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Efﬁcient Estimation of Markov Models Where the Transition Density is Unknown George J. Jiang∗ and John L. Knight† First version: March 2001 This Version: December 2001‡ ∗ George J. Jiang, Finance Department, Eller College of Business & Public Administration, University of Arizona, P.O. Box 210108, Tucson, Arizona 85721-0108 and Finance Area, Schulich School of Business, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3. E-mail: gjiang@eller.arizona.edu. George J. Jiang is also a SOM research fellow at the University of Groningen in The Netherlands. † John L. Knight, Department of Economics, University of Western Ontario, London, Ontario, Canada, email: jknight@julian.uwo.ca. ‡ We wish to thank Alan Rogers for helpful comments along with participants at the New Zealand Economet- ric Study Group, Auckland, March 2001. Both authors acknowledge ﬁnancial support from NSERC, Canada. The usual disclaimer applies. Efﬁcient Estimation of Markov Models Where the Transition Density is Unknown Abstract In this paper we consider the estimation of Markov models where the transition den- sity is unknown. The approach we propose is the empirical characteristic function (ECF) estimation procedure with an approximate optimal weight function. The approximate optimal weight function is obtained through an Edgeworth/Gram-Charlier expansion of the logarithmic transition density of the Markov process. Based on the ECF estimation procedure, we derive the estimating equations which are essentially a system of mo- ment conditions. When the approximation error of the optimal function is arbitrarily small, the new estimation approach, which we term as the method of system of moments (MSM), leads to consistent parameter estimators with ML efﬁciency. We illustrate our approach with examples of various Markov processes. Monte Carlo simulations are per- formed to investigate the ﬁnite sample properties of the proposed estimation procedure in comparison with other methods. JEL Classiﬁcation: C13, C22, C52, G10 Key Words: Markov Process, Efﬁcient Estimation, Empirical Characteristic Function (ECF), Method of System of Moments (MSM), Edgeworth/Gram-Charlier Expansion 1 Introduction The estimation of Markov models, in either continuous or discrete time, can proceed straight- forwardly via the maximum likelihood (ML) estimation method based on the observations of state variables when the transition density is known in closed form. However, when the transition density is unknown alternative estimators to ML estimation need to be considered. Various estimation methods have been proposed and applied in the literature, e.g. the quasi- maximum likelihood (QML) method by Fisher and Gilles (1996) among others, simulation based methods such as the simulated moments estimator (SME) by Dufﬁe and Singleton (1993), the indirect inference approach by Gouriéroux, Monfort and Renault (1993) and the efﬁcient method of moments (EMM) by Gallant and Tauchen (1996) and Gallant and Long (1997) with applications by Andersen, Benzoni, and Lund (2001) and Chernov and Ghy- sels (2000) among others, and the generalized method of moments (GMM) by Hansen and Scheinkman (1995), Liu (1997), Jiang (2000) and Pan (2000), and the C-GMM by Carrasco and Florens (2000a) that extends the GMM procedure to the case of a continuum of mo- ment conditions, as well as the Bayesian method by Jones (1997), etc. A new approximate maximum likelihood (AML) approach to the estimation of continuous-time diffusion mod- els is developed recently in Aït-Sahalia (1999, 2000). The basic idea is to approximate the transition density, after a suitable transformation of the diffusion process, by deriving a se- quence of Hermite approximations to the transition density function and then to maximize the approximate log-likelihood function, which is given by the sum of the logarithms of these approximate densities. A number of estimation methods have also been developed in the frequency domain ex- ploiting the analytical characteristic functions of the state variables. It is noted that, for many Markov processes, while a closed form of the transition density is not available, the associated conditional characteristic function (CCF) of the state variables can often be de- 1 rived analytically. For instance, in the continuous time ﬁnance literature, models speciﬁed within the afﬁne framework often have closed form CCF. This observation opens the door to alternative estimation methods using the characteristic functions. In this regard we have the estimators developed in Chacko and Viceira (1999), Singleton (2001), Jiang and Knight (2001) and Carrasco, Chernov, Florens and Ghysels (2001) for continuous time diffusion and jump-diffusion models. In these approaches the models are estimated by minimizing some weighted distance between the empirical characteristic function (ECF) or joint ECF and their theoretical counterparts. These methods are essentially in the framework of GMM, albeit may involve a continuum of conditional or unconditional moment restrictions. Single- ton (2001) also proposes to proceed with estimation using the likelihood function obtained by Fourier inversion of the conditional characteristic function of the state variables. The question of efﬁciency is paramount in these alternative estimation techniques and it can be shown that those based on the ECF and CCF can have efﬁciency arbitrarily close to ML estimator. However, in the case of the estimator proposed in Singleton (2001) the efﬁciency is related to the optimal choice of the discrete grid of points at which the ECF and CCF are matched with the weight function being the covariance of the ECF. While in the univariate case the choice of these points has received some attention, see Feuerverger and Mureika (1977), Schmidt (1982), Knight and Satchell (1997), Yu (1998), and Singleton (2001), in the multivariate case the choice is indeed an open question. To overcome some of these problems, Jiang and Knight (2001) propose minimizing the integrated MSE (IMSE) between the empirical characteristic function (ECF) and its theoretical counterpart. This thus avoids the choice of points at which the matching is done. The optimal weight function used in the estimation, however, is no longer readily available. The approach we propose in this paper combines the ECF technique along with an ap- proximate optimal weight function. The basic idea stems from the fact that, due to the one- one correspondence between the CCF and the transition density, the ﬁrst-order conditions 2 for ML estimation can indeed be written as the sum of a weighted integral of the difference between ECF and CCF. The optimal weight in this set-up is the inverse Fourier transform of the tth score. Thus by approximating the logarithm of the transition density we can approx- imate the optimal weight function and hence solve the integral to develop the appropriate estimating equations, the solution of which leads to the approximate ML estimator. The es- timation equations turn out to be in the form of a system of moment conditions, which is different from the conventional method of moments (MM) or the generalized method of mo- ments (GMM). We name this new estimation approach as the method of system of moments or MSM. In addition to being exact and parsimonious, the method leads to consistent and asymptotically efﬁcient parameter estimators. When the approximate error of the optimal weight function is arbitrarily small, the asymptotic efﬁciency of the estimator approaches that of ML estimator. The paper is organized as follows. Section 2 will develop the ECF estimation approach with approximate optimal weight functions, detailing the expansion for the transition density for an N-dimensional Markov process and deriving the appropriate estimating equations. In a series of remarks we relate our approach to various existing estimation methods, such as GMM, QML, and the approximate ML estimator of Aït-Sahalia (1999, 2000). We show that the estimating equations are in the form of a system of moment restrictions and specialize the equations explicitly in the univariate case. In Section 3 we illustrate the application of our approach with examples of various Markov processes in both discrete time and continuous time and for both univariate and multivariate cases. In section 4, we perform Monte Carlo simulations to investigate the ﬁnite sample properties of the proposed estimation procedure in comparison with other methods for selected models. A brief conclusion is contained in Section 5 along with some ideas for further research. All proofs are collected in the Appendix. 3 2 ECF Estimation and the Method of System of Moments (MSM) Let xt ∈ RN , t > 0, be a N-dimensional stationary Markov process deﬁned in either a dis- crete time set or a continuous time set with a complete probability space (Ω, F, P ). Suppose that {xt }T +1 represents an observed sample over a discrete set of time from the Markov pro- t=1 cess. Let f (xt+1 |xt ; θ) : RN × RN × Θ → RN denote the measurable transition density for the Markov process and θ ∈ Θ ⊂ RK denote the parameter vector of the data generating process for xt . Following Singleton (2001), a consistent estimator of the parameters based on the empirical characteristic function (ECF) is given by 1 XZ T 0 w(θ, t, r|xt )(eir xt+1 − φ(r, xt+1 |xt ))dr = 0 (1) T t=1 where φ(r, xt+1 |xt ) = E[exp(ir0 xt+1 )|xt ] is the conditional characteristic function (CCF) and w(θ, t, r|xt ) ∈ W (θ, t, r|xt ) with W (θ, t, r|xt ) being the set of “instrument" or “weight" functions as deﬁned in Singleton (2001). Namely, for each “instrument" or “weight" func- tion w(θ, t, r|xt ) : Θ × R+ × RN × RN → CK where CK denotes the complex numbers, we − have w(θ, t, r|xt ) ∈ It and w(θ, t, r|xt ) =w (θ, t, −r|xt ), t = 1, 2, ..., T + 1, where It is the σ-algebra generated by xt . The ECF estimation procedure has been proposed by Feuerverger and Mureika (1977), Schmidt (1982), and Feuerverger and McDunnough (1981b) for i.i.d. cases and Feuerverger and McDunnough (1981b) and Feuerverger (1990) for generic sta- tionary Markov processes using the joint ECF of state variables. As shown in Feuerverger (1990), Singleton (2001) and Jiang and Knight (2001), there exists an optimal “instrument" or “weight" function in the sense that the estimator deﬁned in (1) is also an efﬁcient estimator with the same asymptotic properties as ML estimator. We summarize these results in the following lemma. Lemma 1 Under standard regularity conditions where w(θ, t, r|xt ) ∈ W (θ, t, r|xt ) is a 4 well-deﬁned “instrument” or “weight” function 1 , equation (1) leads to consistent parameter estimators. Furthermore, let f (xt+1 |xt ) be the transition density of the Markov process, the following weight function is optimal 1 Z Z ∂ ln f (xt+1 |xt ) −ir0 xt+1 w(θ, t, r |xt ) = .. e dxt+1 (2) (2π)N ∂θ in the sense that equation (1) will result in parameter estimator with ML efﬁciency. Proof: See appendix: It is noted that the optimal weight function is determined by the logrithm of the tran- sition density or likelihood function of the Markov process. When f (xt+1 |xt ) is explicitly known, the Markov model can be estimated straightforwardly via ML method and it can be shown that the estimation equation (1) will result exactly in conditional ML estimator. However, if f (xt+1 |xt ) is not known explicitly but φ(r, xt+1 |xt ) is, then (1) must be imple- mented with other than the optimal weight function w(θ, t, r|xt ). The solution proposed in Singleton (2001) is to ﬁrst approximate the integral in (1) with a sum over a discrete set of r and then appeal to the GMM theory to ﬁnd the approximate optimal weight function. As shown in both Feuerverger (1990) and Singleton (2001), under certain regularity conditions as the grid of r0 s becomes increasingly ﬁne in RN , the asymptotic efﬁciency of the estimator approaches that of ML estimator. The appeal to GMM to ﬁnd the optiomal weight matrix in essence is a GLS solution but its major drawback is the necessity to choose the vectors r over which the sum and hence the integral is approximated. In the scalar Markov case the choice of the points has been considered in the literature, at least in the i.i.d. case, see Feuerverger and Mureika (1977), Schmidt (1982), Knight and Satchell (1997), Yu (1998), and Singleton (2001). In the N-dimensional case (N ≥ 2) the problem is much more com- plicated and there is virtually no guidance given in the literature. In general, there is an 1 For the ECF based estimation procedures considered in this paper, as in Singleton (2000) we assume that Hansen (1982)’s regularity conditions of the generalized method of moments (GMM) are satisﬁed. 5 obvious trade-off between ﬁner grids and coarser grids. With a too coarse grid, the GMM estimation based on selected set of moments is easier to implement but would achieve lower asymptotic efﬁciency due to the loss of information. While with a too ﬁne grid, the number of moment conditions can be so large that the variance-covariance matrix becomes singu- lar and the implementation of the estimation procedure turns out to be infeasible. Thus, in practice there is a limitation to the actual implementation of very ﬁne grid. In the context of the estimation of mixture distributions, Carrasco and Florens (2000b) provide Monte Carlo evidence of efﬁciency loss relative to the ML estimation for the ECF based GMM estimators using discrete grid. To overcome some of these problems, Jiang and Knight (2001) propose minimizing the integrated MSE (IMSE) between the empirical characteristic function (ECF) and its theoretical counterpart. This thus avoids the choice of points at which the matching is done. The optimal weight function used in the estimation, however, is no longer readily available. The approach we are proposing in this paper is to ﬁrst approximate the ln f (xt+1 |xt ) which will then, via (2), give us an approximate weight function to use in (1) and hence result in a consistent estimator. When the approximation error of the optimal weight function is arbitrarily small, the estimator also has ML efﬁciency. In this paper, we consider series expansions for the log transition density rather than for the density itself. In addition to the fact that the log transition density appears explic- itly in the optimal weight function and thus is the function we aim to approximate, for a number of other reasons better approximations may be obtained by approximating the log transition density and then exponentiating. Since the solution of (1) requires the knowledge of φ(r, xt+1 |xt ) or ln φ(r, xt+1 |xt ), we can use this function to develop approximations to ln f (xt+1 |xt ). The approximation we propose is the multivariate Edgeworth/Gram-Charlier expansion. This expansion will be taken around a multivariate normal density as an initial ﬁrst-order approximation with the mean vector and covariance matrix consistent with the 6 model and readily derived from ln φ(r, xt+1 |xt ). The correction terms in the expansion will involve generalized third, fourth and higher order cumulant products as coefﬁcients of the Hermite polynomials. More generally, however, in order to minimize the number of correc- tion terms, it maybe advantageous to use a ﬁrst-order approximation other than the normal density. The correction terms then involve derivatives of the approximating density function and are not necessarily polynomials. Following McCullagh (1987), via the use of tensor notation, we have the general Gram- Charlier/Edgeworth expansion for the log multivariate density ln f (xt+1 |xt ) given by ln f (xt+1 |xt ) = ln f0 (xt+1 |xt ) (3) 1 + K i,j,k hijk (xt+1 |xt ) 3! 1 + K i,j,k,l hijkl (xt+1 |xt ) 4! 1 + K i,j,k,l,m hijklm (xt+1 |xt ) 5! 1 + [K i,j,k,l,m,n hijklmn (xt+1 |xt ) + K i,j,k K l,m,n hijk,lmn (xt+1 |xt )[10]] 6! +.... where f0 (xt+1 |xt ) is chosen such that its ﬁrst-order and second-order moments agree with those of xt+1 conditional on xt . Upon letting ψ(r, xt+1 |xt ) = ln φ(−ir, xt+1 |xt ) (the cumu- lant generating function), we have the conditional cumulants of various orders ∂ λi = ψ(r, xt+1 |xt ) |r=0 ∂ri ∂2 λi,j = ψ(r, xt+1 |xt )|r=0 ∂ri ∂rj ∂3 K i,j,k = ψ(r, xt+1 |xt ) |r=0 ∂ri ∂rj ∂rk ∂4 K i,j,k,l = ψ(r, xt+1 |xt |r=0 ) ∂ri ∂rj ∂rk ∂rl .... It is noted that Edgeworth series used for approximations to distributions are most conve- 7 niently expressed using cumulants. Moreover, where approximate normality is involved, higher-order cumulants can usually be neglected but not higher-order moments. In the present paper, since we often deal with situations where the log characteristic functions have simpler expression, the cumulants can be more conveniently obtained than the moments. Furthermore, the Hermite polynomial tensors in the general Edgeworth/Gram Charlier ex- pansion of equation (3) are given by2 hi = λi,j (xj − λj ) t+1 hij = hi hj − λi,j hijk = hi hj hk − hi λj,k [3] hijkl = hi hj hk hl − hi hj λk,l [6] + λi,j λk,l [3] hijklm = hi hj hk hl hm − hi hj hk λl,m [10] + hi λj,k λl,m [15] hijklmn = hi ...hn − hi hj hk hl λm,n [15] + hi hj λk,l λm,n [45] − λi,j λk,l λm,n [15] .... In this paper, we let the initial approximating function be the multivariate normal density with mean vector λi and covariance matrix λi,j , i.e. ¯ −1/2 1 f0 (xt+1 ¯xt ) = (2π)−N/2 |λi,j | ¯ exp(− (xi − λi )(xj − λj )λi,j ) t+1 (4) 2 t+1 with xt+1 being a N × 1 vector whose ith element is xi , mean λi and covariance matrix t+1 λi,j , λi,j is the inverse matrix of λi,j and |λi,j | is the determinant of the covariance matrix. For clarity of presentation and ease of notation and without loss of generality, in the following discussion we focus on the Gram-Charlier series expansion and set the truncation 2 In tensor notation it is understood that any index repeated once as a subscript and once as a superscript is interpreted as sums over these repeated scripts, i.e. X hi = λi,j (xj − λj ) = λi,j (xj − λj ) j etc. Also, the numbers in square brackets refer to the number of permutations of the various subscripts. 8 order p = 4, consequently we have, N 1 ¯ ¯ b ln f (xt+1 |xt ) = − ln 2π − ln ¯λi,j ¯ ¯ ¯ 2 2 1 − (xi − λi )(xj − λj )λi,j 2 1 1 + K i,j,k hijk + K i,j,k,l hijkl (5) 3! 4! = ln f (xt+1 |xt ) − ln f ∆ (xt+1 |xt ) where ln f ∆ (xt+1 |xt ) is the approximation error. It is noted that the choice between Edge- worth expansion and Gram-Chalier expansion will have no impact on the following analysis and the derivation of the results presented in Lemma 3 as long as the expansion is based on the cumulants. The only difference between the use of Edgeworth series and that of Gram- Chalier series with the same order is that different cumulants or moments may appear in the estimating equations. Similarly, the difference that the truncation order makes is the highest order of cumulants or moments included in the estimating equations. With a higher trunca- tion order, the expressions in Lemma 3 will be more cumbersome. The Edgeworth series and Gram-Chalier series are formally identical when the expansion order is inﬁnite and the main difference is the different criteria used in collecting terms in a truncated series. The Edgeworth series is often preferred for statistical calculations. According to Blinnikov and Moessner (1998), while both the Edgeworth series and Gram-Chalier series diverge in many situations of practical interest, when ﬁtting weakly non-normal distributions better results can often be achieved with the asymptotic Edgeworth expansion. Let the parameter vector to be estimated be denoted by θ ∈ Θ, then all cumulants and the Hermite tensors are functions of θ. The approximate score function, i.e., the derivative b of the ln f (xt+1 |xt ) is given by: b ∂ ln f (xt+1 |xt ) 1 ∂ ¯ i,j ¯ ∂λi 1 ∂λi,j ¯ ¯ = − ¯ i,j ¯ ¯λ ¯ + hi − (xi − λi )(xj − λj ) ∂θ 2 ¯λ ¯ ∂θ ¯ ¯ ∂θ 2 ∂θ " # 1 ∂K i,j,k ∂hijk + hijk + K i,j,k 6 ∂θ ∂θ 9 " # 1 ∂K i,j,k,l ∂hijkl + hijkl + K i,j,k,l (6) 24 ∂θ ∂θ Using the approximate score in (6), we can deﬁne an approximate optimal weight func- tion from (2) as 1 Z Z ∂ ln f (xt+1 |xt ) −ir0 xt+1 b b ω(θ, t, r |xt ) = .. e dxt+1 (7) (2π)N ∂θ and hence our approximate ML estimator as the solution of (1) with w(θ, t, r|xt ) replaced by ω(θ, t, r|xt ). That is b 1 XZ T 0 ω(θ, t, r|xt )(eir xt+1 − φ(r, xt+1 |xt ))dr = 0 b (8) T t=1 Manipulating the above equation as in the proof of Lemma 1, we have T b 1 X ∂ ln f (xt+1 |xt ) b ∂ ln f (xt+1 |xt ) { (xt+1 |xt ) − E[ (xt+1 |xt )]} = 0 (9) T t=1 ∂θ ∂θ b When the true transition density of the Markov process f (xt+1 |xt ) is known and f (xt+1 |xt ) = b f (xt+1 |xt ), then we have E[ ∂ ln f (xt+1 |xt ) ] = 0. However, this will not necessarily be the case ∂θ b if we approximate f (xt+1 |xt ), i.e. f (xt+1 |xt ) 6= f (xt+1 |xt ). Lemma 2 Under standard regularity conditions, the approximate ML estimator deﬁned in (8) or (9) is consistent and asymptotically normal, i.e. √ d b T (θ − θ) −→ N(0, Ω) (10) where the limiting covariance matrix Ω is given in the appendix. Furthermore, when the ap- proximation error of the optimal weight function is arbitrarily small, the limiting covariance matrix Ω reaches the asymptotic Cramer-Rao lower bound. Proof: See appendix. 10 Furthermore, from the deﬁnition of the hermite polynomials we can readily establish that ∂hi ∂λi,j j ∂λj = (x − λj ) − λi,j ∂θ ∂θ ∂θ − − − j j j − = λi,j (x − λ ) − λi,j λ =hi − z i ∂hijk ∂hi ∂hi ∂λj,k = hj hk [3] − λj,k [3] − hi [3] ∂θ ∂θ ∂θ ∂θ ∂hijkl ∂hi ∂λk,l ∂hi = hj hk hl [4] − hi hj [6] − hj λk,l [12] ∂θ ∂θ ∂θ ∂θ ∂λi,j + λk,l [6] ∂θ Substituting these derivatives into the expansion given by (6) and taking expectations we can derive the appropriate estimating equations, which are stated in the following Lemma. Lemma 3 For an N-dimensional Markov process with known conditional characteristic function associated with an unknown transition density, following the ECF estimation proce- dure with approximate optiomal weight function the use of a Gram-Charlier approximation for the unknown transition density as in (5) results in the following estimating equations. ( 1 X ∂λi T 1 ∂λi,j hi − [(xi − λi )(xj − λj ) − λi,j ] T t=1 ∂θ 2 ∂θ " 1 ∂K i,j,k − − + (hijk − E(hijk |xt ) + 3K i,j,k [(hi hj hk − E[hi hj hk |xt ]) 6 ∂θ # − − ∂λj,k − z i (hj hk − E[hj hk |xt ]) − hi λj,k − hi ] ∂θ " 1 ∂K i,j,k,l − − + (hijkl − E(hijkl |xt )) + K i,j,k,l [4(hi hj hk hl − E[hi hj hk hl |xt ]) 24 ∂θ − − − −4 z i (hi hk hl − E[hj hk hl |xt ]) − 12(hi hj λk,l − E[hi hj λk,l |xt ]) #) − ∂λk,l −12 zi hj λk,l − 6(hi hj − E[hi hj |xt ]) =0 (11) ∂θ If θ is of dimension K then there will be K such equations, the solution of which will lead to approximate ML estimation. Proof : See Appendix. 11 Remark 1 As we have mentioned, there are various approaches to approximate proba- bility density functions. In this paper we use the widely used multivariate Edgeworth/Gram- Charlier expansion for the approximation of the logarithmic transition density. Still many choices remain ﬂexible, such as how many terms should be used in the expansion, what is choice of the initial approximating density, etc. In practice, most of these issues have to be dealt on a case by case basis. One difﬁculty associated with the inﬁnite series expansion of the density or log density function is that it is very likely divergent. In the univariate case, when the state variable is the standardized sum of N independent and identically distributed random variables, Feller (1971) gives conditions ensuring the validity of the Edgeworth ex- pansions for the density and log density. Similar conditions dealing with the multivariate case are given by Barndorff-Nielsen and Cox (1979). Remark 2 Our method is similar but different than the conventional method of moments (MM) or the generalized method of moments (GMM). Our estimating equations turn out to be based also on moment restrictions, but may involve more moments than in the case of MM estimation. Moreover, the dimension of the estimating equations equals that of the parameter vector instead of the number of moments used as in GMM. Remark 3 When applied to the univariate diffusion process, our method is similar to the approximate ML estimator of Aït-Sahalia (2000). The approximate MLE in Aït-Sahalia (2000) is developed based on an alternative expansion of ln f (xt+1 |xt ) after ﬁrst convert- ing the original diffusion process to a unit diffusion process. The method is based on the following estimating equation T X b ∂ ln f (xt+1 |xt ) = 0. t=1 ∂θ · ¸ b ∂ ln f Since it is very likely that E ∂θ (xt+1 |xt ) 6= 0 due to approximation error, the estimators can be inconsistent. Remark 4 (Consistency of QML) If one was merely to approximate f (xt+1 |xt ) by a 12 b normal density, i.e. only the initial approximating density and f (xt+1 |xt ) 6= f (xt+1 |xt ), the approach is essentially the quasi maximum likelihood (QML) estimation. In this case we do have " # b ∂ ln f E (xt+1 |xt ) = 0. ∂θ which shows the consistency of quasi maximum likelihood (QML) estimation. Remark 5 While the estimating equations in the general case are cumbersome, in the univariate case they collapse into the well-known method of moments as we will now illus- trate, albeit the moment restrictions are a system of conventional moment conditions. In the univariate case we essentially just let i = j = k = l = 1 and thus drop the supersript index −1 λ11 = K2 h1 = (xt+1 − K1 )/K2 h11 = h2 = h2 − 1/K2 1 h111 = h3 − 3h1 /K2 1 2 h1111 = h4 − 6h2 /K2 + 3/K2 1 1 with ∂h1 ∂K2 /∂θ ∂K1 /∂θ = −h1 − ∂θ K2 K2 and ∂h1 ∂K2 /∂θ ∂K1 /∂θ h1 = −h2 1 − h1 ∂θ K2 K2 ∂h1 ∂K2 /∂θ ∂K1 /∂θ h2 1 = −h3 1 − h2 1 ∂θ K2 K2 ∂h1 ∂K2 /∂θ ∂K1 /∂θ h3 1 = −h4 1 − h3 1 ∂θ K2 K2 and since E(h1 |xt ) = 0 13 E(h2 |xt ) = K2 1 −1 E[h11 |xt ] = 0 3 E[h111 |xt ] = K3 /K2 and 2 E[h1111 |xt ] = E[h4 |xt ] − 6E[h2 |xt ]/K2 + 3/K2 1 1 4 = K4 /K2 The estimating equations collapse to: T 1 X ∂K1 1 2 h1 + (h1 − 1/K2 )∂K2 /∂θ T t=1 ∂θ 2 " # 1 ∂K3 3 3 3∂K3 + (h1 − K3 /K2 ) − h1 /K2 6 ∂θ ∂θ K3 3 ∂K2 /∂θ ∂K1 /∂θ ∂K2 /∂θ − (h3 − K3 /K2 ) 1 + (h2 − 1/K2 ) 1 − 2h1 2 2 K2 K2 K2 " Ã ! # 2 1 ∂K4 4 (K4 + 3K2 ) ∂K4 2 + h1 − −6 (h − 1/K2 )/K2 24 ∂θ K2 4 ∂θ 1 Ã Ã !! Ã ! 2 K4 4 K4 + 3K2 ∂K2 /∂θ 3 K3 ∂K1 /∂θ − 4 h1 − 4 + 4 h1 − 3 24 K2 K2 K2 K2 ∂K2 /∂θ ∂K1 /∂θ −18(h2 − 1/K2 ) 1 − 12h1 2 =0 (12) K2 K2 Again the derivatives are taken with respect to all elements in the K-dimensional parameter vector θ. Remark 6 The above estimating equations can be readily put into a more recognizable form by combining coefﬁcients on (hj − E(hj |xt )), j = 1, 2, 3, 4(p = 4). Letting Ai be 1 1 jt the appropriate coefﬁcient on (hj − E(hj |xt )), j = 1, 2, 3, 4(p = 4), associated with the 1 1 derivative with respect to the ith element of θ, we have the estimating equations given by: T 1X At gt = 0 T t=1 14 where At is a K × 4(p = 4) matrix with the ith row being associated with ∂ ∂θi and gt is a 4 × 1(p = 4) vector given by h1 h2 − 1/K2 1 g= 3 h3 − K3 /K2 1 2 4 h4 − (K4 + 3K2 )/K2 1 More speciﬁcally, we have ∂K1 ∂K3 /∂θ ∂K2 /∂θ K4 A1t = − + K3 2 + 2 ∂K1 /∂θ ∂θ 2K2 K2 2K2 ∂K2 /∂θ K3 1 ∂K4 3K4 A2t = − ∂K1 /∂θ − + 2 ∂K2 /∂θ 2 2K2 4K2 ∂θ 4K2 ∂K3 /∂θ K3 K4 A3t = − ∂K2 /∂θ − ∂K1 /∂θ 6 2K2 6K2 ∂K4 /∂θ K4 A4t = − ∂K2 /∂θ 24 6K2 Remark 7 In the estimating equation (11) for the multivariate case and (12) for the uni- variate case, we implicitly assume that various moments or cumulants are known in analyti- cal form. When the moments or cumulants exist but are unavailable in analytical form, path simulation of the Markov process can be used to generate the moments or cumulants used in the estimation procedure. The estimation would be in the framework of Dufﬁe and Singleton (1993) and under the regularity conditions there we have both consistency and asymptotic normality of the parameter estimators. 3 Illustrative Examples of Markov Processes Example 1: The Ornstein-Uhlenbeck Process (Equivalence to MLE) The O-U process is a univariate diffusion speciﬁed by the following stochastic differential equation: dxt = β(α − xt )dt + σdwt (13) 15 where wt is a standard Brownian motion. Its discrete time representation is a AR(1) process with Gaussian error, xt+1 = α + (xt − α)e−β + ²t+1 2 where ²t+1 ∼ N(0, 2β (1 − e−2β ). The Ornstein-Uhlenbeck process or the discrete time σ AR(1) process has a normal transition density function given by 1 (xt+1 − α − (xt − α)e−β )2 f (xt+1 |xt ) = √ exp{− } 2πs2 2s2 σ2 PT where s2 = 2β (1 − e−2β ). The conditional likelihood is given by ln L = t=1 ln f (xt+1 |xt ) and maximize the likelihood function leads to the ML estimator. As a member of the afﬁne class of diffusions, the conditional characteristic function of the O-U process is given by r2 σ 2 φ(r, xt+τ |xt ) = exp{ir(α + (xt − α)e−β ) − (1 − e−2β )} 4β or the cumulant generating function −β r2 σ 2 ln φ(−ir, xt+τ |xt ) = r(α + (xt − α)e )+ (1 − e−2β ) 4β The conditional cumulants can be easily derived as K1 = (α + (xt − α)e−β ) σ2 K2 = (1 − e−2β ) 2β and Ki = 0, ∀i ≥ 3 Substituting the cumulants into the estimating equation, we have T 1 X ∂K1 1 2 1 ∂K2 (h1 + (h1 − ) )=0 T t=1 ∂θ 2 K2 ∂θ 16 where h1 = (xt+1 − K1 )/K2 , θ = (α, β, σ). It is straightforward to verify that this is equivalent to the ML estimation. Example 2: The Square-Root Diffusion Process: (Continuous-Time Process) The square- root process is a univariate diffusion speciﬁed by the following stochastic differential equa- tion: √ dxt = β(α − xt )dt + σ xt dwt (14) This is a also member of the afﬁne class of diffusions and has the following conditional characteristic function associated with its transition density ir −(q+1) ire−βτ φ(r, xt+τ |xt ) = (1 − ) exp{ xt } c (1 − ir/c) or the cumulant generating function r re−βτ ln φ(−ir, xt+τ |xt ) = −(q + 1) ln(1 + ) − xt c (1 + r/c) 2αβ where c = 2β/(σ 2 (1−e−βτ )), q = σ2 −1. The following four cumulants are easily derived. K1 = α(1 − e−βτ ) + xt e−βτ ασ 2 xt σ 2 −βτ K2 = (1 − e−βτ )2 + e (1 − e−βτ ) 2β β ασ 4 3xt σ 4 −βτ K3 = (1 − e−βτ )3 + e (1 − e−βτ )2 2β 2 2β 2 3ασ 6 3xt σ 6 −βτ K4 = (1 − e−βτ )4 + e (1 − e−βτ )3 4β 3 β3 From the estimating equations detailed above, it is clear we require the derivatives of these cumulants with respect to the elements in the parameter vector θ, where θ = (α, β, σ 2 ). The necessary derivatives can be easily derived and are included in the Appendix. Other Examples 17 4 Monte Carlo Simulations 4.1 Model I: The continuous-time square-root diffusion process The square-root diffusion process is speciﬁed in (14). The parameter values are set as α = 0.075, β = 0.80, σ = 1.00, which are close to the estimates of interest rate process using historical U.S. 3-month Treasury bill yields. The choice of parameter values gives an integer value of the degree of freedom for the non-central chi-square transition density function and makes it feasible to generate exact sampling path. Thus, there is no approximation error involved in the path simulation and differences between different estimates are entirely due to the different estimation methods. We set two sampling intervals, i.e. ∆ = 1/4, 1 with sample size T = 250, 500. In each sampling path simulation, the ﬁrst 200 observations are discarded to mitigate the start-up effect. The number of replications in the Monte Carlo simualtion is 1000. The estimation methods we consider include the empirical characteristic function based method of system of moments (ECF/MSM) we developed in this paper, the GMM based on the continuous-time model, the GMM based on discretized model, the MLE and QMLE based on continuous- time model. 4.1.1 GMM estimation based on the Continuous-Time Model The same moment conditions as in Chan, Karolyi, Longstaff and Sanders (1992) are used for GMM, except that the moment conditions are exact in the sense that they are derived from the continuous-time model, i.e. ²t ft (θ) = ²2 t − E[²2 |xt−1 ] t 18 where t = 1, 2, ..., T, θ = (α, β, σ) and the lagged variable is used as instrumental variable in the estimation, where ²t = ∆xt − E[∆xt |xt−1 ] with E[∆xt |xt−1 ] = (1 − e−β )(α − xt−1 ) ν 2 −β σ αν 2 E[²2 |xt−1 ] = t (e − e−2β )xt−1 + σ (1 − e−β )2 (15) β 2β These moment conditions correspond to transitions over a unit period and are not subject to discretization bias. 4.1.2 GMM estimation based on the Discretized Model In ﬁnancial economics literature, estimation of the square-root process using GMM often consists in ﬁrst discretizing the continuous-time model, then deriving the moment condi- tions based on the discrete-time model. The moment conditions used in the literature are as follows: ²t ft (θ) = ²2 − σ 2 xt−1 t also with the lagged variable as instrumental variable in the estimation of the squared-root process, where ²t = ∆xt − β(α − xt−1 ). It is clear that these moment conditions are different from those derived from the continuous-time model. 4.1.3 ML estimation based on the Continuous-Time Model Solving from the Kolmogrov backward (or Fokker-Planck) equation or from the conditional characteristic function via Fourier inversion, the transition density function of the square-root process can be obtained as ν f (xt |xt−1 ) = ce−u−ν ( )q/2 Iq (2(uν)1/2 ) (16) u with xt taking nonnegative values, where c = 2β/(σ 2 (1 − e−βτ )), u = cxt−1 e−βt , ν = 2βα cxt , q = σ2 − 1, and Iq (·) is the modiﬁed Bessel function of the ﬁrst kind of order q. 19 The transition density function is non-central chi-square, χ2 [2cxt ; 2q + 2, 2u], with 2q + 2 degrees of freedom and parameter of noncentrality 2u proportional to the current level of the stochastic process. If the process displays the property of mean reversion (β > 0), the process is stationary and its marginal distribution can be derived from the transition density, ω s s−1 −ωxt which is a gamma probability density function, i.e., g(xt ) = x e Γ(s) t where ω = 2β/σ 2 and s = 2αβ/σ 2 , with mean α and variance ασ 2 /2β. 4.1.4 QML estimation based on the Continuous-Time Model The QMLE is based on the conditional mean and variance as well as the unconditional mean and variance of the square-root process. The conditional mean and variance are given in (15) and the unconditional mean and variance are given in section 4.1.3. The simulation results for alternative estimators are reported in Table 1 for different sam- pling intervals and sample sizes. Overall, the ECF/MSM estimator performs closely as well as the ML estimator and consistently better than other estimators. For simulations with dif- ferent sample size and parameter values, the relative performance of alternative estimators is similar to those reported in Table 1. 5 Conclusion In this paper we have developed a new estimator for Markov models by combining an ap- proximation to the transition density along with the ﬁrst-order conditions associated with the ECF estimation approach. The estimator is guaranteed to be consistent and the asymptotic efﬁciency of the estimator approaches that of the exact ML estimator when the approxima- tion error of the optimal function is arbitrarily small. We are currently pursuing an extensive Monte Carlo study to ascertain both the accuracy of the approximation of the transition density as well as that of the estimator itself. 20 Table 1: Monte Carlo Simulation Results of Alternative Estimation Methods Panel A: Sampling Interval ∆ = 1/4, Sample Size = 250 √ parameter estimation method mean median st dev m.s.e (95 percentiles) α ECF/MSM 0.0748 0.0746 0.0043 0.0043 (0.0671 0.0829) (0.075) ML 0.0748 0.0746 0.0043 0.0043 (0.0672 0.0840) QML 0.0748 0.0746 0.0043 0.0043 (0.0672 0.0838) GMM 0.0747 0.0746 0.0044 0.0044 (0.0667 0.0839) dGMM 0.0746 0.0746 0.0044 0.0044 (0.0669 0.0837) β ECF/MSM 0.8925 0.8797 0.2002 0.2205 (0.5535 1.3618) (0.800) ML 0.8958 0.8803 0.1991 0.2208 (0.5486 1.3545) QML 0.8929 0.8790 0.2089 0.2285 (0.5221 1.3691) GMM 0.8840 0.8692 0.2046 0.2211 (0.5465 1.3476) dGMM 0.7718 0.7635 0.1582 0.1606 (0.5015 1.1180) σ ECF/MSM 0.0995 0.0994 0.0050 0.0051 (0.0894 0.1084) (0.100) ML 0.0994 0.0994 0.0051 0.0051 (0.0896 0.1100) QML 0.0712 0.0712 0.0038 0.0291 (0.0642 0.0792) GMM 0.0997 0.0996 0.0055 0.0055 (0.0894 0.1103) dGMM 0.0897 0.0898 0.0048 0.0113 (0.0805 0.0990) Panel B: Sampling Interval ∆ = 1/4, Sample Size = 500 √ parameter estimation method mean median st dev m.s.e (95 percentiles) α ECF/MSM 0.0748 0.0747 0.0031 0.0031 (0.0686 0.0802) (0.075) ML 0.0748 0.0746 0.0031 0.0031 (0.0687 0.0811) QML 0.0748 0.0746 0.0031 0.0031 (0.0688 0.0811) GMM 0.0747 0.0746 0.0031 0.0031 (0.0687 0.0810) dGMM 0.0747 0.0746 0.0031 0.0031 (0.0686 0.0809) β ECF/MSM 0.8508 0.8420 0.1374 0.1465 (0.6133 1.1521) (0.800) ML 0.8517 0.8427 0.1370 0.1464 (0.6094 1.1480) QML 0.8453 0.8337 0.1450 0.1519 (0.5921 1.1606) GMM 0.8428 0.8301 0.1403 0.1466 (0.6016 1.1400) dGMM 0.7427 0.7352 0.1101 0.1241 (0.5497 0.9692) σ ECF/MSM 0.0995 0.0991 0.0034 0.0035 (0.0922 0.1050) (0.100) ML 0.0990 0.0990 0.0035 0.0036 (0.0924 0.1056) QML 0.0709 0.0709 0.0026 0.0292 (0.0659 0.0759) GMM 0.0998 0.0997 0.0037 0.0037 (0.0927 0.1074) dGMM 0.0903 0.0903 0.0032 0.0103 (0.0839 0.0966) 21 Panel C: Sampling Interval ∆ = 1, Sample Size = 250 √ parameter estimation method mean median st dev m.s.e (95 percentiles) α ECF/MSM 0.0749 0.0749 0.0023 0.0023 (0.0705 0.0793) (0.075) ML 0.0749 0.0749 0.0022 0.0022 (0.0709 0.0796) QML 0.0749 0.0749 0.0022 0.0022 (0.0709 0.0796) GMM 0.0750 0.0749 0.0023 0.0023 (0.0708 0.0797) dGMM 0.0748 0.0748 0.0023 0.0023 (0.0706 0.0793) β ECF/MSM 0.8333 0.8236 0.1424 0.1462 (0.5896 1.1490) (0.800) ML 0.8385 0.8246 0.1364 0.1417 (0.5955 1.1497) QML 0.8376 0.8203 0.1473 0.1520 (0.5883 1.1710) GMM 0.8327 0.8200 0.1438 0.1474 (0.5861 1.1631) dGMM 0.5402 0.5383 0.0578 0.2661 (0.4292 0.6599) σ ECF/MSM 0.1003 0.0997 0.0067 0.0067 (0.0881 0.1137) (0.100) ML 0.1002 0.1000 0.0066 0.0066 (0.0883 0.1138) QML 0.0714 0.0711 0.0050 0.0291 (0.0625 0.0818) GMM 0.1001 0.0997 0.0070 0.0070 (0.0874 0.1143) dGMM 0.0697 0.0697 0.0038 0.0306 (0.0623 0.0773) Panel D: Sampling Interval ∆ = 1, Sample Size = 500 √ parameter estimation method mean median st dev m.s.e (95 percentiles) α ECF/MSM 0.0750 0.0749 0.0015 0.0015 (0.0718 0.0779) (0.075) ML 0.0749 0.0748 0.0016 0.0016 (0.0718 0.0781) QML 0.0749 0.0748 0.0016 0.0016 (0.0718 0.0781) GMM 0.0749 0.0749 0.0016 0.0016 (0.0718 0.0781) dGMM 0.0748 0.0748 0.0016 0.0016 (0.0715 0.0780) β ECF/MSM 0.8145 0.8131 0.0991 0.1002 (0.6452 1.0244) (0.800) ML 0.8212 0.8146 0.0977 0.0999 (0.6531 1.0294) QML 0.8173 0.8113 0.1051 0.1065 (0.6329 1.0490) GMM 0.8151 0.8108 0.0998 0.1009 (0.6451 1.0286) dGMM 0.5350 0.5345 0.0413 0.2682 (0.4578 0.6205) σ ECF/MSM 0.0998 0.0996 0.0047 0.0047 (0.0907 0.1090) (0.100) ML 0.0997 0.0996 0.0047 0.0047 (0.0911 0.1093) QML 0.0710 0.0708 0.0036 0.0293 (0.0644 0.0784) GMM 0.0999 0.0997 0.0049 0.0049 (0.0906 0.1098) dGMM 0.0702 0.0701 0.0026 0.0300 (0.0651 0.0754) Note: dGMM is the GMM estimation based on the discretized model, all other methods are based on the continuous-time model. 22 References Aït-Sahalia, Y. (1999), “Transition Densities for Interest Rate and Other Nonlinear Dif- fusions,” Journal of Finance 54, 1361-1395. (2000), “Maximum-liklihood Estimation of Discretely Sampled Diffusions: A Closed Form Approach,” Econometrica, forthcoming, 2001. Andersen, T. G., L. Benzoni, and J. Lund (2001), “Estimating jump-diffusions for equity returns” [Working paper, Northwestern University]. Barndorff-Nielsen, O.E. and D.R. Cox (1979), “Edgeworth and saddle-point approxima- tions with statistical applications," J. Roy. Statist. Soc. B, 41,279-312. Blinnikov,S. and R. Moessner (1998), “Expansions for nearly Gaussian distributions", Astron. Astrophys. Suppl. Ser., 130, 193-205. Carrasco, M., M. Chernov, J. Florens, and E. Ghysels (2001), “Estimating diffusions with a continuum of moment conditions” [Working paper, University of North Carolina]. Carrasco, M., J. Florens (2000a), “Generalization of GMM to a continuum of moment conditions”, Econometric Theory, 16, 797-834. Carrasco, M., J. Florens (2000b), “Efﬁcient GMM estimation using the empirical char- acteristic function”, [Working paper, CREST, Paris]. Chacko, G. and L. M. Viceira (1999), “Spectral GMM Estimation of Continuous-Time Processes,” [Working Paper, Graduate School of Business Administration, Harvard Univer- sity]. Chen, R., L. Scott (1985), “Maximum likelihood estimation for a multifactor equilibrium model of the term structure of interest rates.” Journal of Fixed Income, 3, 14-31. Chernov, M. and E. Ghysels (2000), “A study toward a uniﬁed approach to the joint estimation of objective and risk-neutral measures for the purposes of options valuation”, Journal of Financial Economics, 56, 407-458. 23 Cox, J. C., J. E. Ingersoll and S. A. Ross (1985), “A Theory of the Term Structure of Interest Rates,” Econometrica, 53, 385-407. Dufﬁe, D. and K.J. Singleton (1993), “Simulated moments estimation of Markov models of asset prices”, Econometrica, 61, 929-952. Feller, W. (1971), An Introduction to Probability Theory and Its Applications 2, New York: Wiley. Feuerverger, A. (1990), “An efﬁciency result for the empirical characteristic function in stationary time-series models”, The Canadian Journal of Statistics, 18, 155–161. Feuerverger, A. and P. McDunnough (1981a), “On some Fourier methods for inference.”, Journal of the American Statistical Association, 76, 379-141. Feuerverger, A. and P. McDunnough (1981b), “On the efﬁciency of empirical charateris- tic function procedures”, Journal of the Royal Statistical Society, Series B 43, 20-27. Feuerverger, A. and R. A. Mureika (1977), “The empirical characteristic function and its applications”, The Annals of Statistics, 5, 88–97. Fisher, M. and C. Gilles (1996), “Estimating exponential afﬁne models of the term struc- ture.” [Working Paper]. Fisher, A. and R.A. Mureika (1977), “The empirical characteristic function and its appli- cations”, The Annals of Statistics, 5, 88–97. Gallant, A. R. and J. R. Long (1997), “Estimation stochastic differential equation efﬁ- ciently by minimum Chi-square,” Biometrika, 84, 125-141. Gallant, A. R and G. E. Tauchen (1996), “Which moments to match?”, Econometric Theory, 12, 657–681. Gouriéroux, C. and Monfort, A. and Renault, E. (1993), “Indirect Inference”, Journal of Applied Econometrics, 8, S85–S199. Hansen, L.P. and J.A. Scheinkman (1995), “Back to the Future: Generating Moment Implications for Continuous-time Markov processes,” Econometrica, 63, 767-804. 24 Jiang, G. J. and J. L. Knight (2001), “Estimation of Continuous Time Processes Via the Empirical Characteristic Function,” Journal of Business & Economic Statistics, forthcoming. Jones, C.S. (1997), “Bayesian analysis of the short-term interest rate,” [Working paper, The Wharton School, University of Pennsylvania.] Knight, J. L. and S. E. Satchell (1997), “The Cumulant Generating Function Estimation Method,” Econometric Theory, 13, 170-184. Liu, J. (1997), “Generalized method of moments estimation of afﬁne diffusion pro- cesses.” [Working Paper, Graduate School of Business, Stnaford University]. McCullagh, P. (1987), Tensor Methods in Statistics, Chapman and Hall, London. Pearson, N. D. and T. Sun (1994), “Exploiting the conditional density in estimating the term structure: an application to the Cox, Ingersoll, and Ross model”, Journal of Finance, XLIX(4), 1279-1304. Schmidt, P. (1982), “An Improved Version of the Quandt-Ramsey MGF Estimator for Mixtures of Normal Distributions and Switching Regressions,” Econometrica, 50, 501-524. Singleton, K. J. (2001), “Estimation of Afﬁne Asset Pricing Models Using the Empirical Characteristic Function,” Journal of Econometrics, 102, 114-141. Yu, J. (1998), “Empirical Characteristic Function in Time Series Estimation and a Test Statistic in Financial Modelling,” [unpublished Ph.D dissertation]. 25 Appendix Proof of Lemma 1: From (2), we have Z ∂ ln f 0 (xt+1 |xt ) = eir xt+1 w(θ, t, r|xt )dr ∂θ and Z w(θ, t, r|xt )φ(r, xt+1 |xt )dr Z Z 0 = w(θ, t, r|xt ) eir xt+1 f (xt+1 |xt )dxt+1 dr Z Z 0 = ( w(θ, t, r|xt )eir xt+1 dr)f (xt+1 |xt )dxt+1 Z 0 = E[ eir xt+1 w(θ, t, r|xt )dr] ∂ ln f (xt+1 |xt ) = E[ ] ∂θ Thus the estimating equations (1) lead to T 1 X ∂ ln f ∂ ln f [ (xt+1 |xt ) − E[ (xt+1 |xt )]] = 0. T t=1 ∂θ ∂θ which is equivalent as ML estimation. Proof of Lemma 2: Firstly, from (8) or (9), we note immediately from Singleton (2001) that our estimator is merely a GMM estimator and as such will be asymptotically normally distributed. Secondly, denote equation (8) as T 1X H(θ) = ht (θ) = 0 T t=1 under certain regularity conditions, we have that the asymptotic variance-covariance matrix Ω is given by Ω = D(θ)−1 Σ(θ)D(θ)−1 PT ∂ht (θ) with D(θ) = plim T 1 t=1 ∂θ and Σ(θ) = plimH(θ)H(θ)0 . 26 Proof of Lemma 3: The results in Lemma 3 follow immediately from the substitution of equation (6) into equation (9). Alternatively, the estimating equation (11) can be derived by subsituting the approximating weight function into equation (8) and apply the deﬁnition of cumulants. The Square-Root Diffusion Process: Derivatives of Cumulants with respect to the Pram- eters. ∂K1 = (1 − e−βτ ) ∂α ∂K2 σ2 = (1 − e−βτ )2 ∂α 2β ∂K3 σ4 = (1 − e−βτ )3 ∂α 2β 2 ∂K4 3σ 6 = (1 − e−βτ )4 ∂α 4β 3 ∂K1 = τ (α − xt )e−βτ ∂β ∂K2 ασ 2 = (1 − e−βτ )((1 + 2βτ )e−βτ − 1) ∂β 2β 2 xt σ 2 + 2 e−βτ ((1 + 2βτ )e−βτ − 1 − βτ ) β ∂K3 ασ 4 −βτ 2 = 3 (1 − e ) ((2 + 3βτ )e−βτ − 2) ∂β 2β 3xt σ 4 −βτ + 3 e (1 − e−βτ )((2 + 3βτ )e−βτ − 2 − βτ ) 2β ∂K4 3ασ 6 −βτ 3 = 4 (1 − e ) ((3 + 4βτ )e−βτ − 3) ∂β 4β 3xt σ 6 + 4 e−βτ (1 − e−βτ )2 ((3 + 4βτ )e−βτ − 3 − βτ ) β ∂K1 = 0 ∂σ 2 ∂K2 = K2 /σ 2 ∂σ 2 ∂K3 = 2K3 /σ 2 ∂σ 2 ∂K4 = 3K4 /σ 2 ∂σ 2 27

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Efficient Estimation of Markov Models Where the Transition Density ...

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