Bayesian Risk Management by Sekerke Matt;

Author:Sekerke, Matt;
Language: eng
Format: epub
Publisher: John Wiley & Sons, Incorporated
Published: 2015-09-15T00:00:00+00:00

Now is a and . All of the following results generalize in a straightforward way to the multivariate case, however.

Polynomial Trend Components

The most basic time series model assumes that data evolve according to some kind of trend, which can be assumed without loss of generality to have a polynomial form. Polynomial trends can range in complexity from the one-dimensional, linear case of the local-level model to higher-order trends that might be rationalized as Taylor approximations to more complicated nonlinear trends. A third-order model is generally more than adequate for most applications, capturing the level, rate of increase, and change in the rate of increase for a series.

A pure polynomial trend DLM defines and , where n is the order of the approximation. The matrix En is a vector of length n with 1 in the initial position and zeros otherwise: whereas is an upper triangular matrix of ones with zeros in all other positions; for example,

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