Extensions of Linear Regression Model: Concepts and Applications

1/16: Topic 1.2 Extensions of the Linear Regression Model Microeconometric Modeling William Greene Stern School of Business New York University New York NY USA 1.2 Extensions of the Linear Regression Model

2/16: Topic 1.2 Extensions of the Linear Regression Model Concepts Models Linear Regression Model Quantile Regression Robust Covariance Matrix Bootstrap

3/16: Topic 1.2 Extensions of the Linear Regression Model Regression with Conventional Standard Errors

4/16: Topic 1.2 Extensions of the Linear Regression Model Robust Covariance Matrices The White Estimator i -1 2 i -1 Est.Var[ ]=( ) e ( ) b X X x x X X i i Robust standard errors; (bis not robust ) Robust to: Heteroscedasticty Not robust to: (all considered later) Correlation across observations Individual unobserved heterogeneity Incorrect model specification Robust inference means hypothesis tests and confidence intervals using robust covariance matrices

5/16: Topic 1.2 Extensions of the Linear Regression Model Heteroscedasticity Robust Covariance Matrix Uncorrected Note the conflict: Test favors heteroscedasticity. Robust VC matrix is essentially the same.

6/16: Topic 1.2 Extensions of the Linear Regression Model Bootstrap Estimation of the Asymptotic Variance of an Estimator Known form of asymptotic variance: Compute from known results Unknown form, known generalities about properties: Use bootstrapping Root N consistency Sampling conditions amenable to central limit theorems Compute by resampling mechanism within the sample.

7/16: Topic 1.2 Extensions of the Linear Regression Model Bootstrapping Algorithm 1. Estimate parameters using full sample: b 2. Repeat R times: Draw n observations from the n, with replacement Estimate with b(r). 3. Estimate variance with 1 R R V = [ (r)- ][ (r)- ] b b b b r= 1

8/16: Topic 1.2 Extensions of the Linear Regression Model Application to Spanish Dairy Farms lny=b1lnx1+b2lnx2+b3lnx3+b4lnx4+ Input Units Mean Std. Dev. Minimum Maximum Y Milk production (liters) 131,108 92,539 14,110 727,281 Milk X1 Cows # of milking cows 2.12 11.27 4.5 82.3 X2 Labor # man-equivalent units 1.67 0.55 1.0 4.0 X3 Land Hectares devoted to pasture and crops. Total amount feedstuffs fed to dairy cows (tons) of land 12.99 6.17 2.0 45.1 X4 Feed of 3,924.1 4 57,941 47,981 376,732 N = 247 farms, T = 6 years (1993-1998)

9/16: Topic 1.2 Extensions of the Linear Regression Model Bootstrapped Regression

10/16: Topic 1.2 Extensions of the Linear Regression Model Example: Bootstrap Replications

11/16: Topic 1.2 Extensions of the Linear Regression Model Bootstrapped Confidence Intervals Estimate Norm( )=( 12 + 22 + 32 + 42)1/2

12/16: Topic 1.2 Extensions of the Linear Regression Model

13/16: Topic 1.2 Extensions of the Linear Regression Model Quantile Regression Q(y|x, ) = x, = quantile Estimated by linear programming Q(y|x,.50) = x, .50 median regression Median regression estimated by LAD (estimates same parameters as mean regression if symmetric conditional distribution) Why use quantile (median) regression? Semiparametric Robust to some extensions (heteroscedasticity?) Complete characterization of conditional distribution

14/16: Topic 1.2 Extensions of the Linear Regression Model Estimated Variance for Quantile Regression Asymptotic Theory Based Estimator of Variance of Q-REG x i i i i y u y x x x x = + - - x , ) = , ] = |x Model: , ( Q y x | , [ Q u 0 i i i i = Residuals: u i i i 1 N ( ) 1 1 A C A Asymptotic Variance: 1 N 1 1 B 2 N ] Estimated by i | B = E[f (0) A xx x x i u 1 | u i = 1 i Bandwidth B can be Silverman's Rule of Thumb: 1.06 Min s N ( |.75) ( |.25) Q u Q u i i , u .2 1.349 (1- ) N ] Estimated by (1- ) [ C = xx X X E ( ) 1 For =.5 and normally distributed u, this all simplifies to 2 X X . us 2 But, this is an ideal application for bootstrap pin g.

15/16: Topic 1.2 Extensions of the Linear Regression Model Quantile Regressions = .25 = .50 = .75

16/16: Topic 1.2 Extensions of the Linear Regression Model

17/16: Topic 1.2 Extensions of the Linear Regression Model Coefficient on MALE dummy variable in quantile regressions