Regular Variation of Popular GARCH Processes Allowing for Distributional Asymmetry

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Regular Variation of Popular GARCH Processes Allowing for Distributional Asymmetry Todd Prono 2017-095 Please cite this paper as: Prono, Todd (2017). “Regular Variation of Popular GARCH Processes Allowing for Distributional Asymmetry,” Finance and Economics Discussion Series 2017-095. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.095. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Regular Variation of Popular GARCH Processes Allowing for Distributional Asymmetry1 Todd Prono2 First Version: August 2017 Abstract LinearGARCH(1;1)andthresholdGARCH(1;1)processesareestablishedasregularlyvarying, meaning their heavy tails are Pareto like, under conditions that allow the innovations from the,respective,processestobeskewed. Skewnessisconsideredastylizedfactformany(cid:133)nancial returns assumed to follow GARCH-type processes. The result in this note aids in establishing the asymptotic properties of certain GARCH estimators proposed in the literature. Keywords: GARCH, threshold GARCH, heavy tail, Pareto tail, regular variation. JEL codes:C20, C22, C53, C58. 1Gratitude is owed to Dong Hwan Oh for providing detailed comments. The views expressed in this paper are those ofthe author and do not necessarily re(cid:135)ect those ofthe FederalReserve Board. 2FederalReserve Board. (202) 973-6955,todd.a.prono@frb.gov. 1

1. Introduction Generalized autoregressive conditional heteroskedastic (GARCH) models are a workhorse for conditionalvarianceforecastingin(cid:133)nancialeconomics. ThelinearGARCH(1,1)modelofBollerslev (1986) is a popular choice amongst practitioners, owing, in part, to its (relative) simplicity, but also to its strong forecasting performance, generally, and superior performance, speci(cid:133)cally, on foreign exchangeratereturnsagainstmorecomplicatedalternatives(see; e.g.,HansenandLunde,2005). It is widely recognized that the conditional variance of equity returns tends to be asymmetric.3 This feature, sometimes referred to as a "leverage e⁄ect," is captured by threshold GARCH models, Glosten, Jagannathan, and Runkle (1993), hereafter GJR GARCH, being one-such example. In out-of-sample forecast evaluations using equity returns, GJR GARCH(1;1) is shown to improve uponthelinearGARCH(1;1)speci(cid:133)cation(HansenandLunde, 2005). Asaconsequence, thelinear GARCH(1;1) and GJR GARCH(1;1) models represent (very) popular choices among academics and practitioners alike for characterizing the conditional variance of (cid:133)nancial returns. Linear GARCH processes are shown to be regularly varying (see Basrak, Davis, and Mikosch, 2002), meaning their tails are heavy and Pareto like. Mikosch and Sta…rica…(2000) study the linear GARCH(1;1)caseindetailanddemonstrateittoberegularlyvaryingundertheconditionthatthe model(cid:146)s innovations follow a symmetric distribution.4 They do not consider the GJR GARCH(1;1) speci(cid:133)cation. Table 1 summarizes the skewness statistics on various (very) high frequency equity and foreign exchange rate returns. Evident from the table is that these statistics tend to be large in absolute terms and (highly) statistically signi(cid:133)cant, a tendency su¢ ciently prevalent to render skewness a stylized fact for many (cid:133)nancial returns. Under either the linear GARCH(1;1) or GJR GARCH(1;1) model, skewness in returns necessarily sources to the given model(cid:146)s innovations. Skewness in these innovations con(cid:135)icts with the aforementioned demonstration that a linear GARCH(1;1) process is regularly varying. Moreover (to the best of my knowledge), such a demonstration (regardless of the treatment of the model(cid:146)s innovations) is not extended to the GJR GARCH(1;1) case. As a consequence, this note establishes linear GARCH(1;1) and GJR GARCH(1;1)processesasregularlyvarying,wherethisresultdoesnotdependonthegivenmodel(cid:146)s innovations being symmetrically distributed. Besides being interesting in its own right, this result alsoaidsinestablishingthelarge-samplepropertiesofthelinearGARCH(1;1)estimatorsdiscussed 3That is, tomorrow(cid:146)s variance tends to be higher (all else equal) if today(cid:146)s return is negative. 4Davisand Mikosch (1998)conductan equally-detailed study ofthe linearARCH(1)case,demonstrating itto be regularly varying under the same condition. 2

inMikoschandStraumann(2002),KristensenandLinton(2006),andVaynmannandBeare(2014). 2. Regular Variation Consider the GARCH model of Y = (cid:27) (cid:15) ; (cid:15) i:i:d: D(0; 1); (1) t t t t (cid:24) where (cid:27)2 = !+(cid:11) Y2 I +(cid:11) Y2 I +(cid:12)(cid:27)2 : (2) t 1 t (cid:0) 1 (cid:2) f Y t (cid:0) 1(cid:21) 0 g 2 t (cid:0) 1 (cid:2) f Y t (cid:0) 1 <0 g t (cid:0) 1 Given (1), the general model under consideration follows the Drost and Nijman (1993) de(cid:133)nition of a strong GARCH process. Given (2), if (cid:11) = (cid:11) , the speci(cid:133)c model is GJR GARCH(1;1). Under 1 6 2 the special case where (cid:11) = (cid:11) , the speci(cid:133)c model is linear GARCH(1;1). Recasting (2) as 1 2 (cid:27)2 = !+(cid:27)2 (cid:13) (cid:15)2 +(cid:12) ; (cid:13) = (cid:11) I +(cid:11) I ; t t (cid:0) 1 t (cid:0) 1 t (cid:0) 1 t (cid:0) 1 1 (cid:2) f Y t (cid:0) 1(cid:21) 0 g 2 (cid:2) f Y t (cid:0) 1 <0 g = !+(cid:27)2 A(cid:0) (cid:1) t 1 t (cid:0) represents the GARCH process as a stochastic recurrence equation (SRE), which is important for establishing (Y ; (cid:27) ) as regularly varying. f t t gt Z 2 For a (cid:133)xed and non-negative h, let Y = Y = (Y ;(cid:27) ); :::; Y ;(cid:27) : t t t t+h t+h (cid:16) (cid:17) (cid:0) (cid:1) This section demonstrates that Y is regularly varying with (tail) index (cid:20), or, using shorthand notation, Y is RV((cid:20)). That is, there exists a sequence of constants a such that n f g nP ( Y > a ) 1; n ; n j j (cid:0)! ! 1 where denotes the max norm, a = n1=(cid:20)L(n), and L( ) is slowly-varying at . n j(cid:1)j (cid:1) 1 ASSUMPTION A1: The distribution D has an unbounded support, and E (cid:15) i+(cid:14) < for t j j 1 i 2 and some (cid:14) > 0. (cid:21) ASSUMPTION A2: ! ! > 0, (cid:11) > 0 for j = 1;2, and (cid:12) > 0. (cid:21) j 3

ASSUMPTION A3: E Al < 1 for l 1; i , where i is de(cid:133)ned in A1. t 2 2 (cid:0) (cid:1) (cid:2) (cid:3) The moment existence condition in A1 is (fairly) standard (see; e.g., Lee and Hansen, 1994, Berkes, HorvÆth, and Kokoszka, 2003, and Berkes and HorvÆth, 2004). The novelty of A2 is the strictly positive, lower-bound for ! (see Kristensen and Rahbek, 2005). Establishing regular variation for Y relies on taking a (cid:133)rst-order Taylor Expansion around this lower bound; see (7). t f g Notice, as well, the strict positivity of all model parameters, thus excluding the ARCH(1) case. In order to establish Y as regularly varying in the special case where (cid:12) = 0, see Prono (2016). t f g Under A3, (at least) E Y2 < (see; e.g., Loretan and Phillips, 1994, for empirical evidence t 1 supporting this condition(cid:0) for(cid:1)various stock and foreign exchange rate returns). A3 is su¢ cient for (Y ; (cid:27) ) to be strictly stationary (see; e.g., Mikosch, 1999, Corollary 1.4.38 and Remark 1.4.39). t t f g (Y ; (cid:27) ) is also strong mixing by Carrasco and Chen (2002, Corollary 6) when (cid:11) = (cid:11) , and f t t g 1 2 Carrasco and Chen (2002, Corollary 10), otherwise. A1 and A3 together distinguish (cid:15) as being thinner tailed than (cid:27) . As a consequence, t t f g f g regular variation of Y stems directly from (cid:27) , as is also the case in Davis and Mikosch (1998), t t f g f g Mikosch and Sta…rica…(2000), and Basrak et al. (2002). The generality of A1 and A3 includes the baseline case of a covariance-stationary GARCH(1;1) process, but also covers the higher-moment existence conditions necessary in Mikosch and Straumann (2002), Kristensen and Linton (2006), and Vaynaman and Beare (2014). PROPOSITION. For the GARCH model of (1) and (2), let Assumptions A1(cid:150)A3 hold, and consider Y = (Y ;(cid:27) ); :::; (Y ;(cid:27) ) 0 0 h h (cid:16) (cid:17) for a (cid:133)xed h 0. Then Y is RV((cid:20)). (cid:21) REMARK. In the proof that follows, C denotes a generic constant that can assume di⁄erent values in di⁄erent places. Proof. Since A is (strictly) positive t, t 8 P ((cid:27) > x) cx (cid:20); x ; (3) (cid:0) (cid:24) ! 1 wherec = c(!;(cid:11) ;(cid:11) ;(cid:12)), theprecisevalueofwhichisgiveninGoldie(1991), and(cid:20) (2; (cid:20)], where 1 2 2 4

(cid:20) is an upper bound, is the unique solution to E(A)(cid:20)=2 = 1; by Kesten (1973, Theorem 4). Next, for (cid:18) = (!;(cid:11) ;(cid:11) ;(cid:12)); (cid:18) = (!;(cid:11) ;(cid:11) ;(cid:12)); 1 2 1 2 de(cid:133)ne (cid:27)2 (cid:27)2((cid:18)) = !+(cid:13) Y2 +(cid:12)(cid:27)2 ; t t t 1 t 1 t 1 (cid:17) (cid:0) (cid:0) (cid:0) and (cid:27)2 (cid:27)2((cid:18)) analogously. Also de(cid:133)ne t t (cid:17) (cid:11) max((cid:11) ; (cid:11) ) (cid:13) t: (4) 1 2 t 1 (cid:17) (cid:21) (cid:0) 8 Then @(cid:27) @ (cid:27)2 1 @(cid:27)2 1 1 t = t = (cid:27) 1 t (cid:27) 1 ; (5) @! @! 2 (cid:2) (cid:0)t (cid:2) @! (cid:20) 2 (cid:2) (cid:0)t (cid:2) 1 (cid:12) p (cid:0) where the inequality follows from Lumsdaine (1996, Lemma 1, A1.2). Also, using recursive substitution, t 1 (cid:0) ! ! (cid:27)2 (cid:27)2 = (! !) (cid:12)i+ (cid:27)2 (cid:27)2 (cid:12)t (cid:0) : (6) t (cid:0) t (cid:0) 0 (cid:0) 0 (cid:20) 1 (cid:12) i=0 (cid:0) X (cid:0) (cid:1) Consider a (cid:133)rst-order Taylor Expansion of (cid:27) around ! such that t @(cid:27) (cid:27) = (cid:27) + t (! !) (7) t t @! (cid:0) (cid:13) Y2 +(cid:12)(cid:27)2 t 1 t 1 t 1 +C(cid:27) 1 (cid:20) (cid:0) (cid:0) (cid:27) (cid:0) (cid:0)t t (cid:13) Y2 +(cid:12)(cid:27)2 C t 1 t 1 t 1 (cid:0) (cid:0) (cid:0) (cid:20) (cid:2) (cid:27) (cid:18) t (cid:19) (cid:13) (cid:27)2 +C (cid:15)2 +(cid:12)(cid:27)2 C t 1 t 1 t 1 t 1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:20) (cid:2) (cid:27) (cid:0) t(cid:1) ! (cid:27)2 (cid:11)(cid:15)2 +(cid:12) (cid:15)2 C t 1 t 1 +C t 1 (cid:0) (cid:0) (cid:0) (cid:20) (cid:2) (cid:27) (cid:2) (cid:27) (cid:0) t (cid:1) t ! (cid:15)2 C (cid:27) A +C t 1 ; (cid:20) (cid:2) (cid:18) t (cid:0) 1 t (cid:2) (cid:27) (cid:0) t (cid:19) 5

where the (cid:133)rst inequality relies on (5) , the second on (cid:27) 1 being bounded and (cid:12) > 0, the third on (cid:0)t (6), the fourth on (4), and the (cid:133)fth on (cid:27) > (cid:12)1=2(cid:27) . Consider next t t 1 (cid:0) (cid:27) (cid:27) t 1 t 1: (cid:0) (cid:0) (cid:27) (cid:20) (cid:27) 0 0 For (cid:27) (cid:27) 1 1 (cid:27) (cid:20) (cid:27) 0 0 !+(cid:27)2A 1=2 0 1 (cid:20) (cid:27) (cid:0) 0 (cid:1) 1=2 !+(cid:27) A 0 1 (cid:20) (cid:27) 0 1=2 C A ; 1 (cid:20) (cid:2) where the third inequality follows from the Triangle Inequality, and the fourth from (cid:27) 1 being (cid:0)0 bounded and (cid:12) > 0. Parallel reasoning produces (cid:27) (cid:27) 2 2 (cid:27) (cid:20) (cid:27) 0 0 !+(cid:27)2A 1=2 1 2 (cid:20) (cid:27) (cid:0) 0 (cid:1) (cid:27) C 1 A 1=2 (cid:20) (cid:2) (cid:27) (cid:2) 2 (cid:18) 0(cid:19) 1=2 1=2 C A A : 1 2 (cid:20) (cid:2) (cid:2) Suppose then that (cid:27) (cid:27) t 2 t 2 t 2 C (cid:0) A 1=2 : (8) (cid:27) (cid:0) (cid:20) (cid:27) (cid:0) (cid:20) (cid:2) i 0 0 i=1 Q From (8) follows that (cid:27) (cid:27) t 1 t 1 (9) (cid:0) (cid:0) (cid:27) (cid:20) (cid:27) 0 0 !1=2 (cid:27) + t 2 A 1=2 (cid:20) (cid:27) 0 (cid:18) (cid:27) (cid:0) 0 (cid:19) (cid:2) t (cid:0) 1 t 1 C (cid:0) A 1=2 : i (cid:20) (cid:2) i=1 Q 6

Then Y = (cid:27) ((cid:15) ;1); (cid:27) ((cid:15) ;1); (cid:27) ((cid:15) ;1); :::; (cid:27) ((cid:15) ;1); 0 0 1 1 2 2 h h (cid:16) (cid:17) (cid:27) ((cid:15) ;1); C (cid:27) 0 A ((cid:15) ;1); C (cid:27) 1 A ((cid:15) ;1); :::; C (cid:27) h 1 A ((cid:15) ;1); (cid:20) 0 (cid:2) 0 (cid:2) (cid:27) 0 (cid:2) 1 1 (cid:2) (cid:27) 0 (cid:2) 2 2 (cid:2) (cid:27)(cid:0) 0 (cid:2) h h (cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17) +R (cid:27) E+R; 0 (cid:20) (cid:2) where h 1 E = ((cid:15) ;1); C A ((cid:15) ;1); C A 1=2 A ((cid:15) ;1); :::; C (cid:0) A 1=2 A ((cid:15) ;1) ; 0 (cid:2) 1 1 (cid:2) 1 2 2 (cid:2) i h h (cid:18) (cid:18)i=1 (cid:19) (cid:19) Q with A = (cid:11)(cid:15)2 +(cid:12) h 1; h h 1 (cid:0) 8 (cid:21) given (7) and (9), and R = 0; C (cid:15)2((cid:15) ;1); C (cid:15)2((cid:15) ;1); :::; C (cid:15)2 ((cid:15) ;1); (cid:27) 1 (cid:2) 0 1 (cid:27) 2 (cid:2) 1 2 (cid:27) h (cid:2) h (cid:0) 1 h (cid:16) (cid:17) given (7). Let Z = (cid:27) E+R. Because (cid:27) 1 is bounded h, the tail of R is (cid:145)light(cid:146)relative to 0 (cid:2) (cid:0)h 8 the tail of (cid:27) E. As a consequence, the tail of Z is determined by the tail of (cid:27) E. Then, 0 (cid:2) 0 (cid:2) since E E (cid:20)+" < h and some " > 0, (cid:27) E is RV((cid:20)) by (3) and Basrak et al. (2002, j hj 1 8 0 (cid:2) Corollar(cid:0)y A.2) w(cid:1)ith d = 1, which means that the tail of Z is determined by the tail of (cid:27) . Since 0 Y = (cid:27) D (cid:27) E+R, the tail of Y is also determined by the tail of (cid:27) , which implies, then, 0(cid:2) (cid:20) 0(cid:2) 0 that Y is RV((cid:20)). Let Y2 = Y2;(cid:27)2 ; :::; Y2;(cid:27)2 : 0 0 h h (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) The (general) method of proof behind the Proposition is comparable to those methods used to establish Y2 as RV((cid:20)=2) in Davis and Mikosch (1998, Lemma A.1) and Mikosch and Sta…rica…(2000, Theorem 2.3). In contrast to these two aforementioned results, however, moving to establish Y as RV((cid:20)) does not require a symmetric D. As a consequence, the Proposition is consistent with the empirical features (see Table 1) of many (cid:133)nancial returns to which the model of (1) and (2) gets applied and is complementary to Basrak et al. (2002). Moreover, the Proposition explicitly covers 7

a threshold GARCH(1;1) model under empirically-relevant cases. References [1] Basrak, B., R.A Davis & T. Mikosch (2002) Regular variation of garch processes. Stochastic Processes and Their Applications 99, 95-115. [2] Berkes, I., & L. HorvÆth (2004) The e¢ ciency of the estimators of the parameters in garch processes. Annals of Statistics 32, 633-655. [3] Berkes, I., L. HorvÆth & P. Kokoska (2003) GARCH processes: structure and estimation. Bernoulli 9, 201-227. [4] Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307-327. [5] Campbell, J.Y., A.W. Lo & A.C. MacKinlay (1997) The Econometrics of Financial Markets. Princeton University Press, Princeton. [6] Carrasco, M., &X.Chen(2002)Mixingandmomentpropertiesofvariousgarchandstochastic volatility models. Econometric Theory 18, 17-39. [7] Davis, R.A., & T. Mikosch (1998) The sample autocorrelations of heavy-tailed processes with applications to arch. The Annals of Statistics 26, 2049-2080. [8] Drost, F.C., & T.E. Nijman (1993) Temporal aggregation of garch processes. Econometrica 61, 909-927. [9] Glosten, L.R., R. Jagannathan & D.E. Runkle (1993) On the relation between expected value and the volatility of the nominal excess return on stocks. Journal of Finance 48, 1779-1801. [10] Goldie, C.M. (1991) Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1, 126-166. [11] Hansen, P.R., & A. Lunde (2005) A forecast comparison of volatility models: does anything beat a garch(1,1)? Journal of Applied Econometrics 20, 873-889. 8

[12] Kesten, H. (1973) Random di⁄erence equations and renewal theory for products of random matrices. Acta. Math. 131, 207-248. [13] Kristensen, D., & O. Linton (2006) A closed form estimator for the garch(1,1)-model. Econometric Theory 22, 323-327. [14] Kristensen, D., & A. Rahbek (2005) Asymptotics of the qmle for a class of arch(q) models. Econometric Theory 21, 946-961. [15] Lee, S.W., & B.E. Hansen (1994) Asymptotic theory for the garch(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 29-52. [16] Loretan, M., & P.C.B Phillips (1994) Testing the covariance stationarity of heavy-tailed time series. Journal of Empirical Finance 1, 211-248. [17] Lumsdaine, R.L. (1996) Consistency and asymptotic normality of the quasi-maximum likelihood estimator in igarch(1,1) and covariance stationary garch(1,1) models. Econometrica 64, 575-596. [18] Mikosch, T. (1999) Regular Variation, Subexponentiality and their applications in probability theory.Lecturenotesfortheworkshop"HeavyTailsandQueques,"EURANDOM,Eindhoven, Netherlands. [19] Mikosch, T., & C. Sta…rica…(2000) Limit theory for the sample autocorrelations and extremes of a garch(1,1) process. The Annals of Statistics 28, 1427-1451. [20] Mikosch, T., & D. Straumann (2002) Whittle estimation in a heavy-tailed garch(1,1) model. Stochastic Processes and Their Applications 100, 187-222. [21] Prono, T. (2016) Closed-form estimation of (cid:133)nite-order arch models: asymptotic theory and (cid:133)nite-sample performance. Finance and Economics Discussion Series 2016-083. Washington: Board of Governors of the Federal Reserve System. [22] Vaynman, I. & B.K. Beare (2014) Stable limit theory for the variance targeting estimator, in Y. Chang, T.B. Fomby & J.Y. Park (eds), Essays in Honor of Peter C.B. Phillips, vol. 33 of Advances in Econometrics: Emerald Group Publishing Limited, chapter 24, 639-672. 9

TABLE 1 CHF EUR JPY DJIA SPX freq. obs. skew. obs. skew. obs. skew. obs. skew. obs. skew. 1-min 174,741 0.41 190,338 -1.27 190,058 -1.59 46,557 -1.25 46,551 -1.75 (0.01) (0.01) (0.01) (0.01) (0.01) 5-min 34,973 0.35 38,081 -0.30 38,035 -1.20 9,315 -2.68 9,312 -3.17 (0.01) (0.01) (0.01) (0.03) (0.03) 10-min 17,489 0.55 19,044 -1.26 19,021 -0.75 (0.02) (0.02) (0.02) 15-min 11,660 0.14 12,699 -0.78 12,680 -0.73 (0.02) (0.02) (0.02) 20-min 8,747 -0.05 9,525 -0.50 9,512 -0.49 (0.03) (0.03) (0.03) Notes to Tables 1. All data source to Bloomberg LP. The date range for the Swiss Franc (CHF) spot return series is 1/16/2015(cid:150)7/1/2015. The date range for the Euro (EUR) and Japanese Yen (JPY) spot return series is 1/1/2015(cid:150)7/1/2015. The date range for the Dow Jones Industrial Average (DJIA) and S&P 500 (SPX) spot return series is 7/19/2015(cid:150) 12/31/2015. Skew is an estimate of the (unconditionally) standardized third moment. The standard error for this estimate is in parentheses and is measured against a null of normality, as in Cambell, Lo, and MacKinlay (1997). 10