Simulation-Based Estimation with Applications in S-PLUS to Probabilistic Discrete Choice Models and Continuous-Time Financial Models.

Transcription

1 Simulation-Based Estimation with Applications in S-PLUS to Probabilistic Discrete Choice Models and Continuous-Time Financial Models Eric Zivot Department of Economics, University of Washington and Insightful Corporation Research funded by NSF SBIR Phase II DMI Note: Original PI: Dr. Jiahui Wang, Insightful Corporation Slides and scripts available at

2 1 Efficient Method of Moments Theory 1. Gallant, A. R., and G. Tauchen (1996). Which Moments to Match?, Econometric Theory 2. Gallant, A. R., and J.R. Long (1997). Estimating Stochastic Differential Equations Efficiently by Minimum Chi-Squared, Boimetrika. Applications 1. Andersen, T.G. and J. Lund (1997). Estimating Continuous-Time Stochastic Volatility Models of the Short-term Interest Rate, Journal of Econometrics

3 2. Gallant, A.R., and G. Tauchen (2001). Efficient Method of Moments, manuscript, Dept. of Economics, University of North Carolina 3. Gallant, A.R., and G. Tauchen (2002). Simulated Score Methods and Indirect Inference for Continuous-time Models, forthcoming in the Handbook of Financial Econometrics. 4. Gallant, R. and G. Tauchen (2001). SNP: A Program of Nonparametric Time Series Analysis, Version 8.9, User s Guide, available at ftp.econ.econ.duke.edu 5. Gallant, R. and G. Tauchen (2002). EMM: A Program for Efficient Method of Moments, Version 1.6, User s Guide, available at ftp.econ.econ.duke.edu

4 1.1 Estimating Dynamic Models with Unobserved States Standard statistical methods, both classical and Bayesian, are usually not applicable either because it is not practicable to obtain the density of the state vector or because the integration required to eliminate unobserved states from the likelihood is infeasible. On a case-by-case basis, statistical methods are sometimes available, However, the purpose here is to describe methods that are generally applicable. Simulating the evolution of the state vector is often practicable. The methods described here rely on this. Simulated method of moments methods are described in general, and then the discussion focusses on the efficient method of moments.

5 Outline of lectures Dynamical Systems with Unobserved States Continuous-time diffusion models Example: stochastic volatilty models for interest rates Simulated Method of Moments Relationship between GMM and ML Simulated moments estimation using EMM Asymptotics MA(1) example The SNP Auxiliary Model

6 Projection Estimation Reprojection Simulating from continuous time models Euler smethod EMM estimation of continuous time models One factor generalized CIR model Twofactormodel

7 1.2 General Diffusion Processes Many continuous time financial models describe the evolution of state variables in terms of a system of stochastic differential equations (SDEs) U t = state variables du t = A(U t,ρ)dt + B(U t,ρ)dw t 0 t< A(U t,ρ) = drift B(U t,ρ) = diffusion ρ = model parameters W t = Wiener process The state variables are determined by solving the SDE Z t U t U 0 = A(U s,ρ)ds + B(U s,ρ)dw s 0 0 Observables y t are regarded as discretely sampled observations from part of the above system. Lo (1988 ET ) showed that the likelihood of the observed process y t given ρ is generally not available in closed form. Z t

8 Weekly 3 Month U.S. T-Bill Rate Figure 1: Example: Short-term Interest Rates Weekly (Friday) observations on the 3-month T- Bill rate, 1/5/1962-3/31/1995 n =1, 735 weekly observations

9 1.2.2 Stochastic Volatility Models of the short-term interest rate One factor generalized CIR model dr t = κ(µ r t )dt + σr γ t dw t U t = r t A(U t ) = κ(µ r t ) B(U t,ρ) = σrt γ ρ = (κ, µ, σ, γ) 0 Andersen, T. and J. Lund (1997). Estimating Continuous- Time Stochastic Volatility Models of the Short-term Interest Rate, Journal of Econometrics.

10 Two factor model dv t = (a v + a vv v t + a vr r t )dt +(b 1v + b 1vv v t + b 1vr r t )dw 1t + b 2v dw 2t dr t = (a r + a rr r t )dt +(b 2r + b 2rr r t )exp(v t )dw 2t where U t = (v t,r t ) 0, W t =(W 1t,W 2t ) 0 Ã! av + a A(U t,ρ) = vv v t + a vr r t a r + a rr r t à b1v + b B(U t,ρ) = 1vv v t + b 1vr r t b 2v (b 2r + b 2rr r t )exp(v t ) 0 Identification restrictions! b 1vv =0,b 2v =0,a v = a vv 1. Gallant, R. and G. Tauchen (1998). Reprojecting Partially Observed Systems with Application to Interest Rate Diffusions, Journal of the American Statistical Association.

11 1.3 ML and GMM Estimation MLE y t iid p(y t ρ), ρ is L 1 y = (y 1,...,y n ) 0 = random sample The MLE for ρ solves Equivalently, X n ˆρ mle =argmin ln p(y ρ t ρ) t=1 0 = = nx t=1 nx t=1 ρ ln p(y t ˆρ mle ) s(y t ˆρ mle ) Under regularity conditions, ˆρ mle is consistent and

12 asymptotically efficient ˆρ mle p ρ0 n(ˆρmle ρ 0 ) d N(0,I 1 " 2 ln p(y t ρ I 0 = E 0 ) ρ ρ 0 # 0 ) = E[s(y t ρ 0 )s(y t ρ 0 ) 0 ] GMM Suppose there are K L moment functions m(y t,ρ)=(m 1 (y t,ρ),...,m K (y t,ρ)) 0 such that E ρ0 [m(y t,ρ)] = 0 For example, m 1 (y t,ρ) = y t µ m 2 (y t,ρ) = yt 2 (σ 2 µ 2 )

13 The method of moments estimator of ρ is based on matching sample moments to population moments, and solves m n (ρ) = 1 n nx t=1 m(y t,ρ)=0 If K>L,there is no solution to the above equation. For a symmetric and positive definite weight matrix W, the GMM estimator of ρ solves ˆρ GMM =argmin ρ m n (ρ) 0 Wm n (ρ) Under regularity conditions ˆρ GMM is consistent and asymptotically normally distributed. The efficient GMM estimator uses a weight matrix that is the inverse of a consistent estimate of the asymptotic variance of nmn (ρ 0 ): W E = avar( d nm n (ρ 0 )) 1

14 Remarks: The efficient GMM estimator uses the best weight matrix for a given set of moment conditions. A different question is: What are the moment conditions such that the GMM estimator is efficient in the class of consistent and asymptotically normal estimators? Relationship between MLE and GMM If m n (ρ) = 1 n nx t=1 s(y t ρ) then the GMM estimator is equivalent to MLE. That is, the best moment conditions to use are based on the score of the true model.

15 Remark The efficient GMM weight matrix in this case is a consistent estimate of avar( nm n (ρ 0 )) 1 = I 1 0 = ³ E[s(y t ρ 0 )s(y t ρ 0 ) 0 ] 1

16 1.4 Simulated Method of Moments Notation Random Variables : y t, t =,..., Data : ỹ i, i = L,...,n Simulation : ŷ i (ρ), i = L,...,N Dummy arguments of summation (y t L,...,y t 1, y t ) y t, x t 1 =(y t L,...,y t 1 ) Dummy arguments of integration (put t = 0) (y L,...,y 1,y 0 ) y = y 0, x =(y L,...,y 1 ) Assumption underlying the methodology For ρ in the parameter space, the model generating the data is presumed to be stationary, ergodic and convenient to simulate

17 Consequence For any lag length L and parameter setting ρ there exists a stationary density for observables (y t L,...,y t ) p(y L,...,y 0 ρ) And, an unconditional expectation E ρ ψ(y L,...,y 0 ) can be computed by generating a long simulation and averaging {ŷ t (ρ)} N t= L E ρ ψ(y L,...,y 0 ) 1 N NX t=0 ψ(ŷ t L (ρ),...,ŷ t (ρ))

18 1.4.1 EMM Estimator For a quasi-maximum likelihood estimator of an auxiliary model with conditional density f(ỹ t x t 1,θ) θ n =argmax θ 1 n a sample average satisfies 0= 1 n nx t=1 nx t=1 ln f(ỹ t x t 1,θ) θ ln f(ỹ t x t 1, θ n ), because these are the first order conditions of the optimization problem Therefore a large simulation from the model with parametes ρ should satisfy the estimating equations 0=m(ρ, θ n )= 1 N NX t=1 θ ln f(ŷ t(ρ) ˆx t 1, θ n ) except for sampling variation in θ n. These estimating equations hold exactly in the limit as n and N tend to infinity.

19 The EMM estimator attempts to find the parameter vector ρ that solves these estimating equations in a sensetobemadeprecisebelow Minimum Chi-Squared Estimators If the equations m(ρ, θ n )=0 cannot be solved because the dimension of θ is larger than the dimension of ρ, then Use a nonlinear optimizer to solve ˆρ n = argmin ρ m(ρ, θ n ) 0 ³ Ĩ n 1 m(ρ, θ n ) 0 ³Ĩn 1 = weight matrix

20 I n estimates the asymptotic variance of nm(ρ, θ n ). If f(y x, θ) is a good approximation to the true data generating process p(y x, ρ 0 )thenanadequate estimator is Ĩ n = 1 n NX t=1 0 θ ln f(ỹ t x t 1, θ n ) θ ln f(ỹ t x t 1, θ n ) Otherwise a HAC estimator of the asymptotic variance should be used.

21 1.4.3 Asymptotics Let ρ 0 denote the true value and let θ 0 denote an isolated solution to m(ρ 0,θ)=0. If f(y x, θ) encompasses p(y x, ρ 0 ), then under regularity conditions where ˆρ n p ρ a.s. n(ˆρn ρ 0 ) d N(0, h M 0 0 I 1 0 M 0i 1) lim ˆM n n = M 0 lim n Ĩn = I 0 ˆM n = M(ˆρ n, θ n ),M 0 = M(ρ 0,θ 0 ) M(ρ, θ) = ρ0m(ρ, θ) I 0 = E ρ0 " µ θ ln f(y 0 x 1,θ 0 ) µ θ ln f(y 0 x 1,θ 0 ) Result: The better the auxiliary model approximates the true structural model, the closer ˆρ n is to the maximum likelihood estimator based on the true structural model. 0 #

22 1.4.4 (1 α) 100% Confidence Intervals Wald-type intervals ˆρ i,n ± z α/2 SE(ˆρ i,n ) r hnm SE(ˆρ i,n ) = 0 n In 1 i 1 M n ii Always symmetric - may be misleading Invert EMM objective function q i (ρ i ) = n max θ n ) 0 ³ Ĩ n 1 m(ρ, θ n ) 0 ρ,ρ i fixed = profile objective function {ρ i : q i (ρ) q i (ˆρ i ) χ 2 1 α (1)} Better captures nonlinearity of EMM objective function

23 1.4.5 Specification Test and Diagnostics Under the null hypothesis that p(y L,...,y 0 ρ) isthe correct model L(ˆρ n, θ n ) = n m(ˆρ n, θ n ) 0 ³ Ĩ n 1 m(ˆρn, θ n ) 0 χ 2 (q) q = dim(θ) dim(ρ) and nm(ˆρn, θ n ) d N(0, I 0 M 0 h M 0 0 I 1 0 M 0i 1 M 0 0 ) Remarks: L(ˆρ n, θ n ) is the EMM equivalent of the GMM J statistic. It is an omnibus statistics for misspecification. Large values indicate model misspecification

24 Inspection of the t ratios T n (ˆρ n, θ n ) = Ŝn 1 nm(ˆρn, θ n ) h Ŝ n = diag(ĩ n ˆM n ˆM nĩ 1 0 n ˆM i 1 n ˆM n) 0 can suggest reasons for model failure. Different elements of the score correspond to different characteristics of the data. Large t ratios reveal those characteristics that are not well approximated. 1/2 The t ratios T n (ˆρ n, θ n ) do not account for uncertainty in ˆρ n and are biased downward (toward zero). They are therefore conservative.

25 1.4.6 Example: Estimating MA(1) model using EMM y t = ε t γε t 1, t =1,...,n ε t iid N(0,σ 2 ε) ρ = (γ,σ ε ) 0, ρ 0 =truevalue Note: Easy to simulate from MA(1) given ρ. Auxilary Model If θ < 1 then MA(1) is invertible and has the AR( ) representation y t = X j=1 φ j = γ j φ j y t j + ε t A feasible auxilary model is an AR(p) model y t = φ 1 y t φ p y t p + u t u t iid N(0,σ 2 u) θ = (φ 1,...,φ p,σ 2 u) 0

26 The log-density for the auxilary model is ln f(y t x t 1,θ) = 1 2 ln(2π) 1 2 ln(σ2 u) 1 2σ 2 (y t φ 1 y t 1 φ p y t p ) 2 u If p at rate n 1/3 then AR(p) will encompass the MA(1) data generating process as n. Quasi-maximum likelihood estimation Given a sample of observed data {ỹ t } n t=1 based on ρ 0, quasi-maximum likelihood estimation of θ solves θ n =argmax θ The sample score vector 1 n nx t=1 ln f(ỹ t x t 1,θ) 1 n nx t=1 θ ln f(ỹ t x t 1, θ n )

27 has elements 1 n. 1 n nx t=1 nx t=1 (ỹ t φ 1 ỹ t 1 φ p ỹ t p )ỹ t 1 (ỹ t φ 1 ỹ t 1 φ p ỹ t p )ỹ t p 1 2 σ 2[n + σ 2 ] nx t=1 (ỹ t φ 1 ỹ t 1 φ p ỹ t p ) and by construction 1 n P nt=1 θ ln f(ỹ t x t 1, θ n )=0. For a fixed ρ, let {ŷ t (ρ)} N t=1 denote a simulated sample of size N >> n from the MA(1) model. The sample score of the auxiliary model evaluated using the simulated data is m(ρ, θ n )= 1 N NX t=1 θ ln f(ŷ t(ρ) ˆx t 1, θ n )

28 which has elements 1 N. 1 n NX t=1 nx t=1 (ŷ t (ρ) φ 1 ŷ t 1 (ρ) φ p ŷ t p (ρ))ŷ t 1 (ρ) (ŷ t (ρ) φ 1 ŷ t 1 (ρ) φ p ŷ t p (ρ))ŷ t p (ρ) 1 2ˆσ 2[n + σ 2 ] If ρ = ρ 0 then nx t=1 whereas if ρ 6= ρ 0 then (ŷ t (ρ) φ 1 ŷ t 1 (ρ) φ p ŷ t p (ρ)) m(ρ 0, θ n ) 0 m(ρ, θ n ) 6= 0 Since θ is (p+1) 1 andρ is (2 1) it is not possible to find ˆρ such that m(ˆρ, θ n )=0. Instead ρ is estimated by minimizing the quadratic form ˆρ n =argmin ρ m(ρ, θ n ) 0 ³ Ĩ 1 n m(ρ, θ n )

29 1.4.7 EMM Estimation using SPLUS. Example Data Structural model: MA(1) y t = ε t +0.75ε t 1, ε t iid N(0, 1) ρ 0 =( 0.75, 1) 0 n = 250 Auxilary model: AR(3) y t = µ + φ 1 y t 1 + φ 2 y t 2 + φ 3 y t 3 + u t, u t iid N(0,σ 2 u) θ =(µ, φ 1,φ 2,φ 3,σ 2 u) 0 N =10, 000 dim(θ) dim(ρ) =3

30 1.5 General Purpose Auxiliary Model: SNP SNP is a Semi-NonParametric method, based on an expansion in Hermite functions, for estimation of the one-step-ahead conditional density f(y t x t 1,θ). It consists of applying classical parametric estimation and inference procedures to models derived from nonparametric truncated series expansions. Estimation of SNP models entails using a standard maximum likelihood procedure together with a model selection strategy that determines the appropriate degree of the nonparametric expansion. Under reasonable regularity conditions, the estimator is consistent and efficient. An important usage of SNP models is to act as a score generator for EMM estimation.

31 1.5.1 Assumptions 1. Stationary multivariate data y t =(y 1t,...,y Mt ) 0 Stationarity is assumed so that densities for a stretch of data are time invariant. 2. Markovian structure: the conditional density of y t given the entire past depend only on a finite number of lags x t 1 =(y t L,...,y t 1 ) 3. Smoothness: the density f(y t,x t 1 ) must have derivatives to the order ML/2orhigherandhave tails that are bounded above by P (y t L,...,y t )exp 1 2 LX yt L 2 t=0 where P is a polynomial of large but finite degree.

32 4. Focus is on the estimation of the transition density f(y t x t 1 )=f(y t y t L,...,y t 1 )

33 1.5.2 Overview of SNP Model Location-scale transformation for a multivariate innovation z y = R x z + µ x R is upper triangular z = R 1 x (y µ x ) Innovation density (Hermite expansion) h K (z x) = [P (z, x)] 2 φ(z) R [P (u, x)] 2 φ(u)du φ(z) = N(0,I M ) P (z, x) = polynomial in z of degree K whose coefficients are polynomials of degree K x in x Location function (VAR) µ x = b 0 + Bx t 1 x t 1 = (y t L,...,y t 1 )

34 ARCH scale function vech(r xt 1 )=r 0 + L r X i=1 P (i) y t 1 Lr +i µ xt 2 L r +i Garch scale function vech(r xt 1 ) = r L r X i=1 L X g i=1 P (i) y t 1 Lr +i µ xt 2 L r +i diag(g (i) )R xt 2 L g +i DVEC(1,1) Garch option

35 1.5.3 Notation for Multivariate Polynomial Degree K, dimension M P (z) = KX α =0 a α z α z α = z α 1 1 zα 2 2 zα M M α = α 1 + α α M Example: K =2,M =2 and z = (z 1,z 2 ) 0, α =(α 1,α 2 ) 0 α = 0 α =(0, 0) α = 1 α =(1, 0) 0, (0, 1) 0 α = 2 α =(1, 1) 0, (2, 0) 0, (0, 2) 0 P (z) = a (0,0) + a (1,0) z 1 + a (0,1) z 2 +a (1,1) z 1 z 2 + a (2,0) z a (0,2) z2 2

36 1.5.4 Hermite Expansions: Rationale 1 An unnormalized Hermite polynomial has the form P (z) q φ(z) φ(z) =N(0,I M )=(2π) M/2 exp A function g(z) that satisfies Z µ 1 2 z0 z Z g 2 = g2 (z)dz 1 dz M < is called an L 2 function. Result: The Hermite polynomials are dense in L 2 (, ); i.e., q lim g(z) P (z) φ(z) 2 =0 K where the coefficients {a α } α <K of P (z) arethose that minimize the approximation error g(z) P (z) q φ(z) 2.

37 1.5.5 Hermite Expansions: Rationale 2 Let h(z) be a density function. Because q Z h(z)dz = Z Z h(z)dz 1 dz M =1 q h(z) isinl 2 (, ) and can therefore be approximated by P (z) φ(z) as accurately as desired by taking K large enough. This fact motivates using h K (z) = P 2 (z)φ(z) R P 2 (s)φ(s)ds to approximate h(z), where the division is to guarantee that h K (z) integratesto1.

38 1.5.6 The Main Idea Take h(z) as the parent density and use a locationscale transform y = Rz + µ to generate a location-scale family of densities n P [R 1 (y µ)] o 2 φ[r 1 (y µ)] f(y θ) = det(r) R P 2 (s)φ(s)ds which can be estimated from data {ỹ t } n t=1 maximum likelihood θ =argmax θ The density estimate is ny t=1 f(ỹ t θ) by quasi f(y) =f(y θ) The consistency of the estimator was established by Gallant and Nychka (1987), Semi-Nonparametric Maximum Likelihood Estimation, Econometrica.

39 1.5.7 SNP Density: iid Data Location-scale transform Density y = Rz + µ, z = R 1 (y µ) R = upper triangular f(y θ) P 2 [R 1 (y µ)] N(y µ, Σ) Σ = RR 0 Example: K =2,M=2 Ã! R11 R R = 12 0 R 22 θ = (a (0,0),a (1,0),a (0,1), a (1,1),a (2,0),a (0,2), µ 1,µ 2,R 11,R 12,R 22 ) a (0,0) = 1 (normalization)

40 How well does SNP do at approximating densities? SNP performs well relative to kernel estimators SNP performs well on the Marron and Wand (1992) test suite densities. References 1. Fenton, V.M. and R. Gallant (1996). Convergence Rates of SNP Density Estimators, Econometrica. 2. Fenton, V.M. and R. Gallant (1996). Qualitative and Asymptotic Performance of SNP Density Estimators, Journal of Econometrics.

41 1.5.8 Choice of K Results in Coppejans and Gallant (2000), Cross Validated SNP Density Estimates, indicate that minimizing the BIC seems to work well for determining the degree K of the polynomial P (z) Estimation: Eqivalent to maximum likelihood, but more numerically stable is to minimize the negative of the average log likelihood θ n = argmin s n (θ) = 1 n θ nx t=1 s n (θ) ln f(y t θ) Schwarz criterion (BIC): Choose K to minimize BIC(p) = s n (θ)+ p 2n ln(n) p = number of free parameters in θ

42 1.5.9 SNP Transition Density for Time Series Theideaistomodifythelocationandscaletransforms of the SNP density for iid data to be functions of the past, which is the standard method of modifying a model for iid data for application to time series data. Lastly, the SNP density itself is modified to accomodate non-homogeneous innovations. The steps are as follows. VAR location function: y = Rz + µ xt 1 µ xt 1 = b 0 + Bx t 1 x t 1 = (y t Lu,...,y t 1 ) 0 L u = VAR order b 0 is M 1, Bis M L u. Density f(y θ) P 2 [R 1 (y µ xt 1 )] N(y µ, RR 0 ) K z = 0 Gaussian VAR K z > 0 non-gaussian VAR

43 ARCH-type scale function y = R xt 1 z + µ xt 1 µ xt 1 = b 0 + Bx t 1 vech(r xt 1 ) = r 0 + where Density L r X i=1 L r = ARCH order r 0 is M(M +1)/2 1 P (i) y t 1 Lr +i µ xt 2 L r +i P = [P (1) P (Lr ) ]ism(m +1)/2 L r f(y θ) P 2 [Rx 1 t 1 (y µ xt 1 )] N(y µ, R xt 1 Rx 0 t 1 ) K z = 0,L r > 0 Gaussian VAR-ARCH K z > 0,L r > 0 non-gaussian VAR-ARCH

44 GARCH-type scale function y = R xt 1 z + µ xt 1 µ xt 1 = b 0 + Bx t 1 vech(r xt 1 ) = r 0 + where Density + L r X i=1 L X g i=1 L g = ARCH order P (i) y t 1 Lr +i µ xt 2 L r +i diag(g (i) )R xt 2 L g +i G =[G (1) G (Lg ) ]ism(m +1)/2 L g f(y θ) P 2 [Rx 1 t 1 (y µ xt 1 )] N(y µ, R xt 1 Rx 0 t 1 ) K z = 0,L g > 0 Gaussian VAR-GARCH K z > 0,L r > 0 non-gaussian VAR-GARCH

45 SNP for Non-homogenous Innovations Non-homogeneous innovations are accomodated by letting the polynomial part of the SNP model P (z) = K z X α =0 a α z α have coefficients a α that are polynomials in the conditioning information giving x =(x t Lp,...,x t 1 ) a α (x) = K x X β =0 The polynomial part is denoted a αβ x β P (z, x) = K z X K x X α =0 β =0 a αβ x β z α

46 The SNP density for the non-homogeneous innovations is a Hermite polynomial in z whose coefficients are polynomials in x h K (z x) = P (z, x)φ(z) R P (s, x)φ(s)ds K = (K z,k x ) 0

47 SNP: Putting It All Together The general non-gaussian VAR model with GARCH errors is y = R xt 1 z + µ xt 1 µ xt 1 = b 0 + Bx t 1 vech(r xt 1 ) = r 0 + with density + L r X i=1 X L g i=1 P (i) y t 1 Lr +i µ xt 2 L r +i diag(g (i) )R xt 2 L g +i f(y θ) P 2 [Rx 1 t 1 (y µ xt 1 ),x t 1 ] N(y µ, R xt 1 Rx 0 t 1 ) K z > 0 non-gaussian VAR-GARCH K z > 0,K x > 0 non-gaussian, non-linear VAR-GARCH L p = number of lags of x t 1 in P (z, x t 1 )

48 Data Transformations Standardizing data ȳ = 1 n S = 1 n nx ỹ t t=1 nx t=1 (ỹ t ȳ)(ỹ t ȳ) 0 ỹt = diag(s 1/2 )(ỹ t ȳ) S 1/2 = Choleski factor Gallant and Tauchen suggest using only diagonal elements of S

49 Adjusting for extreme values in x t 1 Problem: Iftheturedensityf(y x) is heavy tailed, then x t 1 will contain extreme observations, which have a strong and undesirable influence on estimates when L r > 0 (ARCH terms). Additionally, if data are highly persistent (like interest rates), fitted SNP models can generate explosive simulations Cure: Replace each component of x by its log spline transform x i = σ tr = [ x i σ tr ln(1 x i σ tr )] x i < σ tr x t σ tr x i σ tr 1 2 [ x i + σ tr +ln(1 σ tr + x i )] σ tr < x i Note: Spline transformation attenuates potential explosive simulations ŷ t (ρ) that may occur during course of EMM estimation

50 Order in Which Transformations are Applied ỹ t (raw data) ỹt (centered and scaled) x t 1 (lags) x t 1 (spline) µ x, R x (location-scale)

51 Asymptotics If the parameters of f(y x, θ) are estimated by quasimaximum likelihood θ n = argmins n (θ) s n (θ) = 1 n θ nx t=1 ln f(ỹ t x t 1,θ) and K =(K z,k x ) 0 grows with the sample size, then the estimator f n (y x) =f(y x, θ) converges almost surely to the true transition density f(y x) insobolevnormasn. Moreover, K can depend on the data. Reference: Gallant, R. and D. Nychka (1987). Seminonparametric Maximum Likelihood Estimation, Econometrica.

52 Model Selection Schwarz criterion (BIC): Choose K to minimize BIC(K) = s n (θ)+ p 2n ln(n) nx s n (θ) = 1 ln f(y t θ) n t=1 p = number of free parameters in θ Suggested Search Order 1. Determine best VAR order L u 2. Determine best ARCH and GARCH orders L r,l g 3. Determine best z ploynomial order K z (start at K z =4) 4. Determine the best x polynomial order K x

53 1.6 Simulation Methods for SDEs SDE du t = A(U t )dt + B(U t )dw t Euler s method: Iterate for small delta ( ) Sum U U 0 = A(U 0 ) + B(U 0 )(W W 0 ) t/ X i=1 + h Ui U (i 1) i = t/ X t/ X i=1 Limit as 0 U t U 0 = i=1 A h U (i 1) i B h U (i 1) i h Wi W (i 1) i Z t 0 A(U s)ds + Z t 0 B(U s)dw s

54 1.6.1 Example: Simulating from interest rate models CIR Model dr t = κ(µ r t )dt + σ r t dw t A(r t ) = κ(µ r t ) B(r t,ρ) = σ r t ρ = (κ, µ, σ) 0 =(0.1, 0.08, 0.06) 0 N = 250 = 1/100

55 Two factor interest rate diffusion dv t = (a v + a vv v t + a vr r t )dt +(b 1v + b 1vv v t + b 1vr r t )dw 1t + b 2v dw 2t dr t = (a r + a rr r t )dt +(b 2r + b 2rr r t )exp(v t )dw 2t b 1vv = 0,b 2v =0,a v = a vv ρ = (a v,a vr,a r,a rr, b 1v,b 1vv,b 1vr,b 2v,b 2r,b 2rr ) = ( 0.18, , 0.19, , 0.69, 0, 0.063, 0, 0.038, 0.17) 0

56 1.7 EMM Estimation of Continuous Time Models for Interest Rates 1. Fit SNP Model to observed data 2. Create simulator functon for continuous time model 3. Estimate model by EMM 4. Check EMM diagnostics