194 lines
8.2 KiB
Matlab
194 lines
8.2 KiB
Matlab
function [dLIK, dlikk, a, Pstar, llik] = univariate_kalman_filter_d(data_index, number_of_observations, no_more_missing_observations, Y, start, last, a, Pinf, Pstar, kalman_tol, diffuse_kalman_tol, riccati_tol, presample, T, R, Q, H, Z, mm, pp, rr)
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% Computes the likelihood of a state space model (initialization with diffuse steps, univariate approach).
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%@info:
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%! @deftypefn {Function File} {[@var{LIK},@var{likk},@var{a},@var{Pstar}, @var{llik} ] =} univariate_kalman_filter_d (@var{data_index}, @var{number_of_observations},@var{no_more_missing_observations}, @var{Y}, @var{start}, @var{last}, @var{a}, @var{Pinf}, @var{Pstar}, @var{kalman_tol}, @var{riccati_tol},@var{presample},@var{T},@var{R},@var{Q},@var{H},@var{Z},@var{mm},@var{pp},@var{rr})
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%! @anchor{univariate_kalman_filter_d}
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%! @sp 1
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%! Computes the likelihood of a state space model (initialization with diffuse steps, univariate approach).
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%! @sp 2
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%! @strong{Inputs}
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%! @sp 1
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%! @table @ @var
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%! @item data_index
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%! Matlab's cell, 1*T cell of column vectors of indices (in the vector of observed variables).
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%! @item number_of_observations
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%! Integer scalar, effective number of observations.
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%! @item no_more_missing_observations
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%! Integer scalar, date after which there is no more missing observation (it is then possible to switch to the steady state kalman filter).
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%! @item Y
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%! Matrix (@var{pp}*T) of doubles, data.
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%! @item start
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%! Integer scalar, first period.
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%! @item last
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%! Integer scalar, last period (@var{last}-@var{first} has to be inferior to T).
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%! @item a
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%! Vector (@var{mm}*1) of doubles, initial mean of the state vector.
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%! @item Pinf
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%! Matrix (@var{mm}*@var{mm}) of doubles, initial covariance matrix of the state vector (non stationary part).
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%! @item Pstar
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%! Matrix (@var{mm}*@var{mm}) of doubles, initial covariance matrix of the state vector (stationary part).
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%! @item kalman_tol
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%! Double scalar, tolerance parameter (rcond, inversibility of the covariance matrix of the prediction errors).
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%! @item riccati_tol
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%! Double scalar, tolerance parameter (iteration over the Riccati equation).
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%! @item presample
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%! Integer scalar, presampling if strictly positive (number of initial iterations to be discarded when evaluating the likelihood).
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%! @item T
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%! Matrix (@var{mm}*@var{mm}) of doubles, transition matrix of the state equation.
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%! @item R
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%! Matrix (@var{mm}*@var{rr}) of doubles, second matrix of the state equation relating the structural innovations to the state variables.
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%! @item Q
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%! Matrix (@var{rr}*@var{rr}) of doubles, covariance matrix of the structural innovations (noise in the state equation).
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%! @item H
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%! Vector (@var{pp}) of doubles, diagonal of covariance matrix of the measurement errors (corelation among measurement errors is handled by a model transformation).
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%! @item Z
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%! Matrix (@var{pp}*@var{mm}) of doubles, matrix relating the states to the observed variables.
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%! @item mm
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%! Integer scalar, number of state variables.
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%! @item pp
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%! Integer scalar, number of observed variables.
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%! @item rr
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%! Integer scalar, number of structural innovations.
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%! @item Zflag
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%! Integer scalar, equal to 0 if Z is a vector of indices targeting the obseved variables in the state vector, equal to 1 if Z is a @var{pp}*@var{mm} matrix.
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%! @item diffuse_periods
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%! Integer scalar, number of diffuse filter periods in the initialization step.
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%! @end table
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%! @sp 2
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%! @strong{Outputs}
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%! @sp 1
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%! @table @ @var
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%! @item dLIK
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%! Double scalar, value of (minus) the likelihood for the first d observations.
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%! @item likk
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%! Column vector (d*1) of doubles, values of the density of each (fist) observations.
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%! @item a
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%! Vector (@var{mm}*1) of doubles, mean of the state vector at the end of the (sub)sample.
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%! @item Pstar
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%! Matrix (@var{mm}*@var{mm}) of doubles, covariance of the state vector at the end of the subsample.
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%! @item llik
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%! Matrix of doubles (d*@var{pp}) used by @ref{DsgeLikelihood_hh}
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%! @end table
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%! @sp 2
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%! @strong{This function is called by:}
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%! @sp 1
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%! @ref{dsge_likelihood}, @ref{DsgeLikelihood_hh}
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%! @sp 2
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%! @strong{This function calls:}
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%! @sp 1
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%! @ref{univariate_kalman_filter_ss}
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%! @end deftypefn
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%@eod:
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%
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% Algorithm:
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%
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% Uses the diffuse univariate filter as described in Durbin/Koopman (2012): "Time
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% Series Analysis by State Space Methods", Oxford University Press,
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% Second Edition, Ch. 5, 6.4 + 7.2.5
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% Copyright © 2004-2021 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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% Get sample size.
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smpl = last-start+1;
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% Initialize some variables.
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dF = 1;
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isqvec = false;
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if ndim(Q)>2
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Qvec = Q;
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Q=Q(:,:,1);
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isqvec = true;
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end
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QQ = R*Q*transpose(R); % Variance of R times the vector of structural innovations.
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t = start; % Initialization of the time index.
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dlikk= zeros(smpl,1); % Initialization of the vector gathering the densities.
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dLIK = Inf; % Default value of the log likelihood.
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oldK = Inf;
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llik = zeros(smpl,pp);
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newRank = rank(Pinf,diffuse_kalman_tol);
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l2pi = log(2*pi);
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s=0;
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while newRank && (t<=last)
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s = t-start+1;
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d_index = data_index{t};
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for i=1:length(d_index)
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Zi = Z(d_index(i),:);
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prediction_error = Y(d_index(i),t) - Zi*a; % nu_{t,i} in 6.13 in DK (2012)
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Fstar = Zi*Pstar*Zi' + H(d_index(i)); % F_{*,t} in 5.7 in DK (2012), relies on H being diagonal
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Finf = Zi*Pinf*Zi'; % F_{\infty,t} in 5.7 in DK (2012), relies on H being diagonal
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Kstar = Pstar*Zi';
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% Conduct check of rank
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% Pinf and Finf are always scaled such that their norm=1: Fstar/Pstar, instead,
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% depends on the actual values of std errors in the model and can be badly scaled.
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% experience is that diffuse_kalman_tol has to be bigger than kalman_tol, to ensure
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% exiting the diffuse filter properly, avoiding tests that provide false non-zero rank for Pinf.
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% Also the test for singularity is better set coarser for Finf than for Fstar for the same reason
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if Finf>diffuse_kalman_tol && newRank % F_{\infty,t,i} = 0, use upper part of bracket on p. 175 DK (2012) for w_{t,i}
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Kinf = Pinf*Zi';
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Kinf_Finf = Kinf/Finf;
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a = a + Kinf_Finf*prediction_error;
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Pstar = Pstar + Kinf*(Kinf_Finf'*(Fstar/Finf)) - Kstar*Kinf_Finf' - Kinf_Finf*Kstar';
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Pinf = Pinf - Kinf*Kinf_Finf';
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llik(s,d_index(i)) = log(Finf) + l2pi;
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dlikk(s) = dlikk(s) + llik(s,d_index(i));
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elseif Fstar>kalman_tol
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llik(s,d_index(i)) = log(Fstar) + (prediction_error*prediction_error/Fstar) + l2pi;
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dlikk(s) = dlikk(s) + llik(s,d_index(i));
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a = a+Kstar*(prediction_error/Fstar);
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Pstar = Pstar-Kstar*(Kstar'/Fstar);
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else
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% do nothing as a_{t,i+1}=a_{t,i} and P_{t,i+1}=P_{t,i}, see
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% p. 157, DK (2012)
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end
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end
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if newRank
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oldRank = rank(Z*Pinf*Z',diffuse_kalman_tol);
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else
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oldRank = 0;
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end
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a = T*a;
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if isqvec
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QQ = R*Qvec(:,:,t+1)*transpose(R);
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end
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Pstar = T*Pstar*T'+QQ;
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Pinf = T*Pinf*T';
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if newRank
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newRank = rank(Z*Pinf*Z',diffuse_kalman_tol);
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end
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if oldRank ~= newRank
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disp('univariate_diffuse_kalman_filter:: T does influence the rank of Pinf!')
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disp('This may happen for models with order of integration >1.')
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end
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t = t+1;
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end
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if (t>last)
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warning(['univariate_diffuse_kalman_filter:: There isn''t enough information to estimate the initial conditions of the nonstationary variables']);
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dLIK = NaN;
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return
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end
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% Divide by two.
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dlikk = .5*dlikk(1:s);
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llik = .5*llik(1:s,:);
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dLIK = sum(dlikk(1+presample:end));
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dlikk = llik; |