Regression with one regressor

[sailakshmisuresh]

Hey david!
I had a doubt while reading the chapter "regression with a single regressor" from Schweser. There was a statement that the variance of the slope(beta) decreases with the variance of the explanatory variable. The explanation given was higher variance of the explanatory (X) variable indicates that there is sufficient diversity in observations (i.e. the sample is representative of the population) and hence lower variability (and higher confidence) of the slope estimate. Well my doubt is how can you say that when the variance of X is higher it represents the population. Could you give an example for better understanding

Thanks in advance!

 

[Nicole Seaman]

Hey david!
I had a doubt while reading the chapter "regression with a single regressor" from Schweser. There was a statement that the variance of the slope(beta) decreases with the variance of the explanatory variable. The explanation given was higher variance of the explanatory (X) variable indicates that there is sufficient diversity in observations (i.e. the sample is representative of the population) and hence lower variability (and higher confidence) of the slope estimate. Well my doubt is how can you say that when the variance of X is higher it represents the population. Could you give an example for better understanding

Thanks in advance!

Hello @sailakshmisuresh

You should find a lot of forum threads by using the search and tag functions in the forum: https://forum.bionicturtle.com/search/?type=post. There are a lot of posts in the forum that already discuss regression.

 

[sailakshmisuresh]

Alright! Thanks

 

[David Harper CFA FRM]

@sailakshmisuresh Without going deep (implicit in Nicole's attempt to save me time is our frankly getting a little tired of supporting competitors, especially as people freely steal our content, but I honestly do like the people at Schweser. Going forward, I'll definitely be prioritizing those who are customers and/or those who make contributions), I would just remind you that, to me, we have an easier route to this relationship.

In regressing the response (R) against the explanatory (E) variable, the slope (beta) coefficient β(R, E) = cov(R,E)/σ^2(E) = ρ(R,E)*σ(R)/σ(E). This relationship is so useful . I very much recommend understanding it; e.g., why is the explanatory variance in the denominator? Given β(R, E) is the slope when R is regressed against E, beta is increasing with correlation, ρ(R,E), and increasing with volatility/variance of the response, σ(R); and beta is decreasing with volatility/variance of the explanatory variance, σ(E). The qualitative statements above indeed are true, but if we just want to get to the directional relationship, I prefer the simple math. I hope that's helpful,

 

[sailakshmisuresh]

Thank you so much David !