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Regression of NFL Scores on Vegas Line – 2007 Regular Season

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Problem Description Odds makers Place a Point Spread (differential) and a Over/Under (total) on all National Football League games Combining these two quantities, we can obtain a prediction for the final score of the game Let P A and P H be the odds makers Predicted scores for the Away and Home teams, respectively Spread [wrt Home Team] (PS)= P A – P H (Negative spreads for Home teams mean they are favored (giving points) Over/Under (OU) = P A + P H P A = (OU+PS)/2 P H = (OU-PS)/2

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Data/Model Description Point Spreads, Over/Under, and Actual Scores obtained for all n=256 NFL games from 2007 season Predicted Scores obtained for each team in each game Regression is fit for each teams actual score (n=512 team games) as a function of predicted score, and home team indicator Residuals checked to see if errors are independent within games for the two teams Tests conducted to determine: If Home Team effect is sufficiently accounted for by odds makers If Odds makers are unbiased in their point predictions If relation between actual and predicted scores is linear

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Week 1 Data Note for the first game: Spread = P A – P H = -6.5 (IND was favored to beat NO by 6.5 Points) Over/Under = P A + P H = 49.5 (Predicted Total Score was 49.5 points) P A = ( (-6.5))/2 = 21.5 P H = ( (-6.5))/2 = 28

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Regression Model

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Regression Results

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Test of No Home Effect and Unbiasedness

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Results of Test of No Home Effects and Unbiasedness No evidence to Conclude that E(Y) P

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Fit of Simple Regression of Actual on Predicted Score Note, we clearly do not reject H 0 that the intercept is 0 and slope is 1, but will use this model to obtain Confidence Intervals for Mean Score and Prediction Intervals for Individual Game Scores at various levels of predicted scores

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Confidence Intervals and Prediction Intervals

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Residual Analysis Are the residuals consistent with the model assumptions: Normally Distributed Histogram, Normal Probability Plot, Wilks-Shapiro Test Linear relation between Actual and Predicted Scores Plot of Residuals versus Fitted, Lack-of-Fit F-test Constant Error Variance Plot of Residuals versus Fitted, Regress |resid| vs fitted Independent (e.g. Within Games and Within Teams Over Time) Correlation between Home/Away within games Non-Independent errors within Teams (Random Team effects) Autocorrelation among errors over time within teams

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Normal Distribution of Residuals Correlation between residuals and their corresponding normal scores =.9952

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Linearity of Regression No evidence to reject the hypothesis of a linear relation between Actual and Predicted scores

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Equal (Homogeneous) Variance - I No overwhelming evidence of unequal variance based on graph

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Equal (Homogeneous) Variance - II No evidence to reject the null hypothesis of equal variance among errors

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Equal (Homogeneous) Variance There is some evidence of unequal variance, but keep in mind the sample size is huge. See plot for how weak the association is

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Independence Between Home/Away Residuals Within Games No Evidence of associations between residuals within games

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Testing For Random Team Effects - I No overwhelming evidence of team random effects

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Testing for Random Team Effects - II No evidence of random Team Effects

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Durbin-Watson Test Within Teams over Weeks Teams 2 and 9 have small DW values (positive autocorrelation). Team 22 displays negative autocorrelation (value above 4-d L ). Most teams show no autocorrelation

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