Question

c) What is the minimum set of conditions necessary for the OLS estimator to be the most efficient unbiased estimator (BLUE) of a parameter? List each of these minimum conditions and explain what they mean in one or two sentences.

b) Choose any two of the conditions and for each one (i) explain what could go wrong in estimating a model should the condition not hold, and (ii) give one real world example (each) of a research design where the condition might not be satisfied.

Answer 1

The following are the minimum conditions necessary for OLS estimators to be BLUE.

As the name suggests BLUE denotes best linear unbiased estimators. If the below depicted five conditions hold true, according to the Gauss Marov Theorem, the OLS estimators are BLUE.

1) Assumption of linearity in parameters.

The regression need not be linear in the explanatory variables used but the cooefficients of these variables have to be linear in nature.

   

(2) is an example of a regression equation linear in parameters. (1) is not linear in parameters.

2)Assumption of normality of residuals

We assume that error terms follow a normal distribution. The expected value of error and variance is as follows

3) No Heteroskedasticity

Heteroskedasticity means that error variances are not constant. It results in violation of the fundamntal assumptions of OLS estimators. The regression would lead to biased results.

4) No multicollinearity

Multicollinearity means that the explanatory variables themselves have a linear relationship between them. In such cases regression would have high R2 values even when originally the variables only explain the relationship partially.

5) No Autocorrelation

Auto correlation means that the error term are related to each other.

Hence for OLS estimators to be BLUE   for i not equal to j

b) If the assumption of no auto correlation is satisfied, OLS estimators would be unbiased but they will no longer have minimum variance as required to be considered BLUE.

This condition is not generally satisfied in time series regressions. Consider regressing present GDP on consumption expenditures of previous years and the current year. It is highly likely that the error terms for this regression will be highly correlated to each other and to the existing explanatory variables. However there are statistical tools to get rid of autocorrelation.

If the assumption of no multicollinearity is not satisfied, this would cause the regression to have high R2 values despite insignificant t values of parameters. This is usually found in cross sectional data.

Consider regressing female wages on female education, education of the other family members and income of the family. It i highly likely that there is a linear relationship between atleast two of the explanatory variables.