Simple Linear Regression
NFL Point Spreads – 2007
Background
• Las Vegas Bookmakers provide a
point spread for each game
• The spread reflects how many
points the home team “gets” from
the visiting team (negative
values mean the home team
“gives” points to visitor)
• If bookmakers are accurate, on
average the actual difference
should equal prediction
• Accurate ? How variable ?
Statistical model
Y   0  1 X  
where :
Y  Actual Difference (Away Team - Home Team)
X  Predicted Difference (Away Team - Home Team)
 0  Mean Actual Difference when Predicted Difference  0 (" Pick ' em" )
1  Change in mean Actual Difference per Unit Increase in Predicted
 ~ NID0,  2  (Assumptio n)
If oddsmakers are accurate (on average),  0  0 and 1  1
Actual Difference (Y) vs Opening Spread (X) HomeAway
60
40
20
0
-20
-40
-60
-30
-25
-20
-15
-10
-5
0
5
10
15
20
25
Summary Statistics /
Regression Equation
Mean
Std Dev
Spread
-2.72
6.23
Actual
-1.69
15.37
df
1
254
255
SS
14008.25
46705.23
60713.48
Regression Statistics
Multiple R
0.4803
R Square
0.2307
Adjusted R Square
0.2277
Standard Error
13.5602
Observations
256
ANOVA
Regression
Residual
Total
Coefficients Standard Error
Intercept
-0.2023
0.9008
Open Spread (HT)
1.0778
0.1235
MS
14008.25
183.88
t Stat
-0.2245
8.7282
F
76.18
P-value
0.0000
P-value Lower 95%Upper 95%
0.8225
-1.9763
1.5718
0.0000
0.8346
1.3209
Actual vs Spread - With Fitted Equation
60
55
50
45
40
35
30
25
20
Actual (AT-HT)
15
10
5
0
-5
-10
-15
-20
-25
-30
-35
-40
-45
-50
-55
-60
-35
-25
-15
-5
Vegas Spread (AT-HT)
5
15
25
OLS Residuals vs Fitted Values
50
40
30
Residuals
20
10
0
-10
-20
-30
-40
-30
-25
-20
-15
-10
-5
Fitted Values
0
5
10
15
20
Histogram of Residuals
40
35
30
25
20
15
10
5
0
-30
-25
-20
-15
-10
-5
0
5
10
15
20
25
30
35
Residuals versus Normal Scores = Z((Rank0.375)/(n+0.25))
50
40
30
20
10
0
-4
-3
-2
-1
0
-10
-20
-30
-40
-50
1
2
3
4
Testing normality of errors (I)
Shapiro - Francia Method (n  5) (see Royston, 1993)
Order Errors : e(1)  e( 2 )  ...  e( n 1)  e( n )
 i  0.375 
Obtain Normal scores for each observatio n : m i   1 

 n  0.25 
~
~
Obtain " c"-Weights : ci 
mi
n
~ 2
m
and u 
1
n
j
j 1
Obtain approximat e " a"-Weights :
~
a n  cn  0.221157u  0.147981u 2  2.071190u 3  4.434685u 4  2.706056u 5
~
a n 1  cn 1  0.042981u  0.293762u 2  1.752461u 3  5.682633u 4  3.582633u 5
~ 2
n

m
i
~ 2
~ 2
 2 m n  2 m n 1
i 1
~2
~2
1  2 a n  2 a n 1
~
~
~
mi
a1   a n
~
~
a 2   a n 1
~
ai 

i  3,..., n  2
Testing normality of errors (Ii)
H 0 : Errors are normally distribute d
H A : Errors are not normally distribute d
2


  a i e(i ) 

Test Statistic : W '   ni 1
2
 e(i )  e
n
~


i 1
Converted to a Z - statistic, where : Z ' 
g (W ' )  
where : g (W ' )  ln( 1  W ' ),
  1.2725  1.0521ln(ln( n))  ln( n) ,

2 


  1.0308  0.26758 ln(ln( n)) 
ln( n) 

P - value  PZ  Z '

Example – NFL Spread errors
H 0 : Errors are normally distribute d
H A : Errors are not normally distribute d
2


  a i e(i ) 
  46568.34  0.997069
Test Statistic : W '   ni 1
2
46705.23
 e(i )  e
n
~


i 1
Converted to a Z - statistic, where : Z ' 
g (W ' )  

 -1.10938
where : g (W ' )  ln( 1  W ' )  -5.83241,
  1.2725  1.0521ln(ln( n))  ln( n)   -5.30441,

2 
  0.475945
  1.0308  0.26758 ln(ln( n)) 
ln( n) 

P - value  PZ  Z '  0.866367
Testing accuracy in mean
H0: 0  0, 1  1
HA: 0 ≠ 0 and/or 1 ≠ 1
Fit Model UnDer H0: Y*=X
Obtain error sum of squares
under Y*
• Compare with error sum of
squares from full model (HA).
•
•
•
•
Testing for Accuracy
^ F
Full Model (H A ) : Y i  -0.2023  1.0778 X i
^ R
Reduced Model (H A ) : Y i  X i
Test Statistic : Fobs 

SSE ( F )    Yi  Y i
i 1 
n
^ F
2

  46705.23


2
^ R

SSE ( R)    Yi  Y i   46818
i 1 

n
SSE ( R)  SSE ( F )  2  (46818  46705.23) 2  56.385  0.307
SSE ( F ) (n  2)
46705.23 254
183.879
P - value : PF2, 254  0.307   0.7359
Do not reject the null hypothesis that  0  0, 1  1