sar_g.m

计量工具箱

function results = sar_g(y,x,W,ndraw,nomit,prior)
% PURPOSE: Bayesian estimates of the spatial autoregressive model
%          y = rho*W*y + XB + e, e = N(0,sige*V), V = diag(v1,v2,...vn) 
%          r/vi = ID chi(r)/r, r = Gamma(m,k)
%          B = N(c,T), 
%          1/sige = Gamma(nu,d0), 
%          rho = Uniform(rmin,rmax), or rho = beta(a1,a2); 
%-------------------------------------------------------------
% USAGE: results = sar_g(y,x,W,ndraw,nomit,prior)
% where: y = dependent variable vector (nobs x 1)
%        x = independent variables matrix (nobs x nvar)
%        W = spatial weight 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.rval  = r prior hyperparameter, default = 4
%            prior.novi  = 1 turns off sampling for vi, producing homoscedastic model            
%            prior.m     = informative Gamma(m,k) prior on r
%            prior.k     = (default: not used)
%            prior.nu    = informative Gamma(nu,d0) prior on sige
%            prior.d0    = default: nu=0,d0=0 (diffuse prior)
%            prior.a1    = parameter for beta(a1,a2) prior on rho see: 'help beta_prior'
%            prior.a2    = (default = 1.0, a uniform prior on rmin,rmax) 
%            prior.eig   = 0 for default rmin = -1,rmax = +1, 1 for eigenvalue calculation of these
%            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.lndet = a matrix returned by sar, sar_g, sarp_g, etc.
%                          containing log-determinant information to save time
%            prior.logm  = 0 for no log marginal calculation, = 1 for log marginal (default = 1)
%-------------------------------------------------------------
% RETURNS:  a structure:
%          results.meth   = 'sar_g'
%          results.beta   = posterior mean of bhat based on draws
%          results.rho    = posterior mean of rho based on draws
%          results.sige   = posterior mean of sige based on draws
%          results.sigma  = posterior mean of sige based on (e'*e)/(n-k)
%          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.rdraw  = r draws (ndraw-nomit x 1) (if m,k input)
%          results.bmean  = b prior means, prior.beta from input
%          results.bstd   = b prior std deviations sqrt(diag(prior.bcov))
%          results.r      = value of hyperparameter r (if input)
%          results.novi   = 1 for prior.novi = 1, 0 for prior.rval 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.resid  = residuals, based on posterior means
%          results.rsqr   = r-squared based on posterior means
%          results.rbar   = adjusted r-squared
%          results.nu     = nu prior parameter
%          results.d0     = d0 prior parameter
%          results.a1     = a1 parameter for beta prior on rho from input, or default value
%          results.a2     = a2 parameter for beta prior on rho from input, or default value
%          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.lndet = a matrix containing log-determinant information
%                          (for use in later function calls to save time)
%          results.mlike = log marginal likelihood (a vector ranging over
%                          rho values that can be integrated for model comparison)
% --------------------------------------------------------------
% NOTES: - use either improper prior.rval 
%          or informative Gamma prior.m, prior.k, not both of them
% - for n < 1000 you should use lflag = 0 to get exact results  
% - use a1 = 1.0 and a2 = 1.0 for uniform prior on rho
% - results.mlike can be used for model comparison (see model_compare.m for a demonstration)
% --------------------------------------------------------------
% SEE ALSO: (sar_gd, sar_gd2 demos) prt
% --------------------------------------------------------------
% 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, last updated 10/2003
% 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;

nobsa = n;

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,rval,mm,kk,rho,sige,rmin,rmax,detval,ldetflag, ...
eflag,order,iter,novi_flag,c,T,inform_flag,a1,a2,logmflag] = sar_parse(prior,k);

% 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);
          if mm~= 0
          rsave = zeros(ndraw-nomit,1);
          end;
          psave = zeros(ndraw-nomit,1);
          ssave = zeros(ndraw-nomit,1);
          vmean = zeros(n,1);
          acc_rate = 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;

switch novi_flag
    
case{0} % we do heteroscedastic model    

hwait = waitbar(0,'sar\_g: 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;          
          xpy = xs'*yss;
          b = xs'*yss + sige*TIc;
          b0 = AI*b;
          bhat = norm_rnd(sige*AI) + b0;  
          xb = xs*bhat;
                    
          % update sige
          nu1 = nobsa + 2*nu; 
          e = (yss - xb);
          d1 = 2*d0 + e'*e;
          chi = chis_rnd(1,nu1);
          sige = d1/chi;

          % update vi
          ev = y - rho*Wy - x*bhat; 
          %chiv = chis_rnd(n,rval+1);  
          chiv = chi2rnd(rval+1,n,1);
          vi = ((ev.*ev/sige) + in*rval)./chiv;
          V = in./vi; 
                        
          % update rval
          if mm ~= 0           
          rval = gamm_rnd(1,1,mm,kk);  
          end;
          
      % we use griddy Gibbs to perform rho-draw
          b0 = (xs'*xs + sige*TI )\(xs'*ys + sige*TIc);
          bd = (xs'*xs + sige*TI)\(xs'*Wys + sige*TIc);
          e0 = ys - xs*b0;
          ed = Wys - xs*bd;
          epe0 = e0'*e0;
          eped = ed'*ed;
          epe0d = ed'*e0;
          logdetx = log(det(xs'*xs + TI));
          rho = draw_rho(detval,epe0,eped,epe0d,n,k,rho,a1,a2,logdetx);

               
    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;
    vmean = vmean + vi;

    if mm~= 0
        rsave(iter-nomit,1) = rval;
    end;         
    end;
                    
iter = iter + 1; 
waitbar(iter/ndraw);         
end; % end of sampling loop
close(hwait);

time3 = etime(clock,t0);

% compute posterior means and log marginal likelihood for return arguments
bmean = mean(bsave);
beta = bmean';
rho = mean(psave);
sige = mean(ssave);
results.sige = sige;
vmean = vmean/(ndraw-nomit);
V = in./vmean;
[nobs,nvar] = size(x);
if logmflag == 1
          xs = matmul(x,sqrt(V));
          ys = sqrt(V).*y;
          Wys = sqrt(V).*Wy;
          AI = inv(xs'*xs + sige*TI);
          b0 = AI*(xs'*ys + sige*TIc);
          bd = AI*(xs'*Wys + sige*TIc);
          e0 = ys - xs*b0;
          ed = Wys - xs*bd;
          epe0 = e0'*e0;
          eped = ed'*ed;
          epe0d = ed'*e0;
 logdetx = log(det(xs'*xs + sige*TI));
 mlike = sar_marginal(detval,e0,ed,epe0,eped,epe0d,nobs,nvar,logdetx,a1,a2);
 yhat = (speye(nobs) - rho*W)\(x*beta);
 e = y - yhat;
elseif logmflag == 0
 xs = matmul(x,sqrt(V));
 ys = sqrt(V).*y;
 yhat = (speye(nobs) - rho*W)\(x*beta);
 e = y - yhat;
end;
% compute R-squared
epe = e'*e;
sige = epe/(n-k);
results.sigma = sige;
ym = y - mean(y);
rsqr1 = epe;
rsqr2 = ym'*ym;
results.rsqr = 1- rsqr1/rsqr2; % r-squared
rsqr1 = rsqr1/(nobs-nvar);
rsqr2 = rsqr2/(nobs-1.0);
results.rbar = 1 - (rsqr1/rsqr2); % rbar-squared

time = etime(clock,timet);


results.meth  = 'sar_g';
results.beta = beta;
results.rho = rho;
results.bdraw = bsave;
results.pdraw = psave;
results.sdraw = ssave;
if logmflag == 1
results.mlike = mlike;
end;
results.vmean = vmean;
results.yhat  = yhat;
results.resid = e;
results.bmean = c;
results.bstd  = sqrt(diag(T));
results.ndraw = ndraw;
results.nomit = nomit;
results.time  = time;
results.time1 = time1;
results.time2 = time2;
results.time3 = time3;
results.nu = nu;
results.d0 = d0;
results.a1 = a1;
results.a2 = a2;
results.tflag = 'plevel';
results.rmax = rmax; 
results.rmin = rmin;
results.lflag = ldetflag;
results.lndet = detval;
results.novi  = novi_flag;
results.priorb = inform_flag;

if mm~= 0
results.rdraw = rsave;
results.m     = mm;
results.k     = kk;
else
results.r     = rval;
results.rdraw = 0;
end;



case{1} % we do homoscedastic model    
    
hwait = waitbar(0,'sar\_g: MCMC sampling ...');
t0 = clock;                  
iter = 1;
xpx = x'*x;
xpy = x'*y;
Wy = W*y;
xpWy = x'*Wy;


          while (iter <= ndraw); % start sampling;
                  
          % update beta   
          AI = inv(xpx + sige*TI);        
          ys = y - rho*Wy;          
          b = x'*ys + sige*TIc;
          b0 = AI*b;
          bhat = norm_rnd(sige*AI) + b0;  
          xb = x*bhat;
          
          % update sige
          nu1 = n + 2*nu; 
          %e = e0 - rho*ed;
          e = (ys - xb);
          d1 = 2*d0 + e'*e;
          chi = chis_rnd(1,nu1);
          sige = d1/chi;
          
          % update rho using griddy Gibbs
          AI = inv(xpx + sige*TI);
          b0 = AI*(xpy + sige*TIc);
          bd = AI*(xpWy + sige*TIc);
          e0 = y - x*b0;
          ed = Wy - x*bd;
          epe0 = e0'*e0;
          eped = ed'*ed;
          epe0d = ed'*e0;
          logdetx = 0.5*log(det(x'*x + sige*TI));
          rho = draw_rho(detval,epe0,eped,epe0d,n,k,rho,a1,a2,logdetx);

               
    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;
    end;
                    
iter = iter + 1; 
waitbar(iter/ndraw);         
end; % end of sampling loop
close(hwait);

time3 = etime(clock,t0);