infonet-economy

This paper develops and applies a novel estimation procedure for quantile regressions with time-varying coefficients based on a fully parametric, multifactor specification. The algorithm recursively filters the multifactor dynamic coefficients with a Kalman filter and parameters are estimated by maximum likelihood. The likelihood function is built on the Skewed-Laplace assumption. In order to eliminate the non-differentiability of the likelihood function, it is reformulated into a non-linear optimization problem with constraints. A relaxed problem is obtained by moving the constraints into the objective, which is then solved numerically with the Augmented Lagrangian Method. In the context of an application to electricity prices, the results show the importance of modelling the time-varying features and the explicit multi-factor representation of the latent coefficients is consistent with an intuitive understanding of the complex price formation processes involving fundamentals, policy instruments and participant conduct.

We demonstrated the value of a well specified dynamic model for quantile estimation by means of an application to electricity price risk. Electricity prices are a commodity in which price formation is nonlinear in its relationship to fundamentals, dynamic in the relative influences of drivers, with further complications introduced by policy interventions for supporting specific technologies and opportunities for participant conduct to be influential at high and low prices. Despite these complications careful consideration of the shape of the supply function with its concave, flat and convex regions, together with the information that is available to market participants day ahead allows plausible expectations for the price dynamics to be considered, and these explain very well the signs and significance of the parameters in the estimated models. Nevertheless, the models need to have a detailed specification with the various quantiles being related to multiple factors through coefficients which have dynamic properties themselves related to some of the exogenous factors. This modelling requirement motivates the development of quantile models that need fully parametric specifications to capture dynamics through exogenous factors and time-varying coefficients.

A novel general methodology has therefore been developed in which time-varying multi factor coefficients are recursively estimated with a Kalman filter using maximum likelihood. Since the likelihood function is non-differentiable, the problem is re-formulated as a non-linear optimization with constraints, and furthermore re-formulated again by moving the constraints into the objective function to solve an augmented Lagrangian method. With careful selection of starting values, maximum likelihood estimates were thereby acquired. As a general approach, we would expect this to be useful in many applications of risk management and quantile estimation where there is dynamic complexity in price formation and plausible exogenous price drivers.