Discovering Quantile Regression: Beyond Ordinary Least Squares

1. CHAPTER 20 BEYOND OLS: QUANTILE REGRESSION Damodar Gujarati Econometrics by Example, second edition

2. BEYOND ORDINARY LEAST SQUARES • Ordinary least squares (OLS) models focus on the mean, E(Y|X) or the conditional expectation function (CEF). • Applied economists increasingly want to know what is happening to an entire distribution, which can be achieved using quantile regression (QR). • Quantiles divide or partition the number of observations in a study into equally sized groups, such as Quartiles (divisions into four groups), quintiles (division into five groups), deciles (division into ten equal groups), and centiles or percentiles (division into 100 equal groups). Damodar Gujarati Econometrics by Example, second edition

3. QUANTILE • The pth value of Y is equal to: • The quantile, Qp, is the inverse of the CDF, F. Damodar Gujarati Econometrics by Example, second edition

4. QUANTILE REGRESSION MODEL (QRM) • 1. First estimate a QR for the 50th quantile (the median). • 2. Then estimate a QR for the 25th (i.e. the first quartile) and the 75th quantile (i.e. the third quartile) simultaneously. • 3. Compare the QRs estimated in (2) with the OLS regression. • 4. One possible reason that the coefficients may differ across quantiles is the presence of heteroscedastic errors. • 5. The estimated quantile coefficients may look different for different quantiles. • To see if they are statistically different, we use the Wald test. We can use the Wald test to test the coefficients of a single regressor across the quantiles or we can use it to test the equality of the coefficients of all the regressors across the various quantiles. Damodar Gujarati Econometrics by Example, second edition

5. QUANTILE REGRESSION MODEL (QRM) • In OLS, we minimize the sum of squared errors: • In QRM, we minimize the absolute sum of errors: Damodar Gujarati Econometrics by Example, second edition