Hidden in plain sight: using household data to measure the shadow economy

Abstract

We develop an estimator of unreported income that relies on more flexible identifying assumptions than those that have been used previously. Assuming only that evaders have a higher consumption–income gap than non-evaders in surveys, our model enables the estimation of both the probability of hiding income and the amount of unreported income for each household. We illustrate the method using Czech and Slovak household budget surveys. Our results are robust to alternative specifications. Furthermore, we show that since the underreported share decreases with reported income, income inequality in these countries may be lower than suggested by the reported income.

Appendices

Supplementary material—alternative model

A representative individual earning income $Y_{i}^{T}$, the size of which depends on her life-cycle characteristics, makes a joint decision whether or not to evade (participating in $s\in \{e,ne\}$, where e is the evading regime and ne is the non-evading regime) and, if evading, to report $Y_{i}^{R}<Y_{i}^{T}$ as her taxable income. If not evading, $Y_{i}^{R}=Y_{i}^{T}$ by definition. Conditional on not evading, the agent gets utility $u(c_{i,ne})$, whereas conditional on participating in the evading sector, the agent gets utility $u(c_{i,e})$ and incurs a cost of evading that is a function of true and reported incomes, i.e., $F_{i}(Y_{i}^{T},Y_{i}^{R})$. Reporting truthfully does not incur additional cost (apart from potential lower utility from consumption). The agent chooses sector s and, conditional on $s=e$, reported income $Y_{i}^{R}$ in order to maximize their payoff. Let $c_{i,e}^{*}$ be the optimal level of consumption associated with being in the evading regime and $c_{i,ne}^{*}$ the optimal level of consumption conditional on being in the non-evading regime. One can then write the payoff of being in the evading regime as:

$$\begin{aligned} U_{i}^{*}=u(c_{i,e}^{*})-F_{i}\left( Y_{i}^{T},Y_{i}^{R*}\right) \,. \end{aligned}$$

(22)

where $Y_{i}^{R*}$ is the optimal degree of reporting in the evading regime. A rational household will choose to be in the evading regime if, and only if, the payoff from doing so is higher than the payoff of being in the non-evading sector, i.e., $u(c_{i,ne}^{*})$. Therefore, the agent i will evade if:

$$\begin{aligned} u(c_{i,e}^{*})-F_{i}\left( Y_{i}^{T},Y_{i}^{R*}\right) >u\left( c_{i,ne}^{*}\right) \,. \end{aligned}$$

(23)

If we assume a logarithmic utility function and a cost of evading function $F_{i}(Y_{i,}^{T}Y_{i}^{R})=\log \left( \frac{Y_{i}^{R}}{Y_{i}^{T}}\right) +f_{i}$, where $f_{i}$ contains household-specific factors affecting costs of evasion that are orthogonal to income, the criterion to enter the shadow sector is:

$$\begin{aligned} \log c_{i,e}^{*}-\log \left( \frac{Y_{i}^{R*}}{Y_{i}^{T}}\right) -f_{i}>\log c_{i,ne}^{*}\,, \end{aligned}$$

(24)

which can be rearranged to:

$$\begin{aligned} \left( \log c_{i,e}^{*}-\log Y_{i}^{R*}\right) -\left( \log c_{i,ne}^{*}-\log Y_{i}^{T}\right) >f_{i}\,. \end{aligned}$$

(25)

Supplementary material—technical appendix

The estimation was done in TSP 5.1 (64-bit) via the command “ml.” This command maximizes the log-likelihood function numerically²⁷, and therefore, choosing appropriate initial values is essential for convergence. The initial values were set by a procedure described in Dutoit (2007). We initially separate the sample through a dummy $I_{i}=1$ if the household i’s gap is above a certain threshold (creating an initial group presumed to be evading) or $I_{i}=0$ if it is below that threshold (initially assumed non-evading group). To obtain initial values of $\varvec{\delta }$, a probit regression of $I_{i}$ on ${\mathbf {Z}}_{i}$ is run. After that we use these estimated values $\left( {\hat{\delta }}\right)$ to obtain initial values of the $\varvec{\beta }$’s by running the following OLS regressions:

$$\begin{aligned} \ln C_{i}-\ln Y_{i}={\mathbf {X}}_{i}\varvec{\beta }_{e}-\sigma _{e,s}\frac{\phi \left( {\mathbf {Z}}_{i}\hat{\varvec{\delta }}\right) }{\varPhi \left( {\mathbf {Z}}_{i}\hat{\varvec{\delta }}\right) }+\varepsilon _{i,e}\,\text{ if } I_{i}=1\,, \end{aligned}$$

(26)

and

$$\begin{aligned} \ln C_{i}-\ln Y_{i}={\mathbf {X}}_{i}\varvec{\beta }_{ne}+\sigma _{ne,s}\frac{\phi \left( {\mathbf {Z}}_{i}\hat{\varvec{\delta }}\right) }{1-\varPhi \left( {\mathbf {Z}}_{i}\hat{\varvec{\delta }}\right) }+\varepsilon _{i,ne}\,\text{ if } I_{i}=0\,. \end{aligned}$$

(27)

Then, we get initial values of $\sigma _{e}$ and $\sigma _{e,s}$ by running the following OLS estimation:

$$\begin{aligned} {\hat{u}}_{e,i}^{2}=\sigma _{e}^{2}-\sigma _{e,s}\frac{\phi \left( {\mathbf {Z}}_{i}\hat{\varvec{\delta }}\right) }{\varPhi \left( {\mathbf {Z}}_{i}\hat{\varvec{\delta }}\right) }\,, \end{aligned}$$

where ${\hat{u}}_{e,i}=\left( \ln C_{i}-\ln Y_{i}\right) -X_{i}{\hat{\beta }}_{e}$, where ${\hat{\beta }}_{e}$ is the estimate of $\beta _{e}$ coming from Equation (26). The initial values of $\sigma _{ne}$ and $\sigma _{ne,s}$ are obtained analogously from:

$$\begin{aligned} {\hat{u}}_{ne,i}^{2}=\sigma _{ne}^{2}-\sigma _{ne,s}\frac{\phi \left( {\mathbf {Z}}_{i}\hat{\varvec{\delta }}\right) }{1-\varPhi \left( {\mathbf {Z}}_{i}\hat{\varvec{\delta }}\right) }\,. \end{aligned}$$

These initial values of $\delta$, $\varvec{\beta }$’s, and $\sigma$’s are then used as starting values for the numerical optimization procedure.

To make the results robust, for each random sample within the Monte Carlo simulation, the initial sample separation is in turn set to the first, second, and third quartiles, as well as the mean of the consumption–income gap of the given Monte Carlo sample. After applying the above procedure to each of these initial splits, we choose the results of the one that yields the highest log likelihood as the final results for the given sample. This process results in the data series from which statistics such as the estimated size of the shadow economy and its standard error are computed.

Supplementary material—coefficient estimates

See Tables 7 and 8.

Table 7 Coefficient estimates—Czech Republic (2008)

Table 8 Coefficients estimates—Slovak Republic (2008)