Case weights: (a) 'Show how case weights can be used to ‘‘adjust’’ the usual formulas for the least squares coefficients and their covariance matrix. How do these case-weighted formulas compare with...

Case weights:

(a) 'Show how case weights can be used to ‘‘adjust’’ the usual formulas for the least squares coefficients and their covariance matrix. How do these case-weighted formulas compare with those for weighted-least-squares regression (discussed in Section 12.2.2.)?

(b) Using data from a sample survey that employed disproportional sampling and for which case weights are supplied, estimate a least-squares regression (1) ignoring the case weights, (2) using the case weights to estimate both the regression coefficients and their standard errors (rescaling the case weights, if necessary, so that they sum to the sample size), and (3) using the case weights but estimating coefficient standard errors with the bootstrap. Compare the estimates and standard errors obtained in (1), (2), and (3).