ECO375H_Slides_2 - Lecture 2: The Simple Regression Model...

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Lecture 2: The Simple Regression Model Junichi Suzuki University of Toronto September 17th, 2009
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Announcement I Problem set 1 is posted online and due September 25th I Two normal questions and two STATA questions I On Friday, Sacha will present the very basic of STATA I On 23rd (Wed), Sacha will have an o¢ ce hour to help students having STATA-related problems
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Some Advice on Problem Set 1 I Start early! Need time to get used to use software! I I Will post some hints on 2.10 in the blackboard on Friday
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Chapter 2 The Simple Regression Model
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I Provide a quick review of the ordinary least squares estimates (OLSE) using cross-sectional data I Theoretical aspects: I De±nition I How to derive I Mechanical and stochastic properties and assumptions needed I Practical aspects: I How to interpret results I How to deal with nonlinearity
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Model
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I Want to quantify the change of x on y with all other factors being equal (e.g., x : training and y : wage) I Suppose all other factors are not observable I One relevant econometric model may be a simple linear regression model: y = β 0 + β 1 x + u I y : dependent variable/regressend I x : independent variable/regressor I u : error term/disturbance I β 0 : intercept parameter/constant term I β 1 : slope parameter
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Assumptions I The average value of u is zero E ( u ) = 0 I Not restrictive I When E ( u ) 6 = 0 , β 0 and u so that ˜ β 0 = β + E ( u ) and E ( ˜ u ) = 0 I The average value of u does not depend on that of x E ( u j x ) = E ( u ) I Restrictive! I Imply that workers±tendency to attend training does not depend on their IQ I The most important assumption in econometrics
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Population Regression Function (PRF) I Under these two assumptions, we can separate the model in two parts: y = β 0 + β 1 x + u = E ( y j x ) + u I E ( y j x ) = β 0 + β 1 x : PRF or systematic part I u : unsystematic part I PRF tells how the average value of y changes as the value of x changes
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2.2: Deriving the OLSE
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Deriving the OLSE I Consider a simple regression model: y = β 0 + β 1 x + u I Suppose you observe n data points: f y i , x i g n i = 1 I β 0 and β 1 that I OLSE is the most popular method for its simplicity and good properties
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Deriving the OLSE I ˆ β 0 , ˆ β 1 ± that solve min β 0 , β 1 n i = 1 ² y i β 0 β 1 x i | {z } ³ 2 residual I minimizes the sum of squared "residual" (SSR) I This is a mere mathematical problem I Solving this problem entails the knowledge of basic calculus
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Deriving the OLSE I At the minimum, its partial derivative w.r.t. β 0 , β 1 must be zero I FOCs are ∂β 0 n i = 1 ( y i β 0 β 1
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ECO375H_Slides_2 - Lecture 2: The Simple Regression Model...

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