function [out] = lambdafoc(theta,y,x,W);
% This is the first order condition of the concetrated log-likelihood where 
%lambda is the variable of concentration. 
%--------------------------------------------------------------
% OUTPUT: out = value of the lambda FOC function given
% INPUT:  theta(1,1) = lambda - first entry of the vector
%               y = (nx1) synthetic data
%               x = (nx1) synthetic data
%               W = (nxn) weight matrix constructed from z's

%Strangly, we've found that variables of interest need to be included as "r
%subscript 1" which in MatLab is "r(1)". We had to do this even when we
%only have one variable. Here we have two variables.

n = size(y,1);
S = eye(n,n)-theta(1).*W;
M = eye(n,n) -x*inv(x'*x)*x';
sig2hat = (1/n)*y'*S'*M*S*y;
out = (1/sig2hat)*y'*W'*M*S*y-trace(W*inv(S));