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725 note 2010_s1_1_linear_updated_feb_9

725 note 2010_s1_1_linear_updated_feb_9 - 1 LINEAR...

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1 | Linear Regressions under Ideal Conditions (I) 1. LINEAR REGRESSION UNDER IDEAL CONDITIONS (I) What do we learn in this section? [1] Regression model. • What is “regression model”? [2] (Strong) Assumptions. • What assumptions for regression models? • How should a sample be collected from population? [3] Ordinary Least Squares (OLS). • This is the popular estimation method for regression models. [4] Goodness of Fit. • Does your estimated regression model explain your sample well? [5] Statistical Properties of the OLS estimator. • Is the OLS estimator unbiased and normal?

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2 | Linear Regressions under Ideal Conditions (I) What do we learn in this section? Continue… [6] Efficiency. • Is the OLS estimator most reliable estimator? • If so, in what sense? [7] Testing Linear Hypothesis. • How can we test the hypotheses related to regression models? [8] Forecasting. • Can we use our regression results for forecasting? [9] Weaker Assumptions. • Does the OLS estimator have good properties under more realistic circumstances?
3 | Linear Regressions under Ideal Conditions (I) [1] What is “Regression Model”? • Interested in the average relation between income ( y ) and education ( x ). • For the people with 12 years of schooling ( x =12), what is the average income ( ( | 12) Ey x )? • For the people with x years of schooling, what is the average income ( ( | ) x )? • Regression model: ( | ) y E y xu , where u is an error term with ( | )0 E ux .

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4 | Linear Regressions under Ideal Conditions (I) Warming-Up Probability Theory (1) Bivariate Distributions Consider two random variables (RV), X and Y, with a joint probability density function (pdf): f ( x , y ) = Pr( X = x , Y = y ). Marginal (unconditional) pdf: f x ( x ) = y f ( x , y ) = Pr( X = x ) regardless of Y ; f y ( y ) = x f ( x , y ) = Pr( Y = y ) regardless of X . Conditional pdf: f ( y | x ) = Pr( Y = y , given X = x ) = (, ) () x f x y f x .
5 | Linear Regressions under Ideal Conditions (I) Conditional Mean and Variance: X , Y : RVs with f ( x , y ). (e.g., Y = income, X = education) • Population of billions and billions: {( x (1) , y (1) ), . ... ( x (b) , y (b) )}. • Average of y ( j ) = E ( y ) = () (,) yy x y yf y yf x y   . • For the people earning a specific education level x , what is the average of y ? ( | ) Ey Xx = ( | ) y yf y x . 22 var( | )[ ( ( | )) | ][ ( | )] ( | ) y y E y E y xx y E y x fy x  .

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6 | Linear Regressions under Ideal Conditions (I) Regression model: • Let ( | ) u y E y x  (deviation from conditional mean). ( | )( | | ) y E y x y E y xE y xu   (regression model). ( | ) Ey x = explained part of y by x . u = unexplained part of y (called disturbance term) with ( | )0 E ux .
7 | Linear Regressions under Ideal Conditions (I) EX: • A population with X (income=\$10,000) and Y (consumption = \$10,000). • Joint pdf: Y \ X 4 8 1 1/2 0 2 1/4 1/4 • Graph for this population: y x 1 2 4/3 4 8 Regression line

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8 | Linear Regressions under Ideal Conditions (I) • Marginal pdf: Y \ X 4 8 f y ( y ) 1 1/2 0 1/2 2 1/4 1/4 1/2 f x ( x ) 3/4 1/4 • Conditional Probabilities of f ( y | x ): Y \ X 4 8 1 2/3 0 2 1/3 1
9 | Linear Regressions under Ideal Conditions (I) • Conditional mean: E ( y | x = 4) = y yf ( y | x = 4) = 1× f ( y =1| x = 4) + 2× f ( y = 2| x = 4) = 1×(2/3) + 2×(1/3) = 4/3.

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725 note 2010_s1_1_linear_updated_feb_9 - 1 LINEAR...

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