Modeling the Clustering Volatility of India s Wholesale Price Index and the Factors Affecting It

Transcription

1 Journal of Managemen and Susainabiliy; Vol. 6, No. 1; 016 ISSN E-ISSN Published by Canadian Cener of Science and Educaion Modeling he Clusering Volailiy of India s Wholesale Price Index and he Facors Affecing I Mohammad Naim Azimi 1 1 Deparmen of Indusrial Economics and Business Adminisraion, Rana Universiy, Kabul, Afghanisan Corresponding: Mohammad Naim Azimi, Deparmen of Indusrial Economics and Business Adminisraion, Rana Universiy, Kabul 55, Afghanisan. vc@ru.edu.af Received: November 9, 015 Acceped: December 11, 015 Online Published: February 5, 016 doi: /jms.v6n1p141 URL: hp://dx.doi.org/ /jms.v6n1p141 Absrac This paper proposes o examine he clusering volailiy of India s Wholesale Price Index hroughou he period 1960 o 014 by applying he ARCH (1) and GARCH (1) model. The pre-condiional requiremen for he compuaion of ARCH (1, 1) required us o perform several oher ess i.e. Dickey Fuller, Ordinary Leas Squared Regression and pos OLS ess for invesigaing he ARCH effec in he firs difference of WPI. The saisical analysis reveals a p-value of for he GARCH mean model which is no significan a 0.05 o explain ha he previous period s volailiy can influence he WPI. The coefficien of WPI a firs difference exhibis a value of less han 1 which is nice in magniude wih a p-value of for ARCH a 0.05 which is significan o explain he volailiy of he WPI. The diagnosic es of auocorrelaion in he residuals reveals ha he residuals are whie noise by exhibiing a corresponding probabiliy value of Since, he overarching objecive of his paper is o examine he clusering volailiy of he aforemenioned variable wih regards o he inernal shocks, here migh have been oher facors of exernal shocks on WPI ha have deliberaely been overlooked in his paper. Keywords: clusering volailiy, ARCH model, GARCH model, WPI, Gaussian disribuion 1. Inroducion Wholesale Price Index (WPI) as a macroeconomic variable has always been he cenral heme of many research papers sudying o invesigae he reason of movemen in he values overime, given rise o criics in which clusering volailiy is a significan economic phenomenon o be debaed. The economic variables exhibied significan flucuaions in he las few decades and i is only recenly ha he researcher became ineresed in he economic consequences of heir volailiy for developing counries (Plosser, 009). Gaunersdorfer & Hommes (007) sae ha changes in price appear o be unpredicable, whereas he magniude of such changes measured by he absolue or squared variables, appear o be predicable in he sense ha he high volailiy ends o follow he high volailiy of eiher signs and low volailiy ends o follow he low volailiy in a ime series process. Mandelbor (1967) was he firs researcher who observed his economic phenomenon in he price of commodiy and laer pioneered by Engle (198) and Bollerslev (1986). The mos common economerics models ha have so far been applied o examine he clusering volailiy or sylized facs of he macroeconomic variables are he ARCH (Auoregressive Condiional Heeroskedasiciy) and GARCH (Generalized Auoregressive Condiional Heeroskedasiciy) by a vas number of researchers (Bera e al., 199; Chung & Wu, 005; Bonomo & Marins, 003; Daal e al., 007; Con, 007; Gonzalez & Gimeno, 01; Chi e al., 010). I is assumed ha he compuaion of ARCH and GARCH model is only a decade ago, bu is iniiaion goes far as back as o Bachelier (1900) who had iniially observed and esed for he price speculaion. WPI is one of he mos changing and volaile variable and is widely used in he Indian cases o measure he economic performance of he counry overime as i has a large coverage of commodiies and is compuaion is as imporan as he oher economic variables (Bhaskara & Singh, 006). The exensive lieraure on economic and financial ime series analysis sugges ha such volaile and gradually changing variables compuaion is based on ARCH and GARCH models (Guo, 006; Nelson, 1991; Zakoian, 1994; Higgins e al., 199; Ding e al., 1993; Engle, 198). This paper will es he clusering volailiy of he wholesale price index of India by applying ARCH (1, 1) model ha a concise discussion of which follows his para. 141

2 Journal of Managemen and Susainabiliy Vol. 6, No. 1; ARCH (1, 1) and Clusering Volailiy According o Miles (008) ARCH is an effecive economeric esing model which is applied on hose specific variables ha exhibi ARCH effec. Given rise o mos of he economic variables wih due o heir flucuaions overime, boh ARCH effec and clusering volailiy can be observedha ends o move overime. Chang & McAleer (015) argue ha using volailiy measures based on he assumpion of consan volailiy over period of ime when he resuling series moves hrough ime is saisically and logically vicimized by auocorrelaion. Tickling he serial correlaion associaed wih he ARCH model requires diagnosic ess o ensure ha he model is nicely fied and he anicipaed research finding is of value o a maximum exen (see, secion..4) which all such issues are aken ino accoun in his paper. The remainder of his paper is organized as follow: Secion, illusraes he daa and esing procedures plus he pre-condiional requiremens for he applicaion of ARCH and GARCH model, Secion 3, presens he daa analysis and research findings, Secion 4, concludes he paper and is followed by acknowledgemen of he auhor and he lis of references.. Daa and Mehod.1 Basic Daa The daa used in his sudy is rerieved from WDI (World Developmen Indicaor) on Wholesale Price Index of India. The observaion is for he period 1960 o 014 and is arranged on annual basis. The iniial research plan on daa collecion was he monhly observaion, bu he concerned source of daa used herein, does no provide monhly daa on Indian WPI (see, Table 1 for descripive saisics).. The Model Following he model of ARCH (1) and GARCH (1) developed by Engle (198) and pioneered by Bollerslev (1986) he daa colleced for his sudy is esed agains he clusering volailiy of he previous periods volailiy on he Wholesale Price Index of India. In our approach o use he ARCH and GARCH model, here are wo fundamenal condiions o fulfill e.g., 1) he exisence of Clusering Volailiyin he residuals ha ends o deermine he period of high volailiy being followed by high volailiy and ha he period of low volailiy is being followed respecively and ) here is an ARCH effec in he residuals. Therefore, o ensure ha our daa fulfills he requiremens, we apply he following sequenial economeric es models o obain an accurae resul:..1 Dickey Fuller Tes As an iniial poin o sar, a Dickey Fuller es is applied o check he saionariy for he series of he model ha includes 1 o k lags a firs difference wih a 5% criical value (Dickey & Fuller, 1979). The equaion used o fi he model is: Δ y = α + β y 1 + δ + ζ 1 Δ y 1 + ζ Δ y ζ k Δ y k + ε (1) Here we es he null hypohesis of D.WPI being non-saionary and has uni roo agains he alernaive being saionary wih no uni roo a 5% criical value. Since, he firs aemp o es he null is wih drif, here are wo oher alernaive ess ha we perform o check he saionariy of he daa a is firs difference: - DF wih ime rend and; - Saionariy wih non-zero bu wih no linear ime rend. The rejecion of null under DF es will allow us o go furher and apply our nex es model. The approach o conclude he es is designed on DF wih drif a 5% criical value (see, Table 1)... OLS Regression Tes To compue he ARCH/GARCH mean model (1, 1), we regress he OLS (Ordinary Leas Square) wih consan-only o faciliae he deerminaion of ARCH effec in he residuals and o es he null being no ARCH effec vs. here is ARCH effec in he residuals (see, Table )...3 Lagrange Muliplier es for ARCH effecs For esing he null, we borrow from he work of Engle (198) o es for ARCH (p) effecs ha can fi on Ordinary Leas Square (simply OLS) regression of uˆ on uˆ,..., uˆ 1 : p u ˆ = γ + γ uˆ γ uˆ + ε () p p 14

3 Journal of Managemen and Susainabiliy Vol. 6, No. 1; 016 Where he saisics is nr and is asympoically disribued X (p) (Baum, 001). Before compuing he LM es for ARCH effecs, we es he residuals colleced from he OLS regression (refers o as D.Ln_WPI ) o see he periods of ranquiliy and periods of high volailiy in he residuals (see, Figure 1 D.Ln_WPI). This ensures ha he model is a good candidae for ARCH / GARCH compuaion...4 ARCH/ GARCH Mean Model The ARCH (Auoregressive Condiional Heeroskedasiciy) and GARCH (Generalized Auoregressive Condiional Heeroskedasiciy) models have been primarily developed o deal wih issues of clusering volailiy in he financial ime series (Nelson, 1991). We use he models for examining wheher he previous periods of volailiy affec he Indian Wholesale Price Index for he purpose of which, we perform all pre-condiional requiremen discussed in secions..1,.. and..3. The prime ARCH model which has been designed and inroduced by Engle (198) was modeled ha he variance of he regression disurbance is a linear funcion of he lagged values of he squared regression disurbance (Glosen e al., 1993). For he mean model of ARCH (m), we can fi he following equaion: Condiional Mean y = X β + ε (3) Condiional Variance m m σ = γ + γ ε + γ ε + + γ ε (4) Theε is he squared residual and heγ i represens he ARCH parameer. ARCH model accouns boh for mean and condiional variance. The variance iself is a funcion of he size for he prior unexpeced innovaions. The GARCH (m, k) model which was developed by Bollerslev (1986) and is applied in his paper can be expressed as of he following equaion: y = X β + ε m m k k σ = γ + γ ε + γ ε + + γ ε + δ σ + δ σ + + δ σ (4) In he GARCH model equaion above, γ i are he ARCH parameers and δ i are accouning for GARCH parameers. The serial correlaion associaed wih he model is esed by Barle Whie Noise mehod for he randomness and normaliy of he residuals. The null is ha serial correlaion is whie noise and randomly disribued agains he alernaive being he serial correlaion is no whie noise and is no randomly disribued (see, Table 7 and Figure 3). 3. Daa Analysis STATA 14 saisical sofware is used o analysis he daa and o apply he saed models for esing he clusering volailiy of he variables hroughou he period 1960 o 014. In his secion, we shall presen all he saisical analysis and findings of he research in a sequenial order and as discussed in secion.. (3) Table 1. Descripive saisics Variable Obs Mean Sd. Dev. Min Max OBS WPI Ln_WPI Observaion represens he ime period from 1960 o 014 annualized daa on Wholesale Price Index of India. Ln_WPI is he log differences of he Wholesale Price Index calculaed by use of excel funcion for he concerned period of ime under sudy. 143

4 Journal of Managemen and Susainabiliy Vol. 6, No. 1; 016 WPI India Wholesale Price Index WPI, D India wholesale Price Log Differenced OBS OBS Figure 1. Line plo The line-plo for original variable (WPI) shows a moving rend by an upward slop which means ha he variable is non-saionary. Since he applicaion of ARCH (auoregressive condiional heeroskedasiciy) and GARCH (generalized auoregressive condiional heeroskedasiciy) model in deermining he volailiy of he residuals in he saed variable requires saionariy (low periods of volailiy following he low periods of volailiy and higher volailiy ends o follow he higher volailiy for prolong period) o reflec he clusering volailiy, he WPI is ploed a is firs difference which is exacly he same as we plo i on is residual which shows he D.WPI ends o follow a clusering volailiy. Table. Dickey fuller es Number of obs = Z() has -disribuion Tes Saisics 1% Criical Value 5% Criical Value 10% Criical Value Z() p-value for Z() = D.WPI Coef. Sd. Err. T P> [95% Conf. Inerval] WPI L _cons The Dickey Fuller es resul shows ha he value of saisics (he absolue value) is more han he criical value a 0.05 (5%) (We ignore he negaive sign of he criical value here) which means ha we can rejec he null hypohesis in he favor of he alernaive hypohesis. Hence, he variable is saionary a firs difference (Noe 1) and his is wha we require o faciliae he applicaion of ARCH / GARCH (1, 1) model. The auocorrelaion associaed wih his model is also esed ha he resul of which, documens ha he auocorrelaion is whie noise wih a p-value > 0.05 (say, , see, Figure 3) and randomly disribued which is desirable for our saisical analysis. To begin wih, we regress he WPI (he variable) a firs difference and coninue our analysis (see, Table 3). Table 3. Linear regression a firs difference of WPI Source SS Df MS Number of obs = 54 Model 0 0. F(0, 53) = 0.00 Residual Prob > F =. R-squared = Toal Adj R-squared = Roo MSE =.3306 D.WPI Coef. Sd. Err. P> [95% Conf. Inerval] _cons

5 Journal of Managemen and Susainabiliy Vol. 6, No. 1; 016 The clusering volailiy of he D.WPI (WPI in firs difference) is documened by ploing i (see, Figure 1 India Wholesale Price Index a firs difference) and he whie noise disribuion of residuals (see, Figure 3 p-value > ) in he regression and he ADF es also proves o be desirable in our sudy on he basis of which, we coninue o es he exisence of ARCH effec or he clusering volailiy of he mean model (1, 1) by esing he following hypohesis: H 0 : Mean Model has no ARCH effec H A : Mean Model has ARCH effec Table 4. LM es for auoregressive condiional heeroskedasiciy (ARCH) lags(p) chi Df Prob > chi The LM es for auoregressive condiional heeroskedasiciy presens a p-value of which is < 0.05 (we normally es a α.05/95% confidence level) and herefore, he null hypohesis can be rejeced agains he alernaive and we know ha he mean model (1, 1) has an ARCH effec. In oher words, on he basis of he p-value being less han 0.05, he mean model (1, 1) has clusering volailiy (see, Engle, 198). Table 5. ARCH family regression Sample: Number of obs = 54 Disribuion: Gaussian Wald chi (.) =. Log likelihood = Prob > chi =. D.Ln_WPI OPG Coef. Sd. Err. Z P> z [95% Conf. Inerval] Ln_WPI OPG _cons ARCH Arch L Garch L _cons Table 5 shows he esimaion of ARCH (1) parameer wih a value of 1.04 and consequenly he GARCH (1) parameer reflecs a value of 0.741, ha our fied GARCH (1, 1) model can be expressed as follow: y = ε σ = ε 1 + σ 1 y = Ln( WPI ) Ln( WPI 1 ) The corresponding probabiliy value of z saisics for ARCH is which is < 0.05 meaning ha he ARCH p-value is significan o explain he volailiy of WPI. On he oher hand, he corresponding value of he z saisics for GARCH is being > 0.05 which means ha his value for GARCH is no significan o explain he volailiy of he WPI. Hence, he previous years volailiy of he Indian Wholesale Price Index (simply, WPI) canno influence he WPI. Keeping he accoun for he serial correlaion, we furher es he following hypohesis: H 0 : There is no serial correlaion H A : There is serial correlaion in his model. Since, he ARCH / GARCH (1, 1) model is compued on Gaussian based approach, we creae anoher residual (say, GR) o execue he auocorrelaion and parial correlaion funcions on 10 lags as below: 145

6 Journal of Managemen and Susainabiliy Vol. 6, No. 1; 016 Table 6. AC and PAC LAG AC PAC Q Prob>Q The able above in AC and PAC (see, Figure for he AC and PAC graphs as well) shows very smaller values boh wih posiive and negaive signs ha are closer o zero ha represen saionariy of he variable GR. Auocorrelaions of D.Ln_WPI Lag Barle's formula for MA(q) 95% confidence bands Parial auocorrelaions of D.Ln_WPI Lag 95% Confidence bands [se = 1/sqr(n)] Figure. AC and PAC On he oher hand, he corresponding probabiliy value of he Q saisics is higher in some lags and lesser han 5% in he following lags which almos shows a serial correlaion in he residuals. Hence, we have o develop and es he following hypohesis for is normal and random disribuion: H 0 : The residuals are random and normally disribued H A : The residuals are no random and no normally disribued Table 7. Cumulaive periodogram whie-noise es Barle s (B) Saisics = Prob > B = For esing he above hypohesis and o check for randomness of he residual (GR) disribuion, he whie noise es developed by Barle (1995) is used and is shown in Table 7. The corresponding probabiliy value of in he es is > 0.05 on he basis of which, we canno rejec he null hypohesis, raher we accep i and furher submi ha he residual disribuion is whie noise and wihin he band (see also, Figure 3). 146

7 Journal of Managemen and Susainabiliy Vol. 6, No. 1; 016 Cumulaive periodogram for R Cumulaive Periodogram Whie-Noise Tes Frequency Barle's (B) saisic = 0.91 Prob > B = Figure 3. Whie noise es The values represened by blue squares are seen o be always wihin he confidence bands direced by he limelines in he graph. The corresponding p-value of he Barle es is which is > 0.05, meaning ha he residuals are normally disribued and are whie noise. In oher word, we can conclude ha he variable in process is no differen from whie noise and never crossed he bands. 4. Conclusion For invesigaing he clusering volailiy of he Indian Wholesale Price Index (WPI), ARCH and GARCH mean model is applied. The daa for a wide range of ime series over 55 years on WPI has been carefully esed ha he resul of which reveals ha he GARCH L1 corresponding probabiliy value of being > 5% is no significan o explain WPI and herefore, we can conclude ha he volailiy of he previous periods of he WPI is no significan o influence he WPI. In addiion o his, he ARCH L1 corresponding probabiliy value of being < 5% is significan o explain volailiy of he WPI under he Gaussian model. Since, he model has almos suffered from auocorrelaion bu he residuals under he Gaussian model are whie noise by which shows ha he residuals are random and normally disribued. Acknowledgemens Wih profound words, I acknowledge he presence of he academic and research members of Rana Universiy and heir effecive commens wih regards o his paper. The auhor acknowledges he consrucive and valuable commens of he anonymous edior and reviewer in relaion o his sudy. References Bachelier, L. (1900). Théorie de la spéculaion. Annales Scienifiques de l École Normale Supérieure, 3, Barle, M. S. (1995). An Inroducion o Sockhasic wih Special Reference o Mehods and Applicaions. Cambridge: Cambridge Universiy Press. Baum, C. F. (001). Saa: The language of choice for ime series analysis? Saa Journal, 1(1), Bhaskara Rao, B., & Singh, R. (006). Demand for money in India: Applied Economics, 38(11), hp://doi.org/ / Bollerslev, T. (1986). Generalized Auoregressive Condiional Heeroskedasiciy. Journal of Economerics, 31(1), hp://doi.org/ /tnn Bonomo, M., Marins, B., & Pino, R. (003). Deb composiion and exchange rae balance shee effec in Brazil: A firm level analysis. Emerging Markes Review. hp://doi.org/ /s (03)00061-x Chang, C. L., & McAleer, M. (015). Economeric analysis of financial derivaives: An overview. Journal of Economerics, 187(), hp://doi.org/ /j.jeconom Chen, S. L., & Wu, J.-L. (005). Long-run money demand revisied: evidence from a non-linear approach. Journal of Inernaional Money and Finance, 4(1), hp://doi.org/ /j.jimonfin Chi, M. M., Rizov, M., & Willenbockel, D. (010). Exchange Rae Volailiy and Expors: New Empirical 147

8 Journal of Managemen and Susainabiliy Vol. 6, No. 1; 016 Evidence from he Emerging Eas Asian Economies. The World Economy, 33(), hp://doi.org/ /j x Con, R. (007). Volailiy clusering in financial markes: Empirical facs and agen-based models. Long Memory in Economics (pp ). hp://doi.org/ / _10 Daal, E., Naka, A., & Yu, J. S. (007). Volailiy clusering, leverage effecs, and jump dynamics in he US and emerging Asian equiy markes. Journal of Banking and Finance, 31(9), hp://doi.org/ /j.jbankfin Dickey, D. A., & Fuller, W. A. (1979). Disribuion of he esimaors for auoregressive ime series wih a uni roo. Journal of Th American Saisical Associaion, 74(366a), hp://dx.doi.org/ / Ding, Z., Granger, C. W. J., & Engle, R. F. (1993). A long memory propery of sock marke reurns and a new model. Journal of Empirical Finance. hp://doi.org/ / (93)90006-d Engle, R. F. (198). Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion. Economerica, 50(4), hp://doi.org/10.307/ Gaunersdorfer, A., & Hommes, C. H. (007). A Nonlinear Srucural Model for Volailiy Clusering. Long Memory in Economics (pp ). hp://doi.org/10.139/ssrn Glosen, L. R., Jagannahan, R., & Runkle, D. E. (1993). On he Relaion beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks. Journal of Finance, 48(5), hp://doi.org/10.307/39067 Gonzalez, C., & Gimeno, R. (01). Financial Analyss Impac on Sock Volailiy. Rerieved from hp://papers.ssrn.com/sol3/papers.cfm?absrac_id= Guo, H. (006). Time-varying risk premia and he cross secion of sock reurns. Journal of Banking and Finance, 30(7), hp://doi.org/ /j.jbankfin Higgins, M. L., Bera, A. K., & ohers. (199). A class of nonlinear ARCH models. Inernaional Economic Review, 33(1), hp://doi.org/10.307/56988 Mandelbro, B. (1967). The Variaion of Some Oher Speculaive Prices. The Journal of Business. hp://doi.org/ /95006 Miles, W. (008). Volailiy clusering in US home prices. Journal of Real Esae Research, 30(1), Rerieved from hp://ares.meapress.com/index/n3v544976h11635.pdf Nelson, D. B. (1991). Condiional Heeroskedasiciy in Asse Reurns: A New Approach. Economerica, 59(), hp://doi.org/10.307/93860 Plosser, C. I. (009). Financial economerics, financial innovaion, and financial sabiliy. Journal of Financial Economerics, 7(1), hp://doi.org/ /jjfinec/nbn014 Schwar, G. (1989). Tes for Uni Roo: A Mone Carlo Invesigaion. Journal of Business and Economics Saisics, 7, Zakoian, J. M. (1994). Threshold heeroskedasic models. Journal of Economic Dynamics and Conrol, 18(5), hp://doi.org/ / (94) Noe Noe 1. The DF es was also applied wih rend and p max an experimen suggesed by Schwar (1989) o ensure he ulimae resul on which we can base our analyical opinion ha is no eiher biased or is suffered by lag lengh p, boh ways resuled in rejecing he null hypohesis and accep he alernaive. Copyrighs Copyrigh for his aricle is reained by he auhor(s), wih firs publicaion righs graned o he journal. This is an open-access aricle disribued under he erms and condiions of he Creaive Commons Aribuion license (hp://creaivecommons.org/licenses/by/3.0/). 148