Eliciting permanent and transitory undeclared work from matched administrative and survey data

Abstract

We study the undeclared work patterns of Hungarian employees in relatively stable jobs, using a panel dataset that matches individual-level self-reported Labour Force Survey data with administrative records of the Pension Directorate for 2001–2006. We estimate the determinants of undeclared work using Heckman-type random-effects panel probit models, and develop a two-regime model to separate permanent and transitory undeclared work, where the latter follows a Markov chain. We find that about 6–7% of workers went permanently unreported for six consecutive years, and a further 4% were transitorily unreported in any given year. The models show lower reporting rates—especially in the permanent segment—among males, high-school graduates, those in agriculture and transport, small firms and various forms of atypical employment. Transitory non-reporting may be partly explained by administrative records missing for technical reasons. The results suggest that (1) the “aggregate labour input method” widely used in Europe can indeed be a simple yet reliable tool to estimate the size of informal employment, although it slightly overestimates the true magnitude of black work and (2) the long-term pension consequences of undeclared work may be substantial because of the high share of permanent non-reporting.

Appendices

Appendix 1

See Fig. 2 and Tables 6, 7 and 8.

Table 6 The percentage share of selected groups among respondents requesting NPID data after the first LFS visit versus later visits

Table 7 The composition of employed LFS respondents inside and outside the merged sample (per cent)

Table 8 Descriptive statistics of the simulated ratios of workers undeclared for exactly k years $\left( {k = 1, 2, \ldots , 6} \right)$ in the 6-year period and the observed distribution in the sample (in per cent)

Appendix 2: Maximum likelihood estimation of the two-regime model

Let ${\text{t}}_{\text{i}}$ denote worker ${\text{i}}$’s first year in the sample (which can take values between 2001 and 2006) and let us use the notations $p_{it}^{\left( k \right)} = \Pr \left( {Q_{it} = k | J_{i} = 0, \varvec{ X}_{it} } \right)$ for the conditional probability of transitory non-reporting (k = 1) and reporting (k = 0), respectively, at time t.

If $\prod\nolimits_{{t = t_{i} }}^{2006} {Q_{it} } = 0$ (i.e. if the person is reported at least once), then for k_t ∊ {0, 1} $( {t = t_{i} , \ldots , 2006} )$:

$$ \Pr \left( {\mathop {\bigcap }\limits_{{t = t_{i} }}^{2006} \left\{ {Q_{it} = k_{t} } \right\} | \left\{ {\varvec{X}_{it} } \right\}} \right) = \left( {1 - \varPhi \left( {\varvec{X}_{i}\varvec{\beta}_{Z} } \right)} \right)*p_{{i,t_{i} }}^{{\left( {k_{{t_{i} }} } \right)}} *\mathop \prod \limits_{{t = t_{i} + 1}}^{2006} p_{it}^{{\left( {k_{t - 1} ,k_{t} } \right)}} $$

(9)

because in this case, by definition, the worker belongs to the transitory regime and then the joint probability of her undeclared pattern can be written as a product of the starting probability $p_{{i,t_{i} }}^{{( {k_{{t_{i} }} } )}}$ and the transition probabilities $p_{it}^{{\left( {k_{t - 1} ,k_{t} } \right)}}$.

If $\prod\nolimits_{{t = t_{i} }}^{2006} {Q_{it} } = 1$ (i.e. if the person is never reported), then

$$ \Pr \left( {\mathop {\bigcap }\limits_{{t = t_{i} }}^{2006} \left\{ {Q_{it} = 1} \right\} | \left\{ {\varvec{X}_{it} } \right\}} \right) = \varPhi \left( {\varvec{X}_{i}\varvec{\beta}_{Z} } \right) + \left( {1 - \varPhi \left( {\varvec{X}_{i}\varvec{\beta}_{Z} } \right)} \right)*p_{{i,t_{i} }}^{\left( 1 \right)} *\mathop \prod \limits_{{t = t_{i} + 1}}^{2006} p_{it}^{{\left( {11} \right)}} , $$

(10)

where the first term shows the contribution of the permanent regime and the second term gives the probability that a person of the transitory regime is undeclared for the whole period.

For workers who entered the sample at the start of their current job (i.e. for whom $C_{{i,t_{i} }} = 1$), Eq. (7) implies that $p_{{i,t_{i} }}^{\left( k \right)} = p_{{i,t_{i} }}^{{\left( {0,k} \right)}}$ and hence the above likelihood calculation is complete for them. However, the majority of our observations are left-censored because C_i,2001 > 1 for most workers who entered our sample in 2001. The missing observations from their work history could be tackled, for instance, using the expectation–maximization (EM) algorithm, or we can follow a computationally less intensive but equally satisfactory approach. Indeed, we first note that p ⁽¹⁾ _it can be calculated recursively by considering the probability of being undeclared in the previous year (p ⁽¹⁾ _{i, t−1} ) and the transition probabilities (p ⁽¹¹⁾ _it and p ⁽⁰¹⁾ _it ):

$$ p_{it}^{\left( 1 \right)} = p_{it}^{{\left( {11} \right)}} *p_{i,t - 1}^{\left( 1 \right)} + p_{it}^{{\left( {01} \right)}} *\left( {1 - p_{i,t - 1}^{\left( 1 \right)} } \right) . $$

(11)

Hence, to obtain p ⁽¹⁾ _i,2001 and p ⁽⁰⁾ _i,2001 —only they are needed in Eqs. (9)–(10) for the likelihood—we can go back to (for example) year 1997, approximate p ⁽¹⁾ _i,1997 as the stationary distribution of the two-state Markov chain whose transition probabilities are fixed at their 1997 levels:

$$ {\text{p}}_{{{\text{i}},1997}}^{\left( 1 \right)} \approx {\text{p}}_{{{\text{i}},1997}}^{{\left( {01} \right)}} /\left( {{\text{p}}_{{{\text{i}},1997}}^{{\left( {10} \right)}} + {\text{p}}_{{{\text{i}},1997}}^{{\left( {01} \right)}} } \right) , $$

(12)

and then calculate recursively the starting values p ⁽¹⁾ _i,2001 and p ⁽⁰⁾ _i,2001 . This final step is only an approximation, because (1) the transition probabilities are slightly time-varying and (2) the Markov chain may not have reached the stationary distribution for some workers by 1997. Nevertheless, since the probability of return to the reported state, ${\text{p}}_{\text{it}}^{{\left( {10} \right)}}$, turns out to be relatively large in our case (see Sect. 4.1), the corresponding Markov chain has good mixing properties, and it seems to be enough to go back 4 years in time to get a satisfactory approximation for ${\text{p}}_{{{\text{i}},2001}}^{\left( 1 \right)}$ in Eqs. (9)–(10). We note that the maximum likelihood estimates turn out to be almost identical, irrespective of whether 1996, 1997 or 1998 is used as the start year in the approximation.