Estimation of β̂1 and β̂2

     Estimate β̂1 and β̂2

We know,

Ŷi =β̂1+β̂2Xi
and, Yi =β̂1+β̂2Xi+ Ûi
Or,   Ûi= Yi−(β̂1+β̂2Xi)
Or,   Ûi= Yi−β̂1−β̂2Xi
Or, ∑Û2i =  ∑(Yi- β̂1-β̂2Xi)2
∑ Û2i =f(β̂1,β̂2).
So, 𝛿∑Û2i/ 𝛿 β̂1=2∑(Yi- β̂1-β̂2Xi)(-1)=0
Or,  ∑(Yi- β̂1-β̂2Xi)=0
Or,  ∑Yi-nβ̂1-β̂2∑ Xi=0
So, ∑Yi=nβ̂1+β̂2∑Xi——————(i)
Again,  𝛿∑Û2i/ 𝛿 β̂2=2∑(Yi- β̂1-β̂2Xi)(-Xi)=0
∑(Yi- β̂1-β̂2Xi)(Xi)=0
∑Yi Xi -β̂1∑Xi -β̂2∑Xi2=0
∑Yi Xi =β̂1∑Xi+β̂2∑Xi2=0
∑XiYi =β̂1∑Xi+β̂2∑Xi2=0————–(ii)
Now, equation (i) x ∑Xi and equation (ii) ᳵ n  and we get—
∑Xi∑Yi =nβ̂1∑Xi+β̂2(∑Xi)2=0 ———(iii)
n∑XiYi =nβ̂1∑Xi+nβ̂2∑Xi2=0———–(iv)
                               or, ∑Xi∑Yi –  n∑XiYi=β̂2(∑Xi)2-n β̂2∑Xi2 (Subtracting (iv) from (iii))
                                  Or,    β̂2[(∑Xi)2-n∑Xi2]= ∑Xi∑Yi –  n∑XiYi

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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