Fixed Effects Individual Slopes: Accounting and Testing for Heterogeneous Effects in Panel Data or Other Multilevel Models

Abstract

Fixed effects (FE) panel models have been used extensively in the past, as those models control for all stable heterogeneity between units. Still, the conventional FE estimator relies on the assumption of parallel trends between treated and untreated groups. It returns biased results in the presence of heterogeneous slopes or growth curves that are related to the parameter of interest (e.g., selection into treatment is based on individual growth of the outcome). In this study, we derive the bias in conventional FE models and show that fixed effects individual slope (FEIS) models can overcome this problem. This is a more general version of the conventional FE model, which accounts for heterogeneous slopes or trends, thereby providing a powerful tool for panel data and other multilevel data in general. We propose two versions of the Hausman test that can be used to identify misspecification in FE models. The performance of the FEIS estimator and the specification tests is evaluated in a series of Monte Carlo experiments. Using the examples of the marital wage premium and returns to preschool education (Head Start), we demonstrate how taking heterogeneous effects into account can seriously change the conclusions drawn from conventional FE models. Thus, we propose to test for bias in FE models in practical applications and to apply FEIS if indicated by the specification tests.

Environmental Inequality and Residential Sorting in Germany: A Spatial Time-Series Analysis of the Demographic Consequences of Industrial Sites

“…As in the con ven tional FE mod els, the FEIS esti ma tor is based on munic i pal i ties exhibiting rel e vant within-var i ance, while we keep those with out within-var i ance as a "con trol group" for exog e nous time shocks by includ ing time dummy variables (as in two-way FE). Obviously, we still rely on the strict exogeneity assump tion of no time-vary ing unob served con found ers being cor re lated with ourcovariatesnetofincludedcon trolsandmunic i pal ity-spe cificeco nomictrends(for more details, see Brüderl and Ludwig 2015;Rüttenauer and Ludwig 2020).…”

“…Conventional two-way fixed-effects (FE) esti ma tors rely on the assump tion of par al lel trends between munic i pal i ties receiv ing new facil i ties (or expe ri enc ing a decline) and those not expe ri enc ing a change, as obser va tions with out var i ance in facil i tiesremainintheeffec tiveesti ma tionsam pleasa"con trolgroup"fortem po ral shocks (Rüttenauer and Ludwig 2020). Still, dif fer ent regimes of eco nomic devel opment likely lead to diverg ing trends in income and the num ber of facil i ties over time.…”

Effekt eines einwöchigen Gesundheitsprogramms auf die psychische Gesundheit von pflegenden Eltern – eine kontrollierte Panelstudie

“…Die beschriebenen Gründe für die Ausfälle der Vorher-Messzeitpunkte sind organisatorischen Gründen geschuldet und dürften unsystematisch sein. Selection-on-slope, d. h. Steigungsunterschiede, könnten kontrolliert werden [36], hätten aber weitere Vorhermesszeitpunkte erfordert. Das bedeutet, dass ein Teil der Interventionseffekte auf Steigungsunterschiede im Wohlbefinden zurückzuführen sein könnte -sofern steigende Wohlbefindenskurven für die Anmeldung zur Intervention maßgeblich wären.…”

Does Marriage Increase Couples’ Life Satisfaction?

“…the fi xed-effects models with individual slopes (FEIS) (Ludwig/Brüderl 2018; Wooldridge 2010), which accounts for the second shortcoming mentioned above by relaxing the parallel trends assumption. The FEIS is a generalized FE estimator that allows for different slopes among subgroups by de-trending for the individual trajectory in the dependent and independent variable (Ludwig/Brüderl 2018;Rüttenauer/Ludwig 2020;Wooldridge 2010). That (within) effects of an independent variable differ across subgroups is a standard fi nding in empirical studies and is usually modelled by the inclusion of interaction effects, such as in an FE model.…”

“…Accounting (partially) for self-selection into marriage, or accounting for mechanism a) above, using a fi xed-effects estimator does presumably underestimate this effect since it does not fully control for the possibility that people with steeper trajectories in life satisfaction; people whose relationship evolves more positively over time, are more likely to be selected as marriage partners. This mechanism b) can only be accounted for when estimating fi xed effects with individual slopes (FEIS, see Ludwig/Brüderl 2018;Rüttenauer/Ludwig 2020;Wooldridge 2010). Our analysis results in a stronger effect of marriage on life satisfaction, which provides evidence that marriage does indeed make people more satisfi ed with life, or mechanism c) above.…”