dynare/matlab/mh_optimal_bandwidth.m

function optimal_bandwidth = mh_optimal_bandwidth(data,n,bandwidth,kernel_function) 
%%  This function gives the optimal bandwidth parameter of a kernel estimator 
%%  used to estimate a posterior univariate density from realisations of a 
%%  Metropolis-Hastings algorithm.  
%% 
%%  * M. Skold and G.O. Roberts [2003], "Density estimation for the Metropolis-Hastings algorithm". 
%%  * Silverman [1986], "Density estimation for statistics and data analysis". 
%%
%%  data            :: a vector with n elements.
%%  bandwidth       :: a scalar equal to 0,-1 or -2. For a value different from 0,-1 or -2 the
%%                     function will return optimal_bandwidth = bandwidth.
%%  kernel_function :: 'gaussian','uniform','triangle','epanechnikov',
%%                     'quartic','triweight','cosinus'.
%%
%%  stephane.adjemian@cepremap.cnrs.fr [07-15-2004] <-- [01/16/2004] 


%% KERNEL SPECIFICATION...
if strcmpi(kernel_function,'gaussian') 
    k    = inline('inv(sqrt(2*pi))*exp(-0.5*x.^2)'); 
    k2   = inline('inv(sqrt(2*pi))*(-exp(-0.5*x.^2)+(x.^2).*exp(-0.5*x.^2))'); % second derivate of the gaussian kernel 
    k4   = inline('inv(sqrt(2*pi))*(3*exp(-0.5*x.^2)-6*(x.^2).*exp(-0.5*x.^2)+(x.^4).*exp(-0.5*x.^2))'); % fourth derivate... 
    k6   = inline('inv(sqrt(2*pi))*(-15*exp(-0.5*x.^2)+45*(x.^2).*exp(-0.5*x.^2)-15*(x.^4).*exp(-0.5*x.^2)+(x.^6).*exp(-0.5*x.^2))'); % sixth derivate... 
    mu02 = inv(2*sqrt(pi)); 
    mu21 = 1; 
elseif strcmpi(kernel_function,'uniform') 
    k    = inline('0.5*(abs(x) <= 1)'); 
    mu02 = 0.5; 
    mu21 = 1/3; 
elseif strcmpi(kernel_function,'triangle') 
    k    = inline('(1-abs(x)).*(abs(x) <= 1)'); 
    mu02 = 2/3; 
    mu21 = 1/6; 
elseif strcmpi(kernel_function,'epanechnikov') 
    k    = inline('0.75*(1-x.^2).*(abs(x) <= 1)'); 
    mu02 = 3/5; 
    mu21 = 1/5;     
elseif strcmpi(kernel_function,'quartic') 
    k    = inline('0.9375*((1-x.^2).^2).*(abs(x) <= 1)'); 
    mu02 = 15/21; 
    mu21 = 1/7; 
elseif strcmpi(kernel_function,'triweight') 
    k    = inline('1.09375*((1-x.^2).^3).*(abs(x) <= 1)'); 
    k2   = inline('(105/4*(1-x.^2).*x.^2-105/16*(1-x.^2).^2).*(abs(x) <= 1)'); 
    k4   = inline('(-1575/4*x.^2+315/4).*(abs(x) <= 1)'); 
    k6   = inline('(-1575/2).*(abs(x) <= 1)'); 
    mu02 = 350/429; 
    mu21 = 1/9;     
elseif strcmpi(kernel_function,'cosinus') 
    k    = inline('(pi/4)*cos((pi/2)*x).*(abs(x) <= 1)'); 
    k2   = inline('(-1/16*cos(pi*x/2)*pi^3).*(abs(x) <= 1)'); 
    k4   = inline('(1/64*cos(pi*x/2)*pi^5).*(abs(x) <= 1)'); 
    k6   = inline('(-1/256*cos(pi*x/2)*pi^7).*(abs(x) <= 1)'); 
    mu02 = (pi^2)/16; 
    mu21 = (pi^2-8)/pi^2;     
else
    disp('mh_optimal_bandwidth :: ');
    error('This kernel function is not yet implemented in Dynare!');        
end     


%% OPTIMAL BANDWIDTH PARAMETER....
if bandwidth == 0;  %  Rule of thumb bandwidth parameter (Silverman [1986] corrected by 
                    %  Skold and Roberts [2003] for Metropolis-Hastings). 
    sigma = std(data); 
    h = 2*sigma*(sqrt(pi)*mu02/(12*(mu21^2)*n))^(1/5); % Silverman's optimal bandwidth parameter. 
    A = 0; 
    for i=1:n; 
        j = i; 
        while j<= n & data(j,1)==data(i,1); 
            j = j+1; 
        end;     
        A = A + 2*(j-i) - 1; 
    end; 
    A = A/n; 
    h = h*A^(1/5); % correction 
elseif bandwidth == -1;     % Adaptation of the Sheather and Jones [1991] plug-in estimation of the optimal bandwidth 
                            % parameter for metropolis hastings algorithm. 
    if strcmp(kernel_function,'uniform')      | ... 
       strcmp(kernel_function,'triangle')     | ... 
       strcmp(kernel_function,'epanechnikov') | ... 
       strcmp(kernel_function,'quartic'); 
       error('I can''t compute the optimal bandwidth with this kernel... Try the gaussian, triweight or cosinus kernels.'); 
    end;         
    sigma = std(data); 
    A = 0; 
    for i=1:n; 
        j = i; 
        while j<= n & data(j,1)==data(i,1); 
            j = j+1; 
        end;     
        A = A + 2*(j-i) - 1; 
    end; 
    A = A/n; 
    Itilda4 = 8*7*6*5/(((2*sigma)^9)*sqrt(pi)); 
    g3      = abs(2*A*k6(0)/(mu21*Itilda4*n))^(1/9); 
    Ihat3 = 0; 
    for i=1:n; 
        Ihat3 = Ihat3 + sum(k6((data(i,1)-data)/g3)); 
    end;     
    Ihat3 = -Ihat3/((n^2)*g3^7); 
    g2      = abs(2*A*k4(0)/(mu21*Ihat3*n))^(1/7); 
    Ihat2 = 0; 
    for i=1:n; 
        Ihat2 = Ihat2 + sum(k4((data(i)-data)/g2)); 
    end;     
    Ihat2 = Ihat2/((n^2)*g2^5); 
    h       = (A*mu02/(n*Ihat2*mu21^2))^(1/5);    % equation (22) in Skold and Roberts [2003] --> h_{MH} 
elseif bandwidth == -2;     % Bump killing... We construct local bandwith parameters in order to remove 
                            % spurious bumps introduced by long rejecting periods.   
    if strcmp(kernel_function,'uniform')      | ... 
       strcmp(kernel_function,'triangle')     | ... 
       strcmp(kernel_function,'epanechnikov') | ... 
       strcmp(kernel_function,'quartic'); 
        error('I can''t compute the optimal bandwidth with this kernel... Try the gaussian, triweight or cosinus kernels.'); 
    end;         
    sigma = std(data); 
    A = 0; 
    T = zeros(n,1); 
    for i=1:n; 
        j = i; 
        while j<= n & data(j,1)==data(i,1); 
            j = j+1; 
        end;     
        T(i) = (j-i); 
        A = A + 2*T(i) - 1; 
    end; 
    A = A/n; 
    Itilda4 = 8*7*6*5/(((2*sigma)^9)*sqrt(pi)); 
    g3      = abs(2*A*k6(0)/(mu21*Itilda4*n))^(1/9); 
    Ihat3 = 0; 
    for i=1:n; 
        Ihat3 = Ihat3 + sum(k6((data(i,1)-data)/g3)); 
    end;     
    Ihat3 = -Ihat3/((n^2)*g3^7); 
    g2      = abs(2*A*k4(0)/(mu21*Ihat3*n))^(1/7); 
    Ihat2 = 0; 
    for i=1:n; 
        Ihat2 = Ihat2 + sum(k4((data(i)-data)/g2)); 
    end;     
    Ihat2 = Ihat2/((n^2)*g2^5); 
    h = ((2*T-1)*mu02/(n*Ihat2*mu21^2)).^(1/5); % Note that h is a column vector (local banwidth parameters). 
elseif bandwidth > 0 
    h = bandwidth; 
else
    disp('mh_optimal_bandwidth :: ');
    error('Parameter bandwidth must be a real parameter value or equal to 0,-1 or -2.'); 
end

optimal_bandwidth = h;