Term premia dynamics in he US and Euro Area: who is leading whom? Nikolay Iskrev Banco de Porugal January 2018 Absrac This aricle examines he dynamic relaionship beween erm premia in euro area and US governmen bond yields. The erm premia are exraced using an affine erm srucure model using daily daa on zero-coupon bond yields. The resuls show srong co-movemen beween changes in he premia, especially a he long end of he yield curves. A furher invesigaion of he causal relaionship beween he euro area and US erm premia reveals ha only a small fracion of he co-movemens can be aribued o one region driving he oher. (JEL: G12, E43) Inroducion While ineres raes a all mauriies play a role in he borrowing and lending decisions of businesses and households, longer-erm raes are ypically he ones ha maer he mos for aggregae spending in he economy. In paricular, long-erm raes play a cenral role when businesses decide wheher o sar new invesmen projecs, households wheher and when o purchase a new home or car, and policy makers in deciding how o finance governmen expendiures. From a heoreical poin of view, longer-erm raes can be seen as risk-adjused averages of expeced fuure shor-erm raes. This link beween shor and long-erm raes explains how he ransmission mechanism of moneary policy usually works changes in he shor-erm ineres rae, which is under cenral banks direc conrol, influence aggregae spending decisions by affecing expecaions abou fuure shor-erm raes and hereby changing longer-erm raes. 1 Acknowledgemens: I would like o hank Isabel Correia, Nuno Alves, Anónio Anunes, Sandra Gomes, Miguel Gouveia, and he seminar paricipans a he Bank of Porugal for helpful commens and discussions. The views expressed here are our own and do no necessarily reflec he views of he Bank of Porugal or he Eurosysem. E-mail: nikolay.iskrev@bporugal.p 1. In he case of he US Federal Reserve, promoing moderae long-erm ineres raes is one of he explicily mandaed goals, alongside maximum employmen and sable prices.
40 The need o accoun for risk makes maers more difficul. Boh he amoun of risk in long-erm bonds and is price change over ime, giving rise o a imevarying erm premium which complicaes he relaionship beween policy raes and long-erm raes. The erm premium represens he compensaion invesors in long-erm bonds require for he risk ha fuure shor raes do no evolve as expeced. Given is imporance, here has been a large amoun of research direced a characerizing he erm premium and he facors affecing is level and dynamics. In his aricle, I sudy he relaionship beween erm premia in he yields of euro area (EA) and US governmen bonds. I is a well-known empirical fac ha ineres raes of governmen bonds of advanced economies end o move closely ogeher, especially a he longer end of he yield curve. One of he objecives here is o esablish wheher his is also rue for he erm premium componens of he yields. To ha end, I esimae affine erm srucure models of he ineres raes for he euro area and he US, and use hem o separae expecaions from erm premia. Then, I measure he degree of co-movemen beween he levels and he changes in he erm premia using linear correlaion coefficiens. The second objecive of he aricle is o explore he evidence for a causal relaionship beween he wo erm premia, ha is, he exen o which we can say ha movemens in he erm premia of one economic area drive he movemens in he erm premia of he oher area. For ha purpose I esimae saic and dynamic versions of indicaors ha have been proposed in he ime series lieraure o measure he srengh and direcion of causal relaionships. The resuls from his analysis show ha here exis ime-varying causal linkages beween he EA and US erm premia. A he same ime, i is found ha only a relaively small fracion of he observed co-movemens can be aribued o one region driving he oher. The res of he aricle is organized in four secions. The firs one presens some basic yield curve conceps and inroduces he expecaions heory of ineres raes. The second secion firs oulines and esimaes an affine erm srucure model, which is used o decompose long-erm yields ino expecaions and erm premia, and hen evaluaes he srengh of comovemen beween euro area and US erm premia. The hird secion describes and esimaes several measures of causaliy beween he erm premia. The las secion offers some concluding remarks. Term srucure of ineres raes This secion inroduces some basic yield curve erminology and presens he expecaions heory of ineres raes, which is in he background of mos modern erm srucure models.
41 Noaion and basic conceps While bonds ypically pay coupons during heir lifeime, economiss prefer o work wih zero-coupon bonds, also known as pure discoun bonds. These are bonds ha promise o pay one euro on a given fuure day he mauriy dae of ha bond. Non-zero coupon bonds can be seen as porfolios of zerocoupon bonds. The ineres raes on he zero-coupon bonds are called yields, and he funcion describing he relaionship beween bond mauriies and heir yields a a given poin in ime is called he yield curve. Zero-coupon bonds are convenien because here exiss a simple relaionship beween he price P (n) a ime and he yield y (n) P (n) a ime of such bonds: = e n y(n), where n is he ime o mauriy measured in years. The yield is he coninuously compounded annualized reurn from holding he zero coupon bond unil mauriy. A a given poin in ime he yield of a bond will depend on is mauriy, and he yield curve is he funcion describing ha relaionship. Figure 1 shows several hisorical yield curves for mauriies beween 3 monhs and 10 years for he euro area and he US. The observaions are from he firs and las monhs in our sample from Ocober 2004 unil Ocober 2017. Also shown are he average curves over he sample period. Several feaures of he figure are worh noing: firs, he curves are upward sloping and have very similar shapes, boh across ime and regions. Upward-sloping yield curves are more common in general alhough hisorically here have been episodes of downward-sloping curves, for insance he US in he early 2000s. Second, boh he EA and US yield curves have shifed downwards over he sample period, and remain below he average curves a he end of he sample. However, while a he beginning of he sample period he levels of he EA and US yield curves are approximaely he same, hey are very differen a he end of he sample, wih he EA yield curve being much lower han he one for he US. Explaining such differences in he shape of he yield curve across ime and economic regions is one of he main objecives of he research on he erm srucure of ineres raes. The expecaions hypohesis The expecaions heory of ineres raes is among he oldes and mos popular models of he erm srucure. 2 In is general form, he expecaion hypohesis posulaes ha long-erm raes and expeced shor-erm raes mus be linked. 3 2. The main ideas behind he expecaions hypohesis can be raced back o he work of Fisher (1896) and Luz (1940). 3. In he lieraure i is common o disinguish beween he pure expecaions hypohesis, which saes ha he long raes are equal o he average expeced shor raes, and he
42 FIGURE 1: EA and US yield curves. The figure shows he EA and US zero-coupon yield curves a he beginning and he end of he sample (Ocober 2004 and Ocober 2017, respecively), as well as he average yield curves across he sample period. Source: ECB, FRB, and own calculaions. The heory is moivaed by he observaion ha invesors choose beween shor and long-erm bonds by comparing he reurn of he long-erm bond o he expeced reurn of an invesmen sraegy of rolling-over a sequence of shor-erm bonds. To undersand he basic inuiion, assume for a momen ha fuure yields are cerain, and consider an invesor who chooses beween wo invesmen sraegies: buying 2-year bonds oday, or buying 1-year bonds oday, he proceeds from which are hen re-invesed in 1-year bonds one year hence. Using he firs sraegy, he invesor has o pay P (2) = e 2 y(2) euros oday o receive one euro in wo years. The price nex year of a 1-year bond is P (1) +1 = e y(1) (1) (1) +1. The price oday of P +1 one-year bonds is P P (1) +1 = e y(1) P (1). Therefore, o receive 1 euro in wo years using he second sraegy, +1 he invesor has o pay e y(1) e y(1) +1 oday. The wo sraegies yield he same expecaions hypohesis which saes ha deviaions of long raes from he average expeced shor raes are consan over ime.
43 reurn and herefore mus require he same iniial invesmen, i.e e 2 y(2) Hence, absence of arbirage requires ha y (2) = e (y(1) +y (1) +1 ) = 1 2 (y(1) + y (1) +1 ) Using he same argumen, we can esablish he following relaionship beween he yield of bonds wih n years o mauriy and he yield on he presen and fuure one-year bonds: y (n) = 1 n ( ) y (1) + y (1) +1 +... + y(1) +n 1 Uncerainy abou fuure shor-erm yields means ha invesmen decisions have o be made on he basis of invesors expecaions abou fuure yields. Furhermore, invesors are averse o risk and will demand a premium for holding riskier long-erm bonds. The classical formulaions of he expecaions hypohesis se he premium o zero or o a non-zero consan. However, numerous sudies esing formulaions of he expecaions hypohesis have found evidence for ime-varying risk premia (see for insance Mankiw e al. (1984), Fama and Bliss (1987), Campbell and Shiller (1991)). This leads o he following more general represenaion of bond yields: y (n) = 1 n h=0 (1) n 1 E y (1) +h + T P (n), (2) where T P (n) denoes he erm premium a ime for bonds wih n years o mauriy. In order o separae he erm premia from he expecaions componen, we need a model for he erm srucure. The nex secion describes and esimaes one such model. Yield decomposiion based on affine erm srucure model In his secion, I use daily zero-coupon yields daa o decompose observed long-erm raes ino expecaion componens and erm premia. To ha end, I esimae a no-arbirage affine erm srucure model of he ineres raes. According o his model, boh he acual yields and he expecaion componens can be expressed as affine funcions of a small number of risk facors, which are modeled as linear processes. Ruling ou arbirage opporuniies imposes resricions on he yields behavior over ime and across differen mauriies. Those resricions faciliae he esimaion of he model in erms of a small number of parameers. A fuller descripion of he affine erm srucure model and is derivaion are presened in he Appendix.
44 Daa and esimaion I esimae he affine erm srucure model using daily zero-coupon yields for he EA and he US. To compue he daily yield curves I use he Svensson (1994) model wih parameer esimaes provided by he ECB and he US Federal Reserve. 4 In he case of he EA he yields are of AAA-raed sovereign bonds, which are comparable in erms of risk properies o he US reasury bonds. 5 Using he esimaed parameers I consruc daily yield curves for mauriies from 1 monh up o 10 years, for he period beween Sepember 2004 and Ocober 2017. 6 The ime series of he EA and US zero-coupon yields for seleced mauriies are presened in Figure 2. I esimae he model oulined above following a procedure developed by Adrian e al. (2013) (ACM henceforh), who show ha he underlying model parameers can be esimaed using a series of linear regressions. Specifically, heir approach akes he risk facors o correspond o he firs few principal componens of he observed bond yields, and models he facors as a sandard vecor auoregressive model. The parameers of he model are hen obained in hree seps using sandard OLS regressions. More deails on he esimaion procedure is provided in he Appendix. Number of risk facors Following he work of Lierman and Scheinkman (1991), i is common in he lieraure o summarize he erm srucure using principal componens of he covariance marix of he zero-coupon yields. Typically, i is found ha he firs hree principal componens are sufficien o capure mos of he variaion in he yields. In oher words, here are hree significan risk facors explaining he shape of he yield curve. These facors are ypically referred o as level, slope and curvaure facors. The reason for hese labels can be undersood by considering he facor loadings displayed in Figure 3. The facor loadings show how sensiive yields a differen mauriies are o changes in each principal componen, or risk facor. In he figure we see ha changes in he firs facor resul in a level shif for he yields of all mauriies. Changes in he second facor move he shor and long mauriies in opposie direcions. 4. The esimaed parameers are downloaded from hp://www.ecb.europa.eu/sas/ financial_markes_and_ineres_raes/euro_area_yield_curves/hml/index.en.hml for he EA and hps://www.federalreserve.gov/pubs/feds/2006/200628/200628abs.hml for he US. The Svensson model is also used by he ECB o produce daily yield curves for he EA, as well as by Gürkaynak e al. (2007) whose zero-coupon yield daa se is commonly used for esimaing erm srucure models for he US. 5. Noe ha he selecion of EA counries wih AAA raing changes over ime. The raings ECB uses are provided by Fich Raing. 6. Official daa for he EA is available from 6 Sepember 2004, while he daa for he US sars in 14 June 1961.
45 FIGURE 2: EA and US zero-coupon yields. The figure shows he ime series of EA and US zero-coupon yields for seleced mauriies. Source: ECB and FRB. FIGURE 3: Risk facors loadings. The figure displays he loadings of bond yields on he firs five principal componens. Source: ECB, FRB, and own calculaions.
46 all 3m 1y 2y 3y 4y 5y 10y # of PCs (a) EA 1 96.439 86.158 89.698 95.056 98.247 99.650 99.855 94.282 2 3.434 12.473 10.147 4.884 1.610 0.208 0.054 5.529 3 0.115 1.184 0.141 0.038 0.139 0.138 0.081 0.168 4 0.009 0.069 0.004 0.021 0.002 0.003 0.009 0.019 5 0.003 0.105 0.011 0.000 0.002 0.001 0.000 0.002 # of PCs (b) US 1 94.685 85.601 89.251 92.898 96.084 98.451 99.573 88.355 2 4.972 11.552 10.101 7.033 3.725 1.164 0.030 11.053 3 0.309 2.296 0.643 0.004 0.165 0.383 0.383 0.537 4 0.032 0.525 0.001 0.064 0.025 0.000 0.014 0.051 5 0.002 0.023 0.004 0.001 0.001 0.002 0.000 0.004 TABLE 1. Percen of he variance explained by he firs 5 principal componens. Source: Own calculaions. Lasly, changes in he hird facor move he shor and long mauriies in he same direcion, leaving he medium-erm mauriies mosly unaffeced. In addiion, he figure shows ha he yields of all mauriies are mosly sensiive only o he firs hree facors, while changes in eiher he fourh or he fifh principal componen have only a minor impac. Figure 3 is based on daa for he EA, bu he resuls wih US daa are very similar. Anoher sandard approach for deermining he number of facors is o compue he fracion of he oal variance of he observed yields explained by each addiional risk facor. As can be seen in Table 1, for boh he EA and he US, he firs hree principal componens are sufficien o capure more ha 99% of he variance of he yields as a whole, as well as he variances of yields a seleced mauriies. These resuls are in line wih he broad consensus in he lieraure ha he firs hree principal componens of he yield curve are sufficien o capure well he dynamics of he erm srucure. However, he ACM esimaes of he US erm premia are based on five pricing facors, and ha is he specificaion underlying he yield curve decomposiion published by he New York Fed. For consisency wih heir approach, here I presen resuls based on a five facor model for boh he EA and US yield curves. 7 7. I should be noed ha he US erm premia esimaes published daily by he New York Fed are esimaed wih a sample saring in 1961, while he esimaes presened in his aricle are obained wih a sample saring in 2004. The main impac his difference has on he resuls is on he level of erm premium, which is higher wih he more recen sample. The dynamics of he erm premia remains almos unchanged. This level effec is due o he fac ha he mean of he shor-erm rae is much higher in he longer sample, which drives he expecaions componen up and he erm premium down.
47 FIGURE 4: 10 year yield decomposiion. This figure plos decomposiions of he EA and US 10-year daily yields ino expecaion componens and erm premia. Source: ECB, FRB, and own calculaions. Term premia esimaes Following ACM, I esimae he parameers of he model using end-of-monh observaions of he zero-coupon yields. Given he esimaed parameers, I can compue he model-implied decomposiion of he fied yields y (n) ino expecaions componen ỹ (n) and erm premium T P (n) for all mauriies
48 and a any poin in ime. In paricular, wih daily observaions of he risk facors, exraced as principal componens of he daily zero-coupon yields, I can decompose he yields ino expecaions componen and erm premia a daily frequency. Figure 4 shows an example wih daily decomposiions of he 10-year bond yields in he EA and US. In he case of he EA yields, for insance, he decomposiion suggess ha he reurn of he 10-year yields ino posiive erriory a he end of 2016 was enirely due o an increase in he erm premium, i.e. he compensaion for holding longer-erm bonds by invesors. In fac, he 10-year yields have racked closely he movemens in he erm premium for mos of he ime since 2012, due o he expecaion componen remaining relaively fla over ha period. On he oher hand, he expecaions componen in he US 10-year yields has been increasing seadily since 2014. This rise in he shor rae expecaions explains o a large exen he observed divergence in he 10-year yields in he wo regions. A he same ime, as can be seen beer in Figure 5, he 10-year erm premia in he EA and he US have followed very similar pahs during he sample period. In boh regions he erm premia reached hisorically low levels in he second half of 2016. Also shown in he figure is he 250-day rolling correlaion beween he wo series. During mos of he period he correlaion is posiive and very srong, ofen in excess of 0.9. However, using correlaion here may be misleading since he wo series appear o be non-saionary. 8 Thus, i is more reasonable o compare changes in he erm premia componens of he respecive bond reurns. Figure 6 shows he changes in he 10-year erm premia in he EA and he US and he 250-day rolling correlaion beween hose series. Again, during mos of he sample period he correlaion is posiive and relaively srong. This is no a feaure of he 10-year erm premia only. Figure 7 shows a hea plo of rolling correlaions beween changes in he EA and US erm premia for all mauriies up o 10 years. The degree of correlaion ends o be sronger for longer mauriies, and is abou as high as for he 10-year premia for all mauriies above 6 or 7 years. On he oher hand, for mauriies of less han 4 years he correlaion ends o be week and is someimes even negaive. 8. This observaion is confirmed by formal uni roo ess he resuls of which are presened in he Appendix.
49 FIGURE 5: 10-year EA and US erm premia. The figure shows 10-year EA and US erm premia and 250-day rolling pairwise correlaions beween he wo series. Source: ECB, FRB, and own calculaions. FIGURE 6: Changes in he 10-year EA and US erm premia. The figure shows he changes in he 10-year EA and US erm premia and 250-day rolling pairwise correlaions beween he wo series. Source: ECB, FRB, and own calculaions.
50 0.8 9y 8y 0.6 7y mauriy 6y 5y 4y 3y 0.4 0.2 0.0 2y 1y 0.2 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 dae FIGURE 7: Rolling correlaions beween changes in he EA and US erm premia. The figure show 250-day rolling pairwise correlaions beween changes in he EA and US erm premia for all mauriies up o 10 years. Source: ECB, FRB, and own calculaions.
51 Deecing and measuring direcionaliy Indicaors The resuls in he previous secion show ha changes in he erm premia in he EA and US are srongly posiively correlaed, especially a he longer end of he yield curve. In his secion I consider he evidence for direcionaliy in he ineracions beween he wo variables. Specifically, I esimae hree indicaors designed o deec and quanify he srengh of causal ineracion in ime series. The indicaors are Granger causaliy, ransfer enropy and direcional connecedness, and are described below. Granger causaliy. Saed simply, he definiion of Granger causaliy is ha a variable X causes a variable Y if a forecas of Y using X is more accurae han a forecas of Y wihou using X. To make his definiion operaional, one needs o specify a forecasing model for Y and ypically his is done using linear vecor auoregressions (VAR). Then, esing for causaliy amouns o comparing he size of he forecas errors of Y from a VAR which includes lags of X o he size of he errors from a VAR wihou hose lags. Transfer enropy. The concep of Granger causaliy can be inerpreed in erms of informaion conen, i.e. he pas of variable X conaining informaion abou he fuure of variable Y, informaion no conained in he pas of Y iself. From his perspecive, one can define a more flexible, i.e. non-linear, model for predicing Y, as well as use a more general measure of informaion han he reducion of forecas error variance, which underlies he sandard approach o esing for Granger causaliy. This is in essence wha he concep of ransfer enropy ries o accomplish. 9 The amoun of informaion from X o Y is measured as he reducion of uncerainy abou he fuure of Y using a model-free measure, namely he enropy of he empirical disribuion of he daa. Direcional connecedness. In a series of papers, Diebold and Yilmaz (2009, 2012, 2015) developed a measure of connecedness for he purpose of assessing he srengh and direcion of inerdependence across financial markes in differen counries. The measure is based on variance decomposiions esimaed from VAR applied o wo or more financial variables. In paricular, he connecedness from X o Y is deermined by he 9. The enropy of a variable is defined as he negaive expeced value of he logarihm of he probabiliy disribuion of ha variable. In he case of a normally disribued variable, he enropy is equivalen o he variance of ha disribuion. Transfer enropy, as a measure of he amoun of informaion ransferred from one ime series process o anoher, was inroduced by Schreiber (2000)
52 share of he forecas error variance of Y due o shocks in X. The idenificaion of he shocks is achieved using he generalized variance decomposiion approach of Pesaran and Shin (1998). Similar o he Granger causaliy measure, he noion of connecedness can be inerpreed in erms of informaion conen, namely, he amoun of addiional informaion abou fuure values of one variable conained in he shocks associaed wih anoher variable. As before, informaion is quanified as he reducion of uncerainy abou he fuure values of he firs variable. Insead of informaion in he second variable iself, connecedness is abou he impac of he shocks associaed wih ha variable. This common inerpreaion suggess ha we can use he following general represenaion of he hree measures: I X Y = 100 ( 1 ) Uncerainy(Y X, Z) U ncerainy(y Z) Noe ha having more informaion canno increase uncerainy. Therefore, U ncerainy(y X, Z) U ncerainy(y Z) is always rue. Equaliy would imply ha X conribues no informaion abou Y, once Z is observed. In ha case I X Y = 0. On he oher exreme, we could have Uncerainy(Y Z) > Uncerainy(Y X, Z) = 0, which means ha observing boh X and Z is equivalen o also observing Y. In ha case we have I X Y = 100. In he case of boh Granger causaliy and ransfer enropy, Y represens fuure values of one observed variable, for example he 10-year EA erm premium, X represens he pas values of he oher observed variable, i.e. he 10-year US erm premium, and Z represens he pas values of he firs observed variable he 10-year EA erm premium. The value of he indicaor in boh cases shows he reducion of uncerainy abou he fuure values of he 10-year EA erm premium as a resul of observing he pas values of he 10-year US erm premium, compared o using only he pas values of he 10- year EA erm premium. The difference beween hese wo indicaors is in how uncerainy is esimaed wih a VAR model and using he forecas error variance in he case of Granger causaliy, and wih a non-parameric esimaor of enropy in he case of ransfer enropy. For he direced connecedness measure, Y is again he fuure values of an observed variable he 10-year EA erm premium bu X represens he fuure values of he shock associaed wih he oher variable, i.e. he 10-year US erm premium, while Z represens he pas values of boh observed variables, EA and US 10-year erm premia. Resuls I esimae he measures of direcionaliy using boh he full sample and rolling-window samples. The full sample resuls are presened in Table 2. Two of he measures he Granger causaliy and he direcional connecedness indicae a sronger causal impac from he US o he EA erm premia (3)
53 changes. The ransfer enropy shows he inverse relaionship, i.e. he EA having sronger impac. All hree measures agree ha he causal influence from one area o he oher is relaively weak. EA US US EA Granger causaliy 1.6 2.9 Transfer enropy 4.4 3.6 Direcional connecedness 4.4 9.0 TABLE 2. Saic indicaors of direcional influence. The values represen he per cen reducion in uncerainy regarding fuure yields in one area, due o he informaion from he pas yields (in he case of Granger causaliy and ransfer enropy) or fuure shocks (in he case of direcional connecedness) from he oher area. The sample is from Sepember 7, 2004 hrough Ocober 31, 2017 Source: Own calculaions. To see how he degree of causaion changes over ime, I perform a rollingwindow analysis using windows wih a lengh of 250 days. The resuls are displayed in Figure 8. They show ha he srengh of causal influence changes over ime, and in some periods he impac from he EA is sronger, while in ohers he influence from he US dominaes. In paricular, all hree measures are consisen in suggesing ha EA has a sronger impac on he US during he period from 2011 hrough 2013, while from he middle of 2013 unil he second half of 2014 he degree of causaliy from US o EA is sronger. The Granger causaliy and direcional connecedness measures also indicae ha influence from he US dominaes ha from he EA in he beginning of he sample from 2006 unil 2008. In he case of ransfer enropy, he EA has somewha sronger impac during ha period. Overall, wih a few excepions, he ransfer enropy measure suggess a relaively more equal degree of causal influence from eiher area, while he oher wo measure show several periods where causal influence from one of he areas clearly dominaes. A he same ime, all hree measures indicae a relaively small causal impac from eiher area o he oher. In erms of informaion ransfer, his means ha here is a relaively small amoun of unique informaion in eiher series ha helps predic he fuure developmens in he oher. Therefore, one of he main reasons for he srong co-movemen beween he series mus be ha hey are boh subjec o influence by a global facor or facors. For insance, inernaional facors driving uncerainy abou fuure inflaion will also affec erm premia in differen markes. Empirical evidence linking he downward slope in inernaional erm premia o declining inflaion uncerainy are discussed by Wrigh (2011).
54 EA US US EA FIGURE 8: Dynamic indicaors of direcional influence. The figure shows 250-day rolling window esimaes of he indicaors. The values represen he per cen reducion in uncerainy regarding fuure yields in one area, due o he informaion from he pas yields (in he case of Granger causaliy and ransfer enropy) or fuure shocks (in he case of direcional connecedness) from he oher area. Source: ECB, FRB, and own calculaions. Concluding remarks This aricle invesigaed he dynamics of erm premia in EA and US governmen bonds. I found ha here is a srong co-movemen beween he premia, especially a he long end of he yield curve, boh in erms of he levels as well he changes in he wo series. Furher analysis of he poenial causal relaionship beween he bond erm premia revealed ha only a small fracion of he join dynamics can be aribued o one region driving he oher. This par of he analysis was based on several differen indicaors which, in conras o measures of co-movemen like correlaion, are non-symmeric and provide informaion abou he direcion of causaliy. While all indicaors sugges he exisence of a ime-varying causal linkages beween EA and US erm premia, hey were found o be relaively weak. Given his evidence, a more plausible explanaion of he srong co-movemen is ha here exis a common global facor ha affecs erm premia in boh regions.
55 References Adrian, Tobias, Richard K. Crump, and Emanuel Moench (2013). Pricing he erm srucure wih linear regressions. Journal of Financial Economics, 110(1), 110 138. Campbell, John Y and Rober J Shiller (1991). Yield spreads and ineres rae movemens: A bird s eye view. The Review of Economic Sudies, 58(3), 495 514. Diebold, Francis X and Kamil Yilmaz (2009). Measuring financial asse reurn and volailiy spillovers, wih applicaion o global equiy markes. The Economic Journal, 119(534), 158 171. Diebold, Francis X and Kamil Yilmaz (2012). Beer o give han o receive: Predicive direcional measuremen of volailiy spillovers. Inernaional Journal of Forecasing, 28(1), 57 66. Diebold, Francis X and Kamil Yilmaz (2015). Financial and Macroeconomic Connecedness: A Nework Approach o Measuremen and Monioring. Oxford Universiy Press, USA. Fama, Eugene F and Rober R Bliss (1987). The informaion in long-mauriy forward raes. The American Economic Review, pp. 680 692. Fisher, Irving (1896). Appreciaion and Ineres: A Sudy of he Influence of Moneary Appreciaion and Depreciaion on he Rae of Ineres wih Applicaions o he Bimeallic Conroversy and he Theory of Ineres, vol. 11. American Economic Associaion. Gürkaynak, Refe S, Brian Sack, and Jonahan H Wrigh (2007). The US Treasury yield curve: 1961 o he presen. Journal of Moneary Economics, 54(8), 2291 2304. Lierman, Rober B and Jose Scheinkman (1991). Common facors affecing bond reurns. The Journal of Fixed Income, 1(1), 54 61. Luz, Friedrich A (1940). The srucure of ineres raes. The Quarerly Journal of Economics, 55(1), 36 63. Mankiw, N Gregory, Lawrence H Summers, e al. (1984). Do Long-Term Ineres Raes Overreac o Shor-Term Ineres Raes? Brookings Papers on Economic Aciviy, 15(1), 223 248. Pesaran, H Hashem and Yongcheol Shin (1998). Generalized impulse response analysis in linear mulivariae models. Economics leers, 58(1), 17 29. Schreiber, Thomas (2000). Measuring informaion ransfer. Physical review leers, 85(2), 461. Svensson, Lars EO (1994). Esimaing and inerpreing forward ineres raes: Sweden 1992-1994. Tech. rep., Naional Bureau of Economic Research. Wrigh, Jonahan H (2011). Term premia and inflaion uncerainy: Empirical evidence from an inernaional panel daase. The American Economic Review, 101(4), 1514 1534.
56 Appendix: Arbirage-free Gaussian affine erm srucure models Affine erm srucure models model zero-coupon bond yields as funcions of a vecor of variables X, called pricing or risk facors, and assumed o follow a Gaussian vecor auoregression (VAR(1)): X = µ + ΦX 1 + ε, v N(0, Σ) (A.1) Le P (n) be he price of a zero-coupon bond wih mauriy n a ime. Assuming ha here is no arbirage implies he exisence of a price kernel M such ha ( ) M = E M +1 P (n 1) (A.2) +1 Assume ha he pricing kernel is exponenially affine, i.e: M = exp ( r 12 λ λ λ Σ ) 1/2 v +1 (A.3) where r = ln(p (1) ) is he coninuously compounded one-period rae, and λ are he marke prices of risk. Boh r and λ are assumed o be affine funcions of he pricing facors r = δ 0 + δ 1 X (A.4) λ = Σ 1 (λ 0 + λ 1 X ) (A.5) Denoe wih rx (n 1) +1 he log of he excess holding reurn of a bond mauring in n periods: rx (n 1) +1 = ln P (n 1) +1 ln P (n) r (A.6) ACM show ha if {rx +1, v +1 } are joinly normally disribued, hen E ( rx (n 1) +1 ) = β (n 1) (λ 0 + λ X ) 1 2 var ( rx (n 1) +1 ) (A.7) ( ) where β (n 1) = cov rx (n 1) +1, v +1 Σ 1. Furhermore, he reurn generaing process for he log excess reurns is rx (n 1) +1 = β (n 1) (λ 0 + λ X ) 1 (β (n 1) Σβ (n 1) + σ 2) 2 + β (n 1) v +1 + e (n 1) +1 (A.8) where e (n 1) +1 is a reurn pricing error assumed o follow an i.i.d. process wih mean 0 and variance σ 2. The above equaion can be wrien in a sacked form
57 for all and n as follows rx = β (λ 0 ι T + λ X _ ) 1 ( B vec(σ) + σ 2 ) ι N ι 2 T + β V + E (A.9) where rx is N T marix of excess reurns, β is K N marix of facor loadings, ι T and ι N are T and N dimensional vecors of ones, X _ = [X 0, X 1,..., X T 1 ] is a K T marix of pricing facors, B = [vec(β (1) β (1) ),..., vec(β (N) β (N) )] is an N K 2 marix, V is a K T marix, and E is an N T marix. A.1. Esimaion ACM show ha he parameers of he model can be obained using a series of linear regressions. We sar by esimaing equaion (A.1) by OLS. The esimaed innovaions ˆv are sacked ino a marix ˆV which is used as a regressor in he esimaion of he reduced-form of (A.9) by OLS: rx = aι T + cx _ + β V + E (A.10) Using he resricions equaion (A.9) imposes on a and c in he equaion above gives us he following esimaes of he risk parameers λ 0 and λ 1 : ( ˆλ 0 = ( ˆβ ˆβ ) 1 ˆβ â + 1 ) 2 (B vec(ˆσ) + ˆσ 2 ι N ) (A.11) ˆλ 1 = ( ˆβ ˆβ ) 1 ˆβĉ (A.12) where ˆσ 2 is compued using he esimaed residuals of (A.10). Lasly, we esimae he shor rae parameers δ 0 and δ 1 by OLS regression of equaion (A.4). A.2. Term premium The affine srucure of he model implies ha he coninuously compounded yield on a n period zero-coupon bond a ime, defined as y (n) = 1 n log P,n is given by y (n) = 1 n ( An + B nx ) (A.13) where he A n and B n parameers are derived recursively using he following sysem of equaions: A n = A n 1 + B n 1 (µ λ 0 ) + 1 ( B 2 n 1 ΣB n 1 + σ 2) δ 0 (A.14) B n = B n 1 (Φ λ 1 ) δ 1 (A.15) A 0 = 0, B 0 = 0 (A.16)
58 The yield in (A.13) includes a compensaion for risk, demanded by risk-averse invesors o inves in a longer-erm bond insead of rolling over a series of shor-erm bonds. Tha is, we can decompose he model-implied yields ino an expecaion componen and a erm premium: y (n) = 1 n n 1 E r +j + T P (n) j=0 (A.17) where he firs erm represens he risk-neural yield, defined as he yield ha would be demanded by invesors which are risk-neural. To obain he riskneural yield we se he price-of-risk parameers λ 0 and λ 1 o zero, and use he recursions in (A.14) and (A.15) o derive he risk-adjused parameers Ãn and B n. The risk-neural yields are compued using: ỹ (n) = 1 n (Ãn + B nx ) (A.18) The erm premium is obained as he difference beween acual (modelimplied) and risk-neural yield T P (n) = y (n) ỹ (n) (A.19) A.3. Uni roo ess EA US level diff. level diff. Dickey-Fuller GLS es -0.18 (-1.95) -6.83 (-1.95) -0.77 (-1.95) -6.97 (-1.95) Phillips-Perron es -1.75 (-3.41 ) -9.04 (-3.41) -3.03 (-3.41) -9.44 (-3.41) TABLE A.1. Tesing for uni roo in he level and differences of he EA and US 10- year erm premium. The null hypohesis for boh ess is ha he process conains a uni roo. The able shows he values of he es saisics and he respecive 5% criical values (in parenhesis). Source: Own calculaions.