How to forecast $Y/std(Y)$ using a linear model?

How do I forecast $$Y/std(Y)$$ using a linear model?

I'm currently forecasting only $$Y$$, and the model is producing results where the larger predictions corresponding to higher variance $$Y$$. For example, if I take all the $$Y$$ where the prediction exceeds the 70th quantile, the variance of these observations is 30%, and if I take all the $$Y$$ where the prediction is lower than the 30th quantile, the variance of these observations is 10%. For my application, this is not desirable, I want the forecast not be able to separate the predictor by its variance.

This is a financial application where I wish to predict risk adjusted returns over the next 24 hours. I have 3 rows of data for AAPL (3 arbitrary days in the last 5 years), 5 rows of data for MSFT (5 arbitrary days in the last 5 years), etc. So it can be viewed as a panel data set that isn't balanced. The first column of data is $$Y$$, which is the return. The other columns are the design matrix $$X$$. It is a single data frame (hence the panel data designation). Ideally I can build a model:

$$Y_{i,t}/std(Y_{i,t}) = \alpha + X_{i,t}\beta$$

Note that $$\alpha$$ and $$\beta$$ shouldn't vary in either the time or cross-sectional dimensions.

Some of the predictors in $$X$$:

• yesterday's log price change
• some technical analysis signals
• some fundamental analysis signals

All predictors are $$~N(0,1)$$

• If the data generating process is such that variance increases with the level of the series, why would you want your predictions not to reflect that? Also, what is $E(std(Y_i))$? Does $std$ by itself not involve the expectation? – Richard Hardy Aug 22 '20 at 16:54
• @RichardHardy It's a financial application, the higher variance represents more risk. I wish to predict risk-adjusted returns. – ABC Aug 22 '20 at 16:56
• GARCH models are often used for financial returns. They allow for time-varying variance. You can have the mean depend on variance, too, in a model known as GARCH-in-mean. This would reflect that higher risk commands higher return. – Richard Hardy Aug 22 '20 at 16:57
• My application isn't time-series, each observation is iid with no heteroskedasticity – ABC Aug 22 '20 at 16:59
• Then I do not quite understand what kind of data you have. You seem to indicate there is heteroskedasticity as some observations have variance of 30%, other 10%? Yet you say each observation is iid with no heteroskedasticity. Could you perhaps edit the post to explain your data and the variable behind it in more detail? – Richard Hardy Aug 22 '20 at 17:01