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Ekonometrika
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Sebagian Materi dapat di download di
ariefyulianto.wordpress.com Software dapat di download di
uap.unnes.ac.id
Konsep dan Aplikasi Teori Ekonomi melaluiPendekatan Kuantitatif
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Referensi1. Damodar N Gujarati. Basic econometrics.
Copyrighted Material. Fourth Edition.2. Damodar N Gujarati. 2006. Dasar-DasarEkonometrika. Jakarta : Penerbit Erlangga.
3. Rainer Winkelmann. 2008. Econometric
Analysis of Count Data. Fifth edition. BerlinHeidelberg : Springer-Verlag
4. Sarwoko. 2008. Dasar-Dasar Ekonometrika.Yogyakarta : Penerbit Andi
5. Badi H. Baltagi. 2008. Econometrics. BerlinHeidelberg : Springer-Verlag
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Kontrak (1)Metode Pembelajaran
Agar dicapai hasil pengajaran yang optimal, maka pada mata kuliah inidigunakan kombinasi metode pembelajaran ceramah dan diskusi didalam kelas, serta observasi mandiri di luar kelas (lapangan).
Sistem Penilaian
Penilaian atas keberhasilan mahasiswa dalam mengikuti dan memahamimateri pada mata kuliah ini didasarkan penilaian selama prosesperkuliahan dan nilai ujian, dengan komposisi sebagai berikut:
a. nilai tugas individu/kelompok, nilai presensi bobot 1
b. nilai mid test bobot 2
c. nilai ujian: bobot 3
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Kontrak (2)Tugas
Tugas pada mata kuliah ini dapat bersifat tugas individu atau tugaskelompok, dan pemberian tugas oleh dosen dilakukan pada saatperkuliahan. Tidak ada toleransi terhadap keterlambatan penyerahan/pengumpulan tugas, kecuali ada alasan yang adapatdipertanggungjawabkan.
Persyaratan Mengikuti KuliahSesuai dengan Tata Tertib Mengikuti Kuliah yang ditetepkan oleh UNNES.
Telah membaca dan membawa sekurang-kurangnya buku referensi utamapada setiap perkuliahan.
Lain-lain:
Toleransi keterlambatan untuk dosen dan mahasiswa adalah 30 menitdari jadual dan yang masuk ke kelas terakhir adalah dosen
Alat komunikasi mahasiswa dimatikan selama perkuliahan
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1. WHAT IS ECONOMETRICS
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econometricsmeans economic measurement
. . . econometrics may be defined as thequantitative analysis of actual economicphenomena based on the concurrent
development of theory and observation, relatedby appropriate methods of inference
Econometrics is concerned with the empirical
determination of economic laws.
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WHY A SEPARATE DISCIPLINE?econometrics is an amalgam of economic theory (makes
statements or hypotheses that are mostly qualitative innature), mathematical economics (to express economictheory in mathematical form (equations) without regard tomeasurability or empirical verification of the theory),economic statistics (collecting, processing, and presentingeconomic data in the form of charts and tables), andmathematical statistics (provides many tools used in thetrade, the econometrician often needs special methods inview of the unique nature of most economic data, namely,
that the data are not generated as the result of a controlledexperiment)
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METHODOLOGY OFECONOMETRICS
1. Statement of theory or hypothesis.
2. Specification of the mathematical model of thetheory3. Specification of the statistical, or econometric,
model
4. Obtaining the data5. Estimation of the parameters of the
econometric model
6. Hypothesis testing7. Forecasting or prediction8. Using the model for control or policy purposes
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To illustrate the preceding steps
1.Statement of Theory or Hypothesis
The fundamental psychological law . . . is
that men [women] are disposed, as a ruleand on average, to increase theirconsumption as their income increases, but
not as much as the increase in their incomemarginal propensity to consume (MPC)
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2. Specification of the Mathematical Model of
Consumption
Y= 1 + 2X0
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3. Specification of the Econometric Model of
Consumption Mathematical Model are exactor deterministic
relationship between consumption and income.But relationships between economic variablesare generally inexact
Y= 1 + 2X+ u(I.3.2)
where u, known as the disturbance, or error,term, is a random (stochastic) variable thathas well-defined probabilistic properties.
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4. Obtaining Data
To estimate the econometric model given
in (I.3.2), that is, to obtain the numericalvalues of 1 and 2, we need data
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5. Estimation of the Econometric Model
For now, note that the statistical techniqueof regression analysis is the main toolused to obtain the estimates
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Y = 184.08 + 0.7064Xi
The hat on the Yindicates that it is anestimate.11 The estimated consumptionfunction (i.e., regression line)
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6. Hypothesis Testing
Statistical inference (hypothesistesting).
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7. Forecasting or Prediction
To illustrate, suppose we want to predictthe mean consumption expenditure for1997. The GDP value for 1997 was 7269.8billion dollars
Y1997 = 184.0779 + 0.7064 (7269.8) =
4951.3167
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8. Use of the Model for Control or Policy
Purposes
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The Eight Components of
Integrated Service Management1. Product Elements
2. Place, Cyberspace, and Time3. Process
4. Productivity and Quality
5. People
6. Promotion and Education
7. Physical Evidence
8. Price and Other User Outlays
Principles of service marketing and management.lovelook, wright
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Marketing management (Philip
Kotler twelfth edition
Product is the first and most importantelement of the marketing mix. Productstrategy calls for making coordinateddecisions on product mixes, product lines,brands, and packaging and labeling.
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2. THE NATURE OFREGRESSION ANALYSIS
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Anatomy of econometric modeling
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Measurement Scales of
Variables Ratio Scale For a variable X, taking two values,
X1 and X2, the ratio X1/X2 and the distance (X2 X1) are meaningful quantities
Interval Scale the distance between two timeperiods, say (20001995) is meaningful, but not
the ratio of two time periods (2000/1995) Ordinal Scale Examples are grading systems(A, B, C grades) or income class (upper, middle,lower).
Nominal Scale Variables such as gender (male,female) and marital status (married, unmarried,divorced, separated) simply denote categories
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TWO-VARIABLE REGRESSION
ANALYSIS:SOME BASIC IDEAS
the simplest possible regression analysis,
namely, the bivariate, or twovariable,regression in which the dependent variable(the regressand) is related to a single
explanatory variable (the regressor)
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A HYPOTHETICAL EXAMPLE
in the table refer to a total population of 60 families in ahypothetical community and their weekly income (X) and weeklyconsumption expenditure (Y), both in dollars. The 60 families aredivided into 10 income groups (from $80 to $260) and the weeklyexpenditures of each family in the various groups are as shown in
the table
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E
(Y
|Xi
) = 1 + 2Xi
where 1 and 2 are unknown but fixed parameters known as theregression coefficients; 1 and 2 are also known as intercept andslope coefficients, respectively. Equation (2.2.1) itself is known as thelinear population regression function. Some alternative expressions
used in the literature are linear population regression modelor simplylinear population regression
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THE MEANING OF THE TERMLINEAR
Linearity in the Variables (a regression function such as E(Y| Xi) =1 + 2X
2i is not a linear function because the variable Xappears
with a power or index of 2. Linearity in the Parameters (E(Y| Xi) = 1 + 2X
2i is a linear (in the
parameter) regression model ; E(Y| Xi) = 1 + 32 x2 , which is
nonlinear in the parameter 2)
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STOCHASTIC SPECIFICATION OF
population regression function (PRF)
family consumption expenditure on the average increases, therelationship between an individual familys consumptionexpenditure and a given level of income?
where the deviation uiis an unobservable random variable takingpositive or negative values. Technically, uiis known as the
stochastic disturbance or stochastic error term.
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THE SIGNIFICANCE OF THE STOCHASTIC
DISTURBANCE TERM (1)
1. Vagueness of theory (The theory, if any, determining the behaviorof Ymay be, and often is, incomplete)
2. Unavailability of data (family wealth as an explanatory variable inaddition to the income variable to explain family consumptionexpenditure. But unfortunately, information on family wealthgenerally is not available
3. Core variables versus peripheral variables (Assume in ourconsumptionincome example that besides income X1, the numberof children per family X2, sex X3, religion X4, education X5, andgeographical region X6 also affect consumption expenditure
4. Intrinsic randomness in human behavior5. Poor proxy variables (The disturbance term umay in this case
then also represent the errors of measurement)
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THE SIGNIFICANCE OF THE STOCHASTIC
DISTURBANCE TERM (2)
1. Principle of parsimony (we would like to keep our regressionmodel as simple as possible
2. Wrong functional form (we do not know the form of thefunctional relationship between the regressand - Dependentvariable and the regressors - independent variable )
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THE SAMPLE REGRESSION FUNCTION (SRF)
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TWO-VARIABLE REGRESSION MODEL: THE
PROBLEM OF ESTIMATION (ordinary least square)
the method of least squares has some very attractivestatistical properties that have made it one of the most
powerful and popular methods of regression analysis
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Sering ditemukan pada data cross section
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Sering ditemukan pada data timeseries
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THE COEFFICIENT OF DETERMINATION r2:
A MEASURE OF GOODNESS OF FIT
The coefficient of determination r2 (two-variable case) or R2(multiple regression) is a summary measure that tells how
well the sample regression line fits the data.
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b
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ANOVAb
8552,727 1 8552,727 202,868 ,000a
337,273 8 42,159
8890,000 9
Regression
Residual
Total
Model
1
Sum of
Squares df Mean Square F Sig.
Predictors: (Constant), Pendapatana.
Dependent Variable: Konsumsib.
Coefficientsa
24,455 6,414 3,813 ,005
,509 ,036 ,981 14,243 ,000
(Constant)
Pendapatan
Model
1
B Std. Error
Unstandardized
Coefficients
Beta
Standardi
zed
Coefficien
ts
t Sig.
Dependent Variable: Konsumsia.
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Notes Alasan menggunakan adjusted R2 karena nilai
R2 bias, setiap tambahan satu variabel padavariabel independent akan meningkat tidakpeduli variabel tersebut berpengaruh signifikanatau tidak
Alasan menggunakan standarized beta mampumengeliminasi perbedaan unit/ukuran padavariabel independent (butir, ekor) namun tidakdapat diketahui multikolinieritas (korelasi antarvar bebas), nilai beta tidak dapatdiinterpretasikan
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CLASSICAL NORMAL LINEAR REGRESSION
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CLASSICAL NORMAL LINEAR REGRESSION
MODEL (CNLRM)
Using the method of OLS we were able to
estimate the parameters 1, 2, and 2.Under the assumptions of the classicallinear regression model(CLRM), we were
able to show that the estimators of theseparameters, 1, 2, and 2,
TWO-VARIABLE REGRESSION: INTERVAL
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TWO-VARIABLE REGRESSION: INTERVAL
ESTIMATION AND HYPOTHESIS TESTING
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Asumsi Klasik Model regresi linier : terspesifikasi benar dan
error term additif
Nilai rata-rata yang diharapkan disturbance errorterm = 0
Kovarian distrubance dengan x = nol
Varian dari variabel residu, disturbance adalahsama atau homokedastisitas Tidak ada otokorelasi antar variabel disturbance Tidak ada korelasi sempurna antar variabel
bebas Variabel error term berdistribusi normal
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Type kesalahanHipotesis o Menerima Ho Menolak Ho
Jika Ho benar Keputusan tepat Kesalahan jenis I
Jika Ho salah Kesalahan jenis II Keputusan tepat
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HYPOTHESIS TESTING:
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THE CONFIDENCE-INTERVAL APPROACH
One-Sided or One-Tail Test Sometimes
we have a strong a priori or theoreticalexpectation (or expectations based onsome previous empirical work) that thealternative hypothesis is one-sided orunidirectional rather than two-sided, as
just discussed. Thus, for ourconsumptionincome example, one couldpostulate that H0: 2 0.3 and H1: 2 >0.3
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MULTICOLLINEARITY:
WHAT HAPPENS IFTHE REGRESSORS
ARE CORRELATED?
What is the nature of
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multicollinearity Model regresi yang baik, seharusnya tidak
terjadi korelasi diantara variabelindependen.
Jika berkorelasi maka variabel tidak
ortogonal (korelasi antar variabelindependent = 0)
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Ciri-Ciri Multikolinieritas (Ghozali,
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2005) Nilai R square yang dihasilkan dari estimasi
model regresi tinggi, namun secara individualvariabel independent banyak yang tidaksignifikan -> dependen
Antar variabel independent memiliki korelasi>0,9
Setiap variabel independent yang dijelaskanoleh variabel independet lainnya. Output nilaitolerance rendah (10
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AUTOCORRELATION:
WHAT HAPPENS IFTHE ERROR TERMS ARE
CORRELATED?
three types of data
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yp
(1) cross section
(2) time series(3) combination of cross section and time
series
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shows a cyclical pattern
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y p
suggests an upward or downward
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linear trend in the disturbances
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indicates no systematic pattern
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nonautocorrelation
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Korelasi
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Korelasi antara x(t) dan y(t) dinamakan
dengan cross-correlation, dirumuskandengan
dytxtytxtC
atau
dtyxtytxtC
xy
xy
)()()()()(
)()()()()(
==
+==
Auto-korelasi
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Korelasi x(t) dengan dirinya sendiri disebut
auto-korelasi
dtxxtxtxtCxx )()()()()( ==
Korelasi
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Contoh
1
t0 1
h(t)
1
t1.5 2.5
x(t)
dtthptxpCxh )()()( =
Korelasih(t)x(t)
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1. Untuk 1.5+p>1 atau p>-0.5
1
t01
h(t)
1.5+p 2.5+p
1
t
x(t)
0)( =pCxh
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Korelasi
1x(t p) h(t)
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3. Untuk 1.5+p1, atau -1.5
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Korelasih(t)x(t)
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1. Untuk 1+p
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Korelasi
1
x(t) h(t-p)
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1
tp 1+p4. Untuk p>2.5
0)(=
pCxh
1
p
y(p)
2.50.5
-p+2.5p-0.5
Autokorelasi
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1
t1+pp
h(t-p)h(t)
1. Untuk 0
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1
t1+pp
2. Untuk 0>p>-1, karena p negatif, maka geser kiri
[ ]ppC
tdtpC
dtthpthpC
hh
p
p
hh
hh
+=
==
=
+
+
1)(
1.1)(
)()()(
1
0
1
0
Autokorelasi
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1
p
y(p)
-1 +1
1+p 1-p
3. Untuk p>1 dan p
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dtxxtCxx )()()(
)()()()0( tCC xxxx
)()(
)()(
txty
tytx
dxtytC
dytxtC
yx
xy
)()()(
)()()(
=
=
( )
( )
( ) )()()(
)()()(
)()()()(
)()()(
)()()()(
tztytx
tztytx
tztxtytx
tztytx
txtytytx
=
+
=+
)()( tCtC yxxy =
ILUSTRASI ANALISIS
REGRESI
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REGRESIApakah Skor Tes Masuk dan Peringkat kelas di
SMU mempengaruhi Nilai Mutu Rata rata
Mahasiswa Tingkat Pertama ?
Variabel Dependen :
NMR (Y)Variabel Independen :
Skor Tes (X1)
Peringkat (X2)
ILUSTRASI ANALISIS
REGRESI
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REGRESI
NMR Skor Tes Peringkat
1.93 565.00 3.002.55 525.00 2.00
1.72 477.00 1.00
2.48 555.00 1.00
2.87 502.00 1.001.87 469.00 3.00
1.34 517.00 4.00
3.03 555.00 1.00
2.54 576.00 2.002.34 559.00 2.00
NMR Skor Tes Peringkat
1.40 574.00 8.00
1.45 578.00 4.001.72 548.00 8.00
3.80 656.00 1.00
2.13 688.00 5.00
1.81 465.00 6.002.33 661.00 1.00
2.53 477.00 1.00
2.04 490.00 2.00
3.20 524.00 2.00
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HASIL ANALISIS
Regression
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Model Summaryb
.691a .478 .417 .4915 2.254
Model1
R R Square
Adjusted
R Square
Std. Error of
the Estimate
Durbin-W
atson
Predictors: (Constant), PERINGKA, SKORTESa.Dependent Variable: NMRb.
ANOVAb
3.762 2 1.881 7.786 .004a
4.107 17 .242
7.869 19
Regression
Residual
Total
Model
1
Sum of
Squares df Mean Square F Sig.
Predictors: (Constant), PERINGKA, SKORTESa.
Dependent Variable: NMRb.
Coefficientsa
1.269 .978 1.298 .212
2.769E-03 .002 .275 1.568 .135 .998 1.002
-.184 .050 -.648 -3.692 .002 .998 1.002
(Constant)
SKORTES
PERINGKA
Model
1
B Std. Error
Unstandardized
Coefficients
Beta
Standardi
zed
Coefficien
ts
t Sig. Tolerance VIF
Collinearity Statistics
Dependent Variable: NMRa.
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PEMERIKSAAN ASUMSI
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2. ASUMSI AUTOKORELASI
Diperoleh nilai d = 2.254
Kaidah Uji Durbin Watson : Disimpulkan tidak ada autokorelasi biladu < d < 4 du, Nilai du dapat dilihat di Tabel
Dengan n = 20 dan k (banyak variable bebas) = 2, diperoleh nilai du = 1.54
dan 4 du = 4 1.54 = 2.46
Karena du = 1.54 < d = 2.254 < 4 du = 2.46 maka dapat diterima bahwa asumsi
nonautokorelasi terpenuhi
Model Summaryb
.691a .478 .417 .4915 2.254
Model1
R R Square
Adjusted
R Square
Std. Error of
the Estimate
Durbin-W
atson
Predictors: (Constant), PERINGKA, SKORTESa.
Dependent Variable: NMRb.
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PEMERIKSAAN ASUMSI
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4. ASUMSI
HETEROSKEDASTISITAS
Plotkan residual terstudentkan dengannilai dugaan.
a. Pilih Graphs > Scatter > Simple.
b. Pilih Define
Pilih Stundentized Residual sebagai Y
axisPilih Unstundardizedpredicted value sebagai X axis
Klik OKPlot antara residual terstudentkan
dengan nilai dugaan berpola
acak, sehingga asumsi
homoskedastisitas terpenuhi
Unstandardized Predicted Value
3.02.52.01.51.0
Stud
entizedResidual
3
2
1
0
-1
-2
INTERPRETASI
VALIDASI MODEL
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VALIDASI MODEL
Koefisien determinasi (R2) = 0.478
Artinya kontribusi pengaruh skor tes dan peringkat terhadap nilai muturata-rata sebesar 47.8%. Sedang sisanya dipengaruhi oleh variabellain yang belum ada dalam model
Bila kita melakukan prediksi besarnya NMR berdasar skor tes danperigkat, maka tingkat akurasinya sebesar 47.8%
Uji F melalui ANOVA Regresi menghailkan p = 0.004
Uji koefisien regresi secara simultan signifikan
Uji t menghasilkan p untuk skor tes dan peringkat masing masing0.135 dan 0.002. Artinya hanya peringkat yang berpengaruhsignifikan terhadap besarnya NMR
INTERPRETASI
Model hasil regresi
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Model hasil regresi
NMR = 1.269 + 0.002769 Skor tes 0.184 Peringkat
1. Penjelasan terhadap fenomenaVariabel yang berpengaruh secara signifikan adalah peringkatdengan koefisien regresi 0.184
Artinya semakin kecil peringkat maka semakin tinggi NMR.
Pada keadaan Skor tes konstan, jika Peringkat meningkat 1tingkat maka NMR akan turun sebesar 0.184
INTERPRETASI
2 Prediksi
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2. Prediksi
Misal terdapat seorang anak dengan Skor tes 550 denganperingkat 4, maka berapa NMR nya?
NMR = 1.269 + 0.002769 (550) 0.184 (4)
= 2.05
Prediksi NMR adalah 2.05
Tingkat akurasi dari hasil prediksi ini adalah sebesar 47.8% (relatifrendah), akan tetapi bersifat general (karena nilai p untuk uji Fpada ANOVA sebesar 0.004
INTERPRETASI
3 Faktor determinan
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3. Faktor determinan
ZNMR = 0.275 ZSkor tes- 0.648 Zperingkat
Variabel yang berpengaruh paling kuat terhadap NMR adalah
peringkat, kemudian Skor tes. (Koefisien standardize Beta terbesarberarti pengaruhnya paling kuat, seandainya seluruh variabelsignifikan). Dalam contoh ini yang signifikan hanya peringkat,sehingga yang berpengaruh secara bermakna terhadap NMR hanyaperingkat.
Coefficients a
1.269 .978 1.298 .212
2.769E-03 .002 .275 1.568 .135 .998 1.002
-.184 .050 -.648 -3.692 .002 .998 1.002
(Constant)
SKORTES
PERINGKA
Model
1
B Std. Error
Unstandardized
Coefficients
Beta
Standardi
zed
Coefficien
ts
t Sig. Tolerance VIF
Collinearity Statistics
Dependent Variable: NMRa.
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HETEROSCEDASTICITY
WHAT HAPPENS IF THE
ERROR VARIANCE IS
NONCONSTANT?
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THE CLASSICAL LINEAR
REGRESSION MODELPRF: Yi = 1 + 2Xi + ui It shows that Yi
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PRF: Yi= 1 + 2Xi+ ui . It shows that Yidepends on both Xiand ui. Therefore,
unless we are specific about how Xiand uiare created or generated, there is no waywe can make any statistical inference about
the Yiand also, as we shall see, about 1and 2. Thus, the assumptions made aboutthe Xivariable(s) and the error term are
extremely critical to the valid interpretation ofthe regression estimates
There are several reasons why the variances of ui
may be variable, some of which are as follows
Following the error-learning models
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As incomes grow, people have more discretionary income2 andhence more scope for choice about the disposition of their income.Hence, 2iis likely to increase with income
As data collecting techniques improve, 2iis likely to decrease Heteroscedasticity can also arise as a result of the presence of
outliers the regression model is correctly specified (ex demand function for a
commodity, if we do not include the prices of commoditiescomplementary to or competing with the commodity in question (theomitted variable bias)
Another source of heteroscedasticity is skewness in the distributionof one or more regressors included in the model
There are several reasons why the variances of ui
may be variable, some of which are as follows
Another source of heteroscedasticity is skewness in the
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ydistribution of one or more regressors included in themodel. Examples are economic variables such asincome, wealth, and education. It is well known that thedistribution of income and wealth in most societies isuneven, with the bulk of the income and wealth beingowned by a few at the top.
Heteroscedasticity can also arise because of (1)incorrect data transformation (e.g., ratio or first differencetransformations) and (2) incorrect functional form (e.g.,linear versus loglinear models)
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what happens to the regression results if theobservations for Chile are dropped from the
analysis
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DETECTION OF
HETEROSCEDASTICITY as in the case of multicollinearity, there are
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as t e case o u t co ea ty, t e e a eno hard-and-fast rules for detectingheteroscedasticity, only a few rules ofthumb (need most economic
investigations. In this respect theeconometrician differs from scientists infields such as agriculture and biology,
where researchers have a good deal ofcontrol over their subjects)
Park Test
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Glejser Test
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Rank spearman
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DUMMY VARIABLEREGRESSION MODELS
model is based on several simplifyingassumptions, which are as follows
The regression model is linear in the parameters The values of the regressors, the Xs, are fixed in repeated
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e a ues o t e eg esso s, t e s, a e ed epeatedsampling.
For given Xs, the mean value of the disturbance uiis zero For given Xs, there is no autocorrelation in the disturbances If the Xs are stochastic, the disturbance term and the (stochastic) Xs are independent or at least uncorrelated The number of observations must be greater than the number of
regressors There must be sufficient variability in the values taken by the
regressors. The regression model is correctly specified There is no exact linear relationship (i.e., multicollinearity) in the
regressors. The stochastic (disturbance) term uiis normally distributed.
four types of variables
ratio scale, interval scale, ordinal scale,
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and nominal scale
known as indicator variables,categorical variables, qualitative
variables, or dummy variables
THE NATURE OF DUMMY
VARIABLES In regression analysis the dependent variable, orregressand is frequently influenced not only by ratio
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regressand, is frequently influenced not only by ratioscale variables (e.g., income, output, prices, costs,
height, temperature) qualitative,or nominal scale, in nature, such as sex, race,
color, religion, nationality, geographical region, politicalupheavals, and party affiliation
As a matter of fact, a regression model may containregressors that are all exclusively dummy, or qualitative,in nature. Such models are called Analysis of Variance(ANOVA) models
Dummy Variables
Dummy variables refers to the technique ofusing a dichotomous variable (coded 0 or 1) to
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using a dichotomous variable (coded 0 or 1) to
represent the separate categories of a nominallevel measure.
The term dummy appears to refer to the factthat the presence of the trait indicated by thecode of 1 represents a factor or collection offactors that are not measurable by any bettermeans within the context of the analysis.
Coding of dummy Variables
Take for instance the race of the respondent
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Take for instance the race of the respondent
in a study of voter preferences Race coded white(0) or black(1)
There are a whole set of factors that are possiblydifferent, or even likely to be different, between voters of
different races
Income, socialization, experience of racial discrimination,
attitudes toward a variety of social issues, feelings ofpolitical efficacy, etc
Since we cannot measure all of those differenceswithin the confines of the study we are doing, weuse a dummy variable to capture these effects.
Multiple categories
Now picture race coded white(0), black(1),Hispanic(2), Asian(3) and Native American(4)
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Hispanic(2), Asian(3) and Native American(4)
If we put the variable race into a regressionequation, the results will be nonsense since thecoding implicitly required in regression assumesat least ordinal level data with approximately
equal differences between ordinal categories. Regression using a 3 (or more) categorynominal variable yields un-interpretable andmeaningless results.
Creating Dummy variables
The simple case of race is already coded correctly Race: coded 0 for white and 1 for black
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ace coded 0 o te a d o b ac Note the coding can be reversed and leads only to changes in sign
and direction of interpretation. The complex nominal version turns into 5 variables:
White; coded 1 for whites and 0 for non-whites
Black; coded 1 for blacks and 0 for non-blacks
Hispanic; coded 1 for Hispanics and 0 for non- Hispanics Asian; coded 1 for Asians and 0 for non- Asians
AmInd; coded 1 for native Americans and 0 for non-nativeAmericans
Regression with Dummy Variables
The dummy variable is then added the regressionmodel
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Interpretation of the dummy variable is usually quitestraightforward.
The intercept term represents the intercept for the omittedcategory
The slope coefficient for the dummy variable represents thechange in the intercept for the category coded 1 (blacks)
iiii eRaceBXBaY +++= ** 21
Regression with only a dummy
When we regress a variable on only thedummy variable we obtain the estimates
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dummy variable, we obtain the estimates
for the means of the depended variable.
ais the mean of Y for Whites and a+B1 isthe mean of Y for Blacks
iii eRaceBaY ++= *1
Omitting a category
When we have a single dummy variable, we have informationfor both categories in the model
Al t th t
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Also note that
White = 1 Black Thus having both a dummy for White and one for Blacks is
redundant.
As a result of this, we always omit one category, whoseintercept is the models intercept.
This omitted category is called the reference category
In the dichotomous case, the reference category is simply thecategory coded 0
When we have a series of dummies, you can see that the reference
category is both the omitted variable.
Suggestions for selecting the
reference category Make it a well defined group other is usually a
poor choice
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poor choice.
If there is some underlying ordinality in thecategories, select the highest or lowest categoryas the reference. (e.g. blue-collar, white-collar,
professional) It should have ample number of cases. The
modal category is often a good choice.
Multiple dummy Variables
The model for the full dummy variable schemefor race is:
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for race is:
Note that the dummy for White has beenomitted, and the intercept ais the intercept forWhites.
iii
iiii
eAmIndBAsianB
HispanicBBlackBXBaY
++
++++=
**
***
54
321
Tests of Significance
With dummy variables, the t tests testwhether the coefficient is different from the
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whether the coefficient is different from the
reference category, not whether it isdifferent from 0.
Thus if a= 50, and B1 = -45, the coefficientfor Blacks might not be significantlydifferent from 0, while Whites are
significantly different from 0
Interaction terms
When the research hypothesizes that differentcategories may have different responses on
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categories may have different responses on
other independent variables, we need to useinteraction terms
For example, race and income interact with each
other so that the relationship between incomeand ideology is different (stronger or weaker) forWhites than Blacks
Creating Interaction terms
To create an interaction term is easy Multiply the category * the independent variable The full model is thus:
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The full model is thus:
a is the intercept for Whites;
(a + B1) is the intercept for Blacks; B2 is the slope for Whites; and (B2 + B3) is the slope for Blacks t-tests for B1 and B3 are whether they are different than a and B2
iii eIncomeRaceBIncomeBRaceBaY ++++= )*(321
Non-Linear Models
Tractable non-linearity
Equation may be transformed to a linear
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Equation may be transformed to a linear
model.
Intractable non-linearity
No linear transform exists
Tractable Non-Linear Models
Several general Types
Polynomial
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Polynomial
Power Functions
Exponential Functions
Logarithmic Functions Trigonometric Functions
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Exponential and Logarithmic
Functions Common Growth Curve Formula
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Estimated with
Note that the error terms are now no longer
normally distributed!
iXb
i eaeY +=
iii ebXaLogY ++=
Logarithmic Functions
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Trigonometric Functions
Sine/Cosine functions
Fourier series
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Fourier series
Intractable Non-linearity
Occasionally we have models that wecannot transform to linear ones.
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For instance a logit model
Or an equilibrium system model( )XBeyP += 1
1)(
11 += tt YcbXY )(
Intractable Non-linearity
Models such as these must be estimatedby other means.
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y
We do, however, keep the criteria ofminimizing the squared error as our
means of determining the best model
Estimating Non-linear models
All methods of non-linear estimationrequire an iterative search for the best
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q
fitting parameter values.
They differ in how they modify and search
for those values that minimize the SSE.