In undeclared work research, the estimation of the magnitude of the phenomenon (i.e., the amount of income and/or the percentage of workers involved) is of major interest. This has been done either using indirect methods or by means of ad hoc surveys such as the Eurobarometer special survey on undeclared work, our motivating study. The extent of undeclared work can be measured by means of two different outcomes: the event of working off-the-book (binary variable) and, when the event occurs, the amount of earnings deriving from the undeclared activity (continuous variable). This setup has been typically modeled via the so called two-part model: a binary choice model for the probability of observing a positive-versus-zero outcome and then, conditional on a positive outcome, a regression model for the positive outcome. We propose an extension of the two-part model that goes in two directions. The first regards the measurement error that, given the very nature of undeclared activities, is most likely to affect both the outcomes of interest. The second is that we generalize the linear regression part of the model to allow individual-level means. We also conduct an extensive simulation study to investigate the performance of the proposed model and compare it with traditional approaches.
AllinghamM and SandmoA (1972) Income tax evasion: A theoretical analysis. Journal of Public Economics, 1, 323–338.
2.
ArezzoMF and AllevaG (2012) Estimating the probability of moonlighting in Italian building industry. In Advanced Statistical Methods for the Analysis of Large Data-Sets, edited by Di CiaccioAgostino, ColiMauro, IbanezJM. Berlin: Springer.
3.
ArezzoMF and GuagnanoG (2018) Responsebased sampling for binary choice models with sample selection. Econometrics, 6. URL https://doi.org/10.3390/econometrics6010012.
4.
BajadaC and SchneiderF (2020) Size, Causes and Consequences of the Underground Economy. An International Perspective. Routledge.
5.
BerksonJ (1950) Are there two regressions?Journal of the American Statistical Association, 45, 164–180. URL https://doi.org/10.1080/01621459.1950.10483349.
6.
CaganP (1958) The demand for currency relative to the total money supply. Journal of Political Economy, 66, 302–328.
7.
CraggJG (1971) Some statistical models for limited dependent variables with application to the demand for durable goods. Econometrica, 39, 829–844.
8.
DuanN, ManningWG, MorrisCN and NewhouseJP (1983) A comparison of alternative models for the demand for medical care. Journal of Business & Economic Statistics, 1, 115–126.
9.
DubinJA, GraetzMJ and WildeLL (1990) The effect of audit rates on the federal individual income tax, 1977–1986. National Tax Journal, 43, 395–409.
10.
EnsteD and SchneiderF (2002) Hiding in the shadows; The growth of the underground economy [Economic Issues 2002/001]. International Monetary Fund.
11.
FeigeE (1989) The Underground Economies, Tax Evasion and Information Distortion. Cambridge University Press.
12.
FeigeE (1996) Overseas holdings of US currency and the underground economy. In Exploring the Underground Economy, edited by PozoS, pages 5–62. W. E. Upjohn Institute for Employment Research.
13.
FeldLP and LarsenC (2005) Black activities in Germany in 2001 and 2004: A comparison based on survey data. Rockwool Foundation Research Unit (Technical Report No 12).
14.
FeldLP and LarsenC (2008) ‘Black’ activities low in Germany in 2006. Technical report, Rockwool Foundation Research Unit.
15.
FeldLP and LarsenC (2009) Undeclared work in Germany 2001–2007—Impact of Deterrence, Tax Policy, and Social Norms: An Analysis Based on Survey Data. Berlin: Springer.
16.
FithianW and HastieT (2014) Local case-control sampling: Efficient subsampling in imbalanced data sets. The Annals of Statistics, 42, 1693–1724. URL https://doi.org/10.1214/14-AOS1220.
17.
FreyB and PommerehneW (1984) The hidden economy: state and prospects for measurement. Review Income Wealth, 30, 1–23.
18.
FreyB and Weck–HannemannH (1984) The hidden economy as an unobserved variable. European Economic Review, 26, 33–53.
HaignerS, JeneweinSFS and WakolbingerF (2013) Driving forces of informal labour supply and demand in Germany. International Labour Review, 152, 507–524.
22.
HassanM and SchneiderF (2016) Size and development of the shadow economies of 157 countries worldwide: Updated and new measures from 1999 to 2013. IZA (Technical Report 10281).
23.
HausmanJ, AbrevayaJ and Scott-MortonFM (1998) Misclassification of the dependent variable in a discrete-response setting. Journal of Econometrics, 87, 239–269.
24.
KaufmannD and KaliberdaA (1996) Integrating the unofficial economy into the dynamics of post socialist economies: A framework of analyses and evidence, pages 81–120. M.E. Sharpe.
25.
KingG and ZengL (2001) Logistic regression in rare events data. Political Analysis, 9, 137–163.
26.
Lacko´M (1998) The Hidden Economies of Visegrad Countries in International Comparison: A Household Electricity Approach. Cambridge University Press.
27.
Lacko´M (1999) Electricity Intensity and the Unrecorded Economy in Post-socialist Countries. Ashgate Publishing Company.
28.
Lacko´M (2000a) Do Power Consumption Data Tell the Story? Electricity Intensity and Hidden Economy in Post-socialist Countries. The MIT Press.
29.
Lacko´M (2000b) Hidden economy—an unknown quantity? Comparative analysis of hidden economies in transition economies 1989–95. Economics of Transition, 8, 117–149.
30.
LeungSF and ShihtiY (1996) On the choice between sample selection and two-part models. Journal of Econometrics, 72, 197–229.
31.
MihaylovaB, BriggsA, O’HaganA and ThompsonSG (2011) Review of statistical methods for analysing healthcare resources and costs. Health Economics, 20, 897–916.
32.
MurphyKM and TopelRH (1985) Estimation and inference in two-step econometric models. Journal of Business & Economic Statistics, 3, 370–379.
33.
PedersenS (2003) The shadow economy in Germany, Great Britain and Scandinavia: A measurement based on questionnaire service. Technical Report 10, The Rockwool Foundation Research Unit.
34.
SchneiderF. (1986) Estimating the size of the danish shadow economy using the currency demand approach: An attempt. The Scandinavian Journal of Economics, 88, 643–668.
35.
SchneiderF (1994) Measuring the size and development of the shadow economy. Can the causes be found and the obstacles be overcome? In Essays on Economic Psychology, edited by Brandsta¨tterH and GüthW, pages 193–212. Berlin: Springer.
36.
SchneiderF and WilliamsCC (2013) The Shadow Economy. The Institute of Economic Affairs.
37.
StekhovenDJ and Bu¨ hlmannP (2012) Missforest-non-parametric missing value imputation for mixed-type data. Bioinformatics, 28, 112–118.
38.
TanziV (1980) The Underground Economy in the United States: Estimates and Implications. Banca Nazionale del Lavoro (Technical Report 135).
39.
TanziV (1983) The Underground Economy in the United States: Annual Estimates, 1930–1980. International Monetary Fund (Technical Report 30).
40.
TanziV (2002) The shadow economy, its causes and its consequences. International Seminar on the Shadow Economy Index in Brazil.
ThomasJ (1999) Quantifying the black economy: Measurement without theory yet again?Economic Journal, 109, 381–389.
43.
WangH (2020) Logistic regression for massive data with rare events. Proceedings of Machine Learning Research, 119, 9829–9836.
44.
WilliamsCC (2009) Formal and informal employment in Europe: Beyond dualistic representations. European Urban and Regional Studies, 16, 147–159.
45.
WilliamsCC (2013) Evaluating cross-national variations in the extent and nature of informal employment in the European Union. Industrial Relations Journal, 44, 479–494.
46.
WilliamsCC and LanskyM (2013). Informal employment in developed and developing economies: Perspectives and policy responses. International Labour Review, 152, 355–380.
47.
WilliamsCC and RodgersP (2013). The Role of Informal Economies in the Post-Soviet World: The End of Transition?Routledge.
48.
WilliamsCC and WindebankJ (1998) Informal Employment in the Advanced Economies: Implications for Work and Welfare. Routledge.
49.
WilliamsCC and HorodnicI (2017) Evaluating the policy approaches for tackling undeclared work in the European union. Environment and Planning C: Politics and Space, 35, 916–936.
50.
WilliamsCC and HorodnicIA (2015a) Evaluating the prevalence of the undeclared economy in central and Eastern Europe: An institutional asymmetry perspective. European Journal of Industrial Relations, 21, 389–406.
51.
WilliamsCC and HorodnicIA (2015b) Rethinking the marginalisation thesis: An evaluation of the socio-spatial variations in undeclared work in the European Union. Employee Relations, 37, 48–65.
52.
WilliamsCC and NadinS (2010) Entrepreneurship and the informal economy: An overview. Journal of Developmental Entrepreneurship, 15, 361–378.
53.
WitteA and WoodburyDF (1985) The effect of tax laws and tax administration on tax compliance: The case of the US individual income tax. National Tax Journal, 38, 1–13.
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