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Currently I am trying to find a prediction interval for $q_{x}(t)$ in the following model:

\begin{align} q_{x}(t) &= 1 - \exp(-\mu_{x}(t))\\ \ln(\mu_{x}(t)) & = \ln(\mu_{x}^{EU}(t)) + \ln(\mu_{x}^{NL}(t)) \\ \ln(\mu_{x}^{EU}(t)) &= A_{x} + B_{x}K_{t} \\ \ln(\mu^{NL}_{x}(t)) &= \alpha_{x} + \beta_{x} \kappa_{t} \end{align} where \begin{align} K_{t+1} &= K_{t} + \theta + \epsilon_{t+1} \\ \kappa_{t+1} &= a\kappa_{t} + \delta_{t+1} \end{align}

The error terms $(\epsilon_{t},\delta_{t})$ are assumed to be independent and are drawn from a bi variate normal distribution with mean $(0,0)$ and covariance matrix $\mathbf C = \begin{bmatrix} \sigma^{2}_{\epsilon} & \rho \sigma_{\epsilon} \sigma_{\delta}& \\ \rho \sigma_{\epsilon} \sigma_{\delta} & \sigma^{2}_{\delta} \end{bmatrix}$

Basically, this is an extension of the Lee-Carter model that allows for coherent forecasting of sub-populations. From an earlier question ( Maximum Likelihood in a time series multi-population model) , I have estimated the parameters necessary to compute prediction intervals.

The problem is that I would like to have a prediction interval for $q_{x}(t)$. If this were a standard lee carter model I know I can get a prediction interval by taking a confidence interval for the AR(1) process and then plug in the resulting limits into the expression for $\mu_{x}(t)$ and $q_{x}(t)$. However when I tried doing this for the multi population model the confidence bands seemed to be too small. Is there perhaps a different way to get the prediction intervals for the bivariate case?

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