The Conditional Heterocedasticity on the Argentine Inflation. An Analysis for the Period from 1943 to 2013

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1 Journal of Mahemaics and Sysem Science 7 (07) doi: 0.765/59-59/ The Condiional Heerocedasiciy on he Argenine Inflaion. An Analysis for he Period from 943 o 03 Juan Carlos Abril and María De Las Mercedes Abril D DAV I D PUBLISHING Universidad Nacional de Tucumán, Faculad de Ciencias Económicas and Consejo Nacional de Invesigaciones Cieníficas y Técnicas (CONICET). Tucumán. Argenina. jabril@herrera.un.edu.ar, mabrilblanco@homail.com Absrac. Numerous economic ime series do no have a consan mean and in pracical siuaions, we ofen see ha he variance of observaional error is subjec o a subsanial variabiliy over ime. This phenomenon is known as volailiy. To ake ino accoun he presence of volailiy in an economic series, i is necessary o resor o models known as condiional heeroscedasic models. In hese models, he variance of a series a a given ime poin depends on pas informaion and oher daa available up o ha ime poin, so ha a condiional variance mus be defined, which is no consan and does no coincides wih he overall variance of he observed series. There is a very large variey of nonlinear models in he lieraure, which are useful for he analysis of any economic ime series wih volailiy, bu we will focus in analyzing our series of ineres using ARCH ype models inroduced by Engle (98) and heir exensions. These models are non-linear in erms of variance. Our objecive will be he sudy of he monhly inflaion daa of Argenina for he period from January 943 o December 03. The daa is officially published by he Naional Insiue of Saisics and Censuses (or INDEC as i is known in Argenina). Alhough i is a very long period in which various changes and inervenions ook place, i can be seen ha cerain general paerns of behavior have persised over ime, which allows us o admi ha he sudy can be appropriaely based on available informaion. Keywords: Inflaion, heerocedasiciy, volailiy, ime series. Inroducion correlaed bu hey are dependen. Thus linear models such as hose belonging o he family of Numerous economic ime series do no have a auoregressive moving average models or ARMA consan mean and in pracical siuaions, we ofen models may no be appropriae o describe hese see ha he variance of observaional error is series. subjec o subsanial variabiliy over ime. This The firs inspecion of series such as he one phenomenon is known as volailiy. we will sudy suggess ha hey do no have a To ake ino accoun he presence of volailiy mean and a consan variance. A sochasic variable in an economic series, i is necessary o resor o in which he variance is consan is said o be models known as condiional heeroscedasic homocedasic as opposed o wha would be a models. In hese models, he variance of a series a heeroscedasic variable. For hose series where a given ime poin depends on pas informaion and here is volailiy, he uncondiional variance can be oher daa available up o ha ime poin, so a consan even when he condiional variance in condiional variance can be defined, which is no some periods is unusually large. consan and does no coincide wih he overall Our objecive will be he sudy of he monhly variance of he observed series. inflaion daa of Argenina for he period from An imporan feaure of any economic ime January 943 o December 03. The daa is series is ha hey are no generally serially officially published by he Naional Insiue of

2 70 The Condiional Heerocedasiciy on he Argenine Inflaion. An Analysis for he Period from 943 o 03 Saisics and Censuses (INDEC). Alhough i is a very long period in which basic changes, baske y An ARCH() model can be expressed as = µ + ε h changes and inervenions ook place even in he ε ~ i.i.d D(0,) INDEC iself, i can be seen ha cerain general σ = h = ω + α z -, paerns of behavior have persised over ime, i= which allows us o admi ha he sudy is adeuaely based on he informaion available. where z = y µ and D ( ) is a probabiliy Mehod densiy funcion wih zero mean and variance eual o one. Research Model Modeling volailiy An ARCH model such as he one jus described adeuaely describes volailiy clusering. Linear models of he ARMA ype, for example, admi ha he disurbances have zero mean and consan variance, usually eual o one (his is euivalen o saying ha hese disurbances are whie noise). Under hese condiions, he condiional variance given over he pas hisory, ha is given F, is consan over ime. There is a very large variey of nonlinear models available in he lieraure o address hese siuaions, bu we will focus on ARCH models or auoregressive condiional heeroscedasic models inroduced by R. Engle (98) and heir exensions. These models are nonlinear in erms of variance. In he analysis of non-linear model, errors (also called innovaions, because hey represen he new par of he series ha canno be prediced from he pas) known as and he model has he form y g ε ε, + ε h ε = g + ε h = µ + ε h, ε, are generally assumed IID ( ) (, ε, ) =, where ( ), g g = = µ represens he condiional mean and ( ) h = h is he condiional variance. ARCH ype models The basic idea of his model is ha a series () The condiional variance of y is an increasing funcion of he suare of he shock ha occurs a ime. Conseuenly, if y is sufficienly large in absolue value, σ and hus y are expeced o be also large in absolue value. I is necessary o ake ino accoun ha even when he condiional variance in an ARCH ype model varies wih ime, ha is, σ = E ( z F ), he uncondiional variance of z is consan and, given ha ω> 0 and a <, we have i= ω { E( z F )} =. i () σ = E (3) a i= In mos pracical applicaions, excess kurosis in an ARCH model, along wih a normal disribuion, is no enough o be able o explain wha a daase like ours. Therefore, we can make use of oher disribuions. For example, we can suppose ha ε follows a Suden disribuion wih mean 0, variance eual o and υ degrees of i y is no serially correlaed bu depends on pas prices hrough a uadraic funcion. freedom, ha is, ε is ST (0,, υ). In his case, he uncondiional kurosis for he ARCH () model is

3 λ ( α ) ( λα ) where λ = 3( υ ) ( υ 4). Due o he addiional coefficien υ, he ARCH () model based on a disribuion will have heavier ails han one based on a normal disribuion. The calculaion of The Condiional Heerocedasiciy on he Argenine Inflaion. An Analysis for he Period from 943 o 03 σ in () depends on pas uadraic residues, z, ha are no observed for = 0,,, +. To iniialize he process, unobserved uadraic residuals are se o a value eual o he sample mean. GARCH ype models While Engle (98) cerainly made he greaes conribuion o financial economerics, ARCH ype models are rarely used in pracice because of heir simpliciy. A good generalizaion of his model is found in he GARCH ype models inroduced by Bollerslev (986). This model is also a weighed average of he pas uadraic residuals. This model is more parsimonious han ARCH models, and even in is simples form, has proven o be exremely successful in predicing condiional variances. I should be noed ha GARCH ype models are no he only exension and here are a leas welve specificaions relaed o hem ha will be he objec of fuure research. Generalized ARCH models (or GARCH models as hey are also known) are based on an infinie ARCH specificaion and allow o reduce he number of parameers o be esimaed by imposing nonlinear consrains on hem. The GARCH(p, ) model is expressed as follows = + i i= j= σ ω + a z β σ. (4) j j As in he case of ARCH models, i is necessary o impose some resricions on σ o ensure ha i is posiive for all. Bollerslev (986) showed ha ensuring ha ω > 0, 0 (for i =,, ) and a i 7 β 0 (for i =,, p) is sufficien o guaranee j ha he condiional variance is posiive. In erms of he esimaion process, we can say ha many auhors have proposed using a Suden disribuion in combinaion wih a GARCH model o adeuaely model heavy ails on economic or financial ime series whose daa are of high freuency, which will be seen below. Sample Firs, we proceed o plo he series. Wihin he sudy of a series, graphic mehods are an excellen way o begin an invesigaion. Afer ploing i, we can immerse ourselves in a deailed sudy of he subjec under consideraion. Among he funcions ha comply wih he graphs ha we will presen are he following:. They make he daa under sudy more visible, sysemaized and synhesized.. They reveal heir variaions and heir hisorical or spaial evoluion. 3. They can show he relaionships beween he various elemens of a sysem or a process and show indicaions of he fuure correlaion beween wo or more variables. In addiion o his, he applicaion of hese mehods suggess new research hypoheses and allows he subseuen implemenaion of saisical models ranging from he simples o hose ha are much more refined, hus achieving a beer analysis of he daa and is flucuaions over ime. Secion (a) of Figure shows he monhly levels of he Consumer Price Index. In secion (b) of he same figure we can see he firs differences of he logarihm of he monhly IPC level. This is wha is popularly known as inflaion and will be he series objec of our work.

4 7 The Condiional Heerocedasiciy on he Argenine Inflaion. An Analysis for he Period from 943 o 03 Afer a careful inspecion of his las secion, we can see ha here are periods where he volailiy is low and can be confused wih he presence of seasonaliy especially wihin he period from January 943 unil he end of 974. Then, here is a clear a period of very imporan volailiy unil lae 977, which is repeaed wih similar characerisics beween 983 and he end of 985. Subseuenly, beween 987 and he end of 99 we have a period of high volailiy. From lae 00, he volailiy in he series under sudy is almos null, and i almos disappears from 009. Figure : (a) Monhly levels of he Consumer Price Index from January 943 o December 03. (b) Firs difference of he logarihm of he level of he Consumer Price Index from January 943 o December 03. Figure shows in secion (a) he auocorrelaion for he inflaion series under sudy, also in secion (b) we can see he parial auocorrelaion funcion. Finally, in secion (c), we have he esimaed densiy funcion which is represened by a red line, compared o he normal densiy funcion which is represened by a green line. From he sudy of his figure we see ha he series is no saionary, alhough i has some seasonal componens and is disribuion is differen from ha of a normal variable, so i is possible ha we mus make heir corresponding esimaes using a Suden disribuion.

5 The Condiional Heerocedasiciy on he Argenine Inflaion. An Analysis for he Period from 943 o Figure : (a) Auocorrelaion funcion of he inflaion series for he period from January 943 o December 03. (b) Parial auocorrelaion funcion of he same series under sudy. (c) Esimaed densiy funcion compared o normal densiy (green line). Daa Analysis and Discussion Differen alernaives were ried wih respec o he modeling of he inflaion series in Argenina for he period beween January of 943 and December of 03. Afer analyzing hem and seeing he values of differen goodness of fi saisics, like he Akaike Crierion, he Schwarz Crierion or he Hannan - Quinn Crierion, we are lef wih a model based on he euaion (), where y is he series under sudy and is explici specificaion is given by where y = µ + ε h, (5) ε has a disribuion wih.67 degrees of freedom. The condiional mean, µ is eual o a general mean µ a seasonal componen and an ARMA (, ) process, which is explicily µ µ γ + ϕ y ν θν (6) where = + + +, ν = ε h and ha saisfies he following condiion = γ γ, γ is a seasonal componen γ (7) ha is, is sum is eual o zero over he previous year. This is achieved wih he inroducion of adeuae dummy variables, Besides ha, he condiional variance in (5) is given by h ( y µ ) βσ, = = ω + α + σ (8) ha is, a GARCH(, ) wih a consan given by ω. In he esimaion process for our case, we could see ha he consan was no significan differen from zero, so we decided o remove i in he final formulaion.

6 74 The Condiional Heerocedasiciy on he Argenine Inflaion. An Analysis for he Period from 943 o 03 Figure 3: Characerisics of he inflaion series for he period beween 943 and 03. (a) Inflaion series. (b) Residues from he inflaion series. (c) Quadraic residues. (d) Sandardized residuals. (e) Condiional mean from he applicaion of he volailiy. (f) Condiional sandard deviaion. (g) Condiional variance. (h) Sandardized residuals compared o a disribuion wih 0, and.67 degrees of freedom. In he (a) he series of inflaion again, in (b) we can see he residuals, in (c) we have he uadraic residuals while in (d) we have are he sandardized residuals of he series. In secion (e) of Figure 3 we show he esimaed condiional mean for he proposed volailiy model, in (f) we see he condiional sandard deviaion, while in (g) we observe he esimaed condiional variance ha arises from he applicaion of he model o our series under sudy. Looking closely a secions (a), (e), (f) and (g) of Figure 3, we can say ha he model adeuaely explains he inflaion series of our counry for he period beween 943 and 03. This same characerisic arises again in secion (h) when comparing he disribuion of he sandardized residuals wih a Suden disribuion wih.67 degrees of freedom. An ineresing fac o noe is ha owards he end of he period under consideraion, in paricular from Ocober or November 004, he condiional variance or volailiy of he series becomes pracically insignifican.

7 The Condiional Heerocedasiciy on he Argenine Inflaion. An Analysis for he Period from 943 o Figure 4: Predicion of he condiional mean versus he inflaion series for he period beween 943 and 03 (op). Condiional variance predicion for he model for he period beween 943 and 03 (boom). A he op of Figure 4 we can see he las en observaions of he series under sudy, highlighed in blue, and he corresponding predicions of he condiional mean. The verical bars correspond o he 95% confidence inerval ha serve o compare he prediced value wih he one we observed. In he lower graphic we ried o predic he condiional variance corresponding o he las en observaions of our series under sudy, however, his was no possible. As we have already expressed, he condiional variance, or volailiy, as of Ocober 004 is eual o zero. This siuaion corroboraes he fac ha from ha dae no volailiy predicions can be made for his series, which is in line wih he beginning of a period of lack of confidence in official saisics. Saisically, his involved he predicion of values differen from hose ha may have been in realiy, which gave rise o an exremely smooh series ha does no coincide wih he res of i nor wih he realiy lived. Figure 5 is similar o Figure 4, bu in he laer he bars corresponding o he confidence inervals were removed and he verical axis scale was changed in order o have a beer observaion of he original series and he prediced condiional mean. One can clearly see a grea coincidence beween boh Figures and he analysis ha we can do is exacly he same for boh cases.

8 76 The Condiional Heerocedasiciy on he Argenine Inflaion. An Analysis for he Period from 943 o 03 Figure 5: Predicion of he condiional mean versus he inflaion series for he period beween 943 and 03 (op). Condiional variance predicion for he model for he period beween 943 and 03 (boom). Final Remarks In his iniial sage of our invesigaion, we se ou o analyze mehods o rea a grea variey of daa wih irregulariies ha happen in ime series. Inegraed auoregressive moving averages models (or ARIMA models) are ofen considered o provide he main basis for modeling in any ime series. However, given he curren sae of research in ime series research, here may be more aracive, and above all more efficien alernaives. Numerous economic ime series do no have a consan mean and also in mos cases phases are observed where relaive calm reigns, followed by periods of imporan changes, ha is, ha he variabiliy changes over ime. Such behavior is wha is called he volailiy. Among he models we have presened are he ones belonging o he ARCH family. The ARCH models or auoregressive models wih condiional heeroscedasiciy were firs presened by Engle in 98 wih he aim of esimaing he variance of inflaion in Grea Briain. The basic idea of his model is ha y is no serially correlaed bu he condiional volailiy or variance of he series depends on he pas reurns by means of a uadraic funcion. However his kind of models are rarely used in pracice because of heir simpliciy. A good generalizaion of his model is found in he GARCH ype models inroduced by Bollerslev (986). This model is also a weighed average of he las uadraic residuals, bu i is more parsimonious han he ARCH ype models and even in is simples form has proven o be exremely successful in predicing condiional variances, so we decided o make use of hem when working wih our daa. Before applying any saisical mehod o he daa under sudy i is fundamenal o observe hem graphically in order o be able o become familiar wih hem. This can have numerous benefis, as we have explained a he beginning of our analysis, as his process will serve as an indicaor of ideas for a more deailed laer sudy. This was he firs sep of our work in which we could see he main feaures of he series and served us o make an appropriae adjusmen of i. Again i is necessary o emphasize ha alhough his is a very long period o analyze; where here were numerous changes such as he base, and inervenions in he INDEC, i is possible o make a very ineresing sudy where he main characerisics of he series are appreciaed.

9 The Condiional Heerocedasiciy on he Argenine Inflaion. An Analysis for he Period from 943 o We decided o fi an appropriae GARCH ype model ha capures he main characerisics of he daa. We saw ha i akes adeuaely o he volailiy of he series bu however, i presens some difficulies when making predicions. I remains for laer work o analyze if we can use anoher specificaion ha akes ino accoun his fac ha capures even more he characerisics of he series ha can do a GARCH model like he one we have seen. Wih his analysis we have been able o esimae ha from Ocober 004, volailiy is eual o zero. This siuaion corroboraes he fac ha from ha dae no volailiy predicions can be made for his series, which is consonance wih he beginning of a period of lack of confidence in he official saisics. Saisically, his gave rise o an exremely smooh series ha does no mach he res nor wih he realiy lived. References Abril, Juan Carlos. (999), Análisis de Series de Tiempo Basado en Modelos de Espacio de Esado, EUDEBA: Buenos Aires. Abril, Juan Carlos (004). Modelos para el Análisis de las Series de Tiempo. Ediciones Cooperaivas: Buenos Aires. Abril, Maria de las Mercedes. (04). El Enfoue de Espacio de Esado de las Series de Tiempo para el Esudio de los Problemas de Volailidad. Tesis Docoral. Universidad Nacional de Tucumán. Argenina. Baillie, R. T. y Bollerslev, T. (989). The Message in Daily Exchange Raes: A Condiional-Variance Tale. Journal of Business and Economic Saisics, 7, Bollerslev, T. (986). Generalized auoregressive condiional heeroscedasiciy. Journal of Economerics, 3, Bollerslev, T. (987). A Condiionally Heeroscedasic Time Series Model for Speculaive Prices and Raes of Reurn. Review of Economics and Saisics, 69, Bollerslev, T., Chou, R. Y. y Kroner, K. F. (99). ARCH Modeling in Finance: A Review of he Theory and Empirical Evidence. Journal of Economerics, 5, Bollerslev, T. y Wooldridge, J. M. (99). Quasi-maximum Likelihood Esimaion and Inference in Dynamic Models wih Time-varying Covariances. Economeric Reviews,, Box, G. E. P. y Jenkins, G. M. (976). Time Series Analysis: Forecasing and Conrol (Revised ediion), Holden-Day Inc.: San Francisco. Engle, R. F. (98). Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of he Unied Kingdom Inflaion. Economerical, 50, Goldfarb, D. (970). A Family of Variable Meric Updaes Derived by Variaional Means. Mahemaics of Compuaion, 4 (09), 3-6. Harvey, A. C. (989). Forecasing, Srucural Time Series Models and he Kalman Filer. Cambridge Universiy Press: Cambridge. Hsieh, D. A. (989). Modeling Heeroskedasiciy in Daily Foreign Exchange Raes. Journal of Business and Economic Saisics, 7, Pagan, A. (996). The Economerics of Financial Markes. Journal of Empirical Finance, 3, 5-0. Palm, F. C. (996). GARCH Models of Volailiy. En Handbook of Saisics, edied by G. Maddala, y C. Rao, Elsevier Science, Amserdam. Palm, F. C. y Vlaar, P. J. G. (997). Simple Diagnosics Procedures for Modeling Financial Time Series. Allgemeines Saisisches Archiv, 8, 85-0