3. Consider the following simple regression modelwhere ~ N(0, ²).Y = Bo+BX+ênDuring the class, we learned that we can estimate 30 and 3₁ using data (zi. yi)

Given information: are constants.To show,

Answered: 3. Consider the following simple…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Related questions

Question

3. Consider the following simple regression model
where ~ N(0, ²).
Y = Bo+BX+ên
During the class, we learned that we can estimate 30 and 3₁ using data (zi. yi)<i<n by
SS29
and Bo=ỹ - 3₁
SS₂2
where = † Σ * and y = | Σ13 and
(b)
n
3₁ =
SSxy=(₁-7)(y-7) and SS₂z = - Σ(2₁-2)²
Hint: Start from
0²
SSXX
Hint: Note that because o.₁ are constants and are known constant value, E[3₁]
31, E[Bo] = Bo, and E[i] = z; and Var(3₁) = Var(30) = Var(zi) = 0. Therefore,
Var(y) = Var(30 + Bizi + c) = Var(e) = 0²
Prove (or show) that
for any 1 ≤ i ≤n. Using the previous statement, show that
= Var(3₁) = Var
12
$S9
SSTT
2² = 0² ( = + 55²) -
SS₂
SS₂
= Var(80) = Var(ỹ - ₁) = Var(y) + Var (3₁)

Expert Solution

Step 1: Given information:

Given information:

$X with bar on top space a n d space Y with bar on top$ are constants.

$sigma squared equals V a r left parenthesis epsilon right parenthesis beta with hat on top subscript 0 equals y with bar on top minus beta with hat on top subscript 1 space x with bar on top V a r left parenthesis beta subscript 1 right parenthesis equals fraction numerator sigma squared over denominator S X X end fraction$

To show,

$V a r left parenthesis beta subscript 0 right parenthesis equals sigma squared open parentheses 1 over n plus fraction numerator x with bar on top squared over denominator S X X end fraction close parentheses$