Lecture8_2005

Lecture8_2005 - Green / Stats Causal Order One of the basic...

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Green // Stats Causal Order One of the basic principles of statistics – or philosophy of science for that matter – is that numbers themselves cannot adjudicate questions of causality. Just because variables X and Y are strongly correlated does not mean that X causes Y or Y causes X. Conversely, it is possible to construct examples in which X causes Y, yet X and Y are uncorrelated. Indeed, it is possible for X to be positively correlated with Y even though X exerts a negative causal effect on Y. A few examples may help to drive these points home. 1. Correlation without causation: National League World Series victories and presidential election outcomes. If one looks at enough variables, meaningless correlations will pop up. 2. Correlation without causation due to reversal of cause and effect: Basketball players tend to be unusually tall. Does joining a basketball team cause one to grow? 3. Causation without correlation: Studies in New Haven have found little or no correlation between voting rates and communication with get-out-the-vote canvassers. Yet the very same experimental studies have shown that this communication substantially increases voter turnout. 4. Positive causation despite negative correlation: Incumbent members of Congress who lose their bids for reelection tend to spend much more than incumbents who win reelection. Given this pattern, should one infer that incumbents hurt their reelection prospects by spending money on their campaigns? Concept of “Observational Equivalence” One of the most important ideas in statistics is the notion that different “data generation processes” can produce the same observable outcomes. In other words, the mere fact that a given model is consistent with what we observe does not rule out the possibility that other models would do equally well, if not better. In order to see how this principle plays out numerically, let’s take a step back and think about how regression results are produced. Recall that the OLS estimate of the effect of X on Y is obtained using the formula ( 29 ( 29 ( 29 ( 29 ( 29 - - = - - - - - = = 2 2 1 1 1 1 ) ( ) , ( ˆ X X Y X X X X N Y Y X X N X Var Y X Cov b
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In other words, the OLS estimate is just the ratio of a covariance and a variance. So, in order to formulate examples of observational equivalence, we need to understand how covariances and variances are shaped. Here are the basic rules of “covariance algebra”: 1. Var(b)=0. Interpretation: the variance of a constant is zero. Algebra: = - - = 0 ) ( 1 1 ) var( 2 b b N b 2. Cov(X,b)=0. Interpretation: there is no covariance between a constant and a variable. Algebra: = - - = 0 ) ( 1 1 ) , ( X b b N b X Cov 3. Cov(aX,bZ)=abCov(X,Z). Comment: covariances are easily factored.
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This note was uploaded on 04/07/2008 for the course STAT 102 taught by Professor Jonathanreuning-schererdonaldgreen during the Fall '05 term at Yale.

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Lecture8_2005 - Green / Stats Causal Order One of the basic...

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