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On the second-order random walk model for irregular locations. (English) Zbl 1199.60276
A new model of second-order random walk \(x_1,\dots,x_n\) is proposed for the case when \(x_i\) should represent a smooth curve at locations \(s_i\), \(s_1<\dots<s_n\). The model is based on the idea to interpret \(x_i\) as observations of a continuous process \(x(t)\) at points \(s_i\) where \(x(t)\) is a weak solution to \( {d^2\over dt^2}x(t)={d\over dt}W(t) \), \(W(t)\) being the standard Wienner process. In fact, a Galerkin approximation is used with some simplification to obtain \(x_i\) as a Gaussian Markov random field with the properties \[ E(x_i\;| \;x_{i-1},x_{i-2})= \left( {d_{i-1}\over d_{i-2}}+1 \right)x_{i-1} -{d_{i-1}\over d_{i-2}} x_{i-2}, \]
\[ \text{Var}(x_i\;| \;x_{i-1},x_{i-2})= {d_{i-1}^2(d_{i-2}-d_{i-1})\over 2} \] The random walk is used as a prior in a semiparametric model for Bayesian analysis of an econometric time series.

60J25 Continuous-time Markov processes on general state spaces
62F15 Bayesian inference
91B84 Economic time series analysis
GMRFLib; Fahrmeir
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