Getting the within Estimator of Cross-level Interactions in Multilevel Models with Pooled Cross-sections: Why Country Dummies (Sometimes) Do Not Do the Job

Multilevel models with persons nested in countries are increasingly popular in cross-country research. Recently, social scientists have started to analyze data with a three-level structure: persons at level 1, nested in year-specific country samples at level 2, nested in countries at level 3. By using a country fixed-effects estimator, or an alternative equivalent specification in a random-effects framework, this structure is increasingly used to estimate within-country effects in order to control for unobserved heterogeneity. For the main effects of country-level characteristics, such estimators have been shown to have desirable statistical properties. However, estimators of cross-level interactions in these models are not exhibiting these attractive properties: as algebraic transformations show, they are not independent of between-country variation and thus carry country-specific heterogeneity. Monte Carlo experiments consistently reveal the standard approaches to within estimation to provide biased est...