GLS estimation with partly known form

I have two scalars $$\hat{\beta}_1$$ and $$\hat{\beta}_2$$ which are unbiased estimators of $$\beta$$. Furthermore, the covariance matrix is given by $$var\begin{pmatrix}\hat{\beta_1}\\ \hat{\beta_2} \end{pmatrix}=\begin{pmatrix}\sigma^2_1&0\\0&\sigma^2_2\end{pmatrix}.$$ The ratio $$\sigma_2/\sigma_1$$ is known. Now, I have to construct a more efficient unbiased estimator of $$\beta$$. Could someone help me out here? I am fairly sure I should use Generalised least squares, but I don't know in what form. Thanks in advance!

Hi: divide the elements in the matrix by $$\sigma^2_1$$. Then, the covariance matrix becomes $$\sigma^2_1 \times \left[ {\begin{array} {cc} 1 & 0 \\ 0 & \frac{\sigma^2_{2}}{\sigma^2_{1}} \\ \end{array} } \right]$$

so that $$\sigma^2_{1}$$ is just a scale factor. Now, you can solve the problem:

minimize $$w_1^2 \times 1 + w_2^2 \times \frac{\sigma^2_2}{\sigma^2_1}$$

subject to $$w_1 + w_2 = 1$$.

The resulting estimator, $$w_1 \times \hat\beta_1 + w_2 \times \hat\beta_2$$ is unbiased and minimum variance. You can use the lagrange multiplier approach to deal with the constraint or substitute $$1-w_1$$ for $$w_2$$ and take the derivative.

ADDENDUM A: Use the second approach.

This results in $$w_1^2 + (1-w_1)^2 \times \frac{\sigma^2_2}{\sigma^2_1}$$

Taking the derivative gives: $$2 \times w_1 - 2 \times (1 - w_1) \times \frac{\sigma^2_2}{\sigma^2_1} = 0.$$. This leads to

$$2 \times w_1 + 2 \times w_1 \times \frac{\sigma^2_2}{\sigma^2_1} = 2 \times\frac{\sigma^2_2}{\sigma^2_1}$$

$$w_1 \times \left (2 + 2 \times \frac{\sigma^2_2}{\sigma^2_1} \right) = 2 \times \frac{\sigma^2_2}{\sigma^2_1}$$

Please check the algebra because I expected something simpler for $$w_1$$.

ADDENDUM B: Actually, for $$\sigma^2_1 = \sigma^2_2$$, $$w_1 = \frac{1}{2}$$ which is correct (give same weight to both estimators) so the algebra is probably fine.