Problem Set - Instrumental Variables

Size: px
Start display at page:

Download "Problem Set - Instrumental Variables"

Transcription

1 Problem Set - Instrumental Variables 1. Consider a simple model to estimate the effect of personal computer (PC) ownership on college grade point average for graduating seniors at a large public university: GP A = β 0 + β 1 P C + u where PC is a binary variable indicating PC ownership. (a) Why might PC ownership be correlated with u? (b) Explain why PC is likely to be related to parents annual income. Does this mean parental income is a good IV for PC? Why or why not? (c) Suppose that, four years ago, the university gave grants to buy computers to roughly one-half of the incoming students, and the students who received grants were randomly chosen. Carefully explain how you would use this information to construct an instrumental variable for PC. (a) More able, more motivated, or more wealthy students may have PC s, all of which may be associated with their GPA. (b) Richer parents can better afford PC s. But they may also (i) better afford other inputs, (ii) be a select group who perform differently, (iii) install vices or virtues that influence GPA. (c) Denote by D = 1 and D = 0 students given grants and not given grants, respectively. Since grants are random, E (ɛ D) = 0. Therefore E (GP A D = 1) = β 0 + β 1 E (P C D = 1) E (GP A D = 0) = β 0 + β 1 E (P C D = 0) E (GP A D = 1) E (GP A D = 0) = β 1 [E (P C D = 1) E (P C D = 0)] β 1 = E (GP A D = 1) E (GP A D = 0) E (P C D = 1) E (P C D = 0) A natural consistent estimator of β 1 is then the Wald estimator, using the sample analogues of the means β 1 = GP A (D=1) GP A (D=0) P C (D=1) P C (D=0) 1

2 2. Suppose that you wish to estimate the effect of class attendance on student performance. A basic model is examscore = β 0 + β 1 attendance + β 2 priorgp A + u where examscore is students score on the exam (from 1 to 6), attendance is the number of seminar meetings attended (from 1 to 12), and priorgp A is the average exam grade last year. (a) Let dist be the distance from the students living quarters to the lecture hall. Do you think dist is uncorrelated with u? (b) Assuming that dist and u are uncorrelated, what other assumption must dist satisfy in order to be a valid IV for attendance? (c) Suppose, we add the interaction term priorgp A attendance. If attendance is correlated with u, then, in general, so is priorgp A attendance. What might be a good IV for priorgp A attendance? (a) If living far away implies lower socioeconomic status, then probably not. If it is mixed, or if we observe most of those relevant dimensions, then perhaps. (b) Relevance (i.e. cov(attendance, dist) 0: Living far away from the lecture hall will likely affect attendance, so it should be a relevant instrument. (c) Use as an instrument priorgp A dist. 3. Consider the model y 1 = z 1 δ 1 + α 1 y 2 + u 1 (1) y 2 = zπ 2 + v 2 (2) where z 1 z. (a) How would you estimate α 1 using 2SLS? What is your instrument? i. Regress y 2 on z and form the linear predictions ŷ 2. ii. Regress y 1 on z 1 and ŷ 2. The instruments are the set of variables in z but not in z 1. 2

3 (b) Consider an alternative estimator of (δ 1, α 1 ): (a) estimate equation (2) by OLS and save the residuals ˆv 2. (b) estimate the following equation by OLS y 1 = z 1 δ 1 + α 1 y 2 + ρˆv 2 + error. Show that the OLS estimates of δ 1 and α 1 from this regression are identical to the 2SLS estimators. (Hint: Use the partitioned regression algebra of OLS. In particular, if ŷ = x 1 ˆβ1 +x 2 ˆβ2 is an OLS regression, ˆβ 1 can be obtained by first regressing x 1 on x 2, getting the residuals, say ẍ 1 and then regressing y on ẍ 1. You must also use the fact that z 1 and ˆv 2 are orthogonal in the sample.) We want to estimate the parameters δ 1 and α 1. Using the hint, the OLSestimates of these parameters can also be obtained by partitioned regression: i. For every variable in x 1 (z 1, y 2 ), regress it onto ˆv 2 and save the residual. Collect all the residuals in ẍ 1. ii. Regress y 1 onto the residuals ẍ 1. By definition, ˆv 2 is orthogonal in sample to z. Therefore, the residuals from regressing z 1 onto ˆv 2 are just z 1. (More precisely, n i=1 z 1iˆv 2i = 0.) Also by definition, y 2 = ŷ 2 + ˆv 2 where ŷ 2 and ˆv 2 are orthogonal in sample, which implies that the residuals from regressing y 2 onto ˆv 2 are simply the first stage fitted values, ŷ 2. In other words, ẍ 1 = (z 1, ŷ 2 ). The 2SLS estimator of β 1 is obtained exactly from the OLS regression of y 1 on (z 1, ŷ 2 ). The intuition is the following: ˆv 2 contains all the endogenous variation in y 2. Since it is also orthogonal (by definition) to all the other included regressors, controlling for ˆv 2 is sufficient to control for the endogeneity bias. This illustrates partitioned regression. There is a much simpler way: y 1 = z 1 δ + αy 2 + ρˆv 2 + error = z 1 δ + αŷ 2 + ρˆv 2 + error where ρ = ρ + α. Since cov (ˆv 2, ŷ 2 ) = cov (ˆv 2, z 1 ) ˆπ 2,1 = 0 in sample, omitting ˆv 2 from the regression does not influence the estimates of δ and α. But then this is just the 2SLS-regression. 3

4 4. (Difficult) Consider the following structural equation y i = β 0 + β 1 x 1i + β 2 x 2i + u i where cov(x 1i, u i ) 0 and cov(x 2i, u i ) = 0. You have an instrumental variable z i for x 1i : cov(z i, x 1i ) 0 and cov(z i, u i ) = 0. Which you would use to estimate the first-stage x 1i = π 0 + π 1 z i + π 2 x 2i + v i (a) Show that plim ˆβ 1 = β 1 but plim ˆβ 2 β 2 when x 2 is included in the first-stage estimation, but not used to construct the instrument ˆx 1 (i.e., ˆx 1 = ˆπ 0 + ˆπ 1 z 1 ) (b) Show that both plim β 1 β 1 and plim β 2 β 2 when x 2 is not included in the first-stage estimation (Hint: Remember that x 1 = ˆx 1 +(x 1 ˆx 1 ) which means that the second stage becomes y i = β 0 + β 1ˆx 1i + β 2 x 2i + u i + β 1 (x 1i ˆx 1i ) }{{} ũ i Hint: The 2SLS estimate of β 1 is ˆβ 1 = n i=1 ˆr 1iy i / n i=1 ˆr2 1i where ˆr 1i = ˆx 1i ˆα 0 ˆα 1 x 2i, the residual from the following regression: ˆx 1i = α 0 + α 1 x 2i + r 1i. The 2SLS estimate of β 2 has a similar expression.) The hint gives us y = β 0 + β 1 x 1 + β 2 x 2 + u = β 0 + β 1 x 1 + β 2 x 2 + [β 1 (x 1 ˆx 1 ) + u] = β 0 + β 1ˆx 1 + β 2 + ũ It also uses partitioned regression (Frisch Waugh Lovell theorem), as in the previous exercise. Define ˆr 1 = ˆx 1 ˆα 0 ˆα 1 x 2 as the residual from a regression of ˆx 1 onto x 2. The theorem then says that the OLSestimate of β 1 from the original regression is equivalent to the estimator from the 4

5 bivariate regression of y on ˆr 1, ie. n ˆβ 1 = Mˆr 1,y i = i=1 ˆr 1iy i n. i=1 ˆr2 1i n i=1 = ˆr 1i (β 0 + β 1ˆr 1i + ũ) n i=1 ˆr2 1i = β 1 + Mˆr 1,ũ so that ˆβ 1 is consistent if and only if Mˆr1,ũ p 0. Inserting from ũ above, ˆβ 1 = β 1 + Mˆr 1,β 1 (x 1 ˆx 1 )+u = β 1 + β 1 Mˆr1,x 1 ˆx 1 + Mˆr 1,u (3) Since Mˆr1,u = ˆπ 1 M z,u p 0 in both (a) and (b), I will disregard this term below. Similarly, we can show that ˆβ 2 = β 2 + β 1 Mˆr2,x 1 ˆx 1 Mˆr2,ˆr 2 (a) In (a), we have that ˆx 1 = ˆπ a 0 + ˆπ a 1z, while the correct first stage is x 1 = π 0 + π 1 z + π 2 x 2 + v. Then, x 1 ˆx 1 = (π 0 ˆπ a 0) + (π 1 ˆπ a 1) z + π 2 x 2 + v. Since we estimate the correct first stage, plim ˆπ a s = π s for s = 0, 1, 2. Then plim (x 1 ˆx 1 ) = π 2 x 2 + v. Therefore, plim ˆβ a 1 = β 1 + β 1 cov (ˆr 1, π 2 x 2 + v) varˆr 1 = β 1 plim ˆβ a 2 = β 2 + β 1 cov (ˆr 2, π 2 x 2 + v) varˆr 2 = β 2 + β 1 cov (ˆr 2, π 2 (ˆx 2 + ˆr 2 )) varˆr 2 = β 2 + β 1 π 2 where the third equality uses that x 2 can be decomposed into the orthogonal parts ˆx 2 + ˆr 2, and that v is independent of ˆr 2. 5

6 (b) To solve (b) exactly is a bit more complicated, but the path is very similar. We first note that plim ˆπ b 1 = π 1 + π 2 cov (z, x 2 ) var (x 2 ) π 1 + B 1 π 1 which implies that plim Mˆr1,x 1 ˆx 1 = B 1 cov (ˆr 1, z) which is obviously nonzero from the definition of ˆr 1, any time B 1 0. This is enough to show that the estimator is inconsistent, and shows that we get a consistent estimate of β 1 only if the instrument z is uncorrelated with x 2 (or, obviously, if x 2 is uncorrelated with x 1, i.e. π 2 = 0, in which case we don t need x 2 in the first OR the second stage). Therefore, if the exclusion restriction holds conditional on x 2, we need to control for x 2 in the first stage! To get the exact bias, we derive the following { plim ˆπ 0 b = π 0 + π 2 E [x 2 ] E [z] cov (z, x } 2) var (x 2 ) plim (x 1 ˆx 1 ) = B 0 B 1 z + π 2 x 2 + v. plim = plim Mˆr1,ˆπ b 0 +ˆπb 1 z ˆα 0 ˆα 1 x 2 = plim (ˆπ b 1) plim Mˆr1,z = (π 1 + B 1 ) cov (ˆr 1, z) which gives the probability limit of the estimator, plim ˆβ 1 b B 1 cov (ˆr 1, z) = β 1 β 1 (π 1 + B 1 ) cov (ˆr 1, z) ( ) π1 = β 1 π 1 + B ( 1 ) π 1 = β 1 π 1 + π 2 cov (z, x 2 ) /var (x 2 ) π 0 + B 0 π 0 where the final equality just replaces B 1 with its definition from above. For ˆβ 2 we proceed similarly. 6

Exercise Sheet 4 Instrumental Variables and Two Stage Least Squares Estimation

Exercise Sheet 4 Instrumental Variables and Two Stage Least Squares Estimation Exercise Sheet 4 Instrumental Variables and Two Stage Least Squares Estimation ECONOMETRICS I. UC3M 1. [W 15.1] Consider a simple model to estimate the e ect of personal computer (P C) ownership on the

More information

ECO375 Tutorial 8 Instrumental Variables

ECO375 Tutorial 8 Instrumental Variables ECO375 Tutorial 8 Instrumental Variables Matt Tudball University of Toronto Mississauga November 16, 2017 Matt Tudball (University of Toronto) ECO375H5 November 16, 2017 1 / 22 Review: Endogeneity Instrumental

More information

Econ 2120: Section 2

Econ 2120: Section 2 Econ 2120: Section 2 Part I - Linear Predictor Loose Ends Ashesh Rambachan Fall 2018 Outline Big Picture Matrix Version of the Linear Predictor and Least Squares Fit Linear Predictor Least Squares Omitted

More information

Econometrics. Week 8. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague

Econometrics. 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 information

1 Motivation for Instrumental Variable (IV) Regression

1 Motivation for Instrumental Variable (IV) Regression ECON 370: IV & 2SLS 1 Instrumental Variables Estimation and Two Stage Least Squares Econometric Methods, ECON 370 Let s get back to the thiking in terms of cross sectional (or pooled cross sectional) data

More information

Final Exam. Economics 835: Econometrics. Fall 2010

Final Exam. Economics 835: Econometrics. Fall 2010 Final Exam Economics 835: Econometrics Fall 2010 Please answer the question I ask - no more and no less - and remember that the correct answer is often short and simple. 1 Some short questions a) For each

More information

Motivation for multiple regression

Motivation 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 information

Dealing With Endogeneity

Dealing With Endogeneity Dealing With Endogeneity Junhui Qian December 22, 2014 Outline Introduction Instrumental Variable Instrumental Variable Estimation Two-Stage Least Square Estimation Panel Data Endogeneity in Econometrics

More information

ECON Introductory Econometrics. Lecture 16: Instrumental variables

ECON 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 information

Problem Set #6: OLS. Economics 835: Econometrics. Fall 2012

Problem Set #6: OLS. Economics 835: Econometrics. Fall 2012 Problem Set #6: OLS Economics 835: Econometrics Fall 202 A preliminary result Suppose we have a random sample of size n on the scalar random variables (x, y) with finite means, variances, and covariance.

More information

In the bivariate regression model, the original parameterization is. Y i = β 1 + β 2 X2 + β 2 X2. + β 2 (X 2i X 2 ) + ε i (2)

In the bivariate regression model, the original parameterization is. Y i = β 1 + β 2 X2 + β 2 X2. + β 2 (X 2i X 2 ) + ε i (2) RNy, econ460 autumn 04 Lecture note Orthogonalization and re-parameterization 5..3 and 7.. in HN Orthogonalization of variables, for example X i and X means that variables that are correlated are made

More information

Wooldridge, Introductory Econometrics, 4th ed. Chapter 15: Instrumental variables and two stage least squares

Wooldridge, Introductory Econometrics, 4th ed. Chapter 15: Instrumental variables and two stage least squares Wooldridge, Introductory Econometrics, 4th ed. Chapter 15: Instrumental variables and two stage least squares Many economic models involve endogeneity: that is, a theoretical relationship does not fit

More information

Econometrics Problem Set 4

Econometrics Problem Set 4 Econometrics Problem Set 4 WISE, Xiamen University Spring 2016-17 Conceptual Questions 1. This question refers to the estimated regressions in shown in Table 1 computed using data for 1988 from the CPS.

More information

ECON 3150/4150, Spring term Lecture 7

ECON 3150/4150, Spring term Lecture 7 ECON 3150/4150, Spring term 2014. Lecture 7 The multivariate regression model (I) Ragnar Nymoen University of Oslo 4 February 2014 1 / 23 References to Lecture 7 and 8 SW Ch. 6 BN Kap 7.1-7.8 2 / 23 Omitted

More information

IV Estimation and its Limitations: Weak Instruments and Weakly Endogeneous Regressors

IV Estimation and its Limitations: Weak Instruments and Weakly Endogeneous Regressors IV Estimation and its Limitations: Weak Instruments and Weakly Endogeneous Regressors Laura Mayoral IAE, Barcelona GSE and University of Gothenburg Gothenburg, May 2015 Roadmap Deviations from the standard

More information

Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16)

Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16) Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16) 1 2 Model Consider a system of two regressions y 1 = β 1 y 2 + u 1 (1) y 2 = β 2 y 1 + u 2 (2) This is a simultaneous equation model

More information

ECON 3150/4150, Spring term Lecture 6

ECON 3150/4150, Spring term Lecture 6 ECON 3150/4150, Spring term 2013. Lecture 6 Review of theoretical statistics for econometric modelling (II) Ragnar Nymoen University of Oslo 31 January 2013 1 / 25 References to Lecture 3 and 6 Lecture

More information

The multiple regression model; Indicator variables as regressors

The multiple regression model; Indicator variables as regressors The multiple regression model; Indicator variables as regressors Ragnar Nymoen University of Oslo 28 February 2013 1 / 21 This lecture (#12): Based on the econometric model specification from Lecture 9

More information

Instrumental Variables

Instrumental Variables Instrumental Variables Department of Economics University of Wisconsin-Madison September 27, 2016 Treatment Effects Throughout the course we will focus on the Treatment Effect Model For now take that to

More information

Notes 6: Multivariate regression ECO 231W - Undergraduate Econometrics

Notes 6: Multivariate regression ECO 231W - Undergraduate Econometrics Notes 6: Multivariate regression ECO 231W - Undergraduate Econometrics Prof. Carolina Caetano 1 Notation and language Recall the notation that we discussed in the previous classes. We call the outcome

More information

IV Estimation and its Limitations: Weak Instruments and Weakly Endogeneous Regressors

IV Estimation and its Limitations: Weak Instruments and Weakly Endogeneous Regressors IV Estimation and its Limitations: Weak Instruments and Weakly Endogeneous Regressors Laura Mayoral IAE, Barcelona GSE and University of Gothenburg Gothenburg, May 2015 Roadmap of the course Introduction.

More information

Ec1123 Section 7 Instrumental Variables

Ec1123 Section 7 Instrumental Variables Ec1123 Section 7 Instrumental Variables Andrea Passalacqua Harvard University andreapassalacqua@g.harvard.edu November 16th, 2017 Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November

More information

Intro to Linear Regression

Intro to Linear Regression Intro to Linear Regression Introduction to Regression Regression is a statistical procedure for modeling the relationship among variables to predict the value of a dependent variable from one or more predictor

More information

Lecture 9: Panel Data Model (Chapter 14, Wooldridge Textbook)

Lecture 9: Panel Data Model (Chapter 14, Wooldridge Textbook) Lecture 9: Panel Data Model (Chapter 14, Wooldridge Textbook) 1 2 Panel Data Panel data is obtained by observing the same person, firm, county, etc over several periods. Unlike the pooled cross sections,

More information

Reference: Davidson and MacKinnon Ch 2. In particular page

Reference: Davidson and MacKinnon Ch 2. In particular page RNy, econ460 autumn 03 Lecture note Reference: Davidson and MacKinnon Ch. In particular page 57-8. Projection matrices The matrix M I X(X X) X () is often called the residual maker. That nickname is easy

More information

WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, Academic Year Exam Version: A

WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, Academic Year Exam Version: A WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, 2016-17 Academic Year Exam Version: A INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This

More information

The returns to schooling, ability bias, and regression

The returns to schooling, ability bias, and regression The returns to schooling, ability bias, and regression Jörn-Steffen Pischke LSE October 4, 2016 Pischke (LSE) Griliches 1977 October 4, 2016 1 / 44 Counterfactual outcomes Scholing for individual i is

More information

Linear IV and Simultaneous Equations

Linear IV and Simultaneous Equations Linear IV and Daniel Schmierer Econ 312 April 6, 2007 Setup Linear regression model Y = X β + ε (1) Endogeneity of X means that X and ε are correlated, ie. E(X ε) 0. Suppose we observe another variable

More information

Lecture 6: Dynamic panel models 1

Lecture 6: Dynamic panel models 1 Lecture 6: Dynamic panel models 1 Ragnar Nymoen Department of Economics, UiO 16 February 2010 Main issues and references Pre-determinedness and endogeneity of lagged regressors in FE model, and RE model

More information

Lecture 8: Instrumental Variables Estimation

Lecture 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 information

Lecture 6: Geometry of OLS Estimation of Linear Regession

Lecture 6: Geometry of OLS Estimation of Linear Regession Lecture 6: Geometry of OLS Estimation of Linear Regession Xuexin Wang WISE Oct 2013 1 / 22 Matrix Algebra An n m matrix A is a rectangular array that consists of nm elements arranged in n rows and m columns

More information

8. Instrumental variables regression

8. Instrumental variables regression 8. Instrumental variables regression Recall: In Section 5 we analyzed five sources of estimation bias arising because the regressor is correlated with the error term Violation of the first OLS assumption

More information

WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, Academic Year Exam Version: A

WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, Academic Year Exam Version: A WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, 2016-17 Academic Year Exam Version: A INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This

More information

Intro to Linear Regression

Intro to Linear Regression Intro to Linear Regression Introduction to Regression Regression is a statistical procedure for modeling the relationship among variables to predict the value of a dependent variable from one or more predictor

More information

Problem Set # 1. Master in Business and Quantitative Methods

Problem Set # 1. Master in Business and Quantitative Methods Problem Set # 1 Master in Business and Quantitative Methods Contents 0.1 Problems on endogeneity of the regressors........... 2 0.2 Lab exercises on endogeneity of the regressors......... 4 1 0.1 Problems

More information

EMERGING MARKETS - Lecture 2: Methodology refresher

EMERGING MARKETS - Lecture 2: Methodology refresher EMERGING MARKETS - Lecture 2: Methodology refresher Maria Perrotta April 4, 2013 SITE http://www.hhs.se/site/pages/default.aspx My contact: maria.perrotta@hhs.se Aim of this class There are many different

More information

UNIVERSIDAD CARLOS III DE MADRID ECONOMETRICS Academic year 2009/10 FINAL EXAM (2nd Call) June, 25, 2010

UNIVERSIDAD CARLOS III DE MADRID ECONOMETRICS Academic year 2009/10 FINAL EXAM (2nd Call) June, 25, 2010 UNIVERSIDAD CARLOS III DE MADRID ECONOMETRICS Academic year 2009/10 FINAL EXAM (2nd Call) June, 25, 2010 Very important: Take into account that: 1. Each question, unless otherwise stated, requires a complete

More information

ECO375 Tutorial 9 2SLS Applications and Endogeneity Tests

ECO375 Tutorial 9 2SLS Applications and Endogeneity Tests ECO375 Tutorial 9 2SLS Applications and Endogeneity Tests Matt Tudball University of Toronto Mississauga November 23, 2017 Matt Tudball (University of Toronto) ECO375H5 November 23, 2017 1 / 33 Hausman

More information

We begin by thinking about population relationships.

We begin by thinking about population relationships. Conditional Expectation Function (CEF) We begin by thinking about population relationships. CEF Decomposition Theorem: Given some outcome Y i and some covariates X i there is always a decomposition where

More information

Econometrics Problem Set 11

Econometrics Problem Set 11 Econometrics Problem Set WISE, Xiamen University Spring 207 Conceptual Questions. (SW 2.) This question refers to the panel data regressions summarized in the following table: Dependent variable: ln(q

More information

Topics in Applied Econometrics and Development - Spring 2014

Topics in Applied Econometrics and Development - Spring 2014 Topic 2: Topics in Applied Econometrics and Development - Spring 2014 Single-Equation Linear Model The population model is linear in its parameters: y = β 0 + β 1 x 1 + β 2 x 2 +... + β K x K + u - y,

More information

STAT 350: Geometry of Least Squares

STAT 350: Geometry of Least Squares The Geometry of Least Squares Mathematical Basics Inner / dot product: a and b column vectors a b = a T b = a i b i a b a T b = 0 Matrix Product: A is r s B is s t (AB) rt = s A rs B st Partitioned Matrices

More information

Warwick Economics Summer School Topics in Microeconometrics Instrumental Variables Estimation

Warwick 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 information

Ability Bias, Errors in Variables and Sibling Methods. James J. Heckman University of Chicago Econ 312 This draft, May 26, 2006

Ability Bias, Errors in Variables and Sibling Methods. James J. Heckman University of Chicago Econ 312 This draft, May 26, 2006 Ability Bias, Errors in Variables and Sibling Methods James J. Heckman University of Chicago Econ 312 This draft, May 26, 2006 1 1 Ability Bias Consider the model: log = 0 + 1 + where =income, = schooling,

More information

Econometrics Problem Set 6

Econometrics Problem Set 6 Econometrics Problem Set 6 WISE, Xiamen University Spring 2016-17 Conceptual Questions 1. This question refers to the estimated regressions shown in Table 1 computed using data for 1988 from the CPS. The

More information

The regression model with one stochastic regressor (part II)

The regression model with one stochastic regressor (part II) The regression model with one stochastic regressor (part II) 3150/4150 Lecture 7 Ragnar Nymoen 6 Feb 2012 We will finish Lecture topic 4: The regression model with stochastic regressor We will first look

More information

Econometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018

Econometrics 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 information

08 Endogenous Right-Hand-Side Variables. Andrius Buteikis,

08 Endogenous Right-Hand-Side Variables. Andrius Buteikis, 08 Endogenous Right-Hand-Side Variables Andrius Buteikis, andrius.buteikis@mif.vu.lt http://web.vu.lt/mif/a.buteikis/ Introduction Consider a simple regression model: Y t = α + βx t + u t Under the classical

More information

Statistics 910, #15 1. Kalman Filter

Statistics 910, #15 1. Kalman Filter Statistics 910, #15 1 Overview 1. Summary of Kalman filter 2. Derivations 3. ARMA likelihoods 4. Recursions for the variance Kalman Filter Summary of Kalman filter Simplifications To make the derivations

More information

Correlation and Linear Regression

Correlation and Linear Regression Correlation and Linear Regression Correlation: Relationships between Variables So far, nearly all of our discussion of inferential statistics has focused on testing for differences between group means

More information

Applied Statistics and Econometrics

Applied 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 information

Econometrics Problem Set 6

Econometrics Problem Set 6 Econometrics Problem Set 6 WISE, Xiamen University Spring 2016-17 Conceptual Questions 1. This question refers to the estimated regressions shown in Table 1 computed using data for 1988 from the CPS. The

More information

Answer Key: Problem Set 6

Answer 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 information

Applied Statistics and Econometrics. Giuseppe Ragusa Lecture 15: Instrumental Variables

Applied Statistics and Econometrics. Giuseppe Ragusa Lecture 15: Instrumental Variables Applied Statistics and Econometrics Giuseppe Ragusa Lecture 15: Instrumental Variables Outline Introduction Endogeneity and Exogeneity Valid Instruments TSLS Testing Validity 2 Instrumental Variables Regression

More information

Econometrics Problem Set 3

Econometrics Problem Set 3 Econometrics Problem Set 3 Conceptual Questions 1. This question refers to the estimated regressions in table 1 computed using data for 1988 from the U.S. Current Population Survey. The data set consists

More information

Econ 1123: Section 2. Review. Binary Regressors. Bivariate. Regression. Omitted Variable Bias

Econ 1123: Section 2. Review. Binary Regressors. Bivariate. Regression. Omitted Variable Bias Contact Information Elena Llaudet Sections are voluntary. My office hours are Thursdays 5pm-7pm in Littauer Mezzanine 34-36 (Note room change) You can email me administrative questions to ellaudet@gmail.com.

More information

4.8 Instrumental Variables

4.8 Instrumental Variables 4.8. INSTRUMENTAL VARIABLES 35 4.8 Instrumental Variables A major complication that is emphasized in microeconometrics is the possibility of inconsistent parameter estimation due to endogenous regressors.

More information

Final Exam - Solutions

Final Exam - Solutions Ecn 102 - Analysis of Economic Data University of California - Davis March 17, 2010 Instructor: John Parman Final Exam - Solutions You have until 12:30pm to complete this exam. Please remember to put your

More information

Econometrics of causal inference. Throughout, we consider the simplest case of a linear outcome equation, and homogeneous

Econometrics of causal inference. Throughout, we consider the simplest case of a linear outcome equation, and homogeneous Econometrics of causal inference Throughout, we consider the simplest case of a linear outcome equation, and homogeneous effects: y = βx + ɛ (1) where y is some outcome, x is an explanatory variable, and

More information

ECON3150/4150 Spring 2015

ECON3150/4150 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 information

Mid-term exam Practice problems

Mid-term exam Practice problems Mid-term exam Practice problems Most problems are short answer problems. You receive points for the answer and the explanation. Full points require both, unless otherwise specified. Explaining your answer

More information

Chapter 6: Linear Regression With Multiple Regressors

Chapter 6: Linear Regression With Multiple Regressors Chapter 6: Linear Regression With Multiple Regressors 1-1 Outline 1. Omitted variable bias 2. Causality and regression analysis 3. Multiple regression and OLS 4. Measures of fit 5. Sampling distribution

More information

4 Instrumental Variables Single endogenous variable One continuous instrument. 2

4 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 information

Regression Analysis. Ordinary Least Squares. The Linear Model

Regression Analysis. Ordinary Least Squares. The Linear Model Regression Analysis Linear regression is one of the most widely used tools in statistics. Suppose we were jobless college students interested in finding out how big (or small) our salaries would be 20

More information

Econometrics Review questions for exam

Econometrics Review questions for exam Econometrics Review questions for exam Nathaniel Higgins nhiggins@jhu.edu, 1. Suppose you have a model: y = β 0 x 1 + u You propose the model above and then estimate the model using OLS to obtain: ŷ =

More information

Instrumental Variables and the Problem of Endogeneity

Instrumental Variables and the Problem of Endogeneity Instrumental Variables and the Problem of Endogeneity September 15, 2015 1 / 38 Exogeneity: Important Assumption of OLS In a standard OLS framework, y = xβ + ɛ (1) and for unbiasedness we need E[x ɛ] =

More information

Basic econometrics. Tutorial 3. Dipl.Kfm. Johannes Metzler

Basic econometrics. Tutorial 3. Dipl.Kfm. Johannes Metzler Basic econometrics Tutorial 3 Dipl.Kfm. Introduction Some of you were asking about material to revise/prepare econometrics fundamentals. First of all, be aware that I will not be too technical, only as

More information

THE AUSTRALIAN NATIONAL UNIVERSITY. Second Semester Final Examination November, Econometrics II: Econometric Modelling (EMET 2008/6008)

THE AUSTRALIAN NATIONAL UNIVERSITY. Second Semester Final Examination November, Econometrics II: Econometric Modelling (EMET 2008/6008) THE AUSTRALIAN NATIONAL UNIVERSITY Second Semester Final Examination November, 2014 Econometrics II: Econometric Modelling (EMET 2008/6008) Reading Time: 5 Minutes Writing Time: 90 Minutes Permitted Materials:

More information

Applied Microeconometrics I

Applied Microeconometrics I Applied Microeconometrics I Lecture 6: Instrumental variables in action Manuel Bagues Aalto University September 21 2017 Lecture Slides 1/ 20 Applied Microeconometrics I A few logistic reminders... Tutorial

More information

ECON Introductory Econometrics. Lecture 6: OLS with Multiple Regressors

ECON Introductory Econometrics. Lecture 6: OLS with Multiple Regressors ECON4150 - Introductory Econometrics Lecture 6: OLS with Multiple Regressors Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 6 Lecture outline 2 Violation of first Least Squares assumption

More information

Applied Quantitative Methods II

Applied Quantitative Methods II Applied Quantitative Methods II Lecture 10: Panel Data Klára Kaĺıšková Klára Kaĺıšková AQM II - Lecture 10 VŠE, SS 2016/17 1 / 38 Outline 1 Introduction 2 Pooled OLS 3 First differences 4 Fixed effects

More information

Homework Set 2, ECO 311, Fall 2014

Homework Set 2, ECO 311, Fall 2014 Homework Set 2, ECO 311, Fall 2014 Due Date: At the beginning of class on October 21, 2014 Instruction: There are twelve questions. Each question is worth 2 points. You need to submit the answers of only

More information

9. Linear Regression and Correlation

9. Linear Regression and Correlation 9. Linear Regression and Correlation Data: y a quantitative response variable x a quantitative explanatory variable (Chap. 8: Recall that both variables were categorical) For example, y = annual income,

More information

Data Analysis and Machine Learning Lecture 12: Multicollinearity, Bias-Variance Trade-off, Cross-validation and Shrinkage Methods.

Data Analysis and Machine Learning Lecture 12: Multicollinearity, Bias-Variance Trade-off, Cross-validation and Shrinkage Methods. TheThalesians Itiseasyforphilosopherstoberichiftheychoose Data Analysis and Machine Learning Lecture 12: Multicollinearity, Bias-Variance Trade-off, Cross-validation and Shrinkage Methods Ivan Zhdankin

More information

ECON3150/4150 Spring 2016

ECON3150/4150 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 information

Unless provided with information to the contrary, assume for each question below that the Classical Linear Model assumptions hold.

Unless provided with information to the contrary, assume for each question below that the Classical Linear Model assumptions hold. Economics 345: Applied Econometrics Section A01 University of Victoria Midterm Examination #2 Version 1 SOLUTIONS Spring 2015 Instructor: Martin Farnham Unless provided with information to the contrary,

More information

Chapter 6. Panel Data. Joan Llull. Quantitative Statistical Methods II Barcelona GSE

Chapter 6. Panel Data. Joan Llull. Quantitative Statistical Methods II Barcelona GSE Chapter 6. Panel Data Joan Llull Quantitative Statistical Methods II Barcelona GSE Introduction Chapter 6. Panel Data 2 Panel data The term panel data refers to data sets with repeated observations over

More information

Specification testing in panel data models estimated by fixed effects with instrumental variables

Specification testing in panel data models estimated by fixed effects with instrumental variables Specification testing in panel data models estimated by fixed effects wh instrumental variables Carrie Falls Department of Economics Michigan State Universy Abstract I show that a handful of the regressions

More information

Lecture 10: Panel Data

Lecture 10: Panel Data Lecture 10: Instructor: Department of Economics Stanford University 2011 Random Effect Estimator: β R y it = x itβ + u it u it = α i + ɛ it i = 1,..., N, t = 1,..., T E (α i x i ) = E (ɛ it x i ) = 0.

More information

STOCKHOLM UNIVERSITY Department of Economics Course name: Empirical Methods Course code: EC40 Examiner: Lena Nekby Number of credits: 7,5 credits Date of exam: Saturday, May 9, 008 Examination time: 3

More information

4 Instrumental Variables Single endogenous variable One continuous instrument. 2

4 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 information

ECONOMETRICS FIELD EXAM Michigan State University May 9, 2008

ECONOMETRICS FIELD EXAM Michigan State University May 9, 2008 ECONOMETRICS FIELD EXAM Michigan State University May 9, 2008 Instructions: Answer all four (4) questions. Point totals for each question are given in parenthesis; there are 00 points possible. Within

More information

Homework Set 2, ECO 311, Spring 2014

Homework Set 2, ECO 311, Spring 2014 Homework Set 2, ECO 311, Spring 2014 Due Date: At the beginning of class on March 31, 2014 Instruction: There are twelve questions. Each question is worth 2 points. You need to submit the answers of only

More information

Empirical Application of Simple Regression (Chapter 2)

Empirical 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 information

Lecture 14 Simple Linear Regression

Lecture 14 Simple Linear Regression Lecture 4 Simple Linear Regression Ordinary Least Squares (OLS) Consider the following simple linear regression model where, for each unit i, Y i is the dependent variable (response). X i is the independent

More information

Multiple Linear Regression

Multiple Linear Regression Multiple Linear Regression Asymptotics Asymptotics Multiple Linear Regression: Assumptions Assumption MLR. (Linearity in parameters) Assumption MLR. (Random Sampling from the population) We have a random

More information

y response variable x 1, x 2,, x k -- a set of explanatory variables

y response variable x 1, x 2,, x k -- a set of explanatory variables 11. Multiple Regression and Correlation y response variable x 1, x 2,, x k -- a set of explanatory variables In this chapter, all variables are assumed to be quantitative. Chapters 12-14 show how to incorporate

More information

Linear Regression with Time Series Data

Linear Regression with Time Series Data u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f e c o n o m i c s Econometrics II Linear Regression with Time Series Data Morten Nyboe Tabor u n i v e r s i t y o f c o p e n h a g

More information

Applied Econometrics (MSc.) Lecture 3 Instrumental Variables

Applied Econometrics (MSc.) Lecture 3 Instrumental Variables Applied Econometrics (MSc.) Lecture 3 Instrumental Variables Estimation - Theory Department of Economics University of Gothenburg December 4, 2014 1/28 Why IV estimation? So far, in OLS, we assumed independence.

More information

MATH 1150 Chapter 2 Notation and Terminology

MATH 1150 Chapter 2 Notation and Terminology MATH 1150 Chapter 2 Notation and Terminology Categorical Data The following is a dataset for 30 randomly selected adults in the U.S., showing the values of two categorical variables: whether or not the

More information

Linear Regression with Multiple Regressors

Linear Regression with Multiple Regressors Linear Regression with Multiple Regressors (SW Chapter 6) Outline 1. Omitted variable bias 2. Causality and regression analysis 3. Multiple regression and OLS 4. Measures of fit 5. Sampling distribution

More information

ECON The Simple Regression Model

ECON The Simple Regression Model ECON 351 - The Simple Regression Model Maggie Jones 1 / 41 The Simple Regression Model Our starting point will be the simple regression model where we look at the relationship between two variables In

More information

Two-Variable Regression Model: The Problem of Estimation

Two-Variable Regression Model: The Problem of Estimation Two-Variable Regression Model: The Problem of Estimation Introducing the Ordinary Least Squares Estimator Jamie Monogan University of Georgia Intermediate Political Methodology Jamie Monogan (UGA) Two-Variable

More information

Linear models and their mathematical foundations: Simple linear regression

Linear models and their mathematical foundations: Simple linear regression Linear models and their mathematical foundations: Simple linear regression Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/21 Introduction

More information

Econ 582 Fixed Effects Estimation of Panel Data

Econ 582 Fixed Effects Estimation of Panel Data Econ 582 Fixed Effects Estimation of Panel Data Eric Zivot May 28, 2012 Panel Data Framework = x 0 β + = 1 (individuals); =1 (time periods) y 1 = X β ( ) ( 1) + ε Main question: Is x uncorrelated with?

More information

Instrumental Variables. Ethan Kaplan

Instrumental Variables. Ethan Kaplan Instrumental Variables Ethan Kaplan 1 Instrumental Variables: Intro. Bias in OLS: Consider a linear model: Y = X + Suppose that then OLS yields: cov (X; ) = ^ OLS = X 0 X 1 X 0 Y = X 0 X 1 X 0 (X + ) =)

More information

An explanation of Two Stage Least Squares

An 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 information

5. Let W follow a normal distribution with mean of μ and the variance of 1. Then, the pdf of W is

5. 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 information

Regression Discontinuity

Regression Discontinuity Regression Discontinuity STP Advanced Econometrics Lecture Douglas G. Steigerwald UC Santa Barbara March 2006 D. Steigerwald (Institute) Regression Discontinuity 03/06 1 / 11 Intuition Reference: Mostly

More information

Chapter 9: Assessing Studies Based on Multiple Regression. Copyright 2011 Pearson Addison-Wesley. All rights reserved.

Chapter 9: Assessing Studies Based on Multiple Regression. Copyright 2011 Pearson Addison-Wesley. All rights reserved. Chapter 9: Assessing Studies Based on Multiple Regression 1-1 9-1 Outline 1. Internal and External Validity 2. Threats to Internal Validity a) Omitted variable bias b) Functional form misspecification

More information