When is ols unbiased

A biased estimator will yield a mean that is not the value of the true parameter of the population. When there are more than one unbiased method of estimation to choose from, that estimator which has the lowest variance is best. An estimator a function that we use to get estimates that has a lower variance is one whose individual data points are those that are closer to the mean. This estimator is statistically more likely than others to provide accurate answers. The OLS estimator is one that has a minimum variance.

The Gauss Markov theorem says that, under certain conditions, the ordinary least squares OLS estimator of the coefficients of a linear regression model is the best linear unbiased estimator BLUE , that is, the estimator that has the smallest variance among those that are unbiased and linear in the observed output variables. The regression model is where:. The OLS estimator of is.

First of all, note that is linear in. In fact, is the product between the matrix and , and matrix multiplication is a linear operation. It can easily be proved that is unbiased, both conditional on , and unconditionally, that is,. We can use the definition of to re-write the OLS estimator as follows: When we condition on , we can treat as a constant matrix.

Therefore, the conditional expectation of is The Law of Iterated Expectations implies that. Now that we have shown that the OLS estimator is linear and unbiased, we need to prove that it is also the best linear unbiased estimator.

When is a scalar i. Now for the implications. Under 1 - 4, OLS is unbiased, and consistent. Repmat Repmat 3, 1 1 gold badge 15 15 silver badges 32 32 bronze badges. In other words, 4 is both impossible to verify and easy to ignore. Or maybe I just misunderstanding you? Could you either eloborate or give a reference.

I just posted an answer to explain why. In other words, "finite-sample" properties are all gone. So simply put, Strict Exogeneity cannot be "easily ignored". Alecos Papadopoulos Alecos Papadopoulos Isn't assuming that the mean is a not a function of the regressors equivalent to assuming homoscedasticity?

This is the crucial assumption that must be made independently of whether we include a constant term or not. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Now live: A fully responsive profile. Version labels for answers. Linked 0. Related Hot Network Questions. Question feed. Cross Validated works best with JavaScript enabled.