Abstract
This article proposes Bayesian methods for microsimulation models and for policy evaluations. In particular, the Bayesian Multinomial Logit and the Bayesian Multinomial Mixed Logit models are presented. They are applied to labour-market choices by single females and single males, enriched with EUROMOD microsimulated information, to evaluate fiscal policy effects. Estimates using the two Bayesian models are reported and compared to the results stemming from a standard approach to the analysis of the phenomenon under consideration. Improvements in model performances, when Bayesian methods are introduced and when random effects are included, are outlined. Finally, ongoing work, based on nonparametric model extensions and on analysis of work choices by couples is briefly described.
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Notes
- 1.
- 2.
For the sake of simplicity, we identify the Multinomial Logit model (MLM) with a general model in which the underlying utilities depend on individual characteristics, choice attributes and/or variables combining individuals and choices. Sometimes, in econometrics, the latter is considered to be a generalization of the MLM combined with a Conditional Logit model.
- 3.
It can be proved that the difference between two standard i.i.d. Gumbel random variables is a Logistic by means of the characteristic function (c.f.). In general, if \(\varepsilon \sim \mathsf {Gumbel}(0,1)\), its characteristic function is \(\phi _{\varepsilon }(t)=\mathsf {E}(e^{it\varepsilon })=\varGamma (1-i t)\). Thus, \(\phi _{\varepsilon _{i}-\varepsilon _{j}}(t) =\phi _{\varepsilon _{i}}(t)\phi _{\varepsilon _{j}}(-t)=\varGamma (1-i t)\varGamma (1+i t)=\varGamma (1-i t)\varGamma (i t) i t\). By Euler’s reflection formula \(\varGamma (z)\varGamma (1-z)=\frac{\pi }{\sin (\pi z)}\) and by property \(i^{2}=-1\), hence \( \phi _{\varepsilon _{i}-\varepsilon _{j}}(t)=\frac{\pi it}{\sin (\pi i t)}=\frac{\pi t}{-i\sin (\pi i t)} \) which is the c.f. for the Logistic distribution.
- 4.
The form of the choice probability under a MLM model follows from the representation of \(\mathsf {Pr}(Y_{ji}|C)\) as \(\mathsf {Pr}(\mathsf {U}_{ji}>\mathsf {U}_{jh}\,\,\forall \,\,h{=}1,\ldots ,I\,\,\,h\ne i)\), which, for \(h\ne i\), reduces to
$$\begin{aligned}\pi _{ji}{=}&\mathsf {Pr}(\mathbf {x}_{ji}^{'}\varvec{\beta }{+}\varepsilon _{ji}>\mathbf {x}_{jh}^{'}\varvec{\beta }{+}\varepsilon _{jh}\,\,\forall \,\,h{=}1,\ldots ,I)=\mathsf {Pr}(\varepsilon _{ji}-\varepsilon _{jh}>\mathbf {x}_{jh}^{'}\varvec{\beta }-\mathbf {x}_{ji}^{'}\varvec{\beta }\,\,\forall \,\,h=1,\ldots ,I) \\&=\int _{\varepsilon }\mathsf {I}(\varepsilon _{ji}-\varepsilon _{jh}>\mathbf {x}_{jh}^{'}\varvec{\beta }-\mathbf {x}_{ji}^{'}\varvec{\beta }\,\,\forall \,\,h=1,\ldots ,I)f(\varvec{\varepsilon }_{j})d\varvec{\varepsilon }_{j} \end{aligned}$$Assuming that the errors are i.i.d. Gumbel distributed and resorting to the substitution \(t=\exp (-\varepsilon _{ji})\)
$$\begin{aligned} \pi _{ji}=&\mathsf {Pr}(Y_{ji}=1|C,\varepsilon _{ji})\mathsf {Pr}(\varepsilon _{ji})=\int _{-\infty }^{\infty }\left( \prod _{h\ne i}e^{-e^{-(\varepsilon _{ji}+\mathbf {x}_{ji}^{'}\varvec{\beta }-\mathbf {x}_{jh}^{'}\varvec{\beta })}}\right) e^{-\varepsilon _{ji}}e^{-e^{-\varepsilon _{ji}}}d\varepsilon _{ji}\\ {}&=\int _{-\infty }^{\infty }\left( \prod _{h}e^{-e^{-(\varepsilon _{ji}+\mathbf {x}_{ji}^{'}\varvec{\beta }-\mathbf {x}_{jh}^{'}\varvec{\beta })}}\right) e^{-\varepsilon _{ji}}d\varepsilon _{ji}=\int _{-\infty }^{\infty }\exp \left\{ \sum _{h}e^{-(\varepsilon _{ji}+\mathbf {x}_{ji}^{'}\varvec{\beta }-\mathbf {x}_{jh}^{'}\varvec{\beta })}\right\} e^{-\varepsilon _{ji}}d\varepsilon _{ji}\\ {}&=\int _{-\infty }^{\infty }\exp \left\{ -e^{-\varepsilon _{ji}}\sum _{h}e^{-(\mathbf {x}_{ji}^{'}\varvec{\beta }-\mathbf {x}_{jh}^{'}\varvec{\beta })}\right\} e^{-\varepsilon _{ji}}d\varepsilon _{ji}=\int _{0}^{\infty }\exp \left\{ -t\sum _{h}e^{-(\mathbf {x}_{ji}^{'}\varvec{\beta }-\mathbf {x}_{jh}^{'}\varvec{\beta })}\right\} dt \\&=\frac{\exp (-t\sum _{h}e^{-(\mathbf {x}_{ji}^{'}\varvec{\beta }-\mathbf {x}_{jh}^{'}\varvec{\beta })})}{-\sum _{h}e^{-(\mathbf {x}_{ji}^{'}\varvec{\beta }-\mathbf {x}_{jh}^{'}\varvec{\beta })}}\Big |_{0}^{\infty }=\frac{1}{\sum _{h}e^{-(\mathbf {x}_{ji}^{'}\varvec{\beta }-\mathbf {x}_{jh}^{'}\varvec{\beta })}}=\frac{\exp \{\mathbf {x}_{ji}^{'}\varvec{\beta }\}}{\sum _{h=1}^{I}\exp \{\mathbf {x}_{jh}^{'}\varvec{\beta }\}} \end{aligned}$$ - 5.
Alternatively, as suggested in [49] p. 143, random coefficients can be simply considered as part of the utility error component, inducing correlations among alternative utilities.
- 6.
Some other priors less informative than the Inverse-Wishart can also be considered following [28].
- 7.
This is one way among others to overcome possible limitations [47] due to the restrictive i.i.d. assumption about the random component of the utility function in the MLM. Hence, MMLM allows a proper representation of choice behaviour.
- 8.
This is a crucial feature if microsimulation models have to evaluate policy impacts properly [9].
- 9.
The fiscal policies considered were the ones measured in the EUROMOD model, applied to real data provided by the Bank of Italy from the SHIW-1998. Thus, data were partially microsimulated by EUROMOD, according to the 1998 Italian fiscal policy.
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Nava, C.R., Carota, C., Colombino, U. (2017). Bayesian Methods for Microsimulation Models. In: Argiento, R., Lanzarone, E., Antoniano Villalobos, I., Mattei, A. (eds) Bayesian Statistics in Action. BAYSM 2016. Springer Proceedings in Mathematics & Statistics, vol 194. Springer, Cham. https://doi.org/10.1007/978-3-319-54084-9_18
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