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I have a panel data set and I have created a model and finally I have obtained some density forecasts.

That is, I run my model for the $y_{it}$ and i obtain predictions for $\hat{y_{i,t+1}}$ , $\hat{y_{i,t+2}}$ ,..., $\hat{y_{i,t+h}}$ etc.

The $\hat{y_{i,t+1}}$ , $\hat{y_{i,t+2}}$ etc, all of them follow a normal distribution, but with different mean and variance. so $\hat{y_{i,t+1}} \sim N(\mu_{1},\sigma_{1}^{2})$ , $\hat{y_{i,t+2}} \sim N(\mu_{2},\sigma_{2}^{2})$, ..., $\hat{y_{i,t+h}} \sim N(\mu_{h},\sigma_{h}^{2})$.

My question is, what is their joint distribution ? And how can I obtain it ?

Edit: The $y_{it}$'s are the daily consumption of a product of individual $i$ at time $t$, so yes, they are highly correlated.

Thank you for your help :)

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