3. In a simple linear regression model:iidwhere ; N(0, 0²).(a) Show thatC₂ =nßo = Ỹ – Â₁ X = Σ k¿Y₁, where kį = =-i=1Note that ₁=1 C₂Y₁, where- IY₁ = Bo + B₁x₁ + Ei, i = 1,..., nXį -STTnn72with Σq = 0, Σqr =1, Σi=1i=1i=1(b) Show that E(30) = Bo.(c) (Graduate Student Only) Show that the distribution of Bo isN (30₂ ‚ ²² [1/1 + ²/²]).nIciBo~No, 02=STI

Answered: (a) Show that Bo=Y-B₁XkYi, where ki…

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. In a simple linear regression model:
iid
where ; N(0, 0²).
(a) Show that
C₂ =
n
ßo = Ỹ – Â₁ X = Σ k¿Y₁, where kį = =-
i=1
Note that ₁=1 C₂Y₁, where
- I
Y₁ = Bo + B₁x₁ + Ei, i = 1,..., n
Xį -
STT
n
n
72
with Σq = 0, Σqr =1, Σ
i=1
i=1
i=1
(b) Show that E(30) = Bo.
(c) (Graduate Student Only) Show that the distribution of Bo is
N (30₂ ‚ ²² [1/1 + ²/²]).
n
Ici
Bo~No, 02
=
STI

Expert Solution

Step 1: we define the model first then we find out the necessary terms and show the part (a) and (b)

Given,

$Y subscript i equals beta subscript 0 plus beta subscript 1 x subscript i plus epsilon subscript i space comma i equals 1 comma 2.... comma n epsilon i space tilde N left parenthesis 0 comma sigma squared space right parenthesis$

(a)

we need first find out the least square estimate of regression coefficent $left parenthesis beta subscript 0 space end subscript comma beta subscript 1 right parenthesis$

We use the least square method to find it,

$Y subscript i equals beta subscript 0 plus beta subscript 1 x subscript i plus epsilon subscript i space comma i equals 1 comma 2 comma.... comma n$

consider,

$S left parenthesis beta subscript 0 comma beta subscript 1 right parenthesis equals stack sum space epsilon subscript i superscript 2 with i equals 1 below and n on top space space space space space space space space space space space space space space space equals sum from i equals 1 to n of left parenthesis Y subscript i minus beta subscript 0 minus beta subscript 1 x subscript i right parenthesis squared$

by least squre criterion , we find $beta subscript 0 space end subscript a n d space beta subscript 1 space space s u c h space t h a t space S left parenthesis beta subscript 0 comma beta subscript 1 right parenthesis space i s space m i n i m u m comma t h a t space i s space l e a s t space s q u a r e space e s t i m a t o r s space o f space beta subscript 0 space end subscript a n d space beta subscript 1 space comma space s a y space beta with hat on top subscript 0 space end subscript a n d space beta with hat on top subscript 1 m u s t space s t i s f y space t h e space f o l l o w i n g space t w o space e q u a t i o n s comma right enclose fraction numerator partial differential S over denominator partial differential beta subscript 0 end fraction end enclose subscript beta with hat on top subscript 0 comma beta with hat on top subscript 1 end subscript equals 0 space space a n d space right enclose fraction numerator partial differential S over denominator partial differential beta subscript 1 end fraction end enclose subscript beta with hat on top subscript 0 comma beta with hat on top subscript 1 end subscript equals 0 N o w space c o n s i d e r comma right enclose fraction numerator partial differential S over denominator partial differential beta subscript 0 end fraction end enclose subscript beta with hat on top subscript 0 comma beta with hat on top subscript 1 end subscript equals 0 rightwards double arrow sum from i equals 1 to n of left parenthesis negative 2 right parenthesis left parenthesis Y subscript i minus beta with hat on top subscript 0 minus beta with hat on top subscript 1 x subscript i right parenthesis equals 0 rightwards double arrow sum from i equals 1 to n of Y subscript i minus n beta with hat on top subscript 0 minus beta with hat on top subscript 1 sum from i equals 1 to n of x subscript i equals 0 rightwards double arrow sum from i equals 1 to n of Y subscript i equals n beta with hat on top subscript 0 plus beta with hat on top subscript 1 sum from i equals 1 to n of x subscript i space space space.................. space........ left parenthesis 1 right parenthesis C o n s i d e r right enclose fraction numerator partial differential S over denominator partial differential beta subscript 1 end fraction end enclose subscript beta with hat on top subscript 0 comma beta with hat on top subscript 1 end subscript equals 0 rightwards double arrow sum from i equals 1 to n of left parenthesis negative 2 right parenthesis left parenthesis Y subscript i minus beta with hat on top subscript 0 minus beta with hat on top subscript 1 x subscript i right parenthesis x subscript i equals 0 rightwards double arrow sum from i equals 1 to n of Y subscript i x subscript i minus beta with hat on top subscript 0 stack sum x subscript i with i equals 1 below and n on top minus beta with hat on top subscript 1 sum from i equals 1 to n of x subscript i superscript 2 equals 0 rightwards double arrow sum from i equals 1 to n of Y subscript i x subscript i equals beta with hat on top subscript 0 stack sum x subscript i with i equals 1 below and n on top plus beta with hat on top subscript 1 sum from i equals 1 to n of x subscript i superscript 2 space space space............................... left parenthesis 2 right parenthesis e q u a t i o n space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis space a r e space c a l l e d space N o r m a l space e q u a t i o n s D i v i d i n g space e q u a t i o n space left parenthesis 1 right parenthesis space b y space n comma space w e space g e t fraction numerator sum from i equals 1 to n of Y subscript i over denominator n end fraction equals n over n beta with hat on top subscript 0 plus beta with hat on top subscript 1 fraction numerator sum from i equals 1 to n of x subscript i over denominator n end fraction space Y with bar on top space equals beta with hat on top subscript 0 plus beta with hat on top subscript 1 x with bar on top space space space................................................................ left parenthesis 3 right parenthesis beta with hat on top subscript 0 equals Y with bar on top minus beta with hat on top subscript 1 x with bar on top w e space s u b b s t i t u t e space t h e space v a l u e space i n space e q u t i n left parenthesis 2 right parenthesis space w e space g e t sum from i equals 1 to n of Y subscript i x subscript i equals beta with hat on top subscript 0 stack sum x subscript i with i equals 1 below and n on top plus beta with hat on top subscript 1 sum from i equals 1 to n of x subscript i superscript 2 space space space sum from i equals 1 to n of Y subscript i x subscript i equals left parenthesis Y with bar on top minus beta with hat on top subscript 1 x with bar on top right parenthesis stack sum x subscript i with i equals 1 below and n on top plus beta with hat on top subscript 1 sum from i equals 1 to n of x subscript i superscript 2 space space space sum from i equals 1 to n of Y subscript i x subscript i equals Y with bar on top stack sum x subscript i with i equals 1 below and n on top negative beta with hat on top subscript 1 x with bar on top stack sum x subscript i with i equals 1 below and n on top plus beta with hat on top subscript 1 sum from i equals 1 to n of x subscript i superscript 2 space space space sum from i equals 1 to n of Y subscript i x subscript i equals n space Y with bar on top space x with bar on top plus beta with hat on top subscript 1 left parenthesis sum from i equals 1 to n of x subscript i superscript 2 minus n x with bar on top squared space right parenthesis H e n c e beta with hat on top subscript 1 equals fraction numerator sum from i equals 1 to n of Y subscript i x subscript i minus n space Y with bar on top space x with bar on top over denominator sum from i equals 1 to n of x subscript i superscript 2 minus n x with bar on top squared space end fraction L e t S subscript x x end subscript equals sum from blank to blank of left parenthesis x subscript i minus x with bar on top right parenthesis squared space space space space space space space equals sum from blank to blank of x subscript i superscript 2 minus n x with bar on top squared A l s o S subscript x y end subscript equals sum from blank to blank of left parenthesis x subscript i minus x with bar on top right parenthesis left parenthesis y subscript i minus y with bar on top right parenthesis space space space space space space space equals sum from blank to blank of left parenthesis x subscript i minus x with bar on top right parenthesis y subscript i space space space space space space space equals sum from blank to blank of x subscript i y subscript i minus x with bar on top sum from blank to blank of y subscript i T h u s space beta with hat on top subscript 1 equals S subscript x y end subscript over S subscript x x end subscript comma space beta with hat on top subscript 0 equals Y with bar on top minus beta with hat on top subscript 1 x with bar on top$

from above

$beta with hat on top subscript 0 equals Y with bar on top minus beta with hat on top subscript 1 x with bar on top space space space space equals fraction numerator sum from space i equals 1 to n of large Y subscript i over denominator n end fraction minus s subscript x y end subscript over s subscript x x end subscript x with bar on top space space space space space equals fraction numerator sum from space i equals 1 to n of large Y subscript i over denominator n end fraction minus fraction numerator sum from i equals 1 to n of Y subscript i left parenthesis x subscript i minus x with bar on top right parenthesis over denominator s subscript x x end subscript end fraction x with bar on top beta with hat on top subscript 0 equals sum from i equals 1 to n of Y subscript i open parentheses fraction numerator 1 over denominator n end fraction minus fraction numerator left parenthesis x subscript i minus x with bar on top right parenthesis over denominator s subscript x x end subscript end fraction x with bar on top close parentheses B u t space w e space h a v e space g i v e n space k subscript i equals 1 over n minus x with bar on top c subscript i space A l s o space N o t e space t h a t space beta with hat on top subscript 1 equals stack sum c subscript i Y subscript i with i equals 1 below and n on top W h e r e comma c subscript i equals fraction numerator left parenthesis x subscript i minus x with bar on top right parenthesis over denominator s subscript x x end subscript end fraction space w i t h space stack sum space with i equals 1 below and n on top c subscript i equals 0 comma stack sum space with i equals 1 below and n on top c subscript i x subscript i equals 1 comma stack sum space with i equals 1 below and n on top c subscript i superscript 2 equals 1 over s subscript x x end subscript F r o m space s o m e space g i v e n space q u a n t i t y space w e space c a n space w r i t e beta with hat on top subscript 0 equals sum from i equals 1 to n of Y subscript i open parentheses fraction numerator 1 over denominator n end fraction minus x with bar on top c subscript i close parentheses space space space space space space space space space.................................. c subscript i equals fraction numerator left parenthesis x subscript i minus x with bar on top right parenthesis over denominator s subscript x x end subscript end fraction u sin g space.. space space k subscript i equals 1 over n minus x with bar on top c subscript i space beta with hat on top subscript 0 equals sum from i equals 1 to n of Y subscript i k subscript i H e n c e space t h e space P r o v e beta with hat on top subscript 0 equals Y with bar on top minus beta with hat on top subscript 1 x with bar on top equals sum from i equals 1 to n of k subscript i Y subscript i$

(b)

$E left parenthesis beta with hat on top subscript 0 space end subscript right parenthesis equals E left parenthesis sum from i equals 1 to n of k subscript i Y subscript i right parenthesis space space space space space space space space space space space equals sum from i equals 1 to n of k subscript i E left parenthesis Y subscript i right parenthesis B u t space E left parenthesis Y subscript i right parenthesis equals beta subscript 0 plus beta subscript 1 x subscript i space space space space space space space space space space equals sum from i equals 1 to n of k subscript i left parenthesis beta subscript 0 plus beta subscript 1 x subscript i right parenthesis E left parenthesis beta with hat on top subscript 0 space end subscript right parenthesis equals beta subscript 0 stack space sum with i equals 1 below and n on top k subscript i plus space beta subscript 1 sum from i equals 1 to n of k subscript i x subscript i space space space space space............ left parenthesis 4 right parenthesis space space B u t space w e space h a v e space g i v e n space k subscript i equals 1 over n minus x with bar on top c subscript i space A l s o space N o t e space t h a t space beta with hat on top subscript 1 equals stack sum c subscript i Y subscript i with i equals 1 below and n on top c subscript i equals fraction numerator left parenthesis x subscript i minus x with bar on top right parenthesis over denominator s subscript x x end subscript end fraction space w i t h space stack sum space with i equals 1 below and n on top c subscript i equals 0 comma stack sum space with i equals 1 below and n on top c subscript i x subscript i equals 1 comma stack sum space with i equals 1 below and n on top c subscript i superscript 2 equals 1 over s subscript x x end subscript stack space sum with i equals 1 below and n on top k subscript i equals stack space sum with i equals 1 below and n on top left parenthesis 1 over n minus x with bar on top c subscript i right parenthesis space space space space space space space space space space equals 1 minus x with bar on top sum from i equals 1 to n of c subscript i sum from i equals 1 to n of k subscript i equals 1 space space space space space space space............... left parenthesis space stack sum space with i equals 1 below and n on top c subscript i equals 0 right parenthesis A n d space sum from i equals 1 to n of k subscript i x subscript i equals stack space sum with i equals 1 below and n on top left parenthesis 1 over n minus x with bar on top c subscript i right parenthesis x subscript i space space space space space space space space space space space space space space space equals fraction numerator stack space sum with i equals 1 below and n on top x subscript i over denominator n end fraction minus x with bar on top sum from i equals 1 to n of c subscript i x subscript i space space space space space space space space space space space space equals x with bar on top minus x with bar on top space space space space space space space space space space space........................................ left parenthesis stack sum space with i equals 1 below and n on top c subscript i x subscript i equals 1 right parenthesis sum from i equals 1 to n of k subscript i x subscript i equals 0 H e n c e space p u t space i n space e q u a t i o n space left parenthesis 4 right parenthesis space w e space c a n space w r i t e space i t space a s E left parenthesis beta with hat on top subscript 0 space end subscript right parenthesis equals beta subscript 0 asterisk times 1 plus space beta subscript 1 asterisk times 0 E left parenthesis beta with hat on top subscript 0 space end subscript right parenthesis equals beta subscript 0 H e n c e space t h e space p r o v e E left parenthesis beta with hat on top subscript 0 space end subscript right parenthesis equals beta subscript 0$

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(a) Show that Bo=Y-B₁XkYi, where ki Note that ₁=1 C₂Y₁, where 9 12 n = C₂ = Xi - ī with Σε=0, Σqr =1, Σ £4-1 STI i=1 i=1 i=1 n n A ~ N(26.0² [= + =]). n с₂ 72 (b) Show that E(30) = Bo- (c) (Graduate Student Only) Show that the distribution of Bo is