# Simultaneous Equation Models As discussed last week, one important form of endogeneity is simultaneity. This arises when one or more of the

Save this PDF as:

Size: px
Start display at page:

Download "Simultaneous Equation Models As discussed last week, one important form of endogeneity is simultaneity. This arises when one or more of the"

## Transcription

1 Simultaneous Equation Models As discussed last week, one important form of endogeneity is simultaneity. This arises when one or more of the explanatory variables is jointly determined with the dependent variable, usually through an equilibrium mechanism. Simultaneous Equations Models (SEMs) differ from those considered previously because in each model there are two or more dependent variables rather than just one. Simultaneous equations models also differ from most of the econometric models we have considered so far because they consist of a set of equations The least squares estimation procedure is not appropriate in these models and we must develop new ways to obtain reliable estimates of economic parameters. The usual method for estimating SEMs is the instrumental variables method, discussed last week.

2 Example The classic example of an SEM is a supply and demand equation for some commodity (e.g. coffee) or input to production (e.g. labour) Consider a simple market supply function: Where is quantity or output, is price and is some observed variable affecting supply of the commodity (e.g. weather). The error term,, contains other factors that affect supply. The equation is an example of a structural equation, i.e. it is derivable from economic theory and has a causal interpretation. The coefficient measures how supply of the product changes when the price changes. If price and quantity are measured in logs, the coefficient gives the price elasticity of supply. Plotting the supply function, we plot output as a function of price, holding and fixed. Changes in either of these two factors lead to shifts in the supply curve; the difference being that is observed, while is not. The crucial assumption for OLS that we make is that the independent variables are independent of the error term. In this case, this assumption does not hold. Assuming that the demand curve is downward sloping (or vertical), then a shift in the supply curve produces a change in both price and quantity. Thus the error term is correlated with price. In addition, the fact that is random means that on the right-hand side of the supply and demand equations we have an explanatory variable that is random. This is contrary to the assumption of fixed explanatory variables that we usually make in regression model analysis. The important thing to remember is that supply and demand interact to jointly determine the market price of a good and the amount of it that is sold An econometric model that explains market price and quantity should therefore consist of two equations, one for supply and one for demand.

3 Demand: (1) Supply: (2) Where is the quantity demanded and is an observed variable affecting the demand for the commodity (e.g. income). In this model the variables p and q are called endogenous variables because their values are determined within the system we have created. The variables and have values that are given to us, and which are determined outside this system. As such, these are exogenous variables. The error terms in the supply and demand equations are assumed to have the usual properties; i.e. they have a constant mean and variance, and are independently distributed

4 A Bad Example An important point to remember when using SEMs is that each equation in the model should have a ceteris paribus, causal interpretation. In the above example, the two equations describe entirely different relationships. - The supply equation describes the behaviour of firms - The demand equation is a behavioural relationship for consumers Each equation has a ceteris paribus interpretation therefore and stands on its own They become linked in the econometric analysis only because the observed price and quantity are determined by the intersection of supply and demand. Consider the following example: Neither of these equations has a sensible ceteris paribus interpretation because housing and saving are chosen by the same individual. - If income increases, a person will generally change the optimal mix of housing expenditures and saving. The first equation however, makes it seem as though we want to know the impact of a change in income, education or age on housing expenditure, holding saving constant. Just because two variables are determined simultaneously does not mean that a SEM is suitable.

5 Simultaneity Bias in OLS Consider the following example: (3) (4) To show that is generally correlated with we can solve for in terms of the exogenous variables and the error terms. Replacing in (4) with the expression in (3) gives, 1 (5) Assuming that 1 we can divide (5) by 1 to obtain, (6) Where / 1 ; / 1 and / 1 Equation (6) expresses in terms of the exogenous variables and the error terms and is called the reduced form equation for. The parameters and are non-linear functions of the structural parameters, and are termed the reduced form parameters. The reduced form error,, is a linear function of the structural error terms, and. Since the are uncorrelated with the, is also uncorrelated with the, hence the reduced form parameters in (6) can be estimated by OLS. - The reduced form equations can be important for economic analysis. These equations relate the equilibrium values of the endogenous variables to the exogenous variables.

6 Equation (6) also tells us that estimation of equation (3) by OLS will result in biased and inconsistent estimates of and. - In equation (3) the issue is whether and are correlated ( and are by assumption uncorrelated) - From (6) we see that and are correlated if and only if and are correlated - Since is a linear combination of and it is generally correlated with When is correlated with because of simultaneity, we say that OLS suffers from simultaneity bias

7 The Instrumental Variables Solution As we saw last time the IV solution of two-stage least squares can be used to solve the problem of endogenous explanatory variables. This is also true for SEMs the major difference being that because we specify a structural equation for each endogenous variable, we can immediately see whether sufficient IVs are available to estimate either equation. Consider the following example: Here we can think of the coffee market as an example, with being say per capita coffee consumption, being the average price per jar and being something like the weather (in Brazil!) that affects supply. It is assumed that is exogenous to both the supply and demand equations The first question to be addressed is: given a random sample on, and, which of the above equations can be estimated, i.e. which is an identified equation? It turns out that the demand equation is identified, but the supply equation is not. - This is indicated for our rules for instruments - We can use as in instrument for price in the demand equation - Because appears in the supply equation however, we cannot use it as an instrument in this equation - In order to estimate the supply equation we would need an observed exogenous variable that shifts the demand curve

8 Considering the more general two-equation model: Where and are the endogenous variables, and are the structural error terms, and and now denote a set of exogenous regressors, and, that appear in the first and second regression respectively, i.e.,,, and,,,. In many cases and will overlap The assumption that and contain different exogenous variables means that we impose exclusion restrictions on the model, i.e. we assume that certain exogenous regressors do not appear in the first equation and others are absent from the second. This allows us to distinguish between the two structural equations. The Rank Condition for Identification of a Structural Equation - The first equation in a two-equation SEM is identified if, and only if, the second equation contains at least one exogenous variable (with a nonzero coefficient) that is excluded from the first equation. - The order condition for identifying the first equation states that at least one exogenous variable is excluded from this regression. This is simple to check once both equations have been specified. - The rank condition requires more: at least one of the exogenous regressors excluded from the first equation must have a non-zero population coefficient in the second equation. This can be tested using a t or F test. - Identification in the second equation is the mirror image of the above

9 Estimation Once we have determined that an equation is identified, we can estimate it by TSLS - The instruments consist of the exogenous variables appearing in either equation Tests for endogeneity, overidentifying restrictions and so on proceed as before It turns out that, when any system with two or more equations is correctly specified and certain additional assumptions hold, system estimation methods (e.g. Three-Stage-Least-Squares) are generally more efficient than estimating by TSLS

10 Systems with More Than Two Equations SEMs can consist of more than two equations Studying the general identification of these models is not straightforward Once an equation has been shown to be identified, it can be estimated by TSLS Consider the following three equation system (7) (8) (9) It is difficult to show that an equation in a SEM with more than two equations is identified It is clear however that (9) is not identified, since all exogenous regressors are included in the equation, leaving no instruments for - we have in terms of last week an unidentified equation Equation (7) on the other hand looks promising; we have three exogenous regressors excluded from the regression,, and, and only two endogenous regressors, and - this equation is therefore overidentified In general, an equation in any SEM satisfies the order condition for identification if the number of excluded exogenous variables from the equation is at least as large as the number of endogenous regressors - As such, the order condition in (8) is also satisfied since we have one excluded exogenous regressor,, and one endogenous regressor, - the equation is exactly identified Identification of an equation depends on the parameters (which we can never know for sure) in the other equations however - For example, if 0 in (9) then (8) is not identified, as is useless as an instrument for

### Econometrics. Week 9. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague

Econometrics Week 9 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 21 Recommended Reading For the today Simultaneous Equations Models Chapter 16 (pp.

### Simultaneous Equations Models. Sanjaya DeSilva

Simultaneous Equations Models Sanjaya DeSilva 1 Reduced Form and Structural Models We will begin with definitions; 1. Exogenous variables are variables that are determined outside of the model. For the

### 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

### 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

### Economics 140A Identification in Simultaneous Equation Models Simultaneous Equation Models

Economics 140A Identification in Simultaneous Equation Models Simultaneous Equation Models Our second extension of the classic regression model, to which we devote two lectures, is to a system (or model)

### Limitations of regression analysis

Limitations of regression analysis Ragnar Nymoen Department of Economics, UiO 8 February 2009 Overview What are the limitations to regression? Simultaneous equations bias Measurement errors in explanatory

### 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

### 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

### Chapter 23: Simultaneous Equation Models Identification

Chapter 23: Simultaneous Equation Models Identification Chapter 23 Outline Review o Demand and Supply Models o Ordinary Least Squares (OLS) Estimation rocedure o Reduced Form (RF) Estimation rocedure One

### As we explained in the textbook discussion of statistical estimation of demand

Estimating and Forecasting Industry Demand for Price-Taking Firms As we explained in the textbook discussion of statistical estimation of demand and statistical forecasting, estimating the parameters of

### 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

### 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

### 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

### Endogenous Variables: These are `jointly dependent' variables; or, those determined within the system of equations. Exogenous Variables: Determined Ou

Notes on Identification Mostly From Gujarati, Basic Econometrics, Chapter 19 Copyright - Jonathan Nagler; April 11, 2001 Un-identified: We cannot determine the values of the parameters of the model based

### 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

University of British Columbia Department of Economics, Macroeconomics (Econ 502) Prof. Amartya Lahiri Handout 7: Business Cycles We now use the methods that we have introduced to study modern business

### 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

### 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

### 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

### CHAPTER 9: SERIAL CORRELATION

Serial correlation (or autocorrelation) is the violation of Assumption 4 (observations of the error term are uncorrelated with each other). Pure Serial Correlation This type of correlation tends to be

### 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

### Econ 371 Exam #4 - Practice

Econ 37 Exam #4 - Practice Multiple Choice (5 points each): For each of the following, select the single most appropriate option to complete the statement. ) The following will not cause correlation between

### 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

### 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

### Simultaneous Equations

Simultaneous Equations 1 Motivation and Examples Now we relax the assumption that EX u 0 his wil require new techniques: 1 Instrumental variables 2 2- and 3-stage least squares 3 Limited LIML and full

### Control Functions and Simultaneous Equations Methods

Control Functions and Simultaneous Equations Methods By RICHARD BLUNDELL, DENNIS KRISTENSEN AND ROSA L. MATZKIN Economic models of agent s optimization problems or of interactions among agents often exhibit

### Problems with OLS Considering :

Problems with OLS Considering : we assume Y i X i u i E u i 0 E u i or var u i E u i u j 0orcov u i,u j 0 We have seen that we have to make very specific assumptions about u i in order to get OLS estimates

### Topic 1 - Introduction to Labour Economics. Professor H.J. Schuetze Economics 370. What is Labour Economics?

Topic 1 - Introduction to Labour Economics Professor H.J. Schuetze Economics 370 What is Labour Economics? Let s begin by looking at what economics is in general Study of interactions between decision

### 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

### 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

### Lecture 16. Endogeneity & Instrumental Variable Estimation (continued)

Lecture 16. Endogeneity & Instrumental Variable Estimation (continued) Seen how endogeneity, Cov(x,u) 0, can be caused by Omitting (relevant) variables from the model Measurement Error in a right hand

### ECONOMETRIC THEORY. MODULE I Lecture - 1 Introduction to Econometrics

ECONOMETRIC THEORY MODULE I Lecture - 1 Introduction to Econometrics Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur 2 Econometrics deals with the measurement

### Instrumental Variables (IV) Instrumental Variables (IV) is a method of estimation that is widely used

Instrumental Variables (IV) Instrumental Variables (IV) is a method of estimation that is widely used in many economic applications when correlation between the explanatory variables and the error term

### 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

### 1. The Classical Linear Regression Model: The Bivariate Case

Business School, Brunel University MSc. EC5501/5509 Modelling Financial Decisions and Markets/Introduction to Quantitative Methods Prof. Menelaos Karanasos (Room SS69, Tel. 018956584) Lecture Notes 3 1.

### Elementary Statistics. Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination

Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination What is a Scatter Plot? A Scatter Plot is a plot of ordered pairs (x, y) where the horizontal axis is used

### 4. Simple regression. QBUS6840 Predictive Analytics. https://www.otexts.org/fpp/4

4. Simple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/4 Outline The simple linear model Least squares estimation Forecasting with regression Non-linear functional forms Regression

### Simple Linear Regression Chapter 11

Simple Linear Regression Chapter 11 Rationale Frequently decision-making situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related

### Notes 10: An Equation Based Model of the Macroeconomy

Notes 10: An Equation Based Model of the Macroeconomy In this note, I am going to provide a simple mathematical framework for 8 of the 9 major curves in our class (excluding only the labor supply curve).

### 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)...

### 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

### CHAPTER 6. SIMULTANEOUS EQUATIONS

Economics 24B Daniel McFadden 1999 1. INTRODUCTION CHAPTER 6. SIMULTANEOUS EQUATIONS Economic systems are usually described in terms of the behavior of various economic agents, and the equilibrium that

### 1. Briefly explain what an indifference curve is and how it can be graphically derived.

Chapter 2: Consumer Choice Short Answer Questions 1. Briefly explain what an indifference curve is and how it can be graphically derived. Answer: An indifference curve shows the set of consumption bundles

### Department of Economics, Session 2012/2013. EC352 Econometric Methods. Exercises from Week 03

Department of Economics, Session 01/013 University of Essex, Autumn Term Dr Gordon Kemp EC35 Econometric Methods Exercises from Week 03 1 Problem P3.11 The following equation describes the median housing

### Chapter 6. Econometrics. 6.1 Introduction. 6.2 Univariate techniques Transforming data

Chapter 6 Econometrics 6.1 Introduction We re going to use a few tools to characterize the time series properties of macro variables. Today, we will take a relatively atheoretical approach to this task,

### Advanced High School Statistics. Preliminary Edition

Chapter 2 Summarizing Data After collecting data, the next stage in the investigative process is to summarize the data. Graphical displays allow us to visualize and better understand the important features

### 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

### Chapter 11: Two Variable Regression Analysis

Department of Mathematics Izmir University of Economics Week 14-15 2014-2015 In this chapter, we will focus on linear models and extend our analysis to relationships between variables, the definitions

### 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

### Regression Analysis Using ArcMap. By Jennie Murack

Regression Analysis Using ArcMap By Jennie Murack Regression Basics How is Regression Different from other Spatial Statistical Analyses? With other tools you ask WHERE something is happening? Are there

### L1: Dealing with Reverse Causation: Simultaneous Equation Modelling

L: Dealing with Reverse Causation: Simultaneous Equation Modelling Prof Gwilym Pryce AQIM Training June 2006 Introduction Social Science Statistics I & II: We have assumed only one dependent variable,

### Introduction to Regression and Data Analysis

Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it

### Partial Equilibrium: Positive Analysis

Partial Equilibrium: Positive Analysis This Version: November 28, 2009 First Version: December 1, 2008. In this Chapter we consider consider the interaction between different agents and firms, and solve

### Economics 345 Applied Econometrics

Economics 345 Applied Econometrics Lab 3: Some Useful Functional Forms for Regression Analysis Prof: Martin Farnham TAs: Joffré Leroux, Rebecca Wortzman, Yiying Yang Open EViews, and open the EViews workfile,

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

### Three-stage least squares

Three-stage least squares Econometric Methods Lecture 7, PhD Warsaw School of Economics Outline 1 Introduction 2 3 Outline 1 Introduction 2 3 3-Stage Least Squares: basic idea 1-stage (OLS): inconsistent

### Chapter 27: Taxation. 27.1: Introduction. 27.2: The Two Prices with a Tax. 27.2: The Pre-Tax Position

Chapter 27: Taxation 27.1: Introduction We consider the effect of taxation on some good on the market for that good. We ask the questions: who pays the tax? what effect does it have on the equilibrium

### Wooldridge, Introductory Econometrics, 4th ed. Chapter 7: Multiple regression analysis with qualitative information: Binary (or dummy) variables

Wooldridge, Introductory Econometrics, 4th ed. Chapter 7: Multiple regression analysis with qualitative information: Binary (or dummy) variables We often consider relationships between observed outcomes

### 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

### where b is the slope of the line and a is the intercept i.e. where the line cuts the y axis.

Least Squares Introduction We have mentioned that one should not always conclude that because two variables are correlated that one variable is causing the other to behave a certain way. However, sometimes

### Wooldridge, Introductory Econometrics, 4th ed. Multiple regression analysis:

Wooldridge, Introductory Econometrics, 4th ed. Chapter 4: Inference Multiple regression analysis: We have discussed the conditions under which OLS estimators are unbiased, and derived the variances of

### Econometric analysis of the Belgian car market

Econometric analysis of the Belgian car market By: Prof. dr. D. Czarnitzki/ Ms. Céline Arts Tim Verheyden Introduction In contrast to typical examples from microeconomics textbooks on homogeneous goods

### 2. Linear regression with multiple regressors

2. Linear regression with multiple regressors Aim of this section: Introduction of the multiple regression model OLS estimation in multiple regression Measures-of-fit in multiple regression Assumptions

### Topic 1: Linear Functions

Topic 1: Linear Functions Reading: Jacques, Chapter 1 1. Definition 2. Inverse Functions 3. Applications: Market Equilibrium National Income Determination in a Macro Economy Definition: What is a function?

### Chapter 14: Production Possibility Frontiers

Chapter 14: Production Possibility Frontiers 14.1: Introduction In chapter 8 we considered the allocation of a given amount of goods in society. We saw that the final allocation depends upon the initial

### 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

### How to use Instrumental Variables

How to use Instrumental Variables Questions that can be answered with experiments: 1) If I give people aspirin, what happens to their body temperature? 2) If I use fertilizer on tomato plants, do I get

### Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur. Lecture - 2 Simple Linear Regression

Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture - 2 Simple Linear Regression Hi, this is my second lecture in module one and on simple

### 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

### The Classical Linear Regression Model

The Classical Linear Regression Model 1 September 2004 A. A brief review of some basic concepts associated with vector random variables Let y denote an n x l vector of random variables, i.e., y = (y 1,

Chapter 9 The IS-LM/AD-AS Model: A General Framework for Macroeconomic Analysis Chapter Outline The FE Line: Equilibrium in the Labor Market The IS Curve: Equilibrium in the Goods Market The LM Curve:

### 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

### The session previously discussed important variables such as inflation, unemployment and GDP. We also alluded to factors that cause economic growth

The session previously discussed important variables such as inflation, unemployment and GDP. We also alluded to factors that cause economic growth enabling us to produce more and more to achieve higher

### CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

### INTRODUCTORY STATISTICS

INTRODUCTORY STATISTICS FIFTH EDITION Thomas H. Wonnacott University of Western Ontario Ronald J. Wonnacott University of Western Ontario WILEY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore

### Questions and Answers on Hypothesis Testing and Confidence Intervals

Questions and Answers on Hypothesis Testing and Confidence Intervals L. Magee Fall, 2008 1. Using 25 observations and 5 regressors, including the constant term, a researcher estimates a linear regression

### 3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### 1 OLS under Measurement Error

ECON 370: Measurement Error 1 Violations of Gauss Markov Assumptions: Measurement Error Econometric Methods, ECON 370 1 OLS under Measurement Error We have found out that another source of endogeneity

### Linear Model Selection and Regularization

Linear Model Selection and Regularization Recall the linear model Y = β 0 + β 1 X 1 + + β p X p + ɛ. In the lectures that follow, we consider some approaches for extending the linear model framework. In

### Using Hierarchical Linear Models to Measure Growth. Measurement Incorporated Hierarchical Linear Models Workshop

Using Hierarchical Linear Models to Measure Growth Measurement Incorporated Hierarchical Linear Models Workshop Chapter 6 Motivation Linear Growth Curves Quadratic Growth Curves Some other Growth Curves

### 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

### MARKET DEMAND &SUPPLY DEMAND

MARKET Market is a place where consumers meet sellers and the trading takes place. The consumers buy products at certain price, so money is exchanged for goods and services. We distinguish types of market

### Mathematical Economics and Econometrics

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION BA ECONOMICS (2011 Admission Onwards) VI Semester Core Course Mathematical Economics and Econometrics QUESTION BANK MODULE I 1 The measurement of economic

### OLS in Matrix Form. Let y be an n 1 vector of observations on the dependent variable.

OLS in Matrix Form 1 The True Model Let X be an n k matrix where we have observations on k independent variables for n observations Since our model will usually contain a constant term, one of the columns

### Demand, Supply, and Market Equilibrium

3 Demand, Supply, and Market Equilibrium The price of vanilla is bouncing. A kilogram (2.2 pounds) of vanilla beans sold for \$50 in 2000, but by 2003 the price had risen to \$500 per kilogram. The price

### MS&E 226: Small Data. Lecture 17: Additional topics in inference (v1) Ramesh Johari

MS&E 226: Small Data Lecture 17: Additional topics in inference (v1) Ramesh Johari ramesh.johari@stanford.edu 1 / 34 Warnings 2 / 34 Modeling assumptions: Regression Remember that most of the inference

### Univariate Regression

Univariate Regression Correlation and Regression The regression line summarizes the linear relationship between 2 variables Correlation coefficient, r, measures strength of relationship: the closer r is

### 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

### Lecture 10: Aggregate Demand and Aggregate Supply I

EC201 Intermediate Macroeconomics EC201 Intermediate Macroeconomics Lecture 10: Aggregate Demand and Aggregate Supply I Lecture Outline: - how to derive the aggregate demand from the IS-LM model; - a preliminary

### In most situations involving two quantitative variables, a scatterplot is the appropriate visual display.

In most situations involving two quantitative variables, a scatterplot is the appropriate visual display. Remember our assumption that the observations in our dataset be independent. If individuals appear

### 1 Logit & Probit Models for Binary Response

ECON 370: Limited Dependent Variable 1 Limited Dependent Variable Econometric Methods, ECON 370 We had previously discussed the possibility of running regressions even when the dependent variable is dichotomous

### Regression Analysis: Basic Concepts

The simple linear model Regression Analysis: Basic Concepts Allin Cottrell Represents the dependent variable, y i, as a linear function of one independent variable, x i, subject to a random disturbance

### Agenda. The IS-LM/AD-AS Model: A General Framework for Macroeconomic Analysis, Part 2. The AD Curve. Aggregate Demand and Aggregate Supply

Agenda Aggregate Demand and Aggregate Supply The IS-LM/AD-AS Model: A General Framework for Macroeconomic Analysis, art 2 13-1 13-2 Aggregate Demand and Aggregate Supply The AD-AS model is derived from

### 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

### HOW EFFECTIVE IS TARGETED ADVERTISING?

HOW EFFECTIVE IS TARGETED ADVERTISING? Ayman Farahat and Michael Bailey Marketplace Architect Yahoo! July 28, 2011 Thanks Randall Lewis, Yahoo! Research Agenda An Introduction to Measuring Effectiveness

### AN INTRODUCTION TO ECONOMETRICS. Oxbridge Economics; Mo Tanweer

AN INTRODUCTION TO ECONOMETRICS Oxbridge Economics; Mo Tanweer Mohammed.Tanweer@cantab.net Econometrics What is econometrics? Econometrics means economic measurement Economics + Statistics = Econometrics

### Econ 100B: Macroeconomic Analysis Fall Problem Set #3 ANSWERS (Due September 15-16, 2008)

Econ 100B: Macroeconomic Analysis Fall 2008 Problem Set #3 ANSWERS (Due September 15-16, 2008) A. On one side of a single sheet of paper: 1. Clearly and accurately draw and label a diagram of the Production