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Discrete-time affine term structure: an econometric formulation Dean Fantazzini∗ Mario A. Maggi† Abstract Discrete-time Affine Term Structure Models can be expressed in AR (1)-ARCH form, but it is not possible to get a non-negative variance equations simply by restricting the parameters. In this paper we resort to a distribution assumption in order to assure the variance to be non-negative. We present a complete formulation for one-factor and multi-factor models with Gamma conditional noise distribution. This way we get a well defined volatility process that avoids any problem both in generating processes and in computing the conditional likelihoods of observations. The log-likelihood function is derived for one- and multi-factor specifications. Moreover, we implement a one-factor estimation both with simulated and US interest rate data. Finally, we compare the estimation results with a standard ATSM with Gaussian disturbances. Keywords: Discrete-time Affine Term Structure Models, ARCH, VAR, Maximum Likelihood Estimation JEL Classification: C32, C51, E43. ∗ PhD student in Economics, Dipartimento di Economia Politica e Metodi Quantitativi, Università di Pavia, [email protected] † Researcher in Mathematical Finance, Dipartimento di Ricerche Aziendali, Università di Pavia, [email protected] 1 1 Introduction Affine term structure models (ATSM) constitute a wide family of term structure models. ATSM are characterized by a defaultfree zero coupon bond (ZCB) evaluation function P which is exponentially linear: Pt (τ ) = exp {A (τ ) − B (τ ) Xt } (1) being: t the current valuation date, τ the time to maturity, A = A (τ ) a real function, B = B (τ ) ∈ IRn a vector function and Xt ∈ IRn a vector collecting the n stochastic risk factors. As Dai and Singleton (2000) clearly show, ATSM constitute a wide class of models encompassing the majority of the most popular and standard models (for instance, the Vasiček (1977) and the Cox, Ingersoll and Ross (1985) models are special cases). In many works the model chosen is a continuous-time one, e.g. see Dai and Singleton (2000), Duarte (2004), Duffee (2002), Duffie and Kan (1996). A strong contribution to the success of continuous-time ATSM comes from the tractability of their pricing formulas and their dynamics, together with the actual possibility to use them empirically. However, multi-factor continuous-time models are quite demanding in terms of estimation procedures. In fact, out of some special cases, ATSM do not admit general closed forms for the likelihood function and for the distribution function moments of bond prices, yields and factors. This leads to use estimation techniques such as the quasimaximum likelihood, the efficient and the simulated methods of moments, for example see Berardi (1995), Chen and Scott (1993), Dai and Singleton (2000), Duffee and Stanton (2000), Duffie and Singleton (1997), Fisher and Gilles, (1996), Liu (1997), Liu and Pan (1997), and Singleton (2001). Discrete-time models are used as well. For instance, Backus 2 et al. (1998) show some simple discrete-time one- and multifactor affine models estimations. Ang and Piazzesi (2003) use a discrete-time VAR model for the term structure. Rudenbusch and Wu (2004) deal with a VAR(1) models that can easily be expressed as a Gaussian ATSM. Ichiue (2003) employs a Gaussian VAR ATSM. In general, it seems that discrete-time models are well suited for term structure models that uses macroeconomic variable as factors. In fact, even if market data can be sampled at different frequencies from low to very high (intraday) frequency, macroeconomic variable are typically monthly data. For this reason, a discrete-time setting may be convenient for macro-finance term structure models. Moreover, their estimation is quite affordable. As Backus et al. (1998) remark, discrete time version of ATSM processes can take negative value with a positive probability. In non-Gaussian affine processes, the factors linearly enter the volatility equation making possible that the model breaks down when a negative volatility value appears. The probability of this event can be kept small enough, but it still remains positive. This fact can produce emergency stops in simulating algorithms or in likelihood computation. In this paper we propose a way out of this stalemate. We model the disturbances of the volatility factor with non-central Gamma distributions with a suitable conditional lower bound, that imply the volatility processes to be non-negative with probability one. We begin from one-factor models where it is simple to set up our model. We estimate a one-factor model with US interest rates data, finding that it behaves well, but it does not really improves over a standard one-factor ATSM. Indeed, our proposal is useful for multi-factor models, where it allows to exploit much more flexibility in describing the volatility and the correlations than Gaussian VAR models or independent factors affine models. In 3 fact, it is possible to adapt to discrete-time the Dai and Singleton (2000) specification that splits the factor vector into volatility factors and other factors. The paper proceeds as follows. Section 2 presents the discretetime formulation of an ATSM in a general AR(1)-ARCH form. In section 3 we introduce the Gamma disturbances hypothesis for one-factor models. These simple models allows to better understand the implications of our assumption. The likelihood function is derived and used in estimation in section 3.1. Section 3.2 applies to US interest rate data the procedures pointed out in section 3.1. Section 4 deals with multi-factor models combining the results obtained for one-factor models with the Dai and Singleton (2000) general affine formulation. The likelihood function is derived for multi-factor processes too. Section 5 concludes. 2 Discrete-time ATSM The dynamics of prices and interest rates in financial markets are driven by micro- and macro-economic variables, such as the expectations, risk attitude and feelings of economic agents, decisions of political economy. We consider the general case of an economy driven by n random factors or state variables stacked in a column-vector Xt ∈ IRn , with t ∈ {0, 1, . . . , T }. We assume the absence of arbitrage and frictions (no transaction costs, bid-ask spread or any type of frictions are in place) on the fixed income market. An n-factor discrete-time ATSM is described by the relations Xt = Xt−1 + K (Θ − Xt−1 ) + εt , t = 1, 2, . . . , T, εt ∼ f [0] , Σ2t , white noise, Σ2t = diag (αi + βi Xt−1 ) , i = 1, . . . , n, 4 (2) (3) (4) with Xt , Θ, εt ∈ IRn , βi ∈ IRn , K, Σ2t ∈ IRn×n , αi ∈ IR and [0] the null vector. We denote Ωt the information set available at time t, i.e. Ωt = {Xz , εz , z = 0, 1, . . . , t}. The relevance of ATSM both in discrete and in continuous time stems from the computational tractability of the zero coupon pricing function. In fact, the zero coupon bond (ZCB) prices are given by the formula Pt (τ ) = exp (Aτ + Bτ Xt ) , (5) where τ = τ1 , τ2 , . . . , τm is the time to maturity. The yield-tomaturity, or simply the spot rate, between t and (t + τ ) is rtτ = − Aτ + Bτ Xt . τ The rate with shortest maturity is the short rate rt = rtτ1 . It can be shown that formula (5) can be recursively derived from a simple difference system (e.g., see. Rudebusch and Wu (2004) for the details). In this paper we deal with the dynamics (2) of risk factors under the historical probability measure. In the following section we present some result for one-factor models and then we extend them to multi-factor discrete-time ATSM. 3 One-factor ATSM Owing to the linear relationship among different maturities rates (e.g., see Duffie and Kan (1996)), in a one-factor ATSM the factor can always be identified with the short rate, so relations (2)– 5 (4) become rt = rt−1 + k (θ − rt−1 ) + εt , εt ∼ f 0, σt2 , white noise, σt2 = α + βrt−1 . t = 1, 2, . . . , T (6) (7) (8) that does not look like a GARCH model. Equation (6) defines an AR (1) process and can be written in a more common way: rt = kθ + (1 − k) rt−1 + εt , t = 1, 2, . . . , T. The volatility process depends on lagged interest rates. Provided that 0 < k < 1, the MA (∞) form of {rt } rt = θ + +∞ X j=0 (1 − k)j εt−j ; moreover, is convenient to write the volatility process {σt2 } as a MA (∞): " # +∞ X (1 − k)j εt−j−1 , σt2 = α + β θ + j=0 with a null weight on the current disturbance. Writing the first term out of the summation, we get σt2 = α + βθ + βεt−1 + β (1 − k) +∞ X j=0 (1 − k)j εt−j−2 . Also the lagged volatility has a MA (∞) representation " # +∞ X j 2 σt−1 =α+β θ+ (1 − k) εt−j−2 . j=0 6 (9) Therefore, relation (9) can be written as β +∞ X j=0 2 (1 − k)j εt−j−2 = σt−1 − α − βθ, so it follows that σt2 has the following AR (1) representation 2 σt2 = k (α + βθ) + (1 − k) σt−1 + βεt−1 . (10) This representation is useful in order to analyze the condition for the non-negativity of the conditional variance of noise. Although equation (10) is quite similar to a GARCH (1, 1) volatility process, the presence of lagged noise, which is not squared, makes more difficult to find non-negativity condition. The idea we present is simple. If the noise process is Gaussian with β 6= 0, equation (10) has no chance to produce non-negative variances with probability one (see also Backus et al. (1998)). So, there exists a noise conditional distribution assuring that the conditional variance is always non-negative? A simple answer is to choose a distribution for εt−1 |Ωt−2 whose support is suitably bounded from below. Equation (10) suggests that this bound can depend on Ωt−2 : k (α + βθ) 1 − k 2 min {εt−1 |Ωt−2 } = a ≧ − + σt−1 , β β this can be written as k (α + βθ) 1 − k min {εt |Ωt−1 } = a ≧ − + var (εt |Ωt−1 ) , β β (11) for t = 1, 2, . . . T, 7 that is, the lower bound linearly depends on the conditional variance. Moreover, the noise must have mean zero and variance σt2 E [εt |Ωt−1 ] = 0, var (εt |Ωt−1 ) = σt2 . (12) (13) There are various distribution with support bounded from below, for example the χ2 , the Beta, the Gamma, the Exponential, the Weibull distributions. Let we suppose that the noise conditional distribution is a non-central gamma with parameters λt > 0 and qt > 0. We choose this distribution because it has enough parameters whose value can be chosen to match conditions (11) to (13), and its moments have easy expressions that do not introduce computational burdens for estimation. Consider the random variable z ∼ G (λ, q), with λ > 0, q > 0. Its density is fz (z) = λq q−1 −λz z e , z ≧ 0, Γ (q) and its mean and variance are E [z] = q q , var (z) = 2 . λ λ In order to get a zero mean distribution, we need that our variable takes negative values too. However, we need a lower bound a, so we displace the random variable z by the amount a: x=z+a λq (x − a)q−1 e−λ(x−a) , x ≧ a, Γ (q) q q E [x] = + a, var (x) = 2 . λ λ fx (x) = 8 We now come back to conditions (11), (12) and (13) which become (remark that we consider condition (11) with the equality sign) h i k(α+βθ) 1−k 2 a = − + σ t t β β qt (14) + at = 0 λqtt = σ 2 t λ2 t From system (14) it is possible to solve for the values of a, λ and q: h i k(α+βθ) 1−k 2 a = − + σ t β β t at λt = − σ 2 t q = a2t2 t σt that is, the parameters of the noise distribution at time t are known functions of the model parameters and of σt2 ∈ Ωt . Summing up, we can formulate the one-factor discrete-time ATSM model as an “affine Γ-ARCH (1, 1)” model: rt = kθ + (1 − k) rt−1 + εt 2 var (ε t ) = σt 2 2 k (α + βθ) + (1 − k) σt−1 + βεt−1 , if β 6= 0, σt = α, otherwise εt − at ∼ h G (λt , qt ) i at = − k(α+βθ) + 1−k σ 2 , λt = − a2t , qt = a2t2 . t β β σt σt (15) For example, a set of parameter values that can mimic a short rate dynamics is k = 0.025, θ = 5, α = 0.002, β = 0.1. (16) A (little) flaw is that the probability of having a negative rate is positive. Some MC simulations with parameter values (16) 9 have shown that this probability is extremely low: with Gaussian disturbances P [rt < 0] ≃ 1.76 × 10−4 , whereas with mean zero Gamma disturbances P [rt < 0] ≃ 0, 1 . However, this problem can be mended by means of the same idea exploited before: choose at suitably as follows k (α + βθ) 1 − k 2 at = − min + σt , rt−1 (17) β β and the short rate will be non-negative with probability one. Equation (17) will be useful in section 4 in order to ensure nonnegative volatility equation for some factor. Given a set of parameter values Ξ = (k, θ, α, β), the conditional distribution of an observation is [(rt |rt−1 , Ξ) − kθ − (1 − k) rt−1 − at ] ∼ G (λt , qt ) , so its conditional likelihood is f (rt |rt−1 , Ξ) = λq t [rt − kθ − (1 − k) rt−1 − at ]qt −1 Γ (qt ) · e−λt [rt −kθ−(1−k)rt−1 −at ] . The sample likelihood and log-likelihood are, respectively, T Y λq t [rt − kθ − (1 − k) rt−1 − at ]qt −1 L (r̄, Ξ) = f (r0 ) Γ (qt ) t=1 · e−λt [rt −kθ−(1−k)rt−1 −at ] , ln L (r̄, Ξ) = ln f (r0 ) + T X t=1 qt ln λt − ln Γ (qt ) + (qt − 1) ·ln [rt − kθ − (1 − k) rt−1 − at ]−λt [rt − kθ − (1 − k) rt−1 − at ] , 1 In different 106 time-steps simulated processes a negative rt never occurred. 10 model ATSM k̂ θ̂ 0.0291181 4.8496 (0.0000) Γ-ARCH true model: Γ-ARCH 0.025 β̂ 0.0255651 0.09082 (0.0000) (0.0654) 0.0290948 4.8497 (0.0000) α̂ (0.0000) 0.0247342 0.09130 (0.0000) (0.0469) (0.0000) 5 0.1 0.002 Table 1: Maximum likelihood parameters’ estimates, (p-values). Sample: T = 2000 time-steps generation of a Γ−GARCH porcess. where [rt − kθ − (1 − k) rt−1 ] is the conditional innovation (εt |Ωt−1 , Ξ). In order to calculate it recursively we need the initial value of the short rate r0 and its volatility σ02 .2 3.1 A test estimation We perform some preliminary test of the ML estimation. We estimate the parameters from some MC simulation of the Γ-ARCH process. We compare the results obtained from two different specifications: the Γ-ARCH and the ATSM ones. Tables 1, and 2 display the results. 2 In our algorithms we equal r0 to the first data and σ02 to the variance unconditional mean given the parameter set. 11 corresponding parameters GARCH(1, 1) parameters in Γ-ARCH ĉ = 0.11756 (0.028222) kθ = 0.125 φ̂ = 0.96885 (0.0053405) (1 − k) = 0.975 ω̂ = 0.011499 ϕ̂ = 0.89524 (0.0033932) k (α + βθ) = 0.1255 (0.01579) ψ̂ = 0.081588 (0.012067) (1 − k) = 0.975 β = 0.1 log lik. = −1978.1 Table 2: GARCH (1, 1) estimation (standard errors). Sample: T = 2000 time-steps generation of a Γ-GARCH porcess. We estimate also a standard AR (1)-GARCH (1, 1) model rt = c + φrt−1 + ǫt , ǫt ∼ N (0, σt2 ) , var (ǫt |Ωt−1 ) = σt2 , (18) 2 σt2 = ω + ϕσt−1 + ψǫ2t−1 . In table 2 the ML estimate are compared to parameters values (remark that in this model the innovation is squared, so its coefficient ψ can not match β). 12 model ATSM Γ-ARCH k̂ θ̂ α̂ β̂ 0.0039782 5.6421 −0.038377 0.042987 (0.3699) (0.2522) (0.0000) (0.0000) 0.0044149 5.6459 −0.039794 0.044950 (0.3406) (0.2226) (0.0000) (0.0000) Table 3: Maximum likelihood parameters’ estimates, (p-values). Sample: monthly US three months T-bill rates, from 1962:1 to 2005:2. 3.2 An application to US interest rates The maximum likelihood algorithm previously tuned up with the artificial data presented in section 3.1 is now applied to actual interest rate data. The data-set contains monthly sampled 3-months T-bill rates from 1962:1 to 2005:2. Table 3 reports the maximum likelihood estimate of a standard ATSM with Gaussian innovation and our Γ-ARCH specification. The p-values of the AR(1) coefficient are quite high, so we performed the likelihood ratio tests on each parameter for the two specifications. All tests reject the null of zero parameter values3 . The results obtained show that the Γ-ARCH specification works slightly better than a standard ATSM. However, the improvement is so weak that can not justify the use of a Γ-ARCH. Anyway, this application allows us to get confidence on our model so we can build up the multi-factor specification we will present next. 3 The results are not reported but they are available from the authors’. 13 4 Multi-factor discrete-time ATSM It is well documented that term structure dynamics might be modelled accounting for more than one factor (e.g., see Litterman and Scheinkman (1991)). Principal component analysis show that two or three factors are enough to describe the term structure dynamics. For this reason, in this section we turn to multi-factor models. Our assumption on the innovation distribution allows us to fully exploit in a discrete-time setting the potential flexibility of different specification of multi-factor ATSM. We come back to the general multi-factor ATSM presented in section 1. The Dai and Singleton (2000) and Rudebusch and Wu (2004) parameters structure can help getting a tractable specification in discrete-time too. The n factors of the model (2)(4) should be divided into two groups: the first m ≦ n that we call volatility factors XtB ∈ IRm , and the others n − m factors XtD ∈ IRn−m . The parameters are partitioned accordingly so that (2)-(4) can be written in the following form: for t = 1, 2, . . . , T, BB B B [0] Θ Xt K = + ΘD K DB K DD XtD BB B B [0] Xt−1 εt K + I − + εt , DB DD D D K K Xt−1 εt εt ∼ g [0] , Σ2t , white noise, Σ2t = diag (αi + βi Xt−1 ) , i = 1, . . . , n, 14 with K BB diagonal, βi = βii 1i , βii ∈ IR, 1 ≦ i ≦ m, βi = βi1 · · · βim [0] ≧ [0] , α ≧ [0] , m + 1 ≦ i ≦ n, i (19) m + 1 ≦ i ≦ n, where 1i is the i-th basis (row) vector. This way, the factor dynamics are Xit = Kii Θi + (1 − Kii ) Xit−1 + εit 1≦i≦m: εit ∼ fiB (0, σit2 ) σ2 = α + β X i ii it−1 it Xit = Ki Θ + (1i − Ki ) Xt−1 + εit m+1≦i≦n: εit ∼ fiD (0, σit2 ) 2 B σit = αi + βi1 · · · βim Xit−1 where Ki stands for the i-th row of K. Thus, the volatility factors are m independent univariate processes like (6)-(8), whereas the other factors are correlated among them and with the volatility factors, moreover, the volatility of the non-volatility factors linearly depends on the volatility factors. If m = 0, i.e. the model is a Gaussian VAR(1), then K has to be lower triangular and the volatility functions become constant. For volatility factors, we recall the univariate non-negativity argument, so we assume that εit |Ωt−1 has a non central Gamma 15 distribution of the kind (15) with Xit = Kii Θi + (1 − Kii ) Xit−1 + εit var (εit ) = σit2 σ 2 = Kii (αi + βii Θi ) + (1 − Kii ) σ 2 it it−1 εit − ait ∼ G (λit , qit ) nh Kii (αi +βii Θi ) a = − min + it βii λit = − ait2 , qit = a2it2 . σ σ it (1−Kii ) 2 σit βii + βii εit−1 i o , Xit−1 , it In a multi-factor model, the non-negativity condition (17) for the volatility factors XtB become necessary in order to guarantee that restrictions (19) produce volatilities that are well defined positive processes for the non-volatility factor XtD as well. No restriction on the distribution of innovations of non-volatility factors innovations are needed. In the rest of the paper we assume that εit |Ωt−1 is Gaussian for m + 1 ≦ i ≦ n. It is straightforward to compute the likelihood function of the multi-factor discrete-time ATSM presented in this paper. In fact the conditional innovation (εt |Ωt−1 ) = Xt − KΘ − (I − K) Xt−1 , has independent components whose distribution is Gamma for the first m components, and Gaussian for the others n − m. The likelihood of an observation, given the set of parameter values Ξ, is n Y L (Xt |Ξ) = Li (εit |Ωt−1 , Ξ) , i=1 16 where λqit [Xit − Kii Θi − (1 − Kii ) Xit−1 − ait ]qit −1 · Γ (qit ) −λit [Xit −Kii Θi −(1−Kii )Xit−1 −ait ] ·e , for 1 ≦ i ≦ m, Li (εit |Ωt−1 , Ξ) = and Li (εit |Ωt−1 , Ξ) = σit 1 √ (Xit − Ki Θ − (1i − Ki ) Xt−1 ) exp − 2σit2 2π 2 ! , for m + 1 ≦ i ≦ n. Therefore the sample log-likelihood is T X n X ln L X̄|Ξ = ln f (X0 ) + Li (εit |Ωt−1 , Ξ) . t=1 i=1 Obviously, in order to recursively calculate the initial values X0 and σ02 are needed. A common issue in multi-factor models is the growing number of parameters to estimate. So it is worth to remark that the number of free parameters in the multi-factor specification we have just presented is: 4m (n − m) (n − m) n2 − m2 + 2 (n + m) volatility factors (like m one-factor models) n Ki for each non1 αi volatility factor: m βi1 · · · βim ΘD total parameters 17 In most fixed income analysis (see also Litterman and Sheinkman (1991)) the number of risk factors does not exceed 4. Therefore, for discrete-time ATSM up to 4 factors the number of parameters is: n\m 0 1 2 3 4 1 3 4 − − − 2 8 9 8 − − 3 15 16 12 12 − 4 24 25 24 21 16 This means that the models we present are affordable both from the computational and from the identification point of view. In fact, the estimation time can be kept reasonable. 5 Conclusions The model we have presented in this paper is intended to be an operative model which is developed paying attention to its econometric implementation. We have extended to discrete-time the idea of splitting the state vector in order to differently deal with volatility and non-volatility factors. This idea was successfully introduced in continuous-time models by Dai and Singleton (2000) and has been extensively applied for financial market study and asset pricing. We think, however, that discrete-time models may be a better choice for analyzing the low-frequency data typical in macro-finance literature, where the macroeconomic variables can not be sampled more than once a month. Simulations and empirical analysis with US interest rate data 18 show that our affine Γ-ARCH model can be a valuable tool for financial applications, since the volatility processes are nonnegative with probability one. Therefore, the main benefits are to avoid any emergency stops in simulating algorithms or in likelihood function computations. Moreover, our model can also preserve the flexibility allowed by multi-factor ATSM. 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