Consider the simple linear regression model Yi = 0 + 1xi + i, where the errors i
ID: 3179501 • Letter: C
Question
Consider the simple linear regression model
Yi = 0 + 1xi + i,
where the errors i are identically and independently distributed as N (0, 2).
(a) If the predictors satisfy x = 0, show that the least squares estimates ˆ0 and ˆ1 are independently
distributed.
(b) Let r be the sample correlation coefficient between the predictor and response. Under what
conditions will we have ˆ1 = r?
(c) Suppose that ˆ1 = r, as in part b), but make no assumptions on x . Give the fitted regression
equation (in terms of x , Y and r) and use it to show that | Yˆ i Y | | x i x | .
1. Consider the simple linear regression model where the errors ei are identically and independently distributed as NC0, o2). 0, show that the least squares estimates Bo and are independently (a) If the predictors satisfy distributed (b) Let r be the sample correlation coefficient between the predictor and response. Under what conditions will we have B1 T? (c) Suppose that B1 r, as in part b), but make no assumptions on Give the fitted regression equation (in terms of T Y and r) and use it to show that This inequality helps explain the meaning of the term "regression" since the fitted value Yi isExplanation / Answer
Answer a:
Let consider x =0,
The OLS estimators of intercept (b0) and slope (b1) can be expressed as
b0=y-b1x ...Eq 1
b1=(x-x)(y-y)/(x-x)2 ...Eq 2
From these equations it is obvious that b0 is dependent on b1. Which means if b1 (estimate of slope) deviates much from true population slope B1 then b0 also deviates from true population intercept B0.
If we consider x =0 then eq 1 and 2 becomes
b0=y
b1=(x)(y-y)/(x)2
By considering x =0, we can see that now b0 is independent of b1(slope) .
Answer b:
we have
r2=b12 ((xi)2/(yi)2) where xi=Xi-x and yi=Yi-Y
=>r=b1 (Sx/Sy) where Sx, Sy are standard deviations of X and Y
=>If independent variable (X) has the same variance of dependent variable (Y) then we have r=b1 (1/1) =b1.
i.e., the slope b1 can be different from correlation coefficient r only when the standard deviation of X and Y vary.
Answer c:
For the given Population regression function Yi=B0+B1Xi, the estimated regression would be
yi=b0+b1Xi where b0 and b1 are OLS estimators for B0 and B1
yi=b0+b1Xi
=(y-b1x)+b1X (By substituting value of b0)
=y +b1(X-x)
=y +r(X-x) (Since b1=r)... This is the fitted reg equation in terms of y,r,x
=>yi-y =r(X-x) ..Eq 3
For inequality , lets study the r (correlation coefficient)
1. r can take values between -1 and 1, which means r can be either positive or negative
2. If r is negative then X and Y are inversely related and if r is positive then X and Y are directly related
Depending on values of r ,eq 3 becomes |yi-y| |X-x|