Media Ekonometrika

8/7/2019 Media Ekonometrika

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Ekonometrika


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Sebagian Materi dapat di download di

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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.

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