# Incorporating intraday data into end-of-day forecast

my target variable is observable intraday but I am interested only in EOD forecasts. I will denote the variable $\ y_{D,24}$ as the reading of interest for day D is recorded at the end of 24th hour. At midnight (start of day), my model could look like

$\ y_{D,24}=f(1,y_{D-i,24}, ...)+\epsilon_D$

At noon, however, it makes sense to add the intraday record of the given variable, hence the modified model is

$\ y_{D,24}=f(1, y_{D-i,24}, ...)+g(y_{D,12})+\epsilon_D$

If I assume that the target variable does not drop intraday (e.g. intraday measurements of total traded volume can only rise during the day), then if the latter model gives a forecast where

$\ \widehat{y_{D,24}} <y_{D,12}$

I already know my model is useless for that day. My question would thus be:

1. Is it possible to estimate the latter model so that forecasts of $\ y_{D,24}$ are always at least as large as $\ y_{D,12}$?
2. One way of dealing with this problem could be modeling $\ y_{D,24} - y_{D,12}$ with a positive-valued distrubition in the error term, i.e. like $\ y_{D,24}-y_{D,12}=g(1, y_{D-i,24}, ...)+\epsilon_{D,pos}$ This, however, still does not ensure the desired quality, as there is no guarantee that $\ g(1, y_{D-i,24}, ...)+E[\epsilon_{D,pos}] >= 0$ under all circumstances. Second, if somehow we managed to ensure this, then a second question would be how to ensure that the model does not always predict $\ \widehat{y_{D,24}} >y_{D,12}$. For example, if the target variable was measured in thousands of contracts, then at one point in the day, the growing volume would be unlikely to make it to another thousand, which should result into $\ \widehat{y_{D,24}} = y_{D,12}$, and this would happen at least for some hours every day (not necessarily the 12th, but some later hours). So once we ensure $\ \widehat{y_{D,24}} >= y_{D,12}$, how do we allow $\ \widehat{y_{D,24}} = y_{D,12}$ to happen "quite frequently"? A wild guess could be something like "if $\ \widehat{y_{D,24}} - y_{D,12}$ is 'sufficiently small', report it as zero".
3. Or is there some other method usually employed for this kind of problems.

Any help is much appreciated, Daniel