I have a equation with the parameters determined from multiple linear regression:
$$ Y = \beta_0 + X_1 \beta_1 + X_2\beta_2 $$
I would like to forecast the distribution of $Y$ numerically, in other words a histogram of the values $Y$ can take on in the future. From the histogram I can then deduce arbitrary prediction intervals.
Both $X_1$ and $X_2$ are random variables with arbitrary distributions. For example $X_1$ could be uniform and $X_2$ could be normal.
My approach Use a Monte Carlo type approach. Randomly sample $X_1$, $X_2$, $\beta_1$ and $\beta_2$ and calculate $Y$, plot histogram.
- Is this a reasonable approach?
- If yes, then what is a reasonable distribution for $\beta_1$ and $\beta_2$?
- If no, how would I go about computing such a distribution.
Many thanks in advance.
Edit Put another way:
I would like to forecast $Y$ using forecast values of $X_1$ and $X_2$. Instead of just producing a point estimate and 95% prediction interval I would like to produce a histogram which I can then use to get arbitrary quantiles of $Y$. I would like the forecast to take into account the fact that $X_1$ and $X_2$ have a given distribution.
My simplest possible question is:
How do I produce a histogram of forecast values of $Y$ that takes account of the fact that my regression is only an estimate and that my forecasts of $X_1$ etc. are random with a given distribution?