I have a forecasting problem and already built a decently working VAR model which provides forecasts as $\hat{Y}_{iT}$, for $i = 1,..n$ and $T$ is forecast time period.
But now I have an additional constraint that the sum of my forecasts should add up to constant,i.e, $\sum_{i=1}^n{\hat{Y}_{iT}} = c$
One easy way to ensure this is, just multiply all my forecasts with the value $[c/\sum{\hat{Y}_{iT}}]$. But I want to objectively look at this and formulated the below objective function:
$$\arg\min_{\alpha_i} \; E \left( \sum_{i=1}^n(Y_{iT} - \alpha_i \ \hat{Y}_{iT} )^2 \right) $$ $$ \text{such that} \quad \sum_{i=1}^n (\alpha_i \ \hat{Y}_{iT}) = c$$
This can be solved easily using the Lagrange multiplier approach but I am not sure if my expectation derivation is correct. This is what I got:
Assuming a simple VAR(1) model, $Y_{iT}= \beta\ Y_{i\ {T-1}} + \epsilon_{iT} $ and $\hat{Y}_{iT}=\hat{\beta}Y_{i\ T-1}$ putting these into the above expectation we end up with,
$$ E\left( \sum_{i=1}^n(\epsilon_{iT}\ + \hat{\beta}Y_{iT-1}(1-\alpha_i))^2 \right) $$ $$= E\left(\sum_{i=1}^n \epsilon_{iT}^2 \right) + E\left(\sum_{i=1}^n \hat{\beta}^2 Y_{iT-1}^2 (1-\alpha_i)^2 \right) + E\left(\sum_{i=1}^n 2\hat{\beta} Y_{iT-1} (1-\alpha_i) \epsilon_{iT} \right) $$ $$ = \sum_{i=1}^n E(\epsilon_{iT}^2) + \sum_{i=1}^n Y_{iT-1}^2 (1-\alpha_i)^2 E(\hat{\beta}^2) + \sum_{i=0}^n2(1-\alpha_i)Y_{iT-1}E(\hat{\beta}\epsilon_{iT}) $$ $$= \sum_{i=1}^n \hat{\sigma_{i}}^2 + \left(\sum_{i=1}^n Y_{iT-1}^2 (1-\alpha_i)^2\right) E(\hat{\beta}^2) + 0 $$
Now, I can calculate the individual $\hat{\sigma_{i}}$ and the $E(\hat{\beta}^2)$ also gets cancelled out when i plug this into the Lagrangian equations.
Does this make sense? Are the derivations right or have I overlooked any assumptions, if so can you point me in the right direction.
