How to simulate ARX process with AR errors

I am trying to simulate the following model

where and

I tried learning from this https://uk.mathworks.com/help/econ/regarima-class.html

But was not helpful, can you help? Thank you

Answer 1

[ABDULAZIZ ALTUN]

The way I did it was by writing my own function.

%% ARX_AR_Error: function description
function [outputs] = ARX_AR_Error(c, S, beta, rho_Reg, rho_Error)
	% c is the sample length
	% S regressor variable
	% rho_Reg the rho of the main regression
	% rho_Error the rho of the error regression

	M_1 =rho_Reg*diag(ones(c-1,1),-1);
	M_2=zeros(c,1);
	M_2(1,1)=rho_Reg;
	ey_M_1=inv(eye(c)-M_1);

	% Generate the AR error
	MdlY = arima('AR',{rho_Error},'Constant',0,'Variance',1);
	E = simulate(MdlY,c);

	outputs=ey_M_1*(S*beta+E+M_2*randn(1));

Therefore, in simulating variables I simply wrote the following code

c=100;
s=randn(c,1);
rho_Reg=0.2;
rho_Error=0.3;
beta=0.5;
Y=ARX_AR_Error(c, s, beta, rho_Reg, rho_Error)

plot(Y)