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algorithm for calculating approximation assumed assumption autoregression column j element conditional distribution conditional on _ date t—1 density derivatives described deterministic dynamic econometrician eigenvalues of F element of F et+l evaluate ex ante real ex post real example extended Kalman filter Garcia and Perron given Halbert White Hamilton heteroskedasticity identity matrix initial guess inside the unit Jth element known with certainty likelihood function log likelihood Markov chain maximum likelihood estimates mean squared error Nicholls and Pagan non-Normal nonlinear Normal distribution numbers observation equation observed at date observed variables optimal forecast optimal inference p x p panel of Figure post real interest Prob(S real interest rate recursion Robert Engle sample scalar sequence smoothed estimates smoothed inference smoothed probabilities standard errors state-space form state-space representation summarized Top panel transition probabilities unconditional mean unconditional variance unit circle unobserved vector vector process Watson and Engle