Appendix: Parameter Optimization
5- Idiosyncratic Variances Update Formulae
Applying some matrix manipulations and discarding terms independent of the idiosyncratic noise covariance matrix, Ψj, the auxiliary function in equation (7) may be rewritten as
Q(Θ, Θ(i)) = −1
To find the new idiosyncratic noise covariance matrix, the auxiliary function above is differentiated with respect to its inverse, Ψ−1j , and equated to zero. Solving for Ψj and set-ting the off-diagonal elements to zeroes result in the updated idiosyncratic noise covariance matrix
This is a maximum since the second derivative ofQ(Θ/Θ(i)) with respect to Ψ−j1is negative.
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Contents
1 Introduction 3
2 Factor Analysed Hidden Markov Models 5
2.1 Optimizing FAHMM Parameters . . . 6 2.1.1 Forward-Backward Algorithm . . . 7 2.1.2 Continuous State Posterior Statistics . . . 8
3 Conditionally Heteroscedastic FAHMMs 8
4 A Switching State-Space Representation 10
4.1 Filtering Algorithm . . . 11 4.2 Smoothing Algorithm . . . 13
5 Viterbi Approximation For Latent Structure Inference 14
6 The EM Algorithm 16
7 Monte Carlo Simulations 19
7.1 Accuracy and Stability of the Estimates . . . 19 7.2 Asymptotic Normality of the Estimates . . . 22 7.3 Model Selection . . . 22
8 Financial Data 24
8.1 The Data . . . 28 8.2 Exploratory Analysis . . . 28 8.3 Dynamic Factor Analysis . . . 31
9 Conclusion 40
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