The Annals of Applied Statistics

Power-weighted densities for time series data

Daniel McCarthy and Shane T. Jensen

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While time series prediction is an important, actively studied problem, the predictive accuracy of time series models is complicated by nonstationarity. We develop a fast and effective approach to allow for nonstationarity in the parameters of a chosen time series model. In our power-weighted density (PWD) approach, observations in the distant past are down-weighted in the likelihood function relative to more recent observations, while still giving the practitioner control over the choice of data model. One of the most popular nonstationary techniques in the academic finance community, rolling window estimation, is a special case of our PWD approach. Our PWD framework is a simpler alternative compared to popular state–space methods that explicitly model the evolution of an underlying state vector. We demonstrate the benefits of our PWD approach in terms of predictive performance compared to both stationary models and alternative nonstationary methods. In a financial application to thirty industry portfolios, our PWD method has a significantly favorable predictive performance and draws a number of substantive conclusions about the evolution of the coefficients and the importance of market factors over time.

Article information

Ann. Appl. Stat. Volume 10, Number 1 (2016), 305-334.

Received: November 2014
Revised: September 2015
First available in Project Euclid: 25 March 2016

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Time series analysis power prior forecasting finance


McCarthy, Daniel; Jensen, Shane T. Power-weighted densities for time series data. Ann. Appl. Stat. 10 (2016), no. 1, 305--334. doi:10.1214/15-AOAS893.

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Supplemental materials

  • Discussion of “Power-weighted densities for time series data”. In McCarthy and Jensen (2015), we show the conjugacy for exponential families under our PWD approach and the Kullback–Leibler optimality of the general PWD setup. We provide additional results for computational cost and simulations comparing additional PWD variants to competing models. An adaptive PWD variant which switches between linear and exponentially decaying weights is also explored.