Estimation of state space models with skewed shocks

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1 Estimation of state space models with skewed shocks Grzegorz Koloch April 30, 2012 Abstract This paper provides analytical formulae for the likelihood function, predictive densities and the filtration densities for linear state space models in which shocks are allowed to follow a skewed distribution. In particular we work with the closed skew normal distribution introduced in Genton (2004), which nests a normal distribution as a special case. Closure of the distribution in the state space setting with respect to all necessary transformations is guaranteed by a simple regularization procedure which does not influence the likelihood function. Presented results allow for estimation, filtration and prediction of such models as vector autoregressions and first order perturbations of DSGE models with skewed shocks. This allows economists to assess in their models asymmetries of observed data, of shocks, impulse responses and forecasts confidence intervals. Keywords: Maximum Likelihood Estimation, State Space Models, Closed Skewed Normal Distribution, DSGE, VAR. JEL: C51, C13, E32 1 Introduction In this paper we provide a maximum likelihood estimator for a linear state space model which allows the shocks in the transition equation to follow a skewed distri- National Bank of Poland, Economic Institute, Bureau of Applied Research, Swietokrzyska 19/21, Warsaw, Poland. Warsaw School of Economics, Institute of Econometrics, Department of Decision Analysis and Support, Al. Niepodleglosci 164, Warsaw, Poland. 1

2 bution 1. Let us consider a following state space model: y t = F x t + Hζ t, x t = Ax t 1 + Bξ t, ζ t N(0, Ψ ζ ), ξ t p ξ (θ ξ ) x 0 N(µ x0, Ψ x0 ) for t T = {1, 2,..., T }, where x t R n and y t R m denote states and observables respectively, R n n A 0, B R n n ξ, F R m n, F R m n ζ, n ξ, n ζ > 0, moreover Ψ ζ R n ζ n ζ, Ψ x0 R n n, and Ψ ζ, Ψ ζ 0. Finally, p ξ (θ ξ ) denotes a probability distribution of martingale difference shocks, which depends on a vector of parameters θ ξ. The usual assumption is that p ξ is a multivariate normal distribution, independent across its dimensions. In such a case the Kalman filter constitutes an optimal filtration procedure. If normality assumption is relaxed, Kalman filter remains an optimal linear filter, see e.g. Simon (2006). I assume that shocks are independent, but, for some values of θ ξ, p ξ is a skewed distribution. In particular, I assume that shocks follow a closed skew normal distribution (csn henceforth), see Genton (2004). The csn distribution is chosen, since it is closed under most transformations required for optimal filtration and ML estimation in the state space setting 2. Let us note, however, that matrix A can be rank deficient 3, which will turn out to be an obstacle in ML estimation as singularity of A precludes propagation of the csn distribution through the state space setting. I provide a regularization procedure which deals with this problem. To allow for prediction, filtration and estimation I provide exact formulae for p(y t Y t 1 ), p(x t Y t 1 ), p(ξ t Y t 1 ), p(ζ t Y t 1 ), p(x t Y t ), p(ξ t Y t ), p(ζ t Y t ) and for the likelihood function p(θ Y t ), where θ = (θ F, θ H, θ ζ, θ A, θ F, θ ξ, θ x0 ). When one goes 1 Measurement shocks are assumed to be normally distributed but extension to skewed measurement errors is straightforward. We prefer to work with normal measurement errors because of the appealing interpretation based on the central limit theorem. 2 Details are provided in section 2 3 Which is a standard case when the state space form represents a first order perturbation of a DSGE model. 2

3 through the article, one should also find it easy to obtain formulae for any type of filtration, eg. for the smoother. Since the state space form represents some very popular macroeconomic tools like VARs and reduced form first order perturbations of DSGE model, provided results allow for extending the agenda of empirical macroeconomics by issues related to skewness in the data. In particular it allows estimating VARs and DSGE models with skewed shocks and getting asymmetric confidence intervals of impulse responses and forecasts, hence to assess asymmetries of eg. inflation risks and finally, without referring to nonlinearities. Remainder of the paper is arranged as follows. Section 2 describes the closed skewed normal distribution and discusses its basic properties, section?? derives the filter and section 3 derives the quasi maximum-likelihood estimation procedure. Section 4 provides exemplary application and the last section concludes. 2 The skewed normal distribution 2.1 Definition Let us denote a density function of a p-dimensional normal distribution with mean 4 µ and positive-definite covariance matrix Σ by φ p (z; µ, Σ). Let us also denote a cumulative distribution function of a q-dimensional normal distribution with mean µ and nonnegative-definite covariance matrix Σ by Φ q (z; µ, Σ). For q > 1 function Φ q does not have a closed form. By R p q, p, q 1, let us denote a space of linear operators from R p to R q. For every M R p q let M denote a determinant of M and let r(m) denote a rank of M. We will define the closed skewed normal, possibly singular, distribution by means of the moment generating function (mgf). Then, under nonsingularity conditions, probability density function (pdf) will be provided. Definition 2.1. (csn distribution mgf ) Let µ R p and ϑ R q, p, q 1. Let Σ R p p and R q q, Σ, 0, and let D R q p. We say that random variable z has a (p, q) dimensional closed skewed normal distribution with parameters 4 All vectors are column vectors throughout the paper. 3

4 µ, Σ, D, ϑ and if moment generating function of z, M z (t), is given by: which henceforth will be denoted by: M z (t) = Φ q(d Σt; ϑ, + DΣD T ) Φ q (0; ϑ, + DΣD T e tt µ+ 1 tt Σt 2 ) z csn p,q ( µ, Σ, D, ϑ, ) Note that matrices Σ and are allowed to be singular. If Σ is not positive definite, i.e. Σ = 0, resulting distribution is called singular. If Σ is positive definite, i.e. Σ > 0, distribution is called nonsingular. The csn distribution is closed in the sense, that it is closed under full rank linear transformations 5. Isomorphic linear transformations transform nonsingular csn variables into nonsigular ones and singular variables into singular ones. Full row, but column rank deficient linear transformations (dimension shrinkage) transform nonsingular csn variables into nonsigular ones and singular variables into singular or nonsingular ones. Full column, but row rank deficient linear transformations (dimension expansion) transform nonsingular csn variables into singular ones, whreas singular variables remain singular. Both singular and nonsingular variables can be transformed into a non-csn distributed variables under a rank deficient transformation. The skewed normal distribution when considered as consisting of both singular and nonsingular cns variables is therefore not closed under arbitrary linear transformations, which has negative consequences for maximum likelihood estimation of state space models with csn shocks when the transition matrix in state space equations is singular, which typically is the case in DSGE modeling. If Σ > 0, a csn random variable z has a probability density function accordin to: Definition 2.2. (csn distribution pdf ) If a random variable z follows a (p, q)- dimensional, p, q 1, closed skewed normal distribution with parameters µ, Σ, D, ϑ and, where µ R p, ϑ R q, Σ R p p, Σ > 0, R p p, 0 and D R q p, 5 Under full rank transformation we mean full row or full column rank transformation and this definition embraces the case when matrix of the transformation is square and has a full rank. In the latter case the transformation is called an isomorphism. When both the row and the column ranks are not full, transformation is called rank deficient. 4

5 than probability density function of z is given by: p(z) = φ p (z; µ, Σ) Φ q(d(z µ); ϑ, ) Φ q (0; ϑ, + DΣD T ) (1) Density function (1) defines a (p, q)-dimensional nonsingular closed skewed normal distribution in the sense that a random variable has (p, q)-dimensional nonsingular closed skewed normal distribution with parameters µ, Σ, D, ϑ and if and only if its density function for every z R p equals p(z) in (1). The probability density function (1) involves a probability distribution function of a q-dimensional normal distribution for, in principle, arbitrarily large q, which entails computational difficulties when working with a likelihood function based on p(z) in (1). Parameters µ, Σ and D have interpretation of location, scale and skewness parameters respectively. Parameters ϑ and are artificial, but inclusion of these additional dimensions allows for closure of the csn distribution under conditioning and marginalization respectively. The q-dimension is also artificial, but it allows for closure for sums and the joint distribution of independent (not necessarily iid) variables. When Σ, D and are scalars, they will be denoted respectively by σ, d and δ. 2.2 Basic properties Let us make use of the following: Remark 2.3. For p = q = 1, ϑ = 0 and = 1 the csn distribution reduces to the Azzalini skewed normal distribution, see Azzalini (1996), Azzalini (1999). Such a case will be denoted by: In the next sections we will find useful the following: z sn( µ, σ, d) (2) Corollary 2.4. Let z csn 1,1 ( µ, σ, d, ϑ, δ) for parameters as in definition (2.2), and assume that δ + d 2 σ 0, then: 2 d σ E(z) = µ + π δ + d2 σ var(z) = σ 2 d 2 σ 2 π δ + d 2 σ ) 3 ( E(z E(z)) 3 = (2 π 2 ) ( 2 π d σ (δ + σd 2 ) 1 2 (3) (4) ) 3 (5) 5

6 It follows that: Remark 2.5. Let z csn 1,1 ( µ, σ, d, ϑ, δ) for parameters as in definition (2.4), then d σ E(z) = 0 if and only if µ =. δ+d2 σ 2 π Which we use in sampling csn variables withe zero mean. We also need the following: Corollary 2.6. Let z csn p,q ( µ, Σ, D, ϑ, ), p, q 1, for parameters as in definition (2.1). Elements of z are independent if and only if matrices Σ and D are diagonal. Since Σ is p p and D is q p, corollary (2.6) implies that it is impossible to have q = 1 while keeping elements of z independent for p > 1, because it has to be the case that q = p > 1 in order for D to be diagonal. This is relevant for state space models with csn distributed iid disturbances eg. shocks in the transition equation, because the state variable, say ξ t, in every period consists of the csn distributed state from the previous period, say ξ t 1, plus the csn-distributed disturbance 6, say u t, and when we add two csn variables we have to add their q- dimensions, so thet the q-dimension of ξ t is the sum of q-dimensions of ξ t 1 and u t, hence, according to corollary (2.6), contribution of u t to q-dimension of ξ t in every period cannot be squeezed to eg. 1, but must equal size of u t, hence q-dimension of ξ t quickly increases which poses serious numerical difficulties. In further sections we will also need the following corollaries ( ): Corollary 2.7. Let z csn p,q ( µ, Σ, D, ϑ, ), p, q 1, for parameters as in definition (2.1). Let also x N(µ x, Σ x ), Σ x > 0, be independent of z, then: z + x csn p,q ( µ + µ x, Σ + Σ x, D Σ( Σ + Σ x ) 1, ϑ, + (D(I Σ( Σ + Σ x ) 1 )) ΣD T ) Corollary 2.8. Let z csn 1,q ( µ, σ, d, ϑ, δ), q 1 and for parameters as in definition (2.1), let also ρ 0 and b R, then: ρz + b csn 1,q (ρ µ + b, ρ 2 σ, 1 d, ϑ, δ) ρ 6 Both state from the previous period and the disturbance are transformed by the linear transformation in state space models, but let us ignore this fact for the present argumentation (or assume this transformations are identities). 6

7 Corollary 2.9. Let z csn p,q ( µ, Σ, D, ϑ, ), p, q 1, for parameters as in definition (2.1), let also A R p p, A > 1, and b R p, then: Az + b csn p,q (A µ + b, A ΣA T, D ΣA 1, ϑ, ) Corollary Let z i csn p,qi ( µ i, Σ i, D i, ϑ i, i ), p, q i 1, i = 1, 2,..., n, for parameters as in definition (2.1), then n i=1 z i csn p, P n i=1 q i ( µ, Σ, D, ϑ, ), where: µ = n µ i, Σ = i=1 n Σ i, D = (Σ 1 D1 T,..., Σ n Dn T ) T (Σ ) 1, i=1 n n ϑ = (ϑ T 1, ϑ T 2,..., ϑ T n ) T, = + D Σ D [ D i Σi ]( Σ ) 1 [ D i Σi ] 1 for = n i=1 i, D = n i=1 D i and Σ = Σi, where operator, arbitrary matrices A and B, is defined as: A B = A 0 0 B 2.3 The regularization procedure The following proposition is central for providing analytical maximum likelihood estimation. i=1 i=1 Proposition 2.1. Let η be distributed according to a csn p,q for some p, q 1 with parameters µ η, Σ η 0, D η, ϑ η and η > 0. Let z t = Gη, A R p,p. Then, z t has a csn (possibly singular) distribution if and only if r(g T ) = r([g T w i ]) for all i = 1, 2,..., q, where r(g) denotes rank of G and w i denotes the i-th row D η. Proposition (2.1) states, that for a csn variable η, variable z t = Gη has a csn distribution if and only if rows of D η are linear combinations of rows of G. In other words, all rows of D η must belong to the image of A T, i.e. to span(a T ). This condition is always satisfied (regardless of D η ) if G has a full column rank. However, for a rank deficient operator G this is a very demanding condition, since D η can be in principle arbitrary. Although proposition (2.1) constitutes a negative result for η as a p-dimensional variable, it has to be stressed that it is assured that elements of η are csn distributed. 7

8 The correspondence between proposition (2.1) and the state space formulation with which we work goes as follows: z t = x t G = [A B] η = (x t 1, ξ t ) T therefore for the csn distribution to propagate, we need G to have full column rank. When one works with medium- or large-size DSGE models, the reduced form representation matrix G can be rank deficient. Since the following argument applies to full rank matrices G, one needs to reformulate the model so that G has full column rank, but the value of the likelihood function is unaffected. This turns out to be simple. If G is rank deficient, than some of the states are linear combination of the remaining ones, which means, that they can be substitutes out from the state-space representation both in the transition and in the measurement equation. This obviously does not affect the value of the likelihood function and, moreover, this can be done automatically, since a priori, i.e. before the reduced form has been calculated, it is difficult to judge which variables will be redundant. Such a regularization allows for further steps to apply. 3 The filter In this section I provide formulae for predictive densities p(y t Y t 1 ), p(x t Y t 1 ), p(ξ t Y t 1 ), p(ζ t Y t 1 ) and for filtration densities p(x t Y t ), p(ξ t Y t ), p(ζ t Y t ). All the densities depend on appropriate elements of the parameter vector θ = (θ F, θ H, θ ζ, θ A, θ F, θ ξ, θ x0 ). First, however, I derive unconditional distribution for states and observables. 3.1 Unconditional distributions In DSGE modeling it usually is the case that n ξ < p, hence Bξ t does not have a distribution function since matrix BΣ ξ B T is singular. Assume that n ξ < p and r(b) = n ξ. In this case Bξ t follows: Bξ t CSN p,q (µ B, Σ B, D B, ϑ B, B ) 8

9 where: µ B = Bµ ξ Σ B = BΣ ξ B T D B = D ξ (B T B) 1 B T (6) ϑ B = ϑ ξ B = ξ which is a (p, n ξ )-dimensional singular closed skewed normal distribution. This is true for every t = 1, 2,..., T. Since normal distribution is a special case of CSN distribution, it can be written that: x 0 CSN p,1 (µ x0, Σ x0, D x0, ϑ x0, x0 ) for 7 : µ x0 = x Σ x0 = Ψ x D x0 = 0 (7) ϑ x0 = 0 x0 = 1 Generally, i.e. for t = 1, 2,..., T, if: x t 1 CSN p,qt 1 (µ xt 1, Σ xt 1, D xt 1, ϑ xt 1, xt 1 ) which is true for t = 1, i.e. for x 0, with q 0 = 1, than Ax t 1 follows: Ax t 1 CSN p,qt 1 (µ A,t 1, Σ A,t 1, D A,t 1, ϑ A,t 1, A,t 1 ) where: µ A,t 1 = Aµ x,t 1 Σ A,t 1 = AΣ x,t 1 A T D A,t 1 = D x,t 1 Σ x,t 1 A T Σ 1 A,t 1 (8) ϑ A,t 1 = ϑ x,t 1 A,t 1 = x,t 1 + D x,t 1 Σ x,t 1 D T x,t 1 D x,t 1 Σ x,t 1 A T Σ 1 A,t 1 AΣ x,t 1D T x,t 1 7 x0 > 0 can be arbitrary. 9

10 which is a (p, q t 1 )-dimensional closed skewed normal distribution, and this distribution is non-singular since it is assumed that r(a) = p. This is true for t = 0 and also, by induction, for every = 1, 2,..., T. Induction is true, since: x t = Ax t 1 + Bξ t where Ax t 1 and Bξ t are independent random variables, from which it follows, that x t is distributed according to: x t CSN p,qt 1 +q(µ xt, Σ xt, D xt, ϑ xt, xt ) this means: x t CSN p,qt (µ xt, Σ xt, D xt, ϑ xt, xt ) for q t = q t 1 + q, where: µ x,t = µ A,t + µ B Σ x,t = Σ A,t + Σ B D x,t = [ Σ A,t DA,t, T Σ B DB T ] T Σ 1 x,t ϑ A,t = [ ϑ T A,t, ϑ T ] T B,t (9) x,t = A,t B + (D A,t D B )(Σ A,t Σ B )(D A,t D B ) T + (D A,t Σ A,t D B Σ B )(Σ x,t ) 1 (Σ A,t DA,t T Σ B DB) T Since q 0 = 1 and q t = q t 1 + q, we have q t = tq + q 0 = tq + 1 and: x t CSN p,tq+1 (µ xt, Σ xt, D xt, ϑ xt, xt ) For aplications we will work with q = 1, which yields q t = t + 1. So far we have derived formulae for distribution of states x t for all t = 1, 2,..., T. Now we will pin down the distribution of observables y t for all t = 1, 2,..., T, which is given by measurement equation: y t = F x t + Hu t where: u t N(0, Ψ u ) for all t = 0, 1, 2,..., T. Since normal distribution is a special case of CSN distribution, it can be written that: u t CSN nu,1(µ u, Σ u, D u, ϑ u, u ) 10

11 for 8 : µ u = 0 Σ u = Ψ u D u = 0 (10) ϑ u = 0 u = 1 In DSGE modeling, since measurement errors are introduced in order to avoid stochastic singularity, it usually is the case that n = n u + n ξ, and since n ξ > 0, it is the case that n u < n, hence Hu t does not have a distribution function since matrix HΨ u H T is singular. Assume that n u < n and r(h) = n u. In this case Hu t follows: Hu t CSN n,1 (µ H, Σ H, D H, ϑ H, H ) where: µ H = Hµ u Σ H = HΨ u H T D H = D u (H T H) 1 H T (11) ϑ H = ϑ u H = u for all t = 1, 2,..., T, which is a (n, 1)-dimensional singular normal distribution. Now, form distribution of x t it follows that: F x t CSN n,tq+1 (µ F,t, Σ F,t, D F,t, ϑ F,t, F,t ) where: µ F,t = F µ x,t Σ F,t = F Σ x,t F T D F,t = D x,t Σ x,t F T Σ 1 F,t (12) ϑ F,t = ϑ x,t 8 u > 0 can be arbitrary. F,t = x,t + D x,t Σ x,t D T x,t D x,t Σ x,t F T Σ 1 F,t F Σ x,td T x,t 11

12 which is a (p, tq + 1)-dimensional closed skewed normal distribution, and this distribution is non-singular since it is assumed that r(f ) = n. From the measurement equation it follows that y t is distributed according to: y t CSN n,tq+1 (µ yt, Σ yt, D yt, ϑ yt, yt ) or: where: y t CSN n,tq+2 (µ yt, Σ yt, D yt, ϑ yt, yt ) µ y,t = µ F,t + µ H Σ y,t = Σ F,t + Σ H D y,t = [ Σ F,t DF,t, T Σ H DH T ] T Σ 1 y,t ϑ y,t = [ ϑ T F,t, ϑ T ] T H,t (13) y,t = F,t H + (D F,t D H )(Σ F,t Σ H )(D F,t D H ) T + (D F,t Σ F,t D H Σ H ) (Σ y,t ) 1 ( Σ F,t DF,t T Σ H DH T ) 3.2 Conditional distributions For t = 1, 2,..., T, let us define an information set: Y t = {y 1, y 2,..., y t } which consists of realized values of observables up to time t. We want to derive the a posteriori distribution (x t Y t ) and we will do so on the basis of the joint distribution (x t, y t Y t 1 ). Assume, that the a posteriori distribution of states x t 1, i.e. conditional distribution of states x t 1 with respect to the information set Y t 1, therefore after observing y t 1, is given by: (x t 1 Y t 1 ) = (µ t 1, Σ t 1, D t 1, ϑ t 1, t 1 ) for some parameters µ t 1, Σ t 1, D t 1, ϑ t 1 and t 1. If so, the a priori, i.e. prior to observing y t, conditional distribution of x t is given by: (x t Y t 1 ) = (Ax t 1 + Bξ t Y t 1 ) = (Ax t 1 Y t 1 ) + Bξ t = = (µ, Σ, D, ϑ, ) 12

13 for: µ = µ A + µ B Σ = Σ A + Σ B D = [ Σ A DA, T Σ B DB T ] T Σ 1 ϑ = [ ϑ T A, 0 T ] T = A B + (D A D H )(Σ A Σ D )(D A D H ) T + (D A Σ A D H Σ H )(Σ) 1 (D A Σ A D H Σ H ) T where: µ A = Aµ t 1 Σ A = AΣ t 1 A T D A = D t 1 Σ t 1 A T Σ 1 A ϑ A = ϑ t 1 = t 1 + D t 1 Σ t 1 D T t 1 D t 1 Σ t 1 A T Σ 1 A AΣ t 1D T t 1 Information contained in the joint distribution (x t, y t Y t 1 ) is equivalent to the information contained in the distribution (x t, e t Y t 1 ) where: e t = y t E(y t Y t 1 ) which is approximated by 9 : e t = y t F µ Distribution of e t conditional on x t andy t 1 is normal: (e t x t, Y t 1 ) = (F x t + Hu t F µ x t, Y t 1 ) = (F (x t µ) + (Hu t Y t 1 )) = (F (x t µ) + Hu t ) = N(F (x t µ), Σ H ) Therefore joint distribution (x t, e t Y t 1 ) = (x t, y t Y t 1 ) is given by: (x t, e t Y t 1 ) = ( µ, Σ, D, ϑ, ) 9 Can be approximated explicitly from state-space eq-s for observed x t 1. 13

14 which parameters are approximated by: µ = µ 0 Σ = Σ ΣF T F T Σ F ΣF T + Σ H D = D ϑ = ϑ 0 (14) = Therefore, the a posteriori distribution is: (x t Y t ) = (µ t, Σ t, D t, ϑ t, t ) for: 4 Likelihood function µ t = µ + ΣF T (F T ΣF T + Σ D ) 1 (Y t F µ) Σ t = Σ ΣF T (F T ΣF T + Σ D ) 1 F Σ T D t = D ϑ t = ϑ t = The likelihood function of the state space model which we work with is given by: T L = p(y 0 ) p(y t Y t 1 ) t=2 Because Hu t is independent of Y t 1, using the measurement equation we see that the conditional a priori distribution of y t Y t 1 is: (y t Y t 1 ) = (F x t + Hu t Y t 1 ) = (F x t Y t 1 + Hu t ) Since distribution of x t Y t 1 is: x t Y t 1 (µ, Σ, D, ϑ, ) 14

15 distribution of y t Y t 1 = F x t Y t 1 + Hu t is: y t Y t 1 CSN n, (µ y, Σ y, D y, ϑ y, y ) for: µ y = µ F + µ H Σ y = Σ F + Σ H D y = [ Σ F DF T, Σ H DH T ] T Σ 1 y ϑ y = [ ϑ T F, ϑ T ] T H (15) y = F H + (D F D H )(Σ F Σ H )(D F D H ) T + (D F Σ F D H Σ H ) (Σ y ) 1 ( Σ F DF T Σ H DH T ) where: µ F = F µ x Σ F = F Σ x F T D F = D x Σ x F T Σ 1 F (16) ϑ F = ϑ x F = x + D x Σ x D T x D x Σ x F T Σ 1 F F Σ xd T x therefore: p(y t Y t 1 ) = φ p (y t ; µ y, Σ y ) Φ q(d y (y t µ y ); ϑ y, y ) Φ q (0; ϑ y, y + D y Σ y D T y ) (17) Numerical calculation of the likelihood function can be carried-out by means of the following sequential procedure: 1. Assume parameters of p(y 1 ) and parameters of p(x 1 Y 1 ). 2. For t = 2, 3,..., T implement the following steps: (a) For t > 2 take parameters of p(x t 1 Y t 1 ), i.e. µ t 1, Σ t 1, D t 1, ϑ t 1 and t 1, from the previous iteration, and for t = 2 take them from the initial assumption about p(x 1 Y 1 ). (b) Calculate parameters of p(x t Y t 1 ), i.e. µ, Σ, D, ϑ and. (c) Calculate parameters of p(y t Y t 1 ), i.e. µ y, Σ y, D y, ϑ y and y. (d) Update value of the likelihood function. (e) Calculate parameters of p(y t Y t ), i.e. µ t, Σ t, D t, ϑ t and t. 15

16 Value of the approximate likelihood function L = p(y θ) is now calculated for given θ, which can be implemented in any numerical optimization routine. References [Azalini and Dalla Valle (1996)] Azzalini, A. and A. Dalla Valle.: The Multivariate Skew-Normal Distribution, Biometrika (1996), Vol. 83, No. 4, pp [Azzalini and Capitanio (1999)] Azzalini, A. and A. Dalla Valle.: Applications of the Multivariate Skew Normal Distribution, Journal of the Royal Statistical Society. Series B (Statistical Methodology) (1999), Vol. 61, No. 3, pp [Genton (2004)] Genton, M. G. (ed.): Skew-elliptical distributions and their applications. A journel beyond normality., A CRC Press Company (2004). [Simon (2006)] Simon, D.: Optimal State Estimation. Kalman, H and Nonlinear Approches, John Willey & Sons (2006). 16