Question 2: Let

Y

be a random variable and

x

be a fixed variable. Sometimes it is useful to measure the independent variable around its mean. In such a case, the model has a centered version:\

=\\\\beta _(0)+\\\\beta _(1)x_(i)-\\\\beta _(1)\\\\bar{x} +\\\\beta _(1)\\\\bar{x} +\\\\epsi _(i)\ y_(i)=\\\\beta _(0)_(+)\\\\beta _(1)(x_(i)-(\\\\bar{x} ))+\\\\beta _(1)(\\\\bar{x} )_(+)\\\\epsi lon_(i),(i

)

=

(

1,2,dots,n)\ =\\\\beta _(0)^(**)+\\\\beta _(1)(x_(i)-(\\\\bar{x} ))+\\\\epsi lon_(i)\ =\\\\beta _(0)+\\\\beta _(1)x_(i)+\\\\epsi _(i)

\

where \\\\beta _(0)^(**)=\\\\beta _(0)+\\\\beta _(1)\\\\bar{x} .

\ Find the value of the parameters

hat(\\\\beta )_(0)^(**)

and

hat(\\\\beta )_(1)

.\ Are these parameters biased or unbiased? Prove.\ Find the variance of the estimated value of parameters

hat(\\\\beta )_(0)^(**)

and

hat(\\\\beta )_(1)

.

Question 2: Let Y be a random variable and X be a fixed variable. Sometimes it is useful to measure the independent variable around its mean. In such a case, the model has a centered version: yi=0+1xi1x+1x+i=0+1(xix)+1x+i(i=1,2,,n)=0+1(xix)+i=0+1xi+i where0=0+1x. 1. Find the value of the parameters ^0 and ^1. 2. Are these parameters biased or unbiased? Prove. 3. Find the variance of the estimated value of parameters ^0 and ^1