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Western University
Economics 2122A/B
Terry Biggs

Dougherty Introduction to Econometrics 4e Instructors Manual 12 PROPERTIES OF REGRESSION MODELS WITH TIME SERIES DATA121Definition and consequences of autocorrelation 121 Consider the model uYXYtttt1321uutpttwhere p1 What may be said about the properties of the OLS estimators of the parameters of the equation for Y tAnswerY and u are not distributed independently because both depend on u t1ttpConsequently OLS will yield inconsistent estimators u is a determinant of Y and hence tptpthrough the first equation all values of Y after time tp This is probably sufficient by way of an answer A more complete one might show explicitly that uYXYptptptpt1321and 22pp1XXXY12312311ptttt3321ppuuuY1231ptttpt33122 Consider the model uYYttt12uuttt1OLSShow thatis given byplimb2OLS2 plimb212Answer TTTuYuYYYY11211tttttttOLS222tttb22TTT222YYY111ttt222tttTT11uYnuuYYuY1111ttttttttnn22tt22TT11222YnYYY1111ttttnn22tt C Dougherty 2011 All rights reserved 2 PROPERTIES OF REGRESSION MODELS WITH TIME SERIES DATA whereandare the sample means of Y and u For sufficiently large t Yut1t1tt with expectations taken at time 0 Henceand 0uEYE0limlimupYp1tt1ttcovuYOLS1ttplimb22varY1tNow covcovcovuuYuYuY1122111ttttttttvarcovuuY1122ttt2 andHence varvaruuuYuYcovcov1utttttt1122ucovuY1tt12222varvarvaruuu11ttt2122cov2varvarvaruYYuYY121212uttttttHence 21222u22var1Y22uut1122and1122varYut21122Hence 2u211OLS22 plimb222111222u211222222221122 122 Detection of autocorrelation123 Perform the BreuschGodfrey and DurbinWatson tests for autocorrelation for the logarithmic demand function that you fitted in Exercise 114 Is there evidence of autocorrelationIf so what are the implications for the statistical tests you performed AnswerThe critical value of chisquared with one degree of freedom at the 01 percent level is 1083 Hence the BreuschPagan test rejects the null hypothesis of no autocorrelation at theC Dougherty 2011 All rights reserved
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