The ordinary least squares estimate of beta in the classical linear regression model Yi nb...........

Economics-I Statistical and Econometric Methods

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The ordinary least squares estimate of β in the classical linear regression model

Y_i = $\small \alpha$ + βX_i+ U_i ; i = 1, 2, ..., n

is $\small \hat{\beta }=\sum_{i=1}^{n}W_{i}Y_{i}$ , where $\small w_{i}=\frac{x_{i}}{\sum_{i=1}^{n}x_{i}^{2}}$

and $\small x_{i}= X_{i}-\bar{X}$

, $\small \bar{X}=\frac{1}{n}$ $\small \sum_{i=1}^{n}X_{i}$

show that if $\small \dpi{100} \fn_cm \small Var(\hat{\beta })=\frac{\hat{\sigma }_{u}^{2}}{\sum_{i=1}^{n}x_{i}^{2}}$ , no other linear unbiased estimator of $\small \beta$ can be constructed with a smaller variance. (All symbols have their usual meaning)