Genetic hybrid tuning of VARMAX and state space algorithms

Abstract

The aim of the study was to monitor the system theoretic exogenous variables augmented state space algorithm of Aoki (State space modelling of time series. Springer, Heidelberg, 1987) and the VARMAX algorithm of Spliid (J Am Stat Assoc 78(384):843–849, 1983) within a geno-mathematical framework towards optimal parametric conditions/search intervals. Both algorithms were implemented as an integrated support library for a general computational platform, the Genetic Hybrid Algorithm (GHA), where some key parameters of the algorithms are defined in a search process utilizing a mixed geno-mathematical search technique. The empirical results of our tests using real economic data from the European stock market are encouraging. Specifically, the information criteria used in the VARMAX-search (Vector Autoregressive Moving Average algorithm with Exogenous variables) algorithm tend to favor parsimonious model representations automatically. Furthermore, the state space algorithm captures almost the same dynamics as the complex VARMAX-model estimated in the study. Both algorithms have encouraging in sample properties. When generating k-steps forecasts out-of-sample, k > 1, the state space algorithm seems to deteriorate faster than the VARMAX algorithm, however. The results suggest that more empirical testing is needed, especially in different situations with different degrees of model order and stationarity conditions, in order to provide more evidence on the suitability of the competing methods in particular cases. We demonstrated that the Genetic Hybrid Algorithm can be used as a generic platform for parametric search in vector valued time series modelling. Efficient procedures for optimal grouping of the individual time series processes and recognition of heteroskedasticity may improve the performance of the algorithms further.

Appendices

Appendix 1

VARMA-search for stock prices are given in Table 7.

Table 7  

Appendix 2: Iterative and symplectic estimates of the RICCATI-equation for the covariance matrix of the state vector

(a) Test period 01/86–11/86.

Iterative estimate:

$$ \left[ \begin{array}{lllll} \hfill{0.977343}&\hfill{-0.029117} & \hfill { - 0.012215} &\hfill{0.071944}&\hfill{0.016743} \\ \hfill{-0.029117}&\hfill{0.932263} & \hfill { - 0.086411} &\hfill{0.075123}&\hfill{0.004801} \\ \hfill{-0.012215}&\hfill{-0.086411}&\hfill {0.918078} & \hfill {0.001735} & \hfill {0.103286} \\ \hfill{0.071944}&\hfill{0.075123}&\hfill {0.001735} & \hfill{0.277619} & \hfill {0.024579} \\ \hfill{0.016743}&\hfill{0.004801}&\hfill {0.103286} & \hfill{0.024579}&\hfill{0.584477} \end{array}\right] $$

Symplectic estimate:

$$ \left[ \begin{array}{lllll} \hfill{0.977347} & \hfill { -0.029117} & \hfill { - 0.012215} &\hfill {0.071951} &\hfill {0.016741} \\ \hfill { - 0.029117} & \hfill {0.932268} & \hfill { - 0.086411} &\hfill {0.075129} & \hfill {0.004807}\\ \hfill { - 0.012214} & \hfill { - 0.086412} &\hfill {0.918082} & \hfill {0.001734} & \hfill {0.103288} \\ \hfill {0.071955} & \hfill {0.075126} & \hfill {0.001735} & \hfill{0.277603} & \hfill {0.024582} \\ \hfill {0.016741} & \hfill {0.004808} & \hfill {0.103290} & \hfill {0.024595} & \hfill {0.584668} \end{array}\right] $$

(b) Test period 01/87–12/88

Iterative estimate:

$$ \left[ {\begin{array}{*{20}c} \hfill {0.993520} & \hfill {0.009682} & \hfill { - 0.008211} \\ \hfill {0.009628} & \hfill {0.979185} & \hfill { - 0.025033} \\ \hfill { - 0.008211} & \hfill { - 0.025033} & \hfill {0.916598} \\ \end{array} } \right] $$

Symplectic estimate:

$$ \left[ { \begin{array}{*{20}c} \hfill { 0.993498} & \hfill { 0.009667} & \hfill { - 0.008201} \\ \hfill { 0.009667} & \hfill { 0.979241} & \hfill { - 0.025054} \\ \hfill { - 0.008204} & \hfill { - 0.025054} & \hfill { 0.916602} \\ \end{array} } \right] $$