other unknown variables. Importantly, the vector z fully
t t
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reflects all available information at time t; so, for instance, one need not consider lags of x .
t
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We assume that the state vector z follows a VAR(1) process:
t
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z =µ+Φz +EΣǫ
t  t−1 t     
µm CΦmm Φmxmt−1 Σmm Σmxǫmt 
= +  +  , (1)
       
       
µx Φxm Φxx x Σxm Σxx ǫx
t−1 t
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where ǫ ∼ N(0,I). We denote the vector of parameters that control the dynamics of state
t
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by Θ=(µ,Φ,Σ). The block representation will be useful for later discussions.
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