How to estimate the unknown parameters of state-space model?

I'm trying to estimate the unknown parameters of state-space model as follows:

in which I decompose the observed variable yt into two unobserved state variables y1t & y2t. The only data I have is yt.

e1t & e2t are jointly normally distributed variables with both means zero and a general covariance matrix, let's say the correlation coefficient is p.

The unknown parameters are , which are to be estimated.

I rewrite it into measurement equations and transition equations as below:

First I tried to use the function 'dssm/ssm' in Econometrics Toolbox and pass it to 'estimate' to solve, but I found that in the function 'dssm/ssm',

e1t & e2t are defined as independent standard Gaussian process, and I cannot find a way to reset this defination. Codes are as follows.

An alternative way seems to define my own Log likelihood function and use MLE to solve, but I don't know the exact way how to write the iterations.

Please help me with a way to solve this out; I have struggled with it for couples of days...Thanks!

A = [1 0 0 NaN; 0 NaN NaN 0; 0 1 0 0;0 0 0 1];
B = [1 0 0; 0 1 0; 0 0 0;0 0 0];
C = [1 1 0 0];
D=[];
StateType = [2 0 0 1];
params0=[0 0 0];
Mdl = dssm(A,B,C,'StateType',StateType);
EstMdl = estimate(Mdl2,yt,params0);

Answer 1

[Harsha Priya Daggubati]

Hi,

I think this thread might give you a good headstart:

https://in.mathworks.com/matlabcentral/answers/125328-estimate-state-space-model-parameters-using-parameter-mapping-function?s_tid=answers_rc1-1_p1_Topic