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Unformatted text preview: Stata Technical Bulletin 19 diagonal matrix with elements W ii = q=f e rro rs # if r% ; q # =f e rro rs # if r& otherwise and R is the design matrix XX . This is derived from formula 3.11 in Koenker and Bassett, although their notation is much different. f e rro rs # refers to the density of the true residuals. There are many things that Koenker and Bassett leave unspecified, including how one should obtain a density estimate for the errors in real data. It is at this point that we offer our contribution. We first sort the residuals and locate the observation in the residuals corresponding to the quantile in question, taking into account weights if they are applied. We then calculate w n , the square root of the sum of the weights. Unweighted data is equivalent to weighted data where each observation has weight 1, resulting in w n = p n . For analytically weighted data, the weights are rescaled so that the sum of the weights is the sum of the observations, resulting in p n again. For frequency weighted data, w n literally is the square of the sum of the weights. We locate the closest observation in each direction such that the sum of weights for all closer observations is . If we run off the end of the dataset, we stop. We calculate w s , the sum of weights for all observations in this middle space. Typically, w s is slightly greater than w n . The residuals obtained after quantile regression have the property that if there are k parameters, then exactly k of them must be zero. Thus, we calculate an adjusted weight w a = w s ; k . The density estimate is the distance spanned by these observations divided by w a . Because the distance spanned by this mechanism converges toward zero, this estimate of the density converges in probability to the true density. References Gould, W. 1992. Quantile regression and bootstrapped standard errors. Stata Technical Bulletin 9: 1921. Koenker, R. and G. Bassett, Jr. 1982. Robust tests for heteroscedasticity based on regression quantiles. Econometrica 50: 4361. Rogers, W. H. 1992. Quantile regression standard errors. Stata Technical Bulletin 9: 1619. sg17 Regression standard errors in clustered samples William Rogers, CRC, FAX 3103937551 Statas hreg , hlogit and hprobit commands estimate regression, maximumlikelihood logit, and maximumlikelihood probit models based on Hubers (1967) formula for individuallevel data and they produce consistent standard errors even if there is heteroscedasticity, clustered sampling, or the data is weighted. The description of this in [5s] hreg might lead one to believe that Huber originally considered clustered data, but that is not true. I developed this approach to deal with cluster sampling problems in the RAND Health Insurance Experiment in the early 1980s (Rogers 1983; Rogers and Hanley 1982; Brook, et al. 1983). What is true is that with one simple assumption, the framework proposed by Huber can be applied to produce the answer we propose. That assumption is that the clusters are drawn as a simple random sample from some population. Thethe answer we propose....
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This note was uploaded on 12/26/2011 for the course ECON 245a taught by Professor Staff during the Fall '08 term at UCSB.
 Fall '08
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