Modul Ajar Statistika Inferensia ke-12: Uji Asumsi Klasik pada Regresi Linier Berganda

Analisa Regresi:
Uji Asumsi Klasik
ARIF RAHMAN
1

Statistika
Statistika adalah cabang ilmu matematika yang
mempelajari metode ilmiah untuk mengumpulkan,
mengorganisasi, merangkum, menyederhanakan,
menyajikan, menginterpretasikan, menganalisa dan
mensintesa data (numerik atau nonnumerik) untuk
menghasilkan informasi dan/atau kesimpulan, yang
membantu dalam penyelesaian masalah dan/atau
pengambilan keputusan.
2

Statistika
3
Mengorganisasi,
Merangkum,
Menyederhanakan,
Menyajikan,
Menginterpretasikan
Menganalisa
Mensintesa
Mengumpulkan data
Menghasilkan informasi dan/atau kesimpulan
Menggeneralisasi
Mengestimasi,
Menguji hipotesa,
Menilai relasi,
Memprediksi
Menyelesaikan masalah Mengambil keputusan

Statistika Inferensia
Statistika inferensia adalah cabang statistika yang
menganalisa atau mensintesa data untuk
menggeneralisasi sampel terhadap populasi,
mengestimasi parameter, menguji hipotesa, menilai
relasi, dan membuat prediksi untuk menghasilkan
informasi dan/atau kesimpulan.
Terdapat banyak alat bantu statistika (statistical tools)
yang dapat dipergunakan untuk menginferensi
populasi atau sistem yang menjadi sumber asal data
sampel
4

Statistika Inferensia
5
Tujuan studi terhadap populasi Observasi atau eksperimen pada sampel
SAMPLING
INFERENSI
Parameter :
N (banyaknya anggota populasi),
μ (rata-rata populasi),
σ (simpangan baku populasi),
π (proporsi populasi)
Statistik :
n (banyaknya anggota sampel),
ẋ (rata-rata sampel),
s (simpangan baku sampel),
p (proporsi sampel)

Tipe Data
Data Nominal, data yang hanya berupa simbol (meski berupa
angka) untuk membedakan nilainya tanpa menunjukkan tingkatan
Data Ordinal, data yang mempunyai nilai untuk menunjukkan
tingkatan, namun tanpa skala yang baku dan jelas antar tingkatan.
Data Interval, data yang mempunyai nilai untuk menunjukkan
tingkatan dengan skala tertentu sesuai intervalnya. Nilai nol hanya
untuk menunjukkan titik acuan (baseline).
Data Rasio, data yang mempunyai nilai untuk menunjukkan
tingkatan dengan skala indikasi rasio perbandingan. Nilai nol
menunjukkan titik asal (origin) yang bernilai kosong (null).
6

Tipe Data
Data Parametrik, data kuantitatif yang mempunyai
sebaran variabel acak mengikuti pola distribusi
probabilitas dengan parameter tertentu (independent
and identically distributed random variables)
Data Nonparametrik, data yang tidak mempunyai
distribusi probabilitas (distribution-free)
7

Tipe Data
Data Diskrit, data hasil pencacahan atau
penghitungan, sehingga biasanya dalam angka
bilangan bulat.
Data Kontinyu, data hasil pengukuran yang
memungkinkan dalam angka bilangan nyata
(meskipun dapat pula dibulatkan)
8

Statistika Alat Bantu Problem Solving
9
Penting memperhatikan
cara memperoleh
data yang akan diolah
Demikian pula
cara mengolah data
juga penting diperhatikan

Statistika Alat Bantu Problem Solving
10
Metode statistika bukan
ramuan sihir
Alat statistika bukan
tongkat sihir

Ketelitian &
Tipe Kesalahan
11

Akurasi dan Presisi
Akurasi (accuracy), kesesuaian hasil pengukuran
terhadap nilai obyek sesungguhnya (bias kecil)
Presisi (precision), tingkat skala ketelitian
pengukuran dari alat pengukur, atau ketersebaran
yang relatif mengumpul (variansi atau deviasi kecil)
12

Akurat dan Presisi
Tidak presisi, akibat pola sebaran sampel
lebih melebar daripada pola sebaran
populasi menyebabkan deviasi yang besar.
Tidak akurat, akibat pergeseran
pemusatan sampel menjauh dari
pemusatan populasi menyebabkan bias
yang besar.
Akurat dan presisi, bias dan deviasi kecil,
membutuhkan sampel sedikit.
13

Kesalahan Pengambilan Kesimpulan
Galat tipe 1 () : kesalahan menyimpulkan karena
menolak hipotesa yang semestinya diterima
Galat tipe 2 () : kesalahan menyimpulkan karena
menerima hipotesa yang semestinya ditolak
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 

Kesalahan Pengambilan Kesimpulan
15
The true state of nature
Decision H0 is true H0 is false
Reject H0 Type I error Exact decision
Fail to reject H0 Exact decision Type II error
The true state of nature
Decision H0 is true H0 is false
Reject H0  1 – 
Fail to reject H0 1 –  

Ukuran Ketelitian Pendugaan
Tingkat keberartian (significance level, ), probabilitas
penolakan data observasi, karena menyimpang signifikan terhadap
sasaran.
Tingkat kepercayaan (confidence coefficient,1-), persentase
data observasi yang diyakini tidak berbeda signifikan dengan target.
Kuasa statistik (power,1-), persentase data observasi yang
diyakini berbeda signifikan dengan target.
Derajat kebebasan (degree of freedom, df=n-k), besaran
yang menunjukkan bebas terhadap bias dari n data observasi.
16

Kekeliruan pada Analisa Regresi
 Tidak ada logika penalaran atau alasan logis yang mendasari hipotesa
variabel bebas mempengaruhi variabel terikat.
 Deskripsi dari variabel bebas tidak mempunyai hubungan kausal
dengan deskripsi dari variabel terikat.
 Pengukuran atau pengumpulan data dilakukan oleh/dari pihak yang
mempunyai konflik kepentingan atau yang tidak punya kewenangan
atas data.
 Variabel bebas dan/atau variabel terikat diukur pada hanya satu obyek
yang bernilai tunggal atau statis.
 Rentang data sampel sangat sempit, namun dipergunakan untuk
menginduksi rentang populasi yang sangat lebar dengan ekstrapolasi.
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 Variabel bebas (x) = tinggi badan anak-anak di desa A (pertumbuhan tiap tahunnya);
Variabel terikat (y) = harga emas (kenaikan tiap tahunnya). Meskipun jika dihitung,
menunjukkan pertumbuhan tinggi badan anak-anak di desa A mempunyai korelasi
kuat dengan kenaikan harga emas.
 Variabel bebas (x) = motivasi kerja; Variabel terikat (y) = kinerja. Deskripsi “motivasi
kerja” adalah mendapatkan pengakuan dari keluarga besar karena menjadi karyawan
pabrik.Deskripsi “kinerja” adalah waktu penyelesaian pekerjaan lebih cepat daripada
batas waktu yang ditargetkan.
 Data yang diukur : Banyaknya pelanggaran yang dilakukan. Ditanyakan pada pelaku
pelanggaran, atau pada pihak yang tidak pernah melihat pelaku pelanggaran.
 Data yang diukur : Aturan pengupahan dari satu perusahaan di satu waktu.
 Rentang data sampel pada 10 < x < 50, namun dipergunakan untuk menduga nilai Y
jika x=200.
18

19

Perbedaan Korelasi dan Regresi
21
Correlation Regression

Perbedaan Korelasi dan Regresi
22
Correlation Regression

Analisa Regresi Linier Sederhana
29

Analisa Regresi Linier Sederhana
30

Asumsi Klasik
(classical linear regression
model assumptions)
34

The Gauss-Markov Theorem
Given Classical Assumptions, the ordinary least
squares (OLS) estimator βk is the minimum variance
estimator from among the set of all linear unbiased
estimators of βk.
In other words, OLS is BLUE
Best Linear Unbiased Estimator
Where Best = Minimum Variance
35

Asumsi Klasik dalam Regresi
1. The regression model is linear, y = Xβ + ε, correctly
specified and has an additive error term or residual, defined
by ei = yi – ŷi.  linearity
2. The number of observations must be greater than the
number of parameters to be estimated, n > k.
3. There must be variability in data value for the response
variable and explanatory variable(s).
4. The X matrix is non-stochastic. X values are fixed in
repeated sampling.
5. All explanatory variables are uncorrelated with the error
term, Cov(xi, ei) = 0.
36

Asumsi Klasik dalam Regresi
6. Observations of the error term are uncorrelated with each
other  no serial correlation or autocorrelation.
7. No explanatory variable is a perfect linear function of any
other explanatory variables.  no perfect multicollinearity
8. The error term is normally distributed, ~N(0,σ2) 
normality
9. The error term has a zero population mean, E(ei)= 0. 
unbiased estimator
10.The error term has a constant variance, Var(ei) = σ2 < . 
homoskedasticity or no heteroskedasticity
37

Uji Asumsi Klasik :
Linearity
38

Uji Asumsi Klasik: linearity
39

1. Select baseline set of levels for each explanatory
variables (x10, x20, ..., xk0).
2. Successively estimate revised response variable (y'i•)
for each explanatory variable (xi•) with the other
explanatory variables held constant at the baseline
level (xj• = xj0 for j ≠ i).
3. Plot the data (xi•, y'i•) and the regression line with the
other explanatory variables held constant.
40
 
)
(
)
(
)
( 0
20
2
2
10
1
1 k
k
k
i
i x
x
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x
x
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y
y 

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
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

 



 

 

41

42

43

Example 1
45
Response Surface Methodology
3
2
1 .
3433
,
0
.
8616
,
1
.
0161
,
1
1574
,
39 x
x
x
y 




Example 1
variables (x10, x20, x30).
x10 = 6
x20 = 6
x30 = 9
46

Example 1
2. Estimate revised response variable (y'i•)
47
x1 x2 x3 y x10 x20 x30 y'1 y‘2 y‘3
1,74 5,30 10,8 25,5 6 6 9 24,815 30,446 29,588
6,32 5,42 9,4 31,2 6 6 9 30,258 31,012 30,676
6,22 8,41 7,2 25,9 6 6 9 29,769 25,059 26,504
10,52 4,63 8,5 38,4 6 6 9 35,678 33,636 33,337
1,19 11,60 9,4 18,4 6 6 9 28,963 23,425 25,210
1,22 5,85 9,9 26,7 6 6 9 26,730 31,866 31,505
4,10 6,62 8,0 26,4 6 6 9 27,211 27,987 28,543
6,32 8,72 9,1 25,9 6 6 9 30,998 25,609 26,509
4,08 4,42 8,7 32,0 6 6 9 28,956 33,848 33,409
4,15 7,60 9,2 25,2 6 6 9 28,247 27,148 27,629
10,15 4,83 9,4 39,7 6 6 9 37,659 35,620 35,082
1,72 3,12 7,6 35,7 6 6 9 29,858 39,568 39,060
1,70 5,30 8,2 26,5 6 6 9 24,922 30,595 30,629
59,43 81,82 115,4 377,5
 
 
678
,
35
)
9
5
,
8
(
3433
,
0
)
6
63
,
4
(
0161
,
1
4
,
38
)
(
)
( 30
34
3
20
24
2
14
14






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








 x
x
x
x
y
y 


Example 1
3. Plot the data (xi•, y'i•) and the regression line.
48
Estimated Regression Equation
3
2
1 .
3433
,
0
.
8616
,
1
.
0161
,
1
1574
,
39 x
x
x
y 




Example 2
51
2
1 .
01253
,
0
.
74427
,
2
26379
,
2 x
x
y 



Example 2
variables (x10, x20).
x10 = 10
x20 = 300
52

Example 2
2. Estimate revised response
variable (y'i•)
53
x1 x2 y x1 x2 y'1 y‘2
2 50 9,95 10 300 13,082 31,904
8 110 24,45 10 300 26,830 29,939
11 120 31,75 10 300 34,005 29,006
10 550 35,00 10 300 31,868 35,000
8 295 25,02 10 300 25,083 30,509
4 200 16,86 10 300 18,113 33,326
2 375 14,38 10 300 13,440 36,334
2 52 9,60 10 300 12,707 31,554
9 100 24,35 10 300 26,856 27,094
8 300 27,50 10 300 27,500 32,989
4 412 17,08 10 300 15,677 33,546
11 400 37,00 10 300 35,747 34,256
12 500 41,95 10 300 39,444 36,461
2 360 11,66 10 300 10,908 33,614
4 205 21,65 10 300 22,840 38,116
4 400 17,89 10 300 16,637 34,356
20 600 69,00 10 300 65,242 41,557
1 585 10,30 10 300 6,730 34,998
10 540 34,93 10 300 31,923 34,930
15 250 46,59 10 300 47,216 32,869
15 290 44,88 10 300 45,005 31,159
16 510 54,12 10 300 51,489 37,654
17 590 56,63 10 300 52,997 37,420
6 100 22,13 10 300 24,636 33,107
5 400 21,15 10 300 19,897 34,871
206 8294 725,82
 
 
868
,
31
)
300
550
(
01253
,
0
00
,
35
)
( 20
24
2
14
14









 x
x
y
y 

Example 2
3. Plot the data (xi•, y'i•) and the regression line.
54
Estimated Regression Equation
2
1 .
01253
,
0
.
74427
,
2
26379
,
2 x
x
y 



Uji Asumsi Klasik :
Normality
55

Uji Asumsi Klasik: normality
The error term is normally distributed, ~N(0,σ2), with
a zero population mean, E(ei)= 0, and a constant
variance, Var(ei) = σ2.
It needs to assume normality for parametric
hypothesis testing based on gaussian distribution.
Normality assumption can be tested using goodness
of fit test or the Bera Jarque normality test.
56

Goodness of Fit Test
 Chi Square goodness of fit test
 Kolmogorov–Smirnov test
 Liliefors test
 Geary test
 Anderson–Darling test
 Shapiro–Wilk test
 Bayesian information criterion
 Cramér–von Mises criterion
 Akaike information criterion
 Kuiper's test
 Moran test
 Hosmer–Lemeshow test
57

The Bera Jarque normality test
A normal distribution is not skewed and is defined to
have a coefficient of kurtosis of 3.
The kurtosis of the normal distribution is 3 so its
excess kurtosis (b2-3) is zero.
Skewness and kurtosis are the (standardised) third
and fourth moments of a distribution.
58

59
 3
2
3
s
m
skewness 
Mo
Me
͞x Mo Me ͞x
͞x
Skewness = 0 Skewness > 0
Skewness < 0
Positive or right skew
Mo
Me
Symmetric
Negative or left skew
 Normal = 0

60
 2
2
4
s
m
kurtosis
Leptokurtik Mesokurtik Platikurtik
Kurtosis > 3 Kurtosis = 3 Kurtosis < 3
Normal
 Normal = 3

The Bera Jarque normality test
Bera and Jarque testing the residuals for normality by testing
whether the coefficient of skewness and the coefficient of
excess kurtosis are jointly zero.
It can be proved that the coefficients of skewness and
kurtosis can be expressed respectively as:
and
The Bera Jarque test statistic is given by
61
 
b
E u
1
3
2 3 2

[ ]
/
  
b
E u
2
4
2 2

[ ]

   
2
~
24
3
6
2
2
2
2
1






 


b
b
T
W

Uji Asumsi Klasik :
No Serial Correlation or
Autocorrelation
62

Uji Asumsi Klasik: no autocorrelation
Observations of the error term are uncorrelated with
each other.
Autocorrelation or serial correlation is a violation of
the classical assumption that assumes uncorrelated
observations of the error term.
Issue: Is et related to et-1? Such would be the case in
a time-series when a random shock has an impact
over a number of time periods.
63

It assumed the error terms are not interrelated, i.e.
Cov (ei , ej) = 0 for i≠j.
If the covariance of two error terms is not equal to
zero, then autocorrelation exists.
This is essentially the same as saying there is no
pattern in the error term.
If there are patterns in the error term from a model,
we say that they are autocorrelated.
64

65

66

67

Autocorrelation can be categorized into 2 kinds:
Pure autocorrelation (autocorrelation that that occurs
when classical assumption, which assumes uncorrelated
observations of the error term, is violated).
Impure autocorrelation (autocorrelation that is caused by
specification errors: omitted variables or incorrect
functional form).
Autocorrelation mostly happens in a data set where
order of observations has some meaning (e.g. time-
series data).
68

The most commonly assumed kind of autocorrelation is first-
order autocorrelation, in which the posterior value of the
error term is a function of the prior value of the error term:
et = ρ.et–1 + ut
where:
et = the error term in the model at time period t
ρ = the first-order autocorrelation coefficient depicting the
functional relationship between observations of the error
term.
ut = a classical (not serially correlated) error term
69

The magnitude of ρ indicates the strength of the
autocorrelation or serial correlation:
If ρ is zero, ρ ≈ 0, there is no autocorrelation
As ρ approaches one in absolute value, |ρ| ≈ 1, there is significant
autocorrelation
For ρ to exceed one is unreasonable, since the error term
effectively would “explode”
As a result of this, we can state that:
–1 < ρ < +1
70

ρ < 0 indicates negative autocorrelation (the signs of
the error term switch back and forth).
ρ > 0 indicates positive autocorrelation (a positive
error term tends to be followed by a positive error
term and a negative error term tends to be followed
by a negative error term).
Positive autocorrelation is more common than
negative autocorrelation. Situations where negative
autocorrelation occurs are not often encountered.
71

 Examples of higher order autocorrelation:
1. Seasonal autocorrelation:
et = .et-4 + ut
2. Second-order autocorrelation:
et = 1.et-1 + 2.et-2 + ut
2. r-th-order autocorrelation:
et = 1.et-1 + 2.et-2 + . . . + r.et-r + ut
72

Some tests to detect autocorrelation or serial correlation
The Graphical Run Test
The Durbin Watson Test
The Breusch-Godfrey Test
The Box-Pierce Q Test
The Cumby-Huizinga test
The Ljung-Box Q Test
The Portmanteau Test
The Lagrange Multiplier Test
73

The Durbin-Watson (DW) is a test for first order
autocorrelation - i.e. it assumes that the relationship is
between an error and the previous one
et = ρ.et–1 + ut
where ut  N(0, u
2).
The DW test statistic actually tests
H0 : =0 and H1 : 0
The test statistic is calculated by
74
)
1
(
2
)
(
)
.
(
2
)
(
)
(
)
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2
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2
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2
1


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
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
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





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T
t
t
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t
t
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t
t
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t
t
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t
t
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t
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e
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e
e
DW

We can also write
DW ≈ 2(1 – ρ) and
where ρ is the estimated correlation coefficient.
Since ρ is a correlation, it implies that –1 < ρ < 1. Subtituting
ρ by DW would give 0 < DW < 4.
If ρ ≈ 0, DW ≈ 2. There is little evidence to reject the null hypothesis if DW is
near 2  the error terms are not autocorrelated
If ρ ≈ 1, DW ≈ 0. There is significant evidence to reject the null hypothesis if
DW is near 0  the error terms are positive autocorrelated
If ρ ≈ -1, DW ≈ 4. There is significant evidence to reject the null hypothesis if
DW is near 4  the error terms are negative autocorrelated
75
2
1
)
(
)
.
(
2
2
2
1
DW
e
e
e
T
t
t
T
t
t
t










 Unfortunately, DW has 2 critical values, an upper critical value (dU) and
a lower critical value (dL), and there is also an intermediate region
(inconclusive) where we can neither reject nor not reject H0.
 The decision procedure is as follows:
If DW < dL reject H0 : ρ = 0, strong positive autocorrelation
If dL < DW < dU inconclusive, weak positive autocorrelation
If dU < DW < 4-dU do not reject H0 : ρ = 0, no autocorrelation
If 4-dU < DW < 4-dL inconclusive, weak negative autocorrelation
If DW > 4-dL reject H0 : ρ = 0, strong negative autocorrelation
76

77
Conditions which Must be Fulfilled for DW to be a Valid Test
1. Constant term in regression
2. Regressors are non-stochastic
3. No lags of dependent variable

Example 3
80
x1 x2 x3 y y pred e
1,74 5,30 10,8 25,5 27,35 -1,851
6,32 5,42 9,4 31,2 32,26 -1,062
6,22 8,41 7,2 25,9 27,35 -1,450
10,52 4,63 8,5 38,4 38,31 0,090
1,19 11,60 9,4 18,4 15,54 2,855
1,22 5,85 9,9 26,7 26,11 0,592
4,10 6,62 8,0 26,4 28,25 -1,853
6,32 8,72 9,1 25,9 26,22 -0,322
4,08 4,42 8,7 32,0 32,09 -0,088
4,15 7,60 9,2 25,2 26,07 -0,868
10,15 4,83 9,4 39,7 37,25 2,448
1,72 3,12 7,6 35,7 32,49 3,212
1,70 5,30 8,2 26,5 28,20 -1,703
59,43 81,82 115,4 377,5 377,50 0,00

Example 3
81
e(t-1) e(t) (e(t))2 (e(t) – e(t-1)) (e(t) – e(t-1))2
-1,851 -1,062 1,129 0,789 0,623
-1,062 -1,450 2,101 -0,387 0,150
-1,450 0,090 0,008 1,540 2,372
0,090 2,855 8,153 2,765 7,644
2,855 0,592 0,350 -2,263 5,123
0,592 -1,853 3,434 -2,445 5,978
-1,853 -0,322 0,104 1,531 2,345
-0,322 -0,088 0,008 0,234 0,055
-0,088 -0,868 0,753 -0,779 0,608
-0,868 2,448 5,991 3,315 10,991
2,448 3,212 10,317 0,764 0,584
3,212 -1,703 2,901 -4,915 24,160
1,70 5,30 1,129 0,789 0,623
35,249 0,148 60,633
720
,
1
249
,
35
633
,
60
)
(
)
(
2
2
2
2
1
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T
t
t
T
t
t
t
e
e
e
DW
748
,
0
)
82
,
0
00
,
1
(
15
20
15
13
82
,
0 
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L
d
778
,
1
)
75
,
1
68
,
1
(
15
20
15
13
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,
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d

Example 4
86
x1 x2 y ypred e
2 50 9,95 8,379 1,571
8 110 24,45 25,596 -1,146
11 120 31,75 33,954 -2,204
10 550 35,00 36,597 -1,597
8 295 25,02 27,914 -2,894
4 200 16,86 15,746 1,114
2 375 14,38 12,450 1,930
2 52 9,60 8,404 1,196
9 100 24,35 28,215 -3,865
8 300 27,50 27,976 -0,476
4 412 17,08 18,402 -1,322
11 400 37,00 37,462 -0,462
12 500 41,95 41,459 0,491
2 360 11,66 12,262 -0,602
4 205 21,65 15,809 5,841
4 400 17,89 18,252 -0,362
20 600 69,00 64,666 4,334
1 585 10,30 12,337 -2,037
10 540 34,93 36,472 -1,542
15 250 46,59 46,560 0,030
15 290 44,88 47,061 -2,181
16 510 54,12 52,561 1,559
17 590 56,63 56,308 0,322
6 100 22,13 19,982 2,148
5 400 21,15 20,996 0,154
206 8294 725,82 725,820 0,000

Example 4
87
e(t-1) e(t) (e(t))2 (e(t) – e(t-1)) (e(t) – e(t-1))2
1,571 -1,146 1,313 -2,717 7,384
-1,146 -2,204 4,858 -1,058 1,120
-2,204 -1,597 2,550 0,607 0,369
-1,597 -2,894 8,373 -1,297 1,682
-2,894 1,114 1,240 4,007 16,058
1,114 1,930 3,724 0,816 0,666
1,930 1,196 1,431 -0,734 0,538
1,196 -3,865 14,938 -5,061 25,616
-3,865 -0,476 0,227 3,389 11,483
-0,476 -1,322 1,749 -0,846 0,716
-1,322 -0,462 0,213 0,860 0,740
-0,462 0,491 0,241 0,953 0,908
0,491 -0,602 0,363 -1,093 1,196
-0,602 5,841 34,116 6,443 41,516
5,841 -0,362 0,131 -6,203 38,476
-0,362 4,334 18,785 4,696 22,054
4,334 -2,037 4,149 -6,371 40,589
-2,037 -1,542 2,376 0,495 0,245
-1,542 0,030 0,001 1,572 2,470
0,030 -2,181 4,756 -2,211 4,889
-2,181 1,559 2,430 3,740 13,985
1,559 0,322 0,104 -1,236 1,529
0,322 2,148 4,613 1,826 3,333
2,148 0,154 0,024 -1,994 3,976
112,705 -1,418 241,537
143
,
2
705
,
112
537
,
241
)
(
)
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2
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T
t
t
T
t
t
t
e
e
e
DW

The Breusch-Godfrey test is a test for r-th order
autocorrelation - i.e. it assumes that the relationship is
between an error and the previous ones
et = ρ1.et–1 + ρ2.et–2 + . . . + ρr.et–r + ut
where ut  N(0, u
2).
The Breusch-Godfrey test statistic actually tests
H0 : ρ1=0 and ρ2=0 and . . . and ρr=0
H1 : 10 or 20 or . . . or r0
89

The Breusch-Godfrey test has three basic steps:
1. Obtain the residuals of the estimated regression
equation:
ei = yi – (β0 + β1x1i + β2x2i + ... + βkxki)
2. Use the first regression and the residuals to form the
residual model in a second regression:
et = b0 + b1x1t + b2x2t + ... + bkxkt +
ρ1.et–1 + ρ2.et–2 + . . . + ρr.et–r + ut
90

The Breusch-Godfrey test has three basic steps:
3. Obtain R2 from this regression.
(degree of freedom, ν = n-r-2) :
H0: (n-r)R2 < χ2 (No autocorrelated errors)
H1: (n-r)R2 > χ2 (Autocorrelated errors)
If the test statistic exceeds the critical value from the
statistical tables, reject the null hypothesis of no
autocorrelation.
91

92
chi-square χ2 distribution

Autocorrelation consequences:
 Autocorrelation does not cause bias in the β coefficient estimates
 Autocorrelation increases the variance of the β coefficient estimates
 Autocorrelation causes the dependent variable to fluctuate in a fashion that the
estimation procedure (OLS) attributes to the independent variables. Hence the
variance of the estimates of β increases. These estimates are still unbiased,
since over-estimation and under-estimation are still as likely
 Autocorrelation causes OLS to underestimate the variances (and
standard errors) of the β coefficients.
 Intuitively - Autocorrelation increases the fit of the model. Hence the estimation of
the variance and standard errors is lower. This can lead the researcher to
conclude a relationship exists when in fact they are unrelated.
 Hence the t-stats and F-stats can not be relied upon for statistical inference.
 Spurious Regressions.
93

Methods to correct or remedy autocorrelation
Use the Generalized Least Squares to restore the
minimum variance property of the OLS estimation.
Use the Newey-West standard errors
Use the Cochrane-Orcutt method
Use the Hildreth-Lu Procedure
Use the first order Autoregressive, AR(1), model
Use the Maximum Likelihood approach
94

Uji Asumsi Klasik :
Homoskedasticity or
No Heteroskedasticity
95

Uji Asumsi Klasik: no heteroskedasticity
There should be homoskedasticity that the error term
has a constant variance, Var(ei) = σ2.
Heteroskedasticity is a violation of the classical
assumption that assumes the observations of the
error term are drawn from distributions that have a
constant variance.
Issue: Does ei differ across levels of the explanatory
variables? Such would be the case in a capacity constraint
when any particular state has an impact over the capacity.
96

 In homoskedasticity the distribution of
the error term has a constant
variance, so the observations are
continually drawn from the same
distribution.
 In the simplest heteroskedastic case,
discrete heteroskedasticity, there
would be two different error term
variances, and therefore, two different
distributions. One distribution is wider
than the other.
97

It assumed all explanatory variables are uncorrelated
with the error term, i.e. Cov(xi, ei) = 0 for each
explanatory variable(s).
If the covariance is not equal to zero, then
heteroskedasticity exists.
pattern in the error term by explanatory variable(s).
If there are patterns from a model, we say that they
are correlated.
98

99

Heteroskedasticity takes on many more complex forms,
however, than the discrete heteroskedasticity case
Perhaps the most frequently specified model of pure
heteroskedasticity relates the variance of the error term to an
exogenous variable Zi as follows:
yi = β0 + β1x1i + β2x2i + ... + βkxki + ei
VAR(ei) = σ2.f(Zi)
where Z, the “proportionality factor”, may or may not be in
the equation
100

 If the error term is homoskedastic
with respect to Zi, the variance of the
distribution of the error term is the
same (constant) no matter what the
value of Zi, as in VAR(ei) = σ2.
 If the error term is heteroskedastic
with respect to Zi, the variance of the
distribution of the error term changes
systematically as function of Zi. In this
example, the variance is an
increasing function of Zi, as in
VAR(ei) = σ2Zi
2
101

102

103

Heteroskedasticity can be categorized into 2 kinds:
Pure heteroskedasticity (heteroskedasticity that occurs
when classical assumption, which assumes constant
variance of the error term, is violated).
Impure heteroskedasticity (heteroskedasticity that is
caused by specification errors: omitted variables or
incorrect functional form).
Heteroskedasticity mostly happens in a data set where any
particular state may loosen/tighten the dispersion of response
variable (e.g. cross-sectional data).
104

Heteroskedasticity can occur in any situations, such as:
When there is significant change in the variable(s) of a time-
series model.
When there is capacity constraint that limit the option
When there is different control or consistency
When there is different behavior in the different strata of
population.
When there are different amounts of measurement errors in
the sample of different periods or different sub-samples.
105

Before using any test for heteroskedasticity, however, ask the following:
1. Are there any obvious specification errors? Is the regression model
already correctly specified?
 Fix those before testing!
2. Is the subject of the research likely to be afflicted with
heteroskedasticity?
 The cross-sectional studies with large variations in the size of the dependent
variable are particularly susceptible to heteroskedasticity
3. Does a graph of the residuals show any evidence of
heteroskedasticity?
 Specifically, plot the residuals against a potential Z proportionality factor
 In such cases, the graph can often show that heteroskedasticity is or is not likely
 Any graph shows an expanding (or contracting) range of the residuals
106

Because heteroskedasticity can take on many forms,
therefore, there is no specific test to test for
heteroskedasticity.
Scientists and researchers do not all use the same
test for heteroskedasticity since heteroskedasticity
takes a number of different forms, and its precise
manifestation in a given equation is almost never
known.
107

Some tests to detect heteroskedasticity
 The Graphical Zpredictor-Sresidual Test
 The Park Test
 The White Test
 The Glejser Test
 The Levene’s test
 The Goldfeld-Quandt Test
 The Brown-Forsythe Test
 The Harrison-McCabe Test
 The Breusch-Pagan Test
 The Cook-Weisberg Test
108

The Park test is a test for heteroskedasticity - i.e. it assumes
that the relationship is between variable(s) and the error
VAR(ei) = σ2Zi
b1  ln(ei
2) = ln(σ2) + b1.ln(Zi) + ui
where Z: the “proportionality factor”
b1 : the slope of logarithmic model of squared residual
ui: a classical (homoskedastic) error term
One difficulty with the Park test is the specification of the Z
factor. The Z factor may one of the explanatory variables, but
not always.
109

The Park test has three basic steps:
equation:
2. Use these residuals to form the logarithmic model of
squared residual in a second regression:
ln(ei
2) = ln(σ2) + b1.ln(Zi) + ui
where yi' = ln(ei
2) and xi' = ln(Zi)
b0 = ln(σ2) and b1 : the slope of linear model
ui : a classical (homoskedastic) error term
110

The Park test has three basic steps:
3. Check to see whether b1 is significant or not. Test the
significance of the b1 coefficient of Z with t-test
(degree of freedom, ν = n-2) :
H0: b1 = 0 (Homoskedastic errors)
H1: b1 ≠ 0 (Heteroskedastic errors)
111

112
Advantages of the Park test:
a. The test is simple.
b. It provides information about the variance structure.
Limitations of the Park test:
a. The distribution of the dependent variable is problematic.
b. It assumes a specific functional form.
c. It does not work when the variance depends on two or more variables.
d. The correct variable with which to order the observations must be
identified first.
e. It cannot handle partitioned data.

113
Student’s t distribution

Example 5
115
x1 x2 x3 y y pred e e2 ln(x1) ln(x2) ln(x3) ln(e2)
1,74 5,30 10,8 25,5 27,35 -1,851 3,428 0,554 1,668 2,380 1,232
6,32 5,42 9,4 31,2 32,26 -1,062 1,129 1,844 1,690 2,241 0,121
6,22 8,41 7,2 25,9 27,35 -1,450 2,101 1,828 2,129 1,974 0,743
10,52 4,63 8,5 38,4 38,31 0,090 0,008 2,353 1,533 2,140 -4,807
1,19 11,60 9,4 18,4 15,54 2,855 8,153 0,174 2,451 2,241 2,098
1,22 5,85 9,9 26,7 26,11 0,592 0,350 0,199 1,766 2,293 -1,049
4,10 6,62 8,0 26,4 28,25 -1,853 3,434 1,411 1,890 2,079 1,234
6,32 8,72 9,1 25,9 26,22 -0,322 0,104 1,844 2,166 2,208 -2,267
4,08 4,42 8,7 32,0 32,09 -0,088 0,008 1,406 1,486 2,163 -4,857
4,15 7,60 9,2 25,2 26,07 -0,868 0,753 1,423 2,028 2,219 -0,284
10,15 4,83 9,4 39,7 37,25 2,448 5,991 2,317 1,575 2,241 1,790
1,72 3,12 7,6 35,7 32,49 3,212 10,317 0,542 1,138 2,028 2,334
1,70 5,30 8,2 26,5 28,20 -1,703 2,901 0,531 1,668 2,104 1,065
59,43 81,82 115,4 377,5 377,50 0,00 38,676 16,426 23,188 28,311 -2,647
Regression equation for Park test
VAR(ei) = σ2Zi
b1
ln(ei
2) = ln(σ2) + b1.ln(Zi) + ui
where Zi = x1i , x2i , x3i

Example 5
116
X
ln(x1)
Y
ln(e2)
X2 XY Y2
0,554 1,232 0,307 0,682 1,518
1,844 0,121 3,399 0,223 0,015
1,828 0,743 3,341 1,357 0,551
2,353 -4,807 5,538 -11,311 23,102
0,174 2,098 0,030 0,365 4,403
0,199 -1,049 0,040 -0,209 1,100
1,411 1,234 1,991 1,741 1,522
1,844 -2,267 3,399 -4,180 5,141
1,406 -4,857 1,977 -6,829 23,587
1,423 -0,284 2,025 -0,404 0,081
2,317 1,790 5,371 4,149 3,205
0,542 2,334 0,294 1,266 5,447
0,531 1,065 0,282 0,565 1,134
16,426 -2,647 27,993 -12,585 70,806
  239
.
7
13
426
.
16
993
.
27
2
2
2







n
x
x
SXX
  241
.
9
13
)
647
.
2
(
426
.
16
585
.
12 











n
y
x
xy
SXY
  267
.
70
13
)
647
.
2
(
806
.
70
2
2
2








n
y
y
SYY
i
i
i
i
i
i
b
i
i
u
x
e
u
x
b
e
x
e









)
ln(
277
.
1
409
.
1
)
ln(
)
ln(
)
ln(
)
ln(
)
VAR(
1
2
1
1
2
2
1
1
2


277
.
1
239
.
7
241
.
9
1 




XX
XY
S
S
b
409
.
1
13
)
426
.
16
)
277
.
1
((
)
647
.
2
(
.
)
ln( 1
2










n
x
b
y


Example 5
117
316
.
5
2
13
))
241
,
9
(
)
277
.
1
((
267
.
70
2
.
1
2










n
S
b
S
s XY
YY
i
i
i u
x
e 

 )
ln(
277
.
1
409
.
1
)
ln( 1
2
48971
.
1
239
.
7
316
.
5
0
277
.
1
0
2
1







XX
S
s
b
t
267
.
70
241
.
9
239
.
7




YY
XY
XX
S
S
S
Degree of freedom: ν = n – 2 = 13 – 2 = 11
P-value = 2 X P(t < -1.48971) = 2 X 0.0822 = 0.1644
Conclusion: P-value (0.1644) > Sign.level (0.05).
There is no evidence to reject H0. The error term
is Homoskedastic with respect to x1.

Example 5
118
X
ln(x2)
Y
ln(e2)
X2 XY Y2
1,668 1,232 2,781 2,054 1,518
1,690 0,121 2,856 0,204 0,015
2,129 0,743 4,534 1,581 0,551
1,533 -4,807 2,349 -7,366 23,102
2,451 2,098 6,007 5,143 4,403
1,766 -1,049 3,120 -1,853 1,100
1,890 1,234 3,572 2,332 1,522
2,166 -2,267 4,690 -4,910 5,141
1,486 -4,857 2,209 -7,218 23,587
2,028 -0,284 4,113 -0,576 0,081
1,575 1,790 2,480 2,819 3,205
1,138 2,334 1,295 2,656 5,447
1,668 1,065 2,781 1,776 1,134
23,188 -2,647 42,789 -3,356 70,806
  430
.
1
13
188
.
23
789
.
42
2
2
2







n
x
x
SXX
  364
.
1
13
)
647
.
2
(
188
.
23
356
.
3 










n
y
x
xy
SXY
  267
.
70
13
)
647
.
2
(
806
.
70
2
2
2








n
y
y
SYY
i
i
i
i
i
i
b
i
i
u
x
e
u
x
b
e
x
e










)
ln(
954
.
0
905
.
1
)
ln(
)
ln(
)
ln(
)
ln(
)
VAR(
2
2
2
1
2
2
1
2
2


954
.
0
430
.
1
364
.
1
1 


XX
XY
S
S
b
905
.
1
13
)
188
.
23
954
.
0
(
)
647
.
2
(
.
)
ln( 1
2










n
x
b
y


Example 5
119
270
.
6
2
13
)
364
.
1
954
.
0
(
267
.
70
2
.
1
2








n
S
b
S
s XY
YY
i
i
i u
x
e 


 )
ln(
954
.
0
905
.
1
)
ln( 2
2
45556
.
0
430
.
1
270
.
6
0
954
.
0
0
2
1





XX
S
s
b
t
267
.
70
364
.
1
430
.
1



YY
XY
XX
S
S
S
P-value = 2 X P(t > 0.45556) = 2 X 0.3288 = 0.6576

Example 5
120
X
ln(x3)
Y
ln(e2)
X2 XY Y2
2,380 1,232 5,662 2,931 1,518
2,241 0,121 5,021 0,271 0,015
1,974 0,743 3,897 1,466 0,551
2,140 -4,807 4,580 -10,286 23,102
2,241 2,098 5,021 4,702 4,403
2,293 -1,049 5,256 -2,404 1,100
2,079 1,234 4,324 2,566 1,522
2,208 -2,267 4,876 -5,007 5,141
2,163 -4,857 4,680 -10,507 23,587
2,219 -0,284 4,925 -0,630 0,081
2,241 1,790 5,021 4,011 3,205
2,028 2,334 4,113 4,733 5,447
2,104 1,065 4,427 2,241 1,134
28,311 -2,647 61,803 -5,913 70,806
  149
.
0
13
311
.
28
803
.
61
2
2
2







n
x
x
SXX
  149
.
0
13
)
647
.
2
(
311
.
28
913
.
5 











n
y
x
xy
SXY
  267
.
70
13
)
647
.
2
(
806
.
70
2
2
2








n
y
y
SYY
i
i
i
i
i
i
b
i
i
u
x
e
u
x
b
e
x
e









)
ln(
001
.
1
977
.
1
)
ln(
)
ln(
)
ln(
)
ln(
)
VAR(
3
2
3
1
2
2
1
3
2


001
.
1
149
.
0
149
.
0
1 




XX
XY
S
S
b
977
.
1
13
)
311
.
28
)
001
.
1
((
)
647
.
2
(
.
)
ln( 1
2










n
x
b
y


Example 5
121
374
.
6
2
13
))
149
.
0
(
)
001
.
1
((
267
.
70
2
.
1
2










n
S
b
S
s XY
YY
i
i
i u
x
e 

 )
ln(
001
.
1
977
.
1
)
ln( 3
2
15306
.
0
149
.
0
374
.
6
0
001
.
1
0
2
1







XX
S
s
b
t
267
.
70
149
.
0
149
.
0




YY
XY
XX
S
S
S
P-value = 2 X P(t < -0.15306) = 2 X 0.4406 = 0.8812

Example 6
125
x1 x2 y ypred e e2 ln(x1) ln(x2) ln(e2)
2 50 9,95 8,379 1,571 2,469 0,693 3,912 0,904
8 110 24,45 25,596 -1,146 1,313 2,079 4,700 0,273
11 120 31,75 33,954 -2,204 4,858 2,398 4,787 1,581
10 550 35,00 36,597 -1,597 2,550 2,303 6,310 0,936
8 295 25,02 27,914 -2,894 8,373 2,079 5,687 2,125
4 200 16,86 15,746 1,114 1,240 1,386 5,298 0,215
2 375 14,38 12,450 1,930 3,724 0,693 5,927 1,315
2 52 9,60 8,404 1,196 1,431 0,693 3,951 0,358
9 100 24,35 28,215 -3,865 14,938 2,197 4,605 2,704
8 300 27,50 27,976 -0,476 0,227 2,079 5,704 -1,483
4 412 17,08 18,402 -1,322 1,749 1,386 6,021 0,559
11 400 37,00 37,462 -0,462 0,213 2,398 5,991 -1,545
12 500 41,95 41,459 0,491 0,241 2,485 6,215 -1,422
2 360 11,66 12,262 -0,602 0,363 0,693 5,886 -1,014
4 205 21,65 15,809 5,841 34,116 1,386 5,323 3,530
4 400 17,89 18,252 -0,362 0,131 1,386 5,991 -2,032
20 600 69,00 64,666 4,334 18,785 2,996 6,397 2,933
1 585 10,30 12,337 -2,037 4,149 0,000 6,372 1,423
10 540 34,93 36,472 -1,542 2,376 2,303 6,292 0,866
15 250 46,59 46,560 0,030 0,001 2,708 5,521 -6,999
15 290 44,88 47,061 -2,181 4,756 2,708 5,670 1,559
16 510 54,12 52,561 1,559 2,430 2,773 6,234 0,888
17 590 56,63 56,308 0,322 0,104 2,833 6,380 -2,265
6 100 22,13 19,982 2,148 4,613 1,792 4,605 1,529
5 400 21,15 20,996 0,154 0,024 1,609 5,991 -3,745
206 8294 725,82 725,820 0,000 115,173 46,058 139,773 3,190
Regression equation for Park test
VAR(ei) = σ2Zi
b1
ln(ei
2) = ln(σ2) + b1.ln(Zi) + ui
where Zi = x1i , x2i , x3i

Example 6
126
X
ln(x1)
Y
ln(e2)
X2 XY Y2
0,693 0,904 0,480 0,626 0,817
2,079 0,273 4,324 0,567 0,074
2,398 1,581 5,750 3,790 2,498
2,303 0,936 5,302 2,155 0,876
2,079 2,125 4,324 4,419 4,516
1,386 0,215 1,922 0,298 0,046
0,693 1,315 0,480 0,911 1,729
0,693 0,358 0,480 0,248 0,128
2,197 2,704 4,828 5,941 7,311
2,079 -1,483 4,324 -3,085 2,201
1,386 0,559 1,922 0,775 0,312
2,398 -1,545 5,750 -3,704 2,387
2,485 -1,422 6,175 -3,534 2,023
0,693 -1,014 0,480 -0,703 1,028
1,386 3,530 1,922 4,893 12,459
1,386 -2,032 1,922 -2,817 4,130
2,996 2,933 8,974 8,787 8,603
0,000 1,423 0,000 0,000 2,024
2,303 0,866 5,302 1,993 0,749
2,708 -6,999 7,334 -18,954 48,987
2,708 1,559 7,334 4,223 2,432
2,773 0,888 7,687 2,461 0,788
2,833 -2,265 8,027 -6,417 5,131
1,792 1,529 3,210 2,739 2,338
1,609 -3,745 2,590 -6,027 14,025
46,058 3,190 100,844 -0,414 127,613
  990
.
15
25
058
.
46
844
.
100
2
2
2







n
x
x
SXX
  292
.
6
25
190
.
3
058
.
46
414
.
0 










n
y
x
xy
SXY
  206
.
127
25
190
.
3
613
.
127
2
2
2







n
y
y
SYY
i
i
i
i
i
i
b
i
i
u
x
e
u
x
b
e
x
e









)
ln(
393
.
0
853
.
0
)
ln(
)
ln(
)
ln(
)
ln(
)
VAR(
1
2
1
1
2
2
1
1
2


393
.
0
990
.
15
292
.
6
1 




XX
XY
S
S
b
853
.
0
25
)
058
.
46
)
393
.
0
((
190
.
3
.
)
ln( 1
2









n
x
b
y


Example 6
127
423
.
5
2
25
))
292
,
6
(
)
393
.
0
((
206
.
127
2
.
1
2










n
S
b
S
s XY
YY
i
i
i u
x
e 

 )
ln(
393
.
0
853
.
0
)
ln( 1
2
67566
.
0
990
.
15
423
.
5
0
393
.
0
0
2
1







XX
S
s
b
t
206
.
127
292
.
6
990
.
15




YY
XY
XX
S
S
S
P-value = 2 X P(t < -0.67566) = 2 X 0.2530 = 0.5060

Example 6
128
X
ln(x2)
Y
ln(e2)
X2 XY Y2
3,912 0,904 15,304 3,536 0,817
4,700 0,273 22,095 1,281 0,074
4,787 1,581 22,920 7,567 2,498
6,310 0,936 39,815 5,906 0,876
5,687 2,125 32,342 12,085 4,516
5,298 0,215 28,072 1,140 0,046
5,927 1,315 35,128 7,793 1,729
3,951 0,358 15,612 1,416 0,128
4,605 2,704 21,208 12,452 7,311
5,704 -1,483 32,533 -8,461 2,201
6,021 0,559 36,253 3,364 0,312
5,991 -1,545 35,898 -9,256 2,387
6,215 -1,422 38,621 -8,839 2,023
5,886 -1,014 34,646 -5,968 1,028
5,323 3,530 28,334 18,789 12,459
5,991 -2,032 35,898 -12,176 4,130
6,397 2,933 40,921 18,762 8,603
6,372 1,423 40,597 9,065 2,024
6,292 0,866 39,584 5,445 0,749
5,521 -6,999 30,487 -38,645 48,987
5,670 1,559 32,148 8,842 2,432
6,234 0,888 38,868 5,534 0,788
6,380 -2,265 40,706 -14,451 5,131
4,605 1,529 21,208 7,041 2,338
5,991 -3,745 35,898 -22,438 14,025
139,773 3,190 795,094 9,784 127,613
  639
.
13
25
773
.
139
094
.
795
2
2
2







n
x
x
SXX
  052
.
8
25
190
.
3
773
.
139
784
.
9 









n
y
x
xy
SXY
  206
.
127
25
190
.
3
613
.
127
2
2
2







n
y
y
SYY
i
i
i
i
i
i
b
i
i
u
x
e
u
x
b
e
x
e









)
ln(
590
.
0
428
.
3
)
ln(
)
ln(
)
ln(
)
ln(
)
VAR(
2
2
2
1
2
2
1
2
2


590
.
0
639
.
13
052
.
8
1 




XX
XY
S
S
b
428
.
3
25
)
773
.
139
)
590
.
0
((
190
.
3
.
)
ln( 1
2









n
x
b
y


Example 6
129
324
.
5
2
25
))
052
,
8
(
)
590
.
0
((
206
.
127
2
.
1
2










n
S
b
S
s XY
YY
i
i
i u
x
e 

 )
ln(
590
.
0
428
.
3
)
ln( 2
2
94492
.
0
639
.
13
324
.
5
0
590
.
0
0
2
1







XX
S
s
b
t
206
.
127
052
.
8
639
.
13




YY
XY
XX
S
S
S
P-value = 2 X P(t < -0.94492) = 2 X 0.1773 = 0.3546

The White test is a test for heteroskedasticity - i.e. it assumes
that the relationship is between variable(s) and the error
quadratic regression of the squared residual that consists of
the square of each x, and the product of each x times every
other x from the original equation
ei
2 = b0 + b1(x1i)2 + b2(x2i)2 + ... + bk(xki)2 +
bk+1(x1i.x2i) + bk+2(x1i.x3i) + ... + b2k-1(x1i.xki) +
... + b½.k(k+1)(x(k-1)i.xki)
130

The White test has three basic steps:
equation:
2. Use these residuals to form the quadratic model of
squared residual in a second regression:
ei
2 = b0 + b1(x1i)2 + b2(x2i)2 + ... + bk(xki)2 +
bk+1(x1i.x2i) + bk+2(x1i.x3i) + ... + b2k-1(x1i.xki) +
... + b½.k(k+1)(x(k-1)i.xki)
131

The White test has three basic steps:
3. Check to see whether quadratic model is significant
or not. Test the significance of the n.R2 (the sample
size, n, times the coefficient of determination, R2)
with chi-square χ2-test (degree of freedom, ν =
number of regressors = ½.k(k+1))
H0: n.R2 < χ2
critical (Homoskedastic errors)
H1: n.R2 > χ2
critical (Heteroskedastic errors)
132

133
chi-square χ2 distribution

134
Advantages of the White test:
a. It does not assume a specific functional form.
b. It is applicable when the variance depends on two or more variables.
Limitations of the White test:
a.It is an large-sample test.
b.It provides no information about the variance structure.
c.It loses many degrees of freedom when there are many regressors.
d.It cannot handle partitioned data.
e.It also captures specification errors.

Example 7
136
x1 x2 x3 y y pred e e2 x12 x22 x32 x1x2 x1x3 x2x3 e4
1,74 5,30 10,8 25,5 27,35 -1,851 3,428 3,028 28,090 116,640 9,222 18,792 57,240 11,749
6,32 5,42 9,4 31,2 32,26 -1,062 1,129 39,942 29,376 88,360 34,254 59,408 50,948 1,274
6,22 8,41 7,2 25,9 27,35 -1,450 2,101 38,688 70,728 51,840 52,310 44,784 60,552 4,415
10,52 4,63 8,5 38,4 38,31 0,090 0,008 110,670 21,437 72,250 48,708 89,420 39,355 0,000
1,19 11,60 9,4 18,4 15,54 2,855 8,153 1,416 134,560 88,360 13,804 11,186 109,040 66,464
1,22 5,85 9,9 26,7 26,11 0,592 0,350 1,488 34,223 98,010 7,137 12,078 57,915 0,123
4,10 6,62 8,0 26,4 28,25 -1,853 3,434 16,810 43,824 64,000 27,142 32,800 52,960 11,794
6,32 8,72 9,1 25,9 26,22 -0,322 0,104 39,942 76,038 82,810 55,110 57,512 79,352 0,011
4,08 4,42 8,7 32,0 32,09 -0,088 0,008 16,646 19,536 75,690 18,034 35,496 38,454 0,000
4,15 7,60 9,2 25,2 26,07 -0,868 0,753 17,223 57,760 84,640 31,540 38,180 69,920 0,567
10,15 4,83 9,4 39,7 37,25 2,448 5,991 103,023 23,329 88,360 49,025 95,410 45,402 35,892
1,72 3,12 7,6 35,7 32,49 3,212 10,317 2,958 9,734 57,760 5,366 13,072 23,712 106,450
1,70 5,30 8,2 26,5 28,20 -1,703 2,901 2,890 28,090 67,240 9,010 13,940 43,460 8,416
59,43 81,82 115,4 377,5 377,50 0,00 38,676 394,726 576,726 1035,960 360,662 522,078 728,310 247,154
Quadratic model of squared residual
ei
2 = b0 + b1(x1i)2 + b2(x2i)2 + b3(x3i)2 + b4(x1i.x2i) + b5(x1i.x3i) + b6(x2i.x3i)

Example 7
137
13 394,726 576,726 1035,960 360,662 522,078 728,310
394,726 28435,940 14215,890 30724,415 17434,014 28097,685 19924,252
576,726 14215,890 39239,948 46312,736 17114,038 20729,897 40540,700
A = X'X = 1035,960 30724,415 46312,736 86244,604 27835,366 41298,355 59564,444
360,662 17434,014 17114,038 27835,366 14215,890 19924,252 20729,897
522,078 28097,685 20729,897 41298,355 19924,252 30724,415 27835,366
728,310 19924,252 40540,700 59564,444 20729,897 27835,366 46312,736
2,57712 0,00801 -0,02334 -0,04105 -0,02487 -0,00143 0,04125
0,00801 0,00144 -0,00119 -0,00038 0,00064 -0,00220 0,00183
-0,02334 -0,00119 0,00641 0,00400 0,00091 0,00112 -0,01095
A-1 = (X'X)-1= -0,04105 -0,00038 0,00400 0,00314 0,00151 -0,00031 -0,00721
-0,02487 0,00064 0,00091 0,00151 0,00274 -0,00240 -0,00241
-0,00143 -0,00220 0,00112 -0,00031 -0,00240 0,00431 -0,00112
0,04125 0,00183 -0,01095 -0,00721 -0,00241 -0,00112 0,01920

Example 7
138
38,676
880,789
1910,925
g = X'Y = 2976,781
793,631
1216,240
2176,259
8,21240
-0,03213
0,44284
β =A-1.g = 0,17661
-0,14725
0,10519
-0,68045
The estimated regression equation of squared residual quadratic model
ei
2 = 8.212 – 0.032(x1i)2 + 0.440(x2i)2 + 0.177(x3i)2 – 0.147(x1i.x2i) + 0.105(x1i.x3i)
– 0.680(x2i.x3i)

Example 7
139
Response Surface Methodology
ei
2 = 8.212 – 0.032(x1i)2 + 0.440(x2i)2 +
0.177(x3i)2 – 0.147(x1i.x2i) +
0.105(x1i.x3i) – 0.680(x2i.x3i)

Example 7
140
β X'Y Β(X'Y)
8,21240 38,676 317,626
-0,03213 880,789 -28,299
0,44284 1910,925 846,242
0,17661 2976,781 525,737
-0,14725 793,631 -116,866
0,10519 1216,240 127,937
-0,68045 2176,259 -1480,825
191,553
   
 
487
.
76
066
.
115
553
.
191
13
676
.
38
553
.
191
)
(
)
(
)
(
)
(
)
(
2
2
1
1
0
2
















 n
y
y
x
y
x
y
y
y
SSR k
k
i 

 

 
 
088
.
132
066
.
115
154
.
247
13
676
.
38
154
.
247
)
(
)
(
2
2
2
2











 n
y
y
y
y
SST i
 
601
.
55
553
.
191
154
.
247
)
(
)
(
)
(
)
(
)
( 1
1
0
2
2









 



 y
x
y
x
y
y
y
y
SSE k
k
i
i 

 


Example 7
141
52780
.
7
57906
.
0
13
57906
.
0
088
.
132
487
.
76
2
2






nR
SST
SSR
R
P-value = P(χ2 > 7.52780) = 0.2748
is Homoskedastic.
H0: n.R2 < χ2
H1: n.R2 > χ2
Sign.level = 0.05
Degree of freedom: ν = ½.k(k+1) = ½.3(3+1) = 6
χ2=12.59

Example 8
145
x1 x2 y ypred e e2 x12 x22 x1x2 e4
2 50 9,95 8,379 1,571 2,469 4 2500 100 6,096
8 110 24,45 25,596 -1,146 1,313 64 12100 880 1,725
11 120 31,75 33,954 -2,204 4,858 121 14400 1320 23,600
10 550 35,00 36,597 -1,597 2,550 100 302500 5500 6,501
8 295 25,02 27,914 -2,894 8,373 64 87025 2360 70,111
4 200 16,86 15,746 1,114 1,240 16 40000 800 1,538
2 375 14,38 12,450 1,930 3,724 4 140625 750 13,867
2 52 9,60 8,404 1,196 1,431 4 2704 104 2,048
9 100 24,35 28,215 -3,865 14,938 81 10000 900 223,150
8 300 27,50 27,976 -0,476 0,227 64 90000 2400 0,051
4 412 17,08 18,402 -1,322 1,749 16 169744 1648 3,057
11 400 37,00 37,462 -0,462 0,213 121 160000 4400 0,046
12 500 41,95 41,459 0,491 0,241 144 250000 6000 0,058
2 360 11,66 12,262 -0,602 0,363 4 129600 720 0,132
4 205 21,65 15,809 5,841 34,116 16 42025 820 1163,932
4 400 17,89 18,252 -0,362 0,131 16 160000 1600 0,017
20 600 69,00 64,666 4,334 18,785 400 360000 12000 352,864
1 585 10,30 12,337 -2,037 4,149 1 342225 585 17,212
10 540 34,93 36,472 -1,542 2,376 100 291600 5400 5,647
15 250 46,59 46,560 0,030 0,001 225 62500 3750 0,000
15 290 44,88 47,061 -2,181 4,756 225 84100 4350 22,623
16 510 54,12 52,561 1,559 2,430 256 260100 8160 5,903
17 590 56,63 56,308 0,322 0,104 289 348100 10030 0,011
6 100 22,13 19,982 2,148 4,613 36 10000 600 21,281
5 400 21,15 20,996 0,154 0,024 25 160000 2000 0,001
206 8294 725,82 725,820 0,000 115,173 2396 3531848 77177 1941,469
ei
2 = b0 + b1(x1i)2 + b2(x2i)2 + b3(x1i.x2i)

Example 8
146
25 2396 3531848 77177
2396 502184 485990145 14846879
A = X'X = 3531848 485990145 847350799652 17764206203
77177 14846879 17764206203 485990145
1,192E-01 -8,904E-04 -6,830E-07 3,323E-05
-8,904E-04 3,753E-05 1,394E-08 -1,515E-06
A-1 = (X'X)-1= -6,830E-07 1,394E-08 1,149E-11 -7,377E-10
3,323E-05 -1,515E-06 -7,377E-10 7,002E-08
115,173
13020,171
g = X'Y = 14225116,948
378285,711
4,99502
0,01142
β =A-1.g = -0,00001
0,00010
ei
2 = 4.99502 + 0.01142(x1i)2 – 0.00001(x2i)2 + 0.00010(x1i.x2i)

Example 8
147
β X'Y Β(X'Y)
4,99502 115,173 575,294
0,01142 13020,171 148,677
-0,00001 14225116,948 -179,972
0,00010 378285,711 37,360
581,359
   
 
761
.
50
597
.
530
359
.
581
25
173
.
115
359
.
581
)
(
)
(
)
(
)
(
)
(
2
2
1
1
0
2

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









 n
y
y
x
y
x
y
y
y
SSR k
k
i 

 

 
 
872
.
1410
597
.
530
469
.
1941
25
173
.
115
469
.
1941
)
(
)
(
2
2
2
2











 n
y
y
y
y
SST i
 
110
.
1360
359
.
581
469
.
1941
)
(
)
(
)
(
)
(
)
( 1
1
0
2
2









 



 y
x
y
x
y
y
y
y
SSE k
k
i
i 

 


Example 8
148
8995
.
0
03598
.
0
25
03598
.
0
872
.
1410
761
.
50
2
2






nR
SST
SSR
R
P-value = P(χ2 > 0.03598) = 0.82556
Conclusion:
P-value (0.82556) > Sign.level (0.05).
There is no evidence to reject H0.
The error term is Homoskedastic.
H0: n.R2 < χ2
H1: n.R2 > χ2
Sign.level = 0.05
Degree of freedom: ν = ½.k(k+1) = ½.2(2+1) = 3
χ2=7.81
ei
2 = 4.99502 + 0.01142(x1i)2 – 0.00001(x2i)2 + 0.00010(x1i.x2i)

The Goldfeld-Quandt Test is a test for heteroskedasticity -
i.e. it assumes that the relationship is between variable(s) and
the error
Error Sum of Squares, SSE, is calculated by the differences
between each observed response variable and predicted
response variable.
SSE = Σei
2 = Σ(yi-ŷi)2 = Σ(yi
2) – (β0Σyi + β1Σx1i yi +
β2Σx2iyi + ... + βkΣxkiyi)
149

The Goldfeld-Quandt Test has the following steps:
1. Order the data by the magnitude of Explanatory
Variable in ascending order.
2. Divide the data into three parts: less(n1), middle(n2)
and greater(n3). There is no rule that specify how
many observations each parts. n1+n2+n3=n the
number of observation
3. Omit the middle part, and drop these n2 observations.
150

The Goldfeld-Quandt Test has the following steps:
4. Obtain the error sum of squares of each part : SSE1
(less part with degree of freedom ν1=n1–k) and
SSE2 (greater part with degree of freedom ν2=n2–k)
5. Assuming the error process is normally distributed,
then calculate the ratio of the error sum of squares:
F=(SSE2/ν2)/(SSE1/ν1).
6. Apply the F-test. F>Fcritical at the right tail or F<Fcritical
at the left tail indicate that the variances are different.
151

Heteroskedasticity consequences:
 Heteroskedasticity does not cause bias in the β coefficient estimates
 Heteroskedasticity increases the variance of the β coefficient estimates
 Heteroskedasticity causes the dependent variable to fluctuate in a fashion that the
estimation procedure (OLS) attributes to the independent variables. Hence the
variance of the estimates of β increases. These estimates are still unbiased,
since over-estimation and under-estimation are still as likely
 Heteroskedasticity causes OLS to underestimate the variances (and
standard errors) of the β coefficients.
 Intuitively - Heteroskedasticity increases the fit of the model. Hence the
estimation of the variance and standard errors is lower. This can lead the
researcher to conclude a relationship exists when in fact they are unrelated.
 Hence the t-stats and F-stats can not be relied upon for statistical inference.
 Spurious Regressions.
152

Methods to correct or remedy heteroskedasticity
View logarithmized data
Redefine the variables
Apply a weighted least squares estimation
Use Heteroskedasticity-consistent standard errors
Use Heteroskedasticity-corrected standard errors
Use Minimum Norm Quadratic Unbiased Estimation
(MINQUE)
153

Uji Asumsi Klasik :
No Multicollinearity
154

Uji Asumsi Klasik: no multicollinearity
No explanatory variable is highly correlated with one
or more other explanatory variables.
Multicollinearity is a violation of the classical
assumption that assumes no explanatory variable is a
perfect linear function of any other explanatory
variables.
Issue: Is xi related to xj? Such would be the case in
clustered samples when any variables are similar
within cluster but different between clusters.
155

It assumed the explanatory variables are not
interrelated, i.e. Cov (xi , xj) = 0 for i≠j.
If the covariance of two explanatory variables is not
equal to zero, then multicollinearity exists.
pattern between explanatory variables.
If an explanatory variable can be explained by a
pattern with respect to one or more other explanatory
variables, we say that there are multicollinearity.
156

Multicollinearity can be categorized into 2 kinds:
Perfect multicollinearity (multicollinearity which an
explanatory variable can be written as a linear
combination of any other explanatory variables).
Impure multicollinearity (multicollinearity which an
explanatory variable is close to being represented by a
linear function of any other explanatory variables).
Multicollinearity mostly happens in clustering or
segmenting where samples are grouped by similarity of some
characteristics (e.g. clustered data).
157

 With perfect multicollinearity, an
explanatory variable can be
completely explained by the
movement of one or more other
explanatory variable(s).
 Perfect multicollinearity can usually
be avoided by careful screening of
the explanatory variables before a
regression is run.
 With imperfect multicollinearity, an
explanatory variable is a strong but
not perfect linear function of one or
more other explanatoy variable(s).
 Imperfect multicollinearity varies in
degree from sample to sample.
158

Multicollinearity can be categorized into another 2
kinds:
Structural multicollinearity (multicollinearity occurs when
we create a model term using other terms. In other words,
it’s a byproduct of the model that we specify rather than
being present in the data itself.).
Impure multicollinearity (multicollinearity is present in the
data itself rather than being an artifact of our model.
Observational experiments are more likely to exhibit this
kind of multicollinearity).
159

Regression coefficient, βi, is the impact of explanatory
variable, xi, has on response variable, y, holding all other
explanatory variable(s) constant.
If x1 is related to x2 then β1 will also capture the impact of
changes in x2.
In other words, interpretation of the regression coefficients or
parameters becomes difficult.
The easiest way to test multicollinearity is to examine the
standard errors of the coefficients.
Reasonable method to relieve multicollinearity is to drop
some highly correlated variables.
160

In the Venn diagrams, the overlapping
area between Y and X(X1, X2) is the
variance explained.
 In case 1, X1 and X2 are related; X1 and Y are
related, but X2 and Y has no relationship.
 In case 2, both X1 and X2 contribute to some
unique variance explained to Y, but they also
have some common variance explained.
 In case 3, again both X1 and X2 contribute
unique variance explained to Y, but X1 and X2
are totally unrelated (orthogonal).
 In case 4, although both X1 and X2 could
predict Y. The variance explained contributed
by X2 has been covered by X1 because X1 and
X2 are too correlated (collinear).
The above cases are not exhaustive. There
are many other possible combinations
between Y and Xs.
161

162
Severe multicollinearity
produces a distribution of
the βs that is centered
around the true β, but that
has a much wider
variance.
Thus the distribution of βs
with multicollinearity is
much wider than
otherwise.

A special case of multicollinearity problem is a dominant
variable.
The dominant variable is an explanatory variable, x, which is
definitionally related to the response variable, y.
 The dominant variable is generally a part of or a complement of the
response variable. For example, if the Y variable is the number of
computers and the X variable is the number of processors.
 The dominant variable is highly correlated with response variable and it
will make other explanatory variables unimportant in determining the
response variable.
 Do not confuse a dominant variable with a highly significant explanatory
variable.
163

First realize that that some multicollinearity exists in
every equation: all variables are correlated to some
degree (even if completely at random)
So it’s really a question of how much multicollinearity
exists in an equation, rather than whether any
multicollinearity exists
164

Some tests to detect multicollinearity
 The Determination Coefficient R2 and t-Test
 The Simple Correlation Coefficients Test
 The Variance Inflation Factor (VIF) Test
 The Farrar–Glauber Test
 The Condition Number test
 The Perturbing the data Test
165

High Determination Coefficient, R2, with all low t-
scores (individual coefficient estimator test)
If this is the case, you have multicollinearity.
If this is not the case, you may or may not have
multicollinearity.
If all the t-scores are significant and in the expected
direction than we can conclude that multicollinearity is not
likely to be a problem.
166

High Simple Correlation Coefficients
If a simple correlation coefficient, rij, between any two
explanatory variables (xi and xj with i≠j) is high in absolute
value, these two particular Xs are highly correlated and this
evidence indicates the potential for multicollinearity.
How high is high?
Some researchers pick an arbitrary number, such as 0.80
A better answer might be that rij is high if it causes unacceptably
large variances in the coefficient estimates in which we’re
interested.
167

Caution in case of more than two explanatory variables:
An explanatory variable is correlated with a group of any other
explanatory variables, acting together simultaneously.
An explanatory variable is correlated with an interaction of any other
explanatory variables.
An explanatory variable is a nonlinear function of any other
explanatory variables
It may cause multicollinearity without any single simple
correlation coefficient being high enough to indicate that
multicollinearity is present.
168

The matrix plot between two individual variables.
The matrix of correlations between two individual
variables
Note that high correlation between the response
variable, y, and one of the explanatory variable, x’s, is
not muticollinearity.
169

the matrix plot between two individual variables
170

the matrix of correlations between two individual variables
171
Correlation X1 X2 ... Xk
X1 1 r12 ... r1k
X2 R21 1 ... r2k
: : : :
Xk rk1 rk2 ... 1

The Variance Inflation Factors, VIF, is calculated from
the following steps:
1. For each explanatory variable (xi), run an OLS
regression that has xi as a function of all the other
explanatory variables in the equation—
For i = 1, this equation would be:
x1 = b0 + b2.x2 + b3.x3 + . . . + bk.xk + u
where
u  N(0, u
2).: a classical (no multicollinearity) error
term
172

The Variance Inflation Factors, VIF, is calculated from
the following steps:
2. Obtain a value of determination coefficient, Ri
2, from
the regression equation of specific explanatory
variable, xi.
3. Calculate the variance inflation factor for βi :
VIF(βi) = 1 / (1 – Ri
2)
4. If VIF > 5, multicollinearity problem is potentially
severe. Repeat for all X’s
173

The Variance Inflation Factors, VIF
 The higher the VIF, the more severe the effects of
mulitcollinearity
 While there is no table of formal critical VIF values, a
common rule of thumb is that if a given VIF is greater
than 5, the multicollinearity is severe
 As the number of independent variables increases, it
makes sense to increase this number slightly
174

Example 9
176
x1 x2 x3 y y pred
1,74 5,30 10,8 25,5 27,35
6,32 5,42 9,4 31,2 32,26
6,22 8,41 7,2 25,9 27,35
10,52 4,63 8,5 38,4 38,31
1,19 11,60 9,4 18,4 15,54
1,22 5,85 9,9 26,7 26,11
4,10 6,62 8,0 26,4 28,25
6,32 8,72 9,1 25,9 26,22
4,08 4,42 8,7 32,0 32,09
4,15 7,60 9,2 25,2 26,07
10,15 4,83 9,4 39,7 37,25
1,72 3,12 7,6 35,7 32,49
1,70 5,30 8,2 26,5 28,20
59,43 81,82 115,4 377,5 377,50
Cor y x1 x2 x3
y 1 0,65385 -0,78581 -0,18627
x1 0,65385 1 -0,15350 -0,14522
x2 -0,78581 -0,15350 1 0,07484
x3 -0,18627 -0,14522 0,07484 1

Example 9
177
i ≠ 1 β X'Y Β(X'Y)
0 9,73992 59,43000 578,84326
2 -0,20244 360,66210 -73,01385
3 -0,43869 522,07800 -229,03100
276,79842
   
 
11189
.
5
68653
.
271
79842
.
276
13
43
.
59
79842
.
276
)
(
)
(
)
(
)
(
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(
2
2
1
1
0
2

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












 n
y
y
x
y
x
y
y
y
SSR k
k
i 

 

 
 
03897
.
123
68653
.
271
72550
.
394
13
43
.
59
72550
.
394
)
(
)
(
2
2
2
2








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

 n
y
y
y
y
SST i
 
92708
.
117
79842
.
276
72550
.
394
)
(
)
(
)
(
)
(
)
( 1
1
0
2
2









 



 y
x
y
x
y
y
y
y
SSE k
k
i
i 

 

For x1, the regression equation:
x1 = b0 + b2.x2 + b3.x3 + u
04155
.
0
03897
.
123
11189
.
5
2



SST
SSR
R 04335
.
1
04155
.
0
1
1
1
1
2





R
VIF
For x1,
no multicollinearity

Example 9
178
i ≠ 1 β X'Y Β(X'Y)
0 5,66436 81,82000 463,45828
1 -0,10323 360,66210 -37,23184
3 0,12408 728,31000 90,36559
516,59203
   
 
62953
.
1
96249
.
514
59203
.
516
13
82
.
81
59203
.
516
)
(
)
(
)
(
)
(
)
(
2
2
1
1
0
2
















 n
y
y
x
y
x
y
y
y
SSR k
k
i 

 

 
 
76391
.
61
96249
.
514
72640
.
576
13
82
.
81
72640
.
576
)
(
)
(
2
2
2
2











 n
y
y
y
y
SST i
 
13437
.
60
59203
.
516
72640
.
576
)
(
)
(
)
(
)
(
)
( 1
1
0
2
2









 



 y
x
y
x
y
y
y
y
SSE k
k
i
i 

 

x2 = b0 + b1.x1 + b3.x3 + u
02638
.
0
76391
.
61
62953
.
1
2



SST
SSR
R 02710
.
1
02638
.
0
1
1
1
1
2





R
VIF
For x2,

Example 9
179
i ≠ 1 β X'Y Β(X'Y)
0 8,92230 115,40000 1029,63293
1 -0,04199 522,07800 -21,92001
2 0,02329 728,31000 16,96056
1024,67348
   
 
27656
.
0
39692
.
1024
67348
.
1024
13
4
.
115
67348
.
1024
)
(
)
(
)
(
)
(
)
(
2
2
1
1
0
2
















 n
y
y
x
y
x
y
y
y
SSR k
k
i 

 

 
 
56308
.
11
39692
.
1024
96000
.
1035
13
4
.
115
96000
.
1035
)
(
)
(
2
2
2
2











 n
y
y
y
y
SST i
 
28652
.
11
67348
.
1024
96000
.
1035
)
(
)
(
)
(
)
(
)
( 1
1
0
2
2









 



 y
x
y
x
y
y
y
y
SSE k
k
i
i 

 

x3 = b0 + b1.x1 + b2.x2 + u
02392
.
0
56308
.
11
27656
.
0
2



SST
SSR
R 02450
.
1
02392
.
0
1
1
1
1
2





R
VIF
For x3,

Example 10
182
Cor y x1 x2
y 1 0,98181 0,49287
x1 0,98181 1 0,37841
x2 0,49287 0,37841 1

Example 10
183
i ≠ 1 β X'Y Β(X'Y)
0 4,48353 206 923.60688
2 0,01132 77177 873.86423
1797.47111
   
 
03111
.
100
44
.
1697
47111
.
1797
25
206
47111
.
1797
)
(
)
(
)
(
)
(
)
(
2
2
1
1
0
2
















 n
y
y
x
y
x
y
y
y
SSR k
k
i 

 

 
 
56
.
698
44
.
1697
2396
25
206
2396
)
(
)
(
2
2
2
2











 n
y
y
y
y
SST i
 
52889
.
598
47111
.
1797
2396
)
(
)
(
)
(
)
(
)
( 1
1
0
2
2









 



 y
x
y
x
y
y
y
y
SSE k
k
i
i 

 

x1 = b0 + b2.x2 + u
14320
.
0
56
.
698
03111
.
100
2



SST
SSR
R 16713
.
1
14320
.
0
1
1
1
1
2





R
VIF
For x1,

Example 10
184
i ≠ 1 β X'Y Β(X'Y)
0 227,55165 8294 1887313,3777
1 12,64664 77177 976030,0846
2863343,4623
   
 
0223
.
111726
44
.
2751617
4623
.
2863343
25
8294
4623
.
2863343
)
(
)
(
)
(
)
(
)
(
2
2
1
1
0
2
















 n
y
y
x
y
x
y
y
y
SSR k
k
i 

 

 
 
56
.
780230
44
.
2751617
3531848
25
8294
3531848
)
(
)
(
2
2
2
2











 n
y
y
y
y
SST i
 
5377
.
668504
4623
.
2863343
3531848
)
(
)
(
)
(
)
(
)
( 1
1
0
2
2









 



 y
x
y
x
y
y
y
y
SSE k
k
i
i 

 

x2 = b0 + b1.x1 + u
14320
.
0
56
.
780230
0223
.
111726
2



SST
SSR
R 16713
.
1
14320
.
0
1
1
1
1
2





R
VIF
For x2,

Multicollinearity consequences:
 The β coefficient estimates will still remain unbiased
 Estimates will still be centered around the true values.
 R2 will be high but the individual coefficients will have high standard errors
 The variances and standard errors of the estimates will increase
 Harder to distinguish the effect of one variable from the effect of another, so much
more likely to make large errors in estimating the βs than without multicollinearity.
 As a result, the estimated coefficients, although still unbiased, now come from
distributions with much larger variances and, therefore, larger standard errors.
 The computed t-scores will fall.
 Variance and standard error are increased.
 A relatively high R2 in an equation with few significant t statistics.
 Thus confidence intervals for the estimates will be very wide, and significance
tests might therefore give inappropriate conclusions.
185

Multicollinearity consequences:
 Estimates will become very sensitive to changes in specification.
 The addition or deletion of an explanatory variable or of a few observations will
often cause major changes in the values of the βs when significant
multicollinearity exists
 For example, if you drop a variable, even one that appears to be statistically
insignificant, the coefficients of the remaining variables in the equation sometimes
will change dramatically
 This is again because with multicollinearity, it is much harder to distinguish the
effect of one variable from the effect of another (holding all else constant)
 The overall fit of the equation and the estimation of the coefficients of
nonmulticollinear variables will be largely unaffected.
 If the multicollinearity occurs in the population as well as the sample, then the
predictive power of the model is unaffected.
186

Methods to correct or remedy multicollinearity
Do nothing:
a. Multicollinearity will not necessarily reduce the t-scores enough to
make them statistically insignificant and/or change the estimated
coefficients to make them differ from expectations
b. the deletion of a multicollinear variable that belongs in an equation
will cause specification bias
Drop a redundant variable:
a. Viable strategy when two variables measure essentially the same
thing
b. Always use theory as the basis for this decision!
187

Methods to correct or remedy multicollinearity
Increase the sample size:
a. This is frequently impossible but a useful alternative to be
considered if feasible
b. The idea is that the larger sample normally will reduce the variance
of the estimated coefficients, diminishing the impact of the
multicollinearity
188

189
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