sar_gv.m

计量工具箱

function results = sar_gv(y,x,W,ndraw,nomit,prior)
% PURPOSE: Bayesian estimates of the spatial autoregressive model
% THIS FUNCTION: estimates the heteroscedasticity parameter r
%                and returns results in results.rdraw for posterior inference
%          y = rho*W*y + XB + e, e = N(0,sige*V), V = diag(v1,v2,...vn) 
%          r/vi = ID chi(r)/r
%          r = Gamma(delta,2)
%          B = N(c,T), 
%          1/sige = Gamma(nu,d0), 
%          rho = Uniform(rmin,rmax) 
%-------------------------------------------------------------
% USAGE: results = sar_gv(y,x,W,ndraw,nomit,prior)
% where: y = dependent variable vector (nobs x 1)
%        x = independent variables matrix (nobs x nvar)
%        W = 1st order contiguity matrix (standardized, row-sums = 1)
%    ndraw = # of draws
%    nomit = # of initial draws omitted for burn-in            
%    prior = a structure variable with:
%            prior.beta  = prior means for beta,   c above (default 0)
%            priov.bcov  = prior beta covariance , T above (default 1e+12)
%            prior.nu    = informative Gamma(nu,d0) prior on sige
%            prior.d0    = default: nu=0,d0=0 (diffuse prior)
%            prior.delta = default: delta = 20
%            prior.rmin  = (optional) min rho used in sampling (default = -1)
%            prior.rmax  = (optional) max rho used in sampling (default = 1)  
%            prior.lflag = 0 for full lndet computation (default = 1, fastest)
%                        = 1 for MC approx (fast for large problems)
%                        = 2 for Spline approx (medium speed)
%            prior.order = order to use with prior.lflag = 1 option (default = 50)
%            prior.iter  = iters to use with prior.lflag = 1 option (default = 30) 
%            prior.dflag = 0 for numerical integration, 1 for Metropolis-Hastings
%                          (default = 1 for nobs <= 1,000, =0 for nobs > 1,000)
%            prior.lndet = a matrix returned by sar, sar_g, sarp_g, etc.
%                          containing log-determinant information to save time
%-------------------------------------------------------------
% RETURNS:  a structure:
%          results.meth   = 'sar_gv'
%          results.rdraw  = r draws (ndraw-nomit x 1) 
%          results.delta  = value of delta (from input)
%          results.bdraw  = bhat draws (ndraw-nomit x nvar)
%          results.pdraw  = rho  draws (ndraw-nomit x 1)
%          results.sdraw  = sige draws (ndraw-nomit x 1)
%          results.vmean  = mean of vi draws (nobs x 1) 
%          results.bmean  = b prior means, prior.beta from input
%          results.bstd   = b prior std deviations sqrt(diag(prior.bcov))
%          results.mlike  = marginal likelihood
%          results.novi   = 1 for prior.novi = 1, 0 for prior.delta input
%          results.nobs   = # of observations
%          results.nvar   = # of variables in x-matrix
%          results.ndraw  = # of draws
%          results.nomit  = # of initial draws omitted
%          results.y      = y-vector from input (nobs x 1)
%          results.yhat   = mean of posterior predicted (nobs x 1)
%          results.nu     = nu prior parameter
%          results.d0     = d0 prior parameter
%          results.time1  = time for eigenvalue calculation
%          results.time2  = time for log determinant calcluation
%          results.time3  = time for sampling
%          results.time   = total time taken  
%          results.rmax   = 1/max eigenvalue of W (or rmax if input)
%          results.rmin   = 1/min eigenvalue of W (or rmin if input)          
%          results.tflag  = 'plevel' (default) for printing p-levels
%                         = 'tstat' for printing bogus t-statistics 
%          results.lflag  = lflag from input
%          results.iter   = prior.iter option from input
%          results.order  = prior.order option from input
%          results.limit  = matrix of [rho lower95,logdet approx, upper95] 
%                           intervals for the case of lflag = 1
%          results.dflag  = dflag value from input (or default value used)
%          results.lndet = a matrix containing log-determinant information
%                          (for use in later function calls to save time)
%          results.acc   = acceptance rate for dof M-H sampling
% --------------------------------------------------------------
% NOTES: purpose of this function is to provide an inference
%        on the hyperparameter r in the heteroscedastic Bayesian sar model
%        Use the results.rdraw to draw this inference
% --------------------------------------------------------------
% SEE ALSO: (sar_gvd demo) 
% --------------------------------------------------------------
% REFERENCES: James P. LeSage, `Bayesian Estimation of Spatial Autoregressive
%             Models',  International Regional Science Review, 1997 
%             Volume 20, number 1\&2, pp. 113-129.
% For lndet information see: Ronald Barry and R. Kelley Pace, 
% "A Monte Carlo Estimator of the Log Determinant of Large Sparse Matrices", 
% Linear Algebra and its Applications", Volume 289, Number 1-3, 1999, pp. 41-54.
% and: R. Kelley Pace and Ronald P. Barry 
% "Simulating Mixed Regressive Spatially autoregressive Estimators", 
% Computational Statistics, 1998, Vol. 13, pp. 397-418.
%----------------------------------------------------------------

% written by:
% James P. LeSage, 4/2002
% Dept of Economics
% University of Toledo
% 2801 W. Bancroft St,
% Toledo, OH 43606
% [e-mail]


% NOTE: some of the speed for large problems comes from:
% the use of methods pioneered by Pace and Barry.
% R. Kelley Pace was kind enough to provide functions
% lndetmc, and lndetint from his spatial statistics toolbox
% for which I'm very grateful.

timet = clock;

% error checking on inputs
[n junk] = size(y);
[n1 k] = size(x);
[n3 n4] = size(W);
time1 = 0;
time2 = 0;
time3 = 0;

results.nobs  = n;
results.nvar  = k;
results.y = y;      

if n1 ~= n
error('sar_g: x-matrix contains wrong # of observations');
elseif n3 ~= n4
error('sar_g: W matrix is not square');
elseif n3~= n
error('sar_g: W matrix is not the same size at y,x');
end;

if nargin == 5
    prior.lflag = 1;
end;

[nu,d0,rho,sige,rmin,rmax,detval,ldetflag,eflag,order,iter,c,T,prior_beta,cc,metflag,delta] = sar_parse(prior,k);

results.delta = delta;

% error checking on prior information inputs
[checkk,junk] = size(c);
if checkk ~= k
error('sar_g: prior means are wrong');
elseif junk ~= 1
error('sar_g: prior means are wrong');
end;

[checkk junk] = size(T);
if checkk ~= k
error('sar_g: prior bcov is wrong');
elseif junk ~= k
error('sar_g: prior bcov is wrong');
end;

results.order = order;
results.iter = iter;

timet = clock; % start the timer

[rmin,rmax,time1] = sar_eigs(eflag,W,rmin,rmax,n);

[detval,time2] = sar_lndet(ldetflag,W,rmin,rmax,detval,order,iter);

% storage for draws
          bsave = zeros(ndraw-nomit,k);
          rsave = zeros(ndraw-nomit,1);
          psave = zeros(ndraw-nomit,1);
          ssave = zeros(ndraw-nomit,1);
          margl = zeros(ndraw-nomit,1);
          vmean = zeros(n,1);
          ymean = zeros(n,1);
          yhat = zeros(n,1);
          acc_rate = zeros(ndraw,1);
          ccsave = zeros(ndraw,1);

% ====== initializations
% compute this stuff once to save time
TI = inv(T);
TIc = TI*c;
iter = 1;

in = ones(n,1);
V = in;
vi = in;
Wy = sparse(W)*y;
vdraw = delta;
%Prior for degrees of freedom is exponential with mean vl0
vl0=delta;
cc = 5;
pswitch = 0;
acc = 0;


hwait = waitbar(0,'sar\_vg: MCMC sampling ...');
t0 = clock;                  
iter = 1;
          while (iter <= ndraw); % start sampling;
                  
          % update beta   
          xs = matmul(x,sqrt(V));
          ys = sqrt(V).*y;
          Wys = sqrt(V).*Wy;
          AI = inv(xs'*xs + sige*TI);		  
          yss = ys - rho*Wys;          
          b = xs'*yss + sige*TIc;
          b0 = AI*b;
          bhat = norm_rnd(sige*AI) + b0;  
          xb = xs*bhat;
          
          % update sige
          nu1 = n + 2*nu; 
          e = (yss - xb);
          d1 = 2*d0 + e'*e;
          chi = chis_rnd(1,nu1);
          sige = d1/chi;

          % update vi
          ev = ys - rho*Wys - xs*bhat; 
          dof = vdraw + 1;
          error2 = ev.*ev;
          tmp = (1/sige)*error2 + vdraw;
          chiv = chis_rnd(n,dof);   
          V = chiv./tmp;
              
          
     if metflag == 1
         % metropolis step to get rho update
          rhox = c_sar(rho,ys,xb,sige,W,detval);
          accept = 0; 
          rho2 = rho + cc*randn(1,1); 
          while accept == 0
           if ((rho2 > rmin) & (rho2 < rmax)); 
           accept = 1;  
           else
           rho2 = rho + cc*randn(1,1);
           end; 
          end;
          rhoy = c_sar(rho2,ys,xb,sige,W,detval);
          ru = unif_rnd(1,0,1);
          if ((rhoy - rhox) > exp(1)),
          p = 1;
          else, 
          ratio = exp(rhoy-rhox); 
          p = min(1,ratio);
          end;
              if (ru < p)
              rho = rho2;
              end;
          rtmp(iter,1) = rho;
      else % we use numerical integration to perform rho-draw
          b0 = AI*xs'*ys;
          bd = AI*xs'*Wys;
          e0 = ys - xs*b0;
          ed = Wys - xs*bd;
          epe0 = e0'*e0;
          eped = ed'*ed;
          epe0d = ed'*e0;
          [rho mlike] = draw_rho(detval,epe0,eped,epe0d,n,k,rho,sige);
      end;
      
     %Random walk Metropolis step for dof
    temp = -log(V) + V;
    nu = 1/vl0 + .5*sum(temp);
    
     vlcan= vdraw +  cc*randn(1,1);
     if vlcan>0
        lpostcan = .5*n*vlcan*log(.5*vlcan) -n*gammaln(.5*vlcan)...
        -nu*vlcan;
        lpostdraw = .5*n*vdraw*log(.5*vdraw) -n*gammaln(.5*vdraw)...
        -nu*vdraw;
        accprob = exp(lpostcan-lpostdraw);
     else
        accprob=0;
     end
     

%accept candidate draw with log prob = laccprob, else keep old draw
   if  rand<accprob
       vdraw=vlcan;
       pswitch=pswitch+1;
       acc = acc + 1;
   end    

      acc_rate(iter,1) = acc/iter;
      % update cc based on std of rho draws
       if acc_rate(iter,1) < 0.4
       cc = cc/1.1;
       ccsave(iter,1) = cc;
       end;
       if acc_rate(iter,1) > 0.6
       cc = cc*1.1;
       ccsave(iter,1) = cc;
       end;
   
               
    if iter > nomit % if we are past burn-in, save the draws
    bsave(iter-nomit,1:k) = bhat';
    ssave(iter-nomit,1) = sige;
    psave(iter-nomit,1) = rho;
    margl(iter-nomit,1) = mlike;
    vmean = vmean + in./V;
    rsave(iter-nomit,1) = vdraw;
    end;
                    
iter = iter + 1; 
waitbar(iter/ndraw);         
end; % end of sampling loop
close(hwait);

vmean = vmean/(ndraw-nomit);
beta = mean(bsave)';
pmean = mean(psave);
results.acc = acc_rate;
results.rdraw = rsave;

yhat = (speye(n) - pmean*W)\(x*beta);