# Simulating a state space model

I want to simulate data from the following model:

$\textbf{z}_k=\textbf{H}\textbf{x}_k+\textbf{v}_k$ $\textbf{v}_k \sim N(\textbf{0},\textbf{R})$

$\textbf{H}$ does not change over time
$\textbf{x}$ is a vector of loadings
$\textbf{R}$ is a diagonal of constants

$\textbf{x}_k=\textbf{F}\textbf{x}_{k-1}+(\textbf{I}-\textbf{F}){\mu} + \textbf{w}_k$ $\textbf{w}_k \sim N(\textbf{0},\textbf{Q})$

$\textbf{I}$ is the identity matrix
$\mu$ is the vector of mean values of $\textbf{x}$
$\textbf{F}$ is diagonal with the AR(1) params which do not change over time
$\textbf{Q}$ is diagonal with the innovation processes for $\textbf{x}$

I have the following code in Matlab

nDates=20000; %number of dates

mats=[1 2 3 4 5 6 7 8 9 10 12 15 20 25 30]'; %maturities
nY=length(mats); %#number of yields

z=zeros(nY,nDates);  %declare vector for yields
x=zeros(3,nDates); %declare vector for factors

R=0.00001;   %standard deviation
I=eye(3); %3*3 identity matrix

v=normrnd(0,R,nY,nDates); %generate residuals
F=[0.9963 0 0; 0 0.9478 0; 0 0 0.774]; %AR(1) matrix
mu=[0.0501; -0.0251;-.0116]; %mean of X
lambda=0.5536;

q = [0.0026^0.5 0 0;0 0.0027^0.5 0; 0 0 0.0035^0.5];
Q=q*q';
rng('default');  % For reproducibility
r = randn(nDates,3);
w= (r*Q)';

B= [ones(nY,1),((1-exp(-lambda*mats))./(lambda*mats)),((1-exp(-lambda*mats))./(lambda*mats))-exp(-lambda*mats)];

X(:,1)=mu;

for t=2:nDates
x(:,t)=F*(x(:,t-1))+(I-F)*mu+w(:,t);
z(:,t)=B*x(:,t)+v(:,t);
end

z(:,1)=[];


It all seems straightforward enough but what test can I so to ensure that it has been implemented correctly?

Ones I have thought of:
Check the correlation of the factors x on their 1 period lags match the values given in matrix F
Check that the variances of v and w are correct
Check that the mean of the simulated variables are correct

I would like to check that the empirical variance of the parameters matches their theoretical equivalent, but I don't know what the theoretical equivalent should be?

Please feel free to suggest further tests that will allow me to know for sure if the implementation is correct.