The size of the underground economy in Germany: a correction of the record and new evidence from the modified-cash-deposit-ratio approach

Abstract

Based on the Ahumada et al. (Rev Income Wealth 53(2):363–371, 2007) critique we revise existing estimates of the size of the German underground economy. Among other things, it turns out that most of these estimates are untenable and that the tax pressure induced size of the German underground economy may be much lower than previously thought. To this extent, German policy and law makers have been misguided during the last three decades. Therefore, we introduce the Modified-Cash-Deposit-Ratio approach, which is not subject to the recent critique and apply it to Germany for the period 1960–2008.

Appendix

Data on currency in circulation outside banks (series TXI300 and printed matter ‘Monatsberichte’), sight deposits held by domestic non-banks (series OU0221) and time deposits of 1 year or less held by domestic non-banks (series OUA192) was collected from the Bundesbank. Data on population was collected from Statistisches Bundesamt (Statistical Yearbook). Data on household consumption (series 13496FCZF, 13496FCZW) and on the exchange rate (series 134RFZF, 163RFZF) was collected from International Financial Statistics online. The consumer price index (2000 = 100) was collected from International Financial Statistics online (series 1346DZF, 1959–1989) and the Bundesbank (series UUFA01, 1990–2008). We used the EViews 5.1 software package for our estimations.

With respect to the estimation procedure proposed by Seitz (1995) we introduced three changes. First, we used annual data because quarterly data for currency in circulation outside banks was not available for the period before 1970. Second, we have used inflation instead of the interest rate to measure the opportunity cost of cash holding. Third, with respect to cointegration we tested alternative methods and found that the Engle-Granger procedure performed best. In particular, we estimated the following error correction currency demand Eq. (19):

$$ \begin{aligned} \Updelta \ln (C_{r} )_{t} & = \mathop {0.163}\limits_{(2.10)} \Updelta \ln (PHC_{r} )_{t} + \mathop {0.476}\limits_{(5.94)} \Updelta \ln (PHC_{r} )_{t - 1} - \mathop {0.016}\limits_{( - 4.83)} \Updelta (INF)_{t} \\ + \mathop {0.172}\limits_{(4.44)} \Updelta (ER)_{t} + \mathop {0.056}\limits_{(4.63)} (D87)_{t} - \mathop {0.664}\limits_{( - 4.73)} [\ln (C_{r} )_{t - 1} - \mathop {12.764}\limits_{( - 4.52)} - \mathop {0.437}\limits_{( - 4.08)} \ln (PHC_{r} )_{t - 1} + \mathop {0.010}\limits_{(2.94)} (INF)_{t - 1} \\ - \mathop {0.165}\limits_{( - 3.80)} (ER)_{t - 1} - \mathop {0.132}\limits_{( - 5.75)} (D87)_{t - 1} - \mathop {0.195}\limits_{( - 9.95)} (D90)_{t - 1} + \mathop {0.082}\limits_{(2.80)} (D91)_{t - 1} - \mathop {0.009}\limits_{( - 2.55)} (Trend)_{t - 1} ] \\ \end{aligned} $$

where C _r is real currency in circulation outside banks, PHC _r denotes real private household consumption, INF denotes inflation, ER denotes the Dollar/EUR (DM) exchange rate, D87 is a dummy that takes the value 1 in 1987, 1988, 1989 and 0 otherwise to capture withholding tax effects (Seitz 1995, p. 11), D90 is a dummy that takes the value 1 from 1990 onwards and 0 otherwise to capture reunification, D91 is a dummy that takes 1 in 1991 and 0 otherwise to capture the reunification shock, Trend is a deterministic time trend, Δ denotes first differences and t denotes the time period.

Relevant t-statistics are given in parenthesis below the coefficients and diagnostic statistics are: Adj. R ² = 0.75, standard error = 0.019521, normality $\chi _{{{\text{Norm}}}}^{2} (2) = 0.71{\text{}}[0.70]$, no residual serial correlation $\chi _{{{\text{SC}}}}^{2} (1) = 0.24{\text{ }}[0.62]$, no autocorrelation in the error term $\chi _{{{\text{ARCH}}}}^{2} (1) = 0.25{\text{ }}[0.61]$, heteroscedasticity $\chi _{{{\text{Hetero}}}}^{2} (1) = 7.82{\text{ }}[0.73]$ and no misspecification $\chi _{{{\text{RESET}}}}^{2} (1) = 0.15{\text{ }}[0.69]$, with p-values given in brackets. To rule out misspecification due to parameter instability, we have applied the cumulative sum of recursive residuals CUSUM (results not displayed) and the CUSUM of squares tests (see Fig. 2). Both tests indicate the absence of parameter instability because the test statistics are within the 5% critical bounds.

Fig. 2

CUSUM of squares test

Actual real currency C _rt and forecasted real currency FC _rt are displayed in logarithms in Fig. 3. The consumer price index (CPI) was then used to transform FC _rt into forecasted nominal currency FC _t , for the period 1960–2006. Data for the years 2007 and 2008 in FC _t corresponds again to nominal actual currency, C _t , to avoid deviations. Hence, FC _t data for the period 1960–2008 is used in Eq. (15) of the main text.

Fig. 3

Actual and forecasted real currency in logarithms, Germany 1960–2006

Regarding the amount of forecasted currency in circulation outside banks, inside Germany, INFC _t in Eq. (18) we used the following procedure. The error correction model in (19) can be expressed in logarithms as:

$$ \ln C_{rt} = \alpha_{0} + \alpha_{1} \ln PHC_{rt} + \alpha_{2} INF_{t} + \alpha_{3} ER_{t} + \varepsilon_{t} , $$

(19)

Reversing logarithms yields:

$$ C_{rt} = PHC_{rt}^{\alpha 1} e^{{(\alpha_{0} + \alpha_{2} INF_{t} + \alpha_{3} ER_{t} + \varepsilon_{t} )}} $$

(20)

Next, we assume that real currency held inside Germany, INC _rt , does not depend on the exchange rate ER, which yields:

$$ INC_{rt} = PHC_{rt}^{\alpha 1} e^{{(\alpha_{0} + \alpha_{2} INF_{t} + \varepsilon_{t} )}} $$

(21)

Dividing (20) by (21) yields:

$$ {\frac{{C_{rt} }}{{INC_{rt} }}} = e^{{(\alpha_{3} ER_{t} )}} $$

(22)

Because C _rt and ER _t are known and α ₃ can be obtained from (19), which is 0.164820, we can calculate INC _rt from (22) and replacing C _rt and INC _rt by FC _rt and INFC _rt yields the relevant values for (18). Note, however, that α ₃ is obtained from an estimation covering the period 1960–1999 and, thus, may not be a good proxy for years after 1999. This should be taken into account with respect to the interpretation of G3 in Table 2.