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Switching VARMA Term Structure Models (1) (2) Alain MONFORT Fulvio PEGORARO First version : February, 2005 This version : April, 2006 [Preliminary and incomplete version] Abstract Switching VARMA Term Structure Models The purpose of the paper is to propose a global discrete-time modeling of the term struc- ture of interest rates able to capture simultaneously the following important features : (i) interest rates with an historical dynamics involving several lagged values, and switching regimes; (ii) a speciﬁcation of the stochastic discount factor (SDF) with time-varying and regime-dependent risk-premia; (iii) the possibility to derive explicit or quasi explicit for- mulas for zero-coupon bond and interest rate derivative prices; (iv) the positiveness of the yields at each maturity. We develop the Switching Autoregressive Normal (SARN) Term Structure model of order p and the Switching Autoregressive Gamma (SARG) Term Structure model of order p. Regime shifts are described by a Markov chain with (histori- cal) state-dependent transition probabilities. In both cases multifactor generalizations are proposed. An empirical application to the U.S. term structure of interest rates, observed from June 1964 to December 1995, is presented. Keywords : Aﬃne Term Structure Models, Stochastic Discount Factor, Car processes, Switching Regimes, VARMA processes, Lags, Positiveness, Derivative Pricing. JEL number : C1, C5, G1 1 CNAM, Chaire de Modelisation Statistique, 292, rue Saint-Martin 75141 Paris cedex 03 (France); E-mail: monfort@cnam.fr. CREST, Laboratoire de Finance-Assurance, Bu- e reau 1121, Timbre J320, 15, Boulevard Gabriel P´ri, 92245 Malakoﬀ Cedex (France); E-mail: monfort@ensae.fr. 2 e e CEREMADE, Universit´ Paris-Dauphine, Place du Mar´chal de Lattre de Tassigny 75775 Paris Cedex 16 (France); E-mail: pegoraro@ceremade.dauphine.fr. CREST, Lab- e oratoire de Finance-Assurance, Bureau 1112, Timbre J320, 15, Boulevard Gabriel P´ri, 92245 Malakoﬀ Cedex (France); E-mail: pegoraro@ensae.fr. 1 INTRODUCTION In this paper we propose a global discrete-time modeling of the term struc- ture of interest rates, which captures simultaneously the following important features : - interest rates with an historical dynamics involving several lagged val- ues, and switching regimes; - a speciﬁcation of the stochastic discount factor (SDF) with time- varying and regime-dependent risk-premia; - the possibility to derive explicit or quasi explicit formulas for zero- coupon bond and interest rate derivative prices; - the positiveness of the yields at each maturity. It is well known in the literature that interest rates show an histori- cal dynamics involving lagged values and switching regimes [see, among the others, Hamilton (1988), Cai (1994), Driﬃll and Sola (1994), Garcia and Perron (1996), Gray (1996), Boudoukh, Richardson, Smith, and Whitelaw (1999), Ang and Bekaert (2002a, 2002b), Christiansen (2002), Christiansen and Lund (2005), Cochrane and Piazzesi (2005)]; indeed, changes in the busi- ness cycle conditions or monetary policy may aﬀect real rates and expected inﬂation and cause interest rates to behave quite diﬀerently in diﬀerent time periods, both in terms of level and volatility. In addition, there is a large empirical literature on bond yields, based in general on the class of Aﬃne Term Structure Models (ATSMs)3 , suggesting that regime switching models describe the term structure of interest rates better than single-regime mod- els [see, for example, Bansal and Zhou (2002), Driﬃll, Kenc and Sola (2003), Evans (2003), Ang and Bekaert (2005), Dai Singleton and Yang (2005)]. This results lead us to propose dynamic term structure models (DTSMs) where the yield curve is driven by a univariate or multivariate factor (xt ) 3 The Aﬃne family of dynamic term structure models (DTSMs) is characterized by the fact that the zero-coupon bond yields are aﬃne functions of Markovian state variables, and it gives closed-form expressions for zero-coupon bond prices which greatly facilitates pricing and econometric implementation [see Duﬃe and Kan (1996), and Dai and Singleton (2003) and Piazzesi (2003) for a survey]. Observe that the Aﬃne Term Structure family is much larger that it has been considered in the literature : indeed, it has been observed recently that the family of Quadratic Term Structure Models (QTSMs) [see Beaglehole and Tenney (1991), Ahn, Dittmar and Gallant (2002), and Leippold and Wu (2002)] is a special case of the Aﬃne class obtained by stacking the factor values and their squares [see Gourieroux and Sufana (2003), Cheng and Scaillet (2005)]. 1 which depends on its p most recent lagged values [Xt , say] and for which all the sensitivity coeﬃcients depend on the present and past values of a latent J-states non homogeneous Markov Chain (zt ) [Zt , say] describing diﬀerent regimes in the economy. Consequently, the joint dynamics of (xt , zt ) is not a Compound Autoregressive (Car) process4 under the historical probability, and allows for nonlinearities which has been documented in the literature [see Ait-Sahalia (1996), Stanton (1997), Ang and Bekaert (2002b)]. The factor (xt ) is considered as an exogenous variable or an endogenous variable: in the second case the factor is a vector of several yields. We consider an exponential-aﬃne SDF with time-varying and regime- dependent risk correction coeﬃcients deﬁned as functions of the present and past values of the factor (xt ) and the regime indicator function (zt ). In our models, both factor risk and regime-shift risk are priced, and this is done by taking into account not just the information at date t, that is (xt , zt ), but a larger information given by (Xt , Zt ). This speciﬁcation leads to stochastic and regime-dependent risk premia. This speciﬁcation is coherent with the recent empirical literature which suggests to deﬁne risk correction coeﬃcients as functions, at the same time, of the factors and their volatilities, in order to well replicate the observed temporal variation of one- period expected excess returns on zero-coupon bonds [see Ahn, Dittmar and Gallant (2002), Dai and Singleton (2002), Duﬀee (2002), Duarte (2004), Dai, Singleton and Yang (2005)]. Moreover, the fact to consider these coeﬃcients as function of (Xt , Zt ) lead to a multi-lag speciﬁcation which generalizes the Markovian of order one speciﬁcations proposed in the literature [see Dai and Singleton (2000), Duﬀee (2002), Duarte (2004), Cheridito, Filipovic and Kimmel (2005), Dai, Le and Singleton (2006)]. At the same time, we want to exploit the tractability of Car models, and obtain explicit or quasi explicit formula for zero-coupon bond and interest rate derivative prices. This result is achieved by matching the historical distribution and the SDF in order to get a Car risk-neutral joint dynamics for (xt , zt ). Moreover, in this paper we deeply use the nice property of the Car family of being able to incorporate lags and switching regimes. It is now well known [see Gourieroux, Monfort and Polimenis (2005), and Darolles, Gourieroux, Jasiak (2006)] that the class of discrete-time aﬃne (Car) models is much larger than the discrete-time counterparts of the continuous-time aﬃne processes [see Duﬃe and Kan (1996), Dai and Singleton (2000), and Duﬃe, Filipovic and Schachermayer (2003)]. 4 A Car (discrete-time aﬃne) process is a Markovian process with an exponential-aﬃne conditional Laplace transform [see Darolles, Gourieroux, Jasiak (2006) for details]. 2 We develop the Switching Autoregressive Normal (SARN) Term Struc- ture model of order p and the Switching Autoregressive Gamma5 (SARG) Term Structure model of order p, and in both cases we propose multifac- tor generalizations : the Switching Vector Autoregressive Normal (SVARN) and the Switching Vector Autoregressive Gamma (SVARG) Term Structure models of order p. Even if the Gaussian family of models does not guarantee the positive- ness of the yields for every time to maturity [see, among the others, Vasicek (1977), Dai and Singleton (2000), Bekaert and Grenadier (2001), Ang and Bekaert (2002), Ang and Piazzesi (2003), Ang, Piazzesi and Wei (2005)], we study the SARN(p) Term Structure model (and its multivariate generaliza- tion), because it extends many standard models, like the ones just mentioned above and the more recent ones like Dai, Singleton and Yang (2005). Indeed, the historical and risk-neutral dynamics of (xt ) depends on several of their lagged values and on several lagged values of the regime-indicator variable (zt ). In this general setting, we are able to derive formulas, for the yield curve and for the price of derivatives, with simple analytical or quasi explicit representations. The second kind of models we propose in the paper, based on the (scalar and vector) Switching Autoregressive Gamma process of order p (which has a Regime-Switching AR(p) representation with a martingale diﬀerence error), implies the positiveness of the yields for each time to maturity, and regard- less of an exogenous or endogenous speciﬁcation for the factor (xt ). The SARG(p) and the SVARG(p) term structure models give the possibility to replicate complex nonlinear (historical and risk-neutral) factor dynamics and provide explicit or tractable formulas for zero-coupon bond and derivative prices. In a related study, Bansal and Zhou (2002) propose an (approximate, scalar and bivariate) discrete-time Cox-Ingersoll-Ross term structure model with regime shifts. We extend their framework, using the exact discrete- time equivalent of the CIR process (with switching regimes) generalized to an autoregressive order p larger than one (the SARG(p) and the SVARG(p) processes), allowing for a non homogeneous historical transition matrix for (zt ), pricing the regime-shift risk, and providing an exact yield to maturity formula [in Bansal and Zhou (2002), (zt ) is an homogeneous Markov chain, the associated risk correction coeﬃcient is assumed equal to zero, and the term structure formula they provide is based on a log-linear approximation 5 The Autoregressive Gamma (ARG) process is a Car process, and the ARG(1) speciﬁ- cation is the discrete-time counterpart of the Cox-Ingersoll-Ross process [see Gourieroux and Jasiak (2006), Cox, Ingersoll, and Ross (1985)]. 3 applied on the fundamental asset pricing equation]. In a recent paper Dai, Le and Singleton (2006) propose a (discrete-time multivariate) conditionally Gaussian term structure model where nonlinear- ities are introduced in the (latent) state-factor (historical and risk-neutral) dynamics by means of stochastic volatility factors, for which the risk-neutral conditional distribution is described by a particular VARG(1) process with conditionally independent components. The switching vector Autoregressive Gamma process we use to describe the risk-neutral dynamics of the factor (xt ), in the SVARG(p) term structure model, presents three generalizations with respect to their Markovian of order one speciﬁcation: a) we consider an autoregressive order p in general larger than one; b) conditionally to the present and past values of xt and zt , there is dependence between the com- ponents of the factor xt+1 ; c) the historical and risk-neutral dynamics of xt+1 is aﬀected by switching regimes. The plan of the paper is as follows. In Section 2, we present the Index- Car(p) processes. This family of processes is developed in a univariate and multivariate setting, with and without Switching Regimes. In particular, we study the (scalar and vector) Autoregressive Gaussian of order p mod- els and the (scalar and vector) Autoregressive Gamma of order p models, under single-regime and regime-switching speciﬁcations. Then, this class of processes is used, following the SDF modeling principle, to the derive the SARN(p) and the SARG(p) discrete-time term structure models, and their multivariate generalizations. In Section 3 we study the SARN(p) and the SVARN(p) Term Structure models, we derive the Generalized Linear Term Structure formulas and we specify the historical and risk-neutral dynamics of the yield curve processes. These results are given for an exogenous or an endogenous factor. Moreover, we discuss the propagation of shocks on the interest rate surface. Section 4 deal with the SARG(p) and the SVARG(p) Term Structure models. Here, regardless the endogenous or exogenous na- ture of the factor (xt ), we derive the Generalized Linear Term Structure formulas and the yield curve processes, and we guarantee the positiveness of the yields for each time to maturity. Finally, the pricing methodology pro- posed in sections 3 and 4, for zero-coupon bonds, is generalized in Section 5 to the case of interest rate derivatives. Section 6 concludes and appendices gather the proofs. 4 2 LAPLACE TRANSFORMS, CAR(p) PROCESSES AND SWITCHING REGIMES It is now well documented [see e.g. Darolles, Gourieroux and Jasiak (2006), Gourieroux and Monfort (2006), Gourieroux, Monfort and Polimenis (2002, 2003), Polimenis (2001)] that the Laplace transform (or moment generating function) is a very convenient mathematical tool in many ﬁnancial domains. It is, in particular, a crucial notion in the theory of Car(p) processes [see Darolles, Gourieroux and Jasiak (2006) for details]. 2.1 Deﬁnition of a Car(p) process ˜ x Deﬁnition 1 [Car(p) process]: A n-dimensional process x = (˜t , t ≥ 0) is a compound autoregressive process of order p [Car(p)] if the distribution of ˜ ˜ x ˜ xt+1 given the past values xt = (˜t , xt−1 , . . .) admits a real Laplace transform of the following type: ˜ ˜ E exp(u xt+1 ) | xt ˜ = Et [exp(u xt+1 )] (1) = exp a1 (u) xt + . . . + ap (u) xt+1−p + ˜ ˜ ˜ ˜ ˜ b(u) , u ∈ Rn , where ai (u), i ∈ {1, . . . , p}, and b(u) are nonlinear functions, and where ap (u) = 0, ∀ u ∈ Rn . The existence of this Laplace transform in a neigh- borhood of u = 0, implies that all the conditional moments exist, and that the conditional expectations and variance-covariance matrices (and all con- x ˜ ˜ ditional cumulants) are aﬃne functions of (˜t , xt−1 , . . . , xt+1−p ). 2.2 Univariate Index-Car(p) process An important class of Car(p) processes are the Index-Car(p) processes, which are built from a Car(1) process. In this section we consider a univariate process xt and the multivariate case will be considered in sections 2.6 and 2.7. Deﬁnition 2 [Univariate Index-Car(p) process]: Let exp[a(u)yt +b(u)] be the conditional Laplace transform of a univariate Car(1) process yt , the process xt admitting a conditional Laplace transform deﬁned by: E exp(uxt+1 ) | xt = exp [a(u)(β1 xt + . . . + βp xt+1−p ) + b(u)] , u ∈ R , (2) 5 is called an Univariate Index-Car(p) process. Note that, if yt is a positive process and if the parameters β1 , . . . , βp are positive, the process xt will be positive. Using the notation β = (β1 , . . . , βp ) and Xt = (xt , xt−1 , . . . , xt+1−p ) , the Laplace transform (2) can be written as: E exp(uxt+1 ) | xt = exp [a(u)β Xt + b(u)] . (3) 2.3 Examples of Univariate Index-Car(p) processes a. Gaussian model If yt is a Gaussian AR(1) process deﬁned by: yt+1 = ν + ρyt + εt+1 where εt+1 is a gaussian white noise distributed as N (0, σ 2 ), the conditional Laplace transform of yt+1 given yt is: σ2 2 E exp(uyt+1 ) | yt = exp uρyt + uν + 2 u . 2 The process is Car(1) with a(u) = uρ and b(u) = uν + σ u2 . The associated 2 Index-Car(p) process has a conditional Laplace transform deﬁned by: σ2 2 E exp(uxt+1 ) | xt = exp uρ(β1 xt + . . . + βp xt+1−p ) + uν + 2 u ; so, using the notation ϕi = ρβi , we see that xt+1 is the Gaussian AR(p) process deﬁned by: xt+1 = ν + ϕ1 xt + . . . + ϕp xt+1−p + εt+1 (4) and its conditional Laplace transform becomes: σ2 2 E exp(uxt+1 ) | xt = exp uϕ Xt + uν + 2 u , (5) where ϕ = (ϕ1 , . . . , ϕp ) . b. Gamma model Let us now consider an autoregressive gamma of order one [ARG(1)] process yt . The conditional Laplace transform is [see Gourieroux and Jasiak (2005) for details]: ρu E exp(uyt+1 ) | yt = exp 1−uµ yt − ν log(1 − uµ) , ρ > 0 , µ > 0 , ν > 0 , 6 and it is well known that, given yt , yt+1 can be obtained by ﬁrst drawing a latent variable Ut+1 in the Poisson distribution P( ρyt ) and, then, drawing µ yt+1 µ in the gamma distribution γ(ν +Ut+1 ). The process yt+1 is positive and the associated Index-Car(p) process xt+1 is also positive. The conditional Laplace transform of this process is: ρu E exp(uxt+1 ) | xt = exp 1−uµ (β1 xt + . . . + βp xt+1−p ) − ν log(1 − uµ) , with βi ≥ 0, for i ∈ {1, . . . , p}, or using the same notation as above: u E exp(uxt+1 ) | xt = exp 1−uµ ϕ Xt − ν log(1 − uµ) . (6) X Similarly, given Xt , xt+1 can be obtained by drawing Ut+1 in P( ϕ µ t ) xt+1 and µ in γ(ν+Ut+1 ). It easily seen that the conditional mean and variance of xt+1 , given xt , are respectively given by νµ + ϕ Xt and νµ2 + 2µϕ Xt ; so, the process xt+1 has the weak AR(p) representation: xt+1 = νµ + ϕ Xt + εt+1 , (7) where εt+1 is a conditionally heteroscedastic martingale diﬀerence, whose conditional variance is νµ2 + 2µϕ Xt ; the process is stationary if and only if ϕ e < 1 [where e = (1, . . . , 1) ∈ Rp ] and, in this case, the process εt+1 has ϕe ﬁnite unconditional variance given by νµ2 + 2νµ2 1−ϕ e . The unconditional νµ mean of xt+1 is given by 1−ϕ e . 2.4 Univariate Switching regimes Car(p) process Let us ﬁrst consider a J-states homogeneous Markov Chain zt+1 , which can take the values ej ∈ RJ , j ∈ {1, . . . , J}, where ej is the j th column of the (J × J) identity matrix. The transition probability, from state ei to state ej is π(ei , ej ) = P r(zt+1 = ej | zt = ei ). It is ﬁrst worth noting that zt+1 is a Car(1) process. Proposition 1 : The Markov chain process zt+1 is a Car(1) process with a conditional Laplace transform given by: E[exp(v zt+1 )| zt ] = exp(az (v, π) zt ) , (8) where J J az (v, π) = log exp(v ej )π(e1 , ej ) , . . . , log exp(v ej )π(eJ , ej ) . j=1 j=1 7 [Proof : straightforward.] Let us now consider a univariate Index-Car(p) process with a conditional Laplace transform given by exp [a(u)β Xt + b(u)], and let us assume that b(u) can be written: b(u) = ˜ b(u) λ where (9) ˜ b(u) = (b1 (u), . . . , bm (u)) and λ = (λ1 , . . . , λm ) . We can generalize this model by assuming that the parameters λi are stochastic and linear functions of Zt = (zt , . . . , zt−p ) . More precisely, we assume that the conditional distribution of xt+1 given xt and zt+1 has a Laplace transform given by: E[exp(uxt+1 )| xt , zt+1 ] = exp a(u)β Xt + ˜ b(u) ΛZt , (10) where Λ is a [m, (p + 1)J] matrix. Note that we assume no instantaneous causality between xt+1 and zt+1 and we admit one more lag in Zt that in Xt [examples given in Section 2.5 show that this assumption may be convenient]; if the process zt is not observed by the econometrician the no instantaneous causality assumption is not really important at the estimation stage since we could rename zt as zt+1 , however it will be useful at the pricing level in order to obtain simple pricing procedures [Dai, Singleton and Yang (2005) also make this kind of assumption]. The joint process (xt+1 , zt+1 ) is easily seen to be a Car(p + 1) process. Proposition 2 : The conditional Laplace transform of (xt+1 , zt+1 ) given xt , zt has the following form: E exp(uxt+1 + v zt+1 ) | zt , xt (11) = exp a(u)β Xt + e1 ⊗ az (v, π) + ˜ b(u) Λ Zt , where e1 is the ﬁrst component of the canonical basis in Rp+1 , and where ⊗ denotes the Kronecker product. [Proof : straightforward.] 8 2.5 Examples of Univariate Switching regimes Car(p) pro- cesses a. Gaussian case Let us start from the AR(p) model (4). Its conditional Laplace transform is given by (5): σ2 2 E exp(uxt+1 ) | xt = exp uϕ Xt + uν + 2 u , 2 and the function b(u) has the form (9) with ˜ b(u) = u, u 2 and λ = (ν, σ 2 ). If λ is replaced by ΛZt , the joint process (xt+1 , zt+1 ) is Car(p + 1) with a conditional Laplace transform given by: E exp(uxt+1 + v zt+1 ) | zt , xt (12) u2 = exp uϕ Xt + u, 2 ΛZt + az (v, π)zt . λ1 More precisely, the dynamics is given by [using the notation Λ = ]: λ2 xt+1 = λ1 Zt + ϕ Xt + (λ2 Zt )1/2 εt+1 , (13) where εt+1 is a gaussian white noise distributed as N (0, σ 2 ), Zt = (zt , . . . , zt−p ) and zt is a Markov chain such that P r(zt+1 = ej | zt = ei ) = π(ei , ej ). In particular, let us consider the case: (1, −ϕ1 , . . . , −ϕp ) ⊗ ν ∗ Λ= (14) e1 ⊗ σ ∗2 ∗ ∗ ∗2 ∗2 and ν ∗ = (ν1 , . . . , νJ ), σ ∗2 = (σ1 , . . . , σJ ), the conditional distribution of xt+1 given xt and zt+1 is the one corresponding to the switching AR(p) model deﬁned by: xt+1 − ν ∗ zt = ϕ1 (xt − ν ∗ zt−1 ) + . . . + ϕp (xt+1−p − ν ∗ zt−p ) + (σ ∗ zt )εt+1 . (15) b. Gamma case Let us now start from the ARG(p) process associated with the condi- tional Laplace transform (6): u E exp(uxt+1 ) | xt = exp 1−uµ ϕ Xt − ν log(1 − uµ) . 9 Here we have ˜b(u) = − log(1 − uµ) and λ = ν. If ν is replaced by ΛZt , where ΛZt > 0, the process xt has, conditionally to the process zt , a weak AR(p) representation given by: xt+1 = µΛZt + ϕ1 xt + . . . + ϕp xt+1−p + ζt+1 , (16) where ζt+1 is a conditionally heteroscedastic martingale diﬀerence. For in- stance, we can take : ˜ ν Λ = e1 ⊗ , (17) µ ˜ ν ˜ ν ˜ ˜ where ν = (˜1 , . . . , νJ ), νj ≥ 0. We have ΛZt = µ zt and, conditionally to the process zt , the process xt has a weak AR(p) representation given by: ˜ xt+1 = ν zt + ϕ1 xt + . . . + ϕp xt+1−p + ζt+1 . (18) ˜ ν It is also possible to consider a Λ of the form (1, −ϕ1 , . . . , −ϕp ) ⊗ µ if J 1 J ν ν min(˜i ) > max(˜i ) i=1 ϕj , since in this case ΛZt = µ ν zt − ˜ i=1 ϕj ˜ ν zt−i ≥ 0. The weak conditional AR(p) representation is then given by: ˜ ˜ ˜ xt+1 − ν zt = ϕ1 (xt − ν zt−1 ) + . . . + ϕp (xt+1−p − ν zt−p ) + ζt+1 . (19) 2.6 Speciﬁcation of multivariate Car(1) processes In order to have simple notations we will consider the bivariate case, but all the results are easily extended to the general case. A bivariate Car(1) process yt = (y1,t , y2,t ) will be deﬁned in a recursive way. We consider two univariate exponential aﬃne Laplace transforms : exp [a1 (u1 )w1,t + b1 (u1 )] , (20) and exp [a2 (u2 )w2,t + b2 (u2 )] . Then, we assume that the conditional distribution of y1,t+1 given (y2,t+1 , y1,t , y2,t ) has a Laplace transform given by : Et [exp(u1 y1,t+1 ) | y2,t+1 , y1t , y2t ] (21) = exp [a1 (u1 )(βo y2,t+1 + β11 y1,t + β12 y2,t ) + b1 (u1 )] 10 and the conditional distribution of y2,t+1 , given (y1,t , y2,t ), has a Laplace transform given by Et [exp(u2 y2,t+1 ) | y1,t , y2,t ] = exp [a2 (u2 )(β21 y1,t + β22 y2,t ) + b2 (u2 )] . (22) Note that, if the Laplace transforms (20) correspond to positive variables and if the parameters βo , β11 , β12 , β21 , β22 are positive the bivariate process yt has positive components. Moreover, we have the following result : Proposition 3 : The bivariate process yt deﬁned by the conditional dy- namics (21), (22) is a bivariate Car(1) process with a conditional Laplace transform given by : E[exp(u1 y1,t+1 + u2 y2,t+1 )| y1,t , y2,t ] = exp {[a1 (u1 )β11 + a2 (u2 + a1 (u1 )βo )β21 ]y1,t (23) +[a1 (u1 )β12 + a2 (u2 + a1 (u1 )βo )β22 ]y2,t +b1 (u1 ) + b2 (u2 + a1 (u1 )βo )} . [Proof : see Appendix 1.] 2.7 Speciﬁcation of multivariate Index-Car(p) processes ˜ We consider a bivariate process xt = (x1,t , x2,t ) and we introduce the no- tations : X1t = (x1,t , . . . , x1,t+1−p ) , X2t = (x2,t , . . . , x2,t+1−p ) . Given the univariate Laplace transforms like (20), a bivariate Index-Car(p) is deﬁned in the following way. Deﬁnition 3 : A bivariate Index-Car(p) dynamics is deﬁned by the condi- tional Laplace transforms: Et [exp(u1 x1,t+1 ) | x2,t+1 , x1,t , x2,t ] = exp [a1 (u1 )(βo x2,t+1 + β11 X1t + β12 X2t ) + b1 (u1 )] , Et [exp(u2 x2,t+1 ) | x1,t , x2,t ] = exp [a2 (u2 )(β21 X1t + β22 X2t ) + b2 (u2 )] , (24) ˜ where the βij are p-vectors. It is easily seen that the process xt is a Car(p) process with a conditional Laplace transform given by (23) in which y1,t is 11 replaced by X1t and y2,t by X2t and the βij by the βij , i.e. ˜ ˜ E exp(u xt+1 ) | xt = exp{[a1 (u1 )β11 + a2 (u2 + a1 (u1 )βo )β21 ] X1t (25) +[a1 (u1 )β12 + a2 (u2 + a1 (u1 )βo )β22 ] X2t + b1 (u1 ) + b2 (u2 + a1 (u1 )βo )} . From the properties of Car(p) processes we get a representation of the form: x1,t+1 = α1 + αo x2,t+1 + α11 X1t + α12 X2t + ε1,t+1 (26) x2,t+1 = α2 + α21 X1t + α22 X2t + ε2,t+1 where the errors terms satisfy : ˜ E[ε1,t+1 | x2,t+1 , xt ] = 0 (27) ˜ E[ε2,t+1 | xt ] = 0; in particular, we get ˜ E[ε1,t+1 | xt ] = 0 ˜ E[ε2,t+1 | xt ] = 0 ˜ Cov(ε1,t+1 , ε2,t+1 ) = E(ε1,t+1 ε2,t+1 | xt ) (28) ˜ ˜ = E ε2,t+1 E(ε1,t+1 | x2,t+1 , xt ) | xt = 0. So, the error terms are non correlated, conditionally heteroscedastic, mar- tingale diﬀerences. In particular, in the stationary case, ε1,t and ε2,t are uncorrelated weak white noises and (26) is a weak recursive VAR(p) repre- ˜ sentation of the process xt . In the rest of the paper we will consider two important particular cases. 12 a) Normal VAR(p) or VARN(p) processes In this case the conditional distributions deﬁned by (20) are gaussian, with aﬃne expectations and ﬁxed variances. In other words: 2 σ1 u 2 a1 (u1 ) = ρ1 u1 , b1 (u1 ) = ν1 u1 + 2 1 (29) 2 σ2 u 2 a2 (u2 ) = ρ2 u2 , b2 (u2 ) = ν2 u2 + 2 2 . Using the notations ϕo = ρ1 βo , ϕ11 = ρ1 β11 , ϕ12 = ρ1 β12 , ϕ21 = ρ2 β21 , ϕ22 = ρ2 β22 , we have the following strong VAR(p) recursive representation ˜ for the process xt = (x1,t , x2,t ) : x1,t+1 = ν1 + ϕo x2,t+1 + ϕ11 X1t + ϕ12 X2t + σ1 η1,t+1 (30) x2,t+1 = ν2 + ϕ21 X1t + ϕ22 X2t + σ2 η2,t+1 , where ηt = (η1,t , η2,t ) is a bivariate gaussian white noise distributed as N (0, I2 ), where I2 denotes the (2 × 2) identity matrix. b) Gamma VAR(p) or VARG(p) processes In this case we have: ρ1 u1 a1 (u1 ) = 1−u1 µ1 , b1 (u1 ) = −ν1 log(1 − u1 µ1 ) (31) ρ2 u2 a2 (u2 ) = 1−u2 µ2 , b2 (u2 ) = −ν2 log(1 − u2 µ2 ) , ˜ and the process xt = (x1,t , x2,t ) has the following weak VAR(p) represen- tation (using the same notation as above, and where all the parameters are positive): x1,t+1 = ν1 µ1 + ϕo x2,t+1 + ϕ11 X1t + ϕ12 X2t + ξ1,t+1 (32) x2,t+1 = ν2 µ2 + ϕ21 X1t + ϕ22 X2t + ξ2,t+1 , where ξ1,t and ξ2,t are non correlated, conditionally heteroscedastic, mar- tingale diﬀerences. The conditional variances of ξ1,t+1 and ξ2,t+1 are given by: V [ξ1,t+1 | xt ] = ν1 µ2 + 2µ1 [ϕo (ν2 µ2 + ϕ21 X1t + ϕ22 X2t ) ˜ 1 + ϕ11 X1t + ϕ12 X2t ] (33) V [ξ2,t+1 | xt ] ˜ = ν2 µ2 + 2µ2 (ϕ21 X1t + ϕ22 X2t ) . 2 It is important to stress that the components of this VARG(p) process are positive. 13 2.8 Switching Multivariate Index-Car processes Switching regimes can be introduced in a multivariate Index-Car(p) model using a method extending the one retained in the univariate case. If we assume that the functions b1 (u1 ), b2 (u2 ) appearing in deﬁnition 3 can be ˜ ˜ written, respectively, as b1 (u1 ) λ1 and b2 (u2 ) λ2 , and if we replace λ1 and λ2 , respectively by Λ1 Zt and Λ2 Zt , we obtain the following conditional Laplace transform for the distribution of (x1,t+1 , x2,t+1 , zt+1 ) given (x1,t , x2,t , zt ): E[exp(u1 x1,t+1 + u2 x2,t+1 + v zt+1 )| x1,t , x2,t , zt ] = exp {[a1 (u1 )β11 + a2 (u2 + a1 (u1 )βo )β21 ] X1t +[a1 (u1 )β12 + a2 (u2 + a1 (u1 )βo )β22 ] X2t ˜ ˜ , +[ e1 ⊗ az (v, π) + b1 (u1 ) Λ1 + b2 (u2 + a1 (u1 )βo ) Λ2 ]Zt (34) where az (v, π) is given in proposition 1. So we obtain a multivariate Car(p+ 1) process. Proposition 4 : The Laplace transform of (x1,t+1 , x2,t+1 , zt+1 ), condition- ally to (x1,t , x2,t , zt ), has the form given in (34) and the process (x1,t , x2,t , zt ) is Car(p + 1). 2.9 Examples of Switching Multivariate Index-Car processes a. Gaussian case Taking 2 σ1 2 2 u a1 (u1 ) = ρ1 u1 , b1 (u1 ) = ν1 u1 + ˜ 2 u1 , b1 (u1 ) = u1 , 21 , 2 σ2 2 2 u a2 (u2 ) = ρ2 u2 , b2 (u2 ) = ν2 u2 + ˜ 2 u2 , b2 (u2 ) = u2 , 22 , λ11 λ21 Λ1 = , Λ2 = , λ12 λ22 and using the notations ϕo = ρ1 βo , ϕ11 = ρ1 β11 , ϕ12 = ρ1 β12 , ϕ21 = ρ2 β21 , 14 ϕ22 = ρ2 β22 , we obtain the Switching VARN(p) model: x1,t+1 = λ11 Zt + ϕo x2,t+1 + ϕ11 X1t + ϕ12 X2t + (λ12 Zt )1/2 η1,t+1 x2,t+1 = λ21 Zt + ϕ21 X1t + ϕ22 X2t + (λ22 Zt )1/2 η2,t+1 , (35) where ηt = (η1,t , η2,t ) is a gaussian white noise distributed as N (0, I2 ), Zt = (zt , . . . , zt−p ) , and where zt is a homogeneous J-states Markov chain with transition probability π(ei , ej ). Note that (35) can also be written as: ˜ 1/2 ˜ ˜ x1,t+1 = λ11 Zt + ϕ11 X1t + ϕ12 X2t + ϕo (λ22 Zt ) η2,t+1 + (λ12 Zt )1/2 η1,t+1 (36) x2,t+1 = λ21 Zt + ϕ21 X1t + ϕ22 X2t + (λ22 Zt )1/2 η2,t+1 , ˜ ˜ ˜ with λ11 = λ11 + ϕo λ21 , ϕ11 = ϕ11 + ϕo ϕ21 , ϕ12 = ϕ12 + ϕo ϕ22 or, with obvious notations ˜ ˜ ˜ (λ12 Zt )1/2 ϕo (λ22 Zt )1/2 ˜ xt+1 = λ Zt + Φ Xt + ηt+1 . (37) 0 (λ22 Zt )1/2 b. Gamma case If we take a1 (u1 ) = ρ1 u1 ˜ 1−u1 µ1 , b1 (u1 ) = −ν1 log(1 − u1 µ1 ), b1 (u1 ) = log(1 − u1 µ1 ), a2 (u2 ) = ρ2 u2 ˜ 1−u2 µ2 , b2 (u2 ) = −ν2 log(1 − u2 µ2 ), b2 (u2 ) = log(1 − u2 µ2 ) , we obtain the positive Switching VARG(p) model x1,t+1 = µ1 Λ1 Zt + ϕo x2,t+1 + ϕ11 X1t + ϕ12 X2t + ξ1,t+1 (38) x2,t+1 = µ2 Λ2 Zt + ϕ21 X1t + ϕ22 X2t + ξ2,t+1 , where ξ1,t and ξ2,t are non correlated, conditionally heteroscedastic, martin- gale diﬀerences, the conditional variances being respectively given by: V [ξ1,t+1 | xt ] = Λ1 Zt µ2 + 2µ1 [ϕo (Λ2 Zt µ2 + ϕ21 X1t + ϕ22 X2t ) ˜ 1 +ϕ11 X1t + ϕ12 X2t ] (39) V [ξ2,t+1 | xt ] ˜ = Λ2 Zt µ2 + 2µ2 (ϕ21 X1t + ϕ22 X2t ) . 2 15 3 SWITCHING AUTOREGRESSIVE NORMAL (SARN) TERM STRUCTURE MODEL OF OR- DER p We ﬁrst consider the case of univariate exogenous factor; the endogenous case and the multivariate cases will be discussed, respectively, in sections 3.7 and 3.8. 3.1 The historical dynamics The ﬁrst set of assumptions of a SARN(p) Term Structure model deals with the historical dynamics. We assume that the historical dynamics of the exogenous factor xt is given by xt+1 = ν(Zt ) + ϕ1 (Zt )xt + . . . + ϕp (Zt )xt+1−p + σ(Zt )εt+1 , (40) where εt+1 is a gaussian white noise with N (0, 1) distribution, Zt = (zt , . . . , zt−p ) , and zt is a J-states non-homogeneous Markov chain such that P (zt+1 = ej | zt = ei ; xt ) = π(ei , ej ; Xt ) (ei is the ith column of the identity matrix IJ ). Equation (40) will be also written xt+1 = ν(Zt ) + ϕ(Zt ) Xt + σ(Zt )εt+1 , (41) where Xt = (xt , . . . , xt+1−p ) , ϕ(Zt ) = (ϕ1 (Zt ), . . . , ϕp (Zt )) . This model can also be rewritten in the following vectorial form: Xt+1 = Φ(Zt )Xt + [ν(Zt ) + σ(Zt )εt+1 ] e1 (42) where ϕ1 (Zt ) . . .. . . ϕp (Zt ) 1 0 ... 0 0 1 ... 0 Φ(Zt ) = . . .. . . . . . 0 ... 1 0 is a (p × p)-matrix, and where e1 is the ﬁrst column of the identity matrix Ip . Note that, since the coeﬃcients ϕi are allowed to depend on Zt and since the Markov chain zt may not be homogeneous, the dynamics of (xt , zt ) is not Car in general. 16 3.2 The Stochastic Discount Factor The second element of a SARN(p) modeling is the SDF. We denote by Mt,t+1 the stochastic discount factor (SDF) between the date t and t + 1 and in order to get time-varying risk-premia we specify it as an exponential aﬃne function of the variables (xt+1 , zt+1 ) but with coeﬃcients depending on the information at time t. More precisely we assume that: Mt,t+1 = exp [−c Xt − d Zt + Γ(Zt , Xt ) εt+1 (43) − 1 Γ(Zt , Xt )2 − δ(Zt , Xt ) zt+1 , 2 ˜ where Γ(Zt , Xt ) = γ(Zt ) + γ (Zt )Xt . Observe that this speciﬁcation extends to the multi-lag case the one proposed by Dai, Singleton, Yang (2005). It is well known that the existence of a positive stochastic discount factor is equivalent to the absence of arbitrage opportunity condition and that the price pt at t of a payoﬀ Wt+1 at t + 1 is given by: pt = E[Mt,t+1 Wt+1 | It ] = Et [Mt,t+1 Wt+1 ] , where the information It , available for the investors at the date t, is given by (xt , zt ). More generally, the price pt,h at t of an asset paying Wt+h at t + h is: pt,h = Et [Mt,t+1 · . . . · Mt+h−1,t+h Wt+h ] . Using the absence of arbitrage assumption for the short-term interest rate between t and t + 1, denoted by rt+1 and known at t, we get: exp(−rt+1 ) = Et (Mt,t+1 ) J = exp [−c Xt − d Zt ] × j=1 π (ei , ej ; Xt ) exp [−δ(Zt , Xt ) ej ] , and assuming the normalization condition: J j=1 π (ei , ej ; Xt ) exp [−δ(Zt , Xt ) ej ] = 1 ∀Zt , Xt , (44) we obtain: rt+1 = c Xt + d Zt . (45) 17 3.3 Risk premia In this paper we will use the following deﬁnition of a risk premium. Deﬁnition 4 : Let pt the price of a given asset at time t. The risk premium of this asset between t and t + 1 is ωt = log(Et pt+1 ) − log pt − rt+1 . Using this deﬁnition we obtain interpretations of the Γ and δ functions appearing in the SDF which generalize that obtained by Dai, Singleton and Yang (2005). Proposition 5 : The risk premium between t and t+1 of an asset providing the payoﬀ exp(−θxt+1 ) at t + 1 is : ωt (θ) = θΓ(Xt , Zt )σ(Zt ) . (46) Therefore, θ, Γ(Xt , Zt ) and σ(Zt ) can be seen respectively as a risk sensi- tivity of the asset, a risk price and a risk measure. [Proof : see Appendix 2.] Proposition 6 : If we consider a digital asset providing one money unit at t + 1 if zt+1 = ej , its risk premium between t and t + 1 is given by : ωt (θ) = δj (Xt , Zt ) , (47) and the j th component of δ can be seen as the risk premium associated with the digital asset.[Proof : see Appendix 2.] We observe that, in general, the magnitude of the risk premium ωt (θ) is not just depending on the currently observed values xt and zt , but it reﬂects the present and past values of both factors, that is, it is a function of the larger information represented by Xt and Zt . 3.4 Risk-Neutral dynamics The assumptions on the historical dynamics and on the SDF imply a risk- neutral dynamics. The probability density function of the one-period con- ditional risk-neutral probability with respect to the corresponding histori- Mt,t+1 Q cal probability is Et (Mt,t+1 ) = exp(rt+1 )Mt,t+1 . Note that using Et as the conditional expectation with respect to this risk-neutral distribution, the Q risk-premium ωt can be written log(Et pt+1 ) − log(Et pt+1 ). Proposition 7 : The risk-neutral dynamics of the process (xt , zt ) is given by: Q ˜ xt+1 = ν(Zt ) + γ(Zt )σ(Zt ) + [ϕ(Zt ) + γ (Zt )σ(Zt )] Xt + σ(Zt )ξt+1 , (48) 18 Q where = denotes the equality in distribution (associated to the probability Q), ξt+1 is (under Q) a gaussian white noise with N (0, 1) distribution, and where Zt = (zt , . . . , zt−p ) , zt being a Markov chain such that: Q(zt+1 = ej | zt ; xt ) = π (zt , ej ; Xt ) exp [(−δ(Zt , Xt )) ej ] . Note that, from (44), these probabilities add to one. [Proof : see Appendix 3.] In order to get a generalized linear term structure we impose that the risk-neutral dynamics is switching regime gaussian Car(p). Using (13), this impose that the dynamics has to satisfy the following speciﬁcation: Q xt+1 = ν ∗ Zt + ϕ∗ Xt + (σ ∗ Zt )ξt+1 , (49) where Zt = (zt , . . . , zt−p ) , with zt a J-states Markov chain such that Q(zt+1 = ej | zt = ei ) = π ∗ (ei , ej ) . (50) From proposition 7, this implies the following restrictions on the histor- ical dynamics and on the SDF: i) σ(Zt ) = σ ∗ Zt : the historical stochastic volatility must be linear in Zt ; ii) ν ∗ Zt − ν(Zt ) γ(Zt ) = : σ ∗ Zt for a given historical stochastic drift ν(Zt ) and stochastic volatility σ ∗ Zt , the coeﬃcient γ(Zt ) belongs to the previous family indexed by the free parameter vector ν ∗ . iii) ϕ∗ − ϕ(Zt ) ˜ γ (Zt ) = : σ ∗ Zt for a given historical stochastic slope parameter ϕ(Zt ) and stochastic volatility σ ∗ Zt the coeﬃcient vector γ (Zt ) belongs to the previous ˜ family indexed by the free parameter vector ϕ∗ . 19 iv) π (zt , ej ; Xt ) δj (Xt , Zt ) = log : π ∗ (zt , ej ) for a given historical transition matrix π (zt , ej ; Xt ), the coeﬃcient δj (Xt , Zt ) depend on zt only and belongs to the previous family in- dexed by the entries π ∗ (zt , ej ) of a transition matrix. Note that condition iv) implies that the risk premia coeﬃcients δj , j ∈ {1, . . . , J}, cannot be all positive [or all negative] since this would imply π (zt , ej ; Xt ) > π ∗ (zt , ej ), ∀j [or π (zt , ej ; Xt ) < π ∗ (zt , ej ), ∀j], which is impossible since J π (zt , ej ; Xt ) = J π ∗ (zt , ej ) = 1. Also note that j=1 j=1 condition iv) implies the normalization condition (44). 3.5 The Generalised Linear Term Structure We have seen in the previous section that the risk-neutral dynamics is deﬁned by relations (49), (50); relation (49) can be rewritten: Q Xt+1 = Φ∗ Xt + ν ∗ Zt + (σ ∗ Zt )ξt+1 e1 (51) where ϕ∗ . . . 1 . . . ϕ∗p 1 0 ... 0 0 1 . . . 0 is a (p × p) − matrix , Φ∗ = . . .. . . . . . 0 ... 1 0 Xt = (xt , . . . , xt+1−p ) , and where e1 is the ﬁrst column of the identity matrix Ip . Denoting by B(t, h) the price at t of a zero-coupon with residual maturity h, we have the following result. Proposition 8 : In the univariate SARN(p) Term Structure model the price at date t of the zero-coupon bond with residual maturity h is : B(t, h) = exp (Ch Xt + Dh Zt ) , for h ≥ 1 , (52) where the vectors Ch and Dh satisfy the following recursive equations : Ch = Φ∗ Ch−1 − c (53) ˜ Dh = −d + C1,h−1 ν ∗ + 1 C1,h−1 σ ∗2 + Dh−1 + F (D1,h−1 ) , 2 2 20 where C1,h−1 denotes the ﬁrst component of the p-dimensional vector Ch−1 , D1,h−1 and D2,h−1 are, respectively, the ﬁrst J-dimensional component and the remaining (pJ)-dimensional component of Dh−1 , i.e. Dh−1 = ˜ (D1,h−1 , D2,h−1 ) , Dh−1 = (D2,h−1 , 0) , and where F (D1,h−1 ) = e1 ⊗az (D1,h−1 , π ∗ ), e being the vector (1, 0, . . . , 0) of size (p + 1) and a is the J-vector 1 z given in proposition 1; σ ∗2 is the vector whose components are the squares of the entries of σ ∗ . The initial conditions are C0 = 0, D0 = 0 (or C1 = −c, D1 = −d). [Proof : see Appendix 4.] For clarity we give again the expression of az (D1,h−1 , π ∗ ) : az (D1,h−1 , π ∗ ) J = log exp(D1,h−1 ej )π ∗ (e1 , ej ) , j=1 J . . . , log exp(D1,h−1 ej )π ∗ (eJ , ej ) . j=1 From proposition 8 we see that the yields to maturity are: 1 R(t, h) = − log B(t, h) h (54) C D = − h Xt − h Zt , h ≥ 1. h h So, they are linear functions of the p-dimensional vector Xt and of the (p + 1)J-dimensional vector Zt . This means that, the term structure at date t depends on the present and past values of xt and zt , and not just on their values in t. Moreover, we observe that there is, in general, instantaneous causality between xt and zt . 3.6 The Switching VARMA yield curve process The result presented in Proposition 8 describes, conditionally to Xt and Zt , the yields as a deterministic function of the time to maturity h, for a ﬁxed date t. Nevertheless, in many ﬁnancial and economic contexts one needs, for instance, also to study the eﬀects of a shock, in the state variables, on the yield curve at diﬀerent future times and for several maturities (e.g.: a Central Bank that needs to set a monetary policy). This means that we are 21 interested in the dynamics of the process RH = [ R(t, h), 0 ≤ t < T, h ∈ H ], for a given set of residual time to maturities H = (1, . . . , H). If we consider a ﬁxed h, the process R = [ R(t, h), 0 ≤ t < T ] can be described by the following proposition. Proposition 9 : For a ﬁxed time to maturity h, the process R = [ R(t, h), 0 ≤ t < T ] is, under the historical probability, a Switching ARMA(p, p − 1) pro- cess of the following type : Ψ(L, Zt ) R(t + 1, h) = Dh (L) Ψ(L, Zt ) zt+1 + Ch (L) ν(Zt ) (55) + Ch (L)[(σ ∗ Zt ) εt+1 ] . where 1 Ch (L) = − (C1,h + C2,h L + . . . + Cp,h Lp−1 ) h 1 Dh (L) = − (D1,h + D2,h L + . . . + Dp+1,h Lp ) h Ψ(L, Zt ) = 1 − ϕ1 (Zt )L − . . . − ϕp (Zt )Lp , are lag polynomials in the lag operator L, and where the AR polynomial Ψ(L, Zt ) applies to t. [Proof : see Appendix 5]. Proposition 10 : For a given set of residual time to maturities H = (1, . . . , H), the stochastic evolution of the yield curve process RH = [ R(t, h), 0 ≤ t < T, h ∈ H ] takes the following particular Switching H-variate VARMA(p, p − 1) representation: R(t + 1, 1) C1 (L) R(t + 1, 2) C2 (L) Ψ(L, Zt ) . . = . (σ ∗ Zt )εt+1 . . . R(t + 1, H) CH (L) D1 (L) C1 (L) D2 (L) C2 (L) + . Ψ(L, Zt ) zt+1 + . ν(Zt ) . . . . . DH (L) CH (L) (56) Similar results are easily obtained in the risk-neutral world. 22 3.7 Endogenous case In the previous sections the factor xt was exogenous. It is often assumed, in term structure models, that the factor xt is the short rate process rt+1 . In this case the previous results remain valid, the only modiﬁcation comes from the absence of arbitrage opportunity condition for rt+1 , which imposes: c = e1 , d = 0 , (57) with e1 the ﬁrst column of the identity matrix Ip ; consequently, the initial conditions in the recursive equations of proposition 8 become: C1 = −e1 , D1 = 0 . (58) Moreover, the Switching ARMA(p, p − 1) representation (55), or its analo- gous in the risk-neutral world, could be used to analyse how a shock on εt , i.e. on rt+1 = R(t, 1), is propagated on the surface [ R(t + τ, h), τ ∈ T , h ∈ H ], where T = {0, . . . , T − t − 1} and H = (1, . . . , H) (for instance when the process zt is exogenous). 3.8 Multi-Factor generalization : the SVARN(p) Term Structure model For sake of notational simplicity we consider the two factor case but an ex- tension to more that two factors is straightforward. The historical dynamics ˜ of xt = (x1,t , x2,t ) is a bivariate SVARN(p) model given by: x1,t+1 = ν1 (Zt ) + ϕo (Zt )x2,t+1 + ϕ11 (Zt ) X1t + ϕ12 (Zt ) X2t + σ1 (Zt )ε1,t+1 (59) x2,t+1 = ν2 (Zt ) + ϕ21 (Zt ) X1t + ϕ22 (Zt ) X2t + σ2 (Zt )ε2,t+1 , where ε1,t and ε2,t are independent standard normal white noises, X1t = (x1,t , . . . , x1,t+1−p ) , X2t = (x2,t , . . . , x2,t+1−p ) , Zt = (zt , . . . , zt−p ) , with zt a J-states non-homogeneous Markov chain such that P (zt+1 = ej | zt = ˜ ˜ ˜ ei ; xt ) = π(ei , ej ; Xt ), and where Xt = (X1t , X2t ) . The recursive form (59) is equivalent to the canonical form : ˜ ˜ ˜ x1,t+1 = ν1 (Zt ) + ϕ11 (Zt ) X1t + ϕ12 (Zt ) X2t + σ1 (Zt )ε1,t+1 + ϕo (Zt )σ2 (Zt )ε2,t+1 (60) x2,t+1 = ν2 (Zt ) + ϕ21 (Zt ) X1t + ϕ22 (Zt ) X2t + σ2 (Zt )ε2,t+1 , 23 ˜ ˜ ˜ where ν1 = ν1 +ϕo ν2 , ϕ11 = ϕ11 +ϕo ϕ21 , ϕ12 = ϕ12 +ϕo ϕ22 or, with obvious notations: ˜ ˜ ˜ ˜ xt+1 = ν (Zt ) + Φ(Zt )Xt + S(Zt )εt+1 , (61) where σ1 (Zt ) ϕo (Zt )σ2 (Zt ) S(Zt ) = 0 σ2 (Zt ) Using the notation ˜ ˜ ˜ Γ(Zt , Xt ) = Γ1 (Zt , Xt ), Γ2 (Zt , Xt ) ˜ ˜ ˜ where Γi (Zt , Xt ) = γi (Zt ) + γi (Zt ) Xt , i ∈ {1, 2} and Γ(Zt , Xt ) = γ(Zt ) + ˜ ˜ ˜ ˜ ˜ ˜ t , Xt )Xt , with γ(Zt ) = [γ1 (Zt ), γ2 (Zt )] , Γ(Zt , Xt ) = [γ1 (Zt ) , γ2 (Zt ) ] , Γ(Z ˜ ˜ the SDF is deﬁned as : ˜ ˜ Mt,t+1 = exp −c Xt − d Zt + Γ(Zt , Xt ) εt+1 (62) 1 ˜ − 2 Γ(Zt , Xt ) ˜ ˜ Γ(Zt , Xt ) − δ(Zt , Xt ) zt+1 . Assuming the normalization condition (44) and the absence of arbitrage opportunity for rt+1 we get: ˜ rt+1 = c Xt + d Zt . (63) It is also easily seen that the risk premium for an asset providing the payoﬀ ˜ ˜ exp(−θ xt+1 ) at t + 1 is ω(θ) = θ S(Zt )Γ(Zt , Xt ) and that the risk premium associated with the digital payoﬀ I(ej ) (zt+1 ) is unchanged. x Proposition 11 : The risk-neutral dynamics of the process (˜t , zt ) is given by: Q ˜ ˜ ˜ ˜ ˜ ˜ xt+1 = ν (Zt ) + S(Zt )γ(Zt ) + [Φ(Zt ) + S(Zt )Γ(Zt , Xt )]Xt + S(Zt )ξt+1 , (64) Q where = denotes the equality in distribution (associated to the probability Q), ξt+1 is (under Q) a bivariate gaussian white noise with N (0, I2 ) distri- bution, and where Zt = (zt , . . . , zt−p ) , with zt a Markov chain such that: ˜ ˜ ˜ Q(zt+1 = ej | zt ; xt ) = π(zt , ej ; Xt ) exp (−δ(Zt , Xt )) ej . [Proof : see Appendix 6.] If we want to obtain a Switching bivariate Car process in the risk-neutral world, we must have using (37) : 24 i) ∗ σ1 (Zt ) = σ1 Zt ∗ σ2 (Zt ) = σ2 Zt ϕo (Zt ) = ϕ∗ , o and, therefore, ∗ ∗ σ1 Zt ϕ∗ σ2 Zt o S(Zt ) = ∗Z 0 σ2 t ii) γ(Zt ) = [S(Zt )]−1 [ν ∗ Zt − ν (Zt )] , ˜ where ν ∗ is a (2 × (p + 1)J)-matrix. iii) ˜ ˜ ˜ Γ(Zt , Xt ) = [S(Zt )]−1 Φ∗ − Φ(Zt ) , where Φ∗ is a (2 × 2p)-matrix. iv) ˜ π(zt , ej ; Xt ) ˜ δj (Xt , Zt ) = log . π ∗ (zt , ej ) The risk-neutral dynamics can be written: Q ˜ x1,t+1 = ν1 Zt + Φ∗ Xt + S1 (Zt )ξt+1 ∗ 1 ∗ (65) Q ∗ ˜ ∗ x2,t+1 = ν2 Zt + Φ∗ Xt + S2 (Zt )ξt+1 , 2 ∗ ∗ where νi , Φ∗ , Si are the ith row of ν ∗ , Φ∗ , S ∗ , with i ∈ {1, 2}, or i ˜ Q ˜ ˜ ∗ ∗ ∗ ∗ Xt+1 = Φ∗ Xt + [ν1 Zt + S1 (Zt )ξt+1 ] e1 + [ν2 Zt + S2 (Zt )ξt+1 ] ep+1 , 25 where e1 (respectively, ep+1 ) is of size 2p, with entries equal to zero except the ﬁrst (respectively, the (p + 1)th ) one which is equal to one, and ∗ Φ11 Φ∗ 12 I ˜ ˜ 0 ˜ Φ∗ = Φ∗ Φ∗ 21 22 ˜ 0 ˜ I where Φ∗ = (Φ∗ , Φ∗ ), Φ∗ = (Φ∗ , Φ∗ ), and where ˜ is a [(p−1)×p ]-matrix 1 11 12 2 21 22 0 of zeros and I˜ is a [(p − 1) × p ]-matrix equal to (Ip−1 , 0), where 0 is a vector of size (p − 1). The term structure is given by the following proposition: Proposition 12 : In the bivariate SVARN(p) Term Structure model the price at date t of the zero-coupon bond with residual maturity h is : ˜ B(t, h) = exp Ch Xt + Dh Zt , for h ≥ 1 (66) where the vectors Ch and Dh satisfy the following recursive equations : ˜ Ch = Φ∗ Ch−1 − c Dh = −d + C1,h−1 ν1 + 1 C1,h−1 (σ1 + ϕ∗2 σ2 ) ∗ 2 2 ∗2 o ∗2 (67) ˜ + Cp+1,h−1 ν2 + 1 Cp+1,h−1 σ2 + Dh−1 + F (D1,h−1 ) , ∗ 2 2 ∗2 ˜ where Dh−1 and F (D1,h−1 ) have the same meaning as in proposition 8, and the initial conditions are C0 = 0, D0 = 0 (or C1 = −c, D1 = −d). [Proof : see Appendix 7.] So, proposition 12 shows that the yields to maturity are: Ch ˜ D R(t, h) = − Xt − h Zt , h ≥ 1. (68) h h In the endogenous case we can take x1t = rt+1 , and x2t = R(t, H) for a given time to maturity H. In this case the absence of arbitrage conditions for rt+1 and R(t, H) imply: (i) C1 = −e1 , D1 = 0 , or c = e1 , d = 0 (69) (ii) CH = − H ep+1 , DH = 0 . 26 ∗ ∗ ˜ ∗ ˜ Using the notations Ch = (C1,h , C1,h , Cp+1,h , C2,h ) , C1,h = (C1,h , 0) , C2,h = ∗ ˜ ˜ ˜ (C2,h , 0) (where the zeros are scalars), and Ch = (C1,h , C2,h ) , it easily seen ˜ that the recursive equation Ch = Φ∗ Ch−1 − c can be written : ˜ Ch = Φ∗ C1,h−1 + Φ∗ Cp+1,h−1 + Ch−1 − c . 1 2 Conditions (i) are used as initial values in the recursive procedure of proposi- ˜ ∗ ∗ tion 10, and conditions (ii) implies restrictions on the parameters Φ∗ , ν1 , ν2 , ∗ , σ ∗ , ϕ∗ , π ∗ (z , e ) which must be taken into account at the estimation σ1 2 o t j stage. 4 SWITCHING AUTOREGRESSIVE GAMMA (SARG) TERM STRUCTURE MODEL OF OR- DER p Like for SARN(p) models, we start the description of the SARG(p) modeling by the case of one exogenous factor. 4.1 The historical dynamics We assume that the Laplace transform of the conditional distribution of xt+1 , given (xt , zt ), is: u E exp(uxt+1 ) | xt , , zt = exp 1−uµ(Xt ,Zt ) [ϕ1 (Zt )xt + . . . + ϕp (Zt )xt−p+1 ] − ν(Zt ) log(1 − uµ(Xt , Zt ))] , (70) where Zt = (zt , . . . , zt−p ) , with zt a J-states non-homogeneous Markov ˜ chain such that P (zt+1 = ej | zt = ei ; xt ) = π(ei , ej ; Xt ), and where Xt = (xt , . . . , xt+1−p ) . Using the notation: u u A[u; ϕ(Zt ), µ(Xt , Zt )] = 1−uµ(Xt ,Zt ) [ϕ1 (Zt ), . . . , ϕp (Zt )] = 1−uµ(Xt ,Zt ) ϕ(Zt ) b[u; ν(Zt ), µ(Xt , Zt )] = − ν(Zt ) log(1 − uµ(Xt , Zt )) , relation (70) can be written: E exp(uxt+1 ) | xt , , zt = exp {A[u; ϕ(Zt ), µ(Xt , Zt )] Xt (71) + b[u; ν(Zt ), µ(Xt , Zt )]} . 27 The process (xt ) can also be written: xt+1 = ν(Zt )µ(Xt , Zt ) + ϕ1 (Zt )xt + . . . + ϕp (Zt )xt+1−p + εt+1 (72) = ν(Zt )µ(Xt , Zt ) + ϕ(Zt ) Xt + εt+1 , where εt+1 is a martingale diﬀerence sequence with conditional Laplace transform given by: E exp(uεt+1 ) | xt , , zt = exp {−u[ν(Zt )µ(Xt , Zt ) + ϕ(Zt ) Xt ] + A[u; ϕ(Zt ), µ(Xt , Zt )] Xt + b[u; ν(Zt ), µ(Xt , Zt )]} = exp {[A[u; ϕ(Zt ), µ(Xt , Zt )] − uϕ(Zt )] Xt + b[u; ν(Zt ), µ(Xt , Zt )] − u ν(Zt )µ(Xt , Zt )} . (73) Note that the dynamics of (xt , zt ) is in general not Car. 4.2 The Stochastic Discount Factor In the SARG(p) model the SDF is speciﬁed in the following way: Mt,t+1 = exp {−c Xt − d Zt + Γ(Zt , Xt )εt+1 + Γ(Zt , Xt ) [ν(Zt )µ(Xt , Zt ) + ϕ(Zt ) Xt ] − A[Γ(Zt , Xt ); ϕ(Zt ), µ(Xt , Zt )] Xt −b[Γ(Zt , Xt ); ν(Zt ), µ(Xt , Zt )] − δ(Zt , Xt ) zt+1 } , (74) ˜ where Γ(Zt , Xt ) = γ(Zt ) + γ (Zt )Xt , or, equivalently Mt,t+1 = exp {−c Xt − d Zt + Γ(Zt , Xt )xt+1 − A[Γ(Zt , Xt ); ϕ(Zt ), µ(Xt , Zt )] Xt −b[Γ(Zt , Xt ); ν(Zt ), µ(Xt , Zt )] − δ(Zt , Xt ) zt+1 } , (75) Assuming the normalisation condition (44), we get that: rt+1 = c Xt + d Zt . (76) 28 4.3 Useful lemmas In the subsequent sections we will use several times the following lemmas. Let us consider the functions: ρu a(u; ρ, µ) = ˜ and ˜ ν, µ) = −ν log(1 − uµ) ; b(u; 1 − uµ we have: Lemma 1 : a(u + α; ρ, µ) − a(α; ρ, µ) = a(u; ρ∗ , µ∗ ) ˜ ˜ ˜ ˜ + α; ν, µ) − ˜ ν, µ) = ˜ ν, µ∗ ) b(u b(α; b(u; ρ µ with ρ∗ = , µ∗ = , (1 − αµ)2 1 − αµ [Proof : see Appendix 8.] Lemma 1 immediately implies lemma 2. Lemma 2 : A[u + α; ϕ(Zt ), µ(Xt , Zt )] − A[α; ϕ(Zt ), µ(Xt , Zt )] = A[u; ϕ∗ (Zt ), µ∗ (Xt , Zt )] b[u + α; ν(Zt ), µ(Xt , Zt )] − b[α; ν(Zt ), µ(Xt , Zt )] = b[u; ν(Zt ), µ∗ (Zt , Xt )] ϕ(Zt ) µ(Xt , Zt ) with ϕ∗ (Zt ) = 2 , µ∗ (Zt , Xt ) = . [1 − αµ(Zt , Xt )] 1 − αµ(Xt , Zt ) 29 4.4 Risk-neutral dynamics The Laplace transform of the risk-neutral conditional distribution of (xt+1 , zt+1 ) is, using the notation Γt = Γ(Xt , Zt ): Q Et [exp(uxt+1 + v zt+1 )] = Et {exp [(u + Γt )xt+1 − A[Γt ; ϕ(Zt ), µ(Xt , Zt )] Xt −b[Zt ; ν(Zt ), µ(Xt , Zt )] + (v − δ(Xt , Zt )) zt+1 ]} = exp {[(A[u + Γt ; ϕ(Zt ), µ(Xt , Zt )] − A[Γt ; ϕ(Zt ), µ(Xt , Zt )]) Xt + b[u + Γt ; ν(Zt ), µ(Xt , Zt )] − b[Γt ; ν(Zt ), µ(Xt , Zt )]]} J × j=1 π(zt , ej ; Xt ) exp [(v − δ(Zt , Xt )) ej ] , (77) and, using lemma 2, (78) can be written: Q Et [exp(uxt+1 + v zt+1 )] = exp{A[u; ϕ∗ (Zt ), µ∗ (Xt , Zt )] Xt + b[u; ν(Zt ), µ∗ (Zt , Xt )]} (78) J × j=1 π(zt , ej ; Xt ) exp [(v − δ(Zt , Xt )) ej ] , ϕ(Zt ) µ(Xt , Zt ) with ϕ∗ (Zt ) = 2 and µ∗ (Zt , Xt ) = . [1 − Γt µ(Zt , Xt )] 1 − Γt µ(Xt , Zt ) So, from (71), we see that the risk-neutral conditional distribution of xt+1 , given (xt , zt ), is in the same class as the historical one and obtained by replacing ϕ(Zt ) with ϕ∗ (Zt ), and µ(Xt , Zt ) with µ∗ (Zt , Xt ). In order to get a generalize linear term structure we impose that the risk-neutral dynamics is a switching regime Gamma Car(p) process. So, using the results in section 2.5.b, we get that ϕ∗ (Zt ) and µ∗ (Zt , Xt ) must be constant, ν(Zt ) = ν ∗ Zt and π (zt , ej ; Xt ) = π ∗ (zt , ej ) exp [(δ(Zt , Xt )) ej ]. Also note that µ∗ must be positive as well as the components of ν ∗ and ϕ∗ . This implies the following constraint on the historical dynamics and on the 30 SDF: µ(Xt , Zt ) = µ∗ [1 − Γ(Xt , Zt )µ(Xt , Zt )] ϕ(Zt ) = ϕ∗ [1 − Γ(Xt , Zt )µ(Xt , Zt )]2 ν(Zt ) = ν ∗ Zt π(zt ,ej ;Xt ) δj (Xt , Zt ) = log π ∗ (zt ,ej ) . ϕ ∗ We see that ϕ(Zt ) = µ∗2 µ(Xt , Zt )2 , so µ(Xt , Zt ) must depend only on Zt , and therefore the same is true for Γ(Xt , Zt ). Finally, we have the constraint: i) µ(Zt ) = µ∗ [1 − Γ(Zt )µ(Zt )] ii) ϕ(Zt ) = ϕ∗ [1 − Γ(Zt )µ(Zt )]2 iii) ν(Zt ) = ν ∗ Zt iv) π (zt , ej ; Xt ) δj (Xt , Zt ) = log ; π ∗ (zt , ej ) ϕ ∗ In particular, since ϕ(Zt ) = µ∗2 µ(Zt )2 , the random vector must be propor- tional to a deterministic vector. Moreover, it is easily seen that the risk premium corresponding to the payoﬀ exp(−θxt+1 ) at t + 1 is: ωt (θ) = {A[−θ; ϕ(Zt ), µ(Zt )] − A[−θ; ϕ∗ , µ∗ ]} Xt + b[−θ; ν ∗ Zt , µ(Zt )] − b[−θ; ν ∗ Zt , µ∗ ] . Like in the gaussian case, we obtain an aﬃne function in Xt also depending on Zt . The risk premium associated with the digital asset providing one money unit at t + 1 if zt+1 = ej , is still given by (47). 31 4.5 The Generalised Linear Term Structure Let us introduce the notations: A∗ (u) = A(u; ϕ∗ , µ∗ ) (79) ˜ Ch = (C2,h , . . . , Cp,h , 0) . As usual, B(t, h) is the price at t of a zero-coupon bond with residual ma- turity h. Proposition 13 : In the univariate SARG(p) Term Structure model the price at date t of the zero-coupon bond with residual maturity h is : B(t, h) = exp (Ch Xt + Dh Zt ) , for h ≥ 1 , (80) where the vectors Ch and Dh satisfy the following recursive equations : Ch = −c + A∗ (C1,h−1 ) + Ch−1 ˜ (81) ∗ log(1 − C ˜ h−1 + F (D1,h−1 ) , ∗) + D Dh = −d − ν 1,h−1 µ ˜ where Dh−1 and F (D1,h−1 ) have the same meaning as in proposition 8; the initial conditions are C0 = 0, D0 = 0 (or C1 = −c, D1 = −d) [Proof : see Appendix 9]. Again, we obtain a generalised linear term structure given by: Ch D R(t, h) = − Xt − h Zt , h ≥ 1, (82) h h and, in the same spirit of propositions 9 and 10 for the univariate SARN(p) model [see section 3.6], it is easy to verify that the processes R = [ R(t, h), 0 ≤ t < T ] and RH = [ R(t, h), 0 ≤ t < T, h ∈ H ] are, respectively, a weak Switching ARMA(p, p − 1) process and a weak H-variate Switching VARMA(p, p − 1) process. In the endogenous case, where xt = rt+1 , the previous results remains valid with C1 = −e1 , D1 = 0. 4.6 Positiveness of the yields Since rt+1 = R(t, 1) = c Xt + d Zt , and since the components of Xt are pos- itive, the short term process will be positive as soon as the components of c and d are nonnegative. The positiveness of rt+1 implies that of R(t, h), at any 32 1 Q date t and time to maturity h, because R(t, h) = − h log Et [exp(−rt+1 − . . . −rt+h )]. This positiveness can also be observed from the recursive equations of proposition 13. Indeed, using the fact that µ∗ and the components of ϕ∗ ∗ and ν ∗ are positive and that 0 < πij < 1, it easily seen that, for any u < 0, the components of A∗ (u) and −ν ∗ log(1 − C1,h−1 µ∗ ) are negative and the result follows. 4.7 Multi-Factor generalization : the SVARG(p) Term Structure model ˜ The bivariate process xt = (x1,t , x2,t ) is a SVARG(p) model deﬁned by the following conditional Laplace transforms: Et [exp(u1 x1,t+1 ) | x2,t+1 , x1,t , zt ] u1 = exp ϕo (Zt )x2,t+1 + ϕ11 (Zt ) X1t + ϕ12 (Zt ) X2t 1 − u1 µ1 (Zt ) −ν1 (Zt ) log(1 − u1 µ1 (Zt ))} , (83) Et [exp(u2 x2,t+1 ) | x1,t , x2,t , zt ] u2 = exp ϕ21 (Zt ) X1t + ϕ22 (Zt ) X2t (84) 1 − u2 µ2 (Zt ) −ν2 (Zt ) log(1 − u2 µ2 (Zt ))} . We will use the notations: ϕo (Zt ) = ϕo,t , [ ϕ11 (Zt ) , ϕ12 (Zt ) ] = ϕ1,t , [ ϕ21 (Zt ) , ϕ22 (Zt ) ] = ϕ2,t , µi (Zt ) = µi,t , νi (Zt ) = νi,t , i ∈ {1, 2} , and using the functions a, ˜ A, B deﬁned in lemma 1 and in section 4.1, we ˜ b, 33 will introduce the notations: a1,t (u1 ) ˜ = a(u1 ; ϕo,t , µ1,t ) b1,t (u1 ) = ˜ 1 ; ν1,t , µ1,t ) , b2,t (u2 ) = ˜ 2 ; ν2,t , µ2,t ) b(u b(u A1,t (u1 ) = A(u1 ; ϕ1,t , µ1,t ) , A2,t (u2 ) = A(u2 ; ϕ2,t , µ2,t ) . With these notations, the Laplace transforms (83) and (84) become respec- tively: Et [exp(u1 x1,t+1 ) | x2,t+1 , x1,t , zt ] (85) ˜ = exp a1,t (u1 )x2,t+1 + A1,t (u1 ) Xt + b1,t (u1 ) , Et [exp(u2 x2,t+1 ) | x1,t , x2,t , zt ] (86) ˜ = exp A2,t (u2 ) Xt + b2,t (u2 ) , ˜ where Xt = (X1t , X2t ) . Moreover, the joint conditional Laplace transform of (x1,t+1 , x2,t+1 ), given (x1,t , x2,t , zt ), is: Et [exp(u1 x1,t+1 + u2 x2,t+1 ) | x1,t , x2,t , zt ] ˜ = exp [A1,t (u1 ) + A2,t (u2 + a1,t (u1 ))] Xt + b1,t (u1 ) + b2,t (u2 + a1,t (u1 )) . (87) The process zt is assumed to be a non-homogeneous Markov chain such that ˜ ˜ P (zt+1 = ej | zt = ei ; xt ) = π(ei , ej ; Xt ). We now introduce the SDF: ˜ Mt,t+1 = exp{−c Xt − d Zt + Γ1t x1,t+1 + Γ2t x2,t+1 ˜ − [A1,t (Γ1t ) + A2,t (Γ2t + a1,t (Γ1t ))] Xt ˜ − [b1,t (Γ1t ) + b2,t (Γ2t + a1,t (Γ1t ))] − δ(Zt , Xt ) zt+1 } , (88) where Γ1t = Γ1 (Zt ) and Γ2t = Γ2 (Zt ). 34 4.8 Risk-neutral dynamics in the multifactor case We can now present, using the lemmas presented above, the joint conditional Laplace transform of (x1,t+1 , x2,t+1 ) in the risk-neutral world in the following proposition. Proposition 14 : The joint conditional Laplace transform of (x1,t+1 , x2,t+1 ) in the risk-neutral world is given by : Q Et [exp(u1 x1,t+1 + u2 x2,t+1 ) | x1t , x2t , zt ] ˜ = exp [A∗ (u1 ) + A∗ [u2 + a∗ (u1 )]] Xt (89) 1,t 2,t 1,t + b∗ [u2 + a∗ (u1 )] + b∗ (u1 ) , 2,t 1,t 1,t where A∗ (u1 ) 1,t = A1 (u1 ; ϕ∗ , µ∗ ) , 1t 1t A∗ [u2 + a∗ (u1 )] = A u2 + a(u1 ; ϕ∗ , µ∗ ); ϕ∗ , µ∗ , 2,t 1,t ˜ ot 1,t 2t 2t b∗ [u2 + a∗ (u1 )] 2,t 1,t = ˜ u2 + a(u1 ; ϕ∗ , µ∗ ); ν2t , µ∗ , b ˜ ot 1,t ∗ 2t b∗ (u1 ) 1,t = ˜1 (u1 ; ν1t , µ∗ ) , b ∗ 1t and with ϕot ∗ ϕ1t ∗ ϕ2t ϕ∗ = ot 2 , ϕ1t = 2 , ϕ2t = (1 − Γ1t µ1t ) (1 − Γ1t µ1t ) {1 − [Γ2t + a1,t (Γ1t )]µ2t }2 µ1t µ2t µ∗ = 1t , µ∗ = 2t . (1 − Γ1t µ1t ) {1 − [Γ2t + a1,t (Γ1t )]µ2t } So, (89) has exactly the same form as (87) with diﬀerent parameters. In other words the risk-neutral dynamics belongs to the same class as the his- torical one. [Proof : see Appendix 10.] In order to have a Car process in the risk-neutral world, we know from section 2.9 that we must have the following constraint between the SDF and the historical dynamics: i) µ1t = µ∗ 1 1 − Γ1t µ1t 35 ii) ϕ1t = ϕ∗ 1 (1 − Γ1t µ1t )2 iii) ∗ ν1 (Zt ) = ν1 Zt iv) ϕot = ϕ∗ o (1 − Γ1t µ1t )2 v) µ2t = µ∗ 2 1 − [Γ2t + a1,t (Γ1t )]µ2t vi) ϕ2t = ϕ∗ 2 (1 − [Γ2t + a1,t (Γ1t )]µ2t )2 vii) ∗ ν2 (Zt ) = ν2 Zt . Moreover, the constraint on the dynamics of the Markov chain are the same as in the gaussian case, namely: viii) ˜ π(zt , ej ; Xt ) ˜ δj (Xt , Zt ) = log . ∗ (z , e ) π t j It is worth noting that, if there is no instantaneous causality between x1,t+1 and x2,t+1 , that is if ϕot = 0, function a1t is also equal to zero and constraint v) and vi) are simpler and become similar to i) and ii). 4.9 The Generalized Linear Term Structure in the multifac- tor case Using the notations: a∗ (u1 ) 1 = a(u1 ; ϕ∗ , µ∗ ) ˜ o 1 A∗ (u1 ) = A(u1 ; ϕ∗ , µ∗ ) 1 1 1 A∗ (u2 ) = A(u2 ; ϕ∗ , µ∗ ) 2 2 2 ˜ Ch = (C2,h , . . . , Cp,h , 0, Cp+2,h , . . . , C2p,h , 0) , 36 we have Proposition 15 : In the bivariate SVARG(p) Term Structure model the price at date t of the zero-coupon bond with residual maturity h is : ˜ B(t, h) = exp Ch Xt + Dh Zt , for h ≥ 1 (90) where the vectors Ch and Dh satisfy the following recursive equations : ˜ Ch = −c + A∗ (C1,h−1 ) + A∗ [Cp+1,h−1 + a∗ (C1,h−1 )] + Ch−1 1 2 1 ∗ D = −d − ν1 log(1 − C1,h−1 µ∗ ) (91) h 1 ˜ ∗ − ν2 log[1 − (Cp+1,h−1 + a∗ (C1,h−1 ))µ∗ ] + Dh−1 + F (D1,h−1 ) , 1 2 ˜ where Dh−1 and F (D1,h−1 ) have the same meaning as in proposition 8; the initial conditions are C0 = 0, D0 = 0 (or C1 = −c, D1 = −d) [Proof : see Appendix 11]. So, proposition 15 shows that, also for the SVARG(p) model, yields to ma- ˜ turity are linear functions of Xt and Zt . In the endogenous case, we can consider as factors the short rate rt+1 and the long rate R(t, H), for a given time to maturity H. Now, if we want to deﬁne a joint historical and risk-neutral dynamics for these vari- ables, compatible with the no-arbitrage opportunity condition, we have to take into account domain restrictions on R(t, H) : given that the support of rt+1 is D1 = (0, + ∞), under A.A.O. the support of R(t, H) has to be DH = [ b, + ∞), for some constant b > 0 [see Gourieroux, Monfort (2006) ˜ for details]. Consequently, the bivariate SVARG(p) process xt , being with support D = D1 × D1 , will be speciﬁed for x1t = rt+1 and x2t = R(t, H) − b, and the results presented for the SVARN(p) case [see section 3.8] will apply also in this case. It is also easily seen that the risk premium of the payoﬀ pt+1 = exp(−θ1 x1,t+1 −θ2 x2,t+1 ) is: ωt (θ1 , θ2 ) = {A2,t [−θ2 + a1,t (−θ1 )] + A1,t (−θ1 ) − A∗ [−θ2 + a∗ (−θ1 )] − A∗ (−θ1 )} Xt 2 1 1 + b2,t [−θ2 + a1,t (−θ1 )] + b1,t (−θ1 ) − b∗ [−θ2 + a∗ (−θ1 )] − b∗ (−θ1 ) , 2,t 1 1,t 37 with ∗ b1,t (u1 ) = −ν1 Zt log(1 − u1 µ∗ ) 1 ∗ b2,t (u2 ) = −ν2 Zt log(1 − u2 µ∗ ) , 2 and the risk premium of the digital asset is still given by relation (47). 5 DERIVATIVE PRICING 5.1 Generalization of the recursive pricing formula In the previous sections we have derived recursive formulas for the zero- coupon bond price B(t, h) in various contexts which share the feature that x the process (˜t , zt ) is Car in the risk-neutral world. In fact the recursive approach can be generalized to other assets. ˜ Let us consider a class of payoﬀs g(Xt+h , Zt+h ), (t, h) varying, for a given g function and let us assume that the price at t of this payoﬀ is of the form: ˜ Pt (g, h) = exp Ch (g) Xt + Dh (g) Zt . (92) It is clear that: ˜ exp Ch (g) Xt + Dh (g) Zt ˜ = Et Mt,t+1 exp Ch−1 (g) Xt+1 + Dh−1 (g) Zt+1 ˜ Q ˜ = exp(−c Xt − d Zt )Et exp Ch−1 (g) Xt+1 + Dh−1 (g) Zt+1 ; so the sequences Ch (g), Dh (g), h ≥ 1, follow recursive equations which does not depend on g and, therefore, are identical to the case g = 1, that is to say to the zero-coupon bond pricing formulas given in the previous sections. The only condition for (92) to be true is to hold for h = 1 and, of course, this initial condition depends on g. ˜ u ˜ ˜ Formula (92) is valid for h = 1 if g(Xt+h , Zt+h ) = exp(˜ Xt+h + v Zt+h ) ˜ ˜ for some vector u and v . Indeed, using the notations ˜ ˜ ˜ ˜ u Xt+1 = u1 xt+1 + u−1 Xt ˜ v Zt+1 = v1 zt+1 + v−1 Zt , 38 with u−1 = (u2 , . . . , up , 0), v−1 = (v2 , . . . , vp , 0), we get: u ˜ ˜ ˜ Pt (˜, v ; 1) = exp(−c Xt − d Zt + u−1 Xt + v−1 Zt ) (93) Q ˜ × Et [exp (u1 xt+1 + v1 zt+1 )] , x which, using the Car representation of (˜t+1 , zt+1 ) under the probability Q, has obviously the exponential linear form (92) and provides the initial conditions of the recursive equations. The standard recursive equations u ˜ u ˜ ˜ provide the price Pt (˜, v ; h) at date t for the payoﬀ exp(˜ Xt+h + v Zt+h ). So we have the following proposition. u ˜ ˜ Proposition 16 : The price Pt (˜, v ; h) at time t of the payoﬀ g(Xt+h , Zt+h ) = exp(˜ X u ˜ t+h + v Zt+h ) has the exponential form (92) where Ch (g) and Dh (g) ˜ follow the same recursive equations as in the zero-coupon bond case with ˜ initial values C1 (g) and D1 (g) given by the coeﬃcients of Xt and Zt in equation (93). ˜ ˜ u ˜ When u and v have complex components, Pt (˜, v ; h) provides the complex u ˜ ˜ Laplace transform Et [Mt,t+h exp(˜ Xt+h + v Zt+h )]. 5.2 Explicit and quasi explicit pricing formulas The explicit formulas for zero-coupon bond prices also immediately provide explicit formulas for some derivatives like swaps. Moreover, the result of ˜ ˜ section 5.1, where u and v have complex components, can be used to price payoﬀs of the form: + u ˜ ˜ u ˜ ˜ exp(˜1 Xt+h + v1 Zt+h ) − exp(˜2 Xt+h + v2 Zt+h ) , like caps, ﬂoors or options on zero-coupon bonds. Let us consider, for in- stance, the problem to price, at date t, a European call option on the zero- coupon bond B(t + h, H − h), then the pricing relation is : pt (K, h) = Et Mt,t+h (B(t + h, H − h) − K)+ (94) = Et Mt,t+h (exp[−(H − h)R(t + h, H − h)] − K)+ , and, substituting here the yield to maturity formula (68), for the SVARN(p) 39 model, or formula (90), for the SVARG(p) model, we can write : + ˜ pt (K, h) = Et Mt,t+h exp[CH−h Xt+h + DH−h Zt+h ] − K ˜ = Et Mt,t+h exp[CH−h Xt+h + DH−h Zt+h ] − K I[−C ˜ H−h Xt+h −DH−h Zt+h <− log K] ˜ = Et Mt,t+h exp[CH−h Xt+h + DH−h Zt+h ] I[−C ˜ H−h Xt+h −DH−h Zt+h <− log K] −KEt Mt,t+h I[−C ˜ H−h Xt+h −DH−h Zt+h <− log K] = Gt (CH−h , DH−h , −CH−h , −DH−h , − log K; h) −KGt ( 0, 0, −CH−h , −DH−h , − log K; h) , (95) where I denotes the indicator function, and where u ˜ ˜ ˜ Gt (˜0 , v0 , u1 , v1 , K; h) u ˜ ˜ = Et Mt,t+h exp[˜0 Xt+h + v0 Zt+h ] I[−˜ u ˜ 1 Xt+h −˜1 Zt+h <K] v denotes the truncated real Laplace transform that we can deduce from the (untruncated) complex Laplace transform. More precisely, we have the fol- lowing formula [see Duﬃe, Pan, Singleton (2000) for details]: u ˜ Pt (˜0 , v0 , h) u ˜ ˜ ˜ Gt (˜0 , v0 , u1 , v1 , K; h) = 2 +∞ 1 u u ˜ v Im[Pt (˜0 + i˜1 y, v0 + i˜1 y; h)] exp(−iyK) − dy π0 y (96) where Im(z) denotes the imaginary part of the complex number z. So, formula (95) is quasi explicit since it only requires a simple (one-dimensional) integration to derives the values of Gt . 40 6 Empirical Analysis 6.1 Introduction The purpose of this section is to propose an empirical analysis of the Gaus- sian term structure models presented in Section 3, using observations on the U. S. term structure of interest rates. We have seen that the Gaussian SVARN(p) Term Structure Models can be characterized by an exogenous or endogenous factor (xt ). In the present empirical analysis we follow an endogenous approach, given that it gives several important advantages coming from the observations we have about the factor, that is, the short rate in the scalar case, or yields at diﬀerent maturities in the multivariate framework. First, thanks to data, we are able to detect stylized facts on interest rates which give us the possibility to justify the autoregressive model with switching regimes we propose for the historical dynamics of (xt ) : indeed, a large empirical literature on bond yields show that interest rates have an historical multi-lag dynamics char- acterized by switching of regimes [see, among the others, Hamilton (1989), Christiansen and Lund (2003), Cochrane and Piazzesi (2005)]. Second, ob- servations about the Gaussian-distributed factor lead to a maximum likeli- hood estimation of historical parameters: in this way, we are able to test hypotheses using likelihood ratio statistics, and rank the models in terms of various information criteria. Finally, the diﬀerence between directly ob- served and estimated factor values determine model residuals that can be used to derive various diagnostic criteria. By a comparison with this multi-lag regime-switching endogenous ap- proach, the classical continuous-time aﬃne term structure approach ` la a Duﬃe and Kan (1996) and Dai and Singleton (2000) has some diﬀerent features. First, the factors are in general assumed not observable and there- fore justiﬁcations for the (historical) factors dynamics, along with a precise econometric analysis of model residuals, are not possible. Second, in order to reconstruct a time series of the latent factors, for an exact maximum like- lihood estimation, prices of some zero-coupon bonds are assumed to be per- fectly observed in order to inverse the pricing equations [see Chen and Scott (1987) and Pearson and Sun (1994)]; this inversion technique depends on the zero-coupon bonds selected values of the parameters, which are not initially available, and therefore the reconstructed time series is model-sensitive [see Collin-Dufresne, Goldstein and Jones (2004) for a discussion]. Third, the class of discrete-time aﬃne (Compound Autoregressive) processes is much 41 larger than the discrete-time counterpart of the continuous-time aﬃne class6 [see Gourieroux, Monfort and Polimenis (2005), and Darolles, Gourieroux and Jasiak (2006)]. We will start the empirical analysis by the single-regime framework, with the estimation of AR(p) and VAR(p) Factor-Based Term Structure models [see Monfort and Pegoraro (2006)], in a scalar (short rate) and bivariate (short rate and spread between the long and short rate) setting. The his- torical parameters are estimated by exact Maximum Likelihood, while the risk-neutral parameters are estimated by nonlinear least squares (NLLS). We observe that the introduction of lags greatly improve the goodness-of-ﬁt of the models, and replicate stylized facts as the increasing shape of the interest rate autocorrelation as a function of the time to maturity. The further step of the empirical analysis concerns the regime-switching framework, that is, the estimation of the SARN(p) and bivariate SVARN(p) term structure models, where the latent variable (zt ) is assumed to be a two-states non-homogeneous Markov chain. As in the single-regime speci- ﬁcations, the factor is the short rate in the scalar case, and the short rate and spread in the bivariate case. The historical parameters are estimated by maximization of the likelihood function calculated using the Kitagawa- Hamilton ﬁlter [see Hamilton (1994)]. 6.2 Description of the Data The CRSP data set on the U. S. term structure of interest rates [treasury zero-coupon bond (ZCB) yields], that we consider in the following appli- cation, covers the period from June 1964 to December 1995 and contains 379 monthly observations for each of the nine maturities : 1, 3, 6 and 9 months and 1, 2, 3, 4 and 5 years [Figure 1 shows a plot of (annualized) monthly ZCB yields of maturity 1, 12 and 60 months]7 . Summary statistics about the above mentioned (annualized) yields are presented in Table 1 : the term structure is, on average, upward sloping and the yields with larger standard deviation, skewness and kurtosis are those with shorter maturities. Moreover, yields are highly autocorrelated with a persistence which is in- creasing with the time to maturity : we call this feature of interest rates as 6 For instance, the discrete-time Gaussian VAR(1) process has a continuous-time equiv- alent if and only if there exists a matrix ϑ such that ϕ = exp(−ϑ), or, any Car(p) process [like the Gaussian VAR(p) process], with p ≥ 2, cannot be the time discretized version of a continuous-time aﬃne process. 7 The same data set is used in the papers of Longstaﬀ and Schwartz (1992) and Bansal and Zhou (2002). We are grateful to Ravi Bansal and Hao Zhou for providing us the data set. 42 the increasing term structure of autocorrelations stylized fact [see Figure 2]. Table 1 : Summary Statistics on U. S. Monthly Yields from June 1964 to December 1995. ACF(k) indicates the empirical autocorrelation between yields R(t, h) and R(t, h − k), with h and k expressed on a monthly basis. Maturity 1-m 3-m 6-m 9-m 1-yr 2-yr 3-yr 4-yr 5-yr Mean 0.0645 0.0672 0.0694 0.0709 0.0713 0.0734 0.0750 0.0762 0.0769 Std. Dev. 0.0265 0.0271 0.0270 0.0269 0.0260 0.0252 0.0244 0.0240 0.0237 Skewness 1.2111 1.2118 1.1518 1.1013 1.0307 0.9778 0.9615 0.9263 0.8791 Kurtosis 4.5902 4.5237 4.3147 4.1605 3.9098 3.6612 3.5897 3.5063 3.3531 Minimum 0.0265 0.0277 0.0287 0.0299 0.0311 0.0366 0.0387 0.0397 0.0398 Maximum 0.1640 0.1612 0.1655 0.1644 0.1581 0.1564 0.1556 0.1582 0.1500 ACF(5) 0.8288 0.8531 0.8579 0.8588 0.8604 0.8783 0.8915 0.8986 0.9053 ACF(10) 0.7278 0.7590 0.7691 0.7699 0.7683 0.7885 0.8021 0.8075 0.8212 ACF(15) 0.5887 0.6164 0.6285 0.6313 0.6395 0.6720 0.6908 0.6987 0.7201 ACF(20) 0.4303 0.4631 0.4880 0.4996 0.5156 0.5742 0.6051 0.6193 0.6431 6.3 Estimated VAR(p) Factor-Based Term Structure Models 6.3.1 Estimation Method The methodology we follow to estimate the parameters of the endogenous VAR(p) term structure models is based on a consistent two-step procedure. In the ﬁrst step, thanks to observations on the n-dimensional endogenous factor (xt ), we estimate the [n(1 + np) + (n(n + 1)/2)]-dimensional vector of parameters θP = [ν , vec(ϕ) , vech(σσ ) ] , characterizing the historical dynamics (xt ), by Maximum Likelihood (ML). In the second step, using observations on yields with maturities diﬀerent from those used in the ﬁrst step and for a given estimates of vech(σσ ), we es- timate the [n(1+np)]-dimensional vector of parameters θQ = [(ν ∗ ) , vec(ϕ∗ ) ] , characterizing the risk-neutral dynamics of (xt ), by minimizing the sum of squared ﬁtting errors between the observed and theoretical yields. More pre- cisely, in the scalar case, we estimate θQ by nonlinear lest squares (NLLS), while, in the multivariate case, these parameters are estimated by con- strained NLLS. The constraints are imposed to satisfy restrictions (69) im- plied by the absence of arbitrage opportunity principle [see Section 3.8]. Given the complete set of nine maturities of our data base, and given a number m of yields used to estimate the vector of historical parameters ∗ θP , we denote by Hm the set of remaining maturities used to estimate the vector of risk-neutral parameters θQ . In the AR(p) Factor-Based case, xt is the one-month yield to maturity R(t, 1) expressed at a monthly frequency, while, in the bivariate VAR(p) Factor-Based case the factor is given by: xt = [R(t, 1), R(t, 60) − R(t, 1)] , 43 where [R(t, 60) − R(t, 1)] is the spread at date t between the ﬁve-year and one-month yield to maturity, expressed at a monthly frequency [see Ang and Bekaert (2002), and Ang, Piazzesi and Wei (2005) for similar speciﬁcations]. The NLLS estimator for the AR(p) case, is determined by : ˆ θQ = Arg minθQ S 2 (θQ ), T (97) 2 S (θQ ) = ˜ h) − R(t, h)]2 , [R(t, ∗ t=p h∈H1 ∗ given the set H1 of maturities used to estimate the risk-neutral parameters; ˜ h) is the observed yield, and R(t, h) is the theoretical yield. R(t, The constrained NLLS estimator, in our bivariate model speciﬁcation, is given by : ˆ 2 θQ = Arg minθQ S (θQ ) T 2 S (θ ) = ˜ [R(t, h) − R(t, h)]2 , Q t=p h∈H2∗ (98) T s. t. ˜ [R(t, 60) − R(t, 60)]2 = 0 , t=p where R(t, h) is the theoretical yield. The constraint in the minimization program (98) guarantees the absence of arbitrage opportunity on the ﬁve- year yield to maturity. 6.3.2 Estimation Results for the AR(p) model Historical Parameter Estimates The maximum value of the mean Log-Likelihood and the values of the es- timated vector of parameters θP = (ν, ϕ1 , . . . , ϕp , σ) of the AR(p) Factor- Based Term Structure models, for p ∈ {1, . . . , 6}, are presented in Tables 2 and 3 [the t-values are given in parenthesis]. We also rank the models in terms of the Akaike Information Criterion (AIC). 44 Table 2 : AR(p) Factor-Based Term Structure models. Maximum value of the mean Log-Likelihood, AIC and parameter estimates of ν and σ 2 . The short rate observations are expressed at a monthly frequency. Parameter estimates are expressed in basis points (bp). We denote with mlogL the mean log-Likelihood of the AR(p) model : mlogL = logL(θP |x1 , . . . , xT −p )/(T − p). (∗∗ ) denotes a parameter signiﬁcant at 0.05; (∗ ) denotes a parameter signiﬁcant at 0.1. The Akaike Information Criterion (AIC) is given by 2mlogL − (2k/(T − p)), with k denoting the dimension of θP . AR(1) AR(2) AR(3) AR(4) AR(5) AR(6) mlogL 5.95657 5.95868 5.96082 5.96134 5.97224 5.97092 AIC 11.8973 11.8961 11.8950 11.8907 11.9071 11.8990 ν 2.3∗∗ bp 2.1∗∗ bp 2.3∗∗ bp 2.1∗∗ bp 1.9∗∗ bp 1.9∗∗ bp [2.6725] [2.4822] [2.6598] [2.4761] [2.1571] [2.1262] 2 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ σ 0.0039 bp 0.0039 bp 0.0039 bp 0.0039 bp 0.0038 bp 0.0038∗∗ bp [13.7483] [13.7301] [13.7118] [13.6937] [13.6754] [13.6571] Table 3 : AR(p) Factor-Based Term Structure models Parameter estimates of (ϕ1 , . . . , ϕp ). (∗∗ ) denotes a parameter signiﬁcant at 0.05; (∗ ) denotes a parameter signiﬁcant at 0.1. AR(1) AR(2) AR(3) AR(4) AR(5) AR(6) ϕ1 0.9580 ∗∗ 0.8798 ∗∗ 0.8861 ∗∗ 0.8912 ∗∗ 0.8814 ∗∗ 0.8806 ∗∗ [65.5620] [17.2393] [17.1525] [17.1688] [17.1628] [16.9714] ϕ2 0.0811 0.1547 ∗∗ 0.1456 ∗∗ 0.1672 ∗∗ 0.1675 ∗∗ [1.5938] [2.2869] [2.0843] [2.4260] [2.3885] ϕ3 −0.0829 ∗ −0.1372 ∗ −0.1595 ∗∗ −0.1586 ∗∗ [−1.6459] [−1.9204] [−2.3048] [−2.2623] ϕ4 0.0608 −0.0790 −0.0798 [1.1455] [−1.1788] [−1.1240] ϕ5 0.1557 ∗∗ 0.1510 ∗∗ [3.1048] [2.4443] ϕ6 0.0053 [0.1232] An examination of the above displayed parameter estimates show, ﬁrst of all, that the historical dynamics of the (one-month to maturity) short rate is not Markovian of order one, given that, in the AR(5) and AR(6) speciﬁ- cations, the parameters (ϕ1 , ϕ2 , ϕ3 , ϕ5 ) are always signiﬁcative. Moreover, the AIC indicates these models as the preferred ones. Another indication of the key role played by the lagged values, in deter- mining well speciﬁed models for the short rate historical dynamics, is given by the Ljung-Box test (for the Gaussian AR(p) short rate model residuals) presented in Table 4. 45 Table 4 : AR(p) Factor-Based Term Structure models. Ljung-Box test for model residuals. (∗∗ ) denotes the null hypothesis accepted at 0.05; (∗ ) denotes the null hypothesis accepted at 0.01. Lags AR(1) AR(2) AR(3) AR(4) AR(5) AR(6) 5 13.2842 ∗ 11.7456 ∗ 9.2398 ∗∗ 9.0748 ∗∗ 0.2110 ∗∗ 0.2933 ∗∗ 10 20.7693 ∗ 19.4151 ∗ 17.4643 ∗∗ 17.3474 ∗∗ 8.8896 ∗∗ 8.7842 ∗∗ ∗ ∗ ∗ ∗∗ 15 33.2058 28.8190 25.5995 25.3238 18.3397 18.2022 ∗∗ 20 43.9331 41.4980 39.3640 38.2511 28.2745 ∗∗ 28.1044 ∗∗ We observe that, the models with small autoregressive orders are not able to pass the test for large lags, denoting a lack of these speciﬁcations in explaining the strong autocorrelation characterizing the short rate [see Table 1]. We are able to explain the short rate persistence also for large lags only when the autoregressive order move to p = 5 and p = 6. Risk-Neutral Parameter Estimates The minimum value of the mean nonlinear least square criterion [S 2 (θQ )/(T − ˆ p)] and the values of the estimated vector of risk-neutral parameters θQ = (ν ∗ , ϕ∗ , . . . , ϕ∗ ), with p ∈ {1, . . . , 6}, are presented in Tables 5 and 6 [the 1 p t-values are given in parenthesis]. Table 5 : AR(p) Factor-Based Term Structure models. Minimum value of the mean NLLS criterion, RMSE, MAE and parameter estimates of ν ∗ . Yields to maturity observations are expressed at a monthly frequency. Parameter estimates are expressed in basis points (bp). (∗∗ ) denotes a parameter signiﬁcant at 0.05; (∗ ) denotes a parameter signiﬁcant at 0.1. AR(1) AR(2) AR(3) AR(4) AR(5) AR(6) S 2 (θQ )/(T − p) ˆ 0.00000054 0.00000051 0.00000050 0.00000048 0.00000047 0.00000046 RMSE 0.000736 0.000716 0.000709 0.000696 0.000687 0.000679 MAE 0.000530 0.000526 0.000528 0.000524 0.000517 0.000509 ν∗ 1.10∗∗ bp 1.51∗∗ bp 1.52∗∗ bp 1.48∗∗ bp 1.48∗∗ bp 1.52∗∗ bp [33.2526] [22.6031] [22.9266] [22.9794] [22.7051] [22.4479] 46 Table 6 : AR(p) Factor-Based Term Structure models. Parameter estimates of (ϕ∗ , . . . , ϕ∗ ). (∗∗ ) denotes a 1 p parameter signiﬁcant at 0.05; (∗ ) denotes a parameter signiﬁcant at 0.1. AR(1) AR(2) AR(3) AR(4) AR(5) AR(6) ϕ∗ 1 0.9899 ∗∗ 0.5076 ∗∗ 0.7333 ∗∗ 0.7758 ∗∗ 0.7382 ∗∗ 0.7037 ∗∗ [1877] [9.6003] [14.2703] [15.4922] [14.2057] [13.3209] ϕ∗ 2 0.4788 ∗∗ −0.0299 0.2291 ∗∗ 0.2947 ∗∗ 0.2998 ∗∗ [9.1313] [−0.4132] [2.8931] [3.6124] [3.6802] ϕ∗ 3 0.2832 ∗∗ −0.3860 ∗∗ −0.1600 ∗∗ −0.1069 [7.5221] [−5.3681] [−2.0834] [−1.3898] ϕ∗ 4 0.3685 ∗∗ −0.1977 ∗∗ 0.0123 [10.2233] [−2.6864] [0.1609] ϕ∗ 5 0.3126 ∗∗ −0.2173 ∗∗ [8.4180] [−2.9386] ϕ∗ 6 0.2961 ∗∗ [7.7697] One may observe the signiﬁcativity of all AR risk-neutral coeﬃcients in the AR(4) and AR(5) model speciﬁcations, and the signiﬁcativity of the coeﬃcients ϕ∗ and ϕ∗ in the AR(6) case. 5 6 6.3.3 Estimation Results for the bivariate VAR(p) model Historical Parameter Estimates As in the scalar case, we present the maximum value of the mean Log- Likelihood and the values of the estimated vector of parameters θP = [ν , vec(ϕ) , vech(σσ ) ] of the bivariate VAR(p) Factor-Based Term Structure models, for an AR order p = 1 and p = 2. These results are presented in Tables 7 and 8 [the t-values are given in parenthesis]. We also rank the models in terms of the Akaike Information Criterion (AIC)8 . 8 We have also estimated the historical parameters of the above mentioned bivariate VAR(p) model, for p larger than 2, but the AIC criterion has indicated the ﬁrst two AR orders as the preferred ones. 47 Table 7 : VAR(p) Factor-Based Term Structure models. Maximum value of the mean Log-Likelihood, AIC and 2 2 parameter estimates of (ν1 , ν2 ) and (σ1 , σ21 , σ2 ). The short rate and long rate observations are expressed at a monthly frequency. Parameter estimates are expressed in basis points (bp). We denote with mlogL the mean log-Likelihood of the VAR(p) model : mlogL = logL(θP |x1 , . . . , xT −p )/(T − p). (∗∗ ) denotes a parameter signiﬁcant at 0.05; (∗ ) denotes a parameter signiﬁcant at 0.1. The Akaike Information Criterion (AIC) is given by 2mlogL − (2k/(T − p)), with k denoting the dimension of θP . VAR(1) VAR(2) mlogL 12.6403 12.6837 AIC 25.2330 25.2984 ν1 0.65 bp 1.32 bp [0.5856] [1.2262] ν2 0.80 bp 0.26 bp [0.8157] [0.2701] 2 σ1 0.0039∗∗ bp 0.0036∗∗ bp [5.94750] [6.02614] σ21 −0.0028∗∗ bp −0.0026∗∗ bp [-6.0995] [-6.2100] 2 σ2 0.0030∗∗ bp 0.0028∗∗ bp [7.6713] [8.0731] Table 8 : VAR(1) and VAR(2) Factor-Based Term Structure models. Parameter estimates of (ϕ1 , ϕ2 ). (∗∗ ) denotes a parameter signiﬁcant at 0.05; (∗ ) denotes a parameter signiﬁcant at 0.1. VAR(1) VAR(2) ϕ1 0.9742∗∗ 0.0719∗∗ 1.3318∗∗ 0.6207∗∗ [59.8835] [2.2174] [15.0111] [7.0095] 0.0091 0.8769∗∗ −0.2744∗∗ 0.4353∗∗ [0.6388] [30.7835] [-3.4988] [5.5601] ϕ2 −0.3648∗∗ −0.5762∗∗ [-3.6117] [-5.8201] 0.2893∗∗ 0.4642∗∗ [3.2397] [5.3020] If we consider the parameter estimates of Tables 7 and 8, we observe that the joint historical dynamics of short rate and spread is not Markovian of order one, given that, in the VAR(2) speciﬁcation, the parameters in the second autoregressive matrix ϕ2 are signiﬁcantly diﬀerent from zero. Moreover, the AIC indicates this model as the preferred one. Table 7 shows also that the constant term (ν1 , ν2 ) is not signiﬁcative for both AR orders. 48 Table 9 : VAR(p) Factor-Based Term Structure models. LBshort denotes the value of the Ljung-Box test statistic for short rate residuals, while LBspread denotes the value of the Ljung-Box test statistic for spread residuals. Q denotes the value of the (adjusted) Portmanteau test statistic for the VAR(1) and VAR(2) model residuals. (∗∗ ) denotes the null hypothesis accepted at 0.05; (∗ ) denotes the null hypothesis accepted at 0.01. VAR(1) VAR(2) Lags LBshort LBspread Q LBshort LBspread Q 5 10.2909 ∗∗ 22.8735 58.9900 5.8051 ∗∗ 12.4876 ∗ 29.9870 10 13.5065 ∗∗ 34.1952 82.6678 9.0209 ∗∗ 19.8705 ∗ 51.2640 ∗ 15 29.2167 ∗ 63.4161 134.2488 23.0114 ∗∗ 35.8770 91.3809 20 36.2600 ∗ 69.7833 165.8618 30.0413 ∗∗ 45.5552 127.9043 With regard to the autocorrelation analysis of model residuals, presented in Table 9, we observe that the ability of the VAR(p) model to explain the serial dependence in the (univariate and joint) short rate and spread his- torical dynamics, improves when we move from the VAR(1) to the VAR(2) speciﬁcation, even if both models are not able to pass the portmanteau test on the bivariate residual vectors. Indeed, for p = 1, the Ljung-Box test on the short rate residuals accepts serial non correlation only at 0.01 for large lags, while, the same test on spread residuals rejects it strongly for all lags. When we consider p = 2, the Ljung-Box, when applied on the short rate residuals, always accepts non serial correlation at 0.05, while, when applied on the spread residuals, it accepts it for ﬁve and ten lags, but rejects it for larger lags. The rejection of the portmanteau test (stronger when p = 1) stresses the diﬃculty of the speciﬁed models to explain the serial dependence in the spread historical dynamics. Risk-Neutral Parameter Estimates We present the minimum value of the mean nonlinear least square crite- ˆ rion [S 2 (θQ )/(T − p)] and the values of the estimated vector of risk-neutral parameters θQ = [(ν ∗ ) , vec(ϕ∗ ) ] , for the bivariate VAR(1) and VAR(2) Factor-Based Term Structure models, in Tables 10 and 11 [the t-values are given in parenthesis]. 49 Table 10 : AR(p) Factor-Based Term Structure models. Minimum value of the mean NLLS criterion, RMSE, ∗ ∗ MAE and parameter estimates of (ν1 , ν2 ). Yields to maturity observations are expressed at a monthly frequency. Parameter estimates are expressed in basis points (bp). (∗∗ ) denotes a parameter signiﬁcant at 0.05; ∗ ( ) denotes a parameter signiﬁcant at 0.1. VAR(1) VAR(2) S 2 (θQ )/(T − p) ˆ 0.00000009 0.00000008 RMSE 0.000297 0.000283 MAE 0.000208 0.000198 ∗ ν1 −0.58∗∗ bp −0.55∗∗ bp [−6.6459] [−4.9423] ∗ ∗∗ ν2 0.72 bp 0.71∗∗ bp [5.7860] [4.5783] Table 11 : VAR(1) and VAR(2) Factor-Based Term Structure models. Parameter estimates of (ϕ∗ , ϕ∗ ). (∗∗ ) 1 2 denotes a parameter signiﬁcant at 0.05; (∗ ) denotes a parameter signiﬁcant at 0.1. VAR(1) VAR(2) ϕ∗ 1 1.0131∗∗ 0.1105∗∗ 1.3154∗∗ 0.6020∗∗ [805.8869] [34.5743] [28.4716] [9.5120] ∗∗ ∗∗ ∗∗ −0.0156 0.9072 −0.2528 0.4142∗∗ [-8.6611] [203.2978] [-3.5778] [4.2509] ϕ∗ 2 −0.3004∗∗ −0.4890∗∗ [-6.5177] [-7.8923] 0.2342∗∗ 0.4839∗∗ [3.3244] [5.0769] We ﬁnd that, also in this bivariate risk-neutral (pricing) framework, the lagged values of the short rate and spread play an important role in the model speciﬁcation. Moreover, one may observe the signiﬁcativity of all risk-neutral AR coeﬃcients in the VAR(2) speciﬁcation. The goodness-of-ﬁt of the VAR(2) Factor-Based Term Structure Model, outperforms the results of the VAR(1) speciﬁcation. In other words, a VAR(2) speciﬁcation for the historical and risk-neutral dynamics of the fac- tor driving term structure shapes, lead to propose a bivariate term structure model which is able to ﬁt yields to maturity better than the VAR(1) (bi- variate endogenous Vasicek) speciﬁcation. 50 6.3.4 The Term Structure of Autocorrelations We have mentioned in Section 6.2 that interest rates are characterized by an autocorrelation which is, for each lag, increasing with the time to maturity. The purpose of this section is to propose this stylized fact as a new test that a well speciﬁed term structure model should be able to overcome, and to verify if the endogenous AR(p) and VAR(p) term structure models presented in the previous sections are able to replicate the above mentioned shape of the term structure of autocorrelations. This test is based on the comparison between the empirical and model- implied term structure of yields autocorrelations, for each of the estimated models, at 5, 10, 15 and 20 lags. The model-implied autocorrelations are calculated on the basis of 10.000 simulations of yields from each of the ﬁtted ∗ yield to maturity formula [estimated using maturities h ∈ H1 in the scalar case, and maturities h ∈ H2 ∗ in the bivariate case], with each replication having a sample size of 379. The replication of yields are based on the simulation of factor values using the estimated historical dynamics. The results, for each of the selected lag length, are respectively presented in Figures 3, 4, 5 and 6. The ﬁrst result we observe is that, for each lag, the model speciﬁcations showing the best ﬁtting of the short-term part (1, 3 and 6 months to ma- turity) of the autocorrelation curve are the scalar AR(5) and AR(6) term structure models, and not the bivariate speciﬁcations. Moreover, for 5 and 10 lags, this dominance is extended to 9 and 12 months to maturity. There- fore, it seems that, if we want to correctly replicate the short-term part of the autocorrelation curve, it is more important to increase the autoregres- sive order in the AR(p) short-rate term structure model, than to move to a bivariate (short rate and spread) setting. In other words, the AR(5) and AR(6) short-rate term structure models lead to a more precise representa- tion of the short-term interest rates persistence, than the proposed bivariate cases. Nevertheless, for maturities larger than 1 year, the best ﬁtting is ob- tained from the bivariate VAR(1) and VAR(2) speciﬁcations, thanks to the information on the long-term part of the yield curve supplied by the spread- factor. In particular, while scalar models replicate an autocorrelation curve which is ﬂat in the long-term part, the VAR(1) and VAR(2) models are able to reproduce the observed increasing shape for each lag and time to maturity. In addition, if we compare the performances of the two bivariate cases, we observe that the introduction of the second lag leads to improve the replication of the short-term part of the autocorrelation curve for 5 and 10 lags, while, for larger lags or maturities, the best ﬁtting is obtained in 51 the VAR(1) setting. 6.4 Estimated SARN(p) Term Structure Models [to be com- pleted] 6.4.1 Estimated Models and Estimation Method In the following sections we will present the parameter estimations of alterna- tive regime-switching short rate historical dynamics, speciﬁed as particular cases of the general relation (40) presented in Section 3.1. In particular, we will ﬁrst estimate the following model [see Hamilton (1988)] : xt+1 − ν zt = ϕ1 [ xt − ν zt−1 ] + . . . + ϕp [ xt+1−p − ν zt−p ] + (σ zt )εt+1 , (99) and, then, we will consider the generalization where also the autoregressive coeﬃcients are function of the regime indicator function : xt+1 − ν zt = (ϕ1 zt−1 ) [ xt − ν zt−1 ] + . . . (100) + (ϕp zt−p ) [ xt+1−p − ν zt−p ] + (σ zt )εt+1 . Model (99) will be called SARN1 (p) model, while, model (100) will be called SARN2 (p) model. In both cases (εt ) is a Gaussian white noise with N (0, 1) distribution, p ∈ {1, . . . , 6}, and (zt ) is a 2-states non-homogeneous Markov chain. In the latter case, the transition probabilities have the following logistic form: P (zt+1 = ei |zt = ei , rt+1 ) = π(ei , ei , rt+1 ) (101) eai +bi rt+1 = , i ∈ {1, 2} . 1 + eai +bi rt+1 The estimations are obtained from the maximization of the likelihood func- tion calculated by means of the Kitagawa-Hamilton ﬁlter [see Hamilton (1994)]. 6.4.2 Estimation Results for the SARN1 (p) model Historical Parameter Estimates The maximum value of the mean Log-Likelihood and the values of the es- timated vector of parameters θP = [ ν1 , ν2 , ϕ1 , . . . , ϕp , σ1 , σ2 , a1 , b1 , a2 , b2 ] of 52 the SARN1 (p) model, for p ∈ {1, . . . , 6}, are presented in Tables 12, 13 and 14 [the t-values are given in parenthesis]. An analysis of the parameter estimates presented in Table 13 shows that, even with the introduction of (non-homogeneous) switching regimes in the parameters, the short rate historical dynamics is not Markovian of order one. Indeed, we have the signiﬁcativity of the AR coeﬃcient ϕ4 in the SARN1 (4), SARN1 (5) and SARN1 (6) models, and the signiﬁcativity of ϕ5 in the SARN1 (5) case. In other words, the important role played by large AR (historical) coeﬃcients, for the AR(p) short rate dynamics of Section 6.3.2, seems to be not just a model misspeciﬁcation eﬀect, induced by the lack of nonlinearities in the modelisation, but a proper feature of the short rate behavior. In additions, the AIC indicate models SARN1 (4) and SARN1 (5) as the preferred speciﬁcations. In Table 12 we observe that (ν1 , ν2 ) and (σ1 , σ2 ) are always signiﬁcantly diﬀerent from zero in each of the estimated models, while Table 14 high- lights the signiﬁcant role played by the lagged short rate in the transition probabilities. Parameter b1 is signiﬁcantly diﬀerent from zero for models SARN1 (4), SARN1 (5) and SARN1 (6), and parameter b2 is signiﬁcant for all the estimated models. In the ﬁrst (low volatility) regime the negative sign and the magnitude taken by b1 implies, when the short rate increases, an increased probability of switching to the second (high volatility) regime, while, in the high volatility regime, the positive sign of b2 induce, as the short rate rises, an increased probability of remaining in this second regime [see Figure 7, and Ang and Bekaert (2002b) for similar results]. Table 12 : SARN1 (p) model. Maximum value of the mean Log-Likelihood, AIC and parameter estimates of 2 2 ν1 , ν2 , σ1 , σ2 . The short rate observations are expressed at a monthly frequency. Parameter estimates are expressed in basis points (bp). We denote with mlogL the mean log-Likelihood of the model : mlogL = logL(θP |x1 , . . . , xT −p )/(T − p). (∗∗ ) denotes a parameter signiﬁcant at 0.05; (∗ ) denotes a parameter signiﬁcant at 0.1. The Akaike Information Criterion (AIC) is given by 2mlogL − (2k/(T − p)), with k denoting the dimension of θP . Vasicek SARN1 (1) SARN1 (2) SARN1 (3) SARN1 (4) SARN1 (5) SARN1 (6) mlogL 5.95657 6.2972 6.3015 6.3012 6.3073 6.3107 6.3085 AIC 11.8973 12.5468 12.5499 12.5439 12.5506 12.5519 12.5419 ν1 0.23∗∗ bp 56∗∗ bp 69∗ bp 64∗∗ bp 56∗∗ bp 54∗∗ bp 54∗∗ bp [2.6725] [4.1940] [1.9038] [2.4589] [5.1030] [4.9714] [4.8997] ν2 − 51∗∗ bp 68∗ bp 63∗∗ bp 50∗∗ bp 48∗∗ bp 47∗∗ bp [3.6951] [1.8661] [2.3859] [4.4237] [4.2362] [4.1762] 2 σ1 0.0039∗∗ bp 0.00078∗∗ bp 0.00078∗∗ bp 0.00078∗∗ bp 0.00090∗∗ bp 0.00078∗∗ bp 0.00078∗∗ bp [13.7483] [7.9280] [7.6578] [7.3951] [9.0231] [9.1997] [9.2470] 2 σ2 − 0.0144∗∗ bp 0.0144∗∗ bp 0.0144∗∗ bp 0.0158∗∗ bp 0.0158∗∗ bp 0.0161∗∗ bp [5.8170] [5.8270] [5.7826] [5.6030] [5.2660] [5.2091] 53 Table 13 : SARN1 (p) model. Parameter estimates of (ϕ1 , . . . , ϕp ). (∗∗ ) denotes a parameter signiﬁcant at 0.05; (∗ ) denotes a parameter signiﬁcant at 0.1. Vasicek SARN1 (1) SARN1 (2) SARN1 (3) SARN1 (4) SARN1 (5) SARN1 (6) ϕ1 0.9580∗∗ 0.9834∗∗ 0.9001∗∗ 0.9040∗∗ 0.8946∗∗ 0.9318∗∗ 0.9294∗∗ [65.5556] [83.7917] [17.7576] [17.9132] [14.7545] [14.7533] [15.0001] ϕ2 − − 0.0911∗ 0.1443∗∗ 0.1788∗∗ 0.1619∗∗ 0.1671∗∗ [1.7924] [2.1623] [2.2788] [2.1545] [2.2149] ϕ3 − − − -0.0591 0.0482 0.0277 0.0274 [-1.2024] [0.6917] [0.4609] [0.4352] ϕ4 − − − − -0.1412∗∗ -0.2549∗∗ -0.2579∗∗ [-2.6212] [-3.2006] [-3.3820] ϕ5 − − − − − 0.1147∗ 0.0953 [1.9014] [1.2282] ϕ6 − − − − − − 0.0202 [0.3462] Table 14 : SARN1 (p) model. Parameter estimates of (a1 , b1 , a2 , b2 ). (∗∗ ) denotes a parameter signiﬁcant at 0.05; (∗ ) denotes a parameter signiﬁcant at 0.1. Vasicek SARN1 (1) SARN1 (2) SARN1 (3) SARN1 (4) SARN1 (5) SARN1 (6) a1 − 5.5958∗∗ 4.0455∗∗ 4.1295∗∗ 6.1772∗∗ 6.1917∗∗ 6.2571∗∗ [3.9982] [3.3300] [3.0575] [4.0186] [4.5137] [4.5461] b1 − -512.1201∗∗ -292.6488 -306.5667 -588.1480∗∗ -617.5216∗∗ -629.2369∗∗ [-2.2888] [-1.4467] [-1.3875] [-2.4228] [-2.7342] [-2.7924] a2 − -4.5652∗∗ -9.1173∗∗ -9.2568∗ -4.3966 -4.1375∗ -4.0951∗ [-2.0169] [-1.9671] [-1.8638] [-1.7952] [-1.9189] [-1.8769] b2 − 912.7172∗∗ 1657.95914∗∗ 1676.5307∗∗ 880.3991∗∗ 770.6290∗∗ 757.2393∗∗ [2.4675] [2.0872] [1.9784] [2.1495] [2.0248] [2.0073] With regard to the ability of the estimated models to explain autocorre- lation in the short rate historical dynamics, we observe that the Ljung-Box test for models residuals [see Table 15] rejects the null hypothesis of non serial correlation for each lag, for the SARN1 (1) model, while the lower values of the test statistics are obtained for the SARN1 (4), SARN1 (5) and SARN1 (6) cases, where the null is accepted for each lag. This means that, also in this regime-switching setting, as in the single-regime case presented in Section 6.3.2, the introduction of lags gives the possibility to explain better the observed short rate autocorrelation. The introduction of switching regimes turns out to be determinant in the explanation of nonlinear serial dependence. Indeed, if we study the presence of serial correlation in squared model residuals, the Ljung-Box test accepts it for univariate and bivariate single-regime models at each lag9 , while the 9 The results are available upon request from the authors. 54 introduction of switching regimes strongly reduces the values of the test statistics, and the test rejects serial correlation for each lag and AR order. Table 15 : SARN1 (p) model. Ljung-Box test for model residuals. (∗∗ ) denotes the null hypothesis accepted at 0.05; (∗ ) denotes the null hypothesis accepted at 0.01. Lags Vasicek SARN1 (1) SARN1 (2) SARN1 (3) SARN1 (4) SARN1 (5) SARN1 (6) 5 13.2842∗ 17.5327 8.8110∗∗ 7.2849∗∗ 1.6654∗∗ 0.6126∗∗ 1.0633∗∗ 10 20.7693∗ 25.1401 14.0125∗∗ 13.4573∗∗ 8.8147∗∗ 7.7319∗∗ 8.8291∗∗ ∗ ∗∗ ∗∗ ∗∗ 15 33.2058 40.4152 25.0287 24.8636 21.1171 19.8278 18.7125∗∗ 20 43.9331 41.8320 26.7724∗∗ 26.3952∗∗ 22.8609∗∗ 21.7085∗∗ 20.0667∗∗ Table 16 : SARN1 (p) model. Ljung-Box test for model squared residuals. (∗∗ ) denotes the null hypothesis accepted at 0.05; (∗ ) denotes the null hypothesis accepted at 0.01. Lags Vasicek SARN1 (1) SARN1 (2) SARN1 (3) SARN1 (4) SARN1 (5) SARN1 (6) 5 74.8243 7.6442∗∗ 2.1786∗∗ 1.5209∗∗ 11.7751∗∗ 7.9440∗∗ 5.7324∗∗ 10 129.0656 15.4161∗∗ 5.3786∗∗ 3.6072∗∗ 14.1805∗∗ 12.1060∗∗ 8.1270∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ 15 147.6571 20.5339 7.4220 6.0125 18.2715 18.4684 15.2539∗∗ 20 160.3003 26.0210∗∗ 9.5002∗∗ 7.4935∗∗ 19.1359∗∗ 20.6802∗∗ 18.8141∗∗ 6.4.3 Estimation Results for the SARN2 (p) model Historical Parameter Estimates The maximum value of the mean Log-Likelihood and the values of the es- timated vector of parameters θP = [ ν1 , ν2 , ϕ1 , . . . , ϕp , σ1 , σ2 , a1 , b1 , a2 , b2 ] of the SARN2 (p) model, for p ∈ {1, . . . , 6}, are presented in Tables 17, 18 and 19 [the t-values are given in parenthesis]. We observe, in this model speciﬁcation where also the AR coeﬃcients are regime-switching, that coeﬃcient ϕ4 is signiﬁcantly diﬀerent from zero in both regimes and for all the estimated models, and that, in the SARN2 (6) case, also the coeﬃcients ϕ5 and ϕ6 are signiﬁcantly diﬀerent from zero in each regime [see Table 17]. These results give a further important indication about the non-Markovian historical dynamics of the short rate. Parameters (ν1 , ν2 ) and (σ1 , σ2 ) are signiﬁcantly diﬀerent from zero in all the estimated models [see Table 17], and the coeﬃcients b1 and b2 , character- izing the state dependence of transition probabilities, are always signiﬁcantly diﬀerent from zero with a magnitude and sign of the same kind as in the SARN1 (p) case [see Table 19]. With regard to the Ljung-Box test for model residuals [see Table 20] and squared model residuals [see Table 21], we ﬁnd, as in the SARN1 (p) 55 model, the important role played by lagged values to explain short rate autocorrelation, and the one played by switching regimes to explain the short rate nonlinear serial dependence. Table 17 : SARN2 (p) model. Maximum value of the mean Log-Likelihood, AIC and parameter estimates of 2 2 ν1 , ν2 , σ1 , σ2 . The short rate observations are expressed at a monthly frequency. Parameter estimates are expressed in basis points (bp). We denote with mlogL the mean log-Likelihood of the model : mlogL = logL(θP |x1 , . . . , xT −p )/(T − p). (∗∗ ) denotes a parameter signiﬁcant at 0.05; (∗ ) denotes a parameter signiﬁcant at 0.1. The Akaike Information Criterion (AIC) is given by 2mlogL − (2k/(T − p)), with k denoting the dimension of θP . Vasicek SARN2 (1) SARN2 (2) SARN2 (3) SARN2 (4) SARN2 (5) SARN2 (6) mlogL 5.95657 6.3049 6.3148 6.3212 6.3270 6.3321 6.3356 AIC 11.8973 12.5569 12.5659 12.5679 12.5687 12.5679 12.5640 ν1 0.23∗∗ bp 78∗∗ bp 50∗∗ bp 51∗∗ bp 38∗∗ bp 46∗∗ bp 52∗∗ bp [2.6725] [8.5840] [19.0153] [20.1164] [6.3938] [10.3287] [15.8177] ν2 − 75∗∗ bp 47∗∗ bp 46∗∗ bp 37∗∗ bp 44∗∗ bp 46∗∗ bp [7.3373] [13.9981] [14.7861] [7.1289] [11.4610] [12.1064] 2 σ1 0.0039∗∗ bp 0.00078∗∗ bp 0.00078∗∗ bp 0.00068∗∗ bp 0.00068∗∗ bp 0.00078∗∗ bp 0.00068∗∗ bp [13.7483] [8.4306] [8.9994] [9.5433] [6.8239] [8.7889] [8.9726] 2 σ2 − 0.0139∗∗ bp 0.0121∗∗ bp 0.0121∗∗ bp 0.0154∗∗ bp 0.0161∗∗ bp 0.0156∗∗ bp [5.9586] [5.2087] [5.3261] [4.8471] [3.9924] [4.7739] 56 Table 18 : SARN2 (p) model. Parameter estimates of (ϕ1 , . . . , ϕp ). (∗∗ ) denotes a parameter signiﬁcant at 0.05; (∗ ) denotes a parameter signiﬁcant at 0.1. zt Vasicek SARN2 (1) SARN2 (2) SARN2 (3) SARN2 (4) SARN2 (5) SARN2 (6) ϕ1 e1 0.9580∗∗ 0.9958∗∗ 1.0867∗∗ 1.0531∗∗ 1.0496∗∗ 1.0611 ∗∗ 1.0663∗∗ [65.5556] [187.8362] [19.6710] [20.7064] [19.0948] [20.5741] [21.5081] e2 − 0.8715∗∗ 0.7625∗∗ 0.7178∗∗ 0.7056∗∗ 0.6964∗∗ 0.7808∗∗ [20.0540] [15.6587] [14.5150] [13.4447] [13.2841] [16.3176] ϕ2 e1 − − -0.1017∗ 0.0473 0.0441 0.0508 0.0269 [-1.8322] [0.6045] [0.4390] [0.6064] [0.3371] e2 − − 0.1970∗∗ 0.2770∗∗ 0.2927∗∗ 0.3009∗∗ 0.1618∗∗ [3.7571] [4.0588] [4.0965] [4.3338] [2.6858] ϕ3 e1 − − − -0.0276 0.0176 -0.0389 -0.0034 [-0.5664] [0.1707] [-0.3909] [-0.0715] e2 − − − -0.1202 ∗∗ 0.0861 0.0602 -0.0467 [-2.1338] [1.2984] [1.0520] [-1.0932] ϕ4 e1 − − − − -0.1322∗∗ -0.1557 ∗∗ -0.3221 ∗∗ [-2.3060] [-2.2900] [-5.3856] e2 − − − − -0.1116∗∗ -0.1814 ∗∗ -0.2013 ∗∗ [-2.6367] [-3.1485] [-4.4420] ϕ5 e1 − − − − − 0.0626 0.3358 ∗∗ [1.1601] [5.4958] e2 − − − − − 0.0875 ∗∗ 0.3858 ∗∗ [1.9589] [7.7233] ϕ6 e1 − − − − − − -0.1228 ∗∗ [-2.7012] e2 − − − − − − -0.1791 ∗∗ [-4.9709] Table 19 : SARN2 (p) model. Parameter estimates of (a1 , b1 , a2 , b2 ). (∗∗ ) denotes a parameter signiﬁcant at 0.05; (∗ ) denotes a parameter signiﬁcant at 0.1. Vasicek SARN2 (1) SARN2 (2) SARN2 (3) SARN2 (4) SARN2 (5) SARN2 (6) a1 − 5.2109∗∗ 5.7561∗∗ 5.8630∗∗ 4.1691∗∗ 4.4126∗∗ 6.1397∗∗ [4.4917] [5.9835] [6.2337] [4.2009] [4.3359] [6.7242] b1 − -471.7369∗∗ -617.2554∗∗ -639.1367∗∗ -322.2644∗∗ -328.7332∗∗ -714.5599∗∗ [-2.3394] [-4.1536] [-4.3501] [-2.3534] [-2.0826] [-5.1708] a2 − -4.8429∗∗ -2.8015∗∗ -2.8419∗ -1.7158 -1.8244 -2.3127∗ [-2.0089] [-2.0076] [-1.9278] [-1.2910] [-1.3822] [-1.9331] b2 − 942.0812∗∗ 477.1256∗∗ 454.7091∗∗ 305.8535∗ 354.1494∗∗ 230.9667∗ [2.3251] [2.5297] [2.1931] [1.7822] [2.0198] [1.7044] 57 Table 20 : SARN2 (p) model. Ljung-Box test for model residuals. (∗∗ ) denotes the null hypothesis accepted at 0.05; (∗ ) denotes the null hypothesis accepted at 0.01. Lags Vasicek SARN2 (1) SARN2 (2) SARN2 (3) SARN2 (4) SARN2 (5) SARN2 (6) 5 13.2842∗ 14.7609∗ 8.4215∗∗ 5.8669∗∗ 1.5441∗∗ 1.3751∗∗ 5.7215∗∗ 10 20.7693∗ 23.1617∗ 18.0274∗∗ 13.9979∗∗ 4.4104∗∗ 8.3853∗∗ 9.1638∗∗ ∗∗ ∗∗ ∗∗ 15 33.2058 34.9264 31.4173 23.9347 12.6148 19.4970 20.4498∗∗ 20 43.9331 35.7057∗ 34.7604∗ 26.9634∗∗ 15.9674∗∗ 24.5028∗∗ 28.9044∗∗ Table 21 : SARN2 (p) model. Ljung-Box test for squared model residuals. (∗∗ ) denotes the null hypothesis accepted at 0.05; (∗ ) denotes the null hypothesis accepted at 0.01. Lags Vasicek SARN2 (1) SARN2 (2) SARN2 (3) SARN2 (4) SARN2 (5) SARN2 (6) 5 74.8243 13.7833∗ 3.1002∗∗ 2.9669∗∗ 4.8844∗∗ 3.3438∗∗ 11.5600∗∗ 10 129.0656 19.0444∗ 7.0689∗∗ 4.0882∗∗ 6.4805∗∗ 13.6854∗∗ 13.4562∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ 15 147.6571 23.4250 9.6184 9.1265 11.8291 21.4573 16.0471∗∗ 20 160.3003 27.5641∗∗ 14.5261∗∗ 10.7503∗∗ 13.8441∗∗ 22.1072∗∗ 19.9279∗∗ 58 7 Conclusions This paper has developed a general discrete-time modeling of the term struc- ture of interest rates able to take into account at the same time several important features : a) interest rates with an historical dynamics involving several lagged values and the present and past values of the (non homoge- neous) regime indicator function (zt ); b) a speciﬁcation of the exponential- aﬃne stochastic discount factor (SDF) with time-varying coeﬃcients imply- ing stochastic risk premia, functions of the present and past values of the factor (xt ) and the regime indicator function (zt ); c) the possibility to de- rive explicit or quasi explicit formulas for zero-coupon bond (the Generalized Linear Term Structure formula) and interest rate derivative prices; d) the positiveness of the yields at each maturity (in the Autoregressive Gamma framework), regardless the endogenous or exogenous nature of the factor (xt ). We have studied, in the Gaussian framework, the SARN(p) and the SVARN(p) Term Structure models, providing a generalization of the recent modelisation proposed by Dai, Singleton and Yang (2005). In the Autore- gressive Gamma setting, we have proposed the SARG(p) and the SVARG(p) Term Structure models, extending several discrete time CIR term structure models like Bansal and Zhou (2002). In the last section of the paper, using monthly observations on the U.S. term structure of interest rates, from June 1964 to December 1995, we have estimated several endogenous Gaussian (single-regime and regime-switching) Term Structure models, with scalar and bivariate factors. We have veriﬁed that, the goodness-of-ﬁt of these models improves when the autoregressive order increases, under both the historical and risk-neutral setting. 59 Appendix 1 Proof of Proposition 3 E[exp(u1 y1,t+1 + u2 y2,t+1 )| y1,t , y2,t ] = E exp(u2 y2,t+1 )E exp(u1 y1,t+1 )| y1,t , y2,t+1 | y1,t , y2,t = exp [a1 (u1 )(β11 y1,t + β12 y2,t ) + b1 (u1 )] Et (u2 + a1 (u1 )βo )y2,t+1 | y1,t , y2,t = exp [a1 (u1 )(β11 y1,t + β12 y2,t ) + b1 (u1 ) +a2 (u2 + a1 (u1 )βo )(β21 y1,t + β22 y2,t ) + b2 (u2 + a1 (u1 )βo )] = exp {[a1 (u1 )β11 + a2 (u2 + a1 (u1 )βo )β21 ]y1,t +[a1 (u1 )β12 + a2 (u2 + a1 (u1 )βo )β22 ]y2,t + b1 (u1 ) + b2 (u2 + a1 (u1 )βo )} . 60 Appendix 2 Proof of propositions 5 and 6 Proof of Proposition 5 : Let us ﬁrst consider an asset providing the payoﬀ exp(−θxt+1 ) at t + 1; the price at t of this asset is pt = Et [Mt,t+1 exp(−θxt+1 )] = exp −rt+1 − θν(Zt ) − θϕ(Zt ) Xt − 1 Γ(Xt , Zt )2 × 2 Et {exp [[Γ(Xt , Zt ) − θσ(Zt )] εt+1 ]} θ2 2 = exp −rt+1 − θν(Zt ) − θϕ(Zt ) Xt − θΓ(Xt , Zt )σ(Zt ) + 2 σ (Zt ) , and Et pt+1 = Et [exp(−θxt+1 )] = exp [−θν(Zt ) − θϕ(Zt ) Xt ] × Et {exp [[−θσ(Zt )] εt+1 ]} θ2 2 = exp −θν(Zt ) − θϕ(Zt ) Xt + 2 σ (Zt ) . Finally, from Deﬁnition 4, the risk premium is: ωt (θ) = θΓ(Xt , Zt )σ(Zt ) . Proof of Proposition 6 : Similarly, if we consider a digital asset providing one money unit at t + 1 if zt+1 = ej , we get: pt = Et [Mt,t+1 I(ej ) (zt+1 )] = exp[−rt+1 ] exp[−δj (Xt , Zt )]π (zt , ej ; Xt ) , and Et pt+1 = Et [I(ej ) (zt+1 )] = π (zt , ej ; Xt ) . Therefore, applying Deﬁnition 4, the risk premium is : ωt (θ) = δj (Xt , Zt ) . 61 Appendix 3 Proof of Proposition 7 The Laplace transform of the one-period conditional risk-neutral probability is: Q Et [exp(uxt+1 + v zt+1 )] 1 = Et {exp[Γ(Xt , Zt ) εt+1 − 2 Γ(Xt , Zt )2 − δ (Zt , Xt )zt+1 +u[ν(Zt ) + ϕ(Zt ) Xt + σ(Zt )εt+1 ] + v zt+1 ]} 1 = exp u[ ϕ (Zt )Xt + Γ(Xt , Zt )σ(Zt )] + uν(Zt ) + 2 u2 σ(Zt )2 × J j=1 π (zt , ej ; Xt ) exp [(v − δ(Zt , Xt )) ej ] 1 = exp u[ ϕ(Zt ) + γ (Zt )σ(Zt )] Xt + u[ν(Zt ) + γ(Zt )σ(Zt )] + 2 u2 σ(Zt )2 × ˜ J j=1 π (zt , ej ; Xt ) exp [(v − δ(Zt , Xt )) ej ] . Therefore, we get the result of Proposition 7. 62 Appendix 4 Proof of Proposition 8 B(t, h) = exp(Ch Xt + Dh Zt ) Q = exp (−rt+1 ) Et [B(t + 1, h − 1)] Q = exp [−c Xt − d Zt ] Et exp Ch−1 Xt+1 + Dh−1 Zt+1 = exp [−c Xt − d Zt ] × Q ˜ Et exp Ch−1 Φ∗ Xt + ν ∗ Zt + σ ∗ Zt ξt+1 e1 + D1,h−1 zt+1 + Dh−1 Zt = exp Φ∗ Ch−1 − c ˜ Xt + −d + C1,h−1 ν ∗ + 1 C1,h−1 σ ∗2 + Dh−1 2 Zt × 2 Q Et exp D1,h−1 zt+1 = exp Φ∗ Ch−1 − c Xt + ˜ −d + C1,h−1 ν ∗ + 1 C1,h−1 σ ∗2 + Dh−1 + F (D1,h−1 ) Zt 2 , 2 and the result follows by identiﬁcation. 63 Appendix 5 Proof of Proposition 9 Using the lag polynomials: 1 Ch (L) = − (C1,h + C2,h L + . . . + Cp,h Lp−1 ) h 1 Dh (L) = − (D1,h + D2,h L + . . . + Dp+1,h Lp ) h Ψ(L, Zt ) = 1 − ϕ1 (Zt )L − . . . − ϕp (Zt )Lp , we get from (54): R(t, h) = Ch (L)xt + Dh (L) zt , and Ψ(L, Zt ) R(t + 1, h) = Ch (L) Ψ(L, Zt ) xt+1 + Dh (L) Ψ(L, Zt ) zt+1 , = Dh (L) Ψ(L, Zt ) zt+1 + Ch (L) ν(Zt ) + Ch (L)[(σ ∗ Zt ) εt+1 ] . 64 Appendix 6 Proof of Proposition 11 The Laplace transform of the one-period conditional risk-neutral distribu- tion is : Q Et [exp(u xt+1 + v zt+1 )] ˜ ˜ 1 ˜ ˜ ˜ = Et {exp[Γ(Xt , Zt ) εt+1 − 2 Γ(Xt , Zt ) Γ(Xt , Zt ) − δ (Zt , Xt )zt+1 ν ˜ ˜ +u [˜(Zt ) + Φ(Zt )Xt + S(Zt )εt+1 ] + v zt+1 ]} ˜ ˜ ˜ ˜ 1 = exp u [ Φ(Zt )Xt + S(Zt )Γ(Xt , Zt )] + u ν (Zt ) + 2 u S(Zt )S(Zt ) u × J ˜ ˜ j=1 π(zt , ej ; Xt ) exp (v − δ(Zt , Xt )) ej ˜ ˜ ˜ ˜ 1 = exp u [ Φ(Zt ) + S(Zt )Γ(Zt , Xt )]Xt + u [ ν (Zt ) + S(Zt )γ(Zt )] + 2 u S(Zt )S(Zt ) u × ˜ J ˜ ˜ j=1 π(zt , ej ; Xt ) exp (v − δ(Zt , Xt )) ej . Therefore, we get the result of Proposition 11. 65 Appendix 7 Proof of Proposition 12 ˜ B(t, h) = exp(Ch Xt + dh Zt ) Q = exp (−rt+1 ) Et [B(t + 1, h − 1)] ˜ Q ˜ = exp −c Xt − d Zt Et exp Ch−1 Xt+1 + Dh−1 Zt+1 ˜ = exp −c Xt − d Zt × Q ˜ ˜ ∗ ∗ Et exp Ch−1 Φ∗ Xt + C1,h−1 (ν1 Zt + S1 (Zt )ξt+1 ) ∗ ∗ ˜ +Cp+1,h−1 (ν2 Zt + S2 (Zt )ξt+1 ) + D1,h−1 zt+1 + Dh−1 Zt = exp ˜ Φ∗ Ch−1 − c Xt + −d + C1,h−1 ν1 + 1 C1,h−1 (σ1 + ϕ∗2 σ2 ) ∗ 2 ∗2 ∗2 2 o ˜ + Cp+1,h−1 ν2 + 1 Cp+1,h−1 σ2 + Dh−1 + F (D1,h−1 ) Zt , ∗ 2 ∗2 2 and the result follows by identiﬁcation. 66 Appendix 8 Proof of Lemma 1 ρ(u + α) ρα ˜ ˜ a(u + α; ρ, µ) − a(α; ρ, µ) = − 1 − (u + α)µ 1 − αµ u = ρ (1 − αµ)2 − uµ(1 − αµ) ρ u = 2 1 − uµ (1 − αµ) 1−αµ ρ∗ u = = a(u; ρ∗ , µ∗ ) ; ˜ 1 − uµ∗ ˜ + α; ν, µ) − ˜ ν, µ) = −ν log(1 − (u + α)µ) + −ν log(1 − αµ) b(u b(α; 1 − (u + α)µ = −ν log 1 − αµ uµ = −ν log 1 − 1 − αµ = −ν log(1 − uµ∗ ) = ˜ ν, µ∗ ) . b(u; 67 Appendix 9 Proof of Proposition 13 B(t, h) = exp(Ch Xt + Dh Zt ) Q = exp [−c Xt − d Zt ] Et exp Ch−1 Xt+1 + Dh−1 Zt+1 ˜ ˜ = exp −c Xt − d Zt + Ch−1 Xt + Dh−1 Zt Q Et exp C1,h−1 xt+1 + D1,h−1 zt+1 ˜ ˜ = exp −c Xt − d Zt + Ch−1 Xt + Dh−1 Zt + A∗ (C1,h−1 ) Xt −ν ∗ Zt log(1 − C1,h−1 µ∗ ) + F (D1,h−1 )Zt , and the result follows by identiﬁcation. 68 Appendix 10 Proof of Proposition 14 The joint conditional Laplace transform of (x1,t+1 , x2,t+1 ) in the risk-neutral world is: Q Et [exp(u1 x1,t+1 + u2 x2,t+1 ) | x1,t , x2,t , zt ] ˜ = exp A2,t [u2 + Γ2t + a1,t (u1 + Γ1t )] Xt + b2,t (u2 + Γ2t + a1,t (u1 + Γ1t )) ˜ + A1,t (u1 + Γ1t ) Xt + b1,t (u1 + Γ1t ) ˜ − A2,t (Γ2t + a1,t (Γ1t )) Xt − b2,t (Γ2t + a1,t (Γ1t )) ˜ − A1,t (Γ1t ) Xt − b1,t (Γ1t ) . Using lemma 2 we get: A2,t [u2 + Γ2t + a1,t (u1 + Γ1t )] − A2,t (Γ2t + a1,t (Γ1t )) = A [u2 + a1,t (u1 + Γ1t ) − a1,t (Γ1t ); ϕ∗ , µ∗ ] , 2t 2t with ϕ2t ϕ∗ = 2t {1 − [Γ2t + a1,t (Γ1t )]µ2t }2 µ2t µ∗ 2t = , {1 − [Γ2t + a1,t (Γ1t )]µ2t } and using lemma 1 A [u2 + a1,t (u1 + Γ1t ) − a1,t (Γ1t ); ϕ∗ , µ∗ ] 2t 2t = A [u2 + a(u1 + Γ1t ; ϕot , µ1,t ) − a(Γ1t ; ϕot , µ1,t ); ϕ∗ , µ∗ ] ˜ ˜ 2t 2t = A u2 + a(u1 ; ϕ∗ , µ∗ ); ϕ∗ , µ∗ ˜ ot 1,t 2t 2t = A∗ [u2 + a∗ (u1 )] (say) 2,t 1,t 69 with ϕot ϕ∗ = ot (1 − Γ1t µ1t )2 µ1t µ∗ = 1t . (1 − Γ1t µ1t ) Similarly, we get: b2,t [u2 + Γ2t + a1,t (u1 + Γ1t )] − b2,t (Γ2t + a1,t (Γ1t )) = ˜ u2 + a(u1 ; ϕ∗ , µ∗ ); ν2t , µ∗ b ˜ ot 1,t ∗ 2t = b∗ [u2 + a∗ (u1 )] (say) , 2,t 1t b1,t (u1 + Γ1t ) − b1,t (Γ1t ) = ˜1 (u1 ; ν1t , µ∗ ) b ∗ 1t = b∗ (u1 ) (say) , 1,t A1,t (u1 + Γ1t ) − A1,t (Γ1t ) = A1 (u1 ; ϕ∗ , µ∗ ) 1t 1t = A∗ (u1 ) (say) , 1,t with ϕ1t ϕ∗ = 1t . (1 − Γ1t µ1t )2 And ﬁnally, the joint conditional Laplace transform of (x1,t+1 , x2,t+1 ) be- comes: Q Et [exp(u1 x1,t+1 + u2 x2,t+1 ) | x1t , x2t , zt ] ˜ = exp [A∗ (u1 ) + A∗ [u2 + a∗ (u1 )]] Xt 1,t 2,t 1,t + b∗ [u2 + a∗ (u1 )] + b∗ (u1 ) , 2,t 1,t 1,t and the result of Proposition 14 is proved. 70 Appendix 11 Proof of Proposition 15 ˜ B(t, h) = exp(Ch Xt + Dh Zt ) ˜ Q ˜ = exp −c Xt − d Zt Et exp Ch−1 Xt+1 + Dh−1 Zt+1 ˜ ˜ ˜ ˜ = exp −c Xt − d Zt + Ch−1 Xt + Dh−1 Zt Q Et exp C1,h−1 x1,t+1 + Cp+1,h−1 x2,t+1 + D1,h−1 zt+1 ˜ ˜ ˜ ˜ ˜ = exp −c Xt − d Zt + Ch−1 Xt + Dh−1 Zt + A∗ (C1,h−1 ) Xt 1 ∗ ˜ −ν1 Zt log(1 − C1,h−1 µ∗ ) + A∗ [Cp+1,h−1 + a∗ (C1,h−1 )] Xt 1 2 1 ∗ −ν2 Zt log[1 − (Cp+1,h−1 + a∗ (C1,h−1 ))µ∗ ] + F (D1,h−1 )Zt , 1 2 and the result follows by identiﬁcation. 71 REFERENCES Ahn, D., Dittmar, R., and R. Gallant (2002) : ”Quadratic Term Struc- ture Models : Theory and Evidence”, Review of Financial Studies, 15, 243- 288. Ang, A., and G. 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