# Monte Carlo Sample input into OLS Regression

I am trying to find the best way to conduct a risk-based regression study. I have distribution data for both X and Y and used a monte-carlo sampling of the distributions to generate a data set. I then conduct an OLS regression on this data.

I discussed this approach to a college who told me its a waste of time doing this as random inputs would only generate random outputs. Is he right about this and the approach is not useful in anyway?

I do not think your approach is a waste of time, with the following caveats:

1. You have the distribution of $X$ and the joint distribution of $Y | X$. If you just have the marginal distribution of $X$ and $Y$ then you don't know how they relate to one another. e.g. they both could be normal distributions but unless you state their correlation, you don't have any kind of useful information.

2. You make multiple independent Monte Carlo samples and construct many regression lines to get a predictive distribution.

If you met these two conditions then what you're doing is essentially like a bootstrapping or Bayesian approach. You can construct a confidence interval around a mean regression line and then make appropriate probabilistic statements about predictions.

If you're making decisions and have a decision cost function, you can feed these multiple regression lines into your cost function.

• Thanks for your reply Robert. Just so I understand. I run the entire process multiple times taking a different a sample set every time from which to run my regression on. I build up a distribution of the returned coefficients and fit results, from which I can create my probabilistic conclusions? – BillyJo_rambler Sep 13 '18 at 10:30
• That's correct. I'm not saying your approach would be the best way of doing it - merely that your results would be meaningful. – Robert Arbon Sep 13 '18 at 14:37
• Dare I ask the question; What would be a better way? – BillyJo_rambler Sep 14 '18 at 15:26