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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 $~N(0,1)$

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    $\begingroup$ 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? $\endgroup$ – Richard Hardy Aug 22 '20 at 16:54
  • $\begingroup$ @RichardHardy It's a financial application, the higher variance represents more risk. I wish to predict risk-adjusted returns. $\endgroup$ – ABC Aug 22 '20 at 16:56
  • $\begingroup$ 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. $\endgroup$ – Richard Hardy Aug 22 '20 at 16:57
  • $\begingroup$ My application isn't time-series, each observation is iid with no heteroskedasticity $\endgroup$ – ABC Aug 22 '20 at 16:59
  • $\begingroup$ 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? $\endgroup$ – Richard Hardy Aug 22 '20 at 17:01

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