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Ekonometrika
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- Software dapat di download di uap.unnes.ac.id
- Konsep dan Aplikasi Teori Ekonomi melalui
Referensi 1. Damodar N Gujarati. Basic econometrics.
Copyrighted Material. Fourth Edition.
2. Damodar N Gujarati. 2006. Dasar-Dasar Ekonometrika. Jakarta : Penerbit Erlangga.
3. Rainer Winkelmann. 2008. Econometric Analysis of Count Data. Fifth edition. Berlin
Kontrak (1)
Metode Pembelajaran Agar dicapai hasil pengajaran yang optimal, maka pada mata kuliah ini digunakan kombinasi metode pembelajaran ceramah dan diskusi di dalam kelas, serta observasi mandiri di luar kelas (lapangan). Sistem PenilaianPenilaian atas keberhasilan mahasiswa dalam mengikuti dan memahami
materi pada mata kuliah ini didasarkan penilaian selama proses
Kontrak (2)
Tugas Tugas pada mata kuliah ini dapat bersifat tugas individu atau tugas kelompok, dan pemberian tugas oleh dosen dilakukan pada saat perkuliahan. Tidak ada toleransi terhadap keterlambatan penyerahan/ pengumpulan tugas, kecuali ada alasan yang adapat dipertanggungjawabkan. Persyaratan Mengikuti KuliahSesuai dengan Tata Tertib Mengikuti Kuliah yang ditetepkan oleh UNNES.
1. WHAT IS ECONOMETRICS
- econometrics means “economic measurement
- . . . econometrics may be defined as the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related
WHY A SEPARATE DISCIPLINE?
- econometrics is an amalgam of economic theory (makes statements or hypotheses that are mostly qualitative in nature), mathematical economics (to express economic theory in mathematical form (equations) without regard to measurability or empirical verification of the theory),
economic statistics (collecting, processing, and presenting
economic data in the form of charts and tables), and
METHODOLOGY OF ECONOMETRICS 1. Statement of theory or hypothesis.
2. Specification of the mathematical model of the theory
3. Specification of the statistical, or econometric, model
4. Obtaining the data
To illustrate the preceding steps
1.Statement of Theory or Hypothesis
The fundamental psychological law . . . is that men [women] are disposed, as a rule
2. Specification of the Mathematical Model of Consumption
- Y = β1 + β2X 0 < β2 < 1 (I.3.1)
3. Specification of the Econometric Model of Consumption
- Mathematical Model are exact or deterministic relationship between consumption and income. But relationships between economic variables are generally inexact
4. Obtaining Data
• To estimate the econometric model given
in (I.3.2), that is, to obtain the numerical
5. Estimation of the Econometric Model
• For now, note that the statistical technique
of regression analysis is the main tool used to obtain the estimates
- Yˆ = −184.08 + 0.7064Xi
- The hat on the Y indicates that it is an estimate.11 The estimated consumption function (i.e., regression line)
6. Hypothesis Testing
• Statistical inference (hypothesis
testing).
7. Forecasting or Prediction
- To illustrate, suppose we want to predict the mean consumption expenditure for 1997. The GDP value for 1997 was 7269.8
8. Use of the Model for Control or Policy
Purposes
The Eight Components of Integrated Service Management
1. Product Elements
2. Place, Cyberspace, and Time
3. Process
4. Productivity and Quality
5. People
6. Promotion and Education
Marketing management (Philip
Kotler twelfth edition
- Product is the first and most important element of the marketing mix. Product strategy calls for making coordinated
Initial public offering
- Emiten • Underwriter • Auditor • Size
2. THE NATURE OF
REGRESSION ANALYSIS
Anatomy of econometric modeling
THE MODERN INTERPRETATION
OF REGRESSION
Regression analysis is concerned with the study of the dependence of one variable, the dependent
variable, on one or more other variables, the explanatory variables,with a view to estimating
and/or predicting the (population) mean or average value of the former in terms of the known or fixed
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 time periods, say (2000–1995) is meaningful, but not the ratio of two time periods (2000/1995)
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 singleA HYPOTHETICAL EXAMPLE
THE MEANING OF THE TERM LINEAR
• Linearity in the Variables (a regression function such as E(Y | X ) =
i
β + β 1 2 2i X is not a linear function because the variable X appears with a power or index of 2.
(E(Y | Xi) = β + β X is a linear (in the
- Linearity in the Parameters 2i 1 2 2 parameter) regression model ; E(Y | Xi) = β + 3β x , which is 1 2 nonlinear in the parameter β ) 2
- the method of least squares has some very attractive
• The coefficient of determination r 2 (two-variable case) or R2
- Alasan menggunakan adjusted R2 karena nilai
- Alasan menggunakan standarized beta mampu
• Using the method of OLS we were able to
estimate the parameters β1, β2, and σ2.
Under the assumptions of the classical- Model regresi linier : terspesifikasi benar dan error term additif
- Nilai rata-rata yang diharapkan disturbance error term = 0
- Kovarian distrubance dengan x = nol
- Varian dari variabel residu, disturbance adalah
• HYPOTHESIS TESTING: GENERAL COMMENTS (Is a
- In the language of statistics, the stated hypothesis is known as the null hypothesis and is denoted by the symbol H0. The null hypothesis is usually tested against an alternative hypothesis (also known as maintained
- Two-Sided or Two-Tail Test To illustrate the confidence-
- Very often such a two-sided alternative hypothesis reflects
• One-Sided or One-Tail Test Sometimes
we have a strong a priori or theoretical expectation (or expectations based on some previous empirical work) that the alternative hypothesis is one-sided or- HYPOTHESIS TESTING: THE TEST-OF-
• Testing the Significance of Regression Coefficients:
- which gives the interval in which ˆ β2 will fall with 1 − α probability, given β2 = β*2. In the language of hypothesis
• Model regresi yang baik, seharusnya tidak
terjadi korelasi diantara variabel independen.- Jika berkorelasi maka variabel tidak
- Nilai R square yang dihasilkan dari estimasi model regresi tinggi, namun secara individual variabel independent banyak yang tidak signifikan -> dependen
- Antar variabel independent memiliki korelasi
- it meant the existence of a “perfect,” or exact, linear relationship among some or all explanatory variables of a regression model
- Yi = β0 + β1Xi + β2X2i + β3X3i + ui
- The data collection method employed, for example, sampling over a limited range of the values taken by the regressors in the population
- Constraints on the model or in the population
- correlation between members of series of observations ordered in time [as in time series data] or space [as in cross-sectional data]
- autocorrelation as “lag correlation of a given series with itself, lagged by a number of time units,’’ whereas he reserves the term serial
- Graphical Method
- Autokorelasi dalam konsep regresi linier berarti komponen error
berkorelasi berdasarkan urutan waktu (pada data timeseries) atau
urutan ruang (pada data cross-sectional). • Contoh data timeseries (terdapat urutan waktu) misalnya pengaruh
biaya iklan terhadap penjualan dari bulan januari hingga bulan desember. Sedangkan data cross-sectional adalah data yang tidak ada urutan waktu, misal pengaruh konsentrasi zat X terhadap kecepatan reaksi suatu senyawa kimia.- Untuk mendeteksi ada atau tidaknya autokorelasi, dapat dilakukan
- Beberapa uji statistik yang sering dipergunakan adalah uji Durbin-Watson atau uji dengan Run Test dan jika data observasi di atas 100 data sebaiknya menggunakan uji Lagrange Multiplier. Beberapa cara untuk menanggulangi masalah autokorelasi adalah dengan mentransformasikan data atau bisa juga dengan
mengubah model regresi ke dalam bentuk persamaan
• Korelasi antara x(t) dan y(t) dinamakan
dengan cross-correlation, dirumuskan dengan • Korelasi x(t) dengan dirinya sendiri disebut
auto-korelasi- Contoh x(t) h(t)
- Masukkan data pada SPSS Data Editor • Pilih Analyze > Regression > Linear
- Regression Model Summary
b
- .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: NMR a.
- Following the error-learning models
- As incomes grow, people have more discretionary income2 and
hence more scope for choice about the disposition of their income.
Hence, σ2i is likely to increase with income - As data collecting techniques improve, σ2i is likely to decrease
- Heteroscedasticity can also arise as a result of the presence of
• the regression model is correctly specified (ex demand function for a
• Another source of heteroscedasticity is skewness in the
distribution of one or more regressors included in the model. Examples are economic variables such asincome, wealth, and education. It is well known that the
distribution of income and wealth in most societies is uneven, with the bulk of the income and wealth being owned by a few at the top.- the problem of heteroscedasticity is likely to be more common in cross-sectional
than in time series data. In cross-sectional
data, one usually deals with members of a
• as in the case of multicollinearity, there are
no hard-and-fast rules for detecting heteroscedasticity, only a few rules of thumb (need most economic- The regression model is linear in the parameters
- The values of the regressors, the X’s, are fixed in repeated sampling.
- For given X’s, the mean value of the disturbance ui is zero
- For given X’s, there is no autocorrelation in the disturbances
• If the X’s are stochastic, the disturbance term and the (stochastic)
- X’s are independent or at least uncorrelated
• The number of observations must be greater than the number of
• ratio scale, interval scale, ordinal scale,
and nominal scale- known as indicator variables,
- In regression analysis the dependent variable, or regressand, is frequently influenced not only by ratio scale variables (e.g., income, output, prices, costs, height, temperature)
• qualitative,or nominal scale, in nature, such as sex, race,
color, religion, nationality, geographical region, political
upheavals, and party affiliation- Dummy variables refers to the technique of using a dichotomous variable (coded 0 or 1) to represent the separate categories of a nominal level measure.
- The term “dummy” appears to refer to the fact that the presence of the trait indicated by the
- 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 possibly
different, or even likely to be different, between voters of- Now picture race coded white(0), black(1),
- If we put the variable race into a regression equation, the results will be nonsense since the coding implicitly required in regression assumes at least ordinal level data – with approximately
- The simple case of race is already coded correctly
- – Race: coded 0 for white and 1 for black
- 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
- The dummy variable is then added the regression model
- Interpretation of the dummy variable is usually quite
- When we regress a variable on only the dummy variable, we obtain the estimates for the means of the depended variable.
- When we have a single dummy variable, we have information for both categories in the model
- 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, whose
- Make it a well defined group – other is usually a poor choice.
- If there is some underlying ordinality in the categories, select the highest or lowest category as the reference. (e.g. blue-collar, white-collar,
- The model for the full dummy variable scheme for race is:
- i
- B Asian B AmInd e 4 i
- With dummy variables, the t tests test whether the coefficient is different from the reference category, not whether it is different from 0.
- When the research hypothesizes that different categories may have different responses on other independent variables, we need to use interaction terms
- For example, race and income interact with each
- To create an interaction term is easy
- – Multiply the category * the independent variable
- – The full model is thus:
- Tractable non-linearity
- – Equation may be transformed to a linear model.
- Intractable non-linearity
- Several general Types
- – Polynomial – Power Functions – Exponential Functions
- Linear
- Parabolic 2 Y a b
• Simple exponents of the Independent
Variable- Common Growth Curve Formula
- Estimated with
- Sine/Cosine functions
- Fourier series
• Occasionally we have models that we
cannot transform to linear ones.- For instance a logit model
• Models such as these must be estimated
by other means.- We do, however, keep the criteria of minimizing the squared error as our
- All methods of non-linear estimation require an iterative search for the best fitting parameter values.
• They differ in how they modify and search
STOCHASTIC SPECIFICATION OF
population regression function (PRF)
family consumption expenditure on the average increases, the relationship between an individual family’s consumption expenditure and a given level of income? where the deviation ui is an unobservable random variable taking
THE SIGNIFICANCE OF THE STOCHASTIC
DISTURBANCE TERM (1)
1. Vagueness of theory (The theory, if any, determining the behavior
of Y may be, and often is, incomplete)2. Unavailability of data (family wealth as an explanatory variable in
addition to the income variable to explain family consumption
expenditure. But unfortunately, information on family wealth generally is not available3. Core variables versus peripheral variables (Assume in our
THE SIGNIFICANCE OF THE STOCHASTIC DISTURBANCE TERM (2)
1. Principle of parsimony (we would like to keep our regression model as simple as possible
2. Wrong functional form (we do not know the form of the functional relationship between the regressand - Dependent variable and the regressors - independent variable )
THE SAMPLE REGRESSION FUNCTION (SRF)
3. TWO-VARIABLE REGRESSION MODEL: THE PROBLEM OF ESTIMATION TWO-VARIABLE REGRESSION MODEL: THE PROBLEM OF ESTIMATION (ordinary least square)
statistical properties that have made it one of the most powerful and popular methods of regression analysis
Sering ditemukan pada data cross section
Sering ditemukan pada data timeseries
2 THE COEFFICIENT OF DETERMINATION r : A MEASURE OF “GOODNESS OF FIT”
(multiple regression) is a summary measure that tells how
well the sample regression line fits the data.
The fundamental psychological law . . . is that men [women] are
disposed, as a rule and on average, to increase their consumption
as their income increases, but not by as much as the increase in their
income,” that is, the marginal propensity to consume (MPC) is greater
than zero but less than one
Variables Entered/Removed b Pendapata n a , Enter Model 1 Variables Entered Variables Removed Method All requested variables entered. a.
Dependent Variable: Konsumsi b.
Model Summary
Model R R Square Adjusted R Square Std. Error of the Estimate R Square Change F Change df1 df2 Sig. F Change Change Statistics
ANOVA
b 8552,727 1 8552,727 202,868 ,000 a 337,273 8 42,159 8890,000 Residual 9 Regression Total Model 1 Sum of Squares df Mean Square F Sig.Predictors: (Constant), Pendapatan a. Dependent Variable: Konsumsi b.
Coefficients
a Unstandardized Standardi zed Coefficien
CONSUMPTION–INCOME RELATIONSHIP
IN THE UNITED STATES, 1982–1996
THE RELATIONSHIP BETWEEN EARNINGS AND EDUCATION
Notes
R2 bias, setiap tambahan satu variabel pada variabel independent akan meningkat tidak peduli variabel tersebut berpengaruh signifikan atau tidak
TWO-VARIABLE REGRESSION MODEL: THE PROBLEM OF ESTIMATION Recall the two-variable PRF CLASSICAL NORMAL LINEAR REGRESSION MODEL (CNLRM)
linear regression model (CLRM), we were
TWO-VARIABLE REGRESSION: INTERVAL ESTIMATION AND HYPOTHESIS TESTING
Asumsi Klasik
HYPOTHESIS TESTING: GENERAL COMMENTS
given observation or finding compatible with some stated hypothesis or not?)
Type kesalahan Hipotesis o Menerima Ho Menolak Ho
Jika Ho benar Keputusan tepat Kesalahan jenis I
Jika Ho salah Kesalahan jenis II Keputusan tepat
HYPOTHESIS TESTING: THE CONFIDENCE-INTERVAL APPROACH
interval approach, once again we revert to the consumption– income example. As we know, the estimated marginal
propensity to consume (MPC), ˆ β2, is 0.5091. Suppose we
postulate that H0: β2 = 0.3 ; H1: β2 = 0.3
the fact that we do not have a strong a priori or theoretical
expectation about the direction in which the alternative
HYPOTHESIS TESTING:
THE CONFIDENCE-INTERVAL APPROACH
SIGNIFICANCE APPROACH
The t Test
MULTICOLLINEARITY: WHAT HAPPENS IF
THE REGRESSORS
What is the nature of
multicollinearity
Ciri-Ciri Multikolinieritas (Ghozali, 2005)
THE NATURE OF MULTICOLLINEARITY
multicollinearity may be due to the
following factors
being sampled
Cara mengobati multikolinieritas
1. Menggabungkan data cross section dan time series
2. Keluarkan satu atau lebih variabel independen yang memp nilai korelasi tinggi (0,94%)
3. Transformasi variabel
AUTOCORRELATION:
WHAT HAPPENS IF
THE ERROR TERMS ARE
three types of data (1) cross section (2) time series
(3) combination of cross section and time
series
indicates that both linear and
quadratic trend terms are present in the disturbances indicates no systematic pattern nonautocorrelation
DETECTING AUTOCORRELATION
Menanggulangi autokorelasi
Korelasi
Korelasi
C ( t ) x ( t ) y ( t ) x ( ) y ( t ) d
Auto-korelasi
C ( t ) x ( t ) x ( t ) x ( ) x ( t ) d
Korelasi
1
1
1.5 2.5 t t 1 Korelasi
1 t 1.5+p 2.5+p 1 h(t)
1 t x(t)
1. Untuk 1.5+p>1 atau p>-0.5
Korelasi
1 t 2.5+p 1.5+p x(t-p) h(t)
2. Untuk 1.5+p<1 dan 1.5+p>0, atau -1.5<p<-0.5
Korelasi
1 2.5+p 1.5+p t x(t-p)
h(t)
3. Untuk 1.5+p<0 dan 2.5+p>1, atau -1.5<p<1.5
C dt t h p t x p xh
) ( ) ( ) ( Korelasi
1
t 2.5+p 1.5+p x(t-p) h(t)4. Untuk 2.5+p<0 atau p<-2.5
) (
C p
x(t) h(t) Korelasi
1
1
1.5
2.5 t p t 1+p C ( p ) h ( t p ) x ( t ) dt
hx
Korelasi
1 t 1+p p x(t) h(t-p)
2. Untuk 1+p>1.5 dan 1+p<2.5, atau 0.5<p<1.5
Korelasi
1 1+p p t x(t) h(t-p)
3. Untuk p<2.5 dan 1+p>2.5, atau 1.5<p<2.5
C dt t x p t h p hx
) ( ) ( ) ( Korelasi
1 t p 1+p x(t) h(t-p)
4. Untuk p>2.5
) (
C p Autokorelasi
1 t 1+p p h(t-p) h(t)
1. Untuk 0<p<1, maka
Autokorelasi
1 1+p p t h(t-p)
h(t)
2. Untuk 0>p>-1, karena p negatif, maka geser kiri
Autokorelasi
3. Untuk p>1 dan p<-1,
1 y(p)
) (
C p hh
Korelasi
C ( t ) C ( t ) xx xx x ( t ) x ( t ) C ( t ) x ( ) x ( t ) d xx
C ( ) C ( t ) xx xx x ( t ) y ( t ) y ( t ) x ( t )
C ( t ) x ( t ) y ( ) d x ( t ) y ( t ) xy
x ( t ) y ( t ) z ( t )
x ( t ) y ( t ) x ( t ) z ( t )
C ( t ) y ( t ) x ( ) d yx y ( t ) x ( t )
x ( t ) y ( t ) z ( t )
ILUSTRASI ANALISIS REGRESI
Apakah Skor Tes Masuk dan Peringkat kelas di SMU mempengaruhi Nilai Mutu Rata – rata Mahasiswa Tingkat Pertama ?
Variabel Dependen :
ILUSTRASI ANALISIS REGRESI NMR Skor Tes Peringkat 1.93 565.00
3.00 2.55 525.00 2.00 1.72 477.00 1.00 2.48 555.00
1.00 NMR Skor Tes Peringkat 1.40 574.00 8.00 1.45 578.00 4.00 1.72 548.00 8.00 3.80 656.00 1.00 2.13 688.00 5.00 1.81 465.00
6.00
LANGKAH -LANGKAH
1. Pilih dependen Variable
2. Pilih Independen Variables
3. Pada pilihan Statistics, aktifkan : Collinearity Diagnostics Durbin Watson
HASIL ANALISIS
.691 a .478 .417 .4915 2.254 Model 1 R R Square Adjusted R Square Std. Error of the Estimate Durbin-W atson Predictors: (Constant), PERINGKA, SKORTES a. Dependent Variable: NMR b.
ANOVA
b 3.762 2 1.881 7.786 .004 a 4.107 17 .242 7.869 Residual19
Regression Total Model 1 Sum of Squares df Mean Square F Sig.PEMERIKSAAN ASUMSI
1. ASUMSI NORMALITAS ERROR Hasil P-P plot menunjukkan pola garis lurus mendekati sudut 45 , sehingga asumsi normalitas sisaan terpenuhi 1.00 Dependent Variab le: NMR
PEMERIKSAAN ASUMSI
2. ASUMSI AUTOKORELASI 1 .691 .478 .417 .4915 2.254 Model R R Square R Square the Estimate atson a Model Summary Adjusted Std. Error of Durbin-W b Kaidah Uji Durbin Watson : Disimpulkan tidak ada autokorelasi bila Diperoleh nilai d = 2.254 b.
a. Dependent Variable: NMR Predictors: (Constant), PERINGKA, SKORTES
Collinearity Diagnostics Dimension a Model Eigenvalue Condition Index (Constant) SKORTES PERINGKA Variance Proportions PEMERIKSAAN ASUMSI 3. ASUMSI MULTIKOLINEARITAS Coefficients a 1.269 .978 1.298 .212 2.769E-03 .002 .275 1.568 .135 .998 1.002
PEMERIKSAAN ASUMSI
3
4. ASUMSI
2 nilai dugaan. Plotkan residual terstudentkan dengan HETEROSKEDASTISITAS 1 Pilih Stundentized Residual sebagai Y a. Pilih Graphs > Scatter > Simple. b. Pilih Define R d e l d u a e si -1 axis d e n tiz
INTERPRETASI
VALIDASI MODEL Koefisien determinasi (R 2 ) = 0.478
Artinya kontribusi pengaruh skor tes dan peringkat terhadap nilai mutu
rata-rata sebesar 47.8%. Sedang sisanya dipengaruhi oleh variabel lain yang belum ada dalam modelBila kita melakukan prediksi besarnya NMR berdasar skor tes dan perigkat, maka tingkat akurasinya sebesar 47.8%
INTERPRETASI Model hasil regresi NMR = 1.269 + 0.002769 Skor tes – 0.184 Peringkat
1. Penjelasan terhadap fenomena Variabel yang berpengaruh secara signifikan adalah peringkat dengan koefisien regresi – 0.184 Artinya semakin kecil peringkat maka semakin tinggi NMR.
INTERPRETASI
2. Prediksi Misal terdapat seorang anak dengan Skor tes 550 dengan peringkat 4, maka berapa NMR – nya? NMR = 1.269 + 0.002769 (550) – 0.184 (4)
= 2.05 Prediksi NMR adalah 2.05
INTERPRETASI
3. Faktor determinan Z = 0.275 Z - 0.648 Z NMR Skor tes peringkat Variabel yang berpengaruh paling kuat terhadap NMR adalah peringkat, kemudian Skor tes. (Koefisien standardize Beta terbesar berarti pengaruhnya paling kuat, seandainya seluruh variabel signifikan). Dalam contoh ini yang signifikan hanya peringkat, sehingga yang berpengaruh secara bermakna terhadap NMR hanya
HETEROSCEDASTICITY
THE CLASSICAL LINEAR
REGRESSION MODEL PRF: Yi = β1 + β2Xi + ui . It shows that Yi depends on both Xi and ui . Therefore, unless we are specific about how Xi and ui are created or generated, there is no way we can make any statistical inference about There are several reasons why the variances of ui may be variable, some of which are as follows
outliers
There are several reasons why the variances of ui may be variable, some of which are as follows
what happens to the regression results if the observations for Chile are dropped from the analysis
DETECTION OF
HETEROSCEDASTICITY
Park Test
Glejser Test Rank spearman
DUMMY VARIABLE
REGRESSION MODELS model is based on several simplifying assumptions, which are as follows
four types of variables
categorical variables, qualitative
THE NATURE OF DUMMY
VARIABLES
Dummy Variables
Coding of dummy Variables
Multiple categories
Hispanic(2), Asian(3) and Native American(4)
Creating Dummy variables
i i i i
Race e BX B a Y
2
1 Regression with only a dummy
Y a Race e * B Omitting a category
Suggestions for selecting the
reference category
Multiple dummy Variables
Y a B
X B B * Black Hispanic *
1 i 2 i 3 i
5 i i
Tests of Significance
Interaction terms
Creating Interaction terms
Y a Race B Income * B B ( Race Income ) e i 1 i 2 3 i
Non-Linear Models
Tractable Non-Linear Models
Polynomial Models
Y a bX e i i i
X b X e i i i i 1 2 Power Functions
X Y ab e i i i Exponential and Logarithmic Functions
Xb Y ae e
i i
Logarithmic Functions Trigonometric Functions
Intractable Non-linearity
Intractable Non-linearity
Estimating Non-linear models