Estimating the Term Structure with Macro Dynamics in a Small Open Economy


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1 Esimaing he Term Srucure wih Macro Dynamics in a Small Open Economy Fousseni ChabiYo Bank of Canada Jun Yang Bank of Canada April 18, 2006 Preliminary work. Please do no quoe wihou permission. The paper represens he views of he auhors and should no be inerpreed as reflecing he views of he Bank of Canada. 1
2 Absrac We sudy he join dynamics of bond yields and macroeconomic variables in a NewKeynesian small open economy model complemened wih a noarbirage erm srucure model. Wih Canadian daa, we are able o sudy he impac of domesic and foreign (US) shocks on he yield curve. The uncondiional variance decomposiion of he yield level show ha he movemen of expeced shor raes is mainly driven by US macroeconomic shocks. The majoriy of he variaion of he yield risk premium are also driven by US macroeconomic shocks. However, he Canadian moneary policy shocks can explain a small proporion of he variaion of he shor o medium yield risk premium. In addiion, he Canadian moneary policy shocks and US aggregae demand shocks explain a majoriy of he variaion of he expeced excess holding period reurns of shor o medium bonds. The expeced excess holding period reurns of long erm bonds are mainly driven by US aggregae supply shocks. 1
3 1. Inroducion This paper invesigaes he economic deerminans of he movemen of he erm srucure of he ineres raes in a small open economy (SOE). We esimae a noarbirage erm srucure model wih he dynamics of macroeconomic variables in a newkeynesian small open economy framework. The macrofinance modeling sraegy developed by Ang and Piazzesi (2003) is implemened wih boh Canadian (proxy for a SOE) and US (proxy for he resofworld) daa. We find ha he US macroeconomic shocks conribue o a larger proporion of he variaion of he yield curve and he yield premium han he Canadian macroeconomic shocks. In addiion, he Canadian moneary policy shocks and US aggregae demand shocks explain a majoriy of he variaion of he expeced excess holding period reurns of shor o medium bonds. The expeced excess holding period reurns of long erm bonds are mainly driven by US aggregae supply shocks. Many finance models have used laen variables o explain erm srucure flucuaions. For example, Lierman and Scheinkman (1991) find ha hree principle facors can explain mos of he variaion in bond reurns, and hey label hese facors "level", "seepness", and "curvaure". Chen and Sco (1993) esimae one, wo, and hreefacor CIR models, and find ha only he hreefacor model can capure he changes in he level, slope and curvaure of he yield curves. Pearson and Sun (1994) esimae an exended wofacor CIR model by uilizing he condiional disribuion of he sae variables. They label he sae variables "shor rae" and "inflaion" even hough hey do no use inflaion daa o esimae hese facors. Recen sochasic volailiy models, such as Balduzzi, Das, Foresi, and Sundaram (1996), Anderson and Lund (1998), and Dai and Singleon (2000), inroduce one or wo sae variables o capure he condiional volailiy of he shorerm ineres rae. Consequenly, hey call hese sae variables "volailiy facors". All of he models described above are developed under he assumpion of noarbirage, and hey can capure some imporan feaures of he shorerm ineres rae by using he laen facors. However, hey fail o explain wha macroeconomic variables direcly affec hese laen variables, and hence deermine he movemen of he erm srucure of ineres raes. In a differen approach, many empirical sudies use Vecor Auoregressive (VAR) models o 1
4 explain he join behavior of he erm srucure of ineres raes and macroeconomic variables. For example, Campbell and Ammer (1993) use a VAR model o sudy he excess sock and bond reurns, and heir resuls show ha sock and bond reurns in US are driven largely by news abou fuure excess sock reurns and inflaion. Evans and Marshall (2001) also use a VAR model o invesigae he impacs of moneary and real shocks on various ineres raes. They find ha he shocks o moneary policy have a pronounced bu ransiory impac on shorerm ineres raes, wih almos no effec on longerm ineres raes. In conras, he shocks o employmen have a longlived impac on ineres raes across he mauriy specrum. The VAR model enables hem o examine he impacs of macroeconomic variables on various ineres raes hrough impulse response funcions. However, here are several disadvanages o using he VAR models o sudy he erm srucure of ineres raes. Firs, one can only sudy he effecs of macroeconomics variables on hose yields of mauriies ha are included in he model. The VAR models do no describe how yields of mauriies no included will respond o changes in he macroeconomic variables. Second, he prediced movemens of he yields wih differen mauriies in he VAR models may no rule ou arbirage, since he unresriced VAR models do no require ha he movemen of various ineres raes provide noarbirage opporuniies. An arbiragebased erm srucure model provides a complee descripion of how he yields of all mauriies respond o he shocks o he underlying sae facors, alhough i canno idenify he sources of hose shocks. In conras, he empirical VAR models can idenify he economic sources of he shocks o he seleced yields, bu hey canno ell how he enire yield curve will respond o hose shocks. Recenly, some auhors have ried o combine he srengh of boh he arbiragebased erm srucure models and he VAR models o describe he movemen of he yield curve. Ang and Piazzesi (2003) incorporae boh macroeconomic variables and laen variables ino a Gaussian diffusion model of he erm srucure of ineres raes. They find ha macro variables explain a significan amoun of he variaion in bond yields, and ha incorporaing macro variables ino he model wih laen variables improves he ouofsample forecas. Oher papers include Dewacher and Lyrio (2004), Rudebush and Wu (2004), Ang, Piazzesi, and Wei 2
5 (2004), Ang, Dong, and Piazzesi (2005), Hördahl, Trisani, and Vesin (2003), Dai and Philippon (2004), and Bakaer, Cho, and Moreno (2003). All hese papers sudy he join dynamics of bond yields and macroeconomic variables in a closed economy framework. In his paper, we invesigae he join dynamics of bond yields and macroeconomic variables in a small open economy framework. In an open economy, he real exchange rae movemens play an imporan role in he ransmission process ha links foreign disurbances o domesic oupu and inflaion movemens. The real exchange rae movemens induce subsiuion effecs beween domesic and foreign goods, hereby influencing aggregae demand and supply. In addiion, he moneary auhoriies may sysemaically adjus shorerm ineres rae according o he real exchange rae movemens (Ball(1999), Clarida e al. (2001), and Svensson(2001)). To undersand he effecs of foreign shocks on he domesic economy, one needs o invesigae he ineracion beween he real exchange rae and domesic oupu, inflaion and ineres rae. We consruc a small scale linear macro model o sudy he dynamics beween domesic and foreign macroeconomic variables. The domesic yield curve is modeled in he affine erm srucural framework wih essenial affine risk premium. The price of risk depends on boh domesic and foreign macroeconomic variables. Dong (2005) incorporaes macro variables as facors in a wocounry erm srucure model. The movemen of he exchange rae is pinned down by noarbirage condiion in he domesic and foreign bond markes. In his seup, he shorerm ineres rae does no response o he movemen of exchange rae. In addiion, he focuses on explaining exchange risk premium insead of idenifying economic deerminans of he movemen of he domesic yield curve. Our main findings are as follows. The variance decomposiion resuls show ha he expeced movemen of he Canadian shor rae is mainly explained by US macroeconomic shocks. In shor horizons, i is mainly driven by US aggregae demand and moneary policy shocks. In long horizon, i is mainly driven by US aggregae supply shocks. The Canadian moneary policy shocks explain from 3060% of he onequarer ahead variaion of risk premium embedded in yields. The explanaory power is reduced o 1025% range in long horizons. On he conrary, he explanaory power of he US macroeconomic shocks increases wih forecasing horizons. They explain up o 75% of 3
6 he uncondiional variaion of risk premium embedded in Canadian yields. The same resul holds for expeced excess holding period reurns of Canadian bonds. The remainder of he paper is organized as follows. Secion 2 and 3 ouline he srucural macroeconomic model he erm srucure model respecively. Secion 4 describes he daa used in he paper. We presen our esimaion mehods and resuls in secion 5. Secion 6 concludes. 2. Macroeconomic Model Our srucural model conains seven equaions. The firs hree equaions are: The resofworld (ROW) aggregae supply equaion, he aggregae demand equaion and moneary policy rule. The fourh equaion characerize he exchange rae dynamic. The las hree equaions are: The SOE aggregae supply equaion, he aggregae demand equaion and moneary policy rule. The ROW is considered as he closed economy wih an assumpion ha he SOE shocks do no affec he ROW, whereas he ROW shocks affec he SOE. As shown in Woodford (2003), he ROW se of equaions can be formulaed wih explici microfoundaions as a general equilibrium model. Indeed, Sevnsson (1998) shows ha he SOE se of equaions can also be obained wih microfoundaions as a general equilibrium model. 2.1 Closed Economy Aggregae Supply The aggregae supply equaion is he generalizaion of he supply equaion developed by Calvo (1983) 1. π = α 0 + α π π 1 +(1 α π ) E π +1 + α g g + ε π (1) where π is he inflaion beween 1 and, g is he oupu gap. ε π is he aggregae supply srucural shock. The aggregae supply dynamics (1) is derived in a pricing framework wih monopolisic compeiion in he inermediae good markes. The AS equaion links inflaion o fuure expeced inflaion and he real marginal cos wih an assumpion ha he oupu gap is proporional o he marginal cos. The endogenous persisence in he AS equaion is obained wih an 4
7 assumpion ha he fracion of priceseers which does no adjus prices opimally indexes heir prices o pas inflaion. The coefficien α y is he Phillips curve parameer Aggregae Demand In a closed economy, he aggregae demand is usually derive from he firs order condiions for a represenaive agen in a general equilibrium model such as Lucas (1978). Since sandard approaches fail o mach he wellknown persisence in he oupu gap. To mach he persisence in he oupu gap and pin down he risk aversion parameer, recen sudies, among ohers, Fuhrer (2000) and Cho and Moreno (2005) derive an alernaive IS equaion from a uiliy maximizing framework wih exernal habi formaion. We follow Fuhrer (2000) and Cho and Moreno (2005) and specify he aggregae demand dynamics as: g = β 0 + β gg β g E g +1 β r r E π +1 + ε g (2) where r is he shorerm ineres rae. The residual ε g is he IS or aggregae demand shock. In equaion (2), he habi formaion specificaion impars endogenous persisence o he oupu gap. The forwardlooking parameer β g depends on he level of habi persisence and he risk aversion parameer Moneary Policy Rule The moneary auhoriy se shorerm ineres rae according o a simple Taylor rule (Taylor(1999)): r = γ MP +(1 ρ )(γ π π + γ y g )+ρ r 1 + ε r. (3) where γ MP is a consan, ρ is he smoohing parameer. ε r is he moneary policy shock. The Cenral Bank reacs o high inflaion and o deviaions of oupu from is rend. The parameer γ π measures he response of he Cenral Bank o inflaion, while γ y describes is reacion o oupu gap flucuaions. 5
8 2.2 Small Open Economy Aggregae Supply The aggregae supply equaion (Phillips curve) describes he shor run inflaion dynamics. The aggregae supply equaion is of he Phillips curve ype esimaed by Svensson (1998): π = α 0 + α π π 1 +(1 α π ) E π +1 + α g g + α q (q q 1 )+ε π (4) where π is inflaion beween 1 and, q isherealexchangeraeandg is he oupu gap. Ahigherq denoes a depreciaion of he SOE currency. This supply ype equaion is derived in Sevensson (1998), from he firs order condiion of an opimizaion problem and hence, wih some microfoundaions. Inflaion depends on lagged inflaion, expeced fuure inflaion, he curren oupu gap and he change in he real exchange rae. I is similar o Fuhrer and Moore (1995) ype Phillips curve in ha inflaion depends on boh lagged inflaion and expeced fuure inflaion. The iming on exchange rae changes reflecs an assumpion of insan passhrough. The resofhe world (hereafer ROW) shocks are ransmied o he SOE inflaion mainly hrough he exchange rae. A zero value of α q can be inerpreed as perfec pricing o marke Aggregae Demand The aggregae demand equaion is an aggregae demand ype of equaion developed by Sevensson (2000): g = β 0 + β g g β g E g +1 β r (r E π +1 )+β q (q q 1 )+β g y + ε g (5) where g is he oupu gap, r is he shorerm ineres rae. ε g is he aggregae demand shock. The aggregae demand equaion is derived, from a firs order condiion consisen wih opimizaion and hence wih some microfoundaions, and discussed in furher deail in Svensson (1989). The oupu gap equaion provides a descripion of he dynamics of aggregae demand, which is assumed o be affeced by movemens in he shor erm real ineres rae, he real exchange rae and he foreign oupu gap. The forward looking erm capures he ineremporal smoohing moives characerizing consumpion. A similar specificaion was recenly used by Giordani (2004) o 6
9 evaluae NewKeynesian models of a SOE. The resofhe world shocks are ransmied o he SOE aggregae demand hrough he exchange rae and he ROW macro variables Moneary Policy Rule We assume ha he moneary auhoriy specifies he shorerm ineres rae according o he following reacion funcion r =(1 ρ)(γ π π + γ g g + γ q (q q 1 )+γ r r )+ρr 1 + ε r (6) The lagged ineres rae capures he well known endency of he moneary auhoriy owards smoohing ineres rae. This formulaion assumes ha he ROW shocks are ransmied o he SOE ineres rae hrough he ROW moneary policy Real Exchange Rae Uncovered Ineres Rae Pariy (UIRP) predics ha high yield currencies should be expeced o depreciae. I also predic ha, ceerius paribus, a real ineres rae increase should appreciae he currency. Neverheless, here appears o be overwhelming empirical evidence agains UIRP (see Hodrick (1987) and Engel (1996)). Furhermore, in New Open Economy Macroeconomic Models, domesic and foreign macrovariables ener he exchange rae equaion in differences: Engel and Wes (2004) assume ha exchange rae is a funcion domesic and foreign macro variables. Given he empirical evidence agains UIRP and ha SOE and he ROW macrovariables ener he real exchange rae equaion, we consider he following exchange rae dynamic: q = q 1 + δ r [(r E 1 π ) (r E 1 π )] + ε q (7) where ε q capures he real exchange rae shock. Boh domesic and foreign srucural shocks are assumed o be independen and idenically disribued wih homoscedisic variances. Bringing ogeher all he macroeconomic equaions: (5), (7), (4), (6), (2), (1), and (3), we 7
10 obain a seven variables sysem: π = α 0 + α π π 1 +(1 α π ) E π +1 + α g g + ε π g = β 0 + β gg β g E g+1 β r r E π +1 + ε g r = γ MP +(1 ρ )(γ π π + γ y g )+ρ r 1 + ε r q = q 1 + δ r [(r E 1 π ) (r E 1 π )] + ε q π = α 0 + α π π 1 +(1 α π ) E π +1 + α g g + α q (q q 1 )+ε π g = β 0 + β g g β g E g +1 β r (r E π +1 )+β q (q q 1 )+β g y + ε g r = γ MP +(1 ρ)(γ π π + γ g g + γ q (q q 1 )+γ r r )+ρr 1 + ε r We summarize our macroeconomic model in marix form: A 11 X = α + B 11 E X +1 + B 12 X 1 + ε wih ε Ã N (0, ΣΣ ) (8) where X = (π,g,r,q, π,g,r ) 0 and ε = ³ ε π, ε g, εr, ε q, επ, ε g, εr The coefficiens of marix A 11, B 11 and B 12 are defined by he srucural equaions of he domesic and foreign counry macroeconomic variables. These coefficiens are given byσσ is diagonal marix wih consan variances. As can be seen in his model, he ROW shocks are ransmied o he SOE mainly hrough he real exchange rae and he ROW macro variables. Under regulariy condiions, he soluion of (8) is based on he Schur decomposiion can be obained numerically following he mehodology described in McCallum (1998). Following an Undeermined Coefficien (UC) approach (see McCallum 1998), he soluion of (8) is: X = c + ΩX 1 + Γε (9) The reduced form (9) implied by he srucural model (8) is a VAR of order 1 wih nonlinear parameers. We now add he erm srucure o he model described by equaion (9). 8
11 3. Adding he Term Srucure Informaion o he Macro Model We follow he sandard dynamic arbiragefree erm srucure lieraure and define he SOE nominal pricing kernel as m +1 =exp( r ) ξ +1 ξ =exp µ r 12 λ0λ λ 0ε +1 where ξ +1 is assumed o follow he lognormal process wih: ξ +1 = ξ exp µ 12 λ0λ λ 0ε +1 where λ are he imevarying marke prices of risk associaed wih he source of uncerainy ε +1 in he economy. The marke price of risk parameer is commonly assumed o be consan in Gaussian models or proporional o he facor volailiies (e.g. Dai and Singleon, 2000). However, recen research has highlighed he benefis in allowing for a more flexible specificaion of he marke price of risk. We herefore decide o parameerize λ as a linear funcion of X λ = λ 0 + λ 1 X (10) where X is defined by (9). (10) relaes he shocks in he underlying macro o ξ +1. The las equaion shows ha he source of uncerainy in he SOE pricing kernel is driven by he shocks in macro variables and shorerm ineres rae. X is a vecor conaining seven variables. Noe ha in a microfounded framework (Bekaer, Cho and Moreno (2005)), he pricing kernel would be linked o consumer preferences raher han being posulaed exogenously as in (10). We prefer (10) because he affine prices of risk specificaion in (10) has been used by, among ohers, Ang and Piazzesi (2003), Dai and Singleon (2002). The laer auhors demonsrae ha he flexible affine price of risk specificaion is able o capure paerns of expeced holding period reurns on bonds ha we observe in daa. The consan risk premium parameer λ 0 is a 7 1 vecor column while he ime varying risk premium parameer λ 1 is a 7 7 marix. We assume ha ime varying risk premium parameer λ 1 is a diagonal marix. This reduces he number of parameers o be esimaed. 9
12 If p (n+1) represens he price a of an n +1period zero coupon bond in he SOE, hen he bond price in he SOE can be compued recursively using he relaionship: p (n+1) h i = E m +1 p (n) +1 (11) wih p (n) =exp ³A n + B 0 nx (12) where A 1 =0and B 0 1 =[0 1 2, 1, ] and: A n+1 = A 1 + A n + B 0 n (c Γλ 0 )+ 1 2 B0 nγγ ³ B 0 n (13) B 0 n+1 = B 0 n (Ω Γλ 1 )+B 0 1 Therefore, he bond yields are an affine funcion of he sae variables: y n = log p(n) n = A n + B 0 nx (14) where A n = A n n and B0 n = B 0 n n Le Y represens he vecor conaining he SOE bond yields. Then, Y = A y + B y X (15) Consequenly he model ha need o be esimaed is he following: X = c + ΩX 1 + Γε (16) We define he oneperiod excess holding period reurn as rx (n) +1 (n 1) +1 =logp P (n) Y = A y + B y X (17) and compue he condiional expeced excess holding period reurns as r = ny n (n 1) y n 1 +1 r (18) ³ E rx (n) +1 = A x n + BnX x 10
13 wih: A x n = 1 ³ 2 B0 n 1ΓΓ B 0 ³ n 1 + B 0 n 1 Γλ0 and Bn x = Γ B 0 n 1λ 1 The excess expeced reurn has wo componens: The firs A x n componen is no ime varying while he second componen BnX x is ime varying. The uncondiional excess expeced holding period reurn can be compued as E (A x n + BnX x ). In he empirical illusraions, we assume ha Canada is he SOE whereas US is he closed economy. 4. Daa We esimae he model wih quarerly Canadian yields and Canadian and US macroeconomic daa. The macroeconomic daa are from 1978:Q4 o 2005:Q3.. The macroeconomic variables include inflaion, oupu gap, ineres rae and he real exchange rae. The real exchange rae is consruced from he nominal exchange rae and CPI indexes of boh counries. The core CPI index daa is also used o compue he inflaion. The inflaion rae is compued as he log difference of he core CPI index beween he end and he beginning of each quarer. We measure oupu gap as he difference beween he real GDP and quadraically derended real GDP. The 3monh Tbill raes are used as he moneary policy insrumens in boh counries. The yield daa are from 1978:Q4 o 2005:Q3, and include zero coupon bond yields of mauriies 2, 4, 8, 12, 20, 28, 40 and 60 quarers. A descripion of he mehodology used o derive he yield curves can be found in Bolder, Johnson, and Mezler (2004). Figure 1 plos he macroeconomic variables and Table 1 presens some sample saisics of macroeconomic variables and bond yields. The able shows ha he average yield curve is slighly upward sloping during he sample. The sandard deviaions of yields generally decrease wih mauriy, and yields are highly persisence. Table 1 also shows ha persisence exiss in he macroeconomic variables. 11
14 5. Esimaion and Resuls 5.1 Esimaion Mehodology We implemen maximum likelihood esimaion echnique o esimae macro srucural parameers and he imevarying risk premium parameers. Because of he esimaion difficuly involved wih high dimension maximizing problem, we use wosep esimaion echnique. In he firs sep, we esimae macro srucural parameers wih boh US and Canadian daa. In he second sep, we fix hese parameers, and esimae he risk premium parameers wih Canadian yield daa. The esimaion resul are presened in Table 2. All he repored sandard errors are based on a 3lag Newey and Wes (1987) consisen covariance esimaor. 5.2 Macro Resuls Parameer Esimaes Table 2 presens he parameer esimaes of he model and heir sandard errors. Our esimaion yielded a saionary unique soluion. Panel A shows he parameer esimaes for Canadian macrovariable dynamics. The firs row of Panel A shows he parameer esimaes of he Canadian Phillips curve. The Phillips curve parameer esimae has he expeced sign, bu no saisically significan from zero. The real exchange rae parameer esimae has he wrong sign and no saisically significan from zero. Using US daa, previous sudies, excep Gali and Gerler (1999) and Bekaer, Cho and Moreno (2005), fail o obain reasonable esimae of he Phillips curve parameer α g. The forwardlooking parameer in he AS equaion is esimaed close o 0.55 which is consisen o previous finding in he lieraure. The second row of Panel A shows ha he parameer esimaes for he Canadian aggregae demand equaion. The real ineres rae parameer esimaes has he wrong sign, and he real exchange rae parameer esimae has he expeced sign. They are no saisically significan. The US oupu gap parameer esimae is posiive and saically significan. The US oupu has a direc posiive effec on Canadian ou pu. The parameer β g is almos indisinguishable from 0.5 implying ha agens pu similar weighs on expeced and pas oupu gap. 12
15 The hird row shows he Canadian shor rae equaion parameers. Canadian shor rae loads posiively on he Canadian inflaion, oupu gap, real exchange rae change, and he US shor rae wih coefficiens of 0.042, 0,042, 0.024, and respecively. They are saisically significan. This suggess ha he Canadian moneary auhoriy responses srongly o US shor rae movemen. A 1% conemporaneous inflaion increase leads o only 4 bp increase in he Canadian shor rae. On he conrary, a 1% US shor rae increase leads o 1.17% increase in he Canadian shor rae. Panel B in able 2 shows parameer esimaes in he exchange rae equaion. The real ineres rae differenial parameer esimae is negaive and saisically significan. I is consisen wih many empirical findings ha he uncovered ineres rae pariy does no hold. Panel C shows he parameer esimaes for he US aggregae demand, aggregae supply and shor rae equaion. All he parameer esimaes have he expeced sign. The firs row shows ha he Phillips curve parameer, 0.001, is small and no significan. Fuhrer and Moore (1995), Ireland (2001) and Cho and Moreno (2005) obained esimaes of similar magniudes. This reflecs he weak link beween derended oupu and inflaion in he daa. This finding is consisen wih he previous lieraure. The second row shows he parameer esimaes for he US oupu gap equaion. The oupu gap can be forecased by he lagged US oupu gap which is consisency wih previous sudies. The US shor rae loads negaively on he US oupu gap bu he coefficien of he US oupu gap in he shor rae equaion is no saisically significan. In he US moneary policy equaion, he smoohing parameer ρ is around 0.74, reflecing he persisence in he shorerm ineres rae. The coefficien of inflaion is around 1.08, suggesing srong response of he FED o inflaion. Figure 2 presens he recovered srucural shocks for Canadian and US. I shows here is no major Canadian (US) AS shocks during he sample period. The Canadian (US) IS shocks exhibis some persisence. The US moneary policy shocks were of small magniude afer his resuls are consisen wih Taylor (1999) and Leeper and Zha (2000). 13
16 5.2.2 Impulse Response of Macro Variables To gauge he effec of he various shocks on Canadian macro variables, we compue impulse response funcions. Figure 3 shows he impulse response funcions of Canadian macroeconomic variables o he srucural shocks. The firs row of graphs in Figure 3 shows he responses of Canadian macro variables o one sandard deviaion Canadian AS shock. The inflaion shock is a negaive echnology or supply shock which decreases he produciviy of firms. As expeced he Canadian aggregae supply shock pushes Canadian inflaion almos 30 bp above is seady sae, bu i soon reurns o is original level, given he forwardlooking naure of he aggregae supply equaion (he coefficien of Canadian inflaion in he Canadian Phillips curve equaion is 0.55). The moneary auhoriy increases he ineres rae by 1 bp following he supply shock. The oupu exhibis a humpshaped decline for few quarers. The real exchange rae depreciaes afer he AS shock. The second row of graphs in Figure 3 shows he responses of Canadian macro variables o one sandard deviaion Canadian IS shock. The IS shock is a demand shock which can also be inerpreed as a preference shock (see Woodford (2003)). The IS shock iniially increases oupu, inflaion and ineres rae. Canadian oupu gap iniially increases abou 50 bp, bu i soon reurns o is seady sae. The IS shock has no iniial impac on he real exchange rae. The hird row of graphs in Figure 3 shows he responses of Canadian macro variables o one sandard deviaion Canadian moneary policy shock. The moneary policy shock reflecs shifs o he ineres rae unexplained by he sae of he economy. Given our moneary ransmission mechanism, he ineres rae increases by 18 bp following he moneary policy shock, bu hen decreases o is seady sae level. The impacs on domesic inflaion and oupu gap are weak. The forh row of graphs in Figure 3 shows he responses of Canadian macro variables o one sandarddeviaionusasshock. Canadianinflaion responses weakly o he US AS shock. The Canadian oupu gap responses negaively o he US AS shock, and say in he negaive region for a long ime. The Canadian ineres rae responses srongly o US AS shock. The fifh row of graphs in Figure 3 shows he responses of Canadian macro variables o one 14
17 sandard deviaion US IS shock. The US IS shock has no effec on Canadian inflaion. I has srong effecsoncanadianinflaion and shorerm ineres rae. The sixh row of graphs in Figure 3 shows he responses of Canadian macro variables o one sandard deviaion US moneary policy shock. Canadian inflaion and oupu gap response weakly o he US moneary policy shock. The Canadian ineres rae responses srongly by increasing 20 bp following he US moneary policy shock. The las row of graphs in Figure 3 shows he responses of Canadian macro variables o one sandard deviaion real exchange rae shock. There is no much response of Canadian inflaion, oupu gap and ineres rae o he real exchange rae shock. 5.3 Yield Resuls Parameer Esimaes Panel D of Table 3 repors he esimaes of he marke prices of risk wih he resricion ha he marix parameer λ 1 is diagonal. The risk premia in λ 1 indicae ha expeced excess reurns vary significanly over ime. All he diagonal elemens of λ 1 are all saisically significan excep for he ime varying componen due o he real exchange rae Impulse Response of Yields Our srucural model allows us o compue impulse response funcions of Canadian bond yields o he 7 srucural shocks. Figure 4 shows he impulse responses of he 1Year, 5Year, and 15Year yields o he srucural shocks. Canadian aggregae supply shock iniially raises he level of all yields. The iniial response is highes for he long yield (15Year yield), a 7 bp1, while he iniial response of he shor yield, 1year yield is small. The US aggregae supply shock iniially raises all yields. The iniial responses o US aggregae supply shock is abou he same for all yields. They peak afer four quarers, hen decreases o heir seady sae. Canadian aggregae demand shock has no impac on all yields, whereas he US aggregae demand shock iniially decreases he 15Year yield by 10 bp. The impac of US aggregae demand shock on Canadian shor and medium yield is small. 15
18 Canadian moneary policy shock iniially raises all he yields bu he iniial response is highes for he shor yield a 15 bp, while he iniial response of he medium and long yields is small. However, he US moneary policy shock has an immediae high posiive impac on all yields, almos 25 bp, hen he responses decline during he firs 5 quarers and reach he seady sae. The real exchange rae shock has no impac on all yields Variance Decomposiion Yield Levels In our model, equaion (15) saes ha he variables in X explain all yield dynamics. To complemen impulse response funcions we presen he analyze based on uncondiional variance decomposiion of yields from equaion (15) and he daa a differen horizons. These decomposiions are based on Cholesky decomposiions of he innovaion variance in he order: X =(π,g,r,q, π,g,r ) 0. We ignore observaion error in he yields when compuing variance decomposiions. The resuls are repored in Table 3 for horizon h =1, 4, quarers. In he column under he heading EH (Expecaion Hypohesis), we compue he proporion of he forecas variance aribuable o Expecaion Hypohesis. In he column under he heading UPRP (Uncondiional Pure Risk Premia), we compue he proporion of he forecas variance aribuable o imevarying risk premia. To compue he proporion of forecas variance aribuable o imevarying risk premia, we follow Ang, Dong and Piazezzi (2005) and pariion he bond coefficien B 0 n on X in equaion (17) ino an Expecaion Hypohesis componen and ino a risk premia componen: B 0 n = B 0 EH n + B 0 RP n where he B 0 EH n bond pricing coefficien is compued by seing λ 1 =0. Since he yield dynamics are given by y n = A n + B 0 nx,wehave y n +h = A n + B 0 EH n X +h + B 0 RP n X +h Le Ω F,h represen he forecas variance of he facors X, a horizon h. The forecas variance of 16
19 he nquarer yield a horizon h is given by Var y+h n 0 = B nω F,h B n = B 0 EH n Ω F,h B EH n {z } (1) +2B 0 EH n Ω F,h Bn RP {z } (2) + B 0 RP n Ω F,h Bn RP. {z } (19) (3) In (19), we ignore he componen (2) whichishecovarianceofheriskpremiawihhesae variables and hen compue he proporion of he variance of yields aribuable o imevarying risk premia as follows: Risk Premia Proporion = B0 RP n Ω F,h Bn RP BnΩ 0 F,h. B n Noe ha he model implied uncondiional pure risk premia proporion is acually a raio (can be higher han % if he risk premia and sae variables are negaively correlaed ). We also compue he proporion of forecas variance aribuable o he expecaion hypohesis as follows: Expecaion Hypohesis Proporion = B0 EH n Ω F,h Bn EH BnΩ 0 F,h. B n Panel A of Table 3 shows he variance decomposiion for yield levels for onequarer ahead horizon. I shows ha risk premia play imporan role in explaining he level of yields. The expecaion hypohesis proporion of he 1Year yield is 104% while he risk premia proporion of he 1Year yield is 39%. As he yield mauriy increases, he expecaion hypohesis proporion decreases (96% for 2Year yield and 9% for 15Year yield) while he risk premia proporion increases (73% for 2Year yield and 109% for 15Year yield). Under he lines Expecaion Hypohesis and Risk Premia, Panel A shows he variance decomposiions for he variance of he expecaion hypohesis componen, B 0 EH n Ω F,h Bn EH and he risk premia variance B 0 RP n Ω F,h Bn RP respecively. For one quarer ahead horizon, Canadian inflaion and oupu gap canno explain he forecas variance of all yields. In he risk premia componens, he Canadian moneary policy shock explains a smaller proporion of he forecas variance of shor and long yields han medium yields. The rang is from 30% for he 1year and 15year yields o 56% for he5year yields. The res are explained by US macroeconomic shocks. Panel B and C repors he variance decomposiion for yield levels for fourquarer and  quarer ahead horizon respecively. The resuls are similar o hose obained in Panel A. The 17
20 uncondiional variance decomposiion resuls show ha he expecaion componen of all yields are oally explained by US macroeconomic shocks. The explanaory power of Canadian moneary policy shock on he risk premium componens declines o a range of 1014%. Yield Spreads We repea he same analysis for yield spreads of mauriy n quarers in excess of he onequarer yield, y n y 1. Table 4 shows ha risk premia maer even more for yield spreads. Panel A, B,and C repors he variance decomposiions of he expecaion hypohesis componen, B 0 EH n Ω F,h Bn EH and he risk premia variance B 0 RP n Ω F,h Bn RP a 1quarer, 4quarer, and  quarer ahead respecively. In he expecaion hypohesis erm, he proporion of forecas variance aribuable o he Canadian moneary policy shock is abou 5673% for all yield spreads while he US macroeconomic shocks explain he res. The Canadian moneary policy shock explains a smaller proporion of he forecas variance of risk premia componens in longer horizons. The US macroeconomic shocks explain abou 7795% of uncondiional risk premia variance. Expeced Excess Holding Period Reurns Table 5 repors he variance decomposiion of expeced excess holding period reurns. The expeced excess holding period is he risk premia. Thus, ime varying risk premia is equivalen o ime varying expeced excess holding period reurns. Panel A, B, and C repors he variance decomposiion of expeced excess holding period reurns a 1quarer, 4quarer, and quarer ahead respecively. I reveals ha US macroeconomic shocks explain a majoriy of he expeced excess holding period reurn variance. The Canadian moneary policy shock and he US aggregae supply shock explain a majoriy of he variaion of he expeced excess holding period reurns of shorerm yields. Bu he US aggregae demand shocks explains up o 95% of he variaion of he expeced excess holding period reurns of long yields Characerizing Excess Reurn In Panel D, we repor he means and sandard deviaions of he approximae excess reurns compued from yield daa and implied by our model. This panel shows ha he sandard deviaion of 18
21 excess reurns compued from he model are nearly idenical o heir approximae counerpars for 4, 8 and 20quarer yields. The model overesimae he sandard deviaion of excess reurns for 40 and 60quarer yields and underesimae he mean of excess reurns for all yields. 6. Conclusion We esimae he join dynamics of macroeconomic variables and bond yields in a small open economy framework complemened wih an affine erm srucure model. Wih Canadian and U.S. daa, we are able o sudy he impac of domesic and foreign (US) shocks on he yield curve. The uncondiional variance decomposiion of he yield level show ha he movemen of expeced shor raes is mainly driven by US macroeconomic shocks. Themajoriyofhevariaionofheyieldrisk premium are also driven by US macroeconomic shocks. However, he Canadian moneary policy shocks can explain a small proporion of he variaion of he shor o medium yield risk premium. In addiion, he Canadian moneary policy shocks and US aggregae demand shocks explain a majoriy of he variaion of he expeced excess holding period reurns of shor o medium bonds. The expeced excess holding period reurns of long erm bonds are mainly driven by US aggregae supply shocks. References [1] Ang, Andrew, Dong, S., and Piazzesi, M., 2005, Noarbirage Taylor rules, Working paper. [2] Ang, Andrew, and Piazzesi, M., 2003, A noarbirage vecor auoregression of erm srucure dynamics wih macroeconomic and laen variables. Journal of Moneary Economics, 50, [3] Ang, Andrew, Piazzesi, M., and Wei, M., 2004, Wha does he yield curve ell us abou GDP growh?, Working paper. [4] Bekaer, Geer, Cho, Seonghoon, and Moreno, Anonio, 2005, NewKeynesian macroeconomics and he erm srucure, Working paper. 19
22 [5] Ball Laurence, 1998, Policy rules for open economies. NBER working paper [6] Clarida, Richard, Galí, J., and Gerler, M., 1999, The science of moneary policy: a new Keynesian perspecive, Journal of Economic Lieraure, 37, [7] Dewacher, Hans, and Lyrio, Macro, 2004, Macro facors and he erm srucure of ineres raes, Working paper. [8] Diebold, Francis X., Rudebusch, Gleen D., and Aruoba, S. Boragan, 2004, The macroeconomy and he yield curve: a dynamic laen facor approach, NBER working paper [9] Evans, Charles L., and Marshall, David, 2001, Economic deerminans of he nominal reasury yield curve, Working paper. [10] Fuhrer, Jeffrey C., Habi formaion in consumpion and is implicaions for monearypolicy models. American Economic Review, 90, [11] Galí, Jordi, and Gerler, M., 1999, Inflaion dynamics: a srucural economeric analysis, Journal of Moneary Economics, 44, [12] Giordani, Paul, Evaluaing NewKeynesian models of a small open economy. Oxford Bullein of economics and Saisics, 66, [13] Hördahl, Peer, Trisani, O., and Vesin, D., 2004, A join economeric model of macroeconomic and erm srucure dynamics, Working paper. [14] Lane, P. R. and Ganelli, G. 2003, Dynamic general equilibrium analysis: he open economy dimension. in Alug S., Chaddha J. and Nolan C. (eds), Dynamic Macroeconomic Analysis, Cambridge Universiy Press, cambridge, pp, [15] Leeper, Eric M., and Zha. T., 2000, Assessing simple policy rules: a view from a complee macro model, Federal Reserve Bank of Alana Working Paper 19. [16] McCallum, Benne T., 1998, Soluions o linear raional expecaions models: a compac exposiion. Economics leers, 6,
23 [17] Rudebusch, Glenn D., and Wu, T., 2003, A macrofiance model of he erm srucure, moneary policy, and he economy, Working paper [18] Svensson, Lars E. O., Open Economy Inflaion Targeing. Journal of Inernaional Economics, 50, [19] Taylor, John B., 1999, A hisorical analysis of moneary policy rules in John B. Taylor, Ed. Moneary Policy Rules, Chicago Universiy of Chicago Press,
24 22
25 Table 1: Descripive Saisics Descripion of macroeconomic variables Mean Variance Skewness Kurosis Canadian Inflaion Canadian Oupu Gap Canadian Shor erm ineres rae Exchange Rae US inflaion US Oupu Gap US Shor erm ineres rae Aurocorrelaions π g r q π g r Lag Lag Descripion of Yields Mean Variance Skewness Kurosis 6 monh Year Years Years Years Years Years Years Auocorrelaions 6 Monh 1 Years 2 Years 3 Years 5 Years 7 Years 10 Years 15 Years Lag Lag Noe: This able shows he summary saisics for macro variables and yields. The sample period is from 1978:Q4 o 2005: Q3 23
26 Table 2: Domesic counry macro dynamics PanelA:SOEMacroDynamics π α π α g α q σ ε π (0.032) (0.041) (0.016) g β g β r β q β g σ ε π (0.027) (0.032) (0.057) (0.003) r γ π γ g γ q γ r ρ σ ε r (0.013) (0.012) (0.002) (0.044) PanelB:ExchangeraeEquaionCoefficiens q δ r σ ε r (0.019) (0.063) Panel C: ROW Macro dynamics π α π α g σ ε π (0.025) (0.012) g β g β r σ ε g (0.019) (0.009) r γ π γ g ρ σ ε r (0.072) (0.002) (0.012) Panel D: Marke Price of Risk π g r q π g r λ (2.475) (0.521) λ (3.625) (10.915) (31.437) (17.612) (24.843) (38.111) Observaion Error Sandard Deviaion σ (n) n =2 n =4 n =8 n =12 n =20 n =28 n =40 n = Noe: This able lis parameer esimaes of he model. Panel A repors parameer values for he domesic counry as in equaions (4), (5) and (6). Sandard errors are in parenhese. Panel B repors he parameer value for he exchange rae dynamic as in equaion (7). Panel C repors parameervalues for he domesic counry as in equaions (1), (2) and (3). Panel D liss marke prices of risk esimaes for he model as in equaion (10)
27 Table 3: Variance Decomposiions:: Yield levels Expecaion Hypohesis Risk Premia Q EHP π g r q π g r UPRP π g r q π g r Panel A: h =1Q Panel B: h =4Q Panel C: h =Q Noe: The able repors uncondiional variance decomposiions of forecas variance for yield levels y n. In each panel, he numbers under he line " Expecaion Hypohesis" repor he variance decomposiions for he componen of he variance yields ha is due o Expecaion Hypohesis, B 0 EH n Ω F,h Bn EH. The numbers under he line "Risk Premia" repor he variance decomposiions for he pure risk premia variance, B 0 RP n Ω F,h Bn RP. The number under he line "EHP" repors he proporion of forecas variance aribuable o Expecaion Hypohesis. The number under he line "UPRP" repors he proporion of forecas variance aribuable o imevarying risk premia. We ignore observaion error for compuing variance decomposiions for yield levels and yield spreads. 25
28 Table 4: Variance Decomposiions:Yield spread levels Expecaion Hypohesis Risk Premia Q EHP π g r q π g r UPRP π g r q π g r Panel A: h =1Q Panel B: h =4Q Panel C: h = Q Noe: The able repors uncondiional variance decomposiions of forecas variance for yield spread levels y n y 1. In each panel, he numbers under he line " Expecaion Hypohesis" repor he variance decomposiions for he componen of he variance yields ha is due o Expecaion Hypohesis, B 0 EH n Ω F,h Bn EH.The numbers under he line "Risk Premia" repor he variance decomposiions for he pure risk premia variance, B 0 RP n Ω F,h Bn RP. The number under he line "EHP" repors he proporion of forecas variance aribuable o Expecaion Hypohesis. The number under he line "UPRP" repors he proporion of forecas variance aribuable o imevarying risk premia. We ignore observaion error for compuing variance decomposiions for yield levels and yield spreads. 26
29 Table 5: Variance Decomposiions: Condiional expeced excess holding period reurns Risk Premia Q UPRP π g r q π g r Panel A: h =1Q Panel B: h =4Q Panel C: h = Q Panel D: Characerizing excess reurns Daa Model Implied mean sd mean sd Noe: Panel A and B of his able³ repors uncondiional variance decomposiions of he condiional expeced excess holding period reurns E rx (n) +1 The numbers under he line "Risk Premia" repor he variance decomposiions for he pure risk premia variance, B 0 RP n Ω F,h Bn RP. The number under he line "UPRP" repors he proporion of forecas variance aribuable o imevarying risk premia. The mauriies are in quarers. 27
30 π Quarers 0.1 π * Quarers 0.1 g 0 g * Quarers Quarers r Quarers 1 r * Quarers 0.8 q Quarers Figure 1: This Figure shows he values of he ime series of he macrovariables for boh foreign and domesic counry. The sample period is from 1978: Q4 o 2005: Q3. 28
31 0.01 ε π 0.02 ε * π π 0 π * Quarers Quarers 0.02 ε g 0.02 ε * g g 0 g * Quarers Quarers 0.01 ε r 0.02 ε * r r 0 r * Quarers Quarers ε q 0.1 q Quarers Figure 2: This Figure shows he values of he ime series of he macrovariable errors for boh foreigh and domesic counry. The sample period is from 1978: Q4 o 2005: Q3. 29
32 Figure 3: The panels show response of he one, four and fouryquarer domesic counry macrovariables o one sandard deviaion shock o macro variables. 30
33 Figure 4: The panels show response of he 1 Year, 5 Year and 15Year yield level o one sandard deviaion shock o macro variables. 31
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