Chapters 4 and 5 - linear regression with one regressor

Chapters 4 and 5 - linear regression with one regressor -...

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Linear Regression with One Regressor (SW Chapters 4 and 5) The class size/test score policy question: What is the effect on test scores of reducing STR by one student/class? Object of policy interest: Test score STR Δ Δ . This is the causal effect on scores of a unit change of class size. This is the slope of the line relating test score and STR 4-1
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4-2 Linear regression allows us to estimate, and make inferences about, population slope coefficients. Ultimately our aim is to estimate the causal effect on Y of a unit change in X – but for now, just think of the problem of fitting a straight line to data on two variables, Y and X . The problems of statistical inference for linear regression are, at a general level, the same as for estimation of the mean or of the differences between two means. Statistical, or econometric, inference about the slope entails: Estimation: What are advantages and disadvantages of OLS? Hypothesis testing: Confidence intervals:
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Linear Regression: Some Notation and Terminology (SW Section 4.1) The population regression line : Test Score = β 0 + 1 STR 1 = slope of population regression line = Test score STR Δ Δ = d Test score d STR = change in test score for a unit change in STR Why are 0 and 1 “population” parameters ? We would like to know the population value of 1 . We don’t know 1 , so must estimate it using data. 4-3
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4-4 The Population Linear Regression Model – general notation Y i = β 0 + 1 X i + u i , i = 1,…, n X is the independent variable or regressor Y is the dependent variable 0 = intercept 1 = slope u i = the regression error The regression error consists of omitted factors, or possibly measurement error in the measurement of Y . In general, these omitted factors are other factors that influence Y , other than the variable X
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This terminology in a picture : Observations on Y and X ; the population regression line; and the regression error (the “error term”): 4-5
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The Ordinary Least Squares Estimator (SW Section 4.2) How can we estimate β 0 and 1 from data? Recall that Y was the least squares estimator of μ Y : Y solves, 2 1 min ( ) n mi i Ym = By analogy, we will focus on the least squares (“ ordinary least squares ” or “ OLS ”) estimator of the unknown parameters 0 and 1 , which solves, 01 2 ,0 1 1 [ ( )] n bb i i i Yb b X = −+ 4-6
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Mechanics of OLS The population regression line: Test Score = β 0 + 1 STR 1 = Test score STR Δ Δ = ?? 4-7
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The OLS estimator solves: 01 2 ,0 1 1 min [ ( )] n bb i i i Yb b X = −+ The OLS estimator minimizes the average squared difference between the actual values of Y i and the prediction (“predicted value”) based on the estimated line. This minimization problem can be solved using calculus (App. 4.2). The result is the OLS estimators of β 0 and 1 .
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This note was uploaded on 09/13/2009 for the course ECONOMICS ECON 435 taught by Professor Pascal during the Spring '07 term at Simon Fraser.

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Chapters 4 and 5 - linear regression with one regressor -...

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