Question

Indicate whether each of the following statements is True or False, and Briefly Justify your answer.

The Gauss-Markov Theorem says that within the class of linear, unbiased estimators, OLS estimators have zero variance.

Answer 1

Greetings for the day,

The Gauss–Markov theorem states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators if the errors in the linear regression model are uncorrelated, have equal variances and the expectation value of zero.

When estimating regression models, we know that the results of the estimation procedure are random. However, when using unbiased estimators, at least on average, we estimate the true parameter. When comparing different unbiased estimators, it is therefore interesting to know which one has the highest precision: being aware that the likelihood of estimating the exact value of the parameter of interest is 00 in an empirical application, we want to make sure that the likelihood of obtaining an estimate very close to the true value is as high as possible. This means we want to use the estimator with the lowest variance of all unbiased estimators, provided we care about unbiasedness. The Gauss-Markov theorem states that, in the class of conditionally unbiased linear estimators, the OLS estimator has this property under certain conditions.

The Gauss-Markov Theorem: OLS is BLUE!

The Gauss-Markov theorem famously states that OLS is BLUE. BLUE is an acronym for the following:

Best Linear Unbiased Estimator

In this context, the definition of “best” refers to the minimum variance or the narrowest sampling distribution. More specifically, when your model satisfies the assumptions, OLS coefficient estimates follow the tightest possible sampling distribution of unbiased estimates compared to other linear estimation methods.

What Does OLS Estimate?

Regression analysis is like any other inferential methodology. Our goal is to draw a random sample from a population and use it to estimate the properties of that population. In regression analysis, the coefficients in the equation are estimates of the actual population parameters.

The notation for the model of a population is the following:

     

The betas (β) represent the population parameter for each term in the model. Epsilon (ε) represents the random error that the model doesn’t explain. Unfortunately, we’ll never know these population values because it is generally impossible to measure the entire population. Instead, we’ll obtain estimates of them using our random sample.

The notation for an estimated model from a random sample is the following:

  

Now we will see the biased and unbiased estimates of sampling distribution to minimise our variance:

In the graph above, we have seen the two graph one with a estimated and another one as a exacted so those, beta represents the true population value. The curve on the right centres on a value that is too high. This model tends to produce estimates that are too high, which is a positive bias. It is not correct on average. However, the curve on the left centres on the actual value of beta. That model produces parameter estimates that are correct on average. The expected value is the actual value of the population parameter. That’s what we want and satisfying the OLS assumptions helps us!

Keep in mind that the curve on the left doesn’t indicate that an individual study necessarily produces an estimate that is right on target. Instead, it means that OLS produces the correct estimate on average when the assumptions hold. Different studies will generate values that are sometimes higher and sometimes lower—as opposed to tending to be too high or too low.

So we can say that the given statement is false because we cannot say that there is no variance