96 lines
2.7 KiB
Matlab
96 lines
2.7 KiB
Matlab
function Ifac = mcmc_ifac(X, Nc)
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% function Ifac = mcmc_ifac(X, Nc)
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% Compute inefficiency factor of a MCMC sample X based on a Parzen Window
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%
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% INPUTS
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% X: time series
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% Nc: # of lags
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%
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% OUTPUTS
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% Ifac: inefficiency factor of MCMC sample
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%
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% SPECIAL REQUIREMENTS
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% none
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% ALGORITHM:
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% Inefficiency factors are computed as
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% \[
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% Ifac = 1 + 2\sum\limits_{i=1}^{Nc} {\hat \rho(i)}
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% \]
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% where $\hat \rho(i)$ denotes the autocorrelation at lag i and the terms
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% of the sum are truncated using a Parzen window.
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%
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% For inefficiency factors, see Section 6.1 of Paolo Giordani, Michael Pitt, and Robert Kohn (2011):
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% "Bayesian Inference for Time Series State Space Models" in : John Geweke, Gary Koop,
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% Herman van Dijk (editors): "The Oxford Handbook of Bayesian
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% Econometrics", Oxford University Press
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%
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% The Parzen-Window is given by
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% \[
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% k(x) = \left\{ {\begin{array}{*{20}{c}}
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% {1 - 6{x^2} + 6|x|^3} \text{ for } 0 \leqslant |x| \leqslant \frac{1}{2}} \\
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% {2(1-|x|^3) \text{ for } \frac{1}{2} \leqslant |x| \leqslant 1} \\
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% {0 \text{ otherwise}}
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% \end{array}} \right.
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% \]
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% See Donald W.K Andrews (1991): "Heteroskedasticity and autocorrelation
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% consistent covariance matrix estimation", Econometrica, 59(3), p. 817-858
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% Copyright © 2015-2017 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <https://www.gnu.org/licenses/>.
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Nc = floor(min(Nc, length(X)/2));
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if mod(Nc,2)
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Nc=Nc-1;
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end
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AcorrXSIM = dyn_autocorr(X(:), Nc);
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%
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%Calculate the Parzen Weight
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Parzen=zeros(Nc+1,1);
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for i=1: Nc/2+1
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Parzen(i)=1 - 6*(i/Nc)^2+ 6*(i/Nc)^3;
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end
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for i=(Nc/2)+1: Nc+1
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Parzen(i)=2 * (1-(i/Nc))^3;
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end
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Parzen=Parzen';
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Ifac= 1+2*sum(Parzen(:).* AcorrXSIM);
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function acf = dyn_autocorr(y, ar)
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% function acf = dyn_autocorr(y, ar)
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% autocorrelation function of y
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%
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% INPUTS
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% y: time series
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% ar: # of lags
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%
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% OUTPUTS
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% acf: autocorrelation for lags 1 to ar
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%
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% SPECIAL REQUIREMENTS
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% none
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y=y(:);
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acf = NaN(ar+1,1);
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acf(1)=1;
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m = mean(y);
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sd = std(y,1);
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for i=1:ar
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acf(i+1) = (y(i+1:end)-m)'*(y(1:end-i)-m)./((size(y,1))*sd^2);
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end
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