The Size of the U.S.A. Shadow Economy

The size of the U.S.A. shadow economy:

A Structural Equation Approach

Adriana AnaMaria ALEXANDRU

PhD Candidate, University Assistant, Department of Statistics and Econometrics

University of Economics, Bucharest, Romania

Scientific researcher

National Scientific Institute in the field of Labour and Social Protection

E-mail:

Ion DOBRE

PhD, University Professor, Department of Economic Cybernetics

University of Economics, Bucharest, Romania

E-mail:

Catalin Corneliu GHINARARU

Scientific researcher I

National Scientific Institute in the field of Labour and Social Protection

Abstract:

This paper aims to estimate the size of the U.S.A. shadow economy (SE) using a Structural Equation Approach using quarterly data for the period 1980-2009. In order to do that, the shadow economy is modeled like a latent variable using a special case of the structural equation models-the MIMIC model.

The model includes tax burden (decomposed in personal current taxes, taxes on production and imports, taxes on corporate income and contributions for government social insurance), government unemployment insurance, unemployment rate, self-employment and government employment as causes of shadow economy and the labour force participation ratio and currency ratio as indicators of the unreported economy.

The results confirm that taxes on corporate income, contributions for government social insurance, unemployment rate and the self-employment arethe main causes of shadow economy.The size of the SE is estimated to be decreasing over the last two decades.

Keywords: shadow economy, MIMIC models, USA

JEL classification codes: C87, E26, H20, H50, O17

Measuring the shadow economy

Studies trying to measure the dimension of shadow economy face the difficulty of how to define it. One commonly used working definition is: all currently unregistered economic activity which contributes to the officially calculated (or observed) Gross National Product[1].Smith (1994) defines it as „market-based production of goods and services, whether legal or illegal that escapes detection in the official estimates of GDP.“

Clandestine by her nature, the shadow economy is difficult to evaluate.In the problem of measuring the dimension of shadow economy, there are various approaches who include using surveys of taxation compliance, using the discrepancy between national income and national expenditure; considering fluctuations in labour force participation rates; the monetary “transactions approach” of Feige (1979); modifications of currency demand equations, where the pioneer was Cagan (1958).One criticism of most of these approaches is that it focus on one cause of hidden economic activity, and one indicator.

In contrast, The MIMIC(“Multiple Indicators, Multiple Causes”) model of Zellner(1970), Goldberger(1972), Jöreskog and Goldberger(1975), Jöreskog and Sörbom(1993)) allows for several indicators variables and several causal variables in forming structural relationships to explain the latent variable. Frey and Weck-Hanneman (1984) estimated underground economy MIMIC models for a range of OECD countries; Aigner etal.(1988) applied a dynamic MIMIC model to U.S. data; and Tedds (1998) used this approach to model the Canadian underground economy.

Modeling the shadow economy

In the process of econometric modelling of U.S.A. shadow economy (SE) we used a different type of models-Structural Equations Models (SEM).The SEM represents statistical relationships among latent(unobserved) and manifest (observed) variables.

Bollen (1989, p.1) presents the fundamental hypothesis for structural equation modelling as:, where is the observed population covariance matrix, is a vector of model parameters, and is the covariance matrix implied by the model. When the quality expressed in the equation holds, the model is said to “fit” the data. Thus, the goal of structural equation modelling is to explain the patterns of covariance observed among the study variables.

A special case of SEM is theMultiple Indicators and Multiple Causes model. It allows to consider the SE as a“latent” variable linked, on the one hand, to a number of observable indicators(reflecting changes in the size of the SE) and on the other, to a set of observedcausal variables, which are regarded as some of the most important determinantsof the unreported economic activity(Dell’Anno, 2003).

Frey and Weck-Hanneman (1984) have been the first economists that consider the dimension of the hidden economy as an “unobservable variable”.

This type of models is composed by two sorts of equations, the structural one and the measurement equations system. The equation that captures the relationships among the latent variable (η) and the causes (Xq) is named “structural model” and the equations that links indicators (Yp) with the latent variable (non-observed economy) is called the “measurement model”.

A MIMIC[2] model of the shadoweconomy is formulated mathematically as follows:

(1)

(2)

where:

is the scalar latent variable(the size of shadow economy);

is the vector of indicators of the latent variable;

is the (vector of causes of ;

and vectors of parameters;

and vectors of scalar random errors;

The and are assumed to be mutually uncorrelated: ().

Since the structural equation model (1) only partially explains the latent variable, the error termrepresents the unexplained component.

The MIMIC model assumes that the variables are measured as deviations from their means and that the error term does not correlate to the causes and.The variance of is abbreviated by and is the covariance matrix of the causes.

The measurement model (2) represents the link between the latent variable and its indicators; the latent unobservable variable is expressed in terms of observable variables.Theircovariance matrix is given by.

Like the MIMIC model’s causes, the indicators are directly measurable andexpressed as deviations from their means, It is assumedthat the error terms in the measurement model do not correlate either to the causes or to the latent variable,. and

Substituting (2) into (1), the MIMIC model can be written as: (3)

where:,. The error term in equation (3) is a (p ×1) vector oflinear combinations of the white noise error terms and from the structural equation and the measurement model,. The covariance matrix is given as, , the diagonal covariance matrix of

The multivariate regression equation (3) has a regressor matrix of rank one, and the error covariance matrix is also constrained, and that is the raison for which it is impossible to obtain cardinal estimates of all of the parameters. The estimation of (1) and (2) requires a normalization of the parameters in (1), and a convenient way to achieve this is to constrain one element of to some pre-assigned value (Giles, Tedds, 2000).

To facilitate the identification of SEM some conditions are available but, unfortunately, none of these are necessary and sufficient conditions (Bollen, 1989). The necessary (but not sufficient) condition, so-called t-rule, enunciates that the number of nonredundant elements in the covariance matrix of the observed variables must be greater or equal to the number of unknown parameters in the model-implied covariance matrix.[3]

A sufficient (but not necessary) condition of identification, is that the number of indicators is two or greater and the number of causes is one or more, provided that is assigned a scale to η (MIMIC rule). For assigning a scale to the latent variable it is needed to fix one λ parameter to an exogenous value. Although several econometric improvements are introduced in the last years,the most important criticism to the MIMIC method is the strong dependence of the outcomes by the (exogenous) choice of the coefficient of scale (λ).

Given an estimate of the vector, and setting the error term to its mean value of zero, equation (2) enables us to “predict” ordinal values for which is the relative size of the hidden economy, at each sample point. Then, if we have a specific value for at some sample point, obtained from some other source, we can convert the within-sample predictions for into a cardinal series. We use an average value from other estimations realized using the currency demand model to calibrate the time-series of the hidden economy.

Giles (1999a) was the first author to “calibrate” such MIMIC model hidden economy results formally, by using the output from a completely separate demand-for-cash model to convert the ordinal predictions into cardinal ones in the context of New Zealand data.

III. Data issues

The variables used in the estimation are defined in Appendix A. The data series are quarterly from 1980:Q1 to 2009:Q2. All the series have been seasonally adjusted.

The series in levels or differences have been tested for unit roots using the Augmented-Dickey Fuller (ADF) test. We test I(2) against I(1) and if we reject I(2), we test I(1) against I(0) as appropriate(Appendix A). All the data has been differentiated for the achievement of the stationarity. While all the variables have been identified like integrated on first order, the latent variable is estimated in the same transformation of independent variables (first difference).

IV. Estimating the MIMIC model of the shadow economy

In our econometrical demarche of estimating the size of shadow economy, the causal variables considered in the model are: tax burden decomposed into personal current taxes (), taxes on production and imports(), taxes on corporate income(), contributions for government social insurance() and government unemployment insurance(), unemployment rate(),self-employment in civilian labour force (),government employment in civilian labour force () called bureaucracy index.

The indicator variables incorporated in the model are: real gross domestic product index (), currency ratio () and civilian labour force participation rate ().

The main elements of tax burden, government unemployment insurance and index of real gross domestic product are expressed as percentages of gross domestic product while government employment, self-employment, unemployment rate and civilian labour force participation rate are calculated like percentages of civilian labour force. A detailed presentation of the variables is realised in (Dobre, Alexandru, 2008)[4].

The identification procedure of the best model starts from the most general model specification (MIMIC 8-1-3)and continues removing the variables which have not structural parameters statistically significant.

The results of estimating the structural equation models, by Maximum Likelihood, using the LISREL 8.8 package, over the period 1990-2009 appear in table 1.The estimates of the coefficients () of the causal variables(x) in the structural equation for the shadow economy (in equation (2) provides the basis for estimating over the sample period.The coefficient of the index of real GDP[5] is normalised to -1 to sufficiently identify the model ().This indicates an inverse relationship between the official and shadow economy.

The results presented in table 1 suggest a negative none statistically significant relationship between the size of the shadow economy and civilian labor force participation rate. Also, between the dimension of the shadow economy and the currency ratio there is a positive none statistically significant relationship.

Although the causal variables have the anticipated signs, many of them lack individual significance. The unemployment rate and social insurance contributions are the only causal variable positively significant in all MIMIC models. The other causal variables from the model are not statistically significant.

The positive sign of the unemployment rate according with the negative one obtained by the civilian labor force participation rate point out the fact that many workers from the official economy go underground when they are laid off. The positive sign of the unemployment rate indicates the existence of a flow of resources from official to shadow economy in recession cycles.

Table 1:Estimated Coefficients[6] of the MIMIC Models

Models / Tax personnal/
GDP / Tax production/
GDP / Tax corporat./
GDP / Social insurance contr./GDP / Governm.unempl.
.insurance/GDP / Unempl.
Rate / Self
Empl. / Index
Bureac. / M1/M2 / Lab.Force
Partic. / Chi-square
(p-value)[7] / RMSEA
(p (p-value)[8] / AGFI[9] / Df[10]
MIMIC
8-1-3 / -0.01
(-0.08) / 0.83
(0.89) / -0.27
(-0.82) / 2.88*
(3.85) / 2.89
(1.89) / 1.06*
(3.73) / 0.98
(1.67) / 0.62
(0.53) / 0.05
(1.11) / -0.01
(-0.89) / 39.18
(0.0017) / 0.11
(0.021) / 0.78 / 17
MIMIC
8-1-2a / -0.01
(-0.08) / 0.42
(0.52) / -0.37
(-1.10) / -1.80*
(-2.85) / -0.41
(-0.28) / 1.48*
(4.99) / 0.85
(1.72) / -0.15
(-0.15) / ----- / -0.02
(-0.79) / 27.66
(0.000) / 0.15
(0.0049) / 0.69 / 8
MIMIC
8-1-2b / -0.01
(-0.08) / 0.83
(0.89) / -0.27
(-0.82) / 2.88*
(3.85) / 2.89
(1.89) / 1.06*
(3.73) / 0.98
(1.67) / 0.62
(0.53) / 0.05
(1.11) / ----- / 10.49+
(0.232) / 0.052+
(0.42) / 0.88 / 8
MIMIC
7-1-3 / -0.01
(0.08) / ----- / -0.27
(-0.81) / 3.06*
(4.22) / 2.89
(1.89) / 1.09*
(3.82) / 1.02
(1.73) / 0.61
(0.51) / 0.05
(1.11) / -0.01
(-0.89) / 35.67
(0.002) / 0.11
(0.021) / 0.79 / 15
MIMIC
6-1-3 / -0.02
(-0.14) / ----- / -0.30
(-0.89) / 2.98*
(4.20) / 2.89
(1.89) / 1.08*
(3.80) / 1.01
(1.73) / ----- / 0.05
(1.11) / -0.01
(-0.89) / 15.50+
(0.28) / 0.041+
(0.53) / 0.90 / 13
MIMIC
6-1-2a / -0.02
(-0.14) / ----- / -0.30
(-0.89) / 2.98*
(4.20) / 2.89
(1.89) / 1.08*
(3.80) / 1.01
(1.73) / ------/ ------/ -0.01
(-0.89) / 7.44+
(0.28) / 0.046+
(0.45) / 0.91 / 6
MIMIC
6-1-2b / -0.02
(-0.14) / ----- / -0.30
(-0.89) / 2.98*
(4.20) / 2.89
(1.89) / 1.08*
(3.80) / 1.01
(1.73) / ------/ 0.05
(1.11) / ------/ 7.02+
(0.31) / 0.038+
(0.49) / 0.91 / 6
MIMIC
5-1-2 / ------/ ----- / -0.29
(-0.88) / 2.97*
(4.20) / 2.85
(1.89) / 1.09*
(3.83) / 1.01
(1.72) / ------/ ------/ -0.01
(-0.89) / 7.44+
(0.19) / 0.065+
(0.33) / 0.90 / 5
MIMIC
4-1-3a / ----- / ----- / ---- / ----- / 2.75
(1.68) / 1.14*
(3.92) / 0.70
(1.12) / -0.24
(-0.19) / 0.05
(1.11) / -0.01
(-0.89) / 26.88
(0.001) / 0.13
(0.011) / 0.81 / 9
MIMIC
4-1-3b / -0.13
(-0.79) / 1.93
(1.68) / -0.99*
(-2.58) / 2.21*
(2.41) / ----- / ---- / ---- / ----- / 0.05
(1.11) / -0.01
(-0.89) / 13.81+
(0.13) / 0.07+
(0.30) / 0.90 / 9
MIMIC
4-1-3c / ----- / ----- / -0.24
(-0.73) / 3.00*
(4.17) / ----- / 1.49*
(7.99) / 1.01
(1.70) / ----- / 0.05
(1.11) / -0.01
(-0.89) / 10.20+
(0.33) / 0.034+
(0.55) / 0.92 / 9
MIMIC
4-1-2 / ----- / ----- / -0.24
(-0.73) / 3.00*
(4.17) / ----- / 1.49*
(7.99) / 1.01
(1.70) / ----- / ------ / -0.01
(-0.89) / 4.41+
(0.35) / 0.03+
(0.50) / 0.93 / 4
MIMIC
3-1-3 / ----- / ---- / -0.15
(-0.46) / 2.93*
(4.03) / ---- / 1.45*
(7.74) / ------/ ------/ 0.05
(1.11) / -0.01
(-0.89) / 8.83+
(0.27) / 0.047+
(0.45) / 0.93 / 7
MIMIC
3-1-2 / ----- / ---- / -0.15
(-0.46) / 2.93*
(4.03) / ---- / 1.45*
(7.74) / ------/ ------/ ------/ -0.01
(-0.89) / 4.05+
(0.26) / 0.055+
(0.37) / 0.93 / 3

V. Obtaining the size of the U.S.A. shadow economy

The econometrical results reveal that the main causes of shadow economy are:taxes on corporate income,contributions for government social insurance, unemployment rate and self-employment. Starting from MIMIC 8-1-3 and removing the variables which have not structural parameters statistically significant, we obtain MIMIC 4-1-2 as the best model.

The MIMIC 4-1-2 model has four causal variables (taxes on corporate income,contributions for government social insurance, unemployment rate and self-employment) and two indicators (index of real GDP and civilian labour force participation rate).

The choice of the model is based on: the statistical significance of parameters, the parsimony of specification, the p-value of chi-square, and the Root Mean Square Error of Approximation (RMSEA) test, adjusted goodness-of-fit index (AGFI).

Fig.1.Path diagram of 4-1-2 MIMIC

Taking into account the reference variable (,) the shadow economy is scaled up to a value in 1990, the base year, and we build an average of several estimates from this year for the U.S.A. shadow economy.

Table 2: Estimates of the size of U.S.A. shadow economy (1990)

Author / Method / Size of Shadow Economy
Johnson et.Al(1998) / Currency Demand Approach / 13.9%
Lacko(1999) / Physical Input(Electricity) / 10.5%
Schneider and Enste(2000) / Currency Demand Approach / 7.5%*
Mean 1990 / 10.6%

*means for 1990-1993

The index of changes of the shadow economy in United States measured as percentage of GDP in the 1990 is linked to the index of changes of real GDP as follow:

Measurement Equation: (4)

The estimates of the structural model are used to obtain an ordinal time series index for latent variable (shadow economy):

Structural Equation: (5)

The index is scaled to take up to a value of 10.6% in 1990 and further transformed from changes respect to the GDP in the 1990 to the shadow economy as ratio of current GDP.These operations are show in the benchmark equation[11]:

(6)

where:

I. is the index of shadow economy calculated by eq. (5).

II. is the exogenous estimate of shadow economy.

III. is the value of index estimated by eq.(5).

IV. is to convert the index of changes respect to base year in shadow economy respect to current GDP.

V. is the estimated shadow economy as a percentage of official GDP.

The shadow economy measured as percentage of official GDP records the value of 13.41% in the first trimester of 1980 and follows an ascendant trend reaching the value of 16.77% in the last trimester of 1982. At the beginning of 1983, the dimension of USA shadow economy begins to decrease in intensity, recording the average value of 6% of GDP at the end of 2007. For the last two year 2008 and 2009, the size of the unreported economy it increases slowly, achieving the value of 7.3% in the second quarter of 2009.

The results of this estimation are not far from the last empirical studies for USA (Schneider 1998, 2000, 2004, 2007, Schneider and Enste 2001).Schneider estimates in his last study, the size of USA shadow economy as average 2004/05, at the level of 7.9 percentage of official GDP.

VI.Conclusions

In this paper, we estimate the size of the shadow economy in the U.S.A. using the MIMIC model. Our results show that the size of the shadow economy varies from thirteen to seventeen percent between 1980 and 1983 and then decreases steadily up to 7 percent of official GDP in 2009.

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