Abstract
The use of state space models and their inference is illustrated using the package <b>SsfPack</b> for Ox. After a rather long introduction that explains the use of <b>SsfPack</b> and many of its functions, four case-studies illustrate the practical implementation of the software to real world problems through short sample programs. The first case consists in the analysis of the well-known (at least to time series analysis experts) Nile data with a local level model. The other case-studies deal with ARIMA and RegARIMA models applied to the (also well-known) Airline time series, structural time series models applied to the Italian industrial production index and stochastic volatility models applied to the FTSE100 index. In all applications inference on the model (hyper-) parameters is carried out by maximum likelihood, but in one case (stochastic volatility) also an MCMC-based approach is illustrated. Cubic splines are covered in a very short example as well.
Highlights
SsfPack (Koopman, Shephard, and Doornik 1999, 2008) is a library of routines for state space modelling and inference written in C and linked to Ox, the efficient matrix language developed by Doornik (2007)1
In order to lighten the exposition of the features of SsfPack, we introduce slight modifications to the state space notation used in this Journal of Statistical Software volume: for t = 1, . . . , n, yt = ct + Ztαt + εt, αt+1 = dt + Ttαt + ηt, εt ∼ NID(0, Ht), ηt ∼ NID(0, Qt), Ct = Eηtεt where yt, ct and εt are (p × 1) vectors, αt, dt and ηt are (m × 1) vectors, Zt is the (p × m) observation matrix, Tt is the (m × m) transition matrix, Ht is (p × p), Qt is (m × m) and Ct is (m × p)
The commercial version of the software, SsfPack Extended, implements exact algorithms to deal with diffuse initial conditions for the state variables and efficient routines to carry out computations in systems with large matrices, in some cases by exploiting the fact that these are typically sparse
Summary
SsfPack (Koopman, Shephard, and Doornik 1999, 2008) is a library of routines for state space modelling and inference written in C and linked to Ox, the efficient matrix language developed by Doornik (2007). One is free for academic use (the same policy as Ox console) and it is referred to as SsfPack Basic, while the other is commercial and its name is SsfPack Extended. The latter version includes the functionalities of SsfPack Basic plus a set of algorithms that are computationally. Only SsfPack Basic will be discussed, but the reader may be interested in knowing that for benefitting from the higher computational efficiency of SsfPack Extended the state space representation of the model must have orthogonal measurement errors (Ht diagonal) and the disturbances in the transition equations must be uncorrelated with the measurement errors (Eηtεt = 0). ◦ // denotes a line comment; ◦ statements end with a semicolon; ◦ & is used for pointers, that is, for passing a variable through its address; ◦ if pointer is a pointer, the pointed variable can be read or assigned using the syntax pointer[0]; ◦ matrix indexing begins with 0; ◦ defines a matrix constant: , switches to the column and ; to the row; ◦ the operator ~ joints two matrices by columns (side by side), while | stacks two matrices; ◦ arithmetic and logic operators on matrices follow linear algebra conventions (e.g., * is matrix product), elementwise operators are preceded by a dot (e.g., .* is Hadamard product)
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