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Estimating and Assessing Distributional Regression

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Analysing Inequalities in Germany

Part of the book series: SpringerBriefs in Statistics ((BRIEFSSTATIST))

Abstract

This chapter sketches out how Structured Additive Distributional Regression relates to other regression models, like classical linear models, quantile regression models and conditional transformation models. In addition, the chapter entails some remarks on covariate selection, model complexity and state space issues.

Essentially, all models are wrong, but some are useful.

George Box (1987)

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Notes

  1. 1.

    Interestingly enough, the advancement of stochastic concepts ran parallel to the appearance of the fictional novel on the literary scene (see, Esposito 2007).

  2. 2.

    For the cross-sectional data predominantly used in this book, OLS generally requires independent and identically distributed error terms that are exogenous. Moreover, the design matrix needs to be void of multicollinearity. Often homoscedasticity and normality are also assumed for inferential purposes. Despite frequently supposing a normality, OLS does not necessarily require any assumption on the nature of the distribution of the error terms though.

  3. 3.

    Best in this context means giving the lowest variance of the estimate.

  4. 4.

    It should be stressed that the requirement that the estimator is unbiased is quintessential to the theorem and other biased estimators have been found which can have better mean-squared error (MSE) properties as they feature lower variance, e.g. estimators from ridge regression.

  5. 5.

    Note that alternatively one could also minimise the negative of the likelihood or related forms thereof like the deviance, where one basically subtracts the likelihood of a given model from another benchmark value thought to represent a saturated model.

  6. 6.

    It should be noted that the term convergence is not used in a strict mathematical sense, like almost sure convergence or stochastic convergence, but rather in a heavily heuristic sense. For a more formally inclined discussion of convergence and estimator properties, see among others White (2001).

  7. 7.

    Fisher criticised Bayesians for assuming that uncertainties can be expressed in form of probabilities (see Gigerenzer et al. 1989, p. 93). While it may be argued if that were true this would prompt inconsistencies all over inferential statistics, it is a fair point that the operationalisation of implicit assumptions can be highly challenging, to say the least.

  8. 8.

    Varying coefficients and more generally mixed models can be seen as a model class that arguably bridges the divide between the frequentist and the Bayesian paradigm. For more information on mixed models the reader is referred to Safken (2015).

  9. 9.

    The direct quote from Marx (1983, p. 189) reads: “Die Gesellschaft besteht nicht aus Individuen, sondern drückt die Summe der Beziehungen, Verhältnisse aus, worin diese Individuen zueinander stehen”.

  10. 10.

    I thus focus on potential labour market experience rather than actual labour market experience. This choice is grounded in the belief that experience is not only derived out of employment spells but also from other life experiences such as caring for children.

  11. 11.

    The reason for not using a continuous variable as in Mincer is that such a continuous variable would feature high point masses rather than a continuous spectrum of the distribution.

  12. 12.

    For example, relating a normal distribution with two parameters to two variables linearly would require the estimation of 6 parameters when including a constant, while even a coarse grid of only ten analogously specified conditional quantiles to approximate the conditional distribution would already require the estimation of 30 parameters.

  13. 13.

    For notational brevity I have excluded a distinct time-specific effect, \(b_t\). Since most economic panel databases for income analysis feature relatively few time periods, this effect can be captured by few more linear effects in the second term of Eq. (3.17).

  14. 14.

    ISEs can either be seen as a fixed effect or a random effect. A fixed effect, in a frequentist setting, is conceived as an ultimately deterministic effect to which the estimator is thought to converge. A random effect, more akin to the Bayesian mode of thought, is conceived as a realisation from a random variable and thus ultimately stochastic in their nature. For sake of simplicity and clarity of contrast, I will assume the estimation of ISEs in the form of fixed effects in the frequentist setting in the following. This is warranted by the fact that fixed effects analysis is without a doubt much more frequent in panel data applications in economics than random effects analysis. The main reason for this is the emphasis put on unbiased estimators and the resultant application of the Hausman test (see Hausman 1978) which, in practice, almost invariably rejects random effect specification on the basis that the expected effects differ significantly from a fixed effect specification for the ISEs. Additionally, it may be argued that random effects are actually somewhere in between including fixed effects and outright excluding all ISEs. In a random effects specification, where the random effects are thought to follow a distribution converging towards a Dirac delta function, the results would be converging towards the results obtained from a specification without ISEs.

  15. 15.

    Most software use transformations of the data, like the within-transformation or first differences to speed up the estimation process dramatically (see StataCorp 2011). By virtue of the transformation the direct estimation of the ISE is evaded and the other linear and nonlinear effects can easily be estimated. However, by evading the estimation of the ISEs, the model complexity is reduced but rather major parts of it are shifted outside the estimation process and potentially forgotten about.

  16. 16.

    In some way, the inclusion of ISEs can be seen as a form of kitchen sink regression, whereby any conceivably relevant variable is lumped into the regression.

  17. 17.

    Especially in the context of income analysis, the possibility of controlling for otherwise unobserved/unobservable factors must seem like divine aid sent by Athena herself to aid the valiant economists given the herculean task of tackling the hydra of the labour market. Accounting for ISEs thus seems to tie down all but one of the biting and hissing heads that influence labour market outcomes such as income. Yet, anyone studying the Greek mythology would know of the often twofold nature of divine aid and should be weary of the potential variety of its implications. Analogously, I believe that one needs to be weary of conditioning on covariates as complex and opaque as ISEs.

  18. 18.

    As I point out in Sect. 2.3, I believe that this iterative loop does not only include analyses with quantitative methods but should explicitly also contemplate insights from qualitative research.

  19. 19.

    Note that I do not conceive state space to be constrained to temporally varying states as is often done in the literature, but rather as a more general concept which allows to capture different economic functions for different populations across time, space or any dimension by which populations may be differentiated. Nonetheless, like in most state space models, I conceive time to be the pivotal dimension. Indeed, based on my research on the development of professorial salaries (Sohn 2016), I am a strong believer in temporal dependence of income structures. This dependence may be conceived in the framework of state space models in general and hidden Markov models in specific.

  20. 20.

    This is very similar to the problem of samples consisting only of WEIRD (Western, Educated, Industrialised, Rich, Democratic Countries) in psychology (see Bellemare et al. 2008).

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    Alexander Silbersdorff

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Silbersdorff, A. (2017). Estimating and Assessing Distributional Regression. In: Analysing Inequalities in Germany. SpringerBriefs in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-65331-0_3

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