Algorithm 878: Exact VARMA likelihood and its gradient for complete and incomplete data with Matlab

ABSTRACT

Matlab functions for the evaluation of the exact log-likelihood of VAR and VARMA time series models are presented (vector autoregressive moving average). The functions accept incomplete data, and calculate analytical gradients, which may be used in parameter estimation with numerical likelihood maximization. Allowance is made for possible savings when estimating seasonal, structured or distributed lag models. Also provided is a function for creating simulated VARMA time series that have an accurate distribution from term one (they are spin-up free). The functions are accompanied by a a simple example driver, a program demonstrating their use for real parameter fitting, as well as a test suite for verifying their correctness and aid further development. The article concludes with description of numerical results obtained with the algorithm.

REFERENCES

• 1: George, B. and Williams, L. 2004. A structured experiment of test-driven development. Inform. Softw. Tech. 46, 5, 337--342.
• 2: Hanna, E., Jonsson, T., and Box, J. E. 2004. An analysis of Icelandic climate since the nineteenth century. Int. J. Climatology 24, 10, 1193--1210.
• 3: Jonasson, K. and Ferrando, S. E. 2008. Evaluating exact VARMA likelihood and its gradient when data are incomplete. ACM Trans. Math Softw. 35, 1.
• 4: Mauricio, J. A. 1997. Algorithm AS 311: The exact likelihood function of a vector autoregressive moving average model. Appl. Statist. 46, 1, 157--171.
• 5: Nielsen, H. B. 2000. UCMINF—An algorithm for unconstrained, nonlinear optimization. Tech, Rep. IMM-REP-2000-19, Department of Mathematical Modelling, Technical University of Denmark.
• 6: Schneider, T. and Neumaier, A. 2001. Algorithm 808: ARfit—A Matlab package for the estimation of parameters and eigenmodes of multivariate autoregressive models. ACM Trans. Math. Softw. 27, 1, 58--65.
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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.3 PROBABILITY AND STATISTICS
      Subjects: Multivariate statistics

Additional Classification:
  G. Mathematics of Computing
  G.3 PROBABILITY AND STATISTICS
      Subjects: Stochastic processes; Time series analysis; Statistical software
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Algorithm design and analysis; Documentation; Efficiency
      Nouns: MATLAB

  I. Computing Methodologies
  I.6 SIMULATION AND MODELING
      I.6.0 General

  J. Computer Applications
  J.2 PHYSICAL SCIENCES AND ENGINEERING
  J.4 SOCIAL AND BEHAVIORAL SCIENCES
      Subjects: Economics

General Terms:
Algorithms, Documentation, Verification

Keywords:
ARMA, Exact likelihood function, VARMA, incomplete data, missing values, vector autoregressive moving average model