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AnindyaBanerjeeFirstSemester2009ECONG26/27/28/299October2009ExerciseSheet11.i)Definethesimplelinearcorrelationcoefficient,r,betweentheobservedvaluesand(fori=1,2,....,n)oftwovariablesXandY.ii)SupposeanordinaryleastsquareslinearregressionisestimatedbetweenthevariablesYandX:.Showthatthecoefficientofdeterminationofthisregression,R2,isequaltor2definedini).Henceprovethat-1≤r≤1.iii)NowsupposeanordinaryleastsquareslinearregressionisundertakenbetweenXandY:Whatistherelationshipbetweenestimatedinthisregressionwithestimatedin(ii)?Canwededucetheestimateoffromtheregressionstatisticsin(ii)?iv)Ingeneral,whyisShowunderwhatcircumstanceswillIllustrateyouranswer.2.Thefollowingaredataon:Y=quitrateper100employeesinmanufacturing;X=unemploymentrateYearYXYearYX19901.36.219972.33.619911.27.919982.53.319921.45.919992.73.319931.45.720002.15.619941.55.020011.96.919951.94.020022.25.619962.63.2i)CalculatearegressionofYonX.ii)Constructa95%confidenceintervalforα.iii)TestthenullhypothesisH0:β=0atthe5%significancelevel.iv)Constructa90%confidenceintervalforσ2,whereσ2isthevarianceoftheregressionerror.v)CalculatetheR2oftheregression.vi)Arethereanyreasonswhytheassumptionsoftheclassicalnormallinearmodelareinvalid?3.AtBirminghamUniversity7economicsundergraduateswererandomlyselectedfromthepopulationandsurveyed.Theywereasked:i)Whatwastheaveragenumberofhoursoflecturesperweekyouattendedlastyear?ii)Whatwastheaveragenumberofhoursperweekyouspentdrinkinglastyear?Thefollowingdatawerecollected:StudentHoursoflecturesHoursofdrinking13.6322.21533.1843.5952.71262.61273.94i)Estimatetheordinaryleastsquaresequation,whereYistheaveragenumberofhoursoflecturesandXrepresentstheaveragenumberofhoursdrinkingperweek.ii)CalculatetheR2oftheregressionandthestandarderroroftheequation.iii)TestthenullhypothesisH0:β=0.4.LetYbeatimedependentvariableandconsiderthelinearregressionmodel:wherei)Le
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