lec34v1_1up

lec34v1_1up - Stat 104 Quantitative Methods for Economists...

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Stat 104: Quantitative Methods for Economists Class 34: Heteroskedasticity and Multicollinearity 1
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The Model and Data Y X = + + β ε 0 1 Given X ’s X X X m 1 2 , , K We assume the following model holds: 2 ~ (0, ) i N independent σ Y X i i i = + + 0 1 2
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Non-Constant Variance or Heteroskedasticity Another of our basic assumptions is that the all have the same distribution and in particular, the same variance. What does a violation of this assumption look like ? i ε heteroskedasticity means the variance of the errors changes. our model assumes “homoskedasticity” 3
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Example: We have data on manufacturing plants for a Fortune 500 company. The data consists of the number of supervisors (Y) and the associated number of supervised workers (X), 200 4 0 super 0 500 1000 1500 2000 worker
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Regression Output 5
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Residual Diagnostics -20 0 20 40 Residuals If you have heteroskedasaticity, your estimates are ok, but your standard errors are incorrect. That’s not good. 6 -60 -40 50 100 150 200 Fitted values
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Good basic solution for heteroskedasticity s One way to think of heteroskedasticity is that the noise in the model increases as the value of |X| increases ssentially there is too much variation in the s Essentially there is too much variation in the model.That is, there is excess variation in the Y variable. s An easy way to reduce the variation in Y is to take the log of it. 7
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The log s By “logging” your data, you are transforming it to a different scale. s The log scale squeezes numbers together, so ere is less variation. However, the model there is less variation. However, the model becomes slightly different to interpret since the scale of the Y variables changes (instead of dollars we are modeling “log dollars”; what exactly are those ?) 8
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The log again- shrinks the scale Y log(Y) 1 0 5 1.609438 10 2.302585 50 3.912023 100 4.60517 1000 6.907755 15000 9.615805 Variance 31446401 10.80543 9
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Plot of Log Y versus X 5 5.5 10 3.5 4 4.5 lsuper 0 500 1000 1500 2000 worker
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New Regression Output WARNING : now can’t compare s or R-sq 11
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New Residual Plot -.2 0 .2 .4 Residuals Are we done or is there anything else wrong ? 12 -.6 -.4 4 4.5 5 5.5 Fitted values
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What transformation should we try ? o back to our Go back to our transformation guide 13
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We will try log(X) 5 5.5 14 3.5 4 4.5 lsuper 5.5 6 6.5 7 7.5 lworker
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New Output 15
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0 .2
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lec34v1_1up - Stat 104 Quantitative Methods for Economists...

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