Lecture 9 Prof. Arkonac's Slides (Ch 6.7 - Ch 7.7) for ECO 4000

Lecture 9 Prof. Arkonac's Slides (Ch 6.7 - Ch 7.7) for ECO 4000

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Introduction to Multiple Regression ECO 4000, Statistical Analysis for Economics and Finance Fall 2010 Lecture 9 Prof: Seyhan Erden Arkonac, PhD 1
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2 The Sampling Distribution of the OLS Estimator (SW Section 6.6) Under the four Least Squares Assumptions, The exact (finite sample) distribution of 1 ˆ has mean 1 , var( 1 ˆ ) is inversely proportional to n ; so too for 2 ˆ . Other than its mean and variance, the exact (finite- n ) distribution of 1 ˆ is very complicated; but for large n 1 ˆ is consistent: 1 ˆ p 1 (law of large numbers) 11 1 ˆˆ () ˆ var( ) E is approximately distributed N (0,1) (CLT) So too for 2 ˆ ,…, ˆ k Conceptually, there is nothing new here!
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3 Multicollinearity, Perfect and Imperfect (SW Section 6.7) Some more examples of perfect multicollinearity The example from earlier: you include STR twice. Second example: regress TestScore on a constant, D , and B , where: D i = 1 if STR ≤ 20, = 0 otherwise; B i = 1 if STR >20, = 0 otherwise, so B i = 1 – D i and there is perfect multicollinearity Would there be perfect multicollinearity if the intercept (constant) were somehow dropped (that is, omitted or suppressed) in this regression? This example is a special case of…
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4 The dummy variable trap Suppose you have a set of multiple binary (dummy) variables, which are mutually exclusive and exhaustive – that is, there are multiple categories and every observation falls in one and only one category (Freshmen, Sophomores, Juniors, Seniors, Other). If you include all these dummy variables and a constant, you will have perfect multicollinearity – this is sometimes called the dummy variable trap . Why is there perfect multicollinearity here ? Solutions to the dummy variable trap : 1. Omit one of the groups (e.g. Senior), or 2. Omit the intercept What are the implications of (1) or (2) for the interpretation of the coefficients?
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Perfect Multicollinearity: . reg ahe age female male Source | SS df MS Number of obs = 7986 -------------+------------------------------ F( 2, 7983) = 164.89 Model | 24300.9748 2 12150.4874 Prob > F = 0.0000 Residual | 588266.294 7983 73.6898777 R-squared = 0.0397 -------------+------------------------------ Adj R-squared = 0.0394 Total | 612567.269 7985 76.7147487 Root MSE = 8.5843 ------------------------------------------------------------------------------ ahe | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | .4415421 .0332389 13.28 0.000 .3763852 .5066989 female | -2.346755 .1950323 -12.03 0.000 -2.729069 -1.964441 male | (dropped) _cons | 4.606864 .9990293 4.61 0.000 2.648505 6.565222 ------------------------------------------------------------------------------ Male ahe = $4.60 Female ahe = $4.60 – $2.34 = $2.26 5
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Perfect Multicollinearity with no intercept: . reg ahe age female male, noconstant Source | SS df MS Number of obs = 7986 -------------+------------------------------ F( 3, 7983) =10270.68 Model | 2270534.96 3 756844.986 Prob > F = 0.0000 Residual | 588266.294 7983 73.6898777 R-squared = 0.7942 -------------+------------------------------ Adj R-squared = 0.7941 Total | 2858801.25 7986 357.976616 Root MSE = 8.5843 ------------------------------------------------------------------------------ ahe | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | .4415421 .0332389 13.28 0.000 .3763852 .5066989 female | 2.260109 .9972543 2.27 0.023 .3052296 4.214988 male | 4.606864 .9990293 4.61 0.000 2.648505 6.565222 ------------------------------------------------------------------------------ Note that difference in female and male coefficients gives the correct female coefficient on the previous output: 2.260109 - 4.606864 = - 2.346755 6
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Summary of multicollinearity; Multicollinearity: (a) perfect multicollinearity: including both male and female in a regression, STATA drops one. (b) imperfect multicollinearity (often this is called as multicollinearity) : occurs when any two or more regressors have high correlation.
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Lecture 9 Prof. Arkonac's Slides (Ch 6.7 - Ch 7.7) for ECO 4000

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