The title is a bit confuse, but what this presumably boils down to is estimating a covariance matrix from data that are itself estimates (here to estimate how the units change together over time). Hence, the model first does its multilevel model, then post stratifies, and then uses those estimates at the units level after first-differencing to estimate a covariance matrix of these differences (to eventually plug them into a random walk model).

I built this based on the SUR description in the manual with `phi_diff`

being my outcome. Yet something is wrong with that part and I can’t quite figure it out. Upon investigation the covariance matrix resulting from `L_Sigma = diag_pre_multiply(L_sigma, L_Omega);`

has several elements in the last columns that are equal to 0 (e.g. -0.0, 0.0, etc) which I think is the issue but I don’t know where this arises from. Thoughts?

Stan code

```
data {
int T;
int<lower = 0> G;
int<lower = 1, upper = G> g[T]; // end points
int<lower = 1, upper = 4> age[G, T];
int<lower = 1, upper = 4> race[G, T];
int<lower = 1, upper = 51> state[G, T];
int<lower = 1, upper = 2> sex[G, T];
int<lower = 0> sm_N[G, T];
int<lower = 0> bi_N[G, T];
int P[4, 4, 51, 2];
}
transformed data{
vector[51] zeros = rep_vector(0.0, 51);
}
parameters {
real alpha[T];
real<lower = 0> sigma_beta[T];
matrix<multiplier = sigma_beta[T]>[4, T] beta;
real<lower = 0> sigma_gamma[T];
matrix<multiplier = sigma_gamma[T]>[4, T] gamma;
real<lower = 0> sigma_delta[T];
matrix<multiplier = sigma_delta[T]>[51, T] delta;
real epsilon[T];
//corr_matrix[51] L_Omega; // prior correlation
//vector<lower=0>[51] tau; // prior scale
cholesky_factor_corr[51] L_Omega;
vector<lower=0>[51] L_sigma;
vector[51] mu;
}
transformed parameters {
matrix[51, T] phi;
vector[51] phi_diff[T - 1];
matrix[51, T] expect_pos = rep_matrix(0, 51, T);
matrix[51,T] total = rep_matrix(0, 51, T);
matrix[51, 51] L_Sigma = diag_pre_multiply(L_sigma, L_Omega);
//matrix[51, 51] Sigma_beta;
for (t in 1:T){
for (b in 1:4){
for (c in 1:4){
for (e in 1:2){
total[,t] += to_vector(P[b, c, :, e]);
expect_pos[,t]
+= to_vector(P[b, c, :, e])
.* to_vector(inv_logit(alpha[t] + beta[b,t] +
gamma[c,t] + delta[:,t] +
{epsilon[t], -epsilon[t]}[e]));
}
}
}
phi[:,t] = expect_pos[:,t] ./ total[:,t];
}
for (t in 2:T) phi_diff[t - 1] = phi[:, t] - phi[:, t - 1];
// covmat
}
model {
// covmat
target += multi_normal_cholesky_lpdf(phi_diff | zeros, L_Sigma);
// mrp model
for (t in 1:T){
sm_N[1:g[t], t] ~
binomial_logit(bi_N[1:g[t], t],
alpha[t] +
beta[age[1:g[t],t], t] +
gamma[race[1:g[t],t], t] +
delta[state[1:g[t],t], t] +
to_vector({epsilon[t], -epsilon[t]}[sex[1:g[t],t]]));
beta[:,t] ~ normal(0, sigma_beta[t]);
gamma[:,t] ~ normal(0, sigma_gamma[t]);
delta[:,t] ~ normal(0, sigma_delta[t]);
}
// mrp priors
alpha ~ normal(0, 2);
epsilon ~ normal(0, 2);
sigma_beta ~ normal(0, 1);
sigma_gamma ~ normal(0, 1);
sigma_delta ~ normal(0, 1);
// covmat priors
// tau ~ normal(0, 2.5);
// L_Omega ~ lkj_corr_cholesky(2);
L_Omega ~ lkj_corr_cholesky(2);
L_sigma ~ normal(0, 2.5);
}
```