Consider a regression model where y is explained by two regressors x1 and x2, i.e. a regression of the form yi = x1i*BETA1 + x2i*BETA2 + ui (1) where i = 1,...,N observations are available.
- Rewrite the model in matrix notation and derive the formula for the OLS estimator.
- Show formally that if two regressors are orthogonal, i.e.SUM(x1i*x2i) = 0, the values obtained for the coefficients BETA1 and BETA2 by estimating (1) are identical to the coefficients obtained by performing an OLS regression of y on only x1 respectively by performing an OLS regression of y on only x2.
- Show that due to the orthogonality of the regressors also the estimated coefficients BETA(hat)1 and BETA(hat)2 are uncorrelated, i.e. show that cov(BETA(hat)1, BETA(hat)2) = 0.