Suppose Mr. Bump observes the selling price and sales volume of milk gallons for 10 randomly selected weeks as follows

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1 Albert Ludwgs Unverst Freburg Department of Emprcal Research and Econometrcs Appled Econometrcs Dr Kestel ummer 9 EXAMPLE IMPLE LINEAR REGREION ANALYI uppose Mr Bump observes the sellng prce and sales volume of mlk gallons for randoml selected weeks as follows Week Weekl sales level, Y* ellng prce, X, $ * Thousand of gallons catter Dagram 5 sales prce Correlaton coeffcent: r ()493 (44) ()56 44 ()488 86

2 WEEK ^ ^ sum Normal equatons: n () n n n ( ( ) ) ()56 (44) () ()49 (44)() 98 ()(493) (44) ()(56) (44) 84 ( 454) Lnear Regresson Model (Ftted model): tandard error of estmate (se): week Actual Estmated error error^ sum ( ) error σ e n n

3 Evews Output Dependent Varable: ALE Method: Least quares Date: 5/8/9 Tme: 8:3 ample: Included observatons: Coeffcent td Error t-tatstc Prob PRICE C R-squared Mean dependent var Adjusted R-squared 7385 D dependent var E of regresson Akake nfo crteron 53 um squared resd chwarz crteron 585 Log lkelhood -3 Hannan-Qunn crter F-statstc Durbn-Watson stat Prob(F-statstc) 84 ANOVA Table: Test for Equalt of Means Between eres Date: 5//9 Tme: : ample: Included observatons: Method df Value Probablt t-test atterthwate-welch t-test* Anova F-test (, 8) Welch F-test* (, 96349) *Test allows for unequal cell varances Analss of Varance ource of Varaton df um of q Mean q Between Wthn Total

4 Resdual Plot ALE Resduals Q-Q Plot of Resduals (Normalt check) 6 4 Quantles of Normal Quantles of REID 4

5 Predctng Y for a gven value of : uppose Mr Bump wshed to forecast the quantt of mlk sold f the prce were set at $63 E Y (454) or 8,44 gallons tandard error of the forecast measures when 63 s ( ) ( 63 44) σ p σ e 7 9 n 84 ( ) 95% predcton nterval for 63 s 844 ± t n-, 5 9 > 844±36(9) > (753, 5) Forecast: ALEF Actual: ALE Forecast sample: Included observatons: Root Mean quared Error 6 Mean Absolute Error 398 Mean Abs Percent Error 7984 Thel Inequalt Coeffcent 587 Bas Proporton Varance Proporton 763 Covarance Proporton ALEF ± E Inference on : tandard error of estmator of σ e σ e 7 σ 3 ( ) 84 5

6 The 95% confdence nterval for s ± c se( ) s: -4,54± t n-, 5 (3) > -454± 698 > (-458, -76) Hpothess Testng H : vs H A : 454 t 48 < -36 Reject Ho 3 p-value: Pr( T 4 8) Pr( T > 4 8) 3 6

7 Albert Ludwgs Unverst Freburg Department of Emprcal Research and Econometrcs Appled Econometrcs Dr Kestel ummer 9 EXAMPLE MULTIPLE LINEAR REGREION ANALYI uppose Mr Bump observes the sellng prce, sales volume of mlk gallons and advertsng epense for randoml selected weeks as follows Week Weekl sales level, Y* ellng prce, X, $ Advertsng**, X * Thousand of gallons; ** hundreds of dollars Correlaton Matr sales prce sales 86 r prce 86 ads ads E-Vews Output of Correlaton matr Covarance Analss: Ordnar Date: 5/8/9 Tme: 8:59 ample: Included observatons: Covarance Correlaton ALE PRICE ALE 336 ADVERTIIME NT PRICE ADVERTIIMENT

8 catter Dagram Y versus X 5 sales prce catter dagram Y versus X sales advertsng 8

9 9 week X X ^ X ^ X ^ X X X X sum Normal equatons: n In matr form: X X X Y ( ) ) ( ) ( X Y X X X X X Y 336 ()488 ) ( n Multple Lnear regresson (Ftted or estmated ) Model

10 tandard error of estmate: week Prce Advertsng Actual Estmated error error^ UM 59 σ e R error ( ) 59 5 n k n k 3 Eplaned Re sdual 5,9,93 Total total 33,6 Corrected or adjusted R A σ u σ r : 5,9/ 7 33,6/9,948 Varables used to eplan r ( ) varance of Y none 336 Prce Prce and advertsement Predcton on gven fed values of and : uppose Mr Bump wshed to forecast the quantt of mlk sold f the prce were set at $5 and advertsng ependtures of $ E Y 5, 64 85(5) 59() 993 or 993 gallons

11 Inferences on Parameters: H : vs H A : 585 t 4,37 >36 Reject Ho 3367 Evews Output Dependent Varable: ALE Method: Least quares Date: 5/8/9 Tme: 8:54 ample: Included observatons: Coeffcent td Error t-tatstc Prob PRICE ADVERTIIMENT C R-squared 9399 Mean dependent var Adjusted R-squared 948 D dependent var E of regresson 5796 Akake nfo crteron 3975 um squared resd 5949 chwarz crteron Log lkelhood Hannan-Qunn crter 384 F-statstc Durbn-Watson stat Prob(F-statstc) 8 ANOVA Table: Test for Equalt of Means Between eres Date: 5//9 Tme: :55 ample: Included observatons: Method df Value Probablt Anova F-test (, 7) Welch F-test* (, 863) 3654 *Test allows for unequal cell varances Analss of Varance ource of Varaton df um of q Mean q Between Wthn Total

12 Resdual Plot ALE Resduals Q-Q plot of resduals (Normalt check) 3 Quantles of Normal Quantles of REID

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