Department of Economics Working Paper Series

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1 Deparmen of Economics Working Paper Series The Grea Moderaion and he Relaionship beween Oupu Growh and Is Volailiy WenSho Fang Feng Chia Universiy Sephen M. Miller Universiy of Connecicu and Universiy of Nevada, Las Vegas Working Paper 7-4 March 7 34 Mansfield Road, Uni 63 Sorrs, CT Phone: (86) Fax: (86) hp:// This working paper is indexed on RePEc, hp://repec.org/

2 Absrac This sudy examines he effec of he Grea Moderaion on he relaionship beween U.S. oupu growh and is volailiy over he period 947 o 6. Firs, we consider he possible effecs of srucural change in he volailiy process. In so doing, we employ GARCH-M and ARCH-M specificaions of he process describing oupu growh rae and is volailiy wih and wihou a one-ime srucural break in volailiy. Second, our daa analyses and empirical resuls sugges no significan relaionship beween he oupu growh rae and is volailiy, favoring he radiional wisdom of dichoomy in macroeconomics. Moreover, he evidence shows ha he ime-varying variance falls sharply or even disappears once we incorporae a one-ime srucural break in he uncondiional variance of oupu saring 98 or 984. Tha is, he inegraed GARCH effec proves spurious. Finally, a join es of a rend change and a one-ime shif in he volailiy process finds ha he one-ime shif dominaes. Journal of Economic Lieraure Classificaion: C3; E3; O4 Keywords: Grea Moderaion, economic growh and volailiy, srucural change in variance, IGARCH

3 . Inroducion Macroeconomic volailiy declined subsanially during he pas years. Kim and Nelson (999), McConnell and Perez-Quiros (), Blanchard and Simon (), Sock and Wason (3), and Ahmed, Levin, and Wilson (4), among ohers, documen his Grea Moderaion in he volailiy of U.S. GDP growh. Moreover, he curren Federal Reserve Board Chairman Bernanke (4) also addressed his issue. Mos research focuses on he causes of he Grea Moderaion such as good policies, srucural change, good luck, or oupu composiion shifs. This paper empirically invesigaes he effec of he Grea Moderaion on he relaionship beween he oupu growh rae and is volailiy. Macroeconomiss have long focused on business cycles and economic growh. Recenly, increasing aenion considers he relaionship beween business cycle volailiy and he long-run rend in growh. Alernaive models give rise o negaive, posiive, or independen relaionships beween he oupu growh rae and is volailiy. For example, he mispercepions heory, proposed originally by Friedman (968), Phelps (969), and Lucas (97), argues ha oupu flucuaions around is naural rae reflec price mispercepions due o moneary shocks, whils he long-run Good policies refer o beer managemen of he economy by moneary policy makers. Srucural change refers o beer invenory managemen. Good luck refers o he reducion in economic shocks (e.g., oil price shocks). Oupu composiion shifs refer o he fall in he volailiy of oupu componens, such as consumpion and invesmen. Bernanke (4) uses a lower-bound fronier on inflaion and oupu volailiies o organize his hinking. Inefficien moneary policy or invenory managemen leaves he economy above he fronier, whereas changes in he volailiy of random shocks will shif he lower-bound fronier. Sock and Wason (3) aribue he Grea Moderaion o good luck, implying ha he fronier shifed oward he origin. Bernanke (4) argues ha a subsanial porion of he Grea Moderaion reflecs beer moneary policy, implying a movemen oward he fronier. The disincion proves imporan, because if Sock and Wason (3) are correc, hen good luck can urn ino bad luck and he fronier can shif back o a more unfavorable rade-off. If Bernanke (4) is correc, hen mainaining good policy can coninue he benefis of he Grea Moderaion. Finally, Blanchard and Simon () and Eggers and Ioannides (6) find ha mos of he decline reflecs a decline in he volailiy of consumpion, invesmen, and/or manufacuring oupu. A leas wo inerpreaions can explain he decline in oupu growh-rae volailiy. For example, McConnell and Perez-Quiros () invoke a sep decrease someime in he mid-98s. Blanchard and Simon () deec a rend decline, emporarily inerruped in he 97s and early 98s. This sudy generally follows he approach of a sep decrease and hen invesigaes is effec on he relaionship beween he oupu growh rae and is volailiy. Secion 4, however, does consider he relaive effec of a rend decline agains a one-ime shif in volailiy.

4 growh rae of poenial oupu reflecs echnology and oher real facors. The sandard dichoomy in macroeconomics implies no relaionship beween he oupu growh rae and is volailiy. Bernanke (983) and Pindyck (99) demonsrae ha irreversibiliy makes invesmen especially sensiive o various forms of risk. Oupu volailiy generaes risk abou fuure demand ha impedes invesmen, leading o a negaive relaionship beween oupu volailiy and growh. Marin and Rogers (997) argue ha learning-by-doing generaes growh whereby producion complemens produciviy-improving aciviies and sabilizaion policy can posiively affec human capial accumulaion and growh. One naural conclusion, herefore, implies ha shor-run economic insabiliy can prove derimenal o human capial accumulaion and growh (Marin and Rogers, ). In conras, Black (987) argues ha echnology comes wih varying levels of risk and expeced reurns ha associae wih he degree of specializaion. More specializaion means more oupu volailiy. Invesmen occurs in specialized echnologies only if expeced reurns sufficienly compensae for associaed risk. Thus, when high expeced reurn echnologies emerge, high oupu volailiy and high growh coexis. Mirman (97) argues ha higher oupu volailiy leads o higher precauionary saving, implying a posiive relaionship beween oupu volailiy and growh. Bean (99) and Sain-Paul (993) show ha he opporuniy cos of produciviy-improving aciviies falls in recessions, implying ha higher oupu volailiy may posiively affec growh. According o Blackburn (999), a relaive increase in he volailiy of shocks increases he pace of knowledge accumulaion and, hence, growh, implying a posiive relaion beween oupu variabiliy and long-erm growh. In a simple sochasic growh model, Blackburn and Galindev (3) illusrae ha differen mechanisms of endogenous echnological change can lead o differen implicaions for 3

5 he relaionship beween oupu variabiliy and growh. Generally, he relaionship more likely exhibis a posiive correlaion, if inernal learning drives echnological change hrough deliberae acions ha subsiue for producion aciviies. The relaionship exhibis a negaive correlaion, if exernal learning drives echnological change hrough non-deliberae acions ha complemen producion aciviy. Blackburn and Pelloni (4) predic ha real shocks generae a posiive correlaion beween oupu variabiliy and growh and nominal shocks produce a negaive relaionship. The saisical evidence also exhibis ambiguiy. The empirical lieraure employs wo approaches. Using cross-counry daa, Kormendi and Meguire (985) and Grier and Tullock (989) find a posiive relaionship beween growh and is sandard deviaion, bu Ramey and Ramey (995), Miller (996), Marin and Rogers (), and Kneller and Young () repor a negaive relaionship. More recenly, Raffery (5) discovers ha unexpeced volailiy reduces growh and expeced volailiy increases i, while he combined effec of expeced and unexpeced volailiy reduces i. Applying generalized auoregressive condiional heeroskedasiciy in mean (GARCH-M) models, Caporale and McKiernan (996, 998) find a posiive relaionship beween oupu volailiy and growh for he U.K. and he U.S., whereas Founas and Karanasos (6) find a posiive relaionship for Germany and Japan. Speigh (999), Grier and Perry (), and Founas and Karanasos (6), however, conclude ha no relaionship exiss in he U.K and he U.S. In conras, Macri and Sinha () and Henry and Olekalns () discover a negaive link beween volailiy and growh for Ausralia and he U.S. The lack of robus evidence on he relaionship beween he oupu growh rae and is volailiy moivaes our analysis. While many empirical sudies employ pos-war daa, no one 4

6 explicily considers he effec of he Grea Moderaion on his relaionship. 3 The volailiy of U.S. GDP growh fell by more han half since he early o mid-98s. Alhough no agreemen exiss on he causes of he Grea Moderaion, he reduced volailiy implies ha empirical models for oupu growh over periods ha span he break may experience model misspecificaion. In addiion o considering he relaionship beween he oupu growh rae and is volailiy, we firs consider he possibiliy ha srucural change affecs he process generaing he volailiy of oupu growh. Deibold (986) firs raised he concern ha srucural changes may confound persisence esimaion in GARCH models. He noes ha Engle and Bollerslev s (986) inegraed GARCH (IGARCH) may resul from insabiliy of he consan erm of he condiional variance, ha is, nonsaionariy of he uncondiional variance. Neglecing such changes can lead o spuriously measured persisence; he sum of he esimaed auoregressive parameers of he condiional variance is heavily biased owards one. Lamoreux and Lasrapes (99) explore Diebold s conjecure and provide confirming evidence ha no accouning for discree shifs in uncondiional variance, he misspecificaion of he GARCH model, can bias upward GARCH esimaes of persisence in variance and, hus, viiaes he usefulness of GARCH when he degree of persisence proves imporan. The longer he sample period is, he higher he probabiliy is ha such changes will occur. Including dummy variables o accoun for such shifs diminishes he degree of GARCH persisence. More recenly, Mikosch and Sărică (4) argue heoreically ha he IGARCH model makes sense when non-saionary daa reflec changes in he uncondiional variance. Hillebrand (5) shows ha in he presence of negleced parameer change-poins, even 3 Grier and Tullock (989), using pooled cross-secion daa on 3 counries beween 95 and 98, invesigae empirical regulariies in pos-war economic growh. They find significan ime-period dummy variables or a rend variable in heir mean growh models for OECD counries and conclude he average growh rae, holding oher variables consan, rises in he pos-96 period. Similarly, Caporale and McKiernan (998) include wo dummies for periods of he Grea Depression and World War II in heir mean equaion over long sample period from 87 o 993. This paper focuses on srucural changes in he variance equaion. 5

7 a single deerminisic change-poin, GARCH inappropriaely measures volailiy persisence. Before carrying ou GARCH esimaions, we perform a horough change-poin sudy of he daa o avoid he spurious effec of almos-inegraion. The idenificaion of change poins will occur endogenously in he daa generaing process. We employ Inclán and Tiao s (994) ieraed cumulaive sums of squares (ICSS) algorihm o deec sudden changes in he variance of oupu growh, as well as he ime poin and magniude of each deeced change in he variance. 4 The algorihm finds one change poin a 98:, wo-year earlier han ha of 984: in McConnell and Perez-Quiros (). Mos analyss argue ha he break dae occurs some ime in he early o mid-98s, bu he exac iming of he decline remains conroversial. For example, Blanchard and Simon () analyze he large decline in U.S. oupu volailiy saring in 98:. This paper employs GARCH-M and ARCH-M models o examine he effec of he Grea Moderaion on he volailiy-growh relaionship over he period 947: o 6:4 wih he break dae of 98:. Our empirical resuls show srong evidence of IGARCH effecs and no evidence of significan links beween volailiy and growh for he U.S. Moreover, he ime-varying variance falls sharply or even disappears once we allow for he srucural break in he uncondiional variance of oupu growh. Tha is, he IGARCH effec proves spurious due o he Grea Moderaion. These resuls prove robus o he alernaive break 984:. Secion discusses he daa and he Grea Moderaion in oupu volailiy. Secion 3 presens he mehodology and he empirical resuls. Secion 4 considers addiional evidence. Finally, Secion 5 concludes. 4 Aggarwal, Inclán, and Leal (999) apply his algorihm o idenify he poins of sudden changes in he variance of reurns in en emerging sock markes, in addiion o Hong Kong, Singapore, Germany, Japan, he U.K., and he U.S. 6

8 . Daa and he Grea Moderaion Oupu growh raes ( quarerly real GDP ( ) equal he percenage change in he logarihm of seasonally adjused ), measured in billions of chained dollars, ha come from he U.S. Bureau of Economic Analysis (BEA) over he period 947: o 6:. A raher dramaic reducion in oupu volailiy in he mos recen wo decades relaive o he previous four produces he mos sriking observaion. McConnell and Perez-Quiros (), applying ess of Andrews (993) and Andrews and Ploberger (994), deec a unique break in he variance of he growh rae in 984: for he sample 953: o 999: wih no break in he mean growh rae. This paper exends he daa from 947: o 6:. Y y As discussed earlier, he mehodology used in his sudy o deec srucural changes in he variance employs he ICSS algorihm described by Inclán and Tiao (994). The analysis assumes ha he ime series of oupu growh displays a saionary variance over an iniial period, and hen a sudden change in variance occurs. The variance hen exhibis saionariy again for a ime, unil he nex sudden change. The process repeas hrough ime, yielding a ime series of observaions wih an unknown number of changes in he variance. Le{ ε } denoe a series of independen observaions from a normal disribuion wih mean zero and uncondiional variance σ. When N variance changes occur in T observaions, < k < k <... < k N < T equal he se of change poins. Le Ck equal he cumulaive sum of he squared observaions from he sar of he series o he k h poin in ime (i.e., = ε, k =,,T). Then, define D as : D k = ( C / C ) k T, k =,..., T wih D D. If no changes k k T / = T = C k k = in variance occur over he sample period, he D k saisic oscillaes around zero. If one or more sudden variance changes exis in he series, hen he D k values drif eiher up or down from zero. 7

9 Criical values based on he disribuion on D k under he null hypohesis of homogeneous variance provide upper and lower boundaries o deec a significan change in variance wih a known level of probabiliy. When he maximum of he absolue value of D k exceeds he criical value, we rejec he null hypohesis of no changes. Le equal he value of k for which max * k k D k occurs. If max k T / D exceeds he predeermined boundary, hen k provides an esimae of he k change poin. The facor T / sandardizes he disribuion. Under he null, asympoically behaves as a Brownian bridge. The criical value of.36 defines he 95 h percenile of he D k asympoic disribuion of max k T / D. Therefore, upper and lower boundaries occur a ±.36 k in he D k plo. Exceeding hese boundaries marks a significan change in variance of oupu. To examine muliple change poins, he ICSS algorihm successively evaluaes of he series, dividing consecuively afer finding a possible change poin. D k a differen pars The procedure idenifies one, and only one, change poin a 98:. Tha is, he shif lass o he end of our sample period wih no breaks in oher periods. Figure plos he series of real GDP and is growh rae and marks he break wih a gray area. We furher conduc srucural sabiliy ess for he uncondiional mean and variance of he growh rae by spliing he sample ino wo sub-periods: 947: o 98:4 and 98: o 6:. For he uncondiional mean, a Wald saisic ess for he equaliy of means for wo differen samples, while a variance-raio saisic ess for he equaliy of he uncondiional variances. Table repors descripive saisics for he daa and he resuls of he srucural sabiliy ess. The mean growh rae in each sub-sample nearly equals he.8358 percen growh rae average for he full 6-year sample. The Wald saisics, disribued as χ (), ha es for srucural change in he mean beween he samples canno rejec he null hypohesis of equaliy of means. In 8

10 conras, a clear decline in he sandard deviaion of he growh rae occurs, equaling.844 percen a quarer in he pre-98 period and.694 percen in he pos-98 period, a decline of 49 percen. The p-values for he variance-raio F-es significanly rejec he null of variance equaliy beween he samples. Economiss call he subsanial drop in he variance of oupu in he pos-98 period as he Grea Moderaion. Skewness saisics suppor symmeric disribuions for he full and pre-98 sample periods, bu no he pos-98 period. Kurosis saisics sugges ha he full and pos-98 sample series exhibi lepokuriciy wih fa ails. Consequenly, Jarque-Bera ess rejec normaliy for hese wo samples, bu canno rejec normal disribuions in he pre-98 sub-sample. The Ljung-Box Q-saisics (LB Q), esing for auocorrelaion of up o 6 lags, indicae serial correlaion in growh for all hree periods. The Lagrange Muliplier saisics (LM) es for ARCH effecs (Engle, 98) up o 6 lags, suggesing a ime-varying variance in oupu growh for he full and pos-98 sample periods and no heeroskedasiciy in he pre-98 period. The ADF uni-roo es implies ha he growh rae exhibis saionariy for each of he hree samples. Useful informaion emerges ha can assis in he model building and verificaion sages. Firs, he evidence of auocorrelaion suggess an auoregressive moving-average (ARMA) model for he mean growh equaion o capure emporal dependence and o generae whie-noise residuals for all he hree periods. Second, he emergence of a ime-varying variance argues for GARCH-M and ARCH-M models o examine he effec of volailiy on growh. The robusness of his conclusion o he Grea Moderaion, however, requires he allowance for a one-ime shif in he uncondiional variance. Third, no heroskedasiciy in he pre-98 sub-period suggess he use of oher volailiy measures such as a moving-sample sandard deviaion o examine he effec for he sub-sample. Fourh, and mos imporanly, he significan decline in he variance combined 9

11 wih no change of he mean growh rae in he pos-98 sub-period may imply a weak relaionship beween volailiy and growh. 3. Mehodology and Empirical Resuls We firs consruc an ARMA model for he growh series o remove any linear dependence in he daa, and hen add oupu volailiy as an explanaory variable in he mean equaion. Based on Schwarz Crierion (SC), an AR() process proves adequae o capure growh dynamics and produces whie-noise residuals for all he hree-sample periods. The mean growh equaion equals he following: y = a + a y + a y + λσ + ε, () where he growh rae y (lny lny ), lny equals he naural logarihm of real GDP, ε equals he whie-noise random error; and σ equals a measure of oupu volailiy. The esimae of λ may exceed or fall below zero and prove significan or insignifican. We, now, derive an operaional measure for oupu volailiy. The descripive saisics show ha ARCH effecs emerge in he full and pos-98 sample periods. The GARCH(,) specificaion proves adequae o represen mos financial and economic ime series (Bollerslev e al., 994). For example, Caporale and McKiernan (996), Speigh (999), Grier and Perry (), Henry and Olekalns (), and Founas and Karanasos (6) apply his process o parameerize he ime varying condiional variance of oupu growh. Caporale and McKiernan (998) and Macri and Sinha (), however, use ARCH() o examine he ime-dependence of he condiional variance. To provide more evidence, we employ boh GARCH(,) and ARCH() models o accoun for periods of high- and low-oupu volailiy for our samples as follows: = α + αε + βσ σ, and () = α + αε + α ε σ, (3)

12 where equaions () and (3) equal he GARCH(,) and ARCH() specificaions, and σ equals he condiional variance of he growh rae, given informaion available a ime -. The presence of he square roo of σ, σ, in he mean equaion of he growh rae makes equaions () and () or (3) a GARCH-M or an ARCH-M model (Engle e al., 987). The condiions ha α, β, and α + β in he GARCH model or α + α < in he ARCH model ensure posiive and < sable condiional variances of ε. The sums, α + β or α + α, measure he persisence of shocks o he condiional variances. Evidence of an inegraed GARCH (IGARCH), or in general, evidence of high persisence proves analogous o a uni roo in he mean of a sochasic process. This persisence may resul from occasional level shifs in volailiy. If β (or α ) equals zero, he process reduces o an ARCH(). When α and β (or α ) boh equal zero, he variance equals a consan. We esimae each of he models using maximum likelihood under normaliy and using he Bernd e al. (974) (BHHH) algorihm. For he pre-98 sub-period, since we do no find ime-varying variance, we also consruc a moving-sample sandard deviaion o measure oupu volailiy for 947: o 98:4 as follows: i i σ, (4) m.5 + m = [( ) (lny + i lny + i ) ] m i= where we selec m = 4, 8, and as he orders of he moving average. The choice of he moving-average order does no affec he resuls, however. Table repors he esimaion resuls of our GARCH-M and ARCH-M models for he full sample wih sandard errors in parenheses, p-values in brackes, and saisics for he sandardized residuals. In he mean equaion, AR() esimaes verify significance a he 5-percen level, lending suppor o he auoregressive specificaion. The coefficien of he condiional sandard deviaion ( λ ) possesses no saisical significance. Each esimae in he variance equaion exceeds zero. The

13 volailiy persisence of.996 in he GARCH and.6837 in he ARCH process, however, proves high. The likelihood raio (LR) ess for α + β in he GARCH and α + α in he ARCH = = process, respecively, do no rejec he null hypohesis of an IGARCH effec a he 5-percen level. I does rejec he IGARCH effec for he ARCH process, however, a nearly -percen level. The fied models adequaely capure he ime-series properies of he daa in ha he Ljung-Box Q-saisics for sandardized residuals (LB Q ) and sandardized squared residuals (LB Q ), up o 6 lags, do no deec remaining auocorrelaion and condiional heeroskedasiciy. The sandardized residuals exhibi symmeric disribuions, bu wih significan excess kurosis. Thus, hey do no exhibi he characerisics of a normal disribuion, as observed in Table. The insignifican esimae of he condiional sandard deviaion ( λ ) in he mean equaion implies no relaionship beween oupu growh and is volailiy. This resul conforms o Friedman s (968), Phelp s (969), and Lucas s (97) mispercepions hypohesis and he previous empirical findings, using GARCH-M models, of Grier and Perry () and Founas and Karanasos (6) for he U.S. and Speigh (999) for he U.K. This finding, however, proves inconsisen wih he discovery of a posiive relaionship by Caporale and McKiernan (996, 998) for he U.K. he U.S., and by Founas and Karanasos (6) for Germany and Japan, as well as he discovery by Macri and Sinha () and Henry and Olekaln () of a negaive relaionship for Ausralia and he U.S. Alhough differen counries, sample periods, daa frequencies, or economeric models may lead o differen findings, wo issues emerge from he empirical resuls. Firs, exising research effors do no limi he phenomenon of he Grea Moderaion o U.S. oupu only. Mills and Wang (3) and Summers (5) find srucural breaks in he volailiy of he oupu growh rae for he G7 counries and Ausralia, alhough he break occurs a differen imes. The

14 well-documened moderaion in he volailiy of GDP growh in he U.S. and oher developed naions suggess ha he finding of (G)ARCH-M effecs and volailiy persisence may prove spurious, since researchers fail o accoun for he srucural change in he variance. 5 Lasrapes (989) shows ha changes in he uncondiional variance should receive consideraion when specifying ARCH models. In his sudy, for insance, he persisence of volailiy in exchange raes decreases afer accouning for hree U.S. moneary policy regime shifs beween 976 and 986, diminishing he likelihood of inegraion-in-variance. Tzavalis and Wickens (995) find srong evidence of a high degree of persisence in he volailiy of he erm premium of bonds. Once hey allow for he moneary regime shif beween 979 and 98, he high persisence in he GARCH(,)-M model disappears. Poerba and Summers (986) noe ha he degree of persisence in variance of a variable imporanly affecs he relaionship beween he variable and is volailiy, for example, sock reurns and heir volailiy. Second, he significan saisical propery of excess kurosis provides a cauionary noe. Kurosis for he sandardized residuals (i.e., ε / σ ) generally falls below ha for he uncondiional sandard deviaion (see Tables and ). According o he disribuional assumpions in he (G)ARCH specificaion, he sandardized residuals should reflec a normal disribuion, if he (G)ARCH model oally accouns for he lepokuric uncondiional disribuion. The sample kurosis in Table for he sandardized residuals indicaes ha (G)ARCH accouns for some, bu 5 Employing he GARCH-M modeling approach, Caporale and McKiernan (998) and Founas and Karanasos (6) use annual real GNP or IP (indusrial producion) daa from he mid-8s o he 99s for he U.S. Japan, and Germany; Macri and Sinha () use Ausralian quarerly GDP index and IP from he lae-95s o he end of 999; Henry and Olekalns () use U.S. quarerly real GNP from 947 o 998; Caporale and McKiernan (996), Speigh (999), and Grier and Perry () use monhly IP from 948 o he mid-99s for he U.K. and he U.S. o examine he relaionship beween oupu growh and is volailiy. The longer he sample period is, he more likely srucural changes in variance occur. Only Caporale and McKiernan (996) creae a dummy variable equaling several periods of high volailiy in heir GARCH process. None of oher sudies consider possible srucural changes in variance. 3

15 no all, of he lepokurosis for he oupu growh rae. 6 Blanchard and Simon () noe ha he disribuion of oupu growh exhibis excess kurosis (or skewness), if large and infrequen shocks occur. This suggess ha he evidence of excess kurosis may also reflec he Grea Moderaion. Thus, we expec o resolve he wo puzzles by modeling he non-saionariy variance arising from he Grea Moderaion. Firs, he high persisence of oupu volailiy decreases afer accouning for he Grea Moderaion, diminishing he likelihood of biasing he sum of he esimaed auoregressive parameers oward one. Second, lepokurosis in he uncondiional disribuion of oupu growh vanishes afer adjusmen for (G)ARCH wih condiional normaliy. To consider he effec of he Grea Moderaion on he variance of oupu in he GARCH-M specificaion, we include a dummy variable in he condiional variance equaion, which equals uniy afer 98:; zero oherwise. To provide more evidence regarding he effec of he Grea Moderaion, we also esimae he variance process wih he break dae 984:. Table 3 repors he new esimaes, showing ha he srucural dummy proves highly significan in he variance equaion in all four cases. The subsanial and significan increase of he value of he maximum log-likelihood indicaes ha including he dummy in he (G)ARCH equaion provides a superior specificaion. The log-likelihood raio es saisics (i.e., χ () = 4.77 for he GARCH-M or χ () = 4.64 for he ARCH-M) prove significan a he -percen level. Based on he log-likelihood values, he wo models perform almos equally well. The Ljung-Box Q-saisics of he sandardized residuals and he squared sandardized residuals show 6 As a general rule, empirical sudies repor he firs- and second-order serial correlaion in he sandardized residuals of he GARCH esimaion based on Ljung-Box diagnosic saisics, bu lack skewness, excess kurosis, and normaliy ess in mos research. We argue ha he higher momens of he sandardized residuals provide imporan diagnosic informaion regarding accurae model specificaion and he rue daa generaing process. Speigh (999) repors excess kurosis and significan Jarque-Bera saisic afer GARCH adjusmen. Oher sudies pay no aenion o he behavior of kurosis before and afer GARCH esimaion. 4

16 no evidence of auocorrelaion and heeroskedasiciy, providing furher suppor for hese specificaions. The coefficiens of skewness and excess kurosis prove insignifican a he 5-percen level. 7 And, hus, he sandardized residuals conform o a normal disribuion. All resuls prove robus o he choice of he alernaive break a 984:, as shown in Table 3. 8 The esimae of λ remains insignifican. Tha is, no relaionship exiss beween growh and volailiy measured by he wo GARCH-M models wih wo differen break daes. Two imporan consequences emerge by allowing for a srucural change in he condiional variance. Firs, a large decline occurs in he esimaed degree of persisence in he condiional variance. Each esimae in he variance equaion in Table 3 falls below ha in he model wihou he dummy in Table. The highly significan LR saisic in Table 3 proves no IGARCH effec. In addiion, he esimaes of α and β or α and α no only fall in size bu also become insignifican in he specificaion ha includes he pos-984 dummy variable. The dummy variable replaces he (G)ARCH effecs. Second, a srong ineracion emerges beween he dummy variable and he excess kurosis, which previously proved significan (see Tables and ). This ineracion now proves insignifican. These resuls sugges ha he saisical evidence for ime-varying variance and for excess kurosis in he growh rae may reflec a shif in he uncondiional variance caused by he Grea Moderaion. Figures -5 plo he condiional variances wih and wihou a dummy for he four models, respecively. The solid line includes he dummy variable while he dashed line 7 Blanchard and Simon () calculae skewness and excess kurosis saisics for he error erm from heir esimaed rolling firs-order auoregressive process, finding significan skewness and excess kurosis only around he early 98s recession. These resuls mach our findings. Tha is, by incorporaing a one-ime shif in eiher 98: or 984: in he GARCH and ARCH variance equaions, we observe insignifican skewness and excess kurosis. 8 We also examine descripive saisics for he daa in he pre- and pos-984 subsamples. The same conclusions emerge as when he break equals 98: (see Table ). The Wald saisic canno rejec he null of equaliy of means beween samples and he variance raio rejecs he null of variance equaliy beween he samples. One minor difference does occur, however. No ARCH effecs exis in eiher he pre- and he pos-984 periods. To save space, we do no repor deailed saisics. 5

17 excludes he dummy variable. One common characerisic appears in he four Figures -- a clear shif in he variance. The high volailiy appears in he period before 98 or 984. The srucural break for he Grea Moderaion in 98: (or 984:) suggess ha we divide he sample ino pre- and pos-98 (or pos-984) groups o esimae he relaionship beween he growh rae and is volailiy separaely for each period. Since he descripive saisics indicae no ime-varying variance in he growh series for hree sub-samples (pre- and pos-984 and pre-98), we consruc a moving-sample sandard deviaion o proxy for oupu volailiy and use OLS o esimae he relaionship. For he pos-98 period, higher-order ARCH ess yield insignifican resuls in Table, suggesing he appropriaeness of a simple ARCH() model. 9 Table 4 presens he resuls. For he wo periods 947: o 98:4 and 947: o 983:4, he coefficien of he auoregressive erm a lag wo proves insignifican. Thus, we repor he esimaion resuls for an auoregressive model wih only one lag. The Ljung-Box diagnosic saisics show no evidence of firs- and second-order auocorrelaion in he residuals for he four sub-samples and he residuals reflec a normal disribuion. The insignifican esimae of λ again verifies our earlier finding ha no relaionship exiss beween he growh rae and is volailiy in he U.S. 4. Furher Evidence This secion considers wo addiional ess. Firs, we examine he possibiliy ha he oupu growh 9 We also experimened wih a GARCH(,) model, bu a negaive and insignifican esimae appears in he variance equaion, violaing he non-negaiviy assumpion. One referee noes ha some inheren limiaions may exis in GARCH-M models for examining he relaionship beween growh and volailiy. Paricularly, he GARCH-M models usually employ high-frequency daa. We use quarerly daa. Bu, he GDP growh in Figure exhibis volailiy clusers. Tha is, cerain ime periods experience high volailiy while oher periods experience low volailiy. This basic characerisic of he daa suggess applying he mehodology o measure volailiy and is effec on growh. Our approach focuses mainly on modeling he non-saionariy variance arising from he Grea Moderaion. In Table 4, he full sample splis and neiher he moving-sample sandard deviaion nor he ARCH process produces a significan effec on growh. 6

18 rae affecs is volailiy, exploring wheher an endogeneiy bias exiss in he GARCH and ARCH processes. Second, we sudy wheher a rend decrease in he volailiy of oupu growh provides a beer specificaion han he one-ime shif considered above. The firs es follows he analysis of Founas and Karanasos (6) while he second es addresses he conclusion of Blanchard and Simon (). Founas and Karanasos (6) recenly find, using annual indusrial producion daa from 86 o 999, ha he oupu growh rae volailiy exhibis no effec on he growh rae, bu he oupu growh rae affecs is volailiy negaively in he U.S. and a bidirecional causaliy beween oupu growh and is volailiy in Germany. The causal relaionship beween he oupu growh rae and is volailiy suggess ha he GARCH-M approach suffers from an endogeneiy bias. Founas and Karanasos (6) include lagged growh in he condiional variance equaion (he level effec) o es for he effec of growh on volailiy in heir GARCH(,)-M model. Following his specificaion, Table 5 repors he GARCH(,)-M esimaion resuls, where we consider his level effec. The insignificance of λ coninues, even while he lagged growh esimae (δ ) proves significan in he variance equaion for eiher break dae, 98: or 984:. All oher esimaes and diagnosic saisics close mirror hose in he models wihou his level effec. The findings ha he oupu growh rae does no depend on changes in is volailiy and ha he oupu growh rae does affec is volailiy negaively prove consisen wih evidence in Founas and Karanasos (6), alhough hey employ he long series of annual oupu daa and we use quarerly daa. Blanchard and Simon () argue ha a rend decline in he volailiy of oupu growh provides a beer explanaion of oupu growh volailiy han does he one-ime shif. Tables 6 and 7 presen he evidence. Table 6 inroduces a ime rend in specificaions of he GARCH and ARCH processes, bu wihou a one-ime shif dummy variable. The coefficien of he ime rend proves 7

19 negaive for boh specificaions, alhough only significanly negaive a he five-percen level in he ARCH process. All oher coefficiens and diagnosic saisics closely mirror hose in he models esimaed wihou he ime rend or he one-ime shif dummy variable (see Table ). One excepion exiss; he LR saisics now prove significan a he five-percen level when he ime rend appears, which rejecs he null hypohesis of an IGARCH. Furhermore, alhough he volailiy persisence falls subsanially wih he ime rend, excess kurosis remains. Thus, he ime rend capures some, bu no all, of he ime-varying propery of he variance. Finally, he volailiy measure remains insignifican in he growh rae equaion, maching he resuls of Tables and 3. Table 7 includes he ime rend and he one-ime shif dummy variable ogeher in he GARCH and ARCH processes. In all four models, he coefficien of he ime rend proves insignifican. Moreover, he coefficien of he one-ime shif dummy variable proves significanly negaive in each specificaion. All remaining coefficiens and diagnosic saisics nearly mach hose in Table 3, including he insignifican coefficien of he variance measure in he growh rae equaion. In summary, he one-ime shif dummy variable dominaes he ime rend across our various ess. Tha is, based on he log-likelihood value, he corresponding specificaions in Tables 3 and 7 do no exhibi significan differences, whereas he corresponding specificaions in Tables and 6 do exhibi significan differences wih hose in Tables 3 and Conclusion This paper examines he effec of he Grea Moderaion on he relaionship beween quarerly real GDP growh rae and is volailiy in he U.S. over he period 947: o 6:. We begin by considering he possible effecs, if any, of srucural change on he volailiy process. Our iniial resuls, based on eiher a GARCH-M or an ARCH-M model of he condiional variance of he 8

20 residuals, find srong evidence of volailiy persisence and excess kurosis in he growh rae. Subsequen analysis reveals ha his conclusion does no remain robus o a one-ime shif in oupu variabiliy due o he Grea Moderaion. Firs, he findings of a ime-varying variance measured by he GARCH-M or ARCH-M model disappear in he specificaions ha include he pos-984 dummy variable. Tha is, he GARCH effec proves spurious. In any case, no GARCH-M effec emerges. Second, excess kurosis vanishes in he specificaions ha include eiher he 98 or he 984 dummy variable in eiher he GARCH or he ARCH process. Boh he daa analysis and he OLS esimaes generally sugges no relaionship beween U.S. oupu volailiy and growh, favoring macroeconomic models ha dichoomize he deerminaion of oupu volailiy and growh. In sum, our resuls add o he conclusion ha he relaionship beween he oupu growh rae and is volailiy in he U.S. proves weak, a bes. The independence beween he oupu growh and is volailiy needs careful inerpreaion. Endogenous growh heory, for example, does no imply any imporance for he second momen. Blackburn and Galindev (3) and Blackburn and Pelloni (4) model he link beween he mean and variance of he oupu growh rae explicily. Differen mechanisms of endogenous echnological change and nominal or real shocks can lead o posiive or negaive relaionship beween growh and volailiy. In his model, Blackburn (999) shows for a linear endogenous learning funcion, he effec of he oupu growh-rae volailiy on he oupu growh rae equals zero. A concave (convex) learning funcion generaes a negaive (posiive) effec. Tha is, an independen relaionship may exis wih or wihou he Grea Moderaion. The discrepancy of our findings from hose in previous sudies highlighs he sensiiviy of he resuls o he counry considered, he ime period examined, he frequency of he daa, and he mehodology employed. This apparen inconclusiveness warrans furher invesigaion of he relaionship beween growh 9

21 and is volailiy. Moreover, since sudies generally focus on developed counries, addiional analysis from developing counries may prove illuminaing. For example, he Asian newly indusrializing counries may provide oally differen scenarios because of heir high growh raes. References Aggarwal, R., Inclán, C. and Leal, R. (999) Volailiy in emerging sock markes, Journal of Financial and Quaniaive Analysis 34, Ahmed, S., Levin, A. and Wilson, B.A. (4) Recen U.S. macroeconomic sabiliy: Good policies, good pracices, or good luck? Review of Economics and Saisics 86, Andrews, W. K. D. (993) Tes for parameer insabiliy and srucural change wih unknown change poin, Economerica 6, Andrews, W. K. D. and Ploberger, W. (994) Opimal ess when a nuisance parameer is presen only under he alernaive, Economerica 6, Bean, C. (99) Endogenous growh and he pro-cyclical behaviour of produciviy, European Economic Review 34, Bernanke, B.S. (983) Irreversibiliy, uncerainy, and cyclical invesmen, Quarerly Journal of Economics 98, Bernanke, B.S. (4) The Grea Moderaion, speech a Easern Economic Associaion, Washingon, February. Bernd, E. K., Hall, B. H., Hall, R. E. and Hausmann, J. A. (974) Esimaion and inference in nonlinear srucural models, Annals of Economic and Social Measuremen 4, Black, F. (987) Business Cycles and Equilibrium, Basil Blackwell, New York. Blackburn, K. (999) Can sabilizaion policy reduce long-run growh? Economic Journal 9,

22 Blackburn, K. and Galindev, R. (3) Growh, volailiy and learning, Economics Leers 79, Blackburn, K. and Pelloni A. (4) On he relaionship beween growh and volailiy, Economics Leers 83, 3-7. Blanchard, O. and Simon, J. () The long and large decline in U.S. oupu volailiy, Brookings Papers on Economic Aciviy 3, Bollerslev, T., Engle, R. F. and Nelson (994) ARCH model in Handbook of Economerics IV, , ed. Engle, R. F. and McFadden, D. C. Amserdam: Elsevier Science. Caporale, T. and McKiernan, B. (996) The relaionship beween oupu variabiliy and growh: Evidence from pos war UK daa, Scoish Journal of Poliical Economy 43, Caporale, T. and McKiernan, B. (998) The Fischer Black hypohesis: Some ime-series evidence, Souhern Economic Journal 63, Diebold, F. (986) Commens on modelling he persisence of condiional variance, Economeric Reviews 5, Eggers, A. and Ioannides, Y. M. (6) The role of oupu composiion in he sabilizaion of US oupu growh, Journal of Macroeconomics 8, Engle, R. F. (98) Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion, Economerica 5, Engle, R. F. and Bollerslev, T. (986) Modelling he persisence of condiional variance, Economeric Reviews 5, -5. Engle, R. F., Lilien, D. and Robins, R. (987) Esimaing ime varying risk premia in he erm srucure: The ARCH-M model, Economerica 55,

23 Founas, S. and Karanasos, M. (6) The relaionship beween economic growh and real uncerainy in he G3, Economic Modelling 3, Friedman, M. (968) The role of moneary policy, American Economic Review 58, -7. Grier, K. B. and Tullock, G.. (989) An empirical analysis of cross-naional economic growh, 95-8, Journal of Moneary Economics 4, Grier, K. B. and Perry, M. J. () The effecs of real and nominal uncerainy on inflaion and oupu growh: Some GARCH-M evidence, Journal of Applied Economerics 5, Henry, O. T. and Olekalns, N. () The effec of recessions on he relaionship beween oupu variabiliy and growh, Souhern Economic Journal 68, Hillebrand, E. (5) Neglecing parameer changes in GARCH models, Journal of Economerics 9, -38. Inclán, C. and Tiao, G. C. (994) Use of cumulaive sums of squares for rerospecive deecion of changes of variance, Journal of he American Saisical Associaion, 89, Kim, C J. and Nelson, C. R. (999) Has he U.S. economy become more sable? A Bayesian approach based on a Markov-Swiching model of he business cycle, Review of Economics and Saisics 8, -. Kneller, R. and Young, G. () Business cycle volailiy, uncerainy and long-run growh, Mancheser School 69, Kormendi, R. and Megquire, P. (985) Macroeconomic deerminans of growh: Cross-counry evidence, Journal of Moneary Economics 6, Lamoreux, C. G. and Lasrapes, W. D. (99) Persisence in variance, srucural change and he GARCH model, Journal of Business and Economic Saisics 68, Lasrapes, W. D. (989) Exchange rae volailiy and U.S. moneary policy: An ARCH applicaion, Journal of Money, Credi, and Banking,

24 Lucas, R. E. (97) Expecaions and he neuraliy of money, Journal of Economic Theory 4, 3-4. Macri, J. and Sinha, D. () Oupu variabiliy and economic growh: The case of Ausralia, Journal of Economics and Finance 4, Marin, P. and Rogers, C.A. (997) Sabilizaion policy, learning by doing, and economic growh, Oxford Economic Papers 49, Marin, P. and Rogers, C.A. () Long-erm growh and shor-erm economic insabiliy, European Economic Review 44, McConnell, M. M. and Perez-Quiros, G. () Oupu flucuaions in he Unied Saes: Wha has changed since he early 98 s? American Economic Review 9, Mikosch, T. and Sărică, C. (4) Non-saionariies in financial ime series, he long-range dependence, and he IGARCH effecs, Review of Economics and Saisics 86, Miller, S. M. (996) A noe on cross-counry growh regressions, Applied Economics 8, 9-6. Mills, T. C. and Wang, P. (3) Have oupu growh raes sabilized? Evidence from he G-7 economies, Scoish Journal of Poliical Economy 5, Mirman, L. (97) Uncerainy and opimal consumpion decisions, Economerica 39, Pindyck, R.S. (99) Irreversibiliy, uncerainy, and invesmen, Journal of Economic Lieraure 9, -48. Poerba, J. M. and Summers L. H. (986) The persisence of volailiy and sock marke flucuaions, American Economic Review 76, 4-5. Raffery, M. (5) The effecs of expeced and unexpeced volailiy on long-run growh: Evidence from 8 developed economies, Souhern Economic Journal 7 (3),

25 Ramey, G. and Ramey V. A. (995) Cross-counry evidence on he link beween volailiy and growh, American Economic Review 85, Sain-Paul, G. (993) Produciviy growh and he srucure of he business cycle, European Economic Review 37, Speigh, A. E. H. (999) UK oupu variabiliy and growh: Some furher evidence, Scoish Journal of Poliical Economy 46, Sock, J. H. and Wason, M. W. (3) Has he business cycle changed? Evidence and explanaions, Moneary Policy and Uncerainy: Adaping o a Changing Economy, proceedings of symposium sponsored by Federal Reserve Bank of Kansas Ciy, Jackson Hole, Wyo., Summers, P. M. (5) Wha caused he Grea Moderaion? Some cross-counry evidence, Economic Review (Third Quarer), Federal Reserve Bank of Kansas Ciy, 5-3. Tzavalis, E. and Wickens, M. R. (995) The persisence in volailiy of he US erm premium , Economics Leers 49,

26 Table : Descripive Saisics for Quarerly Growh of Real GDP 947:-6: 947:-98:4 98:-6: Sample size Mean Sandard deviaion Skewness (.59) (.77) -.593* (.474) Excess kurosis.34* (.384).373 (.455).79* (.4948) Jarque-Bera 7.46* [.].9739 [.644] * [.] LB Q() 6.63* 3.5* 3.96* LB Q(3) 34.54* 6.883* 34.89* LB Q(6) * 5.75* * LM () ** [.653].439 [.8338] 5.763* [.69] LM (3) 8.946* [.4].758 [.64] [.] LM (6).47* [.534] 4.96 [.658] 4.3 [.6468] ADF(n) -.888()* -8.46()* ()* Srucural sabiliy es for uncondiional mean H : full-sample=pre-98 H : full-sample=pos-98 H : pre-98=pos-98 H : full-sample pre-98 H : full-sample pos-98 H : pre-98 pos-98 Wald es Srucural sabiliy es for uncondiional variance H : full-sample=pre-98 H : full-sample < pre-98 F es.6947 [.7] Noe: H : full-sample=pos-98 H : full-sample > pos-98 H : pre-98=pos-98 H : pre-98 > pos [.] [.] Sandard errors appear in parenheses; p-values appear in brackes. The measures of skewness and kurosis are normally disribued as N (,6 / T ) and N (,4 / T ), respecively, where T equals he number of observaions. LB Q(k ) equals a Ljung-Box saisic esing for auocorrelaions in growh up o k lags. LM (k) equals he Lagrange muliplier es for condiional heeroskedasiciy, disribued asympoically as χ ( k ). ADF(n) equals he augmened Dickey-Fuller uni-roo es wih lags n seleced by he SC. The Wald saisic ( ˆ µ ˆ µ ) /( ) ess for srucural change in he mean beween he samples i and j, disribued as i j SD i + SD j χ () F ( df i, df j ). The F saisic equals he variance-raio es beween he samples i and j, asympoically disribued as and df denoes degrees of freedom. * denoes 5-percen significance level. ** denoes -percen significance level. 5

27 Table : GARCH-M Resuls wihou Srucural Break y = a + a y + a y + λσ + ε ; and = α + α ε + β σ σ a a a λ α α β.467* (.648).37* (.69).* (.75).9 (.96).6 (.3).66* (.49).8355* (.434) (3) LB Q (6) Skewness Kurosis Jarque-Bera LR.688 [.647] [.6379].757 [.869] [.3477].388 [.387].8898* [.59] 8.544* [.39].3 [.8839] Maximum log-likelihood funcion value: ARCH-M Resuls wihou Srucural Break y σ = a + a y + a y + λσ + ε ; and = α + α ε + α ε a a a λ α α α.389** (.47).33* (.755).6* (.65).3 (.497).366* (.564).37* (.883).373* (.9) (3) LB Q (6) Skewness Kurosis Jarque-Bera LR.6 [.4575] [.443].33 [.84] [.94] -.93 [.855].35* [.] 5.378* [.5].6977 [.5] Maximum log-likelihood funcion value: Noe: Sandard errors appear in parenheses; p-values appear in brackes; LB Q (k ) and LB Q ( k ) equal Ljung-Box Q-saisics esing for sandardized residuals and squared sandardized residuals for auocorrelaions up o k lags. LR equals he likelihood raio saisic following a χ disribuion wih one degree of freedom ha ess for α + β = or α + α =. * denoes 5-percen significance level. ** denoes -percen significance level. 6

28 Table 3: GARCH-M Resuls wih Srucural Break, 98: y = a + a y + a y + λσ + ε ; and σ α + α ε + β σ + γ Dummy, where Dummy = for 98:; oherwise. = a a a λ α α β γ.447* (.36).634* (.694).757* (.76).59 (.633).3948* (.958).8 (.69).67* (.767) (3) LB Q (6) Skewness Kurosis Jarque-Bera LR [.489] [.4894] [.847] [.5487] [.556] [.5684] [.795] Maximum log-likelihood funcion value: ARCM-M Resuls wih Srucural Break, 98: y = a + a y + a y + λσ + ε ; and σ α + α ε + α ε + γ Dummy, where Dummy = for 98:; oherwise. = -.33* (.65) 7.579* [.] a a a λ α α α γ.4644* (.366).88* (.75).559* (.74).54 (.69).9886* (.843).89* (.844).959 (.5) (3) LB Q (6) Skewness Kurosis Jarque-Bera LR [.499] [.4988] [.859] [.6478] [.6967] [.453] [.65] Maximum log-likelihood funcion value: y GARCH-M Resuls wih Srucural Break, 984: + a y + a + λσ + ε ; and = a y σ α + α ε + β σ + γ Dummy, where Dummy = for 984:; oherwise. = -.89* (.754) 8.59* [.] a a a λ α α β γ.4396* (.96).365* (.78).888* (.67).56 (.6).989* (.546).5 (.893).737 (.486) (3) LB Q (6) Skewness Kurosis Jarque-Bera LR [.5543] [.679] [.7376] [.38] [.683] [.45] [.6494] Maximum log-likelihood funcion value: ARCH-M Resuls wih Srucural Break, 984: y = a + a y + a y + λσ + ε ; and σ α + α ε + α ε + γ Dummy, where Dummy = for 984:; oherwise. = -.936* (.465) * [.] a a a λ α α α γ.436* (.369).59* (.76).86* (.746).533 (.68).68* (.946).93 (.8).83 (.73) (3) LB Q (6) Skewness Kurosis Jarque-Bera LR [.4467] [.5398] [.9349] [.554] [.636] [.544] [.7386] Maximum log-likelihood funcion value: * (.88).9* [.] Noe: Sandard errors appear in parenheses; p-values appear in brackes; LB Q (k ) and LB Q ( k ) equal Ljung-Box Q-saisics, esing for sandardized residuals and squared sandardized residuals for auocorrelaions up o k lags. LR equals he likelihood raio saisic, following a χ disribuion wih one degree of freedom ha ess for α + β = or α + α =. * denoes 5-percen significance level. 7

29 Table 4: Sub-Sample Resuls wih Srucural Break, 98: y = a + a y + a + λσ + ε ; and σ y [( ) (ln ln ) ].5 + m = m Y Y + i + i, where m=8. AR(): 947:-98:4 a a λ.86* (.457).389* (.9) (.6735) (3) LB Q (6) Skewness Kurosis Jarque-Bera.399 [.4949] [.674] 4.8 [.595] 5.36 [.539].69 [.786].7565** [.89] 3.39 [.88] AR()-ARCH-M esimaion: 98:-6: y = a + a y + a y + λσ + ε ; and σ α + α ε = a a a λ α α.6398* (.474).77 (.3).88* (.973) -.4 (.547).578* (.373).3386** (.8) (3) LB Q (6) Skewness Kurosis Jarque-Bera.87 [.645].484 [.8763].835 [.849].683 [.8477].668 [.88] [.743].793 [.574] y σ Sub-Sample Resuls wih Srucural Break, 984: + a y + a y + λσ + ε ; and = a [( ) (ln ln ) ].5 + m = m Y Y + i + i AR(): 947:-983:4 a a.844*.34* (.433) (.839) λ (.6578) (3) LB (6).939 [.5883] 4.66 [.647] AR(): 984:-6: a.4893* (.85).934 [.5333] a a.97** (.39).393* (.66).66 [.8566], where m=8. Q Skewness Kurosis Jarque-Bera [.8655] [.6] [.39] λ (.59) (3) LB (6).56 [.58].549 [.8636].6447 [.453] Q Skewness Kurosis Jarque-Bera [.9479] [.937] [.9939] [.493] Noe: Sandard errors appear in parenheses; p-values appear in brackes; LB Q (k ) and LB Q ( k ) equal Ljung-Box Q-saisics, esing for residuals (sandardized residuals) and squared residuals (squared sandardized residuals) for auocorrelaions up o k lags. * denoes 5-percen significance level. ** denoes -percen significance level. 8