Question

Question 2 (0.5 mark) Consider the multiple regression model containing three independent variables, under Assumptions MLR.1 through MLR.4: y = B. +B,X,+B2x2 +Bzx3+u You are interested in estimating the sum of the parameters on Xı and xz; call this 0 = + B2 (1) Show that Ô, = B1 + B2 is an unbiased estimator of , (ii) Find Varê, in terms of Varhi). Var(82), and Corr1. B2).

Answer 1

solution:

Given that:

consider the multiple regression model containing three independent variables such that the model is given by:

y=$\beta_{_0}$ +$\beta _{_1}$$\chi _{1}$ +$\beta _{2}$$\chi _{2}$ +$\beta _{3}$$\chi _{3}$ +$\upsilon$

also consider that $\Theta _{1}$=$\beta _{1}$ +$\beta _{2 }$

(i) consider the four multiple-linear regression assumptions

1) the model of the population is such that the dependent variable is expressed as the linear combination of explanatory variables and the error term. this is what is exhibited by the given population model. the expression is y=$\beta _{0}$ +$\beta _{1}$$\chi _{1}$ +$\beta _{2}$$\chi _{2}$ +$\beta _{3}$$\chi _{3}$ +$\upsilon$

2)the observations relating to dependent and explanatory variables are random observations.

this is assumed to hold as true in the given case

3)the explanatory variables are independent of each other such that no exact linear relationship among the independent variables holds. that means there is no problem of multi-collinearity

4)the expected value of the disturbance or error term in the population model is zero. that means E(u$E(u /x_1,x_2,x_3)=0$

Given such assumptions , it could be ensured that

$E( \hat{\beta _1})=\beta _1$

$E( \hat{\beta _2})=\beta _2$

Since $\hat{\theta _1}=\hat{\beta _1}+\hat{\beta _2}$

That implies

$\hat{\theta _1}=\hat{\beta _1}+\hat{\beta _2}$

$E(\hat{\theta _1})=E(\beta _1+\beta _2)$

$E(\hat{\theta _1})=E(\beta _1)+E(\beta _2)$

$\theta _1=\beta _1+\beta _2$

This indicates that $\hat{\theta _1}$ is an unbiased estimator of $\theta _1$

ii)

Since $\hat{\theta _1}=\hat{\beta _1}+\hat{\beta _2}$

That implies

$\hat{\theta _1}=\hat{\beta _1}+\hat{\beta _2}$

$Var(\hat{\theta _1})=Var(\hat{\beta _1}+\hat{\beta _2})$

$Var(\hat{\theta _1})=Var(\hat{\beta _1})+Var(\hat{\beta _2})+2Cov(\hat{\beta _1\hat{\beta _2}})$

$Var(\hat{\theta _1})=Var(\hat{\beta _1})+Var(\hat{\beta _2})+2\rho _{\hat{\beta _1,\hat{\beta _2}}}\sigma _{\hat{\beta _1}}\sigma _{\hat{\beta _2}}$

Hence

$Var(\hat{\theta _1})=Var(\hat{\beta _1})+Var(\hat{\beta _2})+2Corr({\hat{\beta _1,\hat{\beta _2}}})\sigma _{\hat{\beta _1}}\sigma _{\hat{\beta _2}}$

$\rho _{\hat{\beta _1,\hat{\beta _2}}}= Corr({\hat{\beta _1,\hat{\beta _2}}})$ and $\rho _{\hat{\beta _1,\hat{\beta _2}}}= Corr({\hat{\beta _1,\hat{\beta _2}}}) = Cov(\hat{\beta _1,\hat{\beta _2}})/\sigma _{\hat{\beta _1}}\sigma _{\hat{\beta _2}}$