# Numerical Approximation of Fisher Information Matrix

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Edoardo Briganti on 25 Oct 2020
Commented: iust on 15 Nov 2020 at 6:53
Dear all,
I need to calculate the asymptotic standard errors of my maximum likelihood estimates. In order to do this, I have to calculate the Fisher Information Matrix. Analyticial derivation of it is possible, however it is a total pain of matrix algebra and matrix differentiation which I don't want to redo whenever I tweak my model and, in turn, my likelihood.
Is there a way to numerically approximate the Fisher Information matrix evaluated at my maximum likelihood estimates?
% Define Concentrated Log-Likelihood:
Log_L = @(r) log_likelihood(r,T,n,dy,d1,d2,W1,W2,X,6); % r is a vector of 2 parameters: r(1) and r(2).
% Maximum Likelihood Estimation (Constrained bivariate optimization)
options = optimoptions('fmincon','display','none');
x0 = .1 + .9*rand(1,2);
lb = [lb1,lb2];
ub = [ub1,ub2];
[sol,~,~,~,~,~,H] = fmincon(Log_L,x0,[],[],[],[],lb,ub,[],options);
% Maximum Likelihood Estimates of parameters which have been concentrated with respect to r(1) and r(2)
[~,hat_beta,hat_omega] = log_likelihood(sol,T,n,dy,d1,d2,W1,W2,X,k);
function [Log_L,beta_wls,omega] = log_likelihood(r,T,n,dy,d1,d2,W1,W2,X,k)
D1 = kron(d1,ones(n,1));
D2 = kron(d2,ones(n,1));
t1 = sum(d1);
t2 = sum(d2);
Z = dy - r(1)*D1.*(kron(eye(T),W1)*dy) - r(2)*D2.*(kron(eye(T),W2)*dy);
beta_ols = X\Z;
eps_ols = Z - X*beta_ols;
omega = diag( 1/(T-k).*sum(reshape(eps_ols,[n,T]).*reshape(eps_ols,[n,T]),2) ); % n parameters to estimate
sigma = kron(eye(T),omega);
beta_wls = ( X'*(sigma\X) )\( X'*( sigma\Z) );
eps_wls = Z - X*beta_wls;
H1 = inv(eye(n) - r(1)*W1);
H2 = inv(eye(n) - r(2)*W2);
% Concentrate (negative) log-Likelihood with respect to r(1) and r(2) (2 parameters to estimate)
Log_L = -( - T/2*log(det(omega)) - t1*log(det(H1)) ...
- t2*log(det(H2)) - 0.5*eps_wls'*(sigma\eps_wls) ) ;
end
You can find my code above. I concentrate my likelihood with respect to 2 parameters. The other parameters are contained in the vector beta_wls (total of n+6 parameters) and in the diagonal matrix omega (n variances).
Best,
Edoardo

#### 1 Comment

iust on 15 Nov 2020 at 6:53
Hi,
I have the same problem and I don't understand what your codes exactly say. My system of equations are dynamic and I use Kalman filter for estimation and then Fisher info matrix becomes a little crazy. Can you help me to find the Fisher info for a dynamic system?
I just added one question, sorry.
Thanks,

John D'Errico on 25 Oct 2020
Edited: John D'Errico on 25 Oct 2020
You cannot perform numerical differentiation? At the very least, you can use tools for numerical differentiation from my derivest set of tools. There is a Hessian matrix tool in there.

#### 1 Comment

Edoardo Briganti on 25 Oct 2020
Hi John,
thanks for your prompt reply. I might use your package to calculate the hessian of my log-likelihood at the values taken by the maximum likelihood estimates. However, the Fisher Information Matrix requires to take expectations of this object: .
Would not this be a problem? Would the numerical approximation of the Hessian be enough to approximate the Fisher Information Matrix?
Best,
Edoardo