Project without external funding

Autoregressive error processes and cubic splines

Project Details

Project duration: 01/2001–12/2003

Abstract

Consider the linear model *y = X beta + u*, where *u = (u*_{1},...,u_{T})' and *u*_{t} = rho u_{t-1} + varepsilon_{t}, *varepsilon*_{t} idependent *N(0, sigma*^{2}) and |rho| < 1. The covariance-matrix of *u* is proportional to * Omega = (p*^{|i-j|}; i,j = 1,...,T) and processes an explicit tridiagonal inverse *Omega*^{-1}. This allows the computation of the aitken-estimator in a very simple way. Equidistant cubic spline interpolation requires the inversion of a tridiagonal matrix *Omega*_{0} whose diagonal elements are equal to 4, while the elements in the side-diagonals are equal to 1. If turns out that thix matrix is of the form a *(Omega*^{-1} + DD'), a = 2 squart 3, rho = -2 + squart 3. The inverse of this matrix can be easily computed by using *Omega* and the Törnquist-Egervary formula. Moreover, it turns out that the elements of this inverse matrix are functions of *a*_{i}, b_{i}; i = 2,...,n+1, where *rho*^{n} = a_{n} + b_{n} rho.