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Ch. 4-5 Linear Regression with One Regressor

# Ch. 4-5 Linear Regression with One Regressor - For...

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1 | One Regressor For Probability and Statistics, many more exercise questions are available from the textbook. The exercise questions in the textbook that you are not responsible for: Chapter 2: (2.16), (2.19) – (2.21), (2.23); and (2.25) – (2.27) from the 3 rd ed. Chapter 3: (3.2) – (3.5), (3.7), (3.9), (3,10), (3.12), (3.13), (3.14), (3,15), (3.16) c,d ; (3.17) – (3.21) .

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2 | One Regressor 3. LINEAR REGRESSION WITH ONE REGRESSOR (CH. 4 - 5) The exercise questions in the textbook that you are not responsible for: Chapter 4: (4.4), (4.8), (4.10); and (4.13), (4.14) from the 3 rd ed. Chapter 5: (5.6), (5.9), (5.11), (5.12), (5.15).
3 | One Regressor [1] Motivation • Wish to estimate the average relation between class size and students’ performances. • Set up the relation by the following equation: 0 ClassSize TestScore ClassSize other factors  , (1) where TestScore means standardized test score and ClassSize means student-teacher ratio. ClassSize measures the average change in TestScore when ClassSize changes by one unit. We wish to estimate ClassSize .

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4 | One Regressor [2] Regression Model with a Single Regressor (1) General linear regression model with a single regressor 01 YX u  , (2) where: Y is the dependent variable (or regressand , or left-hand variable ); X is the independent variable (or regressor, right-hand variable ); β 0 + β 1 X is the population regression line ( function ); β 0 is the intercept of the population regression line; β 1 is the slope (or coefficient ) on the population regression line; and u is the error term .
5 | One Regressor (2) Least Squares Assumptions: Assumption 1: Conditional mean of u given X is zero; E ( u | X ) = 0. u cannot be explained by X . No correlation between u and X . 01 ( | )( | ) E YX E X uX X   . 1 () o X describes the average relation between Y and X . Assumption 2: X and u have finite-fourth moments. E ( x 4 ), E ( u 4 ) and E ( x 2 u 2 ) exist and are finite (< ): Technical assumption for CLT. Generally holds.

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6 | One Regressor Assumption 3: A random sample ( X i , Y i ), i = 1, 2, . .. , n is available. To obtain unbiased and consistent estimates of β 0 and β 1 . Likely to be violated if time-series data are used. The estimators of β 0 and β 1 we will learn could be consistent even if this is violated.
7 | One Regressor

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8 | One Regressor • What can we do with regression analysis? Y = 0 + 1 X + u , where, Y = TestScore and X = ClassSize. Or, Y = teaching evaluation and X = teacher’s beauty. Or, Y = wage and X = beauty. Or, Y = GDP (per capita) growth and X = past per capital GDP.
9 | One Regressor Estimation : Find good approximates of β 1 and β 2 (that is, estimate an average relation between Y & X ). Hypothesis Testing : • A taxpayer claims that ClassSize has no effect on TestScore. • Beauty does not influence teaching evaluation (See the data set named “TeachingRatings”) Forecasting and Prediction : Wish to predict the average test score of the district with average student-teacher ratio = 14.

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10 | One Regressor [3] Ordinary Least Squares (OLS) Estimation - Gauss Let {( X i , Y i )| i = 1, . .. , n } be a random sample.
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Ch. 4-5 Linear Regression with One Regressor - For...

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