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I am trying to implement unsuccessfully a function in matlab, to compute the effective sample size after a MCMC chain, with a posterior with 3 coefficients.

enter image description here

Source: Sims MCMC

$ VAR(1) / Y_t=\mu +AY_{t-1}+ \epsilon_t$

I think that I need to compute $ T\Gamma(0)/\sum_{i=1}^{n}(\Gamma(J)$

If a VAR(1) is stationary $\Gamma(J)=A^j\Gamma(0) $.

So the first step is to estimate the model. Second with the matrixes found, I need to compute $\Gamma(0)=V(y_t)=E(y_t'y_t)=A\Gamma(0)A'+\Omega $. Third compute the effective sample size

This is my try, a very basic code in Matlab to estimate the VAR(1) coefficients with the posterior load in betaProbit.

    Varprobit=betaProbit; % The posterior betas of x0 x1 x2
    Varprobit_1=zeros(ndraws,3);% x0 x1 x2 ,t-1
    for j=2:ndraws 
      Varprobit_1(j,:)=Varprobit(j-1,:);
    end
    epsilons=randn(ndraws,3);% generate Random normal numbers ( errors)
    XVAR1=ones(ndraws,1),Varprobit_1(:,1),Varprobit_1(:,2),...
    Varprobit_1(:,3),epsilons(:,1)];
    XVAR2=[ones(ndraws,1) ,Varprobit_1(:,1),Varprobit_1(:,2),...
    Varprobit_1(:,3),epsilons(:,2)];
    XVAR3=[ones(ndraws,1) ,Varprobit_1(:,1),Varprobit_1(:,2),...
    Varprobit_1(:,3),epsilons(:,3)];
    % Estimate VAR coefficients;
    BetasEc1= inv(XVAR1'*XVAR1)*XVAR1'*Varprobit(:,1);
    BetasEc2= inv(XVAR2'*XVAR2)*XVAR2'*Varprobit(:,2);
    BetasEc3= inv(XVAR3'*XVAR3)*XVAR3'*Varprobit(:,2)   ; 
    A=[BetasEc1(2:4,1)';BetasEc2(2:4,1)';BetasEc3(2:4,1)'];
    OMEGA=eye(3,3);
    Gam=Varprobit'*Varprobit;  
    acfj= VARacfLagj( Gam ,A, OMEGA,j);
    %In this step I need to construct a function to solve Gamma(0)

The key issue is to find a function to solve $\Gamma(0)=V(y_t)=E(y_t'y_t)=A\Gamma(0)A'+\Omega $.

Thanks for your help.

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    $\begingroup$ Did you intend to include some kind of more complete reference or link on [2]? Is this Matlab code? $\endgroup$ – Glen_b Aug 7 '14 at 2:59
  • $\begingroup$ @Glen_b thank you for asking. Yes it is Matlab code. The hyperlink was corrected $\endgroup$ – EAguirre Aug 7 '14 at 3:51
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You are probably no longer looking for an answer. But here it is, anyway.

I am unfamiliar with Matlab, but if you know R, then I think the R package mcmcse has coded the effective sample size similarly. They are also in their vignette using the VAR example as motivation, so you will find a lot of similarities.

It looks like most of their code is also in C++, so if you know that as well, you might be able to translate the logic.

Basically, they are using batch means estimators to estimate the infinite sum variance. This is a strongly consistent estimator, and is much better than using the naive estimator. You can find more information here, including some of the math for the VAR model.

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