# Infer a population, and hence a sampling distribution, from a sample

I'm studying econometrics (i.e. linear regression...) and I have a question about statistical inference concerning the coefficients of the regression.

In order to give ourselves an idea of what the sampling distribution of a coefficient is, there are a bunch of clunky and not very realistic hypotheses to be made. So I was wondering if there is a technique that exists, that would allow us to generate an artificial population, taken the characteristics of the distribution of the sample (let's say it's a very large sample)?

Then all we would have to do is to repeatedly take equally-sized random samples from this artificial population, and there's our sampling distribution, from which we would be able to make all the inference tests we'd like.

Is it possible to do something like this? Thanks

• What you're describing is basically the idea behind bootstrapping. en.wikipedia.org/wiki/Bootstrapping_(statistics) Aug 31, 2016 at 15:57
• Ok, I heard the name before but did not know what it meant; do you know of any good tutorials on the subject? Thanks Aug 31, 2016 at 16:39

The general answer to what you are looking for is a simulation or Monte Carlo simulation. You would just make repeated draws from a generated (known) population, apply your modeling and get the resulting distributions.

However, there are two reasons that this isn't really done (at least for the types of things you are asking about).

First, you still have to make assumptions about the distribution of the variables and how they relate. This doesn't really weaken any of the assumptions that go into things like linear regression.

Second, if you can do it is with theory you can prove it to be true whereas with something like MC you can only say that "it seems likely." In general, if you can prove something, you should prove it.

• If I understand correctly, Monte Carlo simulations are made with an hypothesis about the form of the population linear model. But how do you use it with a real sample of data to determine the sampling distribution? Aug 31, 2016 at 16:38