Consider the following fours different scenarios:\ Constant Variance:

y_(i)=1+x_(i)+\\\\epsi _(i)

\ Strong NonConstant Variance:

y_(i)=1+x_(i)+x_(i)\\\\epsi _(i)

\ Mild NonConstant Variance:

y_(i)=1+x_(i)+\\\\sqrt(x_(i))\\\\epsi _(i)

\ Nonlinearity:

y_(i)=1+cos(\\\\pi (x_(i))/(25))+\\\\epsi _(i)

,\ where

x_(i)=i,\\\\epsi _(i)

are iid standard normal errors, and

i=1,..,50

. For each of\ those scenarios, (a) simulate a data set

(x_(i),y_(i)),i=1,dots,50

, from each model;\ (b) fit a 'misspecified' simple linear regression model

y_(i)=\\\\beta _(0)+\\\\beta _(1)x_(i)+\\\\epsi _(i)

; and (c)\ generate the four diagnostics plots of

plot(lm())

in each case. (d) Comment\ whether the plots are able to identify the corresponding departures from the\ assumptions of the Gauss-Markov linear regression model.

Constant Variance: yi=1+xi+i Strong NonConstant Variance: yi=1+xi+xii Mild NonConstant Variance: yi=1+xi+xii Nonlinearity: yi=1+cos(xi/25)+i, where xi=i,i are iid standard normal errors, and i=1,..50. For each of those scenarios, (a) simulate a data set (xi,yi),i=1,,50, from each model; (b) fit a 'misspecified' simple linear regression model yi=0+1xi+i; and (c) generate the four diagnostics plots of plot(lm()) in each case. (d) Comment whether the plots are able to identify the corresponding departures from the assumptions of the Gauss-Markov linear regression model