I have a question about the least-squares estimator from basic econometrics by Gujarati. It claims that Formula B could be easily and directly verified from Formula A.
Formula A : $K_i=\frac{x_i}{∑(xi)^2}$
Formula B : $∑(K_i)^2=\frac{1}{∑(X_i)^2}$
The symbol is Capital Sigma for summation. I just can't get to the point.
I think the issue comes from an ambiguous phrasing: $$K_i = \frac{X_i}{\sum_i X_i^2}$$ is poorly written because the same index $i$ is used in the numerator and in the summation below it. The same symbol must not be used in the same formula for two different things so it should be written $$K_i = \frac{X_i}{\sum_j X_j^2}$$
Notice that the denominator is a constant number $d = \sum_j X_j^2$, it does not depend on $i$, hence $$\sum_i K_i^2 = \sum_i \left(\frac{X_i}{d}\right)^2 = \frac{1}{d^2}\sum_i X_i^2 = \frac{1}{d^2} \times d = \frac{1}{d} = \frac{1}{\sum_j X_j^2}$$