Proof: The first and last equations are trivial. We now show the second. By the first equation:
Taking the sum of both sides of the equation over all values of i and then squaring both sides, the desired result follows provided
This follows by substituting ŷi by bxi + ai and simplifying.
Proof: We give the proof for the case where both x and y have mean 0 and standard deviation 1. The general case follows via some tedious algebra.
it follows that
and so n – 1 = SST, which means that
Proof: From Property 2, solving for r2, we have the following by Property 1:
Proof: The first assertion is trivial. The second is a consequence of Property 1and 2 since
a) The sums of the y values is equal to the sum of the ŷ values; i.e. =
b) The mean of the y values and ŷ values are equal; i.e. ȳ = the mean of the ŷi
c) The sums of the error terms is 0; i.e. = 0
d) The correlation coefficient of x with ŷ is sign(b); i.e. rxŷ = sign(rxy)
e) The correlation coefficient of y with ŷ is the absolute value of the correlation coefficient of x with y; i.e. = ||
f) The coefficient of determination of y with ŷ is the same as the correlation coefficient of x with y; i.e. =
a) By Theorem 1 of Method of Least Squares
b) That the mean of the ŷi is ȳ follows from (a) since the mean of ŷ = ŷi/n = yi/n = ȳ.
c) This property follows from (a) since
d) First note that by Property 3 of Expectation
Now by Property A of Correlation
Thus, r = 1 if b > 0 and r = -1 if b < 0. If b = 0, r is undefined since there is division by 0. By Property 1 of Method of Least Squares, r = sign(b) = sign(rxy).
e) Using property (b), the correlation of y with ŷ is
By Theorem 1 of Method of Least Squares
As we saw in the proof of (d),
and so putting all the pieces together, we get
By Property 1 of Method of Least Squares, r = sign(b) = sign(rxy), and so
f) This follows from (e)