Question

 

Problem 3 Gauss Markov Theorem

(i) Write down the formula for the standard error of the OLS estimate, , from a multiple linear regression.

(ii) In your answer io part (i), what is ? What is the íurmla for  

(iii) State the Gauss Markov Theorem, including the assumptions (i.e., write E[u|x1,...,xk] not MLR4)

Answer 1

(i)

Standard error of the OLS estimate is given as,

where

is the standard error of the regression

is the total sample variation in , and

is the R-squared from regressing on all other independent variables (and including an intercept).

(ii)

is the R-squared from regressing on all other independent variables (and including an intercept).  Because the R-squared measures goodness-of-fit, a value of close to one indicates that explains much of the variation in all other independent variables in the sample. This means that are highly correlated with all other independent variables.

where n are number of observations and k are number of independent variables

and are the OLS residuals.

(iii)

Gauss Markov Theorem - Under Assumptions given below, are the best linear unbiased estimators (BLUEs) of respectively.

Assumptions -

Assumption MLR.1 (Linear in Parameters) - The model in the population can be written as

where are the unknown parameters (constants) of interest and u is an unobservable random error or disturbance term.

Assumption MLR.2 (Random Sampling) - We have a random sample of n observations, ,  following the population model in Assumption MLR.1.

Assumption MLR.3 (No Perfect Collinearity) - In the sample (and therefore in the population), none of the independent variables is constant, and there are no exact linear relationships among the independent variables.

Assumption MLR.4 (Zero Conditional Mean) - The error u has an expected value of zero given any values of the independent variables. In other words,

Assumption MLR.5 (Homoskedasticity)- The error u has the same variance given any values of the explanatory variables. In other words, .