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Unformatted text preview: 41Linear Regression with One Regressor (SW Chapter 4) Linear regression allows us to estimate, and make inferences about, populationslope coefficients. Ultimately our aim is to estimate the causal effect on Yof a unit change in X but for now, just think of the problem of fitting a straight line to data on two variables, Yand X. 42The 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: oHow should we draw a line through the data to estimate the (population) slope (answer: ordinary least squares). oWhat are advantages and disadvantages of OLS? Hypothesis testing: oHow to test if the slope is zero? Confidence intervals: oHow to construct a confidence interval for the slope? 43Linear Regression: Some Notation and Terminology (SW Section 4.1)Thepopulation regression line:Test Score= + 1STR 1= slope of population regression line = Test scoreSTR= change in test score for a unit change in STRWhy are and 1population parameters? We would like to know the population value of 1. We dont know 1, so must estimate it using data. 44The Population Linear Regression Model general notationYi= + 1Xi+ ui, i= 1,, nXis the independent variableor regressorYis the dependent variable= intercept1= slopeui= 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 X45This terminology in a picture: Observations on Yand X; the population regression line; and the regression error (the error term): 46The Ordinary Least Squares Estimator (SW Section 4.2) How can we estimate and 1from data? Recall that Ywas the least squares estimator of Y: Ysolves, 21min()nmiiYm=By analogy, we will focus on the least squares (ordinary least squares or OLS) estimator of the unknown parameters and 1, which solves, 012,11min[()]nbbiiiYbbX=+47Mechanics of OLS The population regression line: Test Score= + 1STR1= Test scoreSTR= ?? 48The OLS estimatorsolves: 012,11min[()]nbbiiiYbbX=+The OLS estimator minimizes the average squared difference between the actual values of Yiand 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 and 1. 49410Application to the California Test Score Class Sizedata Estimated slope = 1= 2.28 Estimated intercept = = 698.9 Estimated regression line: TestScore= 698.9 2.28STR 411Interpretation of the estimated slope and intercept TestScore= 698.9 2.28STRDistricts with one more student per teacher on average have test scores that are 2.28 points lower. have test scores that are 2....
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This note was uploaded on 05/25/2011 for the course ECON 2007 taught by Professor J during the Spring '11 term at UCL.
 Spring '11
 j

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