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Unformatted text preview: Lecture Note 2 Mechanics and Interpretation of OLS Empirical Methods II (API202A) — Spring 2009 Harvard Kennedy School 1 How Do We Compute the Partial Association We Observe in the Data – i.e. the b β s? Recall our SEQ: y = β + β 1 x 1 + β 2 x 2 + ... + β K x K + μ ‡ for example; K = 1: Test Scores = β + β 1 StudentTeacher Ratio + μ · ‡ for example; K = 2: Salary = β + β 1 Years of Schooling + β 2 Experience + μ · Goal of regression analysis : Infer the value of the β coefficients based on the partial association that we observed between our variables in the sample . The partial association that we observe between variables is represented by our SR: y = b β + b β 1 x 1 + b β 2 x 2 + ... + b β K x K + b μ ‡ for example; K = 1: Test Scores = b β + b β 1 StudentTeacher Ratio + b μ · ‡ for example; K = 2: Salary = b β + b β 1 Years of Schooling + b β 2 Experience + b μ · • We use the OLS method to compute the partial association  i.e. the b β s. • In LN4 we will study under what conditions we can infer a causal effect from the partial association that we observe between variables. 1 LN2—API202A Spring 2009 Harvard Kennedy School 1.1 Computing the b β s by OLS Definitions: Fitted value b y : The part of y that can be associated with the RHS variables in the SR. For K=2: b y = b β + b β 1 x 1 + b β 2 x 2 Thus, y = b y + b μ Therefore, our sample errors b μ represent the part of y that cannot be associated with any RHS variable. b μ = y b y What does OLS do? Finds the values of the b β s that minimize the sample errors b μ . How? By minimizing what is called the sum of the squared residuals (SSR). SSR = ∑ N i =1 b μ 2 i = ∑ N i =1 ( y i b y i ) 2 b μ b μ = ( Y b Y ) ( Y b Y ) (matrix notation) – Intuition for why minimize the sample errors: Extract as much information as possible from our explanatory variables. – Why squared? Notation : We can write the SR in either way (equivalent notations): 1 y = b β + b β 1 x 1 + b β 2 x 2 + b μ y i = b β + b β 1 x 1 ,i + b β 2 x 2 ,i + b μ i 1 Subindex i is a way to denote a single representative observation from the N observations in the sample. 2 LN2—API202A Spring 2009 Harvard Kennedy School How does OLS find the b β s that minimize the SSR? With optimization theory (cal culus/first order conditions): For K=1: min b β , b β 1 SSR = N X i =1 b μ 2 i = N X i =1 ‡ y i b β x 1 ,i b β 1 · 2 first order conditions (FOC): ∂ SSR ∂ b β = 2 N X i =1 ‡ y i b β x 1 ,i b β 1 · = 0 ∂ SSR ∂ b β 1 = 2 N X i =1 ‡ y i b β x 1 ,i b β 1 · x 1 ,i = 0 2 equations and 2 unknowns 2 :→ b β 1 = ∑ N i =1 ( x 1 ,i ¯ x 1 )( y i ¯ y ) ∑ N i =1 ( x 1 ,i ¯ x 1 ) 2 and b β = ¯ y ¯ x 1 b β 1 For any K: The presentation is simpler with matrix notation because we keep track of only one FOC: min b β SSR = ( Y X b β ) ( Y X b β ) ∂ SSR ∂ b β = 2 X ( Y X b β ) = 0→ X...
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This note was uploaded on 04/12/2009 for the course HKS API202A taught by Professor Levy during the Spring '09 term at Harvard.
 Spring '09
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