Jacobs and Carmichael (2002) are comparing various sociological theories that explain why some states have the death penalty and some do not. The investigators have data for 50 states (indexed by i) in years t = 1971, 1981, 1991. The response variable Yit is 1 if state i has the death penalty in year t, else 0. There is a vector of explanatory variables Xit and a parameter vector β, the latter being assumed constant across states and years. Given the explanatory variables, the investigators assume the response variables are independent and log[− log P (Yit = 0|X)] = Xitβ.
(This is a “complementary log log” or “cloglog” model.) After fitting the equation to the data by maximum likelihood, the investigators determine that some coefficients are statistically significant and some are not. The results favor certain theories over others. The investigators say, “All standard errors are corrected for heteroscedasticity by White’s method.... Estimators are robust to misspecification because the estimates are corrected for heteroscedasticity.”
(The quote is slightly edited.) “Heteroscedasticity” means, unequal variances (section 5.4). White’s method is discussed in the end notes to chapter 5: it estimates SEs for OLS when the $’s are heteroscedastic, using equation (5.8). “Robust to misspecification” means, works pretty well even if the model is wrong.
Discuss briefly, answering these questions. Are the authors claiming that parameter estimates are robust, or estimated standard errors? If the former, what do the estimates mean when the model is wrong? If the latter, according to the model, is var(Yit|X) different for different combinations of i and t? Are these differences taken into account by the asymptotic SEs? Do asymptotic SEs for the MLE need correction for heteroscedasticity?