An Alternative Robust Test of Lagrange Multiplier for ARCH Effect

Transcription

1 Inernaional Journal of Mahemaics and Saisics Invenion (IJMSI) E-ISSN: P-ISSN: Volume 5 Issue 8 Ocober. 7 PP-6- An Alernaive Robus Tes of Lagrange Muliplier for ARCH Effec Md. Siraj-Ud-Doulah and Md. Bipul Hossen (Deparmenof Saisics, Faculy of Science, Begum Rokeya Universiy, Rangpur, Bangladesh) Corresponding Auhor: Md. Bipul Hossen ABSTRACT:ARCH model has become very popular in ha i enables he economerician o esimae he variance of a series a a paricular poin in ime. The aim of my paper is as follows: o sudy he ess of ARCH model in ime series daa, o sudy he effec of ouliers in ARCH model, o sudy he es of ARCH model in ime series daa afer using robus mehod and o sudy he es effec of ARCH model in ime series daa by simulaion. In his paper, I propose a new robus -es of ARCH. The usefulness of he proposed mehod is invesigaed by some well-known daa ses as well as Mone Carlo simulaion sudies. KEWORDS: Time Series; Homoscedasiciy; ARCH; Ouliers; LM es; -es of ARCH; Mone Carlo simulaion Dae of Submission: 7--7 Dae of accepance: I. Inroducion Usual knowledge, heeroscedasiciy is a warning of cross-secional daa and auocorrelaion is a noice of ime series daa. In developing he advance ime dependen daa analysis, ime series are ofen plagued by he problem of heeroscedasiciy. Modern analysis of financial daa and economeric daa, such as inflaion raes, exchange raes, volailiy of marke reurn, foreign exchange marke as well as real compensaion-produciviy daa ec, have found rich confirmaion of clusering large and small urbulence, which imply a form of heeroscedasiciy in which he variance of he disurbance depends on he size of he preceding disurbance, auoregressive condiional heeroscedasiciy, or ARCH model as a choice o he ypical ime series reamen []. If he value of any volailiy erm in any paricular period is correlaed wih is own preceding value, hen we say ha ARCH is presen in he variables. An ARCH model considers he variance of he curren error erm o be a funcion of he variables of he previous ime periods volailiy erms [,3]. In oher words, here is a paricular kind of heeroscedasiciy presen in which he variance of he regression volailiy depends on he volailiy in he pas [4]. There are several reasons for ARCH one of hem is when he daa se conains very large and small values or here is evidence of a clumping of a large and small volailiy [5]. There are wo formal graphical ess for deecion of ARCH, which are dependen variable agains index (daa ype, such as monh, year, day and ec.) as well as residuals agains index [, 4, 6, 8]. The graphical mehods are very easy o undersand when here is a clear-ou picure. Bu i may ofen produce ambiguous picures and analyss may come up wih conflicing conclusions. Tha is why more formal es like he analyical es is required. In our reviews, books index [, 4, 6, 7, 8] and journal aricles [, 5, 9,,, ]he mos celebraed es for he deecion of ARCH is Lagrange Muliplier (LM) es []. This es is suiable o deec he order of ARCH. Alhough exremely popular, he LM es is no always an appropriae es for deecion of ARCH. This es suffers a huge se back in he presence of ouliers or oulying observaions. For ha reason we propose a new robus -es o deec he order of ARCH when daa se conains ouliers or free from ouliers. Secion presens ARCH model. Proposed -es of ARCH is presened in secion 3. Examine he ARCH model in secion 4. Secion 5 presens Mone Carlo simulaion sudy. II. ARCH Model Los of economic ime series display periods of volailiy. Condiionally heeroscedasiciy models olerae he condiional variance of a series o depend on he pas undersanding of he error process. A large comprehension of he curren period s disurbance increases he condiional variance in successive periods. For a seady process, he condiional variance will evenually decompose o he long-run variance. As such, ARCH model can deain periods of urbulence and ranquiliy. Mahemaically, a modern version of he k-variable regression model can be expressed in he following forma 6 Page

2 An Alernaive Robus Tes of Lagrange Muliplier for ARCH Effec X Or k X k k k,,,, T To use he condiional informaion, in general, we have o assume ha ~ N, p p Now, we conclude our discussion of ARCH model by considering he aforesaid model, which akes he following form v () p p This is he ARCH (p) Model. Considering an auoregressive process wih Gaussian noise in equaion (), [3] a firs consider ouliers wihin ime series. There are wo ype of ouliers are defined mosly: Addiive Ouliers (AO) where a single observaion is affeced and Innovaion Ouliers (IO) where an unusual observaion in he generaing process affecs all laer observaions and he subsequen series [3]. In his paper, we sugges ha innovaions or disurbances based on he ordinary leas square (OLS) fi of he model should be more convincing and reliable in a diagnosic es for deecion of ARCH effec of he disurbance in ime series daa. III. Propose Tes Condiional variance is a measure of risk. ARCH effec has been included in a regression framework o es he hypohesis of risk-averse agens. Consider he following differen order ARCH model- ARCH (): ARCH (): ARCH (p): v v p p To see if hese are he case, we develop he following model vˆ ˆ vˆ ˆ Where and v are he esimaed values of and respecively. To es he hypohesis H Versus H : :, i. e., There, i. e., There is is no an ARCH ARCH effec effec Now, we would like o es he null hypohesis, he usual es saisic of can be wrien as Where, se ˆ ˆ ˆ mean ˆ vˆ ˆ mean ˆ v refers o he esimaed sandard error v and is he esimaed squared errors of esed ARCH model. I can be shown ha he variable hus defined follows he disribuion wih degrees of freedom and T is he number of observaions. IV. Resuls 4.. Real Daa Examples In he previous secion, we describe heoreical background of ARCH. Now, in his secion, apply some formal graphs and es o deec ARCH. Also apply our newly propose robus -es o deec ARCH. A firs, run he ime series by he OLS o obain he residuals. Then plo he daa wih wo ypes and also apply analyical ess in he daa ses. Le us firs consider he USA inflaion rae daa [3]. Our second daa se is US/UK exchange rae []. Our hird daa se is price of crude oil on he dollar rae, which has aken from [4]. In heir books, [3] considered LM es of ARCH as he only analyical es, alernaively, [] as well as [4] considered graphical aforemenioned wo ess and LM es of ARCH for deecion of ARCH effec in he daa ses. On he basis of hese daa ses he es resuls of he classical suggesed ess and our newly proposed ess are given in he following below- 7 Page

3 U S / U K E x c hange R a e I nf la ion R a e Monhly, /973-/996 early, Residuals U S / U K E x c hange R a e I nf la ion R a e early, Monhly, /973-/ Residuals An Alernaive Robus Tes of Lagrange Muliplier for ARCH Effec Aforesaid figures presen he daa wih ouliers Above-menioned figures express he daa wihou ouliers Figure. Graphical displays of USA inflaion rae daa From Fig. (a-d) he variaion in he series appears o be flucuaing, wih several clusers of large and small movemens for boh siuaions. According o he definiion of graphical ARCH es, here is he srong appearance of heeroscedasiciy. Table : ARCH Tess performance of USA Inflaion Rae daa wih and wihou Ouliers Saisic Value Criical Value (5%) P-Value Wih Ouliers Wihou Ouliers Wih Ouliers Wihou Ouliers Wih Ouliers Wihou Ouliers LM() Propose -Tes() Table shows ha LM es fail o deec he exisence of ARCH when daa se conains oulying observaions. Alernaively, when he daa se free from ouliers, i deec he exisence of ARCH (). Bu, our newly propose -es deec he presence of ARCH () when he daa se hold ouliers or free from ouliers. There is no ARCH effec in he daa se [3]. 4.. US/UK Exchange Rae Daa This daa se holds 86 observaions, a firs we make sure he ouliers by he robus LTS mehod [4]; i idenify 6 ouliers (case 3, 48, 87, 97, 6,,, 56, 59, 87, 5, 9, 7, 45, 46 and 47). Then apply prescribed graphical ess, classical LM es of ARCH as well as our newly proposed -es of ARCH before and afer erasing ouliers, which oucome have publicized below- On op of figures presen he daa wih ouliers Afore-menioned figures express he daa wihou ouliers Figure. Graphical exhibis of US/UK exchange rae daa From Fig. & show he paern are considerably ups and downs flucuaions over he sample period. In Fig. & denoe he periods of wide swings for some ime period and he periods of raher moderae swings in oher ime periods. According o he definiion of graphical ARCH ess, he plos exhibi considerably volailiy clusering. Table : ARCH Tess performance of US/UK Exchange Rae daa wih and wihou Ouliers Saisic Value Criical Value (5%) P-Value Wih Ouliers Wihou Ouliers Wih Ouliers Wihou Ouliers Wih Ouliers Wihou Ouliers LM() Propose -Tes() Oucome presened in Table shows ha he LM es fails o declare he correc conclusion of he daa se when ouliers presen in he daa. On he oher hand, when he daa se free from oulying observaions, i declare ha here is no ARCH effec in he daa. Bu, our newly propose -es announce he righ finding ha here is no ARCH effec when he daa se conain ouliers and free from ouliers. I is worh-menion ha here is srong indicaion of ARCH () []. 8 Page

4 P rice of Crude O il early, P rice of Crude O il early, An Alernaive Robus Tes of Lagrange Muliplier for ARCH Effec 4.3. Price of Crude Oil on Dollar Rae Daa This daa se has 7 observaions, a firs we ensure he ouliers by he robus LTS mehod [4]; i finds ou ouliers (case, 3, 6, 8, 5, 5, 4, 9,, and 6). Then apply recognized graphical ess, classical LM es of ARCH as well as our newly proposed -es of ARCH before and afer removing ouliers, which resuls have shown below- Above figures presen he daa wih ouliers Above-menioned figures express he daa wihou ouliers Figure 3. Graphical exhibis of price of crude oil on he dollar rae daa From Fig. 3 illusrae he behavior of he price of crude oil on dollar rae in macroeconomic analysis. From casual inspecion, Fig. 3 & show he rends o smooh ou long-erm flucuaions. In Fig. 3 & show he paern ranquil alongside periods wih large increase and decrease of volailiy. According o he definiion of graphical ARCH ess, such series display noiceably condiionally heeroscedasic. Resuls presened in Table 3 shows ha LM es fail o idenify he exac inference when he daa se holds ouliers. Conversely, when he daa free from ouliers, i declare here is no ARCH effec in he daa. Bu, when he daa se holds ouliers and free from ouliers, our newly propose -es declare ha here is no ARCH effec in he daa se. I is o be noiceably ha here is srong evidence of ARCH () [4]. Table 3: ARCH Tess performance of Price of Crude Oil on he Dollar Rae Daa wih and wihou Ouliers Saisic Value Criical Value (5%) P-Value Wih Ouliers Wihou Ouliers Wih Ouliers Wihou Ouliers Wih Ouliers Wihou Ouliers LM() Propose -Tes() To sum up he aforesaid discussion we have demonsraed ha for USA inflaion rae, US/UK exchange rae and price of crude oil on he dollar rae daa ses give opposie oucome when ouliers presen in he daa ses. We sudy ha he LM es is very sensiive o ouliers. Overall, we may conclude ha our newly propose -es is perfec o deec he order of ARCH when oulying observaions presen in he daa se or absen. V. Simulaion Sudy In he preceding secion, we consider few real daa ses o see how he ouliers affec he deecion procedure of ARCH effec in a ime series daa. Bu examples are no enough o ge a final answer o a problem. In his secion, we repor a Mone Carlo simulaion ha is design o compare he power performance of he LM es of ARCH as well as newly proposed -es of ARCH. In order o compare he power performance of aforesaid ess, we simulae arificial daa ses in he following way: o simulae homoscedasic and ARCH daa independenly from uniform disribuion based on ime series in five differen sample sizes, i.e. T= 5,,, 5 and respecively. Each experimen is run, imes and he es power of he boh cases are given in Table 4. Table 4. Simulaion Power of ARCH ess under Homoscedasiciy (. 5 Power(in percenage) T-5 T= T= T=5 T= LM() Propose -Tes() ) To invesigae he power of hese wo ess under homoscedasiciy, he LM es, he rejecion power of null hypohesis ( H ) is very much higher han our newly proposed -es when null hypohesis ( H ) is rue. Alernaively, our newly propose -es, he rejecion power of null hypohesis ( H ) is very low han LM es when null hypohesis ( H ) is rue when he daa se. 9 Page

5 An Alernaive Robus Tes of Lagrange Muliplier for ARCH Effec Table 5: Simulaion Power of ARCH ess under ARCH Errors ( H ) Power (in percenage) T-5 T= T= T=5 T= LM() LM() LM(3) LM(4) Propose -Tes() Propose -Tes() Propose -Tes(3) Propose -Tes(4) To examine he simulaed power of hese wo ess under ARCH, from Table 5 shows ha our newly propose -es, he rejecion power of null hypohesis ( H ) is much higher han LM es when alernaive hypohesis ( H ) is rue. Conversely, he performance of LM es is very poor and i power is less han 5% for his simulaion. We demonsrae ha he power of he newly proposed -es of ARCH is very high han LM es of ARCH for homoscedasic and heeroscedasic siuaion. Noice ha he rejecion power of null hypohesis ( H ) of his es is much higher han LM es of ARCH when alernaive hypohesis is rue in differen sample sizes. Therefore, we can say ha he newly proposed -es is appropriae han any oher ess o deec he order of ARCH. VI. Conclusion ARCH is poenially a serious problem specific o ime series daa. If he volailiy is ARCH in a linear model, he OLS esimaor will be consisen bu inefficien under he usual assumpions. Hence, he ess of hypohesis become invalid and give seriously misleading conclusions abou he saisical significance of he esimaed regression coefficiens as well as he model. For hese consequences, i is necessary o deec he exisence of ARCH in a given series of daa. There are differen ypes of ess for he incidence of ARCH. We can apply hese ess o deec he presence of ARCH. Bu hese exising ess have serious drawback. The mos popular and frequenly used es is Lagrange Muliplier (LM) es. This es can correcly idenify he order of ARCH. Bu his es is very much sensiive o oulying observaions. Consequenly, we develop new robus es of ARCH, which is -es of ARCH namely and i deec he order of ARCH correcly. We have seen ha irrespecive of he presence of ouliers or no, he newly proposed -es of ARCH performs much beer han oher ess for differen sample sizes. References [] R.F.Engle, Auoregressive condiional heeroskedasiciy wih esimaes of he variance of Unied-Kingdom inflaion, Economerica, 5, 98, [] D.N. Gujarai, Basic Economerics (Fourh Ediion, McRraw Hill, New ork, ) [3] W.H. Greene, Economeric Analysis (Fifh Ediion, New Jersey: Prenice- Hall, 8) [4] R.S. Pindyck, and D.L Rubinfeld, Economeric Models and Economic Forecass (McGraw Hill, New ork,997) [5] A.H.M.R.Imon,M.S.U. Doulah, and N.A. Hamzah, On he Deecion of ARCH effec in Time Series Daa, Proceedings of Inegraing Mahemaical Sciences Wihin Sociey,Malaysia, 7, [6] T.Bollerslev, R.F. Engle, and D.B.Nelson, Arch models, The Handbook of Economerics (Norh-Holland, 994) [7] W. Enders,, Applied Economeric Time Series (Second Ediion, John Wiley & Sons, Inc, Delhi, 8) [8] J. Johnson, andj. Dinardo, Economeric Mehods (Fourh Ediion, McGraw- Hill, New ork, 997) [9] A. Bera, and M.L. Higgins, ARCH models: properies esimaion and esing, Journal of Economic Surveys, 7,993, [] T. R. Bollerslev, Chou and K. Kroner, ARCH Modeling in Finance, Journal of Economerics, 5,98,5-59. [] R. F. Engle, and V. Ng, Measuring and Tesing he Impac of News on Rober Engle 67 Volailiy. Journal of Finance, 48(5), 993, [] J. Cragg, Esimaion and Tesing in Tesing in Time Series Regression Models wih Heeroscedasic Disurbances, Journal of Economerics,, 98, [3] A.J.Fox, Ouliers in Time Series, Journal of he Royal Saisical Sociey, 34,97, [4] P.J. Rousseeuw, & A. M. Leroy, Robus Regression and Oulier Deecion (New ork, Wiley. 987) Md. Siraj-Ud-Doulah and Md. Bipul Hossen, An Alernaive Robus Tes of Lagrange Muliplier for ARCH Effec. Inernaional Journal of Mahemaics and Saisics Invenion(IJMSI), vol. 5, no. 8, 7, pp. 6. Page