Ekonometrika

Sebagian Materi dapat di download di ariefyulianto.wordpress.com Software dapat di download di uap.unnes.ac.id Konsep dan Aplikasi Teori Ekonomi melalui Pendekatan Kuantitatif

Referensi Damodar N Gujarati. Basic econometrics. Copyrighted Material. Fourth Edition. Damodar N Gujarati. 2006. Dasar-Dasar Ekonometrika. Jakarta : Penerbit Erlangga. Rainer Winkelmann. 2008. Econometric Analysis of Count Data. Fifth edition. Berlin Heidelberg : Springer-Verlag Sarwoko. 2008. Dasar-Dasar Ekonometrika. Yogyakarta : Penerbit Andi Badi H. Baltagi. 2008. Econometrics. Berlin Heidelberg : Springer-Verlag

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 perkuliahan 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

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. Telah membaca dan membawa sekurang-kurangnya buku referensi utama pada setiap perkuliahan. Lain-lain: Toleransi keterlambatan untuk dosen dan mahasiswa adalah 30 menit dari jadual dan yang masuk ke kelas terakhir adalah dosen Alat komunikasi mahasiswa dimatikan selama perkuliahan

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 by appropriate methods of inference Econometrics is concerned with the empirical determination of economic laws.

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 mathematical statistics (provides many tools used in the trade, the econometrician often needs special methods in view of the unique nature of most economic data, namely, that the data are not generated as the result of a controlled experiment)

METHODOLOGY OF ECONOMETRICS Statement of theory or hypothesis. Specification of the mathematical model of the theory Specification of the statistical, or econometric, model Obtaining the data Estimation of the parameters of the econometric model Hypothesis testing Forecasting or prediction Using the model for control or policy purposes

To illustrate the preceding steps Statement of Theory or Hypothesis 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 as much as the increase in their income marginal propensity to consume (MPC)

2. Specification of the Mathematical Model of Consumption Y = β1 + β2X 0 < β2 < 1 (I.3.1) where Y = consumption expenditure and X = income, and where β1 and β2, known as the parameters of the model, are, respectively, the intercept and slope coefficients.

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 Y = β1 + β2X + u (I.3.2) where u, known as the disturbance, or error, term, is a random (stochastic) variable that has well-defined probabilistic properties.

4. Obtaining Data To estimate the econometric model given in (I.3.2), that is, to obtain the numerical values of β1 and β2, we need data

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 billion dollars Yˆ1997 = −184.0779 + 0.7064 (7269.8) = 4951.3167

8. Use of the Model for Control or Policy Purposes

The Eight Components of Integrated Service Management Product Elements Place, Cyberspace, and Time Process Productivity and Quality People Promotion and Education Physical Evidence Price and Other User Outlays Principles of service marketing and management. lovelook, wright

Marketing management (Philip Kotler twelfth edition Product is the first and most important element of the marketing mix. Product strategy calls for making coordinated decisions on product mixes, product lines, brands, and packaging and labeling.

Initial public offering Emiten Underwriter Auditor Size Age

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 (in repeated sampling) values of the latter Contoh : how the average height of sons changes, given the fathers’ heigh ; Distribution in a hypothetical population of heights of boys measured at fixed ages

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) 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

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)

A HYPOTHETICAL EXAMPLE in the table refer to a total population of 60 families in a hypothetical community and their weekly income (X) and weekly consumption expenditure (Y), both in dollars. The 60 families are divided into 10 income groups (from $80 to $260) and the weekly expenditures of each family in the various groups are as shown in the table

E(Y | Xi) = β1 + β2Xi where β1 and β2 are unknown but fixed parameters known as the regression coefficients; β1 and β2 are also known as intercept and slope coefficients, respectively. Equation (2.2.1) itself is known as the linear population regression function. Some alternative expressions used in the literature are linear population regression model or simply linear population regression

THE MEANING OF THE TERM LINEAR Linearity in the Variables (a regression function such as E(Y | Xi) = β1 + β2X2i is not a linear function because the variable X appears with a power or index of 2. Linearity in the Parameters (E(Y | Xi) = β1 + β2X2i is a linear (in the parameter) regression model ; E(Y | Xi) = β1 + 3β2 x2 , which is nonlinear in the parameter β2)

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 positive or negative values. Technically, ui is known as the stochastic disturbance or stochastic error term.

THE SIGNIFICANCE OF THE STOCHASTIC DISTURBANCE TERM (1) Vagueness of theory (The theory, if any, determining the behavior of Y may be, and often is, incomplete) 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 Core variables versus peripheral variables (Assume in our consumptionincome example that besides income X1, the number of children per family X2, sex X3, religion X4, education X5, and geographical region X6 also affect consumption expenditure Intrinsic randomness in human behavior Poor proxy variables (The disturbance term u may in this case then also represent the errors of measurement)

THE SIGNIFICANCE OF THE STOCHASTIC DISTURBANCE TERM (2) Principle of parsimony (we would like to keep our regression model as simple as possible 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

THE COEFFICIENT OF DETERMINATION r 2: 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 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

CONSUMPTION–INCOME RELATIONSHIP THE RELATIONSHIP BETWEEN EARNINGS 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 mengeliminasi perbedaan unit/ukuran pada variabel independent (butir, ekor) namun tidak dapat diketahui multikolinieritas (korelasi antar var bebas), nilai beta tidak dapat diinterpretasikan

TWO-VARIABLE REGRESSION MODEL: THE PROBLEM OF ESTIMATION Recall the two-variable PRF where ˆYi is the estimated (conditional mean) value of Yi . which shows that the ˆui (the residuals) are simply the differences between the actual and estimated Y values

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 able to show that the estimators of these parameters, ˆ β1, ˆ β2, and ˆσ 2,

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 sama atau homokedastisitas Tidak ada otokorelasi antar variabel disturbance Tidak ada korelasi sempurna antar variabel bebas Variabel error term berdistribusi normal

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 hypothesis) denoted by H1 reject or not reject the null hypothesis There are two mutually complementary approaches for devising such rules, namely, confidence interval and test of significance

Type kesalahan Hipotesis o Menerima Ho Menolak Ho Jika Ho benar Keputusan tepat Kesalahan jenis I Jika Ho salah Kesalahan jenis II

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 should move from the null hypothesis.

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 unidirectional rather than two-sided, as just discussed. Thus, for our consumption–income example, one could postulate that H0: β2 ≤ 0.3 and H1: β2 > 0.3

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 testing, the 100(1 − α)% confidence interval established in (5.7.2) is known as the region of acceptance (of the null hypothesis) and the region(s) outside the confidence interval is (are) called the region(s) of rejection (of H0) or the critical region(s). As noted previously, the confidence limits, the endpoints of the confidence interval, are also called critical values

MULTICOLLINEARITY: WHAT HAPPENS IF THE REGRESSORS ARE CORRELATED?

What is the nature of multicollinearity Model regresi yang baik, seharusnya tidak terjadi korelasi diantara variabel independen. Jika berkorelasi maka variabel tidak ortogonal (korelasi antar variabel independent = 0)

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 >0,9 Setiap variabel independent yang dijelaskan oleh variabel independet lainnya. Output nilai tolerance rendah (<0,10) atau VIF >10

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 Model specification An overdetermined model. This happens when the model has more explanatory variables than the number of observations.

Cara mengobati multikolinieritas Menggabungkan data cross section dan time series Keluarkan satu atau lebih variabel independen yang memp nilai korelasi tinggi (0,94%) Transformasi variabel Gunakan model untuk prediksi bukan interpretasi Gunakan center data untuk analisis (data mentah – mean) Sumber imam ghozali, 2006

AUTOCORRELATION: WHAT HAPPENS IF THE ERROR TERMS ARE CORRELATED?

three types of data cross section time series 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 correlation to “lag correlation between two different series. where u and v are two different time series, is called serial correlation

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 dengan menggunakan statistik uji Durbin-Watson. Apabila nilai D-W berada di sekitar angka 2, berarti model regresi kita aman dari kondisi heteroskedastisitas

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 beda umum (generalized difference equation). Selain itu juga dapat dilakukan dengan memasukkan variabel lag dari variabel terikatnya menjadi salah satu variabel bebas, sehingga data observasi menjadi berkurang 1

Korelasi

Korelasi Korelasi antara x(t) dan y(t) dinamakan dengan cross-correlation, dirumuskan dengan

Auto-korelasi Korelasi x(t) dengan dirinya sendiri disebut auto-korelasi

Korelasi Contoh 1 t h(t) 1.5 2.5 x(t)

Korelasi 1 t x(t) 1 t h(t) 1.5+p 2.5+p Untuk 1.5+p>1 atau p>-0.5

Korelasi 2. Untuk 1.5+p<1 dan 1.5+p>0, atau -1.5<p<-0.5 1 x(t-p) h(t) 2. Untuk 1.5+p<1 dan 1.5+p>0, atau -1.5<p<-0.5

Korelasi 3. Untuk 1.5+p<0 dan 2.5+p>1, atau -1.5<p<1.5 1 t x(t-p) h(t) 3. Untuk 1.5+p<0 dan 2.5+p>1, atau -1.5<p<1.5

Korelasi 4. Untuk 2.5+p<0 atau p<-2.5 -p-0.5 p+2.5 1 t 2.5+p x(t-p) h(t) 4. Untuk 2.5+p<0 atau p<-2.5 1 p y(p) -2.5 -0.5 p+2.5 -p-0.5

Korelasi 1 t p 1+p h(t) 1.5 2.5 x(t) Untuk 1+p<1.5 atau p<0.5

Korelasi 2. Untuk 1+p>1.5 dan 1+p<2.5, atau 0.5<p<1.5 1 t x(t) h(t-p) 2. Untuk 1+p>1.5 dan 1+p<2.5, atau 0.5<p<1.5

Korelasi 3. Untuk p<2.5 dan 1+p>2.5, atau 1.5<p<2.5 1 t x(t) h(t-p) 3. Untuk p<2.5 dan 1+p>2.5, atau 1.5<p<2.5

Korelasi 4. Untuk p>2.5 -p+2.5 p-0.5 x(t) h(t-p) 1 t p 1+p 1 p y(p)

Autokorelasi 1 t 1+p p h(t-p) h(t) 1. Untuk 0<p<1, maka

Autokorelasi 2. Untuk 0>p>-1, karena p negatif, maka geser kiri h(t) h(t-p) 1 t p 1+p 2. Untuk 0>p>-1, karena p negatif, maka geser kiri

Autokorelasi 3. Untuk p>1 dan p<-1, y(p) 1 -1 +1 1+p 1-p p

Korelasi

ILUSTRASI ANALISIS REGRESI Apakah 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 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 2.33 661.00 1.00 2.53 477.00 1.00 2.04 490.00 2.00 3.20 524.00 2.00 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 2.87 502.00 1.00 1.87 469.00 3.00 1.34 517.00 4.00 3.03 555.00 1.00 2.54 576.00 2.00 2.34 559.00 2.00

LANGKAH -LANGKAH Masukkan data pada SPSS Data Editor Pilih Analyze > Regression > Linear 1. Pilih dependen Variable 2. Pilih Independen Variables 3. Pada pilihan Statistics, aktifkan : Collinearity Diagnostics Durbin Watson Klik Continue 4. Pada pilihan Plot, aktifkan Normal Probability Plot. Klik Continue 5. Pada Pilihan Save, ~ Predicted Value, aktifkan Unstandardized ~ Residual, aktifkan Studentized 6. Klik OK

HASIL ANALISIS Regression

PEMERIKSAAN ASUMSI 1. ASUMSI NORMALITAS ERROR Hasil P-P plot menunjukkan pola garis lurus mendekati sudut 450, sehingga asumsi normalitas sisaan terpenuhi

PEMERIKSAAN ASUMSI 2. ASUMSI AUTOKORELASI Diperoleh nilai d = 2.254 Kaidah Uji Durbin Watson : Disimpulkan tidak ada autokorelasi bila du < 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

PEMERIKSAAN ASUMSI 3. ASUMSI MULTIKOLINEARITAS Condition Index = 20.639 < 30 Nilai VIF untuk skortes = 1.002 < 10 Nilai VIF untuk peringkat = 1.002 <10 Jadi tidak terdapat multikolinearitas

PEMERIKSAAN ASUMSI 4. ASUMSI HETEROSKEDASTISITAS Plotkan residual terstudentkan dengan nilai dugaan. a. Pilih Graphs > Scatter > Simple. b. Pilih Define Pilih Stundentized Residual sebagai Y axis Pilih Unstundardized predicted value sebagai X axis Klik OK Plot antara residual terstudentkan dengan nilai dugaan berpola acak, sehingga asumsi homoskedastisitas terpenuhi

INTERPRETASI VALIDASI MODEL Koefisien determinasi (R2) = 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% 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 – masing 0.135 dan 0.002. Artinya hanya peringkat yang berpengaruh signifikan terhadap besarnya NMR

INTERPRETASI Model hasil regresi NMR = 1.269 + 0.002769 Skor tes – 0.184 Peringkat Penjelasan terhadap fenomena Variabel yang berpengaruh secara signifikan adalah peringkat dengan koefisien regresi – 0.184 Artinya semakin kecil peringkat maka semakin tinggi NMR. Pada keadaan Skor tes konstan, jika Peringkat meningkat 1 tingkat maka NMR akan turun sebesar 0.184

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 Tingkat akurasi dari hasil prediksi ini adalah sebesar 47.8% (relatif rendah), akan tetapi bersifat general (karena nilai p untuk uji F pada ANOVA sebesar 0.004

INTERPRETASI 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 terbesar berarti pengaruhnya paling kuat, seandainya seluruh variabel signifikan). Dalam contoh ini yang signifikan hanya peringkat, sehingga yang berpengaruh secara bermakna terhadap NMR hanya peringkat.

WHAT HAPPENS IF THE ERROR VARIANCE IS NONCONSTANT? HETEROSCEDASTICITY WHAT HAPPENS IF THE ERROR VARIANCE IS NONCONSTANT?

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 the Yi and also, as we shall see, about β1 and β2. Thus, the assumptions made about the Xi variable(s) and the error term are extremely critical to the valid interpretation of the 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 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 commodity, if we do not include the prices of commodities complementary to or competing with the commodity in question (the omitted variable bias) Another source of heteroscedasticity is skewness in the distribution of 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 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. Heteroscedasticity can also arise because of (1) incorrect data transformation (e.g., ratio or first difference transformations) and (2) incorrect functional form (e.g., linear versus log–linear models)

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 population at a given point in time, such as individual consumers or their families, firms, industries, or geographical subdivisions such as state, country, city, etc

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 investigations. In this respect the econometrician differs from scientists in fields such as agriculture and biology, where researchers have a good deal of control over their subjects)

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 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 ui is normally distributed.

four types of variables ratio scale, interval scale, ordinal scale, 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, 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 As a matter of fact, a regression model may contain regressors 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 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 code of 1 represents a factor or collection of factors that are not measurable by any better means within the context of the analysis.

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 different races Income, socialization, experience of racial discrimination, attitudes toward a variety of social issues, feelings of political efficacy, etc Since we cannot measure all of those differences within the confines of the study we are doing, we use 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) 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 equal differences between ordinal categories. Regression using a 3 (or more) category nominal variable yields un-interpretable and meaningless results.

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 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-native Americans

Regression with Dummy Variables The dummy variable is then added the regression model Interpretation of the dummy variable is usually quite straightforward. The intercept term represents the intercept for the omitted category The slope coefficient for the dummy variable represents the change in the intercept for the category coded 1 (blacks)

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. a is the mean of Y for Whites and a+B1 is the mean of Y for Blacks

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 intercept is the model’s intercept. This omitted category is called the reference category In the dichotomous case, the reference category is simply the category 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. If there is some underlying ordinality in the categories, select the highest or lowest category as 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 scheme for race is: Note that the dummy for White has been omitted, and the intercept a is the intercept for Whites.

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. Thus if a = 50, and B1 = -45, the coefficient for Blacks might not be significantly different from 0, while Whites are significantly 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 other so that the relationship between income and ideology is different (stronger or weaker) for Whites than Blacks

Creating Interaction terms To create an interaction term is easy Multiply the category * the independent variable 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

Non-Linear Models Tractable non-linearity Intractable non-linearity Equation may be transformed to a linear model. Intractable non-linearity No linear transform exists

Tractable Non-Linear Models Several general Types Polynomial Power Functions Exponential Functions Logarithmic Functions Trigonometric Functions

Polynomial Models Linear Parabolic Cubic & higher order polynomials All may be estimated with OLS – simply square, cube, etc. the independent variable.

Power Functions Simple exponents of the Independent Variable Estimated with

Exponential and Logarithmic Functions Common Growth Curve Formula Estimated with Note that the error terms are now no longer normally distributed!

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 Or an equilibrium system 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 means of determining the best model

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 for those values that minimize the SSE.