x
x
ECONOMETRICS
x
x
x
CHAPTER
2
THE MEANING OF REGRESSION
Dependent variable
explained by
Independent variables
Price of iTune
Quantity of iTunes
demanded
Consumer income
Price of CD
Quantity (100s)
Y
Quantity of iTunes
demanded
Price of iTune
X1
Consumer income
X2
Price of CD
X3
6
5
Y = b1 + b2 X
4
3
2
1
.2 .4 .6 .8
1 1.2
Price ($)
Quantity (100s)
6
5
E(Y) = b1 + b2 X
4
3
E(Y) = 550 - 250 X
2
1
.2 .4 .6 .8
Yi = b1 + b2Xi + ei
1 1.2
Price ($)
Xi
0.2
0.4
0.6
1.0
Yi
E(Yi)
ei
Tools
A
Data analysis
B
C
Random Number Generation
1
-0.23002
2
0.389811
Number of variables
OK
3
0.211674
No. of Random No.
Cancel
4
1.31909
Distribution
5
0.785948
Mean =
0
6
0.017634
Stnd deviation =
1
7
-1.3149
8
-1.32496
9
10
11
Help
Random Seed:
Output range:
$A$1:$A$8
fx
49
-1.3149
50
-1.32496
Ctrl
Shift
Enter
51
52
-3
0
20
53
-2
4
15
54
-1
7
10
55
0
13
5
0
56
1
19
-2.5-2 -1.5-1 -0.50 0.5 1 1.52 2.5
57
2
7
Random numbers
58
3
0
59
fx
52
Function Arguments
NORMDIST
X
Mean
Stnd_dev
Cumulative
Formula result =
A53
0
1
Cumulative Cum Norm
53
-3
0
0.001
54
-2
4
0.023
55
-1
7
0.159
56
0
13
0.500
57
1
19
0.841
58
2
7
0.977
59
3
0
0.999
TRUE
OK
Cancel
Population Regression Function (PRF)
Yi = B1 + B2 Xi + ui
• the way the world works
• but we can’t observe this directly
Sample Regression Function (SRF)
Yi = b1 + b2 Xi + ei
• an estimate of the PRF based on a sample
• ordinary least squares (OLS) is method used
Ordinary Least Squares (OLS)
E(Yi ) = Yi = b1 + b2 Xi
ei = Yi – Yi
OLS minimizes:
∑ ei2 = ∑ (Yi – Yi )2
The residual sum of squares (RSS)
∑ Yi
Y =
n
∑ (Xi – X)(Yi – Y)
b2 =
∑ (Xi – X)2
b1 = Y - b2 X
Quantity (100s)
b1 = Y - b2 X
6
5
4
3
2
1
.2 .4 .6 .8
1 1.2
Price ($)
∑ (Xi – X)(Yi – Y)
b2 =
∑ (Xi – X)2
Xi
Yi
E(Yi)
ei
ei2
0.2
0.4
0.6
490
505
336
500
450
400
-10
55
-64
100
3,025
4,096
1.0
318
300
18
324
∑ei2
7,545
Analyze
VAR0001
Regression
VAR0002
Linear
var
1
0.2
Linear Regression
490
2
0.4
505
3
0.6
VAR0001
336
VAR0002
4
1.0
318
5
1.2
249
Dependent
Statistics
Plots
Previous
Next
Independent(s)
Save
Options
6
Method: Enter ▼
OK
Reset
Cancel
Help
Unstandardized Coeffic
Model
1 (Constant)
X1
B
Std. Error
549.837
-250.349
E(Y) = 550 − 250 X
The residuals are
uncorrelated with
the independent variable.
46.833
60.461
Stndardzd
Coeffic
Beta
-.923
t
Sig.
11.740
-4.141
Xi
ei
0.2
0.4
0.6
-10
55
-64
1.0
1.2
18
-1
.001
.026
fx
A
B
1
0.2
-10
2
0.4
55
Array1 A1:A5
3
0.6
-64
Array2 B1:B5
4
1.0
18
5
1.2
-1
6
7
C
Function
Arguments
CORREL
Formula result =
OK
Cancel