Measuring macroeconomic volatility

Transcription

1 FONDATION POUR LES ETUDES ET RECHERCHES SUR LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, by Joël Cariolle Working paper n I14 Innovaive indicaors series March 2012

2 Measuring macroeconomic volailiy: applicaions o expor revenue daa, Joe l Cariolle Summary The lieraure on macroeconomic volailiy covers an exremely wide field, as evidenced by he very broad specrum of indicaors used o grasp i as a phenomenon. In general here seems o be lile discussion abou he choice of indicaor for macroeconomic volailiy, on he grounds ha he differen mehods, based on saionary series, give rise o scores which are closely correlaed. Alhough hese mehods may converge when analysing only he average magniude of disribuion around a reference value, however, hey diverge significanly when one examines eiher asymmery or kurosis (insances of exreme deviaion). This aricle reviews he exising lieraure and ses ou he principal mehods used for calculaing volailiy. These mehods are compared and heir properies analysed based on expor revenue daa for 134 counries from 1970 o In paricular, a disincion is drawn beween measuremens of he magniude of volailiy and measuremens of asymmery and he incidence of exreme deviaions. Foundaion for Inernaional Developmen Sudy and Research, 63 Bvd François Mierrand, Clermon- Ferrand, France. I would like o hank Parick Guillaumon, Michaël Goujon, Jean-Louis Combes and Lauren Wagner for heir pracical suppor and commens. Page 1

3 TABLE OF CONTENTS 1. Inroducion Macroeconomic volailiy and developmen The coss of volailiy... 4 Inernal economic condiions resuling in greaer vulnerabiliy o economic volailiy The origins of macroeconomic volailiy Measures of economic volailiy: a variey of indicaors Economic volailiy as he sandard deviaion of he growh rae of a variable... 9 Economic volailiy as he sandard deviaion of he residual of an economeric regression... 9 Economic volailiy as he sandard deviaion of he cycle isolaed by a saisical filer The quesion of reference values Breakdown of economic series Observaion of series behaviour Theoreical breakdown of series Volailiy as a measure of variabiliy, risk or uncerainy? Parameric approach Esimae based on a linear deerminisic rend Esimae based on a mixed rend Esimae based on a rolling mixed rend Filering approach Calculaing deviaion from he rend Calculaion period for volailiy The magniude of volailiy The asymmery of volailiy Frequency of exreme deviaions Conclusion Bibliography Appendices FIGURES Figure 1. Expor series and specrum densiies for Souh Korea, Argenina, Venezuela, Kenya, Ivory Coas and Burundi Figure 2. Change in expor revenues, simple linear rend Figure 3. Residual of linear rend Figure 4. Change in expor revenues, mixed rend Figure 5. Correlogram of residuals, mixed rend Figure 6. Rolling mixed rend, comparaive change and correlogram of residuals, Argenina Figure 7. Global and rolling mixed rends, comparaive change and correlogram of residuals, Burundi Figure 8. Expor revenues smoohed by HP filer Figure 9. Magniude of volailiy deviaions (as % of rend) for he las five s around a mixed rend Figure 10. Graphical illusraion of correlaions beween magniude of volailiy indicaors Page 2

4 Figure 11 Graphical illusraion of a posiive and negaive asymmeric disribuion of idenical magniude Figure 12. Correlaion beween magniude and asymmery of volailiy indicaors, by reference value Figure 13. Graphic represenaion of he degree of flaening of a disribuion Figure 14. Graphical illusraion of correlaions beween magniude and kurosis of volailiy indicaors, by reference value Figure 15. Correlaion beween asymmery and kurosis of volailiy indicaors, by reference value TABLES Table 1: Specificaion and uni roo es on panel daa Table 2. Correlaion beween magniude of volailiy indicaors Table 3. Descripive saisics of magniude of volailiy indicaors (%) Table 4. Correlaions of coefficiens of asymmery calculaed over he period Table 5. Descripive saisics for kurosis calculaed for he period Table 6. Correlaions in kurosis calculaed over he period Table 7. Correlaions beween kurosis and coefficien of asymmery for each reference value, APPENDICES Appendix A.1 Overview of indicaors of volailiy and heir applicaion in he lieraure Appendix A.2. Mixed rends and correlogram of residuals Appendix A.3. Correlogram of expor revenue cycles smoohed by he HP filer Appendix A.4 Comparaive evoluion of deviaions (in % of rend values) Page 3

5 1. Inroducion The numerous global economic crises of he 20h cenury have made macroeconomic volailiy a key issue in analysing he deerminans of economic growh. The mulipliciy of ways in which i affecs he long-erm growh poenial of economies, is diverse causes and he array of mehods by which i is measured, make economic volailiy a complex and mulidimensional phenomenon. We herefore consider he erm volailiy as a generic erm, combining all he echniques available for measuring economic flucuaions Macroeconomic volailiy and developmen. The lieraure provides an exensive analysis of he coss and consequences of macroeconomic volailiy. Alhough he posiive relaionship beween risk and capial yield may, under cerain condiions, explain a posiive relaionship beween economic volailiy and growh, mos research agrees ha his phenomenon has a negaive impac on long-erm growh and well-being. Indeed, over he long erm, volailiy conribues o a reducion in levels of consumpion, invesmen and facor produciviy, o an increase in he volailiy and unpredicabiliy of economic policy, and o deerioraion in he insiuional environmen. The effecs on performance are even more marked in developing counries, which are ofen subjeced o more significan exernal shocks bu which do no enjoy he inernal condiions ha would allow hem o absorb hem more easily. Hnakovska and Loayza (2005), Aizenman and Pino (2005) and Loayza e al (2007) offer an exhausive overview of he consequences of macroeconomic volailiy and he facors ha cause i. The coss of volailiy Macroeconomic volailiy is a major obsacle o growh. According o esimaes produced by Hnakovska and Loayza (2005), based on a sample of 79 counries, increasing he average value of volailiy by he value of is sandard deviaion resuls in an average loss of 1.3 poins for growh in GDP over he period , and 2.2 poins for he decade Volailiy can, indeed, ac as an obsacle o he key facors in economic and social developmen. An early series of research aricles examined he impac of macroeconomic volailiy on growh from he poin of view of invesmen or producion facor produciviy. Dawe (1996) analyses he effec of volailiy in expors on invesmen and growh. He finds boh a posiive effec of volailiy on invesmen hrough an increase in precauionary savings, and a negaive effec on growh hrough an allocaion of capial o secors wih lower yields. Dehn (2000), on he oher hand, idenifies a significan and negaive impac of shocks in he price of raw maerials on invesmen in developing counries. Guillaumon, Guillaumon-Jeanneney, and Brun (1999) highligh a negaive effec of volailiy in he rae of invesmen on growh, based on a decline in average produciviy. In a similar area, Koren and Tenreyro (2006) empirically es a heoreical model where developmen is accompanied by increased diversificaion of inpus ino he producion sysem, hus reducing he effec of volailiy in world prices on producion facor produciviy and herefore on growh. Finally, Combes and Guillaumon (2002) show ha volailiy in relaion o erms of rade has a Page 4

6 negaive impac on he growh rae of capial produciviy. Macroeconomic volailiy herefore seems o be an obsacle o economic growh insofar as i discourages invesmen decisions, has a negaive effec on facor produciviy and divers capial from he mos producive secors. Oher sudies analyse he impac of macroeconomic volailiy on growh and well-being hrough is effec on he qualiy of economic policy. Easerly e al (1993) show ha posiive shocks in relaion o erms of rade influence he long-erm growh pah of economies, in par hrough an improvemen in economic policy. Ramey and Ramey (1995) show ha he unpredicabiliy of economic policy caused by volailiy in growh raes has a negaive effec on he average growh rae of he economy. Guillaumon and Combes (2002) show ha vulnerabiliy o volailiy in global prices has a negaive effec on he qualiy of economic policy and growh. To ake wo oher examples, boh Faas and Mihov (2006, 2007) and Afonso and Furceri (2010) have emphasised he negaive impac of variabiliy in budge policy on growh in boh OECD counries and developing naions. The negaive effec of macroeconomic volailiy on growh and well-being, based on volailiy in public and privae consumpion, has also been examined in a number of sudies. Aizenman and Pino (2005) and Wolf (2005) poin ou ha in he case of imperfec financial markes, he Sae and individual households are unable o proec hemselves fully agains risks which affec heir revenue and adjus heir consumpion o he vagaries of economic aciviy. The resul is ha volailiy driven by exernal facors, for example in relaion o erms of rade, generaes inernal volailiy in relaion o consumpion, paricularly in developing counries (Aguiar and Gopinah, 2007; Loayza e al, 2007). Economic policy and consumpion herefore appear o be inernal channels by which macroeconomic volailiy is ransmied and may even be magnified, wih a concomian negaive effec on growh and developmen. Inernal economic condiions resuling in greaer vulnerabiliy o economic volailiy. One posiive effec of volailiy on growh has already been menioned (Imbs, 2007; Rancière e al, 2008; Hnakovska and Loayza, 2005). This can be explained primarily by he posiive correlaion beween risk and reurn on invesmen projecs. Hnakovska and Loayza (2005) sugges, however, ha a posiive effec of his kind is dependen on he exisence of risk-sharing mechanisms and respec for righs of ownership, which are in urn suppored by a well-developed financial sysem and high-qualiy insiuions. A counry s vulnerabiliy o macroeconomic volailiy is herefore driven by a number of handicaps, which are eiher srucural or depend on he level of economic developmen. These facors explain why, in general erms, developing counries are more vulnerable o macroeconomic volailiy. Developing counries are more exposed o shocks, and do no always have he mechanisms or inernal condiions in place o enable hem o absorb hem. The size of he populaion, he degree of diversificaion of he economy and he capaciy for operaing a counercyclical economic policy, he exisence of well-developed financial insiuions and insiuional qualiy are herefore deermining facors in he impac of volailiy on growh. The lieraure on economic vulnerabiliy has made a significan conribuion o our undersanding Page 5

7 of he inernal and exernal condiions of vulnerabiliy o shocks (Guillaumon, 2007, 2009a, 2009b, 2010; Cariolle, 2011; Loayza and Raddaz, 2007; Combes and Guillaumon, 2002). The research carried ou disinguishes srucural facors in relaion o vulnerabiliy from more ransiory facors linked o economic policy (or resilience ). As far as srucural facors are concerned, a disincion is drawn beween he magniude and frequency of shocks (commercial or naural) and exposure o such shocks. Facors ha affec exposure o shocks (such as he size of he populaion, he degree of economic diversificaion, disance from global markes and geographical isolaion) increase he propensiy of economies o suffer shocks and he negaive impac of such shocks on growh. A recen sudy by Malik and Temple (2009), on he srucural deerminans of volailiy in relaion o growh, suggess a negaive relaionship beween access o global markes and macroeconomic volailiy. According o he auhors, counries which are isolaed from global markes end o lack diversiy in erms of expors and experience greaer volailiy in relaion o GDP. As o he resilience of paricular counries, alhough his is o some exen dependen on srucural facors, i is linked primarily o economic policy and insiuions. As a resul, developmen sraegies wih a focus on foreign rade, he procyclicaliy and counercyclicaliy of economic policy and he qualiy of governance and democraic insiuions (Rodrik, 1998; 2000) can deermine boh he magniude of volailiy experienced by counries and is effec on heir developmen. The conribuion made by foreign rade o macroeconomic volailiy is one of he hemes mos commonly addressed in he lieraure. Whils he role played by openness o rade has no been clearly esablished (Combes and Guillaumon, 2002), several pieces of research have examined he relaionship beween a counry s degree of specialisaion, is level of developmen and macroeconomic volailiy. Di Giovanni and Levchenko (2010) sudy he exen o which openness o rade can resul in a specialisaion of expor secors which are highly exposed o exernal shocks, and hus o increased economic volailiy. They show ha counries wih a low or moderae comparaive advanage in high-risk expor secors diversify heir economy in order o aenuae he risk affecing heir expor revenues. Conversely, counries wih a very high level of comparaive advanage in hese secors end o specialise in hem, and are hus more exposed o volailiy in relaion o erms of rade, expors and GDP growh per head. Similarly, Koren and Tenreyro (2006, 2007) show ha poor counries specialise in a limied number of secors, wih relaively simple producion echnologies and a limied range of inpus, and are herefore more vulnerable o shocks in global prices. Developmen is herefore suppored by diversificaion ino secors based on more complex echnologies, using a wider variey of inpus, and wih less exposure o macroeconomic volailiy. Van der Ploeg and Poelhekke (2009) find ha growh volailiy, driven by volailiy in global raw maerials prices, is he main deerminan of he naural resource curse. Having conrolled for growh volailiy, hey show ha supplies of naural resources have a posiive and significan effec on economic growh. Developmen is hus generally accompanied by economic diversificaion and by specialising in secors which are less exposed o global volailiy. Conversely, he negaive effec of a low level of economic diversificaion on developmen is assumed o be dependen on a high level of exposure o volailiy in global prices. Page 6

8 Oher sudies have examined he qualiy of insiuions in aenuaing or conribuing o economic volailiy. Acemoglu e al (2003) showed ha poor insiuions resul in poor governance, which in urn conribues o macroeconomic volailiy. Mobarak (2005) finds ha democracy reduces volailiy hrough increased scruiny by ciizens of he managemen of economic policy. These resuls reflec hose found by Rodrik (2000), according o which democraic poliical srucures encourage poliical consensus around poliical responses o exernal shocks. Numerous research aricles have also examined he role of he developmen of financial markes in ransmiing macroeconomic volailiy (Beck e al, 2006; Aghion e al, 2005; Aghion e al, 2004). In general, financial developmen herefore ends o aenuae shocks, alhough i seems o be able o magnify he effec of shocks of moneary origin on volailiy in GDP (Beck e al, 2006). The qualiy of poliical, economic and financial insiuions herefore seems o be boh a source and a vecor of macroeconomic volailiy. The origins of macroeconomic volailiy Anoher se of lieraure examines he sources of macroeconomic volailiy. This research generally draws a disincion beween exernal forms of volailiy (expors, global prices, erms of rade or inernaional ineres raes) and inernal forms (such as economic policy, agriculural producion and naural or climaic disasers). Similarly, i is possible o disinguish beween exogenous sources of macroeconomic volailiy (relaed o inernaional rade, agriculural producion and naural disasers) and endogenous sources (linked o volailiy in economic policy or domesic sociopoliical condiions). Finally, several sudies draw a disincion beween so-called normal flucuaions and crisis flucuaions, he magniude of which exceeds a paricular hreshold (Rancière e al., 2008; Hnakovska and Loayza, 2005). The lieraure on he economic vulnerabiliy of developing counries emphasises he significan conribuion made by he magniude and frequency of exernal and naural shocks o he srucural vulnerabiliy of developing counries (Guillaumon, 2007, 2009a, 2009b, 2010; Cariolle, 2011; Loayza and Raddaz, 2007; Combes and Guillaumon, 2002). Research by Mauro and Becker (2006) idenifies he exernal shocks ha cause growh shocks, such as a deerioraion in relaion o erms of rade and a sudden hal o he movemen of capial. Similarly, Raddaz (2007) examines he sources of volailiy in GDP in he Leas Developed Counries. He ses ou an analysis of he oal and relaive conribuion of exernal shocks o volailiy in GDP based on a breakdown of GDP variance. Raddaz shows ha exernal shocks (erms of rade, price of primary producs, LIBOR and developmen aid) have only a marginal effec on he volailiy of he growh rae of GDP, whils inernal facors relaed o economic policy (level of public defici, inflaion and overvaluaion of exchange raes) make a significan conribuion. These resuls confirm hose of Faas and Mihov (2006, 2007), showing ha economic volailiy resuls in par from volailiy in budge policy. Finally, Hnakovska and Loayza (2005) disinguish beween he effec on growh of normal flucuaions in GDP (posiive or negaive, repeaed and on an average scale) and he effec of crises measured by falls in GDP over a cerain hreshold. The simulaneous inroducion of hese wo volailiy variables in heir regressions shows he imporan and significan effec of a crisis Page 7

9 level of volailiy on growh and he insignifican effec of normal volailiy. The same classificaion of volailiy was used by Rancière e al (2008) and applied o he effec of a financial sysemic risk on growh. They find a posiive relaionship beween financial crisis and growh, which is explained by he leverage effec of companies level of indebedness on heir invesmens. The lieraure on macroeconomic volailiy herefore covers somehing of a wide field, showing he significan ineres in undersanding i and is decisive role on economic performance in relaion o growh. As we will see below, however, here is a diverse range of mehods available for measuring volailiy, alhough here has been no real discussion of he advanages and disadvanages of such mehods since he research carried ou by Gelb (1979) and Tsui (1988) Measures of economic volailiy: a variey of indicaors. Common definiions of volailiy ofen refer o he noion of disequilibrium. Measuring economic volailiy involves evaluaing he deviaion beween he values of an economic variable and is equilibrium value. This equilibrium value, or reference value, in urn refers o he exisence of a permanen sae or rend. In saisical erms, economic volailiy is radiionally measured by he second (sandard deviaion) or someimes a higher momen 1 (Rancière e al, 2008), of he disribuion of a variable around is mean or a rend, which hen represens he equilibrium value (o which he variable ends o reurn quickly afer deviaing in response o a shock). I is frequen for macroeconomic series (GDP, expor revenues, final consumpion) o be non-saionary, i.e. hey flucuae around a rend which iself varies over ime, or for shocks o make he variable deviae from is previous endency over he long erm or permanenly. I hen becomes necessary o use so-called saionarisaion echniques in order o separae he permanen (or rend) componen from he ransiory (or residual) componen of he evoluion of a series (see secion 2.1). The volailiy indicaors obained using hese echniques are inended o reflec he effecs of he episodic variaions of an economic series around a reference value (mean, deerminisic rend or variable over ime). Calculaing volailiy hus relies on wo key quesions: ha of calculaing a reference value or he choice of a saionarisaion mehod and measuring flucuaions around said reference value. A disincion can be drawn beween wo main families of volailiy indicaors: on he one hand, hose which measure he variabiliy of an economic series, i.e. aking ino accoun all of he ransiory variaions of a saisical series, and on he oher, hose which measure economic uncerainy, or he unpredicabiliy of variaions in oal variabiliy (Wolf, 2005). We se ou below a review of he main indicaors of economic volailiy as presened in he lieraure, noing how reference values and flucuaions have been calculaed by various auhors. A summary able of hese indicaors can be found in Appendix A.1. 1Asymmery and kurosis coefficiens of financial values are ofen used as measures of risk in he financial secor. In economics, only Rancière e al (2008) have, o our knowledge, incorporaed he effecs of economic volailiy ino heir research. Page 8

10 Economic volailiy as he sandard deviaion of he growh rae of a variable Mos of he research proposes measuring volailiy on he basis of he sandard deviaion of he growh rae of a variable, which assumes ha said variable is saionary a firs difference. In oher words, his approach pus forward resricive hypoheses as o he behaviour of a series wihou any prior esing. Ramey and Ramey (1995), for example, propose sudying he effec of economic variabiliy using he sandard deviaion of he growh rae of GDP per-capia. Servén (1997) examines he effecs of volailiy on invesmen in sub-saharan Africa and uses wo measures of macroeconomic volailiy, namely he sandard deviaion and coefficien of variaion of several aggregaes 2 (erms of rade, black-marke premium, inflaion, ec.). Acemoglu e al (2003) sudy he effec of insiuional qualiy on macroeconomic volailiy and measure he laer using sandard deviaions of GDP growh raes and erms of rade. Similarly, Di Giovanni and Levchenko (2010), and Van der Ploeg and Poelhekke (2009) examine he effecs of a high level of exposure o exernal shocks and measure macroeconomic volailiy using he sandard deviaion of he growh raes of erms of rade, GDP per inhabian and expors. Raddaz (2007) also uses measures based on he sandard deviaion of he growh rae of several macroeconomic variables (price of primary producs, erms of rade, aid per inhabian, GDP per inhabian and LIBOR) o examine he conribuion of exernal shocks o he volailiy of GDP in African counries. I is hus common o apply measures of volailiy based on variance (sandard deviaion or coefficien of variaion) o differeniaed series such as GDP, erms of rade, expor revenues, prices of goods or inernaional ineres raes. Economic volailiy as he sandard deviaion of he residual of an economeric regression Oher measures of volailiy are based on he residual or explanaory power of economeric regressions. Priche (2000) proposes hree measures of volailiy. The firs is based on he coefficien of deerminaion of a growh-rae regression on a linear emporal rend. The lower he coefficien of deerminaion, he more he explanaory power of he emporal rend is limied and he greaer he level of volailiy. The second measure is based on he difference in growh raes before and afer a break idenified by minimising he sum of he squares of he residuals of a regression on a simple linear rend 3. The auhor also proposes a measure of economic volailiy based on calculaing he sandard deviaion of he residual of a regression of GDP on a mixed deerminisic and sochasic rend, along wih Servén (1998), Combes and Guillaumon (2002), and Guillaumon and Chauve (2007) 4. In a similar approach, Lensink and Morrissey (2006) examine he 2 See secion 3.2 for sandard deviaion and coefficien of variaion formulae. 3 y 1 = a 1I 1 ( < *) + b 1*I 1( < *) + a 11*I( > *)+ b 11 *I 11( > *) +e Where I() is an indicaive funcion, and he break, *, is chosen o minimise he sum of he square of residuals, and * - o 6 and T Amongs oher hings, Servén (1998) uses a specificaion similar o ha used by Combes and Guillaumon (2002), o measure uncerainy, imposing a nulliy consrain on he coefficien associaed wih a second lag. Page 9

11 effec on economic growh of he volailiy of Foreign Direc Invesmens (FDI) as measured by he sandard deviaion of he residual of a regression of FDI on is hree lags and a deerminisic rend. Oher auhors have ended o concenrae more on he effecs of economic uncerainy on invesmen and growh. Ramey and Ramey (1995) propose a measure of he uncerainy componen of volailiy based on he sandard deviaion for predicion error agains he growh rae (he prediced value being obained by a growh rae regression agains a quadraic rend, linear rend, wo GDP lags and he iniial values of he share of invesmen in GDP, populaion and human capial) 5. Servén (1998) and Dehn (2000) in urn examined he effec of economic uncerainy on invesmen using several measures of uncerainy generaed by price volailiy in raw maerials and calculaing he condiional sandard deviaion for predicion error for hese series obained using a GARCH process (1,1). Volailiy measures based on he residual of economeric regressions hus have he meri of being based on a less resricive formalisaion of he process underlying he change in he rend of economic series. Noneheless, i remains o be seen wheher said formalisaion allows proper series saionarisaion, and wheher he inerpreaion of he residual is correc (uncerainy or variabiliy?). Economic volailiy as he sandard deviaion of he cycle isolaed by a saisical filer Finally, several sudies have used he filered value of a saisical series as a reference value. This echnique can be used o disaggregae a series ino rend variaions (long erm) and cyclical variaions (shor erm). This ype of volailiy indicaor is herefore based on cyclical or cycle flucuaions. The filering echnique is differen from he previous wo mehods insofar as i does no formulae he behaviour of a series in advance (order of inegraion, difference-saionariy or rend-saionariy) and filers series on he basis of heir pas and fuure behaviour. Secion 2.3 examines his echnique in more deail. Dawe (1996) hus filers expor series using a moving average based on five s, i.e. on [-2;+2], and bases his volailiy measuremen on he average difference beween he series observed and his moving average. Oher auhors use he Hodrick-Presco (Hodrick and Presco, 1997) filer o calculae heir volailiy indicaor; hey include Chauve and Guillaumon (2007), who use he sandard deviaion of he developmen aid cycle isolaed using his filer (HP), i.e. he sandard deviaion of he difference beween he value smoohed by he filer and he observed value of heir aid variable. Becker and Mauro (2006) propose a shock variable based on he HP filer and idenify decreases in GDP as an even equaing o a decline in filered GDP of over 7%. Hnakovska and Loayza (2005) isolae he cyclical flucuaions in GDP series using he Baxer and King filer (1999) and calculae heir sandard deviaion. Finally, Afonso and Furceri (2010) use he sandard deviaion of he cyclical componen of public spending and ax revenues isolaed using HP and Baxer-King filers. 5The auhors regress he following model for he whole of he sample for each counry: y i 2 = α + α + α + α + ϕ dummy + β y + γy + λ I + δpop + θkhum + ε i 1 i 2 iniial i iniial i iniial i i Page 10

12 The usual indicaors of economic volailiy can herefore be disinguished by he mehod used o calculae he reference value seleced. I is hus possible o draw a disincion beween indicaors based on firs-difference series variance, indicaors based on he variance of he residual of a more complex model of economic series, and hose based on cyclical variance idenified by applying saisical filers. These disincions become more blurred, however, wih regard o he mehod used o calculae deviaions from he reference value, since he lieraure limis he analysis of volailiy o an analysis of he variance of volailiy, i.e. he average magniude of flucuaions 6. In he following secion, we presen an analysis of he various mehods used for calculaing a reference value based on expor daa for he period and illusrae he differences in he analysis depending on he mehod used. In secion hree, we ouline various ways of characerising he flucuaions of a variable around a reference value, showing ha i is possible o quanify volailiy no only by he average magniude of economic flucuaions, bu also in erms of heir asymmery and he occurrence of exreme variaions (or kurosis). 6 Excep for Rancière e al (2008). Page 11

13 2. The quesion of reference values If volailiy refers o he noion of disequilibrium, hen i mus be measured using saionary series. Mos economic aggregaes, however, are no naurally saionary. I herefore becomes necessary o calculae a reference value or rend value, around which series will be saionary, which in urn means idenifying he righ mehod of saionarisaion. Or y = µ + ε,, a non-saionary process wih µ a non-consan erm and ε he residual. Saionarising y consiss of calculaing or esimaing he rend componen µ so ha he residual (or cycle) ε mees he following condiions 7 : (1) E(ε ) = 0; (2) V(ε )= σ ε 2 < for all of ; and (3) Cov(ε ; ε -k) = ρ k. These hree condiions require he residual ε o have a zero mean (1), a variance (2) and a finie auocovariance (3) independen of ime. Volailiy measures are based on he residual (or cycle) ε,, which herefore reflecs only ransiory flucuaions. If he series is poorly saionarised, variaions which are aribuable o a long-erm (or permanen) change in y may be included in he residual, hus breaching condiions (1), (2), and (3). The volailiy measures based on hem would herefore be incorrec, because hey do no correspond wih he definiion given o hem. Saionarisaion is a prerequisie condiion for calculaing volailiy based on non-saionary level series. Calculaing a reference value is herefore a fundamenal sep, since i resuls in idenifying and isolaing he rend or permanen componen in he change of an economic variable, from is ransiory or saionary componen (Dehn, 2000; Hnakovska, 2005). To undersand his issue in more deail, we firs examine he heoreical breakdown of change in saisical series. We hen presen he principles and properies of he usual mehods for calculaing reference values, which we apply o expor daa. We illusrae our analysis using he annual change in expor revenues for 134 (developed and developing) counries over he period from he World Developmen Indicaors 8. The advanage of using hese series is ha expor volailiy is an imporan aspec of macroeconomic volailiy, which is addressed in deph in he lieraure on economic volailiy. Flucuaions in expor revenues may reflec boh he change in domesic (changes in domesic producion condiions, naural disasers, ec.) and inernaional economic condiions (volailiy in global prices). Wihin his framework, we presen sandard echniques for series saionarisaion and calculaing he volailiy of expors applicable o a broad range of developed and developing counries over he period We are referring here o condiions of weak saionariy. 8 hp://daa.worldbank.org/daa-caalog/world-developmen-indicaors Page 12

14 2.1. Breakdown of economic series Observaion of series behaviour Figure 1 shows he changes in expor revenues in consan dollars (2000) in six differen counries (Souh Korea, Argenina, Venezuela, Kenya, Ivory Coas and Burundi) and he specrum densiies associaed wih hem. The specrum of a series is a represenaion of he conribuions of each frequency variaion 9 o he oal variaion of he series. Observing he specrum densiy of a series hen makes i possible o idenify wheher he change in a series is dominaed by variaions over a longer period or shorer period. Examining he specrum is herefore a very useful diagnosic ool if we wish o represen correcly he dynamics of change in a series. A peak a a given frequency indicaes ha a significan proporion of he oal variance in a series can be explained by he variaions in said frequency. Figure 1 shows ha he counries represened had a change in expors dominaed by variaions over a long period, wih an increasing rend (excluding Burundi) and a decreasing specrum densiy. Excep for Souh Korea 10, a large proporion of he series specrum is locaed around variaions in periodiciy of around 20 s (wih a peak in densiy a a frequency of around 0.05). I all appears ha variaions in average periodiciy conribue srongly o oal variabiliy for hese sample counries. In paricular, variaions beween 5 and 7 s (wih a frequency beween 0.1 and 0.3) seem o conribue subsanially o oal variabiliy in series from Venezuela, Ivory Coas and Burundi. This reflecs he conclusions reached by Rand and Tarp (2002), and Aguiar and Gopinah (2007), according o whom developing economies experience greaer volailiy in heir rae of economic growh han developed counries. Finally, an examinaion of he specrum for he change of expors can be used o idenify he exisence of peaks of densiy a high frequencies corresponding o periodiciies of around 2-3 s, which sugges ha he oal variabiliy of he series used can also be explained by flucuaions over shor periods. Examining he changes in expor series for a sample of counries herefore suggess ha, alhough variaions in expor series over a long period explain mos of heir variabiliy, flucuaions over a medium period also play a par. The choice of a reference value is herefore imporan, since his makes i possible o disinguish rend variaions (long/medium periodiciy) from he shor-erm ransiory variaions on which volailiy calculaions are based. A subsequen heoreical breakdown of economic series provides addiional informaion for undersanding changes in saisical series. 9The calculaion for swiching from frequency o period is as follows: F=1/T, where F = frequency and T = period. 10Souh Korea presens a Granger profile (Granger, 1966), wih mos of he power of he specrum close o a zero frequency. Page 13

15 Figure 1. Expor series and specrum densiies for Souh Korea, Argenina, Venezuela, Kenya, Ivory Coas and Burundi. Change in expors (USD, 2000) Expors 5.086e e+11 Souh Korea Expors 4.250e e+10 Argenina Expors 1.140e e+10 Venezuela Expors 1.746e e+09 Kenya Expors e+10 Ivory Coas Expors e+08 Burundi Specrum densiies Log Periodogram Souh Korea Log Periodogram Argenina Log Periodogram Venezuela Frequency Frequency Frequency Log Periodogram Kenya Log Periodogram Ivory Coas Log Periodogram Burundi Frequency Frequency Frequency Theoreical breakdown of series According o Dehn (2000) and Hnakovska (2005), economic series have a rend or permanen (y P ) componen and a cyclical or ransiory (y C ) componen: Page 14

16 y = y P + y C (1) The permanen componen is made up of a deerminisic par (y0+α), wih a emporal rend 11, and a sochasic (ε P ) par: y P = y 0 + α + ε P (2) ε P represen sochasic shocks affecing he series rend on a permanen or prolonged basis. By way of example, produciviy shocks or a change in he preferences of economic agens can affec he change in an economic series over he long erm. The cyclical or ransiory componen comprises a predicable (y CP componen associaed wih srucural facors such as he level of developmen, foreign exchange sysem, he size of he counry, ec.) and an unpredicable (ε C ) componen: y C = y CP + ε C (3) ε C represens unpredicable shocks wih a emporary effec on he series cycle, such as sudden changes in inernaional prices of raw maerials or unforeseen climaic evens. Volailiy as a measure of variabiliy, risk or uncerainy? As Azeinman and Pino (2005) sugges, he lieraure generally sees volailiy as associaed wih economic risk or uncerainy 12. According o he auhors, whils volailiy provides informaion on he observed resuls of a variable, i can also, by exension, provide informaion on possible resuls and hus represen an approximaion of he risk associaed wih i. The volailiy indicaors presened in secion 1.2 would hen be similar o risk indicaors. The auhors emphasise, however, ha such a measure can overesimae risk by also including predicable flucuaions. Pure risk or uncerainy would hen need o be measured by he residual obained from a volailiy predicion model, e.g. condiional variance models such as GARCH models (Dehn e al, 2005; Dehn, 2000; Serven, 1998). Techniques for measuring volailiy would herefore be divided ino wo main families, namely hose which provide measures of he oal variabiliy of a series, and hose which provide measures of uncerainy or risk (Wolf, 2005). Models based on uncerainy indicaors, however, such as GARCH models, generally apply o highfrequency economic daa (daily, monhly or quarerly price changes, for example), whereas here we are examining volailiy in expors, which are repored annually 13. As a resul, his paper does no address uncerainy indicaors bu insead reviews calculaion mehods for he reference values used as he basis for indicaors of volailiy in erms of variabiliy, i.e. based on he variaions of y C in (3). 11Here we are examining he case of a linear deerminisic rend, acceping ha his may ake diverse forms (quadraic and exponenial rends, ec.) 12Ignoring he Knighian disincion beween risk and uncerainy. 13 As in he majoriy of research on macroeconomic volailiy. Page 15

17 2.2. Parameric approach Mos measures of macroeconomic volailiy are based on a univariae parameric approach, which models economic series on he basis of pas change over a given period. Much of he research se ou in he previous secion uses volailiy indicaors based on residual variance. The mos common echniques are presened below. Esimae based on a linear deerminisic rend The radiional approach consiss of developing a volailiy indicaor based on an average deviaion around a linear rend. In realiy, expor series, like all oher acual macroeconomic variables (GDP, expors, ineres raes, ec.) are series dominaed by low frequencies 14 (figure 1), which jusifies modelling hem using a deerminisic rend (linear, polynomial or exponenial). In is simplified (linear) form, his echnique consiss of esimaing he following model: y = α + β + ε (4) Where y is he variable whose volailiy is being measured, α a consan, a linear rend, and ε a zero mean error erm. In his case, he reference value is he rend: yˆ = ˆ α + β ˆ (5) Deviaions from he rend (ε) in principle have no permanen effecs on y. In oher erms, hese deviaions are assumed o be saionary around he rend and can herefore represen he volailiy of y. A measure of volailiy based on ε hus relies on hree key hypoheses: i) ha he series changes a a consan rae over ime, ii) ha he long-erm change in he series is perfecly predicable and iii) ha all deviaions or shocks affecing i are ransiory around he rend. Beveridge and Nelson (1981) highlighed he limiaions of an approach of his kind. 14A series whose variaions over a long period are hose which conribue he mos o oal variance. See Guay and Sain- Aman (1997). Page 16

18 Figure 2. Change in expor revenues, simple linear rend e e e+08 Belize e e+10 Argenina Year Expors (USD, 2000) Linear rend Expors (USD, 2000) Linear rend To illusrae hese limiaions Figure 2 shows he acual change in expors and heir rend as shown in (7) in Belize beween 1980 and 2004, and in Argenina beween 1970 and 2005 (Figure 3 shows he residuals for he esimaes produced). Alhough he change in expor revenues in Belize appears a firs sigh o be linear, he series observed may, in spie of his, depar from is rend over he long erm. This problem is all he more obvious in Argenina, where he growh in expor revenues does no, a firs sigh, follow a linear rend. This example illusraes he limiaions of his echnique for calculaing volailiy. Such a specificaion risks overesimaing he imporance of shocks by wrongly including a par of he non-consan rend in he residual. Figure 3. Residual of linear rend Belize Argenina Residuals -5.00e e Residus -1.00e e e e Page 17

19 Esimae based on a mixed rend Up o his poin, we have assumed ha deviaions from he deerminisic rend are only ransiory. Equaion (2), however, suggess ha i is possible ha residual variaions (ε T ) are no ransiory, and have a permanen effec on he rend followed by a series. I hus becomes possible o esimae he reference value on he basis of a sochasic rend (a random process below), represened by he following firs-order auoregressive AR(1) process, which we can re-wrie as follows: 2 ( ) y = y 1 + ε avec ε ~ N 0, σ (6) In his case, he change in y, is deermined by a successive hisory of random shocks, wih he resul ha a shock occurring in he pas, even he disan pas, has he same effec on he series as a shock in he presen (long-memory process). The series is hen considered o be difference-saionary or ha is rend follows a random pah. The series conains a uni roo. Afer differencing, he series can be rewrien as follows: 2 ( ) y = ε avec ε ~ N 0, σ This hypohesis is neverheless quie srong in he conex of examining macroeconomic variables dominaed by low frequencies (Nelson and Kang, 1981). The analysis of he expor series and heir specrum in secion 2.1 (see Figure 1) would jusify specifying a mixed rend, suggesing he exisence of flucuaions around a rend which is boh deerminisic and sochasic. Namely he process AR(1), which also includes a deerminisic rend: y y = y 0 + α β δ + ε = + + y 1 (7) 1 ε Our aim is o discover wheher he rend as specified in equaion (7) makes i possible o saionarise he expor series for our sample of counries correcly. To do so, we calculae he p-value of he uni roo es using Maddala-Wu panel daa (based on he Phillips Perron es), carried ou in 134 counries, and Fisher saisics on he join nulliy hypohesis for coefficiens associaed wih drif, rend and lag (able 1). Page 18

20 Table 1: Specificaion and uni roo es on panel daa H 0: he series is non-saionary Prob>Chi 2 F-es y i = α i + β i + φ 1 y i 1 + ε i y i = α i + β i + φ 2 y i 1 + ε i Counries (Observaions): 134(3693) The resuls of he ess carried ou hus do no jusify he rejecion of a null hypohesis for uni roo and end o jusify he use of a mixed rend wih drif (cf. equaion (9)). The non-saionariy hypohesis is hus rejeced once he series is differenced for a second ime, and he F-es saisics lead us o rejec he join nulliy hypohesis for he specificaion coefficiens (9). Figure 4 illusraes he rends obained for he case of Belize and Argenina. Figure 4. Change in expor revenues, mixed rend e e e+08 Belize Expors (USD, 2000) Mixed rend e e+10 Argenina Expors (USD, 2000) Mixed rend Figure 5 shows he correlogram for he residuals of equaion (7) esimaed for Belize and Argenina respecively. An examinaion of he correlogram shows whie noise, because he correlaions associaed wih lags are no significanly differen from zero 15. Wih regard o hese wo example counries and hose shown in Appendix A.2, using a mixed rend herefore proves o be more appropriae o saionarise expor revenues. However, an examinaion of Figure 4 and Appendix A.2 suggess ha his mehod of esimaing creaes a rend whose profile appears o be a slighly smoohed and offse version of he change seen in he expors. This rend herefore seems o reproduce in he change in expors observed in -1, which herefore conribues o an arificial creaion of volailiy. This phenomenon is all he more marked in he cases of Argenina (Figure 4), Venezuela, Burundi and Kenya (Appendix A.2). 15The Pormaneau saisics, which are no presened in his aricle, do no make i possible o rejec a null hypohesis for a whie-noise process. They can be supplied o he reader on reques. Page 19

21 Figure 5. Correlogram of residuals, mixed rend Belize Argenina Auocorrelaions Lags Barle's formula for MA(q) 95% confidence bands Auocorrelaions Lags Barle's formula for MA(q) 95% confidence bands Esimae based on a rolling mixed rend The main problem wih a mixed rend as calculaed in he previous sub-secion is ha i is based on a srong hypohesis of consancy over ime of he coefficiens associaed wih he rend of he series. Effecively, his so-called global rend is prediced each for each counry based on coefficiens esimaed for he whole period of daa availabiliy. I also excludes he possibiliy of a change of regime in he deerminisic and sochasic change of he series concerned. Maddala and Kim (1996) emphasise ha imporan changes in he deerminisic componen of he rend aken by a series can lead, wrongly, o a failure o rejec he uni roo hypohesis. I is herefore possible ha he mixed rend we have esimaed aribues o random rend variaions a change in he deerminisic change of he series, which may hen lead o an overesimae of he magniude of flucuaions. Alhough ess do exis (e.g. CUSUM, Max Chow) o idenify breaks in a rend during uni roo ess (Maddala and Kim, 1996), an alernaive and pracical soluion is o produce a rolling esimae of he mixed rend over a shorer period (Guillaumon, 2007), allowing he coefficiens esimaed o change from o and hus reflecing recen changes in he series rend. This rolling mixed rend is calculaed for each counry and each based on esimaing equaion (9) over he period [; -k], raher han over he whole of he period: yˆ T ˆ ˆ T T T = α + β + δ y 1 ˆ (8) and y = ˆ T y + ε where T is he esimaion period for he rend corresponding o (; k). The resuls obained when k = 12 are shown for Argenina in Figure 6 and compared wih hose obained based on he previous global mixed rend. Appendix A.2 shows he predicions and Page 20

22 correlograms for residuals obained using global and rolling mixed rends for oher counry examples. We can see ha he rend hus obained is smooher han wih he previous model and ha i does no have a sawooh profile resuling from consan parameers over ime. Similarly, an examinaion of he correlogram in Figure 6 and Figure 7, and he correlograms in he appendix (Appendix A.2), suggess ha he residuals resuling from his approach are saionary for he counry examples chosen. Figure 6. Rolling mixed rend, comparaive change and correlogram of residuals, Argenina e e Expors (USD, 2000) Rolling mixed rend Global mixed rend Auocorrelaions Lags Barle's formula for MA(q) 95% confidence bands Noneheless, his echnique presens a number of disadvanages. On he one hand, i reduces he ime coverage of volailiy indicaors, since he firs rend value is only available from = 1+k. On he oher hand, a bias may appear as he resul of a limied rend esimaion period. Esimaes based on a rolling mixed rend are no necessarily an ideal soluion where here is a break in he rend of he series. Changes prior o he break may coninue o exhibi ineria when he rolling rend is esimaed, once he break has occurred. Moreover, by proposing an esimae of he rend over a shorer period han he global mixed rend, his echnique may include some long-periodiciy flucuaions in he residual as a resul of being more sensiive o medium-periodiciy flucuaions. This phenomenon is illusraed in paricular by he respecive behaviour of wo mixed rends applied o Burundi beween 1985 and 1995 in Figure 7. The rolling mixed rend ends o underesimae he rend for expors o fall relaive o he global rend over he same period. The choice of period for calculaing he rend is herefore imporan. In his case we have chosen an esimaion period for he rolling rend of 12 s, in order o highligh he advanages and disadvanages of his echnique compared wih he global mixed rend. I could be considered ha a rolling rend calculaed over a period of beween 10 and 20 s represens a reasonable basis wih regard o he specrum densiies presened previously in Figure 1, since medium-erm flucuaions seem o conribue srongly o he oal variabiliy of he series. This choice should be jusified based on a prior examinaion of series behaviour (uni roo ess, graphical examinaion of series and examinaion of heir specrum densiy) and suppored by a sudy of he lieraure on he opic. Page 21

23 Figure 7. Global and rolling mixed rends, comparaive change and correlogram of residuals, Burundi e e e Expors (USD, 2000) Rolling mixed rend Global mixed rend Auocorrelaions Auocorrelaions Lags Barle's formula for MA(q) 95% confidence bands Global mixed rend Reards Barle's formula for MA(q) 95% confidence bands Rolling mixed rend 2.3. Filering approach Some research on economic volailiy uses a saisical filering approach o isolae he cyclical and rend componens of changes in he series (Becker and Mauro, 2006; Chauve and Guillaumon, 2007). The sandard deviaion of he cyclical componen hen becomes an example of a volailiy indicaor. The mos popular filer remains he Hodrick-Presco filer (1997). This can saionarise poenially inegraed series up o order four (King and Rebelo, 1993). The band-pass (BP) filer pu forward by Baxer and King (1999) is also used in he lieraure on economic flucuaions (Hnakovska and Loayza, 2005). Alhough he BP filer mainains he properies of he series more accuraely, his advanage comes a he price of a loss of observaions a he end of he sample 16 (Baxer and King, 1999). Moreover, when applied o our expor daa, boh echniques give exremely similar resuls 17. In his aricle we herefore only presen he resuls obained wih he Hodrick- Presco (HP) filer. The advanage of he filering mehod compared wih hose described above is ha i does no impose a priori any paricular (and someimes arbirary) form (mixed rend, deerminisic, random, ec.) on he behaviour of he series. Moreover, a saisical filering mehod enables changes o he rend over ime, which is a definie advanage over he ime series-based approach we have 16The BP filer is a bilaeral filer, which requires a minimum number of observaions before and afer each filered observaion poin in order o increase he precision of he filering. Baxer and King recommend ignoring he firs 12 and las 12 quarers when filering quarerly series. For annual series, hey recommend ignoring he firs hree and las hree s of he sample. 17Available o he reader on reques. Page 22