Add a comment. Active Oldest Votes. Improve this answer. Rachel Kogan Rachel Kogan 1 1 gold badge 2 2 silver badges 9 9 bronze badges. If you look at the statement of the theorem, it's saying that the variance of some other estimator minus the variance of the OLS estimator is positive semi-definite.
I'm not saying that you're wrong, but it would be nice to have some justification. For example, the maximum likelihood estimator in a regression setup with normal distributed errors is BLUE too, since the closed form of the estimator is identical to the OLS but as a method, ML-estimation is clearly different from OLS. The Gauss—Markov Theorem however tells you that in the class of linear unbiased estimators you don't have too look further than OLS, since every other estimator in this class can not do better under the assumptions.
Olaf Yagmur Olaf Yagmur 41 3 3 bronze badges. The third assumption we make is that the regressors are orthogonal to the error terms. Assumption 3 orthogonality : For each , and are orthogonal, that is,. Proposition If Assumptions 1, 2 and 3 are satisfied, then the OLS estimator is a consistent estimator of. Let us make explicit the dependence of the estimator on the sample size and denote by the OLS estimator obtained when the sample size is equal to By Assumption 1 and by the Continuous Mapping theorem , we have that the probability limit of is Now, if we pre-multiply the regression equation by and we take expected values, we get But by Assumption 3, it becomes or which implies that.
In this section we are going to discuss a condition that, together with Assumptions above, is sufficient for the asymptotic normality of OLS estimators. Assumption 4 Central Limit Theorem : the sequence satisfies a set of conditions that are sufficient to guarantee that a Central Limit Theorem applies to its sample mean. For a review of some of the conditions that can be imposed on a sequence to guarantee that a Central Limit Theorem applies to its sample mean, you can go to the lecture entitled Central Limit Theorem.
In any case, remember that if a Central Limit Theorem applies to , then, as tends to infinity, converges in distribution to a multivariate normal distribution with mean equal to and covariance matrix equal to.
With Assumption 4 in place, we are now able to prove the asymptotic normality of the OLS estimators. Proposition If Assumptions 1, 2, 3 and 4 are satisfied, then the OLS estimator is asymptotically multivariate normal with mean equal to and asymptotic covariance matrix equal to that is, where has been defined above.
As in the proof of consistency, the dependence of the estimator on the sample size is made explicit, so that the OLS estimator is denoted by.
First of all, we have where, in the last step, we have used the fact that, by Assumption 3,. Note that, by Assumption 1 and the Continuous Mapping theorem, we have Furthermore, by Assumption 4, we have that converges in distribution to a multivariate normal random vector having mean equal to and covariance matrix equal to. Thus, by Slutski's theorem, we have that converges in distribution to a multivariate normal vector with mean equal to and covariance matrix equal to.
Assumption 5 : the sequence satisfies a set of conditions that are sufficient for the convergence in probability of its sample mean to the population mean which does not depend on. The OLS estimator is one that has a minimum variance. This article is part of the EconHelp Tutoring Wiki.
Categories : Economics EconHelp. Navigation menu Personal tools Log in. Unbiased estimators are not necessarily consistent , but those whose variances shrink to zero as the sample size grows are consistent. In other words:. We will return to our example in this chapter. We have recently proved the unbiasedness and consistency of OLS estimators.
To illustrate these properties empirically , we will generate replications i. The reason that we choose to generate different samples for each sample size, is to calculate the average and variance of the estimated parameters:. We can see that, the mean of the estimated parameters are close to the true parameter value regardless of sample size. The variance of the estimated parameters decreases with larger sample size, i. More colors in R.
More colors in Python. We see that the histograms of the OLS estimators have a bell-shaped distribution. Under assumption UR. In this chapter we have shown how to derive an OLS estimation method in order to estimate the unknown parameters of a linear regression with one variable.
We have also shown that these estimators under conditions UR. For the linear regression case, the regression appears to be drawn as expected, but for nonlinear models, plotting unsorted data will results in an unreadable plot. Practical Econometrics Preface 1 Introduction 1. Prediction 4. Practical Econometrics and Data Science.
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