Simple Linear Regression

[RobertPaulson]

Just a quick question. I was going through my notes and found this definition regarding least squares estimates.

However, whilst doing a past paper I got stumped by this solution

From the second to third step there seems to be a missing y bar. Is this wrong or am I missing something?
I've gone through all of my notes and there doesn't seem to be anything on it so it's either wrong or it's so simple it's assumed.
If this is in fact right, could someone shed some light on the matter for me?

Thanks so much

 

[royhaas]

If you expand each sum, you will find out they are the same. Only one of the variables needs to be centered.

 

[RobertPaulson]

If you expand each sum, you will find out they are the same. Only one of the variables needs to be centered.

I figured as much, however I'm not great at this type of thing, any chance you could do a quick step by step illustrating this?

 

[royhaas]

\(\displaystyle \sum (x_i-\bar x)(y_i-\bar y) = \sum (x_i-\bar x)y_i- \sum (x_i-\bar x)\bar y\). The second sum is zero.

 

[RobertPaulson]

Ok, I'm starting to see now, I understand the original question now but why is the second sum zero and the first not?

 

[Deleted member 4993]

\(\displaystyle \sum (x_i-\bar x)(y_i-\bar y) = \sum (x_i-\bar x)y_i- \sum (x_i-\bar x)\bar y\). The second sum is zero.

\(\displaystyle \sum (x_i-\bar x)\bar y \ = \ \bar y\cdot \sum x_i - \bar y\cdot\sum \bar x\)

Now can you continue.....