THE STUDY OF THE UNDERGROUND ECONOMY has been pioneered by Vito Tanzi, both through his editorial work (1982) and his empirical studies (1980, 1983). He argues that transactions in the underground economy are carried out in cash and that, since the avoidance of tax is a prime motive for entering the underground economy, the ratio of cash to deposits will be affected by tax variables. His model is

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where (C/M₂) is the ratio of currency holdings to broad money (including time deposits), T is the tax variable, (WS/NI) is the share of wages and salaries in national income, R is the rate of interest on time deposits, Y is real per capita gross national product (GNP) and e is an error term.

All macroeconomic approaches to the measurement of the size of the underground economy involve heroic assumptions, but since those of Tanzi have already been the subject of a comment in these Staff Papers (Acharya 1984, Tanzi 1984), the degree of heroism will not be discussed here. Rather, the object of this comment is to point out some major shortcomings in Tanzi’s econometric analysis. Most important, Tanzi did not test his fitted equations for structural stability, and thereby failed to discover evidence of a structural break in 1945. While equation (1) fits the data for 1930–45, the tax variable is not significant for the period 1946–80; for this latter period a dynamic specification is superior.

Tanzi’s Results

Tanzi reports the parameter estimates for two versions of equation (1), using alternative tax variables T (income tax after credit over adjusted gross income) and TW (weighted average tax rate), his equations being corrected with a first-order Cochrane-Orcutt (CO) transformation for serial correlation. Thus, with t-statistics in parentheses, he reports for 1930–80

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One point to note is that the Durbin-Watson statistics, computed after the CO transformation had been applied, are both in the 0.95 inconclusive range, which suggests that the transformation may not have removed the serial correlation. Moreover, additional diagnostic analysis is needed to test the specification of the model being estimated. In particular, it is important to consider the stability of the model and to apply a common factor (COMFAC) test to see whether Tanzi’s static model with serial correlation is appropriate when tested against a more general dynamic specification of equation (1).¹

Since Tanzi followed the admirable but regrettably rare practice of publishing his data, it has been possible to run the regressions necessary for the additional tests, and the results are presented in Tables 1 and 2.²

^{Table 1.}

Diagnostic Tests on Equation (1) for 1930–45

Note: For notational simplicity, the following abbreviations are used: LCM2 = In C/M₂, LTW = ln(l + TW), LT = ln(l + T), LWS = (In(WS/NI), LR = In R and LY = In Y. The subscript –1 denotes variables lagged one period. Standard errors are given in parentheses and RSS = Residual Sum of Squares. For diagnostic tests, (*), (**), and (***) indicate significance at 0.95, 0.99, and 0.999 probability levels, respectively.

^{Table 1.}

Diagnostic Tests on Equation (1) for 1930–45

(A) Tax variable ln(1 + TW)

1A.1. OLS(Static)

$$\begin{gathered} LCM2=-13.194+0.313LTW+2.582LWS-0.263LR+0.299LY, \\ \left(1.677\right)\left(0.064\right)\left(0.404\right)\left(0.084\right)\left(0.154\right) \end{gathered}$$$RSS=0.028602,R^2=0.972,D-w=1.521$

1A.2. AR1

$$\begin{gathered} LCM2=-13.187+0.273LTW+2.604LWS-0.313LR-0.277LY, \\ \left(1.788\right)\left(0.061\right)\left(0.421\right)\left(0.090\right)\left(0.173\right) \end{gathered}$$$RSS=0.026526,R^2=0.983,D-W=1.776$

1A.3. OLS(Dynamic)

$$\begin{gathered} LCM2=-14.949+0.292LTW+2.851LWS-0.503LR-0.035LY-0.028LCM{2}_{-1}+{0.024LTW}_{-1}+{0.093LWS}_{-1}+{0.300LR}_{-1}-{0.203LY}_{-1},RSS=0.020375,R^2=0.980,D-W=\mathrm{1.607.} \\ \left(7.748\right)\left(0.100\right)\left(0.800\right)\left(0.473\right)\left(0.687\right)\left(0.485\right)\left(0.168\right)\left(1.569\right)\left(0.467\right)\left(0.594\right) \end{gathered}$$

LR test for CO restrictions, CHISQ(4) = 4.221. Test for extra lagged variables, F(5,6) = 0.485

(B) Tax variable ln(14 + T)

1B.1. OLS(Static)

$LCM2=\underset{\left(1.762\right)}{-14.704}+\underset{\left(0.080\right)}{0.321LT}+\underset{\left(0.420\right)}{2.964LWS}\underset{\left(0.082\right)}{-0.373LR}\underset{\left(0.156\right)}{-0.148LY}$$RSS=0.036876,R^2=0.964,D-W=1.448$

1B.2. AR1

$$\begin{gathered} LCM2=\underset{\left(1.936\right)}{-14.034}+\underset{\left(0.086\right)}{0.319LT}\underset{\left(0.454\right)}{+2.815LWS}\underset{\left(0.092\right)}{-0.384LR}\underset{\left(0.184\right)}{-0.191LY} \\ RSS=0.033660,R^2=0.978,D-W=1.922 \end{gathered}$$

IB.3. OLS(Dynamic)

$LCM2=\underset{\left(5.745\right)}{-16.697}+\underset{\left(0.112\right)}{0.361LT}\underset{\left(0.793\right)}{-1.990LWS}\underset{\left(0.358\right)}{-0.257LR}\underset{\left(0.506\right)}{-0.008LY}\underset{\left(0.366\right)}{+0.023LCM{2}_{-1}}\underset{\left(0.116\right)}{-0.183L{T}_{-1}}+\underset{\left(0.425\right)}{1.414LW{S}_{-1}}\underset{\left(0.353\right)}{-0.019L{R}_{-1}}+\underset{\left(0.428\right)}{0.325L{Y}_{-1}},RSS=0.015434,R^2=0.985,D-W=\mathrm{2.056.}$

LR test for CO restrictions, CHISQ(4) = 12.48(*); test for extra lagged variables, F(5, 6) = 1.667

Note: For notational simplicity, the following abbreviations are used: LCM2 = In C/M₂, LTW = ln(l + TW), LT = ln(l + T), LWS = (In(WS/NI), LR = In R and LY = In Y. The subscript –1 denotes variables lagged one period. Standard errors are given in parentheses and RSS = Residual Sum of Squares. For diagnostic tests, (*), (**), and (***) indicate significance at 0.95, 0.99, and 0.999 probability levels, respectively.

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^{Table 1.}

Diagnostic Tests on Equation (1) for 1930–45

(A) Tax variable ln(1 + TW)

1A.1. OLS(Static)

$$\begin{gathered} LCM2=-13.194+0.313LTW+2.582LWS-0.263LR+0.299LY, \\ \left(1.677\right)\left(0.064\right)\left(0.404\right)\left(0.084\right)\left(0.154\right) \end{gathered}$$$RSS=0.028602,R^2=0.972,D-w=1.521$

1A.2. AR1

$$\begin{gathered} LCM2=-13.187+0.273LTW+2.604LWS-0.313LR-0.277LY, \\ \left(1.788\right)\left(0.061\right)\left(0.421\right)\left(0.090\right)\left(0.173\right) \end{gathered}$$$RSS=0.026526,R^2=0.983,D-W=1.776$

1A.3. OLS(Dynamic)

$$\begin{gathered} LCM2=-14.949+0.292LTW+2.851LWS-0.503LR-0.035LY-0.028LCM{2}_{-1}+{0.024LTW}_{-1}+{0.093LWS}_{-1}+{0.300LR}_{-1}-{0.203LY}_{-1},RSS=0.020375,R^2=0.980,D-W=\mathrm{1.607.} \\ \left(7.748\right)\left(0.100\right)\left(0.800\right)\left(0.473\right)\left(0.687\right)\left(0.485\right)\left(0.168\right)\left(1.569\right)\left(0.467\right)\left(0.594\right) \end{gathered}$$

LR test for CO restrictions, CHISQ(4) = 4.221. Test for extra lagged variables, F(5,6) = 0.485

(B) Tax variable ln(14 + T)

1B.1. OLS(Static)

$LCM2=\underset{\left(1.762\right)}{-14.704}+\underset{\left(0.080\right)}{0.321LT}+\underset{\left(0.420\right)}{2.964LWS}\underset{\left(0.082\right)}{-0.373LR}\underset{\left(0.156\right)}{-0.148LY}$$RSS=0.036876,R^2=0.964,D-W=1.448$

1B.2. AR1

$$\begin{gathered} LCM2=\underset{\left(1.936\right)}{-14.034}+\underset{\left(0.086\right)}{0.319LT}\underset{\left(0.454\right)}{+2.815LWS}\underset{\left(0.092\right)}{-0.384LR}\underset{\left(0.184\right)}{-0.191LY} \\ RSS=0.033660,R^2=0.978,D-W=1.922 \end{gathered}$$

IB.3. OLS(Dynamic)

$LCM2=\underset{\left(5.745\right)}{-16.697}+\underset{\left(0.112\right)}{0.361LT}\underset{\left(0.793\right)}{-1.990LWS}\underset{\left(0.358\right)}{-0.257LR}\underset{\left(0.506\right)}{-0.008LY}\underset{\left(0.366\right)}{+0.023LCM{2}_{-1}}\underset{\left(0.116\right)}{-0.183L{T}_{-1}}+\underset{\left(0.425\right)}{1.414LW{S}_{-1}}\underset{\left(0.353\right)}{-0.019L{R}_{-1}}+\underset{\left(0.428\right)}{0.325L{Y}_{-1}},RSS=0.015434,R^2=0.985,D-W=\mathrm{2.056.}$

LR test for CO restrictions, CHISQ(4) = 12.48(*); test for extra lagged variables, F(5, 6) = 1.667

Note: For notational simplicity, the following abbreviations are used: LCM2 = In C/M₂, LTW = ln(l + TW), LT = ln(l + T), LWS = (In(WS/NI), LR = In R and LY = In Y. The subscript –1 denotes variables lagged one period. Standard errors are given in parentheses and RSS = Residual Sum of Squares. For diagnostic tests, (*), (**), and (***) indicate significance at 0.95, 0.99, and 0.999 probability levels, respectively.

^{Table 2.}

Diagnostic Tests on Equation (1) for 1946–80

Note: For notation see Note to Table 1. The stability tests were based on a comparison of the RSS obtained from fitting each equation separately to 1930–45 and 1946–80 with the RSS obtained from fitting an equation to the whole period 1930–80. Superscript i indicates D-W statistic is in inconclusive region at 0.95 probability level; h is Durbin’s test for AR(1) serial correlation in dynamic equations containing the lagged dependent variable.

^{Table 2.}

Diagnostic Tests on Equation (1) for 1946–80

(A) Tax variable ln(l + TW)

2A.1. OLS(Static)

$LCM2=\underset{\left(1.273\right)}{-0.605}+\underset{\left(0.075\right)}{0.214LTW}\underset{\left(0.293\right)}{-0.399LWS}\underset{\left(0.040\right)}{-0.130LR}\underset{\left(0.107\right)}{-0.253LY},$

$RSS=0.032853,R^2=0.969,D-W=0.951(**)stability,F(5,41)=16.89(***)$

2A.2. AR1

$LCM2=\underset{\left(1.578\right)}{-3.657}+\underset{\left(0.086\right)}{0.076LTW}+\underset{\left(0.354\right)}{0.438LWS}\underset{\left(0.045\right)}{-0.056LR}\underset{\left(0.157\right)}{-0.213LY,}$

$RSS=0.019832,R^2=0.790,D-W=1.878stability,F(5,41)=7.411(***)$

2A.3. OLS(Dynamic)

$LCM2=\underset{\left(0.170\right)}{0.359}+\underset{\left(0.058\right)}{0.041LTW}+\underset{\left(0.257\right)}{0.178LWS}+\underset{\left(0.036\right)}{0.010LR}-\underset{\left(0.152\right)}{0.308LY}+\underset{\left(0.710\right)}{0.582LCM{2}_{-1}}+\underset{\left(0.06\right)}{0.057LT{W}_{-1}}-\underset{\left(0.255\right)}{0.520LE{S}_{-1}}-\underset{\left(0.040\right)}{0.063L{R}_{-1}}+\underset{\left(0.125\right)}{0.255L{Y}_{-1}},RSS=0.006668,R^2=0.994,D-W=\mathrm{2.402.}$

Stability, F(10.31) = 19.32(***). LR test for CO restrictions, CHISQ(4) = 38.15(***). Test for extra lagged variables, F(5,25) = 19.63(***)

(B) Tax variable ln(1 + T)

2B.1. OLS(Static)

$LCM2=\underset{\left(1.306\right)}{0.582}+\underset{\left(0.316\right)}{0.236LT}-\underset{\left(0.028\right)}{0.208LR}-\underset{\left(0.096\right)}{0.166LY},$

$RSS=0.034382,R^2=0.968,D-W={1.300}^{\prime }stability,F(5,41)=14.81(***)$

2B.2. AR1

$LCM2=\underset{\left(1.644\right)}{-3.349}+\underset{\left(0.080\right)}{0.035LT}+\underset{\left(0.378\right)}{0.419LWS}-\underset{\left(0.046\right)}{0.056LR}-\underset{\left(0.172\right)}{0.242LY,}$

$$\begin{gathered} RSS=0.020155,R^2=0.775,D-W=\mathrm{1.905.} \\ Stability,F(5,41)=8.118(***) \end{gathered}$$

2B.3. OLS(Dynamic)

$LCM2=\underset{\left(0.724\right)}{0.865}-\underset{\left(0.059\right)}{0.043LT}+\underset{\left(0.261\right)}{0.277LWS}-\underset{\left(0.037\right)}{0.017LR}-\underset{\left(0.148\right)}{0.117LY}+\underset{\left(0.165\right)}{{0.650LCM2}_{-1}}+\underset{\left(0.063\right)}{{0.127LT}_{-1}}-\underset{\left(0.227\right)}{{0.698LWS}_{-1}}-\underset{\left(0.040\right)}{{0.064LR}_{-1}}+\underset{\left(0.130\right)}{0.139L{Y}_{-1}},RSS=0.006728,R^2=0.994,D-W=2.115$

h = –1.567; stability, F(10,31)= 18.06(***); LR test for CO restrictions, CHISQ(4) = 38.40(***); test for extra lagged variables, F(5.25) = 20.55(***)

Note: For notation see Note to Table 1. The stability tests were based on a comparison of the RSS obtained from fitting each equation separately to 1930–45 and 1946–80 with the RSS obtained from fitting an equation to the whole period 1930–80. Superscript i indicates D-W statistic is in inconclusive region at 0.95 probability level; h is Durbin’s test for AR(1) serial correlation in dynamic equations containing the lagged dependent variable.

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^{Table 2.}

Diagnostic Tests on Equation (1) for 1946–80

(A) Tax variable ln(l + TW)

2A.1. OLS(Static)

$LCM2=\underset{\left(1.273\right)}{-0.605}+\underset{\left(0.075\right)}{0.214LTW}\underset{\left(0.293\right)}{-0.399LWS}\underset{\left(0.040\right)}{-0.130LR}\underset{\left(0.107\right)}{-0.253LY},$

$RSS=0.032853,R^2=0.969,D-W=0.951(**)stability,F(5,41)=16.89(***)$

2A.2. AR1

$LCM2=\underset{\left(1.578\right)}{-3.657}+\underset{\left(0.086\right)}{0.076LTW}+\underset{\left(0.354\right)}{0.438LWS}\underset{\left(0.045\right)}{-0.056LR}\underset{\left(0.157\right)}{-0.213LY,}$

$RSS=0.019832,R^2=0.790,D-W=1.878stability,F(5,41)=7.411(***)$

2A.3. OLS(Dynamic)

$LCM2=\underset{\left(0.170\right)}{0.359}+\underset{\left(0.058\right)}{0.041LTW}+\underset{\left(0.257\right)}{0.178LWS}+\underset{\left(0.036\right)}{0.010LR}-\underset{\left(0.152\right)}{0.308LY}+\underset{\left(0.710\right)}{0.582LCM{2}_{-1}}+\underset{\left(0.06\right)}{0.057LT{W}_{-1}}-\underset{\left(0.255\right)}{0.520LE{S}_{-1}}-\underset{\left(0.040\right)}{0.063L{R}_{-1}}+\underset{\left(0.125\right)}{0.255L{Y}_{-1}},RSS=0.006668,R^2=0.994,D-W=\mathrm{2.402.}$

Stability, F(10.31) = 19.32(***). LR test for CO restrictions, CHISQ(4) = 38.15(***). Test for extra lagged variables, F(5,25) = 19.63(***)

(B) Tax variable ln(1 + T)

2B.1. OLS(Static)

$LCM2=\underset{\left(1.306\right)}{0.582}+\underset{\left(0.316\right)}{0.236LT}-\underset{\left(0.028\right)}{0.208LR}-\underset{\left(0.096\right)}{0.166LY},$

$RSS=0.034382,R^2=0.968,D-W={1.300}^{\prime }stability,F(5,41)=14.81(***)$

2B.2. AR1

$LCM2=\underset{\left(1.644\right)}{-3.349}+\underset{\left(0.080\right)}{0.035LT}+\underset{\left(0.378\right)}{0.419LWS}-\underset{\left(0.046\right)}{0.056LR}-\underset{\left(0.172\right)}{0.242LY,}$

$$\begin{gathered} RSS=0.020155,R^2=0.775,D-W=\mathrm{1.905.} \\ Stability,F(5,41)=8.118(***) \end{gathered}$$

2B.3. OLS(Dynamic)

$LCM2=\underset{\left(0.724\right)}{0.865}-\underset{\left(0.059\right)}{0.043LT}+\underset{\left(0.261\right)}{0.277LWS}-\underset{\left(0.037\right)}{0.017LR}-\underset{\left(0.148\right)}{0.117LY}+\underset{\left(0.165\right)}{{0.650LCM2}_{-1}}+\underset{\left(0.063\right)}{{0.127LT}_{-1}}-\underset{\left(0.227\right)}{{0.698LWS}_{-1}}-\underset{\left(0.040\right)}{{0.064LR}_{-1}}+\underset{\left(0.130\right)}{0.139L{Y}_{-1}},RSS=0.006728,R^2=0.994,D-W=2.115$

h = –1.567; stability, F(10,31)= 18.06(***); LR test for CO restrictions, CHISQ(4) = 38.40(***); test for extra lagged variables, F(5.25) = 20.55(***)

Note: For notation see Note to Table 1. The stability tests were based on a comparison of the RSS obtained from fitting each equation separately to 1930–45 and 1946–80 with the RSS obtained from fitting an equation to the whole period 1930–80. Superscript i indicates D-W statistic is in inconclusive region at 0.95 probability level; h is Durbin’s test for AR(1) serial correlation in dynamic equations containing the lagged dependent variable.

A visual inspection of In CIM₂, ln(1 + T) and ln(1 + TW) in the chart below shows that, while the three variables appear to move in a similar fashion between 1930 and 1945, their paths diverged during the period 1946–80, with C/M₂ falling gently throughout the period while both LT and LTW show more irregular fluctuations. For this reason, equations were fitted to 1930–45 and 1946–80 separately to test for a structural break in 1945–46.

Results of the Diagnostic Tests

The diagnostic statistics are presented in Tables 1 and 2, where the F-tests for stability are all highly significant, showing that all the regressions are unstable and exhibit evidence of a structural break in 1945, though the CO transformed (AR1) equations appear to be less unstable than the OLS regressions.³

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Movements in In C/M ₂, In(l + TW), ln(l + T), 1930–80

Citation: IMF Staff Papers 1986, 004; 10.5089/9781451930696.024.A006

Source: Data on C, M₂, TW, and T may be found in Tanzi (1983, Table 1).

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For the period 1930–45 the tax variables are statistically significant in all equations, with the estimated parameters being of the same order of magnitude as those presented by Tanzi for 1930–80. The results of the COMFAC tests are somewhat mixed, with the general dynamic model rejecting the restrictions implied by the CO transformation for LT but not for LTW. Since the F-tests for the contribution of the extra lagged variables in the dynamic equations are insignificant, one may conclude that on balance the data do not reject the CO transformed equations.

However, a very different picture emerges when the results for the period 1946–80 are considered. While the tax variables are statistically significant in the static OLS equations, these exhibit evidence of serial correlation. When the CO transformation is applied to produce the AR1 regressions, the serial correlation is removed, but the tax variables become statistically insignificant. Further, the COMFAC test results are highly significant, and since the F-tests for the contribution of the extra lagged variables in the dynamic models are also highly significant, one is led to a rejection of the AR1 regressions in favor of the dynamic models. Finally, the F-tests for the effects of dropping the tax variables from the dynamic equations are 2.578 for LT and 2.741 for LTW, which are not statistically significant.⁴

Conclusions

The regressions reported here suggest that the apparently significant results presented by Tanzi for 1930–80 are a statistical artifact constructed from a significant static relationship for 1930–45 and nonsignificant static misspecification of a dynamic relationship for 1946–80. As a result, his estimation of the size of the underground economy for 1946–80, the period in which the tax variables do not seem to have affected In C/M₂, must be extremely suspect.