The most familiar fixed effects (FE) and random effects (RE) panel data treatments for count data were proposed by Hausman, Hall and Griliches (HHG) (1984). The Poisson FE model is particularly simple and is one of a small few known models in which the incidental parameters problem is, in fact, not a problem. The same is not true of the negative binomial (NB) model. Researchers are sometimes surprised to find that the HHG formulation of the FENB model allows an overall constant - a quirk that has also been documented elsewhere. We resolve the source of the ambiguity, and consider the difference between the HHG FENB model and a "true" FENB model that appears in the familiar index function form. The familiar RE Poisson model using a log gamma heterogeneity term produces the NB model. The HHG RE NB model is also unlike what might seem the natural application in which the heterogeneity term appears as an additive common effect in the conditional mean. We consider the lognormal model as an alternative RENB model in which the common effect appears in a natural index function form.
Fixed Effects Regression Models Allison Pdf 43
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In addition to this synthesis of the inter-disciplinary methodological literature on FE and RE models (information that, whilst often misunderstood, is not new), we present an original simulation study showing how various forms of these models respond in the presence of some plausible model mis-specifications. The simulations show that estimated standard errors are anti-conservative when random-slope variation exists but a model does not allow for it. They also show the robustness of estimation results to mis-specification of random effects as Normally distributed, when they are not; substantial biases are confined to variance and random effect estimates in models with a non-continuous response variable.
Both the Mundlak model and the within-between random effects (REWB) models (Eqs. 2 and 3 respectively) are easy to fit in all major software packages (e.g. R, Stata, SAS, as well as more specialist software like HLM and MLwiN). They are simply random effects models with the mean of \(x_it\) included as an additional explanatory variable (Howard 2015).
We hope the discussion above has convinced readers of the superiority of the REWB model, except perhaps when the within and between effects are approximately equal, in which case the standard RE model (without separated within and between effects) might be preferable for reasons of efficiency.Footnote 8 Even then, the REWB model should be considered first, or as an alternative, since the equality of the within and between coefficients should not be assumed. As for FE, except for simplicity there is nothing that such models offer that a REWB model does not.
The only difference between RE and FE lies in the assumption they make about the relationship between υ [the unobserved time-constant fixed/random effects] and the observed predictors: RE models assume that the observed predictors in the model are not correlated with υ while FE models allow them to be correlated.
Further, using the REWB model as if it were a FE model leads researchers to use it without taking full advantage of the benefits that RE models can offer. The RE framework allows a wider range of research questions to be investigated: involving time-invariant variables, shrunken random effects, additional hierarchical (e.g. geographical) levels and, as we discuss in the next section, random slopes estimates that allow relationships to vary across individuals, or allow variances at any level to vary with variables. As well as yielding new, substantively interesting results, such actions can alter the average associations found. Describing the REWB, or Hybrid, model as falling under a FE framework therefore undersells and misrepresents its value and capabilities.
These results support the strong critique by Barr et al. (2013) that not to include random slopes is anticonservative. On the other hand, Matuschek et al. (2017) counter that analytical models should also be parsimonious, and fitting models with many random effects quickly multiplies the number of parameters to be estimated, particularly since random slopes are generally given covariances as well as variances. Sometimes the data available will not be sufficient to estimate such a model. Still, it will make sense in much applied work to test whether a statistically significant coefficient remains so when allowed to vary randomly. We discuss this further in the conclusions.
Datasets often have structures that span more than two levels. A further advantage of the multilevel/random effects framework over fixed effects is its allowing for complex data structures of this kind. Fixed effects models are not problematic when additional higher levels exist (insofar as they can still estimate a within effect), but they are unable to include a third level (if the levels are hierarchically structured), because the dummy variables at the second level will automatically use up all degrees of freedom for any levels further up the hierarchy. Multilevel models allow competing explanations to be considered, specifically at which level in a hierarchy matters the most, with a highly parsimonious specification (estimating a variance parameter at each level).Footnote 12
In sum, even substantial violations of the Normality assumption of the higher-level random effects do not have much impact on estimates in the fixed part of the model, nor the standard errors. Such violations can however affect the random effects estimates, particularly in models with a non-continuous response.
Third, and in contrast to much of the applied literature, we argue that researchers should not use a Hausman test to decide between fixed and random effects models. Rather, they can use this test, or models equivalent to it, to verify the equivalence of the within and between relationships. A lack of equality should be in itself of interest and worthy of further investigation through the REWB model.
Note though that, in the longitudinal setting, between effects will only be fully controlled if those effects do not change over time (this is the case with the REWB/Mundlak models as well, unless such heterogeneity is explicitly modelled).
That is, the random effects were in all cases uncorrelated. We also generated binary data based on similar models (both random intercept-only and random intercept, random slope models), using a logit link. In all cases, \(\sigma_\upsilon 0^2\) and \(\sigma_\upsilon 1^2\) were set to 4, and (for the Normally distributed data) the variance of \(\epsilon_it\) to 1. The overall intercept \(\beta_0\) and the overall slope \(\beta_1\) were also set to 1. The \(x_it\) data were drawn from a Normal distribution with a mean of 0 and a variance of 0.25^2.
We then fitted three different models to each simulated dataset: a fixed effects model (with naïve and clustered standard errors), a random intercepts-only model, and a random intercepts-random slopes model.
We conducted the simulations in R. For fitting multilevel models we used the package lme4 (Bates et al. 2015). For deriving clustered standard errors from the fixed effects models, we used the plm package (Croissant and Millo 2008). We caught false or questionable convergences and simply removed them, simulating a new dataset instead (this should not bias the results, although it should be noted as an advantage of FE is that it is unlikely to show convergence problems due to being estimated by OLS). We tried multiple runs of simulations, and found stable results beyond about 200 simulations per DGP.
Despite much interest in how parenthood contributes to the gender pay gap, prior research has rarely explored firms' roles in shaping the parenthood pay penalty or premium. The handful of studies that investigated parenthood's effects within and across firms generally compared parents and their childless peers at a given time and failed to account for unobserved heterogeneity between the two groups. Such comparisons also cannot inform how having children may alter individuals' earnings trajectories within and across firms. Using 26 rounds of the National Longitudinal Survey of Youth 1979 and fixed-effects models, we examine how being a mother or father is linked to earnings growth within and across firms. We find that women's pay decreases as they become mothers and that the across-employer motherhood penalty is larger than the within-employer penalty. By contrast, fatherhood is associated with a pay premium, and the within-employer fatherhood premium is considerably greater than the across-employer one. We argue that these results are consistent with the discrimination explanation of the motherhood penalty and fatherhood premium. Because employers are likely to trust women who become mothers while working for them more than new recruits who are mothers, their negative bias against mothers would be more salient when evaluating the latter, which could result in a larger between-organizational motherhood penalty. Conversely, employers' likely greater trust in existing workers who become fathers than fathers they hire from elsewhere may amplify their positive bias favoring fathers in assessing the former, which could explain the greater within-firm fatherhood premium.
The outcome of interest for our study is the reported hourly pay of respondents' jobs (in cents). Because of the skewedness of earnings distribution, we take the natural log of the hourly earnings. Like most studies of the motherhood pay penalty and fatherhood pay premium, we use fixed-effects models to predict log hourly earnings (Budig and England 2001; Gangl and Ziefle 2009; Glauber 2008; Killewald 2013; Yu and Kuo 2017). Because the selection into parenthood is unlikely to be random, regression models that compare earnings between parents and their otherwise similar peers face the problem that other unobserved factors that contribute to the two groups' childbearing decisions, such as devotion to employment careers, may also account for differences in pay. Fixed-effects models, by contrast, enable us to take into account all unmeasured characteristics that do not vary across observations for a given subject, be it a person, an occupation, or an employing organization (Allison 2009). Such models can better address the problem of unobserved heterogeneity. 2ff7e9595c
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