Course Econometrics I
|
|
- Audrey Henderson
- 6 years ago
- Views:
Transcription
1 Course Econometrics I 4. Heteroskedasticity Martin Halla Johannes Kepler University of Linz Department of Economics Last update: May 6, 2014 Martin Halla CS Econometrics I 4 1/31
2 Our agenda for today Consequences of Heteroskedasticity (H) H-robust inference Testing for H Weighted least squares estimation Form of H know; obs. with higher variance less weight Feasible generalized least squares estimation Form of H is estimated The linear probability model revisited Martin Halla CS Econometrics I 4 2/31
3 Definition of H The homoskedasticity assumption states that the variance of the error term, u, conditional on the explanatory vars, is constant. Var(u x 1, x 2,..., x k ) = σ 2 Put differently, the error has the same variance given any value of the explanatory vars. Whenever the variance of u changes across different values of the explanatory vars., H is present. More generally, we have H, when the error variance differs across units i Var(u i ) = σ 2 i. For example, in a savings eq. H is present, if the variance of u affecting savings increases with income (i. e. there is a higher variance among high income individuals). Martin Halla CS Econometrics I 4 3/31
4 Consequences of H H does not cause bias or inconsistency in OLS. Goodness-of-fit measures are also unaffected. However, usual OLS t-statistics do not have a t distribution. The same applies to F statistics and LM statistics. This is not solved by a larger sample size. We have a problem with statistical inference. Further, OLS is not BLUE; there are more efficient estimators. Intuitively, OLS puts too much weight on obs. with a high error variance. Solutions: Simplest solution: H-robust inference. Given that we know the form of H (or we can estimate it), we can derive a more efficient estimator. Martin Halla CS Econometrics I 4 4/31
5 H-robust inference I Good news: OLS is still useful in the presence of H Econometricians developed in the 1980s methods to adjust s.e. and t, F and LM statistics such that they are valid in the presence of H of unknown form. That means, we can provide valid inference that works in any case. The derivation of this adjusted testing statistics is quite technical, but the application is easy. Martin Halla CS Econometrics I 4 5/31
6 H-robust inference I White (1980, ECM) has shown that under MLR.1 through MLR.4, Var( ˆβ j ) = n i=1 ˆr2 ijû2 i SSR 2 j (1) is a valid estimator for Var( ˆβ j ), where ˆr ij denotes the i th residual from regressing x j on all other indep. vars, and SSR j is the sum of squared residual from this regression. The square root of (1) is the H-robust s.e. for ˆβ j. Sometimes also called White, Huber, Eicker s.e. (or some hyphenated combination of these names); or simply robust s.e. Robust s.e. can be either larger or smaller than usual s.e. Based on (1) we can derive a H-robust t statistic: t = (estimate hyp. value)/s.e. This can easily be done in Stata. Martin Halla CS Econometrics I 4 6/31
7 Example with H-robust inference. reg lwage marrmale marrfem singfem educ exper expersq tenure tenursq, robust Linear regression Number of obs = 526 F( 8, 517) = Prob > F = R-squared = Root MSE = Robust lwage Coef. Std. Err. t P> t [95% Conf. Interval] marrmale marrfem singfem educ exper expersq tenure tenursq _cons In Stata robust standard errors can be easily obtained by the option robust. Martin Halla CS Econometrics I 4 7/31
8 H-robust inference II (I) (II) Coeff. Normal Coeff. Robust s.e. s.e. marrmale 0.213*** (0.055) 0.213*** [0.057] marrfem *** (0.058) *** [0.059] singfem * (0.056) [0.057] educ 0.079*** (0.007) 0.079*** [0.007] exper 0.027*** (0.005) 0.027*** [0.005] exper *** (0.001) *** [0.001] tenure 0.029*** (0.007) 0.029*** [0.007] tenure * (0.001) * [0.001] Constant 0.321** (0.100) 0.321** [0.109] R-squared N In this case usual and robust s.e. are very similar. However, in other cases it might change important conclusions. Martin Halla CS Econometrics I 4 8/31
9 Testing for H Why test? We could simply use only robust s.e.? Under H OLS is not BLUE. Usual t statistics have exact t distrib. under the CLM assumps. Robust s.e./statistics are only valid in large samples. There are many different tests. General idea: test assump. MLR.5: H 0 : Var(u x) = σ 2 If we cannot reject this null hyp., we conclude H is not a problem. Since Var(u x) = E(u 2 x), we can write H 0 : E(u 2 x) = σ 2. That means, we can test whether u 2 is related (in expected value) to any of the x: u 2 = δ 0 + δ 1 x 1 + δ 2 x δ k x k + v The null hyp. of homoskedasticity is H 0 : δ 1 =... = δ k = 0. Implementation with the estimate of u 2, the squared residual û 2. Martin Halla CS Econometrics I 4 9/31
10 The Breusch-Pagan test for H 1. Estimate your model by OLS and obtain û 2 i for each i. 2. Run û 2 = δ 0 + δ 1 x δ k x k + v and save the R-squared (Rû 2 ) Form either the F statistic or the LM statistic F = R 2 û 2 /k (1 R 2 û 2 )/(n k 1) LM = n R 2 û 2 where k is equal to the no. of regressors and n the no. of obs. 4. If the respective p-value is sufficiently small, reject the null-hyp. of homoskedasticity. (If you suspect H only in a sub-set of your indep. vars, you can modify step 2.) Martin Halla CS Econometrics I 4 10/31
11 Breusch-Pagan test for H an example I * Breusch-Pagan test for heteroskedasticity (for Example 8.4) * 1.) Estimate the model by OLS and obtain the squared OLS residuals: qui reg price lotsize sqrft bdrms predict u, resid gen u2=u^2 * 2.) Run the following regression and keep the R-squared: qui reg u2 lotsize sqrft bdrms * 3a.) Form either the F (or the LM) statistic and compute the p-value: * F statistic: display (e(r2)/(1- e(r2))*(84/3)) * P-value display 1-F(3,88,e(F)) » A p-value of suggests strong evidence against homoskedasticity. Martin Halla CS Econometrics I 4 11/31
12 Breusch-Pagan test for H an example II * 3b.) Alternatively we can compute the LM statistic * LM statistic: display e(r2)*e(n) * P-value display 1-chi2(3, ) » Again, strong evidence against the null-hyp. of homoskedasticity. Martin Halla CS Econometrics I 4 12/31
13 Breusch-Pagan test for H an example III Let us consider a model with log transformations of the some variable: drop u u2 qui reg lprice llotsize lsqrft bdrms predict u, resid gen u2=u^2 reg qui u2 llotsize lsqrft bdrms display (e(r2)/(1- e(r2))*(84/3)) display e(f) display 1-F(3,88,e(F)) » Now we fail to reject the null hypothesis of homoskedasticity. Martin Halla CS Econometrics I 4 13/31
14 The White Test (special case) for H 1. Estimate your model by OLS and obtain the residuals and the fitted values and compute also theirs squares (û 2 i, ŷ2 i ). 2. Run û 2 = δ 0 + δ 1 ŷ + δ 2 ŷ 2 + v and save the R-squared (R 2 û 2 ). 3. Form either the F statistic of the LM statistic F 2,n 3 = R 2 û 2 /2 (1 R 2 û 2 )/(n 3) where n is the no. of obs. LM = n R 2 û 2 4. If the respective p-value is sufficiently small, reject the null-hyp. of homoskedasticity. (In the original form of the test you include in step 2. all indep vars x j, their squares x 2 j, and all their cross-products x jx h for j h.) Martin Halla CS Econometrics I 4 14/31
15 The special case of the White Test an example I * Special case of the White Test for heteroskedasticity (see Example 8.5) * 1.) Estimate the model by OLS and obtain the residuals and the fitted values. qui reg lprice llotsize lsqrft bdrms predict u, resid predict fitted, xb /*... and compute also their squares */ gen u2=u^2 gen fitted2=fitted^2 Martin Halla CS Econometrics I 4 15/31
16 The special case of the White Test an example II * 2.) Run the following regression and keep the R-squared: qui reg u2 fitted fitted2 * 3.) Form either the F (or the LM) statistic and compute the p-value: display (e(r2)/(1- e(r2))*(88/2)) display 1-F(2,88,e(F)) * The p-value of provide little evidence against homoskedasticity. * LM statistic display e(r2)*e(n) display 1-chi2(2, ) Martin Halla CS Econometrics I 4 16/31
17 Weighted least squares estimation I Before robust s.e. were available, econometricians used a weighted least squares (WLS) estimation in the presence of H. WLS requires the knowledge of the functional form of the variance. Idea: if we can specify H (as a function of the x), the WLS estimation transforms the estimation model such that we get homoskedastic errors. Under a correct specification of the variance, WLS is more efficient than OLS, and leads to new t and F statistics with correct distributions. WLS is an example for a generalized least squares (GLS) estimation. We can also estimate the form of H before we apply WLS; this procedure is called feasible GLS (FGLS). Martin Halla CS Econometrics I 4 17/31
18 Weighted least squares estimation II Let x denote our RHS vars and assume Var(u x) = σ 2 h(x), (2) where h(x) is some known function that determines H. Of course, σ 2 is unknown, but we will estimate it. For instance, consider the simple savings function sav i = β 0 + β 1 inc i + u i, (3) where assume that the variance of the error is proport. to income Var(u i inc i ) = σ 2 inc i. (4) That means, as income increases the variability in savings increases. We can use this idea to estimate an eq. with heteroskedastic errors, y i = β 0 + β 1 x i β k x ik + u i, (5) and transform it into an eq. that has a homoskedastic error term. Martin Halla CS Econometrics I 4 18/31
19 Weighted least squares estimation III We simply divide the original equation by h i : y i / h i = β 0 / h i + β 1 (x i1 / h i ) β k (x ik / h i ) + (u i / h i ), (6) or y i = β 0 x i0 + β 1 x i β k x ik + u i, (7) where x i0 = 1/ h i and the other starred vars denote the corresponding original vars divided by h i. Note, since Var(u i x i ) = E(u 2 i x i), we can write ( E (u i / h i ) 2) = 1 E(u 2 i ) = 1 (σ 2 h i ) = σ 2, (8) h i h i which means that the error term of the transformed eq. is homoskedastic. Given that the original eq. fulfills MLR.1-4; this eq. fulfills MLR.1-5; savings eq.: sav i / inc i = β 0 (1/ inc i ) + β 1 inci + u i ) Martin Halla CS Econometrics I 4 19/31
20 Weighted least squares estimation IV The OLS estimator gives equal weight to all obs. and minimizes: n = (y i β 0 β 1 x i... β k x k ) 2. (9) i=1 In the transformed model from above we minimize: n ( ) y i β 0 β 1 x i... β k x k hi hi hi hi i=1 n 1 = (y i β 0 β 1 x i... β k x k ) 2 h i=1 i n = w i (y i β 0 β 1 x i... β k x k ) 2. i=1 (10) WLS gives less weight (w i = 1/h i ) to obs. with a higher error var. Martin Halla CS Econometrics I 4 20/31
21 Weighted least squares estimation Example 8.6 * WLS (where we assume that h=inc): * 1. option using transformed vars gen cons_wls = 1/(inc)^(1/2) gen sav_wls = sav/(inc)^(1/2) gen inc_wls = inc/(inc)^(1/2) reg sav_wls inc_wls cons_wls, nocons sav_wls Coef. Std. Err. t P> t [95% Conf. Interval] inc_wls cons_wls * 2. option using Stata s weight option reg sav inc [aw = 1/inc] (sum of wgt is e-02) sav Coef. Std. Err. t P> t [95% Conf. Interval] inc _cons Martin Halla CS Econometrics I 4 21/31
22 Weighted least squares estimation V What are the properties of WLS if our choice for h(x) is incorrect? Just like OLS, WLS still provides an unbiased and consistent estimator. Note, OLS is the special case where we erroneously assumed h(x) = 1. However, the test statistics are no longer valid. Wooldridge argues that even a wrong specification of (strong) H might be better than complete ignorance (by OLS). In case of averaged data (e. g. on a firm-level or country-level) you should always use WLS with 1/h i = m i, where m i is the number of underlying individuals in the ith aggregate unit. Idea: Larger aggregate units have a smaller error variance, and receive a higher weight. Martin Halla CS Econometrics I 4 22/31
23 Feasible GLS Usually, the exact form of H is not obvious. How do we find the function h(x i )? Feasible GLS (FGLS) suggests to use an estimate of h i, denoted as ĥi, in the GLS transformation. FGLS is sometimes also called estimated GLS. Of course, there are many ways to model H. For instance, we could assume that Var(u x) = σ 2 exp(δ 0 + δ 1 x δ k x k ) That means, h(x) = exp(δ0 + δ 1 x δ k x k ) The exponential func. guarantees positive values (for estimated variances). Next slide outlines a corresponding feasible GLS procedure Martin Halla CS Econometrics I 4 23/31
24 A feasible GLS procedure to correct for H 1. Estimate your model and obtain the residual, û 2. Create log(û 2 ) 3. Run the regression of log(û 2 ) on x, and obtain the fitted values, ĝ 4. Exponentiate the fitted values: exp(ĝ) ĥ 5. Estimate your model by WLS, using weights 1/ĥ However, note FGLS is not unbiased. It is only consistent and asymptotically more efficient than OLS. Martin Halla CS Econometrics I 4 24/31
25 Feasible GLS estimation Example 8.7 * 1.) Estimate the model and obtain the residual: qui reg cigs lincome lcigpric educ age agesq restaurn predict u, resid * 2.) Create the log of the squared residual: gen lu2 = log(u^2) * 3.) Run the following regression and obtain the fitted values: qui reg lu2 lincome lcigpric educ age agesq restaurn predict fitted, xb * 4.) Exponentiate the fitted values: gen h = exp(fitted) * 5.) Estimate the model by WLS, using weights $1/h$ reg cigs lincome lcigpric educ age agesq restaurn [aw = 1/h] (sum of wgt is e+01)... cigs Coef. Std. Err. t P> t [95% Conf. Interval] lincome lcigpric educ age agesq restaurn _cons Martin Halla CS Econometrics I 4 25/31
26 The linear probability model revisited Problem: A LPM generally contains H. Solution I: Simply compute robust s.e. Solution II: Estimate the variance and use WLS Var(y x) = p(x))[1 p(x)] Estimate by ĥi = ŷ i (1 ŷ i ). Martin Halla CS Econometrics I 4 26/31
27 The LPM revisited Solution I (Example 8.8) inlf Coef. Std. Err. t P> t [95% Conf. Interval] nwifeinc educ exper expersq age kidslt kidsge _cons Robust inlf Coef. Std. Err. t P> t [95% Conf. Interval] nwifeinc educ exper expersq age kidslt kidsge _cons Martin Halla CS Econometrics I 4 27/31
28 The LPM revisited Solution II Estimating the LPM by weighted least squares 1.) Estimate the model by OLS and obtain the fitted values, ŷ i 2.) Determine whether all ŷ i are inside the interval [0,1] If so, proceed to step 3.). If not, some adjustment is needed to bring all ŷi into the unit interval 3.) Construct the estimated variances in ĥ = ŷ i(1 ŷ i ) 4.) Estimate the following eq. by WLS, using 1/ĥ y = β 0 + β 1 x β k x k + u. Martin Halla CS Econometrics I 4 28/31
29 Estimating the LPM by WLS Example 8.9 (part I) * 1.) Estimate the model by OLS and obtain the fitted values. qui reg PC hsgpa ACT parcoll predict fitted, xb * 2.) Determine whether all fitted values are inside the interval [0,1] sum fitted Variable Obs Mean Std. Dev. Min Max fitted * 3.) Construct the estimated variances. gen h = fitted*(1-fitted) Martin Halla CS Econometrics I 4 29/31
30 Estimating the LPM by WLS Example 8.9 (part II) * 4.) Estimate the following eq. by WLS, using $1/h$ reg PC hsgpa ACT parcoll [w=1/h] (analytic weights assumed) (sum of wgt is e+02) Source SS df MS Number of obs = F( 3, 137) = 2.22 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = PC Coef. Std. Err. t P> t [95% Conf. Interval] hsgpa ACT parcoll _cons Martin Halla CS Econometrics I 4 30/31
31 Estimating the LPM by WLS Example 8.9 (part III) Estimation by OLS: PC Coef. Std. Err. t P> t [95% Conf. Interval] hsgpa ACT parcoll _cons Estimation by WLS: PC Coef. Std. Err. t P> t [95% Conf. Interval] hsgpa ACT parcoll _cons There are no important diffs. The only significant RHS var is parcoll, and in both case the estimated prob. of PC ownership is about 22 percent higher if at least one parent has attended college. Martin Halla CS Econometrics I 4 31/31
Econometrics Multiple Regression Analysis: Heteroskedasticity
Econometrics Multiple Regression Analysis: João Valle e Azevedo Faculdade de Economia Universidade Nova de Lisboa Spring Semester João Valle e Azevedo (FEUNL) Econometrics Lisbon, April 2011 1 / 19 Properties
More informationHeteroskedasticity. Part VII. Heteroskedasticity
Part VII Heteroskedasticity As of Oct 15, 2015 1 Heteroskedasticity Consequences Heteroskedasticity-robust inference Testing for Heteroskedasticity Weighted Least Squares (WLS) Feasible generalized Least
More informationMultiple Regression Analysis: Heteroskedasticity
Multiple Regression Analysis: Heteroskedasticity y = β 0 + β 1 x 1 + β x +... β k x k + u Read chapter 8. EE45 -Chaiyuth Punyasavatsut 1 topics 8.1 Heteroskedasticity and OLS 8. Robust estimation 8.3 Testing
More informationIntroductory Econometrics
Based on the textbook by Wooldridge: : A Modern Approach Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna December 11, 2012 Outline Heteroskedasticity
More informationECO375 Tutorial 7 Heteroscedasticity
ECO375 Tutorial 7 Heteroscedasticity Matt Tudball University of Toronto Mississauga November 9, 2017 Matt Tudball (University of Toronto) ECO375H5 November 9, 2017 1 / 24 Review: Heteroscedasticity Consider
More informationHeteroskedasticity (Section )
Heteroskedasticity (Section 8.1-8.4) Ping Yu School of Economics and Finance The University of Hong Kong Ping Yu (HKU) Heteroskedasticity 1 / 44 Consequences of Heteroskedasticity for OLS Consequences
More informationWarwick Economics Summer School Topics in Microeconometrics Instrumental Variables Estimation
Warwick Economics Summer School Topics in Microeconometrics Instrumental Variables Estimation Michele Aquaro University of Warwick This version: July 21, 2016 1 / 31 Reading material Textbook: Introductory
More informationIntermediate Econometrics
Intermediate Econometrics Heteroskedasticity Text: Wooldridge, 8 July 17, 2011 Heteroskedasticity Assumption of homoskedasticity, Var(u i x i1,..., x ik ) = E(u 2 i x i1,..., x ik ) = σ 2. That is, the
More informationMultiple Regression Analysis
Multiple Regression Analysis y = 0 + 1 x 1 + x +... k x k + u 6. Heteroskedasticity What is Heteroskedasticity?! Recall the assumption of homoskedasticity implied that conditional on the explanatory variables,
More informationECON2228 Notes 7. Christopher F Baum. Boston College Economics. cfb (BC Econ) ECON2228 Notes / 41
ECON2228 Notes 7 Christopher F Baum Boston College Economics 2014 2015 cfb (BC Econ) ECON2228 Notes 6 2014 2015 1 / 41 Chapter 8: Heteroskedasticity In laying out the standard regression model, we made
More informationLab 11 - Heteroskedasticity
Lab 11 - Heteroskedasticity Spring 2017 Contents 1 Introduction 2 2 Heteroskedasticity 2 3 Addressing heteroskedasticity in Stata 3 4 Testing for heteroskedasticity 4 5 A simple example 5 1 1 Introduction
More informationGraduate Econometrics Lecture 4: Heteroskedasticity
Graduate Econometrics Lecture 4: Heteroskedasticity Department of Economics University of Gothenburg November 30, 2014 1/43 and Autocorrelation Consequences for OLS Estimator Begin from the linear model
More informationEconometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018
Econometrics I KS Module 2: Multivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: April 16, 2018 Alexander Ahammer (JKU) Module 2: Multivariate
More information5. Let W follow a normal distribution with mean of μ and the variance of 1. Then, the pdf of W is
Practice Final Exam Last Name:, First Name:. Please write LEGIBLY. Answer all questions on this exam in the space provided (you may use the back of any page if you need more space). Show all work but do
More informationHeteroskedasticity Example
ECON 761: Heteroskedasticity Example L Magee November, 2007 This example uses the fertility data set from assignment 2 The observations are based on the responses of 4361 women in Botswana s 1988 Demographic
More informationIntroduction to Econometrics. Heteroskedasticity
Introduction to Econometrics Introduction Heteroskedasticity When the variance of the errors changes across segments of the population, where the segments are determined by different values for the explanatory
More informationAn explanation of Two Stage Least Squares
Introduction Introduction to Econometrics An explanation of Two Stage Least Squares When we get an endogenous variable we know that OLS estimator will be inconsistent. In addition OLS regressors will also
More informationEconometrics - 30C00200
Econometrics - 30C00200 Lecture 11: Heteroskedasticity Antti Saastamoinen VATT Institute for Economic Research Fall 2015 30C00200 Lecture 11: Heteroskedasticity 12.10.2015 Aalto University School of Business
More informationEconometrics I KS. Module 1: Bivariate Linear Regression. Alexander Ahammer. This version: March 12, 2018
Econometrics I KS Module 1: Bivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: March 12, 2018 Alexander Ahammer (JKU) Module 1: Bivariate
More informationCourse Econometrics I
Course Econometrics I 3. Multiple Regression Analysis: Binary Variables Martin Halla Johannes Kepler University of Linz Department of Economics Last update: April 29, 2014 Martin Halla CS Econometrics
More informationMultiple Regression: Inference
Multiple Regression: Inference The t-test: is ˆ j big and precise enough? We test the null hypothesis: H 0 : β j =0; i.e. test that x j has no effect on y once the other explanatory variables are controlled
More informationECON Introductory Econometrics. Lecture 5: OLS with One Regressor: Hypothesis Tests
ECON4150 - Introductory Econometrics Lecture 5: OLS with One Regressor: Hypothesis Tests Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 5 Lecture outline 2 Testing Hypotheses about one
More informationLecture 8: Instrumental Variables Estimation
Lecture Notes on Advanced Econometrics Lecture 8: Instrumental Variables Estimation Endogenous Variables Consider a population model: y α y + β + β x + β x +... + β x + u i i i i k ik i Takashi Yamano
More information4 Instrumental Variables Single endogenous variable One continuous instrument. 2
Econ 495 - Econometric Review 1 Contents 4 Instrumental Variables 2 4.1 Single endogenous variable One continuous instrument. 2 4.2 Single endogenous variable more than one continuous instrument..........................
More information1 Linear Regression Analysis The Mincer Wage Equation Data Econometric Model Estimation... 11
Econ 495 - Econometric Review 1 Contents 1 Linear Regression Analysis 4 1.1 The Mincer Wage Equation................. 4 1.2 Data............................. 6 1.3 Econometric Model.....................
More informationTopic 7: Heteroskedasticity
Topic 7: Heteroskedasticity Advanced Econometrics (I Dong Chen School of Economics, Peking University Introduction If the disturbance variance is not constant across observations, the regression is heteroskedastic
More informationEcmt 675: Econometrics I
Ecmt 675: Econometrics I Assignment 7 Problem 1 a. reg hours lwage educ age kidslt6 kidsge6 nwifeinc, r Linear regression Number of obs = 428 F( 6, 421) = 3.93 Prob > F = 0.0008 R-squared = 0.0670 Root
More informationLecture 8: Heteroskedasticity. Causes Consequences Detection Fixes
Lecture 8: Heteroskedasticity Causes Consequences Detection Fixes Assumption MLR5: Homoskedasticity 2 var( u x, x,..., x ) 1 2 In the multivariate case, this means that the variance of the error term does
More informationEconometrics I Lecture 7: Dummy Variables
Econometrics I Lecture 7: Dummy Variables Mohammad Vesal Graduate School of Management and Economics Sharif University of Technology 44716 Fall 1397 1 / 27 Introduction Dummy variable: d i is a dummy variable
More information4 Instrumental Variables Single endogenous variable One continuous instrument. 2
Econ 495 - Econometric Review 1 Contents 4 Instrumental Variables 2 4.1 Single endogenous variable One continuous instrument. 2 4.2 Single endogenous variable more than one continuous instrument..........................
More informationProblem Set #5-Key Sonoma State University Dr. Cuellar Economics 317- Introduction to Econometrics
Problem Set #5-Key Sonoma State University Dr. Cuellar Economics 317- Introduction to Econometrics C1.1 Use the data set Wage1.dta to answer the following questions. Estimate regression equation wage =
More informationHeteroskedasticity. Occurs when the Gauss Markov assumption that the residual variance is constant across all observations in the data set
Heteroskedasticity Occurs when the Gauss Markov assumption that the residual variance is constant across all observations in the data set Heteroskedasticity Occurs when the Gauss Markov assumption that
More informationProblem set - Selection and Diff-in-Diff
Problem set - Selection and Diff-in-Diff 1. You want to model the wage equation for women You consider estimating the model: ln wage = α + β 1 educ + β 2 exper + β 3 exper 2 + ɛ (1) Read the data into
More informationApplied Statistics and Econometrics
Applied Statistics and Econometrics Lecture 5 Saul Lach September 2017 Saul Lach () Applied Statistics and Econometrics September 2017 1 / 44 Outline of Lecture 5 Now that we know the sampling distribution
More informationthe error term could vary over the observations, in ways that are related
Heteroskedasticity We now consider the implications of relaxing the assumption that the conditional variance Var(u i x i ) = σ 2 is common to all observations i = 1,..., n In many applications, we may
More informationStatistical Inference with Regression Analysis
Introductory Applied Econometrics EEP/IAS 118 Spring 2015 Steven Buck Lecture #13 Statistical Inference with Regression Analysis Next we turn to calculating confidence intervals and hypothesis testing
More informationHeteroskedasticity. (In practice this means the spread of observations around any given value of X will not now be constant)
Heteroskedasticity Occurs when the Gauss Markov assumption that the residual variance is constant across all observations in the data set so that E(u 2 i /X i ) σ 2 i (In practice this means the spread
More informationEconometrics. 9) Heteroscedasticity and autocorrelation
30C00200 Econometrics 9) Heteroscedasticity and autocorrelation Timo Kuosmanen Professor, Ph.D. http://nomepre.net/index.php/timokuosmanen Today s topics Heteroscedasticity Possible causes Testing for
More informationHeteroskedasticity ECONOMETRICS (ECON 360) BEN VAN KAMMEN, PHD
Heteroskedasticity ECONOMETRICS (ECON 360) BEN VAN KAMMEN, PHD Introduction For pedagogical reasons, OLS is presented initially under strong simplifying assumptions. One of these is homoskedastic errors,
More informationEmpirical Application of Simple Regression (Chapter 2)
Empirical Application of Simple Regression (Chapter 2) 1. The data file is House Data, which can be downloaded from my webpage. 2. Use stata menu File Import Excel Spreadsheet to read the data. Don t forget
More informationChapter 8 Heteroskedasticity
Chapter 8 Walter R. Paczkowski Rutgers University Page 1 Chapter Contents 8.1 The Nature of 8. Detecting 8.3 -Consistent Standard Errors 8.4 Generalized Least Squares: Known Form of Variance 8.5 Generalized
More informationApplied Statistics and Econometrics
Applied Statistics and Econometrics Lecture 6 Saul Lach September 2017 Saul Lach () Applied Statistics and Econometrics September 2017 1 / 53 Outline of Lecture 6 1 Omitted variable bias (SW 6.1) 2 Multiple
More informationHeteroskedasticity. We now consider the implications of relaxing the assumption that the conditional
Heteroskedasticity We now consider the implications of relaxing the assumption that the conditional variance V (u i x i ) = σ 2 is common to all observations i = 1,..., In many applications, we may suspect
More informationWooldridge, Introductory Econometrics, 2d ed. Chapter 8: Heteroskedasticity In laying out the standard regression model, we made the assumption of
Wooldridge, Introductory Econometrics, d ed. Chapter 8: Heteroskedasticity In laying out the standard regression model, we made the assumption of homoskedasticity of the regression error term: that its
More informationEconometrics. Week 8. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 8 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 25 Recommended Reading For the today Instrumental Variables Estimation and Two Stage
More informationRecent Advances in the Field of Trade Theory and Policy Analysis Using Micro-Level Data
Recent Advances in the Field of Trade Theory and Policy Analysis Using Micro-Level Data July 2012 Bangkok, Thailand Cosimo Beverelli (World Trade Organization) 1 Content a) Classical regression model b)
More informationEconometrics. Week 4. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 4 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 23 Recommended Reading For the today Serial correlation and heteroskedasticity in
More informationECON Introductory Econometrics. Lecture 7: OLS with Multiple Regressors Hypotheses tests
ECON4150 - Introductory Econometrics Lecture 7: OLS with Multiple Regressors Hypotheses tests Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 7 Lecture outline 2 Hypothesis test for single
More informationECON3150/4150 Spring 2016
ECON3150/4150 Spring 2016 Lecture 4 - The linear regression model Siv-Elisabeth Skjelbred University of Oslo Last updated: January 26, 2016 1 / 49 Overview These lecture slides covers: The linear regression
More informationControl Function and Related Methods: Nonlinear Models
Control Function and Related Methods: Nonlinear Models Jeff Wooldridge Michigan State University Programme Evaluation for Policy Analysis Institute for Fiscal Studies June 2012 1. General Approach 2. Nonlinear
More informationTHE MULTIVARIATE LINEAR REGRESSION MODEL
THE MULTIVARIATE LINEAR REGRESSION MODEL Why multiple regression analysis? Model with more than 1 independent variable: y 0 1x1 2x2 u It allows : -Controlling for other factors, and get a ceteris paribus
More informationBinary Dependent Variables
Binary Dependent Variables In some cases the outcome of interest rather than one of the right hand side variables - is discrete rather than continuous Binary Dependent Variables In some cases the outcome
More informationLecture 19. Common problem in cross section estimation heteroskedasticity
Lecture 19 Learning to worry about and deal with stationarity Common problem in cross section estimation heteroskedasticity What is it Why does it matter What to do about it Stationarity Ultimately whether
More information1. The shoe size of five randomly selected men in the class is 7, 7.5, 6, 6.5 the shoe size of 4 randomly selected women is 6, 5.
Economics 3 Introduction to Econometrics Winter 2004 Professor Dobkin Name Final Exam (Sample) You must answer all the questions. The exam is closed book and closed notes you may use calculators. You must
More informationPlease discuss each of the 3 problems on a separate sheet of paper, not just on a separate page!
Econometrics - Exam May 11, 2011 1 Exam Please discuss each of the 3 problems on a separate sheet of paper, not just on a separate page! Problem 1: (15 points) A researcher has data for the year 2000 from
More information0. Introductory econometrics
0. Introductory econometrics 0.1 Structure of economic data Cross-sectional data: Data wic are collected from units of te underlying population at a given time period (wic may vary occasionally) (te arrangement
More informationRegression with Qualitative Information. Part VI. Regression with Qualitative Information
Part VI Regression with Qualitative Information As of Oct 17, 2017 1 Regression with Qualitative Information Single Dummy Independent Variable Multiple Categories Ordinal Information Interaction Involving
More informationLab 07 Introduction to Econometrics
Lab 07 Introduction to Econometrics Learning outcomes for this lab: Introduce the different typologies of data and the econometric models that can be used Understand the rationale behind econometrics Understand
More informationLab 10 - Binary Variables
Lab 10 - Binary Variables Spring 2017 Contents 1 Introduction 1 2 SLR on a Dummy 2 3 MLR with binary independent variables 3 3.1 MLR with a Dummy: different intercepts, same slope................. 4 3.2
More informationMeasurement Error. Often a data set will contain imperfect measures of the data we would ideally like.
Measurement Error Often a data set will contain imperfect measures of the data we would ideally like. Aggregate Data: (GDP, Consumption, Investment are only best guesses of theoretical counterparts and
More information. *DEFINITIONS OF ARTIFICIAL DATA SET. mat m=(12,20,0) /*matrix of means of RHS vars: edu, exp, error*/
. DEFINITIONS OF ARTIFICIAL DATA SET. mat m=(,,) /matrix of means of RHS vars: edu, exp, error/. mat c=(5,-.6, \ -.6,9, \,,.) /covariance matrix of RHS vars /. mat l m /displays matrix of means / c c c3
More informationECON3150/4150 Spring 2015
ECON3150/4150 Spring 2015 Lecture 3&4 - The linear regression model Siv-Elisabeth Skjelbred University of Oslo January 29, 2015 1 / 67 Chapter 4 in S&W Section 17.1 in S&W (extended OLS assumptions) 2
More informationLab 6 - Simple Regression
Lab 6 - Simple Regression Spring 2017 Contents 1 Thinking About Regression 2 2 Regression Output 3 3 Fitted Values 5 4 Residuals 6 5 Functional Forms 8 Updated from Stata tutorials provided by Prof. Cichello
More informationECON Introductory Econometrics. Lecture 16: Instrumental variables
ECON4150 - Introductory Econometrics Lecture 16: Instrumental variables Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 12 Lecture outline 2 OLS assumptions and when they are violated Instrumental
More informationHandout 12. Endogeneity & Simultaneous Equation Models
Handout 12. Endogeneity & Simultaneous Equation Models In which you learn about another potential source of endogeneity caused by the simultaneous determination of economic variables, and learn how to
More informationProblem 4.1. Problem 4.3
BOSTON COLLEGE Department of Economics EC 228 01 Econometric Methods Fall 2008, Prof. Baum, Ms. Phillips (tutor), Mr. Dmitriev (grader) Problem Set 3 Due at classtime, Thursday 14 Oct 2008 Problem 4.1
More information(a) Briefly discuss the advantage of using panel data in this situation rather than pure crosssections
Answer Key Fixed Effect and First Difference Models 1. See discussion in class.. David Neumark and William Wascher published a study in 199 of the effect of minimum wages on teenage employment using a
More informationECON3150/4150 Spring 2016
ECON3150/4150 Spring 2016 Lecture 6 Multiple regression model Siv-Elisabeth Skjelbred University of Oslo February 5th Last updated: February 3, 2016 1 / 49 Outline Multiple linear regression model and
More informationAt this point, if you ve done everything correctly, you should have data that looks something like:
This homework is due on July 19 th. Economics 375: Introduction to Econometrics Homework #4 1. One tool to aid in understanding econometrics is the Monte Carlo experiment. A Monte Carlo experiment allows
More informationEconometrics Midterm Examination Answers
Econometrics Midterm Examination Answers March 4, 204. Question (35 points) Answer the following short questions. (i) De ne what is an unbiased estimator. Show that X is an unbiased estimator for E(X i
More informationPractice exam questions
Practice exam questions Nathaniel Higgins nhiggins@jhu.edu, nhiggins@ers.usda.gov 1. The following question is based on the model y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + u. Discuss the following two hypotheses.
More informationAnswer all questions from part I. Answer two question from part II.a, and one question from part II.b.
B203: Quantitative Methods Answer all questions from part I. Answer two question from part II.a, and one question from part II.b. Part I: Compulsory Questions. Answer all questions. Each question carries
More informationHandout 11: Measurement Error
Handout 11: Measurement Error In which you learn to recognise the consequences for OLS estimation whenever some of the variables you use are not measured as accurately as you might expect. A (potential)
More informationIntroductory Econometrics
Introductory Econometrics Violation of basic assumptions Heteroskedasticity Barbara Pertold-Gebicka CERGE-EI 16 November 010 OLS assumptions 1. Disturbances are random variables drawn from a normal distribution.
More informationRegression #8: Loose Ends
Regression #8: Loose Ends Econ 671 Purdue University Justin L. Tobias (Purdue) Regression #8 1 / 30 In this lecture we investigate a variety of topics that you are probably familiar with, but need to touch
More information1 Independent Practice: Hypothesis tests for one parameter:
1 Independent Practice: Hypothesis tests for one parameter: Data from the Indian DHS survey from 2006 includes a measure of autonomy of the women surveyed (a scale from 0-10, 10 being the most autonomous)
More informationAnswer Key: Problem Set 6
: Problem Set 6 1. Consider a linear model to explain monthly beer consumption: beer = + inc + price + educ + female + u 0 1 3 4 E ( u inc, price, educ, female ) = 0 ( u inc price educ female) σ inc var,,,
More informationECON2228 Notes 10. Christopher F Baum. Boston College Economics. cfb (BC Econ) ECON2228 Notes / 48
ECON2228 Notes 10 Christopher F Baum Boston College Economics 2014 2015 cfb (BC Econ) ECON2228 Notes 10 2014 2015 1 / 48 Serial correlation and heteroskedasticity in time series regressions Chapter 12:
More informationProblem Set #3-Key. wage Coef. Std. Err. t P> t [95% Conf. Interval]
Problem Set #3-Key Sonoma State University Economics 317- Introduction to Econometrics Dr. Cuellar 1. Use the data set Wage1.dta to answer the following questions. a. For the regression model Wage i =
More informationECON Introductory Econometrics. Lecture 13: Internal and external validity
ECON4150 - Introductory Econometrics Lecture 13: Internal and external validity Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 9 Lecture outline 2 Definitions of internal and external
More informationIntroductory Econometrics. Lecture 13: Hypothesis testing in the multiple regression model, Part 1
Introductory Econometrics Lecture 13: Hypothesis testing in the multiple regression model, Part 1 Jun Ma School of Economics Renmin University of China October 19, 2016 The model I We consider the classical
More informationEconometrics Homework 1
Econometrics Homework Due Date: March, 24. by This problem set includes questions for Lecture -4 covered before midterm exam. Question Let z be a random column vector of size 3 : z = @ (a) Write out z
More informationFortin Econ Econometric Review 1. 1 Panel Data Methods Fixed Effects Dummy Variables Regression... 7
Fortin Econ 495 - Econometric Review 1 Contents 1 Panel Data Methods 2 1.1 Fixed Effects......................... 2 1.1.1 Dummy Variables Regression............ 7 1.1.2 First Differencing Methods.............
More informationAutocorrelation. Think of autocorrelation as signifying a systematic relationship between the residuals measured at different points in time
Autocorrelation Given the model Y t = b 0 + b 1 X t + u t Think of autocorrelation as signifying a systematic relationship between the residuals measured at different points in time This could be caused
More informationProblem Set 10: Panel Data
Problem Set 10: Panel Data 1. Read in the data set, e11panel1.dta from the course website. This contains data on a sample or 1252 men and women who were asked about their hourly wage in two years, 2005
More informationECON2228 Notes 2. Christopher F Baum. Boston College Economics. cfb (BC Econ) ECON2228 Notes / 47
ECON2228 Notes 2 Christopher F Baum Boston College Economics 2014 2015 cfb (BC Econ) ECON2228 Notes 2 2014 2015 1 / 47 Chapter 2: The simple regression model Most of this course will be concerned with
More informationECO220Y Simple Regression: Testing the Slope
ECO220Y Simple Regression: Testing the Slope Readings: Chapter 18 (Sections 18.3-18.5) Winter 2012 Lecture 19 (Winter 2012) Simple Regression Lecture 19 1 / 32 Simple Regression Model y i = β 0 + β 1 x
More informationECON2228 Notes 10. Christopher F Baum. Boston College Economics. cfb (BC Econ) ECON2228 Notes / 54
ECON2228 Notes 10 Christopher F Baum Boston College Economics 2014 2015 cfb (BC Econ) ECON2228 Notes 10 2014 2015 1 / 54 erial correlation and heteroskedasticity in time series regressions Chapter 12:
More informationQuestion 1 carries a weight of 25%; Question 2 carries 20%; Question 3 carries 20%; Question 4 carries 35%.
UNIVERSITY OF EAST ANGLIA School of Economics Main Series PGT Examination 017-18 ECONOMETRIC METHODS ECO-7000A Time allowed: hours Answer ALL FOUR Questions. Question 1 carries a weight of 5%; Question
More informationWeek 3: Simple Linear Regression
Week 3: Simple Linear Regression Marcelo Coca Perraillon University of Colorado Anschutz Medical Campus Health Services Research Methods I HSMP 7607 2017 c 2017 PERRAILLON ALL RIGHTS RESERVED 1 Outline
More informationModels, Testing, and Correction of Heteroskedasticity. James L. Powell Department of Economics University of California, Berkeley
Models, Testing, and Correction of Heteroskedasticity James L. Powell Department of Economics University of California, Berkeley Aitken s GLS and Weighted LS The Generalized Classical Regression Model
More informationLECTURE 10. Introduction to Econometrics. Multicollinearity & Heteroskedasticity
LECTURE 10 Introduction to Econometrics Multicollinearity & Heteroskedasticity November 22, 2016 1 / 23 ON PREVIOUS LECTURES We discussed the specification of a regression equation Specification consists
More information2. (3.5) (iii) Simply drop one of the independent variables, say leisure: GP A = β 0 + β 1 study + β 2 sleep + β 3 work + u.
BOSTON COLLEGE Department of Economics EC 228 Econometrics, Prof. Baum, Ms. Yu, Fall 2003 Problem Set 3 Solutions Problem sets should be your own work. You may work together with classmates, but if you
More informationFixed and Random Effects Models: Vartanian, SW 683
: Vartanian, SW 683 Fixed and random effects models See: http://teaching.sociology.ul.ie/dcw/confront/node45.html When you have repeated observations per individual this is a problem and an advantage:
More informationHeteroskedasticity and Autocorrelation
Lesson 7 Heteroskedasticity and Autocorrelation Pilar González and Susan Orbe Dpt. Applied Economics III (Econometrics and Statistics) Pilar González and Susan Orbe OCW 2014 Lesson 7. Heteroskedasticity
More informationFTE Employment before FTE Employment after
1. (25 points) In 1992, there was an increase in the (state) minimum wage in one U.S. state (New Jersey) but not in a neighboring location (eastern Pennsylvania). The study provides you with the following
More informationMotivation for multiple regression
Motivation for multiple regression 1. Simple regression puts all factors other than X in u, and treats them as unobserved. Effectively the simple regression does not account for other factors. 2. The slope
More informationLecture#12. Instrumental variables regression Causal parameters III
Lecture#12 Instrumental variables regression Causal parameters III 1 Demand experiment, market data analysis & simultaneous causality 2 Simultaneous causality Your task is to estimate the demand function
More informationSection I. Define or explain the following terms (3 points each) 1. centered vs. uncentered 2 R - 2. Frisch theorem -
First Exam: Economics 388, Econometrics Spring 006 in R. Butler s class YOUR NAME: Section I (30 points) Questions 1-10 (3 points each) Section II (40 points) Questions 11-15 (10 points each) Section III
More informationMediation Analysis: OLS vs. SUR vs. 3SLS Note by Hubert Gatignon July 7, 2013, updated November 15, 2013
Mediation Analysis: OLS vs. SUR vs. 3SLS Note by Hubert Gatignon July 7, 2013, updated November 15, 2013 In Chap. 11 of Statistical Analysis of Management Data (Gatignon, 2014), tests of mediation are
More information