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by: Golden Bernhard


Marketplace > University of Florida > Statistics > STA 6208 > BAS DESIGN ANLY EXPER
Golden Bernhard
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Class Notes
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This 5 page Class Notes was uploaded by Golden Bernhard on Friday September 18, 2015. The Class Notes belongs to STA 6208 at University of Florida taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/206577/sta-6208-university-of-florida in Statistics at University of Florida.




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Date Created: 09/18/15
British Spirits Example Simple Linear Regression Over Time Response Variable Y Per capita consumption of Spirits Predictor Variable X Indexed Pricetoincome ratio Time Period Annual Data 18701938 Source J Durbin and GS Watson Testing for Serial Correlation in Least Squares Regression H Biometrika 38 June 1951 pp 159177 Model Y o 1Xa Step 1 Plot the consumption versus the indexed pricetoincome ratio Consumption vs PriceIncome 22 2 39amp 5 18 m 5 3 8 147 8 o 12 r 075 08 085 09 095 1 105 11 115 12 125 Pricetolncome Ratio Step 2 Fit a simple linear regression model Coef cients Standard Error t Stat Pvaue Lower 95 Upper 95 Intercept 516 0258 2001 0000 464 567 priceinc 314 0238 4317 0000 362 266 Thus the tted equation is Y 516 314X Step 3 Obtain a histogram of the residuals I copied residuals to original spreadsheet Histogram 12 gt 10 is 8 s g 6 e 4W LI 2 W 0 1 lt0 lt0 lt0 lt0 9 9 Q Residuals The rst bin 0 cases represents the number less than 7025 the second bin 9 cases represents the number between 7025 and 7020 and so on The distribution is centered at 0 but not particularly mound shaped Step 4 Plot the residuals versus Y I copied these values to original spreadsheet residuals vs tted values 03 0 027 t 017 o o quotk 99 o 1 1 1 1 1 n 12 13914 16 617 18 19 2 21 22 01 9 9 99 027 o 390 s39 O3 Residuals Note that there is some evidenceof nonconstant error variance but I veseen much Worse in praetiee Step 5 Plotihe residuals Versus X This was automatically printed by PHStat butII rescaled it price39in c Residual Plat 095 priceincr This is a miiror image of the residuals versus tted Values rescaled Step 6 Plot the residuals versus year residuals 03 02 7 V W 01 7 g o 0quot oquot 9 residuals 1860 1880 193 920 9 1940 1960 1 0 v 02 7 quot 3quot o 0 3 Residuals close in time are very similar displaying clearly that there is positive autocorrelation among residuals This is by far the most serious violation of model assumptions from the 4 graphs Step 7 Conduct the DurbinWatson test for Positively correlated errors H0 Errors are not positively correlated over time H A Errors are positively correlated over time 69 281 81702 Test Statistic DW H n 26 11 DurbinWatson Calculations Sum of Squared Difference of Residuals 0099348772 Sum of Squared Residuals 1401388918 IDurbinWatson Statistic 0070893076 Decision Rule n69 observations kl predictor 0c005 signi cance level Conclude HA if dL lt 158 Conclude H0 ide gt 164 Here we clearly conclude in favor of H A There is serious autocorrelation among errors Step 8 Regression statistics and the Analysis of Variance for completeness Tests are not appropriate after step 7 Regression Statistics Multiple R 0849290817 R Square 0721294892 Adjusted R Square 0717135114 Standard Error 0144624523 Observations 69 ANOVA df SS MS F Signi cance F Regression 1 3626825052 3626825052 1733974597 293887E20 Residual 67 1401388918 0020916253 Our estimate of the standard error of the random errors 6 is 8501446 The model explains 7213 of the variation in consumption R207213 Note that the Fstatistic is highly signi cant We would certainly conclude there is an association it s actually negative based on the 95 con dence interval for 51 from step 1 The interval was 362266 which is entirely below 0 However it has been found that the standard errors of regression coefficients can be biased downward when errors are not independent This leads us to believe this con dence interval is probably too narrow


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