TESTING FOR SKEWNESS IN AR CONDITIONAL VOLATILITY MODELS FOR FINANCIAL RETURN SERIES

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1 WORKING PAPER 01: TESTING FOR SKEWNESS IN AR CONDITIONAL VOLATILITY MODELS FOR FINANCIAL RETURN SERIES Panagiois Manalos and Alex Karagrigoriou Deparmen of Saisics, Universiy of Örebro, Sweden & Deparmen of Mahemaics and Saisics, Universiy of Cyprus, Cyprus ECONOMETRICS AND STATISTICS ISSN

2 TESTING FOR SKEWNESS IN AR CONDITIONAL VOLATILITY MODELS FOR FINANCIAL RETURN SERIES Panagiois Manalos and Alex Karagrigoriou Deparmen of Saisics, Universiy of Örebro, Sweden & Deparmen of Mahemaics and Saisics, Universiy of Cyprus, Cyprus ABSTRACT In his paper a es procedure is proposed for he skewness in auoregressive condiional volailiy models. The size and he power of he es are invesigaed hrough a series of Mone Carlo simulaions wih various models. Furhermore, applicaions wih financial daa are analyzed in order o explore he applicabiliy and he capabiliies of he proposed esing procedure. Keywords: ARCH /GARCH model, kurosis, NoVaS, skewness. JEL Classificaion Codes: C01, C1, C15

3 1. INTRODUCTION The momens and he variance of economic variables such as sock index reurns and exchange rae changes have been he subjec of a vas amoun of works in financial lieraure wih mos represenaive modeling examples he generalized auoregressive condiional heeroscedasiciy (GARCH) models. Since he inroducion of he ARCH/GARCH model by Engle (198) and Bollerslev (1986), numerous new models are being inroduced and invesigaed. All hese models are esimaed and allow for ime-varying volailiy. A naural quesion ha arises is he following: Wha abou for imevarying skewness or kurosis? I is specifically known ha he excess kurosis makes exreme observaions more likely han in he normal case, which means ha he marke gives higher probabiliy o exreme observaions han in he normal disribuion. Moreover, he presence of a negaive skewness has also ineresing and pracical implicaions: The effec of accenuaing he lef-hand side of he disribuion, inerpreed as ha, he marke gives higher probabiliy o decreases han increases in asse pricing. 3

4 Some of he works ha sudy ha sock reurn disribuions exhibi negaive skewness and excess kurosis, among ohers, are : Harvey and Siddique, 1999; Premarane and Bera, 000, Jondeau and Rockinger (000) and León, Rubio and Serna, (00). Harvey and Siddique (1999) presen a way o joinly esimae ime-varying condiional variance and skewness. Premarane and Bera (000) have suggesed capuring asymmery and excess kurosis wih he Pearson ype IV disribuion. Similarly, Jondeau and Rockinger (000) employ a condiional generalized Suden- disribuion o capure condiional skewness and kurosis. Finally, León, Rubio and Serna, (00) joinly esimae ime-varying volailiy, skewness and kurosis using a Gram-Charlier series expansion. However, researchers before applying various models should ask he following quesion for capuring he kurosis and skewness: Are he skewness and kurosis explained by he second momen of hose variables, ha is, an ARCH/GARCH effec? Or is i because he underline disribuion of he error erm is no normal? The purpose of he presen work is o answer he skewness par of he quesion, by deriving a skewness es for series ha have ARCH/GARCH effec. Such a es, in a easy and comprehensive way, can discriminae hese

5 wo causes, namely he ARCH/GARCH or he non-normal underline disribuion. The es is developed wih he implemenaion of he Normalising and Variance-Sabilizing Transformaion, known as NoVaS ransformaion, proposed and developed by Poliis in a series of papers (003, 007). The remainder of he paper is organized as follows. In Secion we evaluae he variance, he skewness and he kurosis of he NoVaS ransformed series and propose he new es procedure for he skewness of he error erm. In Secion 3, we presen our simulaions and use hem o esablish he size and power properies of he proposed es. Secion provides an applicaion on financial daa and a brief summary wih he main conclusions.. SKEWNESS, KURTOSIS & THE SKEWNESS TEST Le us consider he general model 1 y E y (.1) wih 1 ~ f(0, h ), where f is he unknown densiy funcion of condiional on he se of pas informaion 1. Le also he error sequence of random variables following he ARCH(q) model given by: be a 5

6 q e h h ai i i1 (.) where 0, a 0 and he innovaions i e are i.i.d N 0,1 sandard normal assumpion is used because he quaniy. Noe ha he S q a i1 i i (.3) can be viewed as he sudenized sequence. Moreover i helps in undersanding he NoVaS ransformaion. I was he widely repored failure of he ARCH (and GARCH) models o predic squared reurns one of he moivaions for he inroducion of he Normalizing and Variance Sabilizing Transformaion, known as NoVaS, for financial reurns series (see Poliis, 003; 007). More specifically, Poliis (007) defined and analyzed he properies of he following NoVaS ransformaion: z a, k 1 0 i i i1 s a a for k 1, k,, T (.) where s 1 1 k k1, 0, ai 0 for all i0 and ai 1. k i0 6

7 Equaion (.) describes he proposed NoVaS ransformaion under which he daa series is mapped o he new seriesz a,. The order k ( 0) and he vecor of nonnegaive parameers have o be chosen by he praciioner wih he win goals of normalizaion/variance sabilizaion in mind. Addiionally, Poliis (007) inroduced also he so called Simple NoVaS ransformaion for choosing he order k ( 0) and he parameers a i in (.). The algorihm for he simple NoVaS is: 1. Le 0 and ake a a 1 k 1 i for all 0 i k.. Pick he order k ( 0) such ha he absolue value of he excess kurosis of za, is minimized. Noe ha he simple NoVaS algorihm works in such a way ha he coefficien a is always adjused wih he neares ineger k which gives he minimum excess kurosis. Observe ha due o sep (1) of he algorihm, (.) reduces o: z a, a k i i0 ( W ), a for k 1, k,, T. (.5) Le z, W, a be he new NoVaS ransformed series. The following resul provides he firs 3 momens of he W a, series (he proof is given in he 7

8 appendix) and is used in Lemma where we provide he momens of new NoVas ransformed series z,. Lemma 1. Assume ha he sequence. Then, he firs momen of he W a, series is: a 1, is variance saionary and m E E W k am. (.6) If in addiion he sequence he second and hird momens of W a, are: a, 1 ( 1) is fourh-order saionary wih m E hen E W k a m k k a m (.7) and 3 a, E( W ) am akm m. (.8) Lemma : Under he same assumpions as in Lemma 1, he simple NoVaS Transformaed series has zero mean and variance 1. Furhermore, he hird and fourh momens for a general lag k are given by: E skew E z k a, 3 3 EW k a, E E and 8

9 kur z a E z a am m,, ka m Proof: Under he assumpion ha. is variance saionary and m E and wih he use of (.6) we have ha he mean of he ransformed series is E z a, E E. E W E( W ), a, a Due o he fac ha a ha a E z., 0 EW and based on (.1) where 0, 0 The second momen of simple NoVaS ransformed series is given by: E we have E z E E 1 1 ( ) a, EWa, k ae (.9) Where he las equaliy holds for a 1 k 1 variable z a, has mean zero and variance one.. Thus, he NoVaS ransformed For he fourh momen of he simple NoVaS for general k lags recall he assumpion ha is fourh-order saionary wih m E and m E E z. Then, a, EWa, E. (.10) 9

10 Using (.7) and aking a 1 k 1 a, we have E W am ka m. (.11) Combining (.9) (.11) we have ha he kurosis of he simple NoVaS is: kur z am m a, ka m (.1) Resriced (.1) o 3, or near o 3 (by he NoVaS algorihm), i is easy o derive ha: m 3k m k. (.13) Finally, for general k lags he hird momen of he simple NoVaS is: E z E. (.1) 3 3 a, 3 EWa, Using (.13) and (.1) we ge: a3k m E( W ) ak m m k 3 a, 1 (.15) 1 1 m m ak m k k k k 3. (.16) Subsiuing (.16) ino (.1) we finally obain he hird momen of he simple NoVaS ransformed series: 10

11 E z E k a, 3 3 EW k a, E E. (.17) Expression (.17) saes he main resul of our work because i reveals he relaion beween he skewness of he simple NoVaS and ha of he original series. More specifically, i saes ha he skewness of he simple NoVaS ransformed series z, is k k imes he skewness of he original unransformed series. Le us focus on he hird momen of he unransformed series in ARCH/GARCH form: E E( e ) E( h ) skew h (.18) 3 3 3/ e 3/ where e, h as in (.). e I is easily seen from (.18) ha if skew 0 hen he skewness of he z e ransformed series skew is also zero. On he oher hand, if skew 0 hen since (.18) is no zero, so is he skewness of he NoVaS ransformed series z skew. This las observaion can be used o derive a skewness es for he error erm in (.1) as follows: 11

12 1. NoVaS ransform he error sequence.. Calculae he es saisic: k SKEWz T skew k (.19) where skew is he sample skewness of he NoVaS ransformed series given by: 1 skew T 1 T T T i1 i1 z z z z 3 3. (.0) Observe ha he es saisic (.0) under he null hypohesis follows a chisquare disribuion wih one degree of freedom 1. The null hypohesis is h herefore, rejeced for values of he es saisic ha exceed he 100(1 ) percenile 1;a. 3. SIMULATIONS In his secion we provide he characerisics of he Mone Carlo experimen. We calculae he esimaed size by observing he number of imes he correc null hypohesis is rejeced in repeaed samples. By varying facors such as he number of observaions (500, 1000, 000 and 3000) we obain a succession of esimaed percenages of correc rae of selecion under differen condiions. 1

13 The Mone Carlo experimen has been performed by generaing daa according o he following GARCH(1,1) daa generaing processes:, 1 e h h h i 1. Models 1-5 are used o esimae he size of he es while, for he power we use Models 6 and 7. Noe also ha for Models 5-7 for generaing he innovaions e, we use he generalised lambda disribuion suggesed by Ramberg and Schmeiser (197), which is an exension of Tukey's lambda disribuion. The inverse disribuion funcion formula is 3 1 F u 1 u 1 u / (.1) where 1 and represen he locaion and he scale parameers and 3 and joinly deermine he shape of he disribuion. Under his formulaion we are allowed o sudy he skewness es under differen shapes. Table 1 summarizes he models wih differen values of he parameers considered. The number of replicaions per model used is 5, iniial observaions are discarded o remove iniializaion effecs.the calculaions were performed using GAUSS 8. 13

14 Table 1: Differen models used for he performance of he Skewness es Innovaions e 1 3 Model 1 0,0 0,9 i.i.d N 0,1 Model 0,5 0,70 i.i.d N 0,1 Model 3 0,75 0,75 i.i.d N 0,1 Model 0,60 0,30 i.i.d N 0,1 Model 5 0,5 0,5 -disr. 1 df Model 6 0,0 0,9 0,75 0,75 Model 7 0,75 0,75 skew=-0.30 kur=3.0 skew=-0.0 kur= The seps of he simulaion procedure are as follows: 1. Generae he GARCH(1,1) models as described before.. Use he Simple NoVaS o Normalize he generaing series. A loop wih lags from 1 o 35 are used, ha is, 35 differen Simple NoVaS ransformaions are used. Each ime we calculae he kurosis and he saisic for he kurosis: Kur T kurosis 3 / (.) Calculae he p-value of he saisic and he lag ha gives he maximum p-value, plus 3, is he lag k ha we use for he final (seleced) Simple NoVaS ransformaion. 1

15 3. Calculae he skewness, is usual saisic and he p-value (using for he original series: Skew T skewness /6 (.3) while for he NoVaS Normalized series we use he k adjused saisic: 1 ) SKEW z k Skew (.) k In his secion we use simple graphical mehods (Davidson and MacKinnon 1998), like he P-value plo and he Size-Power curve o sudy he size and he power of he proposed es. These graphs are based on he empirical disribuion funcion (EDF) of he P-values which is denoed by Fˆ x j. For he P-value plos, if he disribuion used o compue he p s erms is correc, each of he p s erms should be disribued uniformly on (0,1) and he resuling graph should be close o he 5 o line. For judging how reasonable he resuls are, we require ha he esimaed size should be bounded wihin he 95% confidence inerval of he nominal size (wo do lines). For example, if one considers a nominal size of 5%and 5000 Mone Carlo replicaion, a resul is defined as reasonable if he esimaed size lies beween.38% and 5.61%. All Size Figures show on he righ hand side he runcaed (up o 10% nominal level) P-value plos for he acual size of he Skewness ess. 15

16 3.1 The size and he power of he es Figure 1 shows he resuls for Models 1 and and n=500. The Dash line is he p-value for he skewness of he original series and Model 1 while he Do line refers o Model. In boh cases he es over-rejecs he null hypohesis. Noice ha he higher value of he he smaller he over-rejecion rae. On he oher hand he proposed skewness es behaves well; he esimaed size of he es is inside he 95% confidence inerval (wo do lines); see he he righ hand runcaed figure. Figure 1: Size of he es; Models 1 and ; n=500 Observaions Figure shows he resuls for Models 3 & wih n=500. The skewness es for he original series has esimaed size ha is far way from he nominal size (approx. 65% & 70%). Noice also ha he higher he value of he smaller he degree of over-rejecion. 16

17 On he oher hand, our skewness is wihin he 95% confidence bounds and rejecs he null hypohesis abou he righ proporion of ime. For n=1000, 000 & 3000 we noice ha he GARCH dynamic increasing he over-rejecion for he skewness es in he original series. In fac, he higher he number of he observaions he higher he over-rejecion. Again our proposed skewness es is very robus for he sample size. Figure : Size of he es; Models 3 and ; n=500 Observaions Finally, Figure 3 shows he resuls for Model 5 when he innovaions are -disribued wih 1 df. Dash line represens he skewness es for 1000, while he Do for 000 and Do-Dash for 3000 observaions. The over-rejecion is more han 80% larger han he esimaed size, which is even higher han he one for he normal innovaions. Once more he new es is robus and rejecs abou he righ proporion of he ime.. 17

18 Figure 3 Model ,000 and 3000 Observaions Le us consider now he power of he es. In Figures and 5 he solid curves represen he esimaed power of he new es for n=500 (lef par) and n=000 (righ par). The Do-Dash curves are for n=1000 (lef par) and n=3000 (righ par). For he original series ess we have respecively he Dash and Do curves. Observe ha he sample effec is significan. Indeed, he higher he sample size, he higher he power. The new es is so good ha has higher power ha he highes power a he 0% nominal level wih n=3000 observaions (Figure 10, righ hand side). For model 6 wih balanced GARCH parameers, recall ha ess based on he original series have size higher han 70%. Here, i urns ou ha he power is almos less han he size! 18

19 Figure Power of he Tes; Model 6 Figure 5: Power of he es; Model 7 In summary he size resuls clearly show ha he Skewness es of he original series over-rejecs he null hypohesis and is affeced by boh he sample size and he parameers of he GARCH process. On he oher hand he proposed es is very robus in all cases examined wih esimaed size very near o he nominal one. The power resuls of he ess are also very impressive indicaing he appropriaeness of he proposed es. 19

20 . APPLICATION TO FINANCIAL DATA The Hisorical Prices in Fig. 6 & 7 are for he Swedish OMX and he German DAX index (source: hp://finance.yahoo.com). In Fig. 8 & 9, we analyze he closing prices for BP and DELL. In each figure he 1s graph represens he original series and he nd he NoVaS ransformed series. The solid curve is a sandard normal variable ha helps or comparaive purposes. Figure 6a: OMX Index Sockholm (6 Jul 00-9 Nov-010) (1633 observaions) Figure 6b The cdf and pdf curve of he OMX series Figure 6b shows ha he original series has posiive skewness ( , p- value=0.0069) bu afer he NoVaS ransformaion (min=-3.131, max=3.071) he skewness is negaive ( , p-value= ). 0

21 Figure 7a DAX (GDAXI) German Index (1 Feb Dec 006) Figure 7b The cdf and pdf curve of he DAX series Figure 7b for he DAX index shows ha boh he original and he NoVaS ransformed series (min=-.071, max=3.305) have negaive skewness (before=-0.168, p-value =0.0007, afer=-0.61, p-value=1.9557e-008). 1

22 Figure 8a DELL (1 Jan Nov 010) Figure 8b The cdf and pdf curve of he DELL series Figure 8b for he closing prices for DELL is a represenaive example of he NoVaS ransformaion. I shows ha he negaive skewness ( , p-value = ) of he original series afer he NoVaS ransformaion (min= , max=3.67) becomes significan equal o zero ( , p-value 0.53).

23 Figure 9a BP (1 Jan Nov 010) Figure 9b The cdf and pdf curve of he BP series Figure 9b for he closing prices of BP shows ha neiher he original series nor he ransformed NoVaS series (min=-3.37, max=3.5) shows any significan skewness (original=-0). 3

24 5. CONCLUSIONS In his work we derived a skewness es for series ha have ARCH/GARCH effec. A es, in a easy and comprehensive way, can disinguish beween he causes for skewness, namely wheher here is an ARCH/GARCH effec or he underline disribuion is non-normal. The proposed es shows exremely good resuls in an exensive simulaion sudy no only in erms of he size bu also in erms of he power. The es has also been applied o a number of financial daa wih saisfacory resuls. This work provides he ground for invesigaing he complemenary problem of he kurosis which also can been used for similar purposes in financial reurn series.

25 REFERENCES Bollerslev, T. (1986). Generalized auoregressive condiional heeroscedasiciy, J. of Economeics. 31, Engle, Rober F. (198). Auoregressive condiional heeroskedasiciy wih esimaes of he variance of UK inflaion. Economerica, 50, Harvey, C R. & Siddique, A, Auoregressive Condiional Skewness, Journal of Financial and Quaniaive Analysis, Cambridge Universiy Press, 3(0), Jondeau, E. & Rockinger, M., 000. Condiional Volailiy, Skewness, and Kurosis: Exisence and Persisence, Working papers 77, Banque de France. Leon, A., G. Rubio and G. Serna (00), Auoregressive Condiional Volailiy,Skewness and Kurosis, WP-AD 00-13, Insiuo Valenciano de Invesigaciones Economicas. Poliis, D. N. (003). A normalizing and variance-sabilizing ransformaion for financial ime series, In: Recen Advances and Trends in Nonparameric Saisics, M. G. Akrias and D. N. Poliis, (eds.), Elsevier (Norh Holland), Poliis, D. N. (007). Model-free versus Model-based Volailiy Predicions, Journal of Financial Economerics, 5, Premarane, G. and Bera, A. (000). Modeling Asymmery and Excess Kurosis in Sock Reurn Daa, Working Paper , Universiy of Illinois. Ramberg, J. and Schmeiser, B. (197). An Approximae Mehod for Generaing Asymmeric Random Variables, Communicaions of he Associaion for Compuing Machinery, 17,

26 APPENDIX we derive below pars of he proof of Lemma 1: I) The firs momen of W a, Firs recall ha for he Simple NoVaS 0 and consider he Simple NoVaS wih one lag, namely, Recall ha W a a. (A.0), a 1 is variance saionary and m E a E W, E a a 1 am. For wo lags he mean is E Wa E W a,,. Then he mean is: 3am, while for hree lags is am and finally he general formula of he mean for k lags is: a 1 E W k am (A.1), II) The second momen of W a, For he Simple NoVaS model wih one lag, by squaring boh sides of (A.0) and hen aking expecaions we have: If, a 1 1 E W a E a E a E E. is -order saionary wih m E and m E hen we have: 6

27 a, E W a m a m. In a similar fashion we can show ha for he Simple NoVaS model wih wo lags we have a, 3 3 E W a m a m. Also for hree lags we have a, 6 E W a m a m and finally for he a, 1 ( 1) k lags: E W k a m k k a m. (A.) III) The hird momen of W a, Following he same principle as in he case of he nd momen, i is easy o show ha, he general formula for k lags for he hird momen is: E( W ) a k 1 E a k k 1 E E a k 1 E. (A.3) 3, a Equaion (A.3) for a 3 a, 1 k 1 and wih m E and m E becomes: E( W ) am akm m. (A.) 7