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  Ekonometrika

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  • 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 Penilaian

Penilaian 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 Kuliah

Sesuai 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 single

  A 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

      

    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 available

      3. 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)

    • the method of least squares has some very attractive

      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”

    • The coefficient of determination r 2 (two-variable case) or R2

      

    (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

    • Alasan menggunakan adjusted R2 karena nilai

      R2 bias, setiap tambahan satu variabel pada variabel independent akan meningkat tidak peduli variabel tersebut berpengaruh signifikan atau tidak

    • Alasan menggunakan standarized beta mampu

      TWO-VARIABLE REGRESSION MODEL: THE PROBLEM OF ESTIMATION Recall the two-variable PRF CLASSICAL NORMAL LINEAR REGRESSION MODEL (CNLRM)

    • • Using the method of OLS we were able to

      estimate the parameters β1, β2, and σ2.

      Under the assumptions of the classical

      linear regression model (CLRM), we were

      TWO-VARIABLE REGRESSION: INTERVAL ESTIMATION AND HYPOTHESIS TESTING

      

    Asumsi Klasik

    • 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

    • • HYPOTHESIS TESTING: GENERAL COMMENTS (Is a

      given observation or finding compatible with some stated hypothesis or not?)

    • 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

      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

    • Two-Sided or Two-Tail Test To illustrate the confidence-

      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

    • Very often such a two-sided alternative hypothesis reflects

      

    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

    • 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-

    SIGNIFICANCE APPROACH

    • Testing the Significance of Regression Coefficients:

      The t Test

    • which gives the interval in which ˆ β2 will fall with 1 − α probability, given β2 = β*2. In the language of hypothesis

      MULTICOLLINEARITY: WHAT HAPPENS IF

    THE REGRESSORS

      What is the nature of

    multicollinearity

    • • Model regresi yang baik, seharusnya tidak

      terjadi korelasi diantara variabel independen.
    • Jika berkorelasi maka variabel tidak

      Ciri-Ciri Multikolinieritas (Ghozali, 2005)

    • 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

    THE NATURE OF MULTICOLLINEARITY

    • 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

      multicollinearity may be due to the

    following factors

    • 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

      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

    • 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
    shows a cyclical pattern

    suggests an upward or downward linear trend in the disturbances

      

    indicates that both linear and

    quadratic trend terms are present in the disturbances indicates no systematic pattern nonautocorrelation

    DETECTING AUTOCORRELATION

    • 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

      

    Menanggulangi autokorelasi

    • 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

      Korelasi

    • • Korelasi antara x(t) dan y(t) dinamakan

      dengan cross-correlation, dirumuskan dengan

      C ( t ) x ( t ) y ( t ) x ( ) y ( t ) d

         

        

      

    Auto-korelasi

    • • Korelasi x(t) dengan dirinya sendiri disebut

      auto-korelasi

       C ( t ) x ( t ) x ( t ) x ( ) x ( t ) d

            

      Korelasi

    • Contoh x(t) h(t)

      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+pC ( p ) h ( t p ) x ( t ) dt

       

      hx

      Korelasi

      1 t 1+p p x(t) h(t-p)

      

    2. Untuk 1+p&gt;1.5 dan 1+p&lt;2.5, atau 0.5&lt;p&lt;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&lt;p&lt;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 xxx ( 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

    • Masukkan data pada SPSS Data Editor • Pilih Analyze &gt; Regression &gt; Linear

      1. Pilih dependen Variable

      2. Pilih Independen Variables

      3. Pada pilihan Statistics, aktifkan : Collinearity Diagnostics Durbin Watson

    HASIL ANALISIS

    • Regression Model Summary

      b

      .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 Residual

    19

    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

    • .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.

        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 &gt; Scatter &gt; 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 model

        Bila 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

      • 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

        outliers

      • • the regression model is correctly specified (ex demand function for a

        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

        distribution of one or more regressors included in the model. Examples are economic variables such as

        income, 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.

        what happens to the regression results if the observations for Chile are dropped from the analysis

      • 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

        

      DETECTION OF

      HETEROSCEDASTICITY

      • • 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

        Park Test

        Glejser Test Rank spearman

        

      DUMMY VARIABLE

      REGRESSION MODELS model is based on several simplifying assumptions, which are as follows

      • 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

        

      four types of variables

      • ratio scale, interval scale, ordinal scale,

        and nominal scale
      • known as indicator variables,

        categorical variables, qualitative

      THE NATURE OF DUMMY

        

      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

      • 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

        Coding of dummy Variables

      • 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

        

      Multiple categories

      • Now picture race coded white(0), black(1),

        Hispanic(2), Asian(3) and Native American(4)

      • 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

        

      Creating Dummy variables

      • 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
        Regression with Dummy Variables

      • The dummy variable is then added the regression model
      • Interpretation of the dummy variable is usually quite

        

      i i i i

      Race e B

        X B a Y

         

        2

        1 Regression with only a dummy

      • When we regress a variable on only the dummy variable, we obtain the estimates for the means of the depended variable.

        Y a Race e * B    Omitting a category

      • 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

        Suggestions for selecting the

      reference category

      • 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,

        Multiple dummy Variables

      • The model for the full dummy variable scheme for race is:

        Y a B

        X B B * Black Hispanic *

      • i

             1 i 2 i 3 i

      • B Asian B AmInd e 4 i

         

      5 i i

        

      Tests of Significance

      • With dummy variables, the t tests test whether the coefficient is different from the reference category, not whether it is different from 0.

        

      Interaction terms

      • 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

        Creating Interaction terms

      • To create an interaction term is easy
        • – Multiply the category * the independent variable
        • – The full model is thus:

        Y a Race B Income * B B ( Race Income ) e i     1 i 2 3i

        Non-Linear Models

      • Tractable non-linearity
        • – Equation may be transformed to a linear model.

      • Intractable non-linearity

        Tractable Non-Linear Models

      • Several general Types
        • – Polynomial – Power Functions – Exponential Functions

        

      Polynomial Models

      • Linear

        Y a bX e i i i   

      • Parabolic
      • 2 Y a b

          

        X b X e i i i i     1 2 Power Functions

        • • Simple exponents of the Independent

          Variable

          X Y ab e   i i i Exponential and Logarithmic Functions

        • Common Growth Curve Formula

          Xb Y ae e

           

        i i

        • Estimated with

          Logarithmic Functions Trigonometric Functions

        • Sine/Cosine functions
        • Fourier series

          Intractable Non-linearity

        • • Occasionally we have models that we

          cannot transform to linear ones.
        • For instance a logit model

          Intractable Non-linearity

        • • Models such as these must be estimated

          by other means.
        • We do, however, keep the criteria of minimizing the squared error as our

          

        Estimating Non-linear models

        • All methods of non-linear estimation require an iterative search for the best fitting parameter values.
        • • They differ in how they modify and search

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