Consider the following model:yi=α+βxi+i;i= 1,2,···,n(4)wherexiis fixed in repeated sampling, and the random disturbance termisatisfies the usualassumptions ofE(i) = 0V(i) =σ2∀iE(ij) = 0∀i6=j(5)Let ˆi, ˆαandˆβdenote the OLS residuals and the parameter estimators, respectively. The re-gression model of (4) is fitted to the data: (x1,y1),···, (xn,yn), giving least squares estimates ˆα1
andˆβ. Given a new valuexn+1for the explanatory variable, we wish to predictyn+1. It can beassumed thatyn+1=α+βxn+1+n+1wheren+1satisfies the usual assumptions of a white noise so that:E(n+1) = 0V(n+1) =σ2E(in+1) = 0∀i= 1,2,···,n(6)The natural predictor foryn+1is:ˆyn+1= ˆα+ˆβxn+1Let the prediction error be denoted byen+1.In Class 3, we saw that this prediction error could be expressed as:en+1= (α−ˆα) + (β−ˆβ)xn+1+n+1(7)and that the mean of the distribution ofen+1was 0. Answer the following questions.a) Show that this prediction error can also be expressed as:en+1=n∑i=1dii+n+1wheredi=−[1n+(xn+1−x)(xi−x)∑nj=1(xj−x)2],withx=1nn∑i=1xi.(8)[50%]b) Using the result ina), show that the variance of the prediction error is given by:V(en+1) =σ2[1 +1n+(xn+1−x)2∑nj=1(xj−x)2].(9)Discuss the practical relevance of this result