Unemploymen rae and U.S. shadow economy: an analysis based on spline models Adriana AnaMaria Alexandru Deparmen of Saisics and Economerics Academy of Economic Sudies Buchares 5-7, Calea Dorobanilor, Buchares, ROMANIA Naional Scienific Insiue for Labour and Social Proecion 6-8 Povernei Sree, Buchares, ROMANIA adrianaalexandru@yahoo.com Absrac The paper aims o invesiae he relaionship beween unemploymen rae and shadow economy wih USA daa usin spline models. The shadow economy is esimaed as percenae of official, usin MIMIC model. The size of he shadow economy as % of official is esimaed usin a MIMIC model wih four causal variables (axes on corporae income, conribuions for overnmen social insurance, unemploymen rae and selfemploymen and wo indicaors (index of real and civilian labour force paricipaion rae.the size of he shadow economy (SE is esimaed o be decreasin over he las wo decades. In order o evaluae he naure of he relaionship beween he wo variables, we have esimaed cubic B-spline, naural cubic B- spline and smoohin models.usin an F-es, we compare he smoohin spline o a lobal linear fi and he resuls indicae a sufficienly linear relaionship. Finally, we have compared he local polynomial models wih he spline model; he smoohin spline model closely maches he lineariy beween he size of he shadow economy and he unemploymen rae. We exend he classical Okun law, in order o esimae he relaionship beween rowh rae of official economy, unemploymen rae and he size of he shadow economy. The resuls reveal a sinifican direc relaionship beween shadow economy and he unemploymen rae and an indirec relaion beween shadow economy and rowh of official secor. Keywords shadow economy, unemploymen rae, MIMIC models, spline models, Okun law, U.SA. I. INTRODUCTION The relaionship beween he shadow economy and he level of unemploymen is one of major ineres. People work in he shadow economy because of he increased cos ha firms in he formal secor have o pay o hire a worker. The increased cos comes from he ax burden and overnmen reulaions on economic aciviies. In discussin he rowh of he shadow economy, he empirical evidence suess wo imporan facors: (a reducion in official workin hours, (b he influence of he unemploymen rae. Ense [6] poins ou ha he reducion of he number of workin hours below worker's preferences raises he quaniy of hours worked in he shadow economy. Early reiremen also increases he quaniy of hours worked in he shadow economy. In Ialy, Berola and Garibaldi [6] presen he case ha an increase in payroll axaion can have effec on he supply of labour and he size of he shadow economy. An increase in ax and social securiy burdens no only reduces official employmen bu ends o increase he shadow labour force. This is because an increase in payroll ax can influence he decision o paricipae in official employmen. Also, Boeri and Garibaldi [7] show a sron posiive correlaion beween averae unemploymen rae and averae shadow employmen across 0 Ialian reions durin 995-999. The paper examines he possible relaionship beween unemploymen rae and he size of he shadow economy usin a nonparameric analysis based on spline models. Also, a reexaminaion of he classical Okun s law is provided in he paper, showin he relaionship beween unemploymen and official economy in he presence of shadow economy. II. ESTIMATING THE SIZE OF THE U.S. SHADOW ECONOM II.. Daa and Mehodoloy II... Daa issues The variables used in he esimaion are defined in appendix A. The daa series are quarerly, seasonally adjused coverin he period 980:Q o 009:Q. The series in levels or differences have been esed for uni roos usin he Aumened-Dickey Fuller (ADF es and PP ess. All he daa has been differeniaed for he achievemen of he saionariy (appendix, uni roo analysis. While all he variables have been idenified like ineraed on firs order, he laen variable is esimaed in he same ransformaion of independen variables (firs difference. II.. Mehodoloy The size of he U.S. shadow economy is esimaed as % of official usin a paricular ype of srucural equaions models-mimic model. The MIMIC model- Muliple Indicaors and Muliple Causes model (MIMIC model, allows o consider he SE as a Issue 5, Volume 5, 0 939
laen variable linked, on he one hand, o a number of observable indicaors (reflecin chanes in he size of he SE and on he oher, o a se of observed causal variables, which are rearded as some of he mos imporan deerminans of he unrepored economic aciviy [0]. The model is composed by wo sors of equaions, he srucural one and he measuremen equaions sysem. The equaion ha capures he relaionships amon he laen variable (η and he causes ( q is named srucural model and he equaions ha links indicaors ( p wih he laen variable (non-observed economy is called he measuremen model. A MIMIC model of he hidden economy is formulaed mahemaically as follows: =λη+ε ( η = γ + ξ ( where: η is he scalar laen variable(he size of shadow economy; =,... is he vecor of indicaors of he laen ( p variable; =,... is he vecor of causes of η ; ( q λ( p and ( q ε ( p and ( q γ vecors of parameers; ξ vecors of scalar random errors; The ε ' s and ξ are assumed o be muually uncorrelaed. Subsiuin ( ino (, he MIMIC model can be wrien as: = Π+ z (3 ' where: Π =λγ, z =λξ + ε. The esimaion of ( and ( requires a normalizaion of he parameers in (, and a convenien way o achieve his is o consrain one elemen of λ o some pre-assined value ([7]-[8]. The possible causes of shadow economy considered in he model are: ax burden decomposed ino personal curren axes (, axes on producion and impors(, axes on corporae income( 3, conribuions for overnmen social insurance( 4 and overnmen unemploymen insurance( 5, unemploymen rae( 6, self-employmen in civilian labour force ( 7, overnmen employmen in civilian labour force ( 8 called bureaucracy index. The indicaor variables incorporaed in he model are: real ross domesic produc index (, currency raio M M ( and civilian labour force paricipaion rae ( 3. The variables used ino he esimaion of he shadow economy are also quarerly and seasonally adjused coverin he period 980-009. All he daa has been differeniaed for he achievemen of he saionariy. In order o esimae he MIMIC model, by Maximum Likelihood, usin he LISREL 8.8 packae, we normalized he coefficien of he index of real ( λ = o sufficienly idenify he model. This indicaes an inverse relaionship beween he official and shadow economy. In order o idenify he bes model, we have sared wih MIMIC model 8--3 and we have removed he variables which have no srucural parameers saisically sinifican. A deailed descripion and implemenaion of he MIMIC model for he USA shadow economy is provided in [5]. II.. Empirical resuls In order o esimae he size of he shadow economy, we have idenified he bes model as MIMIC 4-- wih four causal variables (axes on corporae income, conribuions for overnmen social insurance, unemploymen rae and selfemploymen and wo indicaors (index of real and civilian labour force paricipaion rae. Takin ino accoun he reference variable (, Re al Re al he shadow economy is scaled up o a value in, he base year, and we build an averae of several esimaes from his year for he U.S.A. shadow economy (able I. The index of chanes of he shadow economy ( η in Unied Saes measured as percenae of in he is linked o he index of chanes of real as follow: Measuremen Equaion: ~ η ~ η = I: Esimaes of he size of U.S.A. shadow economy ( Auhor Mehod Size of Shadow Economy Johnson e. Al(998 Currency Demand 3.9% Approach Lacko(999 Physical Inpu(Elecriciy 0.5% Schneider and Ense(000 Currency Demand 7.5%* Approach Mean 0.6% *means for -993 The esimaes of he srucural model are used o obain an ordinal ime series index for laen variable (shadow economy: Srucural Equaion: ~ η = 0.4 3 + 3.00 4 +.49 6 +.0 7 (4 (5 Issue 5, Volume 5, 0 940
The index is scaled o ake up o a value of 0.6% in and furher ransformed from chanes respec o he in he o he shadow economy as raio of curren : ~ * η η ˆ η ~ = (6 I. ~ η eq.(; η * = η is he index of shadow economy calculaed by II. 0.6% is he exoenous esimae of shadow economy; ~ η III. is he value of index esimaed by (5; IV. is o conver he index of chanes respec o base year in shadow economy respec o curren ; ηˆ V. is he esimaed shadow economy as a percenae of official. shadow economy % of off. 8 6 4 0 8 6 4 shadow economy as % of off. 0 980 98 984 986 989 99 993 995 998 000 00 004 007 009 Fi.. The size of he shadow economy in Unied Saes as % of official The shadow economy measured as percenae of official records he value of 3.4% in he firs rimeser of 980 and follows an ascendan rend reachin he value of 6.77% in he las rimeser of 98. A he beinnin of 983, he dimension of USA shadow economy beins o decrease in inensiy, recordin he averae value of 6% of a he end of 007. For he las wo year 008 and 009, he size of he unrepored economy i increases slowly, achievin he value of 7.3% in he second quarer of 009. The resuls are no far from he las empirical sudies for USA ([6], [9].Schneider esimaes in his las sudy, he size of USA shadow economy as % of, a he level of 7.9% in 005, respecively 8% in 006. III. THERE IS A LINK BETWEEN SHADOW ECONOM AND UNEMPLOMENT RATE IN THE CASE OF UNITED STATES? In many empirical sudies, is has been found ha ax burden is he bies causes of shadow economy. Also he size of shadow economy is influenced by he level of unemploymen. An increase in unemploymen raes reduces he proporion of workers employed in he formal secor his leads o hiher labor paricipaion raes in he informal secor. The raphical evoluion of he shadow economy versus unemploymen rae reveal he exisence of a sron posiive relaionship beween he wo variables, quanified by a value of abou 0.80 of correlaion coefficien. shadow economy % of off. 8 6 4 0 8 6 4 0 0 980 98 984 986 989 99 993 995 998 000 00 004 007 009 shadow economy as % of off. unemploymen rae Fi.. Shadow economy vs. Unemploymen rae in Unied Saes Giles ([7], [8] saes ha he effec of unemploymen on he shadow economy is ambiuous (i.e. boh posiive and neaive. An increase in he number of unemployed increases he number of people who work in he black economy because hey have more ime. On he oher hand, an increase in unemploymen implies a decrease in he shadow economy. This is because he unemploymen is neaively relaed o he rowh of he official economy (Okun s law and he shadow economy ends o rise wih he rowh of he official economy III.. A Nonparameric Analysis of he Relaionship beween Unemploymen Rae and he Size of he Shadow Economy Usin Spline Models The paper aim o invesiae he relaionship beween shadow economy esimaed usin he MIMIC model and unemploymen rae usin a nonparameric analysis based on spline models. The unemploymen rae is expressed in %, aken from U.S. Bureau of Saisics, Labour Force Saisics from Curren Populaion Survey. Insead of assumin ha we know he funcional form for a reression model, a beer alernaive is o esimae he funcional form from he daa, replacin lobal esimaes wih 0 8 6 4 unemploymen rae(% Issue 5, Volume 5, 0 94
local esimaes. In he erms of local esimaion, he saisical dependency beween wo variables is described no wih a sinle parameer such as a mean or a slope coefficien, bu wih a series of local esimaes. Like local polynomial reression (LPR, spline smoohers are anoher nonparameric echnique used wih scaerplos. In any spline model, i mus be seleced he number of knos and he kno placemen [3]. Sandard pracice is o place knos a evenly spaced inervals in he daa. Bu he quesion of how o selec he number of knos remains and has an imporan effec on he Spline fi. One mehod is o use a visual rial. Four knos is he sandard sarin poin. If he fi appears rouh, knos are added. If he fi appears overly nonlinear, knos are subraced. The second mehod is o use Akaike Informaion Crierion o selec he number of knos. The opimal number of knos is reurned by he lowes AIC value. Thus far, LPR esimaes have revealed a linear dependency beween he size of he shadow economy and he unemploymen rae. I will we ineresin o invesiae he naure of he relaionship beween he wo variables, usin boh cubic B-splines and naural cubic B-splines o esimae he nonparameric fi. For he boh spline models, i has been used 4 knos i we will evaluae wheher his is he opimal number of knos, usin Akaike Crierion. Analyzin he raphics of boh funcions, here is a lile difference beween cubic B-splines and he naural cubic B- splines. II: AIC values for differin number of knos Naural Cubic Spline Spline knos 477.0573 478.7849 3 knos 478.00 478.7849 4 knos 477.0685 478.7849 5 knos 479.030 478.7093 6 knos 480.0504 479.769 7 knos 475.998 477.6035 8 knos 477.590 479.86 9 knos 477.97 475.646 Analysin he values of Akaike Informaion Crierion for several knos we observe ha he opimal number of knos for he Naural Spline akin ino accoun he lowes AIC value is 7 knos, while for he cubic Spline he opimal number of knos is 9. Fi.3. The choice of he opimal number of knos for he boh models Fi.. Cubic B-spline and naural spline fi o size of he shadow economy (% of off. In order o selec he number of knos, we use for boh models he Akaike Informaion Crierion (AIC; he opimal number of knos is reurned by he lowes AIC value. For he boh spline models, i has been esimaed several models wih -9 knos. In order o esimae he saisical relaionship beween wo variables, boh splines and local polynomial reression can provide such an esimae wih few assumpions abou funcional form. A common criicism of he boh mehods is ha i is easy o have a surfei of local parameers, which produces overly nonlinear esimaes ha overfi daa [3]. Penalized splines are a nonparameric reression echnique ha minimizes he possibiliy of overfiin. Smoohin splines operae wih penalized esimaion, placin a penaly on he number of local parameers used o esimae he nonparameric fi. Like linear reression models, he spline esimae fˆ minimises he sum of squares beween y and he nonparameric esimae, f x : ( i n SS( f = [ y f ( x] (7 i= Issue 5, Volume 5, 0 94
The main problem is ha he esimae of f ha minimises (7 use oo many parameers. The penalised esimaion soluion is o aach a penaly for he number of parameers x n used o esimae f : λ [ f ( x] dx, named rouhness x penaly ha have wo componens: λ, he smoohin parameer and he second, he ineraed squared second derivaive of f (x. Furher, he spline esimae become: n x n SS( f, λ = [ y f ( x] + λ [ f ( x] dx (8 i= While small values of λ will inerpolae he daa and lare values reurns a leas squares fi, inermediae values does no offer an inerpreable effec on he amoun of he smoohin applied o he daa. I is proposed a ransformaion of λ ino an approximaion of he derees of freedom; by selecin he derees of freedom, i is chosen he number of effecive local parameers used in he spline esimae. The penalized splines named also smoohin splines differ from he sandard splines by he fac ha he number of knos have lile influence over how smooh he fi is since he value of λ conrols now he qualiy of he fi. In order o see how differen derees of freedom (, 4, 8 and affec he fi, i has been esimae he relaionship beween he size of he shadow economy and he unemploymen rae usin smoohin splines. The resuls reveal ha he fi wih derees of freedom is idenical o a linear reression; for he model wih 4 and 8 derees of freedom we have he same paern of lineariy found wih oher spline fis. For he fi wih derees of freedom, we have considerable variabiliy, caused by oo many parameers and we can conclude ha he daa are over fied. x In order o es hypohesis abou he naure of he relaionship beween he size of he shadow economy and he unemploymen rae, we compare he smoohin spline model o a model wih only a consan o es wheher he effec of he unemploymen rae is sinificanly differen from zero. If RSS and RSS are he residual sum of squares from a resriced model and he spline model respecively, he F ( RSS RSS/( df df res res es= Fdf res dfres, n df (9 res RSS /( n dfres Applyin he F-es, we find ha he relaionship beween he wo variables is hihly sinifican as he es saisic is 7.78 on 3 and 5 derees of freedom (p=.e-6. We also es he spline model aains a lobal linear fi, and he value of F-es of.9597 on and 4 derees of freedom is no saisically sinifican(p=0.476. The resuls of he es indicae ha he relaionship beween he size of he shadow economy and he unemploymen rae is sufficienly linear and he lobal linear fi is adequae. Finally, we provide a comparison of he nonparameric reression models (fi.5. The firs wo nonparameric models are he loess and lowess smoohers. In Dobre, Alexandru [9] we have esimaed he boh models idenifyin he opimal value of span a he value of 0.4. Beween he wo LPR smoohers, he lowess esimae provides a beer fi of he daa. In he lower lef panel is he naural cubic B-spline wih 7 knos, chosen by he AIC values. This model displays noiceable undersmoohin of he esimae. Finally, we esimae a smoohin spline usin 4 derees of freedom seleced hrouh visual rial. The smoohin spline closely maches he lineariy beween he size of he shadow economy and he unemploymen rae. Fi.4. Smoohin spline fi o shadow economy daa Fi. 5. Comparison of smooher fis Issue 5, Volume 5, 0 943
III.. A re-examinaion of Okun s law in presence of shadow economy The Okun s law relaes decreases in he unemploymen rae o increases in oupu rowh. We wan o es if he shadow economy has any sinifican effec on his empirical evidence. We o on he hypohesis ha a lower rowh rae of official from poenial oupu is associaed wih hiher deviaions of he unemploymen rae from is "naural" level. The increase in unemploymen leads o an increase in he number of laborers who work in he unofficial labour marke. In fi.(appendix, we presen he sinifican saisical relaionships amon rowh rae of official, chanes in unemploymen rae and rowh of shadow economy for he case of Unied Saes coverin he period 980-009. The esimaes obained based on he sandard relaion iven by Okun s law are presened in he followin able: = 0 u (0 α + ε where: off = ( indicaes he difference of rowh rae (80 09 of he official ross domesic produc ( from i averae calculaed over he period 970 o 008; η shad η = ( indicaes he difference of shadow (80 09 economy( shad off from i averae calculaed over he period 980 o 009, u id he firs difference of unemploymen rae, ε are residuals i.i.d. III. Esimaion oupu of reression: = 0 α u + ε Furhermore, we use a modified version of Okun s law by includin he shadow economy: = + η + ( α u β IV. Esimaion oupu of reression: α u = + β η + ε ε The economeric resuls reveal ha we have a sinifican neaive relaionship on he one hand, beween he rowh rae of official economy and he level of unemploymen, ha confirm he Okun s law, and on he oher hand, beween he rowh rae of official oupu and he size of he shadow economy. We deduce herefore, ha shadow economy ends o cushion he effecs of chanes in unemploymen on he official. In order o invesiae he impac of shadow economy on he unemploymen rae, we develop a srucural relaionship, akin ino accoun also he rowh rae of official : shad = γ + λu + ε ( off where: ( is he firs difference of annual rowh rae of he off official ross domesic produc; is he firs difference of he shadow economy; shad u is he firs difference of unemploymen rae; ε residuals; The esimaes show an inverse relaionship beween chanes in unemploymen and he rowh rae of official oupu. Issue 5, Volume 5, 0 944
V. Esimaion oupu of reression: shad = c+ γ + λu + ε off unemploymen rae has a saisically sinifican effec on he size of he U.S.A. shadow economy. We also es he spline model aains a lobal linear fi and he resuls indicae ha he relaionship beween he size of he shadow economy and he unemploymen rae is sufficienly linear and he lobal linear fi is adequae. Finally, we have compared he local polynomial reression models (loess and lowess esimaed in Dobre, Alexandru[5] wih spline models (naural cubic B-spline and smoohin spline. From he four ypes of models ha we have applied, he smoohin spline model closely maches he lineariy beween he size of he shadow economy and he unemploymen rae. We exend he classical Okun s law, in order o esimae he relaionship beween rowh rae of official economy, unemploymen rae and he size of he shadow economy. The resuls reveal a sinifican direc relaionship beween shadow economy and he unemploymen rae and an indirec relaion beween shadow economy and rowh of official secor. Moreover, we can conclude ha employmen in he shadow economy consiues a form of labor marke ransiion beween or raher from unemploymen back ino formal employmen. The parameer γ of he equaion shows an inverse relaionship beween he rowh of he official economy shad ( (. On he and rowh of he shadow economy off oher-hand, he parameer λ shows a direc relaionship beween chanes in unemploymen and he rowh of he shadow economy. The coefficiens are saisically sinifican (prob.<5% and he deree of deerminaion in he model is hih, 75% of he variaion of shadow economy is explained by he wo exoenous variables unemploymen rae and rowh rae of official. Our esimaions show ha he presence of he shadow economy acs as a buffer as i absorbs some of he unemployed workers from he official economy ino he shadow economy. IV. CONCLUSIONS The main oal of he paper is o invesiae he naure of he relaionship beween unemploymen rae and he size of he shadow economy of he USA daa usin spline models.the shadow economy is esimaed as percenae of official, usin MIMIC model. The resuls show ha he size of he shadow economy varies from hireen o seveneen percen beween 980 and 983 and hen decreases seadily up o 7 percen of official in 009. We invesiae he naure of he relaionship beween he wo variables, usin cubic B-splines and naural cubic B- splines o esimae he nonparameric fi. The raphics of boh spline models reveals a lile difference beween he wo funcions. Usin an F-es, we compare he smoohin spline model o a model wih only a consan, and we conclude ha Issue 5, Volume 5, 0 945
Appendix. A. Uni-roo analysis The daa sources are: Bureau of Economic Analysis (BEA, Bureau of Labor Saisics Daa (BLS and Federal Reserve Banks. CAUSES Source Uni roo analysis Personal curren axes/ BEA I( Taxes on producion and impors/ BEA I( 3 Taxes on corporae income/ BEA I( 4 Conribuions for overnmen social insurance/ BEA I( 5 Governmen unemploymen insurance BEA I( 6 Unemploymen rae BLS I( 7 Self-employmen/Civilian labour force BLS I( 8 Index of bureaucracy BLS I( Level Firs Difference ADF la PP la ADF la PP la T&C -.474 4 -. 7-3.83* 0-3.4* 7 C -.493 4 -.00 6-3.79* 0-3.40* 7 None -0.88-6.76 6-3.79* 0-3.40* 7 T&C -3.543 0-3.83 6 -.4* 0 -.8* C -.9 0-3.86 6 -.7* 0 -.3* None 0.89 0 0.55 5 -.3* 0 -.37* T&C -4.9* 3-3.45 6-0.93* 0-0.93* 4 C -4.4* 3-3.44 6-0.98* 0-0.98* 4 None -.8 0 -.3 5-4.4* -.0* 4 T&C -.3 6 -.0 0-5.99* 5-8.57* 8 C -.73 4-3.40 0-4.36* 3-4.03* 9 None 0.6 4.08 0-4.35* 3-3.75* 9 T&C -.63 -.76 4-4.44* -6.49* 3 C -3. -.07 4-4.00* -6.36* 3 None -0.94-0.38 4-4.33* -6.37* 3 T&C -.03 -.4 6-4.40* 0-4.69* 3 C -.4 -.69 6-4.7* 0-4.5* 3 None -0. 0.03 7-4.7* 0-4.53* 3 T&C -.44 0 -.8 4-9.68* -.79* 3 C -0.90 0-0.5 5 -.03* 0 -.44* 8 None -0.7 0-0.9 9 -.05* 0 -.4* 8 T&C -.69 0 -.67 3 -.73* 0 -.38* 7 C -.88 0 -.94 3 -.0* 0 -.99* 7 None -.7 0 -.0 5 -.37* 0 -.68* 7 INDICATORS M / M Reserve Federal Banks Index of Real BEA I( 3 Civilian labor force paricipaion rae Re al Q Real Gross Domesic Produc, Chained Dollars. Billions of chained (500 dollars. Seasonally adjused a annual raes/ I( BLS I( T&C -. -.43 8-3.0-6.59* 7 C -.69-0.90 8-3.03* -6.5* 7 None -0.59-0.78 8-3.0* -6.48* 7 T&C -.7 -.35 4-5.43* -8.7* 4 C.4.03 5-5.6* -8.44* 4 None 4.63 9.68 5 -.39* -4.45* 6 T&C -0.47-0.66-0.9* -0.59* 0 C -.03 0 -.08 3-5.98* -0.08* 4 None. 0.5 4-5.8* -0.0* 5 ( ( ( 3 ( 4 ( 5 ( 6 ( 7 ( 8 ( ( ( 3 Issue 5, Volume 5, 0 946
Noe: T&C represens he mos eneral model wih a drif and rend; C is he model wih a drif and wihou rend; None is he mos resriced model wihou a drif and rend. Numbers in brackes are la lenhs used in ADF es (as deermined by SCH se o maximum o remove serial correlaion in he residuals. When usin PP es, numbers in brackes represen Newey-Wes Bandwih (as deermined by Barle-Kernel. Boh in ADF and PP ess, uni roo ess were performed from he mos eneral o he leas specific model by eliminain rend and inercep across he models (Kairciolu, 009. *, ** and *** denoe rejecion of he null hypohesis a he %, 5% and 0% levels respecively. Tess for uni roos have been carried ou in E-VIEWS 6.0. Fi..Growh of official, Chanes in unemploymen and Growh of Shadow Economy Issue 5, Volume 5, 0 947
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