Trace likelihood and shrinkage in estimation of the forecasting models

I. Svetunkov, N. Kourentzes, 2015, 27th European Conference on operational Research, Glasgow.

The standard approach for estimating forecasting models is usually based on the likelihood function using one-step-ahead forecasting error. But several empirical studies have shown over the years that minimizing the sum of squared errors for multiple steps-ahead forecasts may lead to more accurate parameters estimation. Although it is intuitive that aligning the forecasting objective with the optimisation cost function is beneficial, we propose a statistically justified theoretical rationale to do this. We also demonstrate the difference between the traditional multi-steps-ahead objective function and the objective function obtained from the proposed trace likelihood function. In addition it has been argued in the literature that in the absence of a true model, it is preferable to use an extended likelihood, using multiple steps-ahead forecast error. We extend this finding by proving that maximising multi-steps-ahead likelihood is equivalent to single-step-ahead optimisation with parameter shrinkage. Therefore, maximising the proposed likelihood both incorporates the forecasting objective in the estimation and overcomes estimation limitations due to sampling or model form. We validate our theoretical findings by conducting experiments on real data and showing the advantage of the proposed approach in comparison to the standard likelihood function maximization.

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