1. As a researcher, you have n independent and identically distributed observations on yi , x1i and x2i , where i = {1,2,3,...,n}. The population from which the sample has been drawn is large. The population model can be described as follows.

For each observation, the dependent variable, yi is a linear function of two regressors and an error term.

yi = x1i x2i +ui

The error term has a standard normal distribution: ui N(0,1).

Three unobserved variables, v ji for j = {1,2,3} all have standard normal distributions so vi j N(0,1).

The regressors, x1i and x2i are related to the three unobserved variables as follows:

x1i = v1i +v3i

x2i = v2i v3i

(a) Explaining your reasoning carefully, as n gets larger:

i. what will the sample variance of each of the observed variables tend to?

ii. what will the covariance of the two regressors tend to?

iii. what will the OLS coefficient estimates tend to in regression model (1)?

yi = 0 +1x1i +2x2i +ui (1)

iv. what will the R 2 of regression model (1) tend to?

v. what will the OLS coefficient estimates in regression model (2) tend to?

x1i = 0 +1x2i +1i (2)

vi. what will the OLS coefficient estimates in regression model (3) tend to? [1 marks] x2i = 0 +1x1i +2i (3)

vii. given that 1i and 2i are the residuals from OLS estimation of regression models (2) and (3) respectively, what will the OLS coefficient estimates in regression model (4) tend to?

yi = 0 +11i +22i +i (4)

viii. what will the R 2 equal in regression model (4) tend to?

ix. if you impose the linear restriction 1 = 2 = on regression model (1) by suitably transforming the regressors, what would the OLS estimate of tend to?