Z: Understanding Unbiased Estimators in Regression Analysis

[wow007051]

first of all...what is an unbiased estimator??
how to check whether a reggression provide an unbiased estimator?

thanks!

 

[operationsres]

www.talkstats.com
stats.stackexchange.com.

The following three events can cause biased estimators:

1) Omitted variable bias.
2) cov(error,regressors) $\not=$ 0
3) cov(regressor1, regressor2) $\not=$ 0
4) Model mis-specification (eg not including a squared term when you should - do a RAMSEY RESET test).

 

[Dickfore]

An unbiased estimator is a sample function:
$$Z_n = f(X_1, \ldots, X_n)$$
such that, for an i.i.d. sample with a parameter of the distribution θ that we are trying to estimate, has the property:
$$\mathrm{E}\left[Z_n \right] = \theta$$
If this does not hold for a finite n, but is true as $n \rightarrow \infty$, then we say that the estimator is asymptotically unbiased.

In general, if the function f is some non-polynomial function, it is very hard to check the bias of the estimator. If, on the other hand, the estimator is a (symmetric) polynomial of degree p (pth moment), we may use some rules for the expectation values. For example, the mean:
$$\bar{X}_n \equiv \frac{1}{n} \, \sum_{k = 1}^{n}{X_k}$$
has the property:
$$\mathrm{E} \left[\bar{X}_n \right] = \frac{1}{n} \, \sum_{k = 1}^{n}{\mathrm{E} \left[ X_k \right]} = E \left[ X \right]$$
is the unbiased estimator of the mathematical expectation of the random variable X.

 

[wow007051]

can i say if a regression with very low r^2 it doesn't provide unbiased estimator?

 

[wow007051]

can i say if a regression with very low r^2 it doesn't provide unbiased estimator?

 

[Ray Vickson]

can i say if a regression with very low r^2 it doesn't provide unbiased estimator?

No. Bias and r^2 are not really related.

RGV