Limitations of regression analysis

Size: px
Start display at page:

Download "Limitations of regression analysis"

Transcription

1 Limitations of regression analysis Ragnar Nymoen Department of Economics, UiO 8 February 2009

2 Overview What are the limitations to regression? Simultaneous equations bias Measurement errors in explanatory variables In both cases the explanatory variable is not exogenous in the econometric sense Main reference is G Ch 15.1 and 15.2;. B Ch 8.1, 10.1 and 10.2;K: Ch 9.3,10.2

3 What are the limitations to regression analysis? It is not linearity in variables, as we have seen it is not linearity in parameters, although we have only covered the linear regression model here Remember that by rst estimating the linear model we can use the results to estimate parameters that are non-linear functions of the estimated model s parameters (the delta method or its equivalent in the Bårdsen method) If the model is non-linear in the parameter from the outset, can use Non-Linear Least Squares to t the best non-linear curve to the data. Greene Ch 11, not in the syllabus to this course. It si not con ned to single equation, as we seen with the SURE estimator. The real limitation to the regression model is when the regression function does not contain the parameter of interest

4 A simple Keynes model Let Y t denote GDP in period t D 1, 2,..., T. C t is endogenous expenditure and let X t denote exogenous expenditure. Assume that C t depends on GDP, then our example model is Y t D C t C X t (1) C t D b 1 C b 2 Y t C " t, 0 < b 2 < 1 (2) " t is a random disturbance term. We assume that it is white noise uncorrelated with X t. For simplicity we assume normality " t N.0, 2 " /. The parameter of interest is the marginal propensity to consume b 2.

5 The reduced form of the model (1) and (2) de nes a simultaneous equations model. Solution for the two endogenous variables: Y t D 11 C 12 X t C 1t (3) C t D 21 C 22 X t C 2t (4) 11 D b 1 12 D 1 1t D 1 " t 21 D b 1 21 D b 2 2t D 1 " t

6 The distribution of Y and C The Reduced Form written more compactly Y t D yt C 1t (5) C t D ct C 2t (6) where 1t 2t N 2 0, y cy cy 2 c j X t. (7) The conditional distributions of the stochastic variables 1t and 2t are binormal with zero expectations and variance matrix: 2 y cy j X t. cy 2 c

7 Conditional distribution of C It follows that Y t and C t are normally distributed with the same covariance matrix as. 1t 2t / 0 and expectations yt D 11 C 12 X t, ct D 21 C 22 X t. It also follows (Lect 1) that the conditional distribution of C t is normal with conditional expectation: E [C t j Y t ] D ct c y yt C c y Y t (8) D 21 C 22 X t c y. 11 C 12 X t / C c y Y t D. 21 c y 11 / C. 22 c y 12 /X t C c y Y t

8 We see that The macro model implies (8) as the conditional expextation for C t. It is the valid regression model of C t on Y t and can be estimated with full e cency by OLS. It will not deliver an estimate of the marginal propensity to consume, b 2! In sum: The regression function implied by (1) and (2) is (8), not the regression of C t on Y t and a constant. And the regression function (8) is not helpful for the estimation for the parameter of interest b 1 (in fact since c y D 1 it estimates the identity in this special case) )

9 Simultaneity bias in the macro model example Suppose we estimate the consumption function by OLS regardless. We will estimate some parameter. What is it? P P Ct.Y t NY / Ct.Y t NY / Ob 2 D P D.Yt NY / 2 P Yt.Y t NY / where NY D 1/T P Y t. Ob 2 D 1 P Yt.Y t NY / X fb1 C b 2 Y t C " t g t.y t NY / (9) D P "t.y t NY / b 2 C P.Yt NY / 2 We must evaluate the term P "t.y t NY / P.Yt NY / 2 in the light of the model.

10 Since Y t depends on the shocks " t to consumption, and C t depends on Y t, then " t and Y t are correlated. This correlation will not go away as T grows. Using the RF expression for Y t, the denominator can be written as 1 X.Yt NY / 2 D 1 X 12.X t NX / C. 1t N 1 / 2 T T Take probability limits: plim 1 T X.Yt NY / 2 D D plim 1 T X 2 12.X t NX / 2 C 2 12 plim 1 T X.Xt NX /. 1t N 1 / C plim 1 T X.1t N 1 / 2 D 2 12 Var.X t/ C 2 y

11 plim b O2 b 2 D D plim 1 P T "t.y t NY / plim 1 P T.Yt NY / 2 Cov." t, Y t / 2 12 Var.X t/ C 2 y From the Reduced Form we also have Cov." t, Y t / D E [" t yt ] D E [" t 1 " t ] D 1 Var[" t ] 2 12 Var.X t/ C 2 y D D 2 " 1 2 Var.X t / C 2 "

12 The inconsistency of OLS, plim b O2 b 2 D D 2 " 1 2 Var.Xt / C 2 ". / 2 " Var.X t / C 2 " D./ C 1 Var.X t / 2 " The bias is positive Large variance in X t relative to " t reduces the biases. But it does not kill the bias. The reason is that OLS assumes the wrong model for C t, one with Cov.Y t, " t / D 0. It is not here.

13 Example with an expectations variable Assume the simple regression model (in Greene s notation again): y i D 1 C 2 x i C " i, i D 1, 2,..., n. (10) with all the classical assumptions holding. If xi is an expectations variable that we as econometricians cannot observe or cannot measure without error, we can still try to estimate 1 and 2 using the observable (actual) where x i. We then need to make assumptions about the properties of the di erence u i D x i x i. (11)

14 Assumptions: u i is random, zero mean, variance 2 u Cov.u i, " i / D 0 Cov.u i, x i / D 0 Both u i and " i have the classical properties The model that we estimate becomes: But with y i D 1 C 2 x i C i (12) i D " i 2 u i (13) E [x i i ] D E [.x i C u i /." i 2 u i /] D 2 2 u

15 OLS gives and we have plim P i.x i Nx/ O 2 b 2 D 2 C P.xi Nx/ 2 plim O 2 2 D plim 1 P T i.x i Nx/ plim 1 P T.xi Nx/ 2 we already have that 2 2 u goes into the numerator. The denominator is more work (like in the sim eq case) but intuitively it must boil down to the sum of the variances of xi and u i, hence plim ( O 2 2 / D 2 2 u Var.x i / C 2 u

16 plim O 2 D 2 1 C 2 u Var.x i / < 2 if 2 is positive. It can be shown that by taking the inverse regression, x i on y i, gives an overestimation, so OLS de nes a bound around the true parameter. Measurement errors in y i : No bias problem, but potential for heteroscedasticity. Solution to both classes of bias problems exempli ed here: Replace OLS with other estimators. IV, 2SLS as we shall see.

Chapter 2. Dynamic panel data models

Chapter 2. Dynamic panel data models Chapter 2. Dynamic panel data models Master of Science in Economics - University of Geneva Christophe Hurlin, Université d Orléans Université d Orléans April 2010 Introduction De nition We now consider

More information

problem arises when only a non-random sample is available differs from censored regression model in that x i is also unobserved

problem arises when only a non-random sample is available differs from censored regression model in that x i is also unobserved 4 Data Issues 4.1 Truncated Regression population model y i = x i β + ε i, ε i N(0, σ 2 ) given a random sample, {y i, x i } N i=1, then OLS is consistent and efficient problem arises when only a non-random

More information

Chapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem

Chapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem Chapter Vector autoregressions We begin by taking a look at the data of macroeconomics. A way to summarize the dynamics of macroeconomic data is to make use of vector autoregressions. VAR models have become

More information

Instrumental Variables & 2SLS

Instrumental Variables & 2SLS Instrumental Variables & 2SLS y 1 = β 0 + β 1 y 2 + β 2 z 1 +... β k z k + u y 2 = π 0 + π 1 z k+1 + π 2 z 1 +... π k z k + v Economics 20 - Prof. Schuetze 1 Why Use Instrumental Variables? Instrumental

More information

Note 2 to Computer class: Standard mis-specification tests

Note 2 to Computer class: Standard mis-specification tests Note 2 to Computer class: Standard mis-specification tests Ragnar Nymoen September 2, 2013 1 Why mis-specification testing of econometric models? As econometricians we must relate to the fact that the

More information

Preparation course Msc Business & Econonomics

Preparation course Msc Business & Econonomics Preparation course Msc Business & Econonomics The simple Keynesian model Tom-Reiel Heggedal BI August 2014 TRH (BI) Keynes model August 2014 1 / 19 Assumptions Keynes model Outline for this lecture: Go

More information

1 Another method of estimation: least squares

1 Another method of estimation: least squares 1 Another method of estimation: least squares erm: -estim.tex, Dec8, 009: 6 p.m. (draft - typos/writos likely exist) Corrections, comments, suggestions welcome. 1.1 Least squares in general Assume Y i

More information

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

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

CAPM, Arbitrage, and Linear Factor Models

CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose mean-variance e cient portfolios. By equating these investors

More information

Lecture 15. Endogeneity & Instrumental Variable Estimation

Lecture 15. Endogeneity & Instrumental Variable Estimation Lecture 15. Endogeneity & Instrumental Variable Estimation Saw that measurement error (on right hand side) means that OLS will be biased (biased toward zero) Potential solution to endogeneity instrumental

More information

= C + I + G + NX ECON 302. Lecture 4: Aggregate Expenditures/Keynesian Model: Equilibrium in the Goods Market/Loanable Funds Market

= C + I + G + NX ECON 302. Lecture 4: Aggregate Expenditures/Keynesian Model: Equilibrium in the Goods Market/Loanable Funds Market Intermediate Macroeconomics Lecture 4: Introduction to the Goods Market Review of the Aggregate Expenditures model and the Keynesian Cross ECON 302 Professor Yamin Ahmad Components of Aggregate Demand

More information

16 : Demand Forecasting

16 : Demand Forecasting 16 : Demand Forecasting 1 Session Outline Demand Forecasting Subjective methods can be used only when past data is not available. When past data is available, it is advisable that firms should use statistical

More information

Financial Risk Management Exam Sample Questions/Answers

Financial Risk Management Exam Sample Questions/Answers Financial Risk Management Exam Sample Questions/Answers Prepared by Daniel HERLEMONT 1 2 3 4 5 6 Chapter 3 Fundamentals of Statistics FRM-99, Question 4 Random walk assumes that returns from one time period

More information

Instrumental Variables & 2SLS

Instrumental Variables & 2SLS Instrumental Variables & 2SLS y 1 = β 0 + β 1 y 2 + β 2 z 1 +... β k z k + u y 2 = π 0 + π 1 z k+1 + π 2 z 1 +... π k z k + v Economics 20 - Prof. Schuetze 1 Why Use Instrumental Variables? Instrumental

More information

Instrumental Variables Regression. Instrumental Variables (IV) estimation is used when the model has endogenous s.

Instrumental Variables Regression. Instrumental Variables (IV) estimation is used when the model has endogenous s. Instrumental Variables Regression Instrumental Variables (IV) estimation is used when the model has endogenous s. IV can thus be used to address the following important threats to internal validity: Omitted

More information

Chapter 3: The Multiple Linear Regression Model

Chapter 3: The Multiple Linear Regression Model Chapter 3: The Multiple Linear Regression Model Advanced Econometrics - HEC Lausanne Christophe Hurlin University of Orléans November 23, 2013 Christophe Hurlin (University of Orléans) Advanced Econometrics

More information

y t by left multiplication with 1 (L) as y t = 1 (L) t =ª(L) t 2.5 Variance decomposition and innovation accounting Consider the VAR(p) model where

y t by left multiplication with 1 (L) as y t = 1 (L) t =ª(L) t 2.5 Variance decomposition and innovation accounting Consider the VAR(p) model where . Variance decomposition and innovation accounting Consider the VAR(p) model where (L)y t = t, (L) =I m L L p L p is the lag polynomial of order p with m m coe±cient matrices i, i =,...p. Provided that

More information

Solución del Examen Tipo: 1

Solución del Examen Tipo: 1 Solución del Examen Tipo: 1 Universidad Carlos III de Madrid ECONOMETRICS Academic year 2009/10 FINAL EXAM May 17, 2010 DURATION: 2 HOURS 1. Assume that model (III) verifies the assumptions of the classical

More information

Chapter 4: Vector Autoregressive Models

Chapter 4: Vector Autoregressive Models Chapter 4: Vector Autoregressive Models 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie IV.1 Vector Autoregressive Models (VAR)...

More information

A Note on Parametric and Nonparametric Regression in the Presence of Endogenous Control Variables

A Note on Parametric and Nonparametric Regression in the Presence of Endogenous Control Variables DISCUSSION PAPER SERIES IZA DP No. 2126 A Note on Parametric and Nonparametric Regression in the Presence of Endogenous Control Variables Markus Frölich April 2006 Forschungsinstitut zur Zukunft der Arbeit

More information

E 4101/5101 Lecture 8: Exogeneity

E 4101/5101 Lecture 8: Exogeneity E 4101/5101 Lecture 8: Exogeneity Ragnar Nymoen 17 March 2011 Introduction I Main references: Davidson and MacKinnon, Ch 8.1-8,7, since tests of (weak) exogeneity build on the theory of IV-estimation Ch

More information

Heteroskedasticity and Weighted Least Squares

Heteroskedasticity and Weighted Least Squares Econ 507. Econometric Analysis. Spring 2009 April 14, 2009 The Classical Linear Model: 1 Linearity: Y = Xβ + u. 2 Strict exogeneity: E(u) = 0 3 No Multicollinearity: ρ(x) = K. 4 No heteroskedasticity/

More information

Introduction to Macroeconomics TOPIC 2: The Goods Market

Introduction to Macroeconomics TOPIC 2: The Goods Market TOPIC 2: The Goods Market Annaïg Morin CBS - Department of Economics August 2013 Goods market Road map: 1. Demand for goods 1.1. Components 1.1.1. Consumption 1.1.2. Investment 1.1.3. Government spending

More information

University of Ljubljana Doctoral Programme in Statistics Methodology of Statistical Research Written examination February 14 th, 2014.

University of Ljubljana Doctoral Programme in Statistics Methodology of Statistical Research Written examination February 14 th, 2014. University of Ljubljana Doctoral Programme in Statistics ethodology of Statistical Research Written examination February 14 th, 2014 Name and surname: ID number: Instructions Read carefully the wording

More information

What is the interpretation of R 2?

What is the interpretation of R 2? What is the interpretation of R 2? Karl G. Jöreskog October 2, 1999 Consider a regression equation between a dependent variable y and a set of explanatory variables x'=(x 1, x 2,..., x q ): or in matrix

More information

Econometrics Simple Linear Regression

Econometrics Simple Linear Regression Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight

More information

Introduction to Path Analysis

Introduction to Path Analysis This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this

More information

IMPACT EVALUATION: INSTRUMENTAL VARIABLE METHOD

IMPACT EVALUATION: INSTRUMENTAL VARIABLE METHOD REPUBLIC OF SOUTH AFRICA GOVERNMENT-WIDE MONITORING & IMPACT EVALUATION SEMINAR IMPACT EVALUATION: INSTRUMENTAL VARIABLE METHOD SHAHID KHANDKER World Bank June 2006 ORGANIZED BY THE WORLD BANK AFRICA IMPACT

More information

SYSTEMS OF REGRESSION EQUATIONS

SYSTEMS OF REGRESSION EQUATIONS SYSTEMS OF REGRESSION EQUATIONS 1. MULTIPLE EQUATIONS y nt = x nt n + u nt, n = 1,...,N, t = 1,...,T, x nt is 1 k, and n is k 1. This is a version of the standard regression model where the observations

More information

IAPRI Quantitative Analysis Capacity Building Series. Multiple regression analysis & interpreting results

IAPRI Quantitative Analysis Capacity Building Series. Multiple regression analysis & interpreting results IAPRI Quantitative Analysis Capacity Building Series Multiple regression analysis & interpreting results How important is R-squared? R-squared Published in Agricultural Economics 0.45 Best article of the

More information

ECON 142 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE #2

ECON 142 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE #2 University of California, Berkeley Prof. Ken Chay Department of Economics Fall Semester, 005 ECON 14 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE # Question 1: a. Below are the scatter plots of hourly wages

More information

Variances and covariances

Variances and covariances Chapter 4 Variances and covariances 4.1 Overview The expected value of a random variable gives a crude measure for the center of location of the distribution of that random variable. For instance, if the

More information

1 Teaching notes on GMM 1.

1 Teaching notes on GMM 1. Bent E. Sørensen January 23, 2007 1 Teaching notes on GMM 1. Generalized Method of Moment (GMM) estimation is one of two developments in econometrics in the 80ies that revolutionized empirical work in

More information

Outline of model. Factors of production 1/23/2013. The production function: Y = F(K,L) ECON 3010 Intermediate Macroeconomics

Outline of model. Factors of production 1/23/2013. The production function: Y = F(K,L) ECON 3010 Intermediate Macroeconomics ECON 3010 Intermediate Macroeconomics Chapter 3 National Income: Where It Comes From and Where It Goes Outline of model A closed economy, market-clearing model Supply side factors of production determination

More information

DEPARTMENT OF ECONOMICS. Unit ECON 12122 Introduction to Econometrics. Notes 4 2. R and F tests

DEPARTMENT OF ECONOMICS. Unit ECON 12122 Introduction to Econometrics. Notes 4 2. R and F tests DEPARTMENT OF ECONOMICS Unit ECON 11 Introduction to Econometrics Notes 4 R and F tests These notes provide a summary of the lectures. They are not a complete account of the unit material. You should also

More information

Chapter 3 A Classical Economic Model

Chapter 3 A Classical Economic Model Chapter 3 A Classical Economic Model what determines the economy s total output/income how the prices of the factors of production are determined how total income is distributed what determines the demand

More information

1 Short Introduction to Time Series

1 Short Introduction to Time Series ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The

More information

Comparing Features of Convenient Estimators for Binary Choice Models With Endogenous Regressors

Comparing Features of Convenient Estimators for Binary Choice Models With Endogenous Regressors Comparing Features of Convenient Estimators for Binary Choice Models With Endogenous Regressors Arthur Lewbel, Yingying Dong, and Thomas Tao Yang Boston College, University of California Irvine, and Boston

More information

Regression analysis in practice with GRETL

Regression analysis in practice with GRETL Regression analysis in practice with GRETL Prerequisites You will need the GNU econometrics software GRETL installed on your computer (http://gretl.sourceforge.net/), together with the sample files that

More information

INVESTMENT DECISIONS and PROFIT MAXIMIZATION

INVESTMENT DECISIONS and PROFIT MAXIMIZATION Lecture 6 Investment Decisions The Digital Economist Investment is the act of acquiring income-producing assets, known as physical capital, either as additions to existing assets or to replace assets that

More information

1 The Problem: Endogeneity There are two kinds of variables in our models: exogenous variables and endogenous variables. Endogenous Variables: These a

1 The Problem: Endogeneity There are two kinds of variables in our models: exogenous variables and endogenous variables. Endogenous Variables: These a Notes on Simultaneous Equations and Two Stage Least Squares Estimates Copyright - Jonathan Nagler; April 19, 1999 1. Basic Description of 2SLS ffl The endogeneity problem, and the bias of OLS. ffl The

More information

10. Fixed-Income Securities. Basic Concepts

10. Fixed-Income Securities. Basic Concepts 0. Fixed-Income Securities Fixed-income securities (FIS) are bonds that have no default risk and their payments are fully determined in advance. Sometimes corporate bonds that do not necessarily have certain

More information

2. What are the theoretical and practical consequences of autocorrelation?

2. What are the theoretical and practical consequences of autocorrelation? Lecture 10 Serial Correlation In this lecture, you will learn the following: 1. What is the nature of autocorrelation? 2. What are the theoretical and practical consequences of autocorrelation? 3. Since

More information

From the help desk: Bootstrapped standard errors

From the help desk: Bootstrapped standard errors The Stata Journal (2003) 3, Number 1, pp. 71 80 From the help desk: Bootstrapped standard errors Weihua Guan Stata Corporation Abstract. Bootstrapping is a nonparametric approach for evaluating the distribution

More information

Matrix Algebra and Applications

Matrix Algebra and Applications Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2 - Matrices and Matrix Algebra Reading 1 Chapters

More information

Empirical Methods in Applied Economics

Empirical Methods in Applied Economics Empirical Methods in Applied Economics Jörn-Ste en Pischke LSE October 2005 1 Observational Studies and Regression 1.1 Conditional Randomization Again When we discussed experiments, we discussed already

More information

Introduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by

Introduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes

More information

What s New in Econometrics? Lecture 8 Cluster and Stratified Sampling

What s New in Econometrics? Lecture 8 Cluster and Stratified Sampling What s New in Econometrics? Lecture 8 Cluster and Stratified Sampling Jeff Wooldridge NBER Summer Institute, 2007 1. The Linear Model with Cluster Effects 2. Estimation with a Small Number of Groups and

More information

Maximum Likelihood Estimation of an ARMA(p,q) Model

Maximum Likelihood Estimation of an ARMA(p,q) Model Maximum Likelihood Estimation of an ARMA(p,q) Model Constantino Hevia The World Bank. DECRG. October 8 This note describes the Matlab function arma_mle.m that computes the maximum likelihood estimates

More information

15.062 Data Mining: Algorithms and Applications Matrix Math Review

15.062 Data Mining: Algorithms and Applications Matrix Math Review .6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

More information

Simple Linear Regression Inference

Simple Linear Regression Inference Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

More information

ELEC-E8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems

ELEC-E8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Minimum Mean Square Error (MMSE) MMSE estimation of Gaussian random vectors Linear MMSE estimator for arbitrarily distributed

More information

Econometric Analysis of Cross Section and Panel Data Second Edition. Jeffrey M. Wooldridge. The MIT Press Cambridge, Massachusetts London, England

Econometric Analysis of Cross Section and Panel Data Second Edition. Jeffrey M. Wooldridge. The MIT Press Cambridge, Massachusetts London, England Econometric Analysis of Cross Section and Panel Data Second Edition Jeffrey M. Wooldridge The MIT Press Cambridge, Massachusetts London, England Preface Acknowledgments xxi xxix I INTRODUCTION AND BACKGROUND

More information

Regression III: Advanced Methods

Regression III: Advanced Methods Lecture 5: Linear least-squares Regression III: Advanced Methods William G. Jacoby Department of Political Science Michigan State University http://polisci.msu.edu/jacoby/icpsr/regress3 Simple Linear Regression

More information

So, using the new notation, P X,Y (0,1) =.08 This is the value which the joint probability function for X and Y takes when X=0 and Y=1.

So, using the new notation, P X,Y (0,1) =.08 This is the value which the joint probability function for X and Y takes when X=0 and Y=1. Joint probabilit is the probabilit that the RVs & Y take values &. like the PDF of the two events, and. We will denote a joint probabilit function as P,Y (,) = P(= Y=) Marginal probabilit of is the probabilit

More information

Topic 5: Stochastic Growth and Real Business Cycles

Topic 5: Stochastic Growth and Real Business Cycles Topic 5: Stochastic Growth and Real Business Cycles Yulei Luo SEF of HKU October 1, 2015 Luo, Y. (SEF of HKU) Macro Theory October 1, 2015 1 / 45 Lag Operators The lag operator (L) is de ned as Similar

More information

Multiple Linear Regression in Data Mining

Multiple Linear Regression in Data Mining Multiple Linear Regression in Data Mining Contents 2.1. A Review of Multiple Linear Regression 2.2. Illustration of the Regression Process 2.3. Subset Selection in Linear Regression 1 2 Chap. 2 Multiple

More information

Calculate the holding period return for this investment. It is approximately

Calculate the holding period return for this investment. It is approximately 1. An investor purchases 100 shares of XYZ at the beginning of the year for $35. The stock pays a cash dividend of $3 per share. The price of the stock at the time of the dividend is $30. The dividend

More information

An introduction to Value-at-Risk Learning Curve September 2003

An introduction to Value-at-Risk Learning Curve September 2003 An introduction to Value-at-Risk Learning Curve September 2003 Value-at-Risk The introduction of Value-at-Risk (VaR) as an accepted methodology for quantifying market risk is part of the evolution of risk

More information

The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.

The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium

More information

The Simple Keynesian Theory. The Simple Keynesian Theory of Income Determination. The Simple Keynesian Theory. The Simple Keynesian Theory

The Simple Keynesian Theory. The Simple Keynesian Theory of Income Determination. The Simple Keynesian Theory. The Simple Keynesian Theory of Income Determination Some Basic Definitions Endogenous variables Output Consumer Spending To become endogenous variables Investment spending Net exports Interest rates Inflation 1 2 Some Basic Definitions

More information

Introduction to Dynamic Models. Slide set #1 (Ch 1.1-1.6 in IDM).

Introduction to Dynamic Models. Slide set #1 (Ch 1.1-1.6 in IDM). 1 Introduction Introduction to Dynamic Models. Slide set #1 (Ch 1.1-1.6 in IDM). Ragnar Nymoen University of Oslo, Department of Economics We observe that economic agents take time to adjust their behaviour

More information

LOGIT AND PROBIT ANALYSIS

LOGIT AND PROBIT ANALYSIS LOGIT AND PROBIT ANALYSIS A.K. Vasisht I.A.S.R.I., Library Avenue, New Delhi 110 012 amitvasisht@iasri.res.in In dummy regression variable models, it is assumed implicitly that the dependent variable Y

More information

Chapter 5 Estimating Demand Functions

Chapter 5 Estimating Demand Functions Chapter 5 Estimating Demand Functions 1 Why do you need statistics and regression analysis? Ability to read market research papers Analyze your own data in a simple way Assist you in pricing and marketing

More information

The Real Business Cycle Model

The Real Business Cycle Model The Real Business Cycle Model Ester Faia Goethe University Frankfurt Nov 2015 Ester Faia (Goethe University Frankfurt) RBC Nov 2015 1 / 27 Introduction The RBC model explains the co-movements in the uctuations

More information

On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information

On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information Finance 400 A. Penati - G. Pennacchi Notes on On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information by Sanford Grossman This model shows how the heterogeneous information

More information

MULTIVARIATE PROBABILITY DISTRIBUTIONS

MULTIVARIATE PROBABILITY DISTRIBUTIONS MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined

More information

Dynamics of Small Open Economies

Dynamics of Small Open Economies Dynamics of Small Open Economies Lecture 2, ECON 4330 Tord Krogh January 22, 2013 Tord Krogh () ECON 4330 January 22, 2013 1 / 68 Last lecture The models we have looked at so far are characterized by:

More information

Department of Economics and Related Studies Financial Market Microstructure. Topic 1 : Overview and Fixed Cost Models of Spreads

Department of Economics and Related Studies Financial Market Microstructure. Topic 1 : Overview and Fixed Cost Models of Spreads Session 2008-2009 Department of Economics and Related Studies Financial Market Microstructure Topic 1 : Overview and Fixed Cost Models of Spreads 1 Introduction 1.1 Some background Most of what is taught

More information

INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition)

INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) Abstract Indirect inference is a simulation-based method for estimating the parameters of economic models. Its

More information

Forecast covariances in the linear multiregression dynamic model.

Forecast covariances in the linear multiregression dynamic model. Forecast covariances in the linear multiregression dynamic model. Catriona M Queen, Ben J Wright and Casper J Albers The Open University, Milton Keynes, MK7 6AA, UK February 28, 2007 Abstract The linear

More information

Panel Data Econometrics

Panel Data Econometrics Panel Data Econometrics Master of Science in Economics - University of Geneva Christophe Hurlin, Université d Orléans University of Orléans January 2010 De nition A longitudinal, or panel, data set is

More information

Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy. Blue vs. Orange. Review Jeopardy Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

More information

Structural Econometric Modeling in Industrial Organization Handout 1

Structural Econometric Modeling in Industrial Organization Handout 1 Structural Econometric Modeling in Industrial Organization Handout 1 Professor Matthijs Wildenbeest 16 May 2011 1 Reading Peter C. Reiss and Frank A. Wolak A. Structural Econometric Modeling: Rationales

More information

Partial r 2, contribution and fraction [a]

Partial r 2, contribution and fraction [a] Multiple and partial regression and correlation Partial r 2, contribution and fraction [a] Daniel Borcard Université de Montréal Département de sciences biologiques January 2002 The definitions of the

More information

Mgmt 469. Fixed Effects Models. Suppose you want to learn the effect of price on the demand for back massages. You

Mgmt 469. Fixed Effects Models. Suppose you want to learn the effect of price on the demand for back massages. You Mgmt 469 Fixed Effects Models Suppose you want to learn the effect of price on the demand for back massages. You have the following data from four Midwest locations: Table 1: A Single Cross-section of

More information

C(t) (1 + y) 4. t=1. For the 4 year bond considered above, assume that the price today is 900$. The yield to maturity will then be the y that solves

C(t) (1 + y) 4. t=1. For the 4 year bond considered above, assume that the price today is 900$. The yield to maturity will then be the y that solves Economics 7344, Spring 2013 Bent E. Sørensen INTEREST RATE THEORY We will cover fixed income securities. The major categories of long-term fixed income securities are federal government bonds, corporate

More information

Sections 2.11 and 5.8

Sections 2.11 and 5.8 Sections 211 and 58 Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I 1/25 Gesell data Let X be the age in in months a child speaks his/her first word and

More information

Least Squares Estimation

Least Squares Estimation Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David

More information

Econometrics Modeling and systems estimation

Econometrics Modeling and systems estimation André K. Anundsen and Claudia Foroni, (Norges Bank), and Ragnar Nymoen (Department of Economics) ECON 4160 Econometrics Modeling and systems estimation TEACHING PLAN Autumn 2015 Lectures and computer classes:

More information

5. Linear Regression

5. Linear Regression 5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4

More information

Chapter 5: The Cointegrated VAR model

Chapter 5: The Cointegrated VAR model Chapter 5: The Cointegrated VAR model Katarina Juselius July 1, 2012 Katarina Juselius () Chapter 5: The Cointegrated VAR model July 1, 2012 1 / 41 An intuitive interpretation of the Pi matrix Consider

More information

FACULTY WORKING PAPER NO. 1021

FACULTY WORKING PAPER NO. 1021 330 3385 1021 COPY 2 STX '«««.2 3 J FACULTY WORKING PAPER NO. 1021 The Use of Linear Approximation to Nonlinear Regression Analysis Anil K. Bera an*** 6? r i ce anc Bus nes; Bureau of Economic and Business

More information

1. Suppose that a score on a final exam depends upon attendance and unobserved factors that affect exam performance (such as student ability).

1. Suppose that a score on a final exam depends upon attendance and unobserved factors that affect exam performance (such as student ability). Examples of Questions on Regression Analysis: 1. Suppose that a score on a final exam depends upon attendance and unobserved factors that affect exam performance (such as student ability). Then,. When

More information

Correlated Random Effects Panel Data Models

Correlated Random Effects Panel Data Models INTRODUCTION AND LINEAR MODELS Correlated Random Effects Panel Data Models IZA Summer School in Labor Economics May 13-19, 2013 Jeffrey M. Wooldridge Michigan State University 1. Introduction 2. The Linear

More information

Clustering in the Linear Model

Clustering in the Linear Model Short Guides to Microeconometrics Fall 2014 Kurt Schmidheiny Universität Basel Clustering in the Linear Model 2 1 Introduction Clustering in the Linear Model This handout extends the handout on The Multiple

More information

The aspect of the data that we want to describe/measure is the degree of linear relationship between and The statistic r describes/measures the degree

The aspect of the data that we want to describe/measure is the degree of linear relationship between and The statistic r describes/measures the degree PS 511: Advanced Statistics for Psychological and Behavioral Research 1 Both examine linear (straight line) relationships Correlation works with a pair of scores One score on each of two variables ( and

More information

Lectures 8, 9 & 10. Multiple Regression Analysis

Lectures 8, 9 & 10. Multiple Regression Analysis Lectures 8, 9 & 0. Multiple Regression Analysis In which you learn how to apply the principles and tests outlined in earlier lectures to more realistic models involving more than explanatory variable and

More information

3.1 Least squares in matrix form

3.1 Least squares in matrix form 118 3 Multiple Regression 3.1 Least squares in matrix form E Uses Appendix A.2 A.4, A.6, A.7. 3.1.1 Introduction More than one explanatory variable In the foregoing chapter we considered the simple regression

More information

FORECASTING DEPOSIT GROWTH: Forecasting BIF and SAIF Assessable and Insured Deposits

FORECASTING DEPOSIT GROWTH: Forecasting BIF and SAIF Assessable and Insured Deposits Technical Paper Series Congressional Budget Office Washington, DC FORECASTING DEPOSIT GROWTH: Forecasting BIF and SAIF Assessable and Insured Deposits Albert D. Metz Microeconomic and Financial Studies

More information

ADVANCED FORECASTING MODELS USING SAS SOFTWARE

ADVANCED FORECASTING MODELS USING SAS SOFTWARE ADVANCED FORECASTING MODELS USING SAS SOFTWARE Girish Kumar Jha IARI, Pusa, New Delhi 110 012 gjha_eco@iari.res.in 1. Transfer Function Model Univariate ARIMA models are useful for analysis and forecasting

More information

Cointegration. Basic Ideas and Key results. Egon Zakrajšek Division of Monetary Affairs Federal Reserve Board

Cointegration. Basic Ideas and Key results. Egon Zakrajšek Division of Monetary Affairs Federal Reserve Board Cointegration Basic Ideas and Key results Egon Zakrajšek Division of Monetary Affairs Federal Reserve Board Summer School in Financial Mathematics Faculty of Mathematics & Physics University of Ljubljana

More information

Time Series and Forecasting

Time Series and Forecasting Chapter 22 Page 1 Time Series and Forecasting A time series is a sequence of observations of a random variable. Hence, it is a stochastic process. Examples include the monthly demand for a product, the

More information

Economics 326: Duality and the Slutsky Decomposition. Ethan Kaplan

Economics 326: Duality and the Slutsky Decomposition. Ethan Kaplan Economics 326: Duality and the Slutsky Decomposition Ethan Kaplan September 19, 2011 Outline 1. Convexity and Declining MRS 2. Duality and Hicksian Demand 3. Slutsky Decomposition 4. Net and Gross Substitutes

More information

The Multiple Regression Model: Hypothesis Tests and the Use of Nonsample Information

The Multiple Regression Model: Hypothesis Tests and the Use of Nonsample Information Chapter 8 The Multiple Regression Model: Hypothesis Tests and the Use of Nonsample Information An important new development that we encounter in this chapter is using the F- distribution to simultaneously

More information

Random Vectors and the Variance Covariance Matrix

Random Vectors and the Variance Covariance Matrix Random Vectors and the Variance Covariance Matrix Definition 1. A random vector X is a vector (X 1, X 2,..., X p ) of jointly distributed random variables. As is customary in linear algebra, we will write

More information

Econometrics II. Lecture 9: Sample Selection Bias

Econometrics II. Lecture 9: Sample Selection Bias Econometrics II Lecture 9: Sample Selection Bias Måns Söderbom 5 May 2011 Department of Economics, University of Gothenburg. Email: mans.soderbom@economics.gu.se. Web: www.economics.gu.se/soderbom, www.soderbom.net.

More information

Panel Data: Linear Models

Panel Data: Linear Models Panel Data: Linear Models Laura Magazzini University of Verona laura.magazzini@univr.it http://dse.univr.it/magazzini Laura Magazzini (@univr.it) Panel Data: Linear Models 1 / 45 Introduction Outline What

More information

ANNUITY LAPSE RATE MODELING: TOBIT OR NOT TOBIT? 1. INTRODUCTION

ANNUITY LAPSE RATE MODELING: TOBIT OR NOT TOBIT? 1. INTRODUCTION ANNUITY LAPSE RATE MODELING: TOBIT OR NOT TOBIT? SAMUEL H. COX AND YIJIA LIN ABSTRACT. We devise an approach, using tobit models for modeling annuity lapse rates. The approach is based on data provided

More information