%function chain,state=markov(T,n,s0,V);%function chain,state=markov(T,n,s0,V);% chain generates a simulation from a Markov chain of dimension% the size of T% T is transition matrix% n is number of periods to simulate% s0 is initial state (initial probabilities)% V is the quantity corresponding to each state% state is a matrix recording the number of the realized state at time t% Original author: Tom Sargent% Comments added by Qiang Chenr c=size(T); % r is # of rows, c is # of columns of Tif nargin = 1; % nargin refers to number of arguments in. So only T is provided in this case V=1:r; s0=1; n=100;end;if nargin = 2; % both T and n are provided V=1:r; s0=1;end;if nargin = 3; % T, n and S0 are provided V=1:r;end;% check if the transition matrix T is squareif r = c; disp(error using markov function); disp(transition matrix must be square); return; % break the program and returnend;% check if each row of T sums up to 1for k=1:r; if sum(T(k,:) = 1; disp(error using markov function) disp(row ,num2str(k), does not sum to one); % num2str converts numbers to a string. disp( it sums to :); disp( sum(T(k,:) ); disp(normalizing row ,num2str(k),); T(k,:)=T(k,:)/sum(T(k,:); end;end;v1 v2=size(V);if v1 = 1 | v2 =r % | means or disp(error using markov function); disp(state value vector V must be 1 x ,num2str(r),) if v2 = 1 &v2 = r; disp(transposing state valuation vector); V=V; % change it to a column vector else; return; end; endif s0 r; disp(initial state ,num2str(s0), is out of range); disp(initial state defaulting to 1); s0=1;end;% The simulation of Markov chain formally starts from here%T%rand(uniform);X=rand(n-1,1); % generate (n-1) random numbers drawn from uniform distribution on 0,1, each number to be used in one simulation.s=zeros(r,1); % initiate the state vector s to be a rx1 zero vectors(s0)=1; % change the s0th element of s to 1cum=T*triu(ones(size(T); % triu(ones(size(T) generates an upper triangular matrix with all elements equal to 1% cum is a rxr matrix whose ith column is the cumulative sum from the 1st column to the ith column % the ith row of cum is the cumulative distribution for the next period given the current state. for k=1:length(X); % length(X) returns the size of the longest dimension of X. k indicates the kth simulation. state(:,k)=s; % state is a matrix recording the number of the realized state at time k ppi=0 s*cum; % this is the conditional cumulative distribution for the next period given the current state s s=(X(k)ppi(1:r); % compares each element of ppi(2:r+1) or ppi(1:r) with a scalar X(k), and % returns 1 if the inequality holds and 0 otherwise % this formula assigns 1 when both inequalities hold, and 0 otherwise end;chain=V*state;