Transcript The Multiple Regression Model

The Multiple Regression Model
Hill et al Chapter 7
A model of the effects of
advertising on revenue.
tr  1  2 p  3a
2= the change in tr ($1000)
when p is increased by one
unit ($1), and a is held
constant
trt  E (trt )  et  1  2 pt  3at  et
The assumptions of the model
MR1. yt  1   2 xt 2 
  K xtK  et , t  1,
MR2. E ( yt )  1  2 xt 2 
,T
  K xtK  E (et )  0 .
MR3. var(yt) = var(et) = .
2
MR4. cov(yt, ys) = cov(et, es) = 0
MR5. The values of xtk are not random and are not exact
linear functions of the other explanatory variables.
MR6. yt ~ N  (1   2 xt 2 
  K xtK ),  2   et ~ N (0,  2 ) ,
The Gauss-Markov Theorem: For the multiple regression
model, if assumptions MR1-MR5 hold, then the least squares
estimators are the Best Linear Unbiased Estimators (BLUE) of
the parameters in a multiple regression model.
The estimators
b1  y  b2 x2  b3 x3
b2
y x   x     y x   x



  x   x     x x 
b3
y x   x     y x   x



  x   x     x x 
* *
t t2
*2
t3
*2
t2
* *
t t3
*2
t3
*2
t2
*2
t2
yt*  yt  y ,
* *
t2 t3
x

* *
t3 t2

* *
t t3
* *
t2 t3
2
* *
t t2
*2
t3
* *
t2 t3
x
2
xt*2  xt 2  x2 , xt*3  xt 3  x3
Sampling properties
eˆt2
ˆ 
T K
2
var(b2 ) 
r23 
bk  k
t
~ t T  K 
se(bk )

2
x

x
(1

r


 t2 2
23 )
2
  x  x  x  x 
 x  x   x  x 
t2
2
t3
3
2
t2
2
t3
3
2
Interval estimates and
significance tests
P bk  tcse(bk )  k  bk  tcse(bk )  1  
H 0 : k  0
H1 :  k  0
bk
t
~ t( T  K )
se  bk 
Measuring Goodness of Fit
y
  yˆt  y 
2
SSR
R 

SST   yt  y 2
2
SSE
eˆt2
1
 1
2
SST
  yt  y 
x
y
SSE /(T  K )
R 1
SST /(T  1)
2
x
Example: Measuring Advertising
Effectiveness
ˆ t  104.79  6.642 pt  2.984at
tr
•
•
•
•
tr: revenue (thousand $)
p: price ($)
a: advertising (thousand $)
conclusions
– demand is elastic
– advertising has a positive effect on sales
Interval Estimates and tests of
significance
T  52
K 3
b1  104.79
ˆ  b1   6.483
se  b1   var
ˆ  b2   3.191
b2  6.642 se  b2   var
b3  2.984
ˆ  b3   0.1669
se  b3   var
[bk  tcse(bk ), bk  tcse(bk )]
tc = 2.01
A 95% interval estimate for 2 is given by
(13.06,  0.23)
6.642
t
 2.08
3.191
2.984
t
 17.88
0.1669
Does advertising break-even?
1. H 0 : 3  1
2. H1 : 3  1
3. If the null hypothesis is true,
b3  1
t
~ t T  K 
se(b3 )
4. Significance level is  = .05, we reject H0 if t  tc = 1.68
5. The value of the test statistic is:
b  3 2.984  1
t 3

 11.89
se  b3 
0.1669
Goodness of fit
Source
Explained
Unexplained
Total
R2  1 
R2  1
Sum of
Squares
11776.18
1805.168
13581.35
DF
2
49
51
eˆt2
  yt  y 
2
 1
1805.168
 0.867
13581.35
SSE /(T  K )
36.8
1
0.8617
SST /(T  1)
266.3