1. Regression of NFL Scores on Vegas Line – 2007 Regular Season

2. 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

3. 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

4. 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 = (49.5 + (-6.5))/2 = 21.5 P H = (49.5 - (-6.5))/2 = 28

6. Regression Model

7. Regression Results

9. Test of No Home Effect and Unbiasedness

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

11. 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

13. Confidence Intervals and Prediction Intervals

15. 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

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

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

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

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

20. 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

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

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

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

24. 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