I want to do this time-varying ordered logit model or state space model

model like this :

Y^* = \beta_t X_{it} + \epsilon_{it}

\beta_t = \beta_{t-1} + w_t

\epsilon_{it} is a logistic fuction and Y^* is a latent variable. X_{it} and \beta_{t} also the vector

and I want to add a constant coefficient like this :

Y^* = \beta_t X_{it} + \lambda Z_{it}+\epsilon_{it}

Is there exist the identification problem ?

And how can I wirte this Z_{it} in the matrix to do multivariate regression to speed up the running which I means the w_t can be multivariate prior? because I use non-centered parameter in my code.

this is my original stan code

```
functions {
real partial_sum(
int[] slice_y,
int start, int end,
matrix x1,matrix x2, vector[] thresh,
vector beta1, vector[] beta2, int[] g
)
{
real lp = 0;
for(i in start:end)
lp += ordered_logistic_lpmf(slice_y[i-start+1] |x1[i]*beta1+x2[i]*beta2[g[i]] , thresh[g[i]]);
return lp;
}
real induced_dirichlet_lpdf(vector c, vector alpha, real phi) {
int K = num_elements(c) + 1;
vector[K - 1] sigma = inv_logit(phi - c);
vector[K] p;
matrix[K, K] J = rep_matrix(0, K, K);
// Induced ordinal probabilities
p[1] = 1 - sigma[1];
for (k in 2:(K - 1))
p[k] = sigma[k - 1] - sigma[k];
p[K] = sigma[K - 1];
// Baseline column of Jacobian
for (k in 1:K) J[k, 1] = 1;
// Diagonal entries of Jacobian
for (k in 2:K) {
real rho = sigma[k - 1] * (1 - sigma[k - 1]);
J[k, k] = - rho;
J[k - 1, k] = rho;
}
return dirichlet_lpdf(p | alpha)
+ log_determinant(J);
}
}
data {
int<lower=1> N; // Number of observations
int<lower=1> K; // Number of ordinal categories
int<lower=1> D;
int<lower=1> DN;
array[N] int<lower=1, upper=K> y; // Observed ordinals
matrix[N,D-DN] x1;
matrix[N,DN] x2;
int g[N];
int<lower=1> P; //P different threshold
}
parameters {
vector[D-DN] beta1;
vector[DN] beta21;
real<lower=0> sigma;
vector<lower=0>[DN] Omega;
ordered[K - 1] thresh[P];
vector[DN] yita[P-1];
}
transformed parameters{
vector[DN] beta2[P];
beta2[1] = beta21;
for (i in 2: P){
beta2[i] = beta2[i-1] + Omega.*yita[i-1];
}
}
model {
beta1~ normal(0,10);
beta21 ~normal(0,10);
thresh[1] ~ induced_dirichlet(rep_vector(1, K), 0);
sigma ~ normal(0,1);
Omega ~ normal(0,1);
for (i in 1: (P-1)){
(thresh[i+1]-thresh[i])~ normal(0,sigma);
yita[i] ~ normal(0,1);
}
target += reduce_sum(partial_sum, y, 1, x1,x2, thresh, beta1,beta2, g);
}
```