Question: Consider the multiple regression model containing three independent variables, under Assumptions MLR.1 through MLR.4:

You are interested in estimating the sum of the parameters on \(x_{1}\) and \(x_{2}\); call this \(\theta_{1}=\beta_{1}\) \(+\beta_{2}\) (i) Show that \(\hat{\theta}_{1}=\hat{\beta}_{1}+\hat{\beta}_{2}\) is an unbiased estimator of \(\theta_{1}\).

(ii) Find \(\operatorname{Var}\left(\hat{\theta}_{1}\right)\) in terms of \(\operatorname{Var}\left(\hat{\beta}_{1}\right), \operatorname{Var}\left(\hat{\beta}_{2}\right)\), and \(\operatorname{Corr}\left(\hat{\beta}_{1}, \hat{\beta}_{2}\right)\)

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