114 lines
4.2 KiB
Matlab
114 lines
4.2 KiB
Matlab
function [ytrend,ycycle]=one_sided_hp_filter(y,lambda,x_user,P_user,discard)
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% function [ytrend,ycycle]=one_sided_hp_filter(y,lambda,x_user,P_user,discard)
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% Conducts one-sided HP-filtering, derived using the Kalman filter
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%
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% Inputs:
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% y [T*n] double data matrix in column format
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% lambda [scalar] Smoothing parameter. Default value of 1600 will be used.
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% x_user [2*n] double matrix with initial values of the state
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% estimate for each variable in y. The underlying
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% state vector is 2x1 for each variable in y.
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% Default: use backwards extrapolations
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% based on the first two observations
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% P_user [n*1] struct structural array with n elements, each a two
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% 2x2 matrix of intial MSE estimates for each
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% variable in y.
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% Default: matrix with large variances
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% discard [scalar] number of initial periods to be discarded
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% Default: 0
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%
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% Output:
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% ytrend [(T-discard)*n] matrix of extracted trends
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% ycycle [(T-discard)*n] matrix of extracted deviations from the extracted trends
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%
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% Algorithms:
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%
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% Implements the procedure described on p. 301 of Stock, J.H. and M.W. Watson (1999):
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% "Forecasting inflation," Journal of Monetary Economics, vol. 44(2), pages 293-335, October.
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% that states on page 301:
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%
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% "The one-sided HP trend estimate is constructed as the Kalman
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% filter estimate of tau_t in the model:
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%
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% y_t=tau_t+epsilon_t
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% (1-L)^2 tau_t=eta_t"
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%
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% The Kalman filter notation follows Chapter 13 of Hamilton, J.D. (1994).
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% Time Series Analysis, with the exception of H, which is equivalent to his H'.
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% Copyright © 2010-2015 Alexander Meyer-Gohde
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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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if nargin < 2 || isempty(lambda)
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lambda = 1600; %If the user didn't provide a value for lambda, set it to the default value 1600
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end
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[T,n] = size (y);% Calculate the number of periods and the number of variables in the series
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%Set up state space
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q=1/lambda; % the signal-to-noise ration: i.e. var eta_t / var epsilon_t
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F=[2,-1;
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1,0]; % state transition matrix
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H=[1,0]; % observation matrix
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Q=[q,0;
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0,0]; % covariance matrix state equation errors
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R=1; % variance observation equation error
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for k=1:n %Run the Kalman filter for each variable
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if nargin < 3 || isempty(x_user) %no intial value for state, extrapolate back two periods from the observations
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x=[2*y(1,k)-y(2,k);
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3*y(1,k)-2*y(2,k)];
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else
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x=x_user(:,k);
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end
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if nargin < 4 || isempty(P_user) %no initial value for the MSE, set a rather high one
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P= [1e5 0;
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0 1e5];
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else
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P=P_user{k};
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end
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for j=1:T %Get the estimates for each period
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[x,P]=kalman_update(F,H,Q,R,y(j,k),x,P); %get new state estimate and update recursion
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ytrend(j,k)=x(2);%second state is trend estimate
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end
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end
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if nargout==2
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ycycle=y-ytrend;
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end
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if nargin==5 %user provided a discard parameter
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ytrend=ytrend(discard+1:end,:);%Remove the first "discard" periods from the trend series
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if nargout==2
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ycycle=ycycle(discard+1:end,:);
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end
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end
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end
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function [x,P]=kalman_update(F,H,Q,R,obs,x,P)
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% Updates the Kalman filter estimation of the state and MSE
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S=H*P*H'+R;
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K=F*P*H';
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K=K/S;
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x=F*x+K*(obs -H*x); %State estimate
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Temp=F-K*H;
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P=Temp*P*Temp';
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P=P+Q+K*R*K';%MSE estimate
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end
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