### Equality restrictions in the social sciences and impacts on estimators of covariates

*Marco Giesselmann, Wolfgang Jagodzinski*

Building: Law Building

Room: Breakout 3 - Law Building, Room 104

Date: 2012-07-10 01:30 PM – 03:00 PM

Last modified: 2012-01-12

#### Abstract

It has become a common practice in survey research to represent groups and collectives by means of dummy variables. In comparative survey research, for example, country-specific influences are often estimated in this way. If a survey includes L countries and we want to find out whether the average life satisfaction in each of the remaining L-1 countries significantly differs from the life satisfaction in a baseline country C0 we represent each country by a country dummy variable (CDV) and regress them on the life satisfaction. Researchers who are interested in micro-level relationships or micro/macro interactions often use CDVs for the elimination of composition effects. Another well-known field of application is cohort analysis. Here dummy variables represent generations, age groups, and/or periods.

While the classical discussion in cohort analysis focused on the aggregate-level effects of dummy variables, developments during the last decades have expanded the approach in two directions. First, simple macro-level models have been replaced by multi-level, mostly two-level models. Researchers nowadays are not only interested in the variation between groups but also in the variation between individuals. Second, models have become more complex insofar, as ordinal or metric variables are included besides dichotomous variables. In this research note we want to examine the interplay between dummy variables and metric variables in two-level models. More precisely we want to find out to what extent the effect of a metric variable on a given dependent variable is affected by the dummy variables in the model.

The latter question in turn is motivated by noticing that the specification of a full set of dummy-variables often leads to identification problems in presence of covariates. To make estimation of such models feasible, it is common to collapse several macro-units on one dummy variable. While the mathematics of an estimation with such â€˜equality restrictionsâ€™ are well-known, its methodological implications are still somehow vague. Therefore, we want to explain how estimators in a DV-model with equality restrictions are constructed and have to be interpreted from a technical viewpoint. By proving a substantial insight in what an equality restriction actually implies, we also want to reveal, why often results differ depending on the collapsing strategy. Additionally, it will be substantively clarified, why especially the choice of a minimal set (with only two macro-units collapsed on one dummy variable) will generate arbitrary results.

While the classical discussion in cohort analysis focused on the aggregate-level effects of dummy variables, developments during the last decades have expanded the approach in two directions. First, simple macro-level models have been replaced by multi-level, mostly two-level models. Researchers nowadays are not only interested in the variation between groups but also in the variation between individuals. Second, models have become more complex insofar, as ordinal or metric variables are included besides dichotomous variables. In this research note we want to examine the interplay between dummy variables and metric variables in two-level models. More precisely we want to find out to what extent the effect of a metric variable on a given dependent variable is affected by the dummy variables in the model.

The latter question in turn is motivated by noticing that the specification of a full set of dummy-variables often leads to identification problems in presence of covariates. To make estimation of such models feasible, it is common to collapse several macro-units on one dummy variable. While the mathematics of an estimation with such â€˜equality restrictionsâ€™ are well-known, its methodological implications are still somehow vague. Therefore, we want to explain how estimators in a DV-model with equality restrictions are constructed and have to be interpreted from a technical viewpoint. By proving a substantial insight in what an equality restriction actually implies, we also want to reveal, why often results differ depending on the collapsing strategy. Additionally, it will be substantively clarified, why especially the choice of a minimal set (with only two macro-units collapsed on one dummy variable) will generate arbitrary results.