# intuitively explain the equation of $Var(b_1)$ in OLS model

Above is the capture from the Econometrics slide from "Hill,Griths and Lim (2018) Principles of Econometrics".

I have no problem the see the blue font sentence separately, it's all about the numerator and denominator issues. But, if I think deeply about its intuitions, it bugs me.

The first blue sentence could interpret as more coverage of $$x_i$$ can reduce the variance

then I expand the equation, larger $$\sum(x_i-\bar{x})^2=\sum x_i^2-n\bar{x}^2$$ means Larger the value of$$\sum x_i^2$$. It's reduce variance, and this contradict to second blue sentence Clearly second blue is right, so what's go wrong with my reasoning about the larger $$\sum x_i^2$$ .

Please enlighten me, Thank you so much

• Welcome to economics.se, please next time do not include picture of texts or equations you should rewrite them so the whole question is searchable
– 1muflon1
Oct 11, 2021 at 13:27

You should be able to confirm the claim after dividing the numerator and denominator by $$\sum x_i^2$$. Then, ceteris paribus, a greater $$\sum x_i^2$$ increases the denominator and reduces the variance of the random variable $$b_1$$.

I don't like the blue sentences. More rigorously, it should state that, if denominator is fixed, an increase in numerator gives a larger variance.

That is, if $$\sum_{i=1}^{n}x_{i}^{2}$$ increases, meanwhile, $$\bar{x}$$ increases too so that the denominator remains the same, then variance becomes smaller. Without any statement about the denominator (more precisely, about $$\bar{x}$$), you don't know whether the variance becomes larger or smaller. What if $$\bar{x}=0$$, will increase in $$\sum_{i=1}^{n}x_{i}^{2}$$ affect the variance？