Bölüm 3
6.1
The Simple Linear
Regression Model
6.2
Regression Output--A
SAS
Model: MODEL1
Dependent Variable: Y
T-1
̂
y
y
̂ 2
SSE
K-1
T-K
SSR/(K-1)
Analysis of Variance
SSR
Source
DF
Sum of
Squares
Model
Error
C Total
1
38
39
25221.22299
54311.33145
79532.55444
Root MSE
Dep Mean
C.V.
37.80536
130.31300
29.01120
Mean
Square
25221.22299
1429.24556
F Value
Prob>F
17.647
0.0002
SST
R-square
Adj R-sq
0.3171
0.2991
R2
adjusted R2
Parameter Estimates
̂
Variable
DF
Parameter
Estimate
Standard
Error
T for H0:
Parameter=0
Prob > |T|
INTERCEP
X
1
1
40.767556
0.128289
22.13865442
0.03053925
1.841
4.201
0.0734
0.0002
Explaining Variation in yt
ê
y
ŷ  b1  b2 x
Toplam
Sapma
y

y
ŷ
y
Açıklanamayan
yˆ  y
6.3
Açıklanabilen
Explaining Variation in yt
6.4
degerine ait sapmanın açıklanması
^
yt = b1 + b2xt + et
Açıklanan Degişken
^y = b + b x
t
1
2 t
Açıklanamayan Sapma
^e = y  ^y = y  b b x
t
t
t
t
1
2 t
Explaining Variation in yt
^
^
yt = yt + et
using y as baseline
^
^
yt  y = y t  y + e t
T
T
(yty) = 
t=1
2
t=1
6.5
Karesini alıp toplarsak
T

2
2
^
^
(y y) + e
t
SST = SSR + SSE
BKT= AKT+KKT
t=1
t
cross
product
term
drops
out
Total Variation in yt
SST = total sum of squares
BKT=bütün kareler toplamı
SST,yt degerinin sapmasını yani y
Etrafında dagılımı ölçer
T
SST
= (yt y)
t=1
2
6.6
6.7
Explained Variation in yt
SSR = regression sum of squares
AKT=Açıklanan Kareler Toplamı
^y = b + b x
t
1
2 t
^
Fitted yt values:
SSR tahmin edilen
Etrafındaki dagılımını ölçer
T
SSR
^yt nin
= (yt y)
^
t=1
2
y
(Yt) Açıklanamayan
Sapmaya dair
6.8
SSE = error sum of squares (Açıklanamayan hata)
KKT=Açıklanamayan Kalıntı Kareler Toplamı
^
^
et = ytyt = yt b1  b2xt
SSE gerçek degerler ile yt Tahmin yt^ degerleri
Arasındaki farklılıgı ölçer
T
SSE
T
= (yt yt) = 
t=1
^ 2
t=1
^e 2
t
Analysis of Variance Table
Table 6.1 Analysis of Variance Table
Source of
Sum of
Mean
Variation
DF
Squares
Square
Explained
1
SSR
SSR/1
Unexplained T-2
SSE SSE/(T-2)
^ 2]
[= 
Total
T-1
SST
6.9
Coefficient of Determination
yt degerinden sapmanın standard
ölçümü:
2
0< R <1
2
R =
SSR
SST
6.10
Coefficient of Determination
SST = SSR + SSE
SST
ile bölersek
SST
SST
=
SSR SSE
+
SST SST
1 =
2
R =
SSR
SST
SSR + SSE
SST SST
= 1
SSE
SST
6.11
Coefficient of Determination
6.12
sadece descriptive (tanımlayıcı
Ölçüm yapar Kalite ölçüm birimi
degildir measure.Yani regresyonun
Ölçüm kalitesini göstermez
2
R
maximizing R2 yani en yüksek Deger
tek başına buna bakmak dogru degıldir
6.13
Regression Output--B
Excel
SUMMARY OUTPUT
̂
Regression Statistics
Multiple R
0.563132517
R Square
0.317118231
Adjusted R Square
0.299147658
Standard Error
37.80536423
Observations
40
ANOVA
Intercept
X Variable 1
SSR=AKT
T
df
Regression
Residual
Total
R2
SSE=KKT
SS
MS
F
Sig. F
1 25221.22 25221.22 17.6465 0.00015495
38 54311.33 1429.246 varyansı
39 79532.55
SST=BKT
Coefficients
St. Error
t Stat
P-value Lower 95% Upper 95%
40.76755647 22.13865 1.841465 0.07337 -4.0498079 85.584921
0.128288601 0.030539 4.200777 0.00015 0.06646511 0.1901121
b2
Se(b2)
For hypothesis testing
Interval
Estimate
Regression Computer Output
6.14
Typical computer output of regression estimates:
Table 6.2 Computer Generated Least Squares Results
(1)
(2)
(3)
(4)
(5)
Parameter Standard
t-stat
Variable
Estimate
Error
p-value
INTERCEPT 40.7676 22.1387
1.841
0.0734
X
0.1283
0.0305
4.201
0.0002
Regression Computer Output
b1 = 40.7676
b2 = 0.1283
se(b1) =
^ 1) = 490.12
var(b
se(b2) =
^ 2) = 0.0009326 = 0.0305
var(b
t =
t =
b1
se(b1)
b2
se(b2)
=
=
= 22.1287
40.7676
22.1287
= 1.84
0.1283
0.0305
= 4.20
6.15
Regression Computer Output
6.16
Sources of variation in the dependent variable:
Table 6.3 Analysis of Variance Table
Sum of
Mean
Source
DF
Squares
Square
AKT=Explained
1 25221.2229 25221.2229
KKT=Unexplained 38
54311.3314 1429.2455
BKT=Total
39 79532.5544
R-square: 0.3171
Regression Computer Output
SST = (yty) = 79532
2
^
SSR = (yty) = 25221
2
^
SSE = e = 54311
2

t
SSE /(T-2) = ^2
2
R =
SSR
SST
= 1
= 1429.2455
SSE
= 0.317
SST
6.17
6.18
2
R =
SSR
SST
= 1
SSE
= 0.317
SST
Yorumu?
Y degerinin ortalamasının etrafındaki farklılık
regresyon modeli ile yüzde 31.7% oranında
açıklana bilmiştir
Yada bir diger ifade ile modelimiz Y degeri
üzerinde yüzde 31.7 oranında açıklayıcı gücü
vardır.
6.19
Reporting Regression Results
yt = 40.7676 + 0.1283xt
(s.e.) (22.1387) (0.0305)
yt = 40.7676 + 0.1283xt
(t) (1.84) (4.20)
Reporting Regression Results
6.20
2
R = 0.317
This R2 value may seem low but it is
typical in studies involving cross-sectional
data analyzed at the individual or micro level.
A considerably higher R2 value would be
expected in studies involving time-series data
analyzed at an aggregate or macro level.
Functional Forms
6.21
The term linear in a simple
regression model does not mean
a linear relationship between
variables, but a model in which
the parameters enter the model
in a linear way.
Linear vs. Nonlinear
6.22
Linear Statistical Models:
yt = 1 + 2xt + et
yt = 1 + 2 ln(xt) + et
ln(yt) = 1 + 2xt + et
yt = 1 + 2x2t + et
Nonlinear Statistical Models:
yt = 1 +
3
2xt +
et
3
yt
yt = 1 + 2xt + exp(3xt) + et
= 1 + 2xt + et
Linear vs. Nonlinear
y
6.23
nonlinear
relationship
between food
expenditure and
income
food
expenditure
0
income
x
Useful Functional Forms
Look at
each form
and its
slope and
elasticity
1.
2.
3.
4.
5.
6.
Linear
Reciprocal
Log-Log
Log-Linear
Linear-Log
Log-Inverse
6.24
Useful Functional Forms
6.25
Linear
yt = 1 + 2xt + et
slope: 2
xt
elasticity: 2 y
t
6.26
Useful Functional Forms
Reciprocal
yt = 1 + 2 xt + et
1
slope:
1
2 2
xt
elasticity:
1
2 x y
t
t
y
Reciprocal Models
6.27
y  1   2 1 / x 
2  0
2  0
x
Useful Functional Forms
Log-Log
(Constant Elasticity Model)
ln(yt)= 1 + 2ln(xt) + et
yt
slope: 2 x
t
elasticity: 2
6.28
6.29
Log-Log Models
y
ln y   1   2 lnx   2  1
ln y   1   2 lnx  0   2  1
x
y
6.30
Log-Log Models
 2  1
ln y   1   2 lnx 
 2  1
 1  2  0
x
Useful Functional Forms
Log-Linear
ln(yt)= 1 + 2xt + et
slope: 2 yt
elasticity: 2xt
6.31
y
6.32
Log-Linear Models
ln y   1   2 x
2  0
2  0
x
Useful Functional Forms
6.33
Linear-Log
yt= 1 + 2ln(xt) + et
slope:
1
_
2
xt
1_
elasticity: 2 y
t
y
6.34
Linear-Log Models
y  1   2 ln x 
2  0
2  0
x
Useful Functional Forms
Log-Inverse
ln(yt) = 1 - 2 x + et
1
t
yt
slope: 2 2
xt
1
elasticity: 2 x
t
6.35
y
6.36
Log-Inverse Model
ln y   1   2 1 / x 
2  0
x
What Functional Form?
1.
2.
3.
4.
E (et) = 0
2
var (et) = 
cov(ei, ej) = 0
2
et ~ N(0,  )
6.37
Economic Models
1.
2.
3.
4.
Demand Models
Supply Models
Production Functions
Cost Functions
6.38
Economic Models
6.39
Demand Models
* quality demanded (yd) and price (x)
* constant elasticity
ln(yt )= 1 + 2ln(x)t + et
d
Economic Models
6.40
Supply Models
s
(y )
* quality supplied
and price (x)
* constant elasticity
ln(yt )= 1 + 2ln(xt) + et
s
Economic Models
6.41
Production Functions
* output (y) and input (x)
* constant elasticity
Cobb-Douglas Production Function:
ln(yt)= 1 + 2ln(xt) + et
6.42
Economic Models
Cost Functions
* total cost (y) and output (x)
yt = 1 +
2
2x t +
et
Economic Models
6.43
Cost Functions
* average cost (x/y) and output (x)
(yt/xt) = 1/xt + 2xt + et/xt