31) The estimate of the slope parameter for a regression model without the intercept term is given by β= [sum(X*Y)/Sum(X^2)] which is mean corrected, thus option d is wrong.
32) The difference b/w model 1 and model 2 is that in model 2 all the dependent and independent variables have been divided by N which is a variable, if the division would have been done by a parameter value or a constant then the model two models would have been comparable, but in this case since the change in dependent and independent variables is done using a variable hence the models are not comparable.
33) The estimate of slope is given by β=Cov(x,y)/Var(x),
Here put Y'=w1*Y and X'=w2*X,
Thus Cov(X',Y')=w1*w2*Cov(X,Y)
Var(X)=(w2)^2*[Var(X)].
therefore β'=(w1*w2)*Cov(X,Y)/(w2^2)*Var(X)=(w1/w2)*{Cov(X,Y)/Var(X)}=(w1/w2)*β. Hence you can eliminate option b. In a similar manner you can find out each of the estimates and you will observe that only c is correct. Also you can eliminate option d as in this case there is a change in scale and r^2 is independent of that. Or refer to pg 168 of Gujarati.
"I don't ride side-saddle. I'm as straight as a submarine"
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