Lecture3_2005

Lecture3_2005 - Prof. Green Intro Stats Regression with...

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Prof. Green Intro Stats Regression with Experimental Data Regression ranks among the most useful tools in statistics. Use #1: Predicting outcomes . Often a mindless, theory-free activity. Can be fun and/or profitable. Example: predicting sales based on leading indicators. Use #2: Estimating causal parameters . Requires assumptions about how the independent variable is related to unobserved disturbances that influence the dependent variable. Easy to produce numbers; producing convincing results requires ingenuity, a strong research design, or a gullible audience. Terminology Dependent variable : outcome or response variable. Usually denoted Y and put on Y- axis. Independent variable : treatment variable or regressor. Usually denoted X and put on X- axis. One “regresses Y on X.” Standard Linear Regression Model Y = a + bX +U Note that a (intercept) and b (slope) are parameters . These are typically unknown to the researcher. Interpretation: the term a is the expected value of Y when X=0. The term b is the rate at which the expected value of Y changes for each one-unit increase in X. X and Y are observed variables. Note that the unobserved variable U is called the “disturbance term” or “error term.” The variance of U is another parameter in the model. This model is linear in the parameters – no exponents or quotients. If the U variable were zero for every observation, we would find a straight line relationship when we plot Y against X.
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Example Suppose the true regression model were: Y = 2 – 3 X + U a B X U Y good 2 -3 1 0 -1 perfectly controlled experiment 2 -3 2 0 -4 (no disturbance variance) 2 -3 3 0 -7 2 -3 4 0 -10 2 -3 5 0 -13 a B X U Y okay 2 -3 1 1 -3 Problem: X and U are correlated 2 -3 2 4 -2 (note that the observed 2 -3 3 7 -8 relationship between X and Y 2 -3 4 10 -9 Is flat when it should be negative) 2 -3 5 13 -13 a B X U Y bad 2 -3 1 1 0 Problem: X and U are correlated 2 -3 2 4 0 (note that the observed 2 -3 3 7 0 relationship between X and Y 2 -3 4 10 0 Is flat when it should be negative) 2 -3 5 13 0 Here’s what the three sets of data look like. The black line correctly reflects the true regression line. The red line distorts the true underlying regression because X and U are correlated. The green points fall more or less along the true regression line.
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X Y-Data 5 4 3 2 1 0 -2 -4 -6 -8 -10 -12 -14 Variable Y okay Y good Y bad Scatterplot of Y good, Y bad, Y okay vs X
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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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Lecture3_2005 - Prof. Green Intro Stats Regression with...

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