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? [2 marks]

ii. what will the covariance of the two regressors tend to? [1 marks]

iii. what will the OLS coeffificient estimates tend to in regression model (1)? [1 marks]

yi = 0 +1x1i +2x2i +ui

(1)

iv. what will the R2 of regression model (1) tend to? [1 marks]

v. what will the OLS coeffificient estimates in regression model (2) tend to? [1 marks]

x1i = 0 +1x2i +1i

(2)

vi. what will the OLS coeffificient 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 coeffificient estimates in regression model (4) tend to?

[1 marks]

yi = 0 +1

1i +2

2i +i

(4)

viii. what will the R2 equal in regression model (4) tend to? [1 marks]

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? [1 marks]