CHANGE POINT TREND ANALYSIS OF GNI PER CAPITA IN SELECTED EUROPEAN COUNTRIES AND ISRAEL

Transcription

1 The 9 th Iteratioal Days of Statistics ad Ecoomics, Prague, September 0-, 05 CHANGE POINT TREND ANALYSIS OF GNI PER CAPITA IN SELECTED EUROPEAN COUNTRIES AND ISRAEL Lia Alatawa Yossi Yacu Gregory Gurevich Abstract The mai goal of this study is to preset ad apply recetly developed oparametric chage poit detectio ad estimatio techiques for cofirmig sigificat chages i short-term ad log-term treds of livig stadards i selected Europea coutries ad Israel. We aim also to coduct the comparative chage poit aalysis of livig stadards treds betwee OECD average, selected coutries ad Israel. The commo idex of livig stadards is the Gross Natioal Icome (GNI). Thus, GNI adjusted for purchasig power parity (PPP) per capita as well as the real GNI i local (atioal) currecy coverted ito U.S. dollars (utilizig the Atlas Method) per capita were ivestigated for the cosidered coutries over a period of last few decades. The chage poit aalysis reveals a complex patter of chage. I particular, we foud sigificat chages i short-term treds of livig stadards i most cosidered coutries. The growth of livig stadards i cosidered coutries has decreased last years i compariso with several previous years. However, we did ot foud sigificat chages i log-term treds of livig stadards. The results were supported also by traditioal statistical methods. Key words: Chage poit aalysis, Noparametric testig, Livig stadards treds, Atlas method JEL Code: C4, C, N0 Itroductio The stadard of livig icludes may factors such as icome, quality ad availability of employmet, class disparity, poverty rate, quality ad affordability of housig, hours of work required to purchase ecessities, gross domestic product, iflatio rate etc. The livig stadards are closely related to quality of life ad ofte used to compare geographic areas or distict poits i time. Most popular idex for comparig livig stadards is icome-measures ability of citizes to satisfy their eeds ad wats, hece livig stadards. Gross Natioal

2 The 9 th Iteratioal Days of Statistics ad Ecoomics, Prague, September 0-, 05 Icome (GNI) is the value added of all residet producers, essetially icome eared. Limitatios here are the use of the exchage rate, it is hard to get a idea of real prices for utilities ad food i the coutry as they differ across atios with the exchage rate. Therefore, it is commo to use the Atlas Method to covert the real GNI i local (atioal) currecy ito U.S. dollars utilizig the Atlas coversio factor. The Atlas coversio factor for ay year is the average of a coutry s exchage rate for that year ad its exchage rates for the two precedig years, adjusted for the differece betwee the rate of iflatio i the coutry ad iteratioal iflatio; the objective of the adjustmet is to reduce ay chages to the exchage rate caused by iflatio. I additio, it is acceptable also to use the real GNI adjusted for purchasig power parity (PPP) - whe exchage rates are equalized to the price of idetical goods ad services i differet ecoomies. This a measure of the real, iflatio adjusted purchasig power of the people i a coutry. Sice some coutries have more populatio tha others we use i our study two aforemetioed idexes (GNI, Atlas Method ad GNI, PPP) per capita that provides us with a much more accurate represetatio of the purchasig power of the average idividual i the ecoomy. The mai goal of this study is to preset ad apply recetly developed oparametric chage poit detectio ad estimatio techiques for cofirmig sigificat chages i short-term ad log-term treds of livig stadards i selected Europea coutries ad Israel. The study also aims to coduct the comparative chage poit aalysis of livig stadards treds betwee OECD average, selected coutries ad Israel. Utilizig preseted chage poit techiques we studied aual growth tedecies of the cosidered idexes over a period of last few decades. The obtaied results were supported also by traditioal statistical methods. This paper proceeds as follows. I Sectio, we itroduce the cosidered data ad the chage poit detectio ad estimatio methodology. I Sectio, a step by step applicatio of the proposed chage poit techiques is outlies, as well as traditioal statistical tools for two represetative examples. This Sectio also presets fial computatioal results of the proposed statistical aalysis for all the data cosidered, accompaied by short commets. The fial sectio cocludes the paper. Data ad Methodology. Database The followig two idicators related to Ecoomy ad Growth topic of the World Bak ope data were used i this study: GNI per capita, Atlas method (curret US$) ad GNI per capita,

3 The 9 th Iteratioal Days of Statistics ad Ecoomics, Prague, September 0-, 05 PPP (curret iteratioal $). These idicators were cosidered for the selected coutries over a period of Methodology: The Chage Poit Detectio ad Estimatio Techiques The issues of the so-called retrospective AMOC (at most oe chage) chage poit problem arise i experimetal ad mathematical scieces (epidemiology, quality cotrol, etc.). Various biostatistical ad egieerig applicatios cause to cosider differet forms of the classical chage poit problem, i.e., detectio ad estimatio of a sigle chage poit i the distributios of a sequece of idepedet radom variables. These problems are directly coected to process capability ad also importat i educatio, ecoomics, climatology ad other fields (James et al., 987; Gombay ad Horvath, 994, Gurevich ad Vexler, 005, 00, Gurevich et al., 0). Let X, X,..., X be a sample of idepedet oe-dimesioal observatios. I the formal cotext of hypotheses testig, the detectio chage poit problem is to test for () H 0, the ull:, X,..., X ~ F0 X versus H, the alterative: X i ~ F, X j ~ F, i,...,, j,...,, where F 0, F, F are distributio fuctios that correspod to desity fuctios f 0, f, f. The ukow parameter, is called a chage poit. Followig certai applied aspects of quality cotrol studies, the literature assumes mostly the fuctio F is equal to F 0 (e.g., James et al., 987; Gombay ad Horvath, 994). Gurevich ad Vexler (00) stated a geeral case whe F ca be differet from the ull distributio F 0. I accordace with the statistical literature, the problem () has bee ivestigated i parametric, semiparametric or oparametric forms, depedig o assumptios made o the distributio fuctios F 0, F ad F. I the parametric case of (), the distributio fuctios F 0, F ad F are assumed to have kow forms that ca cotai certai ukow parameters (e.g., Gurevich, 007; Gurevich ad Vexler, 00). I the semiparametric case of (), assumptios regardig the distributio fuctios F 0, F ad F are oly used to evaluate values of some parameters ivolvig i the test statistics costructio (e.g., Gurevich, 006; Gurevich ad Vexler, 00). Thus, i this case, the ull distributio of test statistics does ot deped of the assumptios related to the fuctios F 0, F ad F. I the oparametric case of (), the fuctios F 0, F, F are assumed to be completely ukow (e.g., Bhattacharyya ad Johso, 968; Se ad Srivastava, 975; Pettitt, 979; Gombay, 00; Gurevich, 009). 3

4 The 9 th Iteratioal Days of Statistics ad Ecoomics, Prague, September 0-, 05 The parametric ad semiparametric cases of the chage poit problem () have bee dealt with extesively i both the theoretical ad applied literature. However, we would like to avoid assumptios regardig distributios of cosidered observatios. Therefore, i this study we cocetrate o the oparametric form of the problem (). Whe the problem () is stated oparametrically i the cotext of oe-dimesioal idepedet radom observatios X,...,, X X, there is o uiversal powerful methodology for this subject. I this case, the commo compoets of chage poit detectio policies have bee proposed to be based o sigs ad/or raks ad/or U-statistics (e.g., Bhattacharyya ad Johso, 968; Pettitt, 979; Gombay, 00; Gurevich, 006, 009; Gurevich ad Vexler, 00). I additio, relatively recetly proposed empirical likelihood methodology was also successfully applied to the chage poit problem () (e.g., Vexler ad Gurevich, 00). Bhattacharyya ad Johso (968) cosidered the problem () with the ukow distributios x F x 0, F ( x) F ( x ), where 0 is ukow ad suggested F rejectig H 0 for large values of the statistics or J M () k, k k where M k k J U, (3) k, k k, the umber of observatios amog the last -k that exceeds the media of all observatios. That is, M k k - ad - k k k, U k, k I X i X j i, is the statistic of the media test for two samples of size jk, ( I is the idicator fuctio) is the Ma- Whitey statistic for two samples of size k - ad -k. For the same problem, without ay aalysis, Se ad Srivastava (975) suggested to reject H 0 for large values of the statistics D max M k k, k k ( k )( k ) / 4( ), (4) U ( k )( k ) / / ( k )( k )( ) / D max k, k k. (5) Cosiderig the same problem, Pettitt (979) proposed rejectig H 0 for large values of the statistic 4

5 The 9 th Iteratioal Days of Statistics ad Ecoomics, Prague, September 0-, 05 k K max U k, k ( k )( k ) /. (6) Thus, the statistics K ad D have a similar structure. Gombay (00) studied the asymptotic behavior of U-statistics. Gurevich (009) aalyzed the problem (), whe F0 F, F are ukow ad for all x, F x) F ( ) (that is, after a possible chage the observatios are ( x stochastically larger tha before the chage). The author modified the statistics J, D ad K (give by (3), (5) ad (6), respectively), ad suggested to reject H 0 for large values of the statistics k, k ( )( ) /, (7) k MK U k k MD U ( k )( k ) / k, k. (8) k ( k )( k )( ) / Sice the cosideratio of the operator sum istead of the operator max i defiitio of statistics (7) ad (8) allows evaluatig a accurate asymptotic behavior of these statistics uder the ull hypothesis H 0, these modificatios were itroduced to preset the followig asymptotic results where MK lim PH x x 0, (9) S MD lim PH x x 0 S, (0) x is the cumulative distributio fuctio of the stadard ormal distributio, x, k S ( k )( k ) ( k )( k r) k k r, S ( ) k k r ( k )( k r) ( k )( k r). Mote Carlo experimets preseted i Gurevich (009) showed that the rate of covergece of the asymptotic results (9), (0) is fast ad these results provide accurate approximatios for a level of sigificace of the tests based o the statistics (7), (8) for sample sizes commoly observed i practice. Moreover, a Broad Mote Carlo study preseted i Gurevich ad Vexler (00) cofirms that the oparametric chage poit tests based o the statistics J, J, D, D, K, MK, MD are powerful ad robust for various stochastically ordered alteratives. For / it seems that the test based o the statistics K ad MK have a higher power tha 5

6 The 9 th Iteratioal Days of Statistics ad Ecoomics, Prague, September 0-, 05 that based o the statistics D ad MD, but for that is close to edges (, ) this property is reversed. Whe the two-sided statemet (after a possible chage the observatios are stochastically larger or stochastically smaller tha before the chage) is assumed, the absolute values of the statistics (7), (8) as well as the absolute values uder the operator max i the statistics (4)-(6) should be cosidered. Whe H 0 is rejected, the issue of estimatig the ukow parameter ca be stated. While the literature o the chage poit relies maily o testig the hypotheses (), rather scat work has bee doe o the problem of estimatig the chage poit. Gurevich ad Raz (00) cosidered several oparametric chage poit estimators as the maximizig poit of the statistics (5), (6). They coducted a broad Mote Carlo study comparig the behavior of these estimators ad ivestigatig their properties. Simulatio results preseted i Gurevich ad Raz (00) cofirm the efficiecy of the proposed estimators eve for small ad average sample sizes. However, the performace of the estimators based o the statistic (5) seems to be slightly better tha that based o the statistic (6). Thus, for both of oe-sided alterative (whe oe assumes the observatios after a possible chage are stochastically larger or smaller tha before the chage) ad two-sided alterative the authors have recommeded the followig estimator based o the statistic (5): ˆ arg max U ( k )( k ) / / ( k )( k )( ) /. () k, k k Applicatio For the sake of the presetatio's simplicity we firstly itroduce i this Sectio the detailed aalysis of OECD s aual growth of GNI per capita, Atlas method (curret US$) betwee the years (log term) ad 00-0 (short term). After that, we preset the fial results (without all itermediate computatioal details) of the similar aalysis for the rest of the data.. A aalysis of OECD s aual growth of GNI per capita, Atlas method The followig Figure depicts the OECD s aual growths (i percet) of GNI per capita, Atlas method (curret US$) betwee the years 980-0, idicatig their sample mea ( ) ad stadard deviatio ( ). The chage poit tests described i Sectio. for a two-sided alterative based o statistics (7), (8) were applied. The followig values of the test statistics were straightforwardly obtaied: MK / S 0.87, MD / S.35, where S, S are defied 6

7 The 9 th Iteratioal Days of Statistics ad Ecoomics, Prague, September 0-, 05 i equatio (0). Therefore, by (9)-(0), two sided p-values of the tests based o statistics (7), (8) are equal to 0.38, 0.8, respectively. Sice both of p-values are larger tha 0.05, the coclusio is that there was ot a sigificat chage i distributio of the observatios. Fig. : OECD's aual growth of GNI per capita, Atlas method (980-0) Source: ow research. Similarly, the OECD s aual growths (i percet) of GNI per capita, Atlas method (curret US$) betwee the years 00-0 are preseted i Figure. For these observatios, the followig values of the test statistics were straightforwardly obtaied: MK / S.67, MD / S.6. By equatios (9)-(0), oe sided p-values of the tests based o statistics (7), Fig. : OECD's aual growth of GNI per capita, Atlas method (00-0) Source: ow research. (8) are equal to , , respectively. Cosequetly, sice the both of p-values are less tha 0.05 the coclusio is that there was a sigificat chage i the distributio of the 7

8 The 9 th Iteratioal Days of Statistics ad Ecoomics, Prague, September 0-, 05 observatios. Straightforwardly, usig the equatio (), ˆ 008 was obtaied for the cosidered data. Applyig the Wilcoxo test ad the Studet s t-test for two samples of observatios (the cosidered OECD s aual growths i the years ad 008-0), the p-values of the tests are foud to be less tha Thus, the fial coclusio is that at 008 there was a chage i OECD s aual growths of GNI per capita ad these aual growths after the chage are sigificatly smaller tha before the chage. The followig Tables, preset fial results (without all computatioal details) of the aalysis outlied above for aual growths (i percet) of the GNI per capita, Atlas method ad GNI Log term (980-0) Short term ( 00-0) Coutry Year of Total mea Mea before Mea after Year of Total mea Mea before Mea after Israel OECD members Turkey Greece Slovak Republic Czech Republic per capita, PPP for several cosidered coutries over idicated time periods. Table : Average aual growth of GNI per capita, Atlas method (Total mea), estimated chage poit (year of ), average aual growth of GNI per capita, Atlas method from the begiig of the period to year of (Mea before ) ad from year of to ed of the period (Mea after ) Source: ow research. Table demostrates that there was ot a chage i log-term tred of aual growth of GNI per capita, Atlas method i all cosidered coutries. However, at 008 there was a sigificat chage i short-term tred ad aual growth of GNI per capita, Atlas method i all cosidered coutries has sigificatly decreased last years comparig with several previous years. Table : Average aual growth of GNI per capita, PPP (Total mea), estimated chage poit (year of ), average aual growth of GNI per capita, PPP from the begiig of the period to year of (Mea before ) ad from year of to ed of the period (Mea after ) Coutry Medium term (990-0) Year of Total mea Mea before Mea after Israel OECD members

9 The 9 th Iteratioal Days of Statistics ad Ecoomics, Prague, September 0-, 05 Turkey Greece Slovak Republic Czech Republic Source: ow research. Table shows that there was ot a sigificat chage i medium-term tred of aual growth of GNI per capita, PPP i Israel, Turkey ad Slovak Republic. However, the aual growth of GNI per capita, PPP i Greece, Czech Republic ad OECD members has sigificatly decreased. Results of the similar chage poit aalysis of GNI per capita, Atlas method (curret US$) ad GNI per capita, PPP (curret iteratioal $) i Israel relative to Frace, OECD members, USA, Caada ad Germay are preseted i followig Tables 3,4. Table 3. Aual GNI per capita, Atlas method Israel relative to Source: ow research. Coutry Log term (980-03) Year of Total mea Mea before Mea after Frace OECD members USA Caada Germay Aalyzig the results provided i Table 3 it ca be cocluded that betwee the years of 99 to 998 the aual GNI per capita, Atlas method i Israel has sigificatly icreased with respect to the cosidered coutries. Table 4. Aual GNI per capita, PPP Israel relative to Source: ow research. Coutry Medium term (990-03) Year of Total mea Mea before Mea after Frace OECD members USA Caada Germay Table 4 demostrates that there was ot a sigificat chage i medium term tred of the aual GNI per capita, PPP i Israel with respect to the cosidered coutries. 9

10 The 9 th Iteratioal Days of Statistics ad Ecoomics, Prague, September 0-, 05 Coclusio A recetly developed oparametric chage poit detectio ad estimatio methods have bee preseted. The step by step applicatio of this techique has bee outlied as well as traditioal statistical tools to study a tred behavior of the GNI per capita, Atlas method (curret US$) ad GNI per capita, PPP (curret iteratioal $) i Israel ad other selected coutries. The preseted aalysis cofirms the practical applicability of the chage poit methodology that ca be useful for forecastig a short-term ad log-term ecoomic growth. Refereces Bhattacharyya, G. K., Johso, R. A. (968). Noparametric tests for shift at a ukow time poit. Aals of Mathematical Statistics, 39, Gombay, E., Horvath, L. (994). A applicatio of the maximum likelihood test to the chage-poit problem, Stochastic Processes ad their Applicatios, 50, 6-7. Gombay, E. (00). U-statistics for Chage uder Alteratives. Joural of Multivariate Aalysis, 78, Gurevich, G., Vexler, A. (005). Chage Poit Problems i the Model of Logistic Regressio, Joural of Statistical Plaig ad Iferece, 3, Gurevich, G. (006). Noparametric AMOC Chagepoit Tests for Stochastically Ordered Alteratives. Commuicatios i Statistics - Theory ad Methods, 35, Gurevich, G. (007). Retrospective Parametric Tests for Homogeeity of Data. Commuicatios i Statistics-Theory ad Methods, 36, Gurevich, G. (009). Asymptotic distributio of Ma-Whitey type statistics for oparametric chage poit problems. Computer Modellig ad New Techologies, 3, 8-6. Gurevich, G., Vexler, A. (00). Retrospective chage poit detectio: from parametric to distributio free policies. Commuicatios i Statistics-Simulatio ad Computatio, 39, Gurevich, G., Raz, B. (00). Mote Carlo aalysis of chage poit estimators. Joural of Applied Quatitative Methods, 5, Gurevich, G., Hadad, Y., Ofir, A., Ohayo, B. (0). Statistical aalysis of temperature chages i Israel: a applicatio of chage poit detectio ad estimatio techiques. Global Nest Joural, 3, 5-8. James, B., James, K.L., Siegmud, D. (987). Tests for a chage-poit. Biometrika, 74,

11 The 9 th Iteratioal Days of Statistics ad Ecoomics, Prague, September 0-, 05 Pettitt, A.N. (979). A o-parametric approach to the chage-poit problem. Applied Statistics, 8, Se, A., Srivastava, M. S. (975). O tests for detectig chage i mea. Aals of Statistics, 3, Vexler, A., Gurevich, G. (00). Desity-based empirical likelihood ratio chage poit detectio policies. Commuicatios i Statistics-Simulatio ad Computatio, 39, Cotact Lia Alatawa SCE-Shamoo College of Egieerig Bialik Sts. 56, Beer Sheva 8400, Israel liaal@ac.sce.ac.il Yossi Yacu SCE-Shamoo College of Egieerig Bialik Sts. 56, Beer Sheva 8400, Israel yossy@ac.sce.ac.il Gregory Gurevich SCE-Shamoo College of Egieerig Bialik Sts. 56, Beer Sheva 8400, Israel gregoryg@sce.ac.il