# Lecture_3 - Linear Regression with One Regressor Linear...

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1 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

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2 Estimation: How should we draw a line through the sample data to estimate the (population) slope (answer: ordinary least squares). Inference: Testing: How to test if the slope is zero? Confidence intervals for the slope The problems of statistical inference for linear regression are similar to estimation of the mean
3 Linear Regression: Some Notation and Terminology The population regression line or the (linear) conditional expectation function : E(Avg Test Score|STR) = 0 + 1 STR 1 = slope of population regression line = avg. chg. in test score for 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.

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Conditional Expectation 4 Is the function of X that best predicts Y Recall that Y was the best predictor of Y in population: it solves. Now instead of predicting Y by a constant, we use any function of X. The function that does the trick: E(Y|X=x) We will make the assumption that this is linear in a given set of X‟s. This will be our OLS assumption 1
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, note by defn. of CEF E(u|X) = 0 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

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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 How can we estimate 0 and 1 from data? Analogy principle We will focus on the least squares (“ ordinary least squares ” or OLS ”) estimator of the unknown parameters 0 and 1 , which solves the same problem as the (linear) CEF in the sample: 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 Two properties of OLS Sample mean of residuals is zero (mimics E(u) = 0 in population) Sample covariance of residuals and predicted values is 0 (mimics the population equation: cov(X, u) = 0)

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12 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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Lecture_3 - Linear Regression with One Regressor Linear...

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