Robust critical values for unit root tests for series with conditional heteroscedasticity errors: An application of the simple NoVaS transformation

Transcription

1 WORKING PAPER 01: Robus criical values for uni roo ess for series wih condiional heeroscedasiciy errors: An applicaion of he simple NoVaS ransformaion Panagiois Manalos ECONOMETRICS AND STATISTICS ISSN hp://wwworuse/insiuioner/handelshogskolan-vid-orebro-universie/forskning/publikaioner/working-papers/ Örebro Universiy Swedish Business School Örebro SWEDEN

2 Robus criical values for uni roo ess for series wih condiional heeroscedasiciy errors: An applicaion of he simple NoVaS ransformaion Panagiois Manalos Deparmens of Saisics, Swedish Business School a Orebro Universiy, Sweden 1

3 ABSTRACT In his paper, we inroduce a se of criical values for uni roo ess ha are robus in he presence of condiional heeroscedasiciy errors using he normalizing and variancesabilizing ransformaion (NoVaS) in Poliis (007) and examine heir properies using Mone Carlo mehods In erms of he size of he es, our analysis reveals ha uni roo ess wih NoVaS-modified criical values have acual sizes close o he nominal size For he power of he es, we find ha uni roo ess wih NoVaS-modified criical values eiher have he same power as, or slighly beer han, ess using convenional Dickey Fuller criical values across he sample range considered Keywords: Criical values, normalizing and variance-sabilizing ransformaion, uni roo ess JEL Classificaion Codes: C01, C1, C15

4 1 Inroducion Recenly, here has been grea ineres by researchers and praciioners in he simulaneous presence of high-persisence and condiional heeroscedasiciy The main reason is heir occurrence in many economic ime series, including sock index and exchange rae series, boh a common subjec for uni roo ess Alongside burgeoning ineres in he second momen (or variance), he firs difference of economic variables here, sock index reurns and exchange rae changes has also been he subjec of a sizeable modelling lieraure in finance and relaed disciplines, wih he mos represenaive approach being he auoregressive condiional heeroscedasiciy (ARCH) and generalized ARCH (GARCH) family of models Subsequen o he seminal work on ARCH/GARCH by Engle (198) and Bollerslev (1986), he inroducion of new models has coninued apace, a common feaure being ha hey esimae and allow for ime-varying volailiy A number of suiable surveys of GARCH models are available, wih Giraiis, Leipus and Surgailis (006) providing a useful reference of exising work across he various models ha have predicive abiliy for squared reurns (or variance) Of his body of work, Panula (1989) is one of he firs sudies invesigaing nonsaionary univariae auoregressive models wih a regular uni roo and a firs-order ARCH error, deriving he asympoic disribuion of he leas squares esimaor (LSE) of he uni roo, he same as ha given by Dickey and Fuller (1979) Subsequenly, Kim and Schmid (1993) employed Mone Carlo simulaion o show ha he Dickey Fuller (DF) ess ended o over-rejec in he presence of GARCH errors However, hey also found ha he problem was no very serious, he excepion being nearinegraion of he GARCH process errors where he volailiy parameer was no small Ling 3

5 and Li (1998a) laer invesigaed an auoregressive inegraed moving average (ARIMA) model wih GARCH (p, q) errors and derived he asympoic disribuion of he maximum likelihood esimaion (MLE) They found ha he asympoic disribuion of he MLE of various uni roos involves a series of bivariae Brownian moions and ha he MLE of uni roos is more efficien han he corresponding LSE when GARCH innovaions are presen Using hese asympoic disribuions, Ling and Li (1998b) laer consruced some new uni roo ess and showed, using Mone Carlo simulaion, ha hese ess could provide beer performance han DF ess While here are differences across hese and oher sudies, hey all sugges ha uni roo ess require a se of robus criical values ha apply in he presence of GARCH errors For his purpose, his paper provides a simple and easy-o-apply algorihm using he normalizing and variance-sabilizing ransformaion (NoVaS) in Poliis (007) o produce robus criical values for ensuring DF ess have appropriae size and adequae power in he presence of condiional heeroscedasiciy However, because he subjec of uni roo ess is large, we limi he analysis here o invesigaing and demonsraing our procedure as i applies o uni roo ess in he absence of serial correlaion The remainder of he paper is organized as follows Secion presens he simple NoVaS and he uni roo es applying in he absence of serial correlaion In Secion 3, we presen our simulaions and use hese o esablish he size and power properies of he ess Secion 4 provides a brief summary and he main conclusions 4

6 NoVaS ransformaion and he uni roo es 1 Simple NoVaS ransformaion Poliis (007) firs defined and analysed he properies of he NoVaS ransformaion Le us consider he NoVaS more closely and hereby observe he moivaion for our sudy Le be a sequence of random variables following he ARCH(q) model q e h h ai i i 1 (11) where 0, a i 0, 1 i q and he innovaions e are iid N 0,1 quaniy Noe, we use he innovaions e as a sandard normal because we can inerpre he S q i 1 a i i (1) as a sudenized sequence Moreover, his provides an inuiive explanaion of he following NoVaS (3) quaniy u a, W a, for k 1, k,, T (13) k a i i W, s 1 a0 a where i 1, s 1 1 k k 1, 0, ai 0 for all i 0 and k i 0 a i 1 Equaion (13) describes he NoVaS ransformaion proposed by Poliis (007) under which he daa series is mapped o he new series u a, 5

7 The praciioner needs o selec he order k ( 0) and he vecor of non-negaive parameers wih he win goals of normalizaion/variance-sabilizaion in mind Wih a focus on ha primary goal of normalizaion and variance sabilizaion, Poliis (007) inroduced he so-called simple NoVaS ransformaion for choosing he order k ( 0) of he parameers a i in (3) Manalos (010) subsequenly modified he algorihm for he simple NoVaS as follows Le 0 and hen ai 1 k 1 for all 0 i k Selec he order k ( 0) such ha he p-value of he Jarque Bera saisic for he u, NoVaS-ransformed series is maximized In our analysis, we underake his in a loop for k = 1 o 5 Consider now ha ua W, a is he new NoVaS-ransformed innovaion Then he ransformed series has he following mean E( ) Eu ( a, ) (14) E W a, a I is no difficul o show ha, based on he assumpion ha he innovaions e are iid N 0,1 (see equaion (11)), ha (14) becomes zero, ha is E( ) Eu ( a, ) 0 E Wa, (15) As shown in he Appendix, he variance of he Simple NoVaS ransformed series is Var u E( ) E( ), a E ua EW E( ) a, ( ) 1 (16) The skewness of he innovaions e are iid N 0,1 is zero Lasly, as also shown in he Appendix, he kurosis is 6

8 kur z 4 4 E ua E( ) E W, a k 3 a ( ) k, a E u E E W (17) As Equaions (15) and (16) provide zero skewness and kurosis close o hree, he ransformed series exhibis he same properies as an iid N 0,1 variable DF uni roo ess in he absence of serial correlaion I is well known (Hamilon 1994) ha when esing for uni roos i is imporan o specify he null and alernaive hypoheses appropriaely based on he ype of he daa a hand For example, if he daa do no display any increasing or decreasing rend, hen we should no include a ime rend in he es regression Moreover, he inclusion of deerminisic erms will also influence he asympoic disribuion of he uni roo es saisics Consequenly, given he imporance of he correc alernaive hypohesis and he rend ype, he daa will deermine he form of he es regression used For his reason, and also because non-rending financial series, such as ineres and exchange raes and spreads, exhibi non-zero means and GARCH errors, he firs model we consider is he following The es regression is 1 Model 1: Consan erm in he regression y c y 1 (18) and he hypoheses o be esed are H0 : 1 y I (1) wihou a drif (19) H A: 1 y I(0) wih non-zero mean The es saisic we use is ha proposed by Dickey Fuller: (110) ˆ 1 SE( ˆ) 7

9 We remind ourselves here ha follows an unknown ARCH/GARCH process ha we could esimae, bu which we do no require for he purpose of our es procedure The second model we consider is as follows The es regression is Model : Time rend and consan erm in he regression y c y 1 b (111) In his case, we assume ha he daa under he null hypohesis, wih an example being a GDP series, follow a simple random walk wih drif Tha is y c y 1, (11) whereas under he alernaive hypohesis i is an AR(1) model wih 1 and rend However, even here we use he DF es saisic ˆ 1 Noe ha his formulaion is SE( ˆ) appropriae for ime series such as asse prices, ha is, rending ime series wih several possible kinds of GARCH errors 3 NoVaS criical values for DF uni roo ess in he presence of GARCH errors The underlying idea and mehod we use in our procedure o produce robus criical values is simple and idenical o he manner in which Dickey and Fuller (1979) produced heir criical values for he uni roo es, ha is, by using Mone Carlo mehods we can approximae he finie-sample disribuion of he uni roo es Model 1 Consider a sequence of T observaions y1, y,, yt for which we wish o es if here is a uni roo in he series In addiion, we assume ha our observed daa do no display any increasing or decreasing rend while a he same ime exhibiing a non-zero mean 8

10 The firs sep is o esimae he es regression y c y 1, (113) and calculae he DF es saisic ˆ 1 SE( ˆ) We perform he simple NoVaS ransformaion on he esimaed residuals from (113) ˆ y cˆ ˆy, ha i is, W a ˆ a ˆ 1 k, a 0 i i i 1 Based on he simple NoVaS properies (5, 6, 7), u W, a * z, is a good approximaion of he error erm in (13), where z, are T iid random numbers from he sandard normal disribuion The nex sep is o calculae he new series under he null hypohesis * T y u 1 We hen esimae he regression y c y e and aferwards he DF es saisic * * * * 1 * ˆ* 1 SE ˆ * ( ) Repeaing seps 3 4 a large number of imes, N MC (in our Mone Carlo simulaions, N = 1000), we are able o approximae he finie-sample disribuion of he DF uni roo es By aking he (1 ) quinile of he approximae finie-sample disribuion of *, we obain he -level criical values ( c * ) and finally rejec Ho if T s c * Model A Mone Carlo esimae of he p-value for esing is * * P We use his o sudy he size and power of he uni roo es (wih he assisance of he p-value plo) In he case of a rend 9

11 The firs sep is o esimae he es regression y c y 1 b, (114) and calculae he DF es saisic ˆ 1 SE( ˆ) ˆ We perform he simple NoVaS ransformaion on he esimaed residuals from (114), such ha y cˆ ˆy b ˆ (115) 1 Tha is, W a ˆ a ˆ, a 0 i i i 1 k The nex sep is o calculae u W, a * z, where z are as previously T iid random numbers from he sandard normal disribuion The nex sep is o calculae he new series under he null hypohesis y ( cˆ u ) * T 1 We hen esimae he regression y c y b e and aferwards he DF es * * * * * 1 saisic * ˆ* 1 SE ˆ * ( ) Seps 6 8 are he same as in Model 1 3 Simulaions 31 Mone Carlo In his secion, we perform a simple Mone Carlo experimen by generaing daa for he size of he es for Model 1 using y y 1 (116) For he power es, we generae daa using 10

12 y y 0,1 0,99y 1 (117) The daa-generaing process for he size of he es for Model is 01 y 1, (118) For he power es, we generae daa using y 0,1 0,99y (119) We sudy he performance of he NoVaS criical values wih error erms iid N(0,1) We assume whie noise and in all oher cases he errors follow a GARCH (1, 1) model wih he following daa -generaing process e h h 0, 001 0,199 0,800h 1 1 e ~ i i dn(0,1) (10) For each model, we perform 5,000 replicaions for he calculaion of size and 1,000 replicaions for he power funcions Our resuls are based on 600, 1100, and 100 replicaions, where he firs 100 ime series observaions in each replicaion is discarded o avoid possible iniial value effecwe calculae he esimaed size of he es by couning how many imes we rejec he null hypohesis in repeaed samples under condiions where he null is rue 3 The p-value plos The convenional way of reporing he resuls of a Mone Carlo experimen is o abulae he proporion of imes we rejec he null hypohesis in repeaed samples under condiions where he null is rue Concerning he significance levels used when judging he properies of he ess, differen sudies have advocaed boh larger and smaller significance levels For example, Maddala (199) suggess significance levels of as much as 5% in diagnosic esing, while MacKinnon (199) suggess going in he oher direcion 11

13 To address his problem, in his sudy we use mainly graphical mehods ha poenially provide more informaion abou he size and power of he ess We use he simple graphical mehods developed and illusraed in Davidson and MacKinnon (1998) as hey are relaively easy o inerpre We employ he p-value plo o assess he size of he ess and size power curves o evaluae he power of he ess The graphs of he p-value plos and size power curves are based on he empirical disribuion funcion (EDF) of he p-values, denoed Fˆ x j For he p-value plos, if he disribuion being used o compue he p s erms is correc, each of he p s erms should be disribued uniformly abou (0,1) Therefore, he resuling graph should be close o he 45 o line The p-value plos also make i possible and easy o disinguish beween ess ha sysemaically over-rejec or under-rejec he null hypohesis and hose ess ha rejec he null hypohesis near he righ amoun of ime To judge he reasonableness of he resuls, we use a 95% confidence inerval for he acual size ( 0 ) given 0(1 0), where N is he number of Mone Carlo 0 N replicaions We consider any resuls ha lie beween hese bounds as saisfacory For example, if we consider 5,000 Mone Carlo replicaions and a nominal size of 5%, we define a resul as reasonable if he esimaed size lies beween 439% and 561% Figure 1 Size of he uni roo ess of Models 1 and for 500 observaions wih whie noise Figure 1a Model 1 Figure 1b Model 1

14 Solid line is uni roo es wih NoVaS criical values; do dash line is uni roo es wih DF criical values 33 Whie noise case We begin our sudy of he behaviour of our es procedure wih he simple case of whie noise errors, ha is, series wihou GARCH errors We show only he resuls for 500 observaions given here is no difference beween hese and hose for 1,000 or,000 observaions As shown in Figure 1a, in Model 1 he uni roo es behaves well, wih boh he DF and NoVaS criical values indicaing esimaed sizes ha are inside he confidence inerval lines As we observe he same behaviour for he uni roo in Model, as shown in Figure 1b, we conclude ha he NoVaS criical values work as well as we expec for he case of whie noise Figure Power of he uni roo ess of Model 1 for 500 observaions wih whie noise Figure a Model 1 Figure b Model 13

15 Solid line is uni roo es wih NoVaS criical values; do dash line is uni roo es wih DF criical values Figure depics he power of he ess The firs hing we observe is ha he uni roo ess display he same power when using boh he DF and NoVaS criical values I is also obvious ha here is a sample effec (he lowes curves are for 500 observaions, followed by 1,000 observaions, and he highes curves are for,000 observaions) In sum, he uni roo ess wih NoVaS criical values assuming only whie noise error work equally as well as he uni roo ess wih convenional DF criical values 34 GARCH case 341 The size of he ess In his subsecion, we presen he resuls of our Mone Carlo experimen concerning he size of he uni roo es given GARCH errors As shown in Figure 3a for Model 1 and Figure 3b for Model for 500, 1,000 and,000 observaions, he p-value plos make i possible o easily disinguish ha uni roo ess wih DF criical values sysemaically over-rejec he null 14

16 hypohesis, whereas uni roo ess wih NoVaS criical values rejec he null hypohesis a abou he righ amoun of ime Once again, we recall ha we use he consan erm in he regression for Model 1 and a ime rend and consan erm in he regression in Model As shown, he uni roo es wih DF criical values over-rejecs by almos hree imes he nominal size, ha is, for a 5% nominal size and Model 1, we esimaed sizes of 1535% for 500 observaions, 149% for 1,000 observaions and for 1435% for,000 observaions We observe a similar picure for he uni roo es wih DF criical values for Model, where for a 5% nominal size we esimaed sizes of 1485 % for 500 observaions, 166% for 1,000 observaions, and 159% for,000 observaions Overall, he uni roos ess wih NoVaS criical values unlike hose wih DF criical values work well in he presence of GARCH errors Figure 3 Size of he uni roo ess for 500, 1,000 and,000 observaions wih GARCH errors Figure 3a Model 1 Figure 3b Model 15

17 Solid line is uni roo es wih NoVaS criical values; doed, do dash, and dashed lines are uni roo ess wih DF criical values for 500, 1,000, and,000 observaions, respecively 34 The power of he ess In his subsecion, we analyse he power of he Wald and boosrap ess using sample sizes of 500, 1,000, and,000 observaions We esimae he power funcion by calculaing he rejecion frequencies for 1,000 replicaions using Equaion (117) for Model 1 and Equaion (119) for Model We employ he size power curves o compare he esimaed power funcions of he alernaive es saisics We follow he same process o evaluae he EDFs denoed by F x j using he same sequence of random numbers used o esimae he size of he ess Figure 4 displays he resuls using he size power curves As shown, he uni roo ess wih he DF criical values are now superior o he uni roo ess wih he NoVaS criical values We also again observe a sample effec, such ha he larger he sample, he greaer he power of he ess To ensure a fair comparison of he power of he uni ess wih he DF and NoVaS criical values, we apply he size power curves on a correc size-adjused basis Figure 4 plos he size power curves and he esimaed power funcions agains he nominal size By ploing he esimaed power funcions agains he rue size, ha is F x j agains F x j, we obain size power curves on a correc size-adjused basis As he uni roo ess wih DF and NoVaS criical values now share he same power, we can see hey are even beer for a small nominal size (less han 15%), as shown in Figure 5 The main finding for our power invesigaion is ha uni roo ess wih NoVaS criical values generally perform adequaely in boh he whie noise and GARCH error cases Figure 4 Power of he uni roo ess for 500, 1,000, and,000 observaions wih GARCH errors 16

18 Figure 4a Model 1 Figure 4b Model Solid line is uni roo es wih NoVaS criical values; doed, do dash, and dashed lines are uni roo ess wih DF criical values for 500, 1,000, and,000 observaions, respecively Figure 5 Size-adjused power of he uni roo ess for 500, 1,000, and,000 observaions wih GARCH errors Figure 5a Model 1 Figure 5b Model 17

19 Solid line is uni roo es wih NoVaS criical values; doed, do dash, and dashed lines are uni roo ess wih DF criical values for 500, 1,000, and,000 observaions, respecively 4 Brief summary and conclusion In his secion, we summarize he resuls of our invesigaion The purpose of his sudy has been o sudy he uni roo es in he presence of condiional heeroscedasiciy errors wih newly proposed criical values modified using he normalizing and variance-sabilizing ransformaion (NoVaS) mehod The main conclusion is ha NoVaS-modified criical values are robus, wih respec o boh whie noise and condiional heeroscedasiciy errors Moreover, given hey have an acual size ha lies close o heir nominal size and adequae power, i makes sense ha we would selec our NoVaS-modified criical values ahead of he convenional DF criical values, especially in he presence of condiional heeroscedasiciy errors In summarizing our mehodology, we sudied he esimaed size and power of he uni roo es in he presence of condiional heeroscedasiciy errors using wo ses of criical values: namely, convenional DF values and our newly proposed NoVaS-modified values In erms of he size of he ess, we used Mone Carlo mehods o invesigae heir properies using 5,000 replicaions per model, wih 500, 1,000, and,000 observaions We employed p- value plos o invesigae he size of he ess and found ha he uni roo ess perform beer wih NoVaS-modified criical values across all of our samples Imporanly, when we consider he power resuls using he size power curves, when considered on a correc sizeadjused basis, he uni roo ess wih NoVaS-modified criical values share he same power as ess using DF criical values or a leas he difference is very small across all of our samples 18

20 5 References T Bollerslev, Generalized auoregressive condiional heeroscedasiciy, J of Econome 31(1986), pp Davidson, R and JG MacKinnon (1998) Graphical mehods for invesigaing he size and power of es saisics The Mancheser School, 66, 1 6 Dickey, D A, and Fuller WA (1979) Disribuion of he Esimaors for Auoregressive Time Series wih a Uni Roo J Amer Sais Assoc 74, Engle, Rober F (198) Auoregressive condiional heeroskedasiciy wih esimaes of he variance of UK inflaion Economerica, 50, L Giraiis, R Leipus and D Surgailis, D Recen advances in ARCH modelling, in Long Memory in Economics, A Kirman and G Teyssiere, eds, Springer, Berlin, 006 J Hamilon, Time Series Analysis, Princeon Universiy Press, 1994 Kim, K and Schmid, P (1993) Uni Roo Tess wih Condiional Heeroskedasiciy J Economerics 59, Ling, S and Li, WK (1998a) Limiing Disribuions of Maximum Likelihood Esimaors for Unsable ARMA Models wih GARCH Errors Ann Sais 6,

21 Ling, S and Li, WK (1998b) Esimaion and Tesing for Uni Roo Processes wih GARCH(1,1) Errors Technical repor Deparmen of Saisics, Universiy of Hong Kong MacKinnon, J G (199) Model Specificaion Tess and Arificial Regressions, Journal of Economic Lieraure, 30, Maddala, G S (199) Inroducion o Economerics, Second Ediion, New York, Maxwell Macmillan Panula, SG (1989) Esimaion of Auoregressive Models wih ARCH Errors Sankhya B, 50, Poliis, DN (007) Model-free vs model-based volailiy predicion, J Financial Economerics, 5(3), pp

22 6 Appendix 61 Mean, variance, kurosis and skewness of NoVaS Le us consider he model y E y 1 wih 1 ~ f(0, h ), where f is he unknown densiy funcion of he condiional on he se of pas informaion 1 The firs momen he mean of NoVaS Firs we remind ourselves ha for he simple NoVaS, 0 Now for he simple NoVaS wih one lag, ha is, W a a, such ha we always ake accoun ha is variance saionary and, a 1 m E, hen he mean is E W a a ae( ) ae( ) ae( ) am (11), a 1 1 For wo lags, he mean is E Wa, 3am, while for hree lags i is E Wa, 4am, and finally for he general k lags, he mean is E W k 1 am (1) a, The second momen of NoVaS For he simple NoVaS model wih one lag, we have W a a By aking he squares of his and hen he expecaion, we have, a 1 E W E a a, a 1 E a a a E a a a E W a E a E a E E (13) 4 4, a 1 1 1

23 If is fourh-order saionary wih 4 m4 E and m E, hen we have E W a E a E a E E a m a m (14) 4 4, a In he same manner, we can show ha he simple NoVaS model wih wo lags has a, 4 E W 3 a m 3 a m (15), for hree lags a, 4 E W 4 a m 6 a m (16) and finally for he general k lags E W k 1 a m k( k 1) a m (17) a, 4 Var W E( W ) E W Now he variance is, a, a, a Consequenly, we have a, 4 Var( W ) k 1 a m k( k 1) a m ( k 1) a m (18) Var( W ) am ka m m am a m a m m (19) a, Kurosis Consider ha z W, a is he new NoVaS ransformed series Then he kurosis of his series is kur z E z E( ) E W 4 4, a ( ), a E z E E W (130) We can see from (1) ha he mean of he simple NoVaS is E W k ae I, a 1 ( ) is easy o see ha for a 1 k 1 hen (1) becomes E W, a E( ) (131)

24 Based on his, we have he kurosis of he new ransformed series as kur z E (13) EW 4 ( ) a, Recall ha we derived he second momen of he NoVaS for he general k lags in (17) Now given we expec ha W, a is iid normal, and E 4, hen EW kur W E E W 4 a, 4, a 3 ( ) 3, a EWa, (133) Consequenly, wih for a simple analysis of (133) wih he subsiuion of he EW, a, and he righ-hand side of (17), he kurosis of he NoVaS ransformed series z 3 k( k 1) a becomes kur 1 3 k 1 a (134) Moreover, for a 1 k 1, (134) becomes z 3ka 3 k ( k 1) k kur 3 1 3a k 1 3 k 1 k (135) The las par of his equaion is he kurosis of he simple NoVaS ransformed series 3