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Unformatted text preview: Lecture Note 4 Under What Conditions Can We Infer Causality from OLS? Empirical Methods II (API202A) — Spring 2009 Harvard Kennedy School We have studied so far: – The concept of causality (our goal) and how we represent it with the SEQ ( β 1 ) – The concept of partial association (what we observe), and how we represent it with the SR ( b β 1 ), and how we compute it with OLS. In this LNs we will dive into the question: Under what conditions can we infer causality from the partial association (OLS)? – We have already developed part of the intuition to answer this question and now we will formalize it. 1 LN4—API202A Spring 2008 Harvard Kennedy School 1 Is Our Intuition Correct? Does b β 1 represent the causal effect of class size on students’ performance? That is, can we infer β 1 from b β 1 ? SEQ: Test Scores = β + β 1 StudentTeacher Ratio + μ SR: Test Scores = b β + b β 1 StudentTeacher Ratio + b μ Our intuition: Average Family Income (AFI) is negatively correlated with Studentper Teacher Ratio (STR). As a result, b β 1 captures the effect of STR on scores and part of the effect of AFI on scores (confounding effect). Therefore, we cannot interpret b β 1 as the causal effect of STR on scores. Is our intuition correct? Let’s see... Note : A useful way to think about causality is to assume that we have data for the entire population of interest. That is, to assume there is no sample randomness. We will see later how things change if we lift this assumption....in practice, nothing really changes. 2 LN4—API202A Spring 2008 Harvard Kennedy School i) We start thinking about whether our intuition is correct from the formula that com putes b β 1 by expressing it in demeaned terms (K=1). b β 1 = ∑ N i =1 ( x 1 ,i ¯ x 1 )( y i ¯ y ) ∑ N i =1 ( x 1 ,i ¯ x 1 ) 2 = ∑ N i =1 x d 1 ,i · y d i ∑ N i =1 ( x d 1 ,i ) 2 ii) Let’s replace y d i with the SEQ written in demeaned terms: y d i = β 1 x d 1 ,i + μ i b β 1 = ∑ N i =1 x d 1 ,i · ( β 1 x d 1 ,i + μ i ) ∑ N i =1 ( x d 1 ,i ) 2 = β 1 ∑ N i =1 x d 1 ,i · x d 1 ,i ∑ N i =1 ( x d 1 ,i ) 2  {z } β 1 · 1 + ∑ N i =1 x d 1 ,i · μ i ∑ N i =1 ( x d 1 ,i ) 2  {z } Cov ( X 1 ,μ ) V ar ( X 1 ) iii) Thus, we can express b β 1 as: iii) ( Advanced topic ) We can rewrite our SEQ to get a closer look at the bias term: SEQ: Test Scores = β + β 1 StudentTeacher Ratio + β 2 AFI + ε  {z } μ and expanding the covariance term we can write b β 1 as: 1 # " ˆ ! b β 1 = β 1 + Cov ( X 1 ,μ ) V ar ( X 1 )  {z } Bias ’ & $ % b β 1 = β 1 + β 2 Cov ( X 1 ,AFI ) V ar ( X 1 )  {z } due to AFI omitted + Cov ( X 1 ,ε ) V ar ( X 1 )  {z } Bias 1 Note that Cov ( X 1 ,β 2 AFI + ε ) = β 2 Cov ( X 1 ,AFI ) + Cov ( X 1 ,ε )....
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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
 LEVY

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