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Lecture 4 Prof Arkonac's slides (Ch 4) for ECO 4000

# Lecture 4 Prof Arkonac's slides (Ch 4) for ECO 4000 -...

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Linear Regression with One Regressor ECO 4000, Statistical Analysis for Economics and Finance Fall 2010 Lecture 4 Prof: Seyhan Arkonac, PhD 1

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2 Linear Regression with One Regressor 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 .
3 Estimation: How should we draw a line through the data to estimate the (population) slope (answer: ordinary least squares). What are advantages and disadvantages of OLS? Hypothesis testing: How to test if the slope is zero? Confidence intervals: How to construct a confidence interval for the slope? 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:

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4 Linear Regression: Some Notation and Terminology (SW Section 4.1) The population regression line : Test Score = 0 + 1 STR 1 = slope of population regression line = Test score 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.
5 The Population Linear Regression Model – general notation Y i = 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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6 This terminology in a picture : Observations on Y and X ; the population regression line; and the regression error (the “error term”):
7 The Ordinary Least Squares Estimator (SW Section 4.2) How can we estimate 0 and 1 from data? Recall that Y was the least squares estimator of 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 min [ ( )] n b b i i i Y b b X

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8 Mechanics of OLS The population regression line: Test Score = 0 + 1 STR 1 = Test score STR = ??
9 The OLS estimator solves: 01 2 , 0 1 1 min [ ( )] n b b i i i Y b 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 0 and 1 .

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11 Application to the California Test Score – Class Size data Estimated slope = 1 ˆ = – 2.28 Estimated intercept = 0 ˆ = 698.9 Estimated regression line: TestScore = 698.9 – 2.28 STR

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12 Interpretation of the estimated slope and intercept TestScore = 698.9 – 2.28 STR Districts with one more student per teacher on average have test scores that are 2.28 points lower.
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Lecture 4 Prof Arkonac's slides (Ch 4) for ECO 4000 -...

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