Econometrics-I-22

2126 part 22 semi and nonparametric estimation ols vs

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™    21/26
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Part 22: Semi- and Nonparametric Estimation OLS vs. Least Absolute Deviations ---------------------------------------------------------------------- Least absolute deviations estimator ............... Residuals Sum of squares = 1537.58603 Standard error of e = 6.82594 Fit R-squared = .98284 Adjusted R-squared = .98180 Sum of absolute deviations = 189.3973484 --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- |Covariance matrix based on 50 replications. Constant| -84.0258*** 16.08614 -5.223 .0000 Y| .03784*** .00271 13.952 .0000 9232.86 PG| -17.0990*** 4.37160 -3.911 .0001 2.31661 --------+------------------------------------------------------------- Ordinary least squares regression ............ Residuals Sum of squares = 1472.79834 Standard error of e = 6.68059 Standard errors are based on Fit R-squared = .98356 50 bootstrap replications Adjusted R-squared = .98256 --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| -79.7535*** 8.67255 -9.196 .0000 Y| .03692*** .00132 28.022 .0000 9232.86 PG| -15.1224*** 1.88034 -8.042 .0000 2.31661 --------+------------------------------------------------------------- ™    22/26
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Part 22: Semi- and Nonparametric Estimation Quantile Regression p Q(y| x ,a )  =   b ¢ x , a  = quantile p Estimated by linear programming p Q(y| x ,.50)  =   b ¢ x , .50 Ł  median regression p Median regression estimated by LAD (estimates same  parameters as mean regression if symmetric conditional  distribution) p Why use quantile (median) regression? n Semiparametric n Robust to some extensions (heteroscedasticity?) n Complete characterization of conditional distribution ™    23/26
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Part 22: Semi- and Nonparametric Estimation Quantile Regression ™    24/26
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Part 22: Semi- and Nonparametric Estimation ™    25/26 ( 29 1 1 Model: , ( | , ) , [ , ] 0 ˆ ˆ Residuals: u 1 Asymptotic Variance: = E[f (0) ] Estimated by - - = + α = α = = Asymptotic Theory Based Estimator of Variance of Q - REG x | x A C A A xx i i i i i i i i i i i u y u Q y Q u y N β x β x -β x [ ] 1 .2 1 1 1 ˆ 1 | | B B 2 Bandwidth B can be Silverman's Rule of Thumb: ˆ ˆ ( |.75) ( |.25) 1.06 , 1.349 (1- ) (1- ) [ ] Estimated by = < - ÷ α α α α x x C = xx N i i i i i i u u N Q u Q u Min s N E N ( 29 1 2 For =.5 and normally distributed u, this all simplifies to . 2 - π α But, this is an ideal application for bootstrapping X X . X X u s
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Part 22: Semi- and Nonparametric Estimation a  = .25 a  = .50 a  = .75     26/26
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