Econometric Theory/Properties of OLS Estimators

OLS estimators have the following properties:

1. Linear
2. Unbiased
3. Efficient: it has the minimum variance
4. Consistent

Linear

OLS estimators are linear functions of the values of Y (the dependent variable) which are linearly combined using weights that are a non-linear function of the values of X (the regressors or explanatory variables). So the OLS estimator is a "linear" estimator with respect to how it uses the values of the dependent variable only, and irrespective of how it uses the values of the regressors.

Unbiased

Suppose that the population size is 100 for anything that we are studying. We use samples of size 10 to estimate the $\alpha$ and $\beta$ of the population. Everytime we use a different sample (a different set of 10 unique parts of the population), we will get a different $\alpha$ and $\beta$.

With the OLS method of getting $\alpha$ and $\beta$, we get a situation wherein after repeated attempts of trying out different samples of the same size, the mean (average) of all the $\alpha$ and $\beta$ from the samples will be equal to the actual $\alpha$ and $\beta$ of the population as a whole.

Basically, this means that if you do the exercise over and over again with different parts of the population, and then you find the mean for all the answers you get, you will have the correct answer (or you will be very close to it).

A biased estimator will yield a mean that is not the value of the true parameter of the population.

Efficient: Minimum variance

This property is what makes the OLS method of estimating $\alpha$ and $\beta$ the best of all other methods. When there are more than one unbiased method of estimation to choose from, that estimator which has the lowest variance is best. (Variance is a measure of how far the different $\alpha$ and $\beta$ are from their mean; the variance is the average distance of an element from the average.)

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.

This property is simply a way to determine which estimator to use.

• An estimator that is unbiased but does not have the minimum variance is not good.
• An estimator that has the minimum variance but is biased is not good
• An estimator that is unbiased and has the minimum variance of all other estimators is the best (efficient).

The OLS estimator is an efficient estimator.

Consistent

A consistent estimator is one which approaches the real value of the parameter in the population as the size of the sample, n, increases.