A simple econometrics question

[JohnMcClane]

I was just wondering about this and was hoping if someone knew the answer to these question. Let's say we are trying to estimate a co-efficient parameter b1 in a multiple regression model. We use some estimator and this estimator satisfies all of the assumptions it is supposed to satisfy. Now, we get an estimate for b1. Let's say this estimate equals 2. Now, is there anyway we can tell how far off this estimate is from the parameter value? In addition, let us assume that the estimator is consistent. Is there anyway to measure the relationship between the sample size and the variance of the estimator?

This is not a homework question. I am just curious about these issues.

[Gecko]

hmmm, use asymptotics. i.e., run monte carlo's and see if your simulations (from data) fit with your theory.

[^__^]

The coefficients in multiple regression is automatically determined, if you are using the least square framework, the estimated value is independent of the assumptions imposed on the residual's distribution (the distribution is chosen for inference and fitting purpose). Of course, there are other kinds of regressions, but I believe in a multiple setting, the estimator should be a vector valued one, i.e. all estimates for the coefficients should be determined at the same time. We are fitting the whole model to explain all the data. Any superb estimating process for any separate coefficients would unlikely be a good model, unless it estimates all other coefficients with some accuracy.

Your 2nd question is very good and incredibly hard. In genreal, theories of statistics on point estimation and inference on parameters are pretty much settled on an asymptotic basis, i.e. as n -> infinity. This is only the asymptotic performance of the estimators, not finite sample performance. In practice however, I believe any econometrician and statistician would try taking finite sample issue into account. Intuitively, this says not only do the limits matter, but also the speeds of convergence of the estimators. In theory (or maybe in some practical problems), even with markov chain monte carlo, we can encounter this problem (I dont think McMc works well for economic data, by our economic intuition). We sometimes have no idea how large the sample needs to be. Similar reasoning applies to your question on consistency, which is a convergence in probability issue, also asymptotic.

If we are lucky, two estimators have the same distribution (at least the same family), we can sort of find a sensable way to proceed (but certainly not perfectly rigorous). If we have different distributions, the issue becomes very tricky. I clearly reached my limit here and dont know what to say... Some perhaps relevant references if you are interested:
google these to find out the journals:

Geometrizing rates of convergence I, II, III, David Donoho

P.S. your question ( I think it is a big problem ) is nowhere near simple.