Maximum Likelihood Estimation of VARMA Models Using a State-Space EM Algorithm

Maximum Likelihood Estimation of VARMA Models Using a State-Space EM Algorithm. Metaxoglou, K. & Smith, A. Journal of Time Series Analysis, 28(5):666–685, Wiley Online Library, 2007.

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We introduce a state‐space representation for vector autoregressive moving‐average models that enables maximum likelihood estimation using the EM algorithm. We obtain closed‐form expressions for both the E‐ and M‐steps; the former requires the Kalman filter and a fixed‐interval smoother, and the latter requires least squares‐type regression. We show via simulations that our algorithm converges reliably to the maximum, whereas gradient‐based methods often fail because of the highly nonlinear nature of the likelihood function. Moreover, our algorithm converges in a smaller number of function evaluations than commonly used direct‐search routines. Overall, our approach achieves its largest performance gains when applied to models of high dimension. We illustrate our technique by estimating a high‐dimensional vector moving‐average model for an efficiency test of California's wholesale electricity market.

@article{metaxoglou2007maximum,
  title={Maximum Likelihood Estimation of VARMA Models Using a State-Space EM Algorithm},
  author={Metaxoglou, Konstantinos and Smith, Aaron},
  journal={Journal of Time Series Analysis},
  volume={28},
  number={5},
  pages={666--685},
  year={2007},
	url={https://files.asmith.ucdavis.edu/2007_JTSA_MS_VARMA.pdf},
	keywords={econometrics},
	abstract={We introduce a state‐space representation for vector autoregressive moving‐average models that enables maximum likelihood estimation using the EM algorithm. We obtain closed‐form expressions for both the E‐ and M‐steps; the former requires the Kalman filter and a fixed‐interval smoother, and the latter requires least squares‐type regression. We show via simulations that our algorithm converges reliably to the maximum, whereas gradient‐based methods often fail because of the highly nonlinear nature of the likelihood function. Moreover, our algorithm converges in a smaller number of function evaluations than commonly used direct‐search routines. Overall, our approach achieves its largest performance gains when applied to models of high dimension. We illustrate our technique by estimating a high‐dimensional vector moving‐average model for an efficiency test of California's wholesale electricity market.},
  publisher={Wiley Online Library}
}

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