Econ 623 Econometrics II Topic 7: ARCH Models 1 Forms and Properties of ARCH-type Models 1.1 ARCH(q) model (Engle, 1982) It models the conditional variance as a deterministic function of past returns. Suppose y is the return of a (cid:133)nancial asset. t y = (cid:22)+(cid:27) " t t t : (cid:27)2 = (cid:11)+(cid:12) y2 + +(cid:12) y2 (cid:26) t 1 t 1 (cid:1)(cid:1)(cid:1) q t q (cid:0) (cid:0) A special ARCH(q) model that we will consider here is, iid y = (cid:27) " ; " N(0;1) t t t t : (cid:24) (cid:27)2 = (cid:11)+(cid:12) y2 + +(cid:12) y2 ( t 1 t 1 (cid:1)(cid:1)(cid:1) q t q (cid:0) (cid:0) 1 Properties: E(y I ) = 0 t t 1 (cid:15) j (cid:0) E(y2 I ) = (cid:27)2 = (cid:11)+(cid:12) y2 + +(cid:12) y2 (cid:15) tj t(cid:0)1 t 1 t(cid:0)1 (cid:1)(cid:1)(cid:1) q t(cid:0)q y2 AR(q) (cid:15) t (cid:24) Since (cid:27)2 is time variant, the model is called the Autoregressive Conditional Het- (cid:15) t eroskedasticity (ARCH) model. According to the model, the conditional variance is completely determined by (cid:15) the lagged squared returns. 2 The kurtosis of ARCH is bigger than 3. In particular, when q = 1, the kurtosis (cid:15) of y is given by 3(1 (cid:12)2)=(1 3(cid:12)2) > 1 as long as (cid:12) = 0. So ARCH models t (cid:0) 1 (cid:0) 1 1 6 have fatter tails than the normal distribution. If (cid:12) ; ;(cid:12) ;(cid:11) > 0, and (cid:12) + + (cid:12) < 1; the ARCH(q) model is covariance 1 q 1 q (cid:15) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) stationary. If so, E(y2) = (cid:11)=(1 (cid:12) (cid:12) ) t (cid:0) 1 (cid:0)(cid:1)(cid:1)(cid:1)(cid:0) q E(y y ) = 0 for any t = s: t s (cid:15) 6 E(y2 y2) = 0 for some t = s: (cid:15) t s 6 6 In most empirical studies, we need many lags in the variance equation (ie q must (cid:15) be large). 3 1.2 GARCH(p,q) (Bollerslev, 1986) It models the conditional variance as a deterministic function of past returns and conditional variance y = (cid:27) " t t t ; (cid:27)2 = (cid:11) + q (cid:12) y2 + p (cid:11) (cid:27)2 (cid:26) t 0 j=1 j t(cid:0)j j=1 j t(cid:0)j P P Properties: E(y I ) = 0, E(y2 I ) = (cid:27)2 (cid:15) tj t(cid:0)1 tj t(cid:0)1 t y2 = (cid:11) + ((cid:11) + (cid:12) )y2 + + ((cid:11) + (cid:12) )y2 + w (cid:11) w (cid:11) w ; (cid:15) t 0 1 1 t(cid:0)1 (cid:1)(cid:1)(cid:1) m m t(cid:0)m t (cid:0) 1 t(cid:0)1 (cid:0) (cid:1)(cid:1)(cid:1) (cid:0) p t(cid:0)p where m = max p;q ; w = y2 (cid:27)2 f g t t (cid:0) t y2 ARMA(max p;q ;p) (cid:15) t (cid:24) f g According to the model, the conditional variance is the weighted average of (cid:15) the lagged squared returns (ARCH term) and the lagged conditional variance (GARCH term). 4 If (cid:11) ; ;(cid:11) > 0; (cid:12) ; ;(cid:12) > 0; (cid:11) > 0 and ((cid:11) +(cid:12) )+ +((cid:11) +(cid:12) ) < 1; the 1 p 1 q 1 1 m m (cid:15) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) GARCH(p,q) model is covariance stationary. E(y2) = (cid:11)=(1 (cid:11) (cid:11) t (cid:0) 1 (cid:0) (cid:1)(cid:1)(cid:1) (cid:0) m (cid:0) (cid:12) (cid:12) ) 1 m (cid:0)(cid:1)(cid:1)(cid:1)(cid:0) In the GARCH (1,1) model, E(y2 I ) = (cid:11) + ((cid:11) + (cid:12) )y2 + w (cid:11) w ; so (cid:15) tj t 0 1 1 t(cid:0)1 t (cid:0) 1 t(cid:0)1 ((cid:11) +(cid:12) ) governs the persistence of volatilities. 1 1 To describe the real (cid:133)nancial time series, a parsimonious GARCH model is often (cid:15) doing the job as well as an ARCH model with a long lag length. In the empirical study, the GARCH(1,1) speci(cid:133)cation has been found to be adequate in most application. In (cid:133)nancial data, often ((cid:11) +(cid:12) ) is very found to be close to 1 in the 1 1 GARCH(1,1) model. Therefore, the volatility process is persistent, suggesting volatility is predictable. 5 1.3 IGARCH (Engle and Bollerslev, 1986) Volatility is as persistent as a unit root y = (cid:27) " t t t (cid:27)2 = (cid:11) + q (cid:12) y2 + p (cid:11) (cid:27)2 ; 8 tq (cid:11)0 + jp=1 (cid:12)j =t(cid:0)j1 j=1 j t(cid:0)j < j=1 j Pj=1 j P Properties: : P P A shock to the conditional variance remains forever. (cid:15) Unconditional expected value of y2 is in(cid:133)nite. Hence, the IGARCH model is not (cid:15) t covariance stationary. It is parsimonious compared with GARCH(p,q). (cid:15) 6 1.4 GJR-GARCH (Glosten, Jaganathan and Runkle, 1993) Conditional variance responds to good news in di⁄erent ways to bad news. y = (cid:27) " t t t (cid:27)2 = (cid:11) + q (cid:12) y2 +(cid:13)y2 d + p (cid:11) (cid:27)2 ; 8 dt =01 if yi=1 0; bad news (y < 0) has t 1 (cid:15) (cid:0) larger impact on volatility than good news (y 0): t 1 (cid:0) (cid:21) It is also known as the threshold ARCH (TARCH) model. (cid:15) 7 1.5 EGARCH (Nelson, 1991) Model the log-conditional volatility to ensure volatility is positive. y = (cid:27) " t t t ; ln((cid:27)2) = (cid:11) + p (cid:11) ln((cid:27)2 )+ q ((cid:12) yt i +(cid:13) yt i) ( t 0 j=1 j t(cid:0)j i=1 i (cid:27)t(cid:0)(cid:0)i i(cid:27)t(cid:0)(cid:0)i (cid:12) (cid:12) Properties: P P (cid:12) (cid:12) (cid:12) (cid:12) Conditional variance depends on both the size and the sign of lagged residuals. (cid:15) (cid:12) captures the asymmetric e⁄ect. When (cid:12) < 0; bad news (y < 0) has larger i i t i (cid:15) (cid:0) impact on volatility than good news (y 0): t i (cid:0) (cid:21) In the volatility equation, ln((cid:27)2) but (cid:27)2 is modeled such that no restriction on (cid:15) t t the parameters is needed. 8 1.6 Absolute Value ARCH (Schwert, 1989) p (cid:27) = (cid:11) + (cid:12) y + (cid:11) (cid:27) t 0 i t i i t i j (cid:0) j (cid:0) i=1;q i=1 X X 1.7 NARCH (non-linear ARCH) (Higgin and Bera, 1992) q (cid:27)(cid:13) = (cid:11) + (cid:12) y (cid:13) + (cid:11) (cid:27) (cid:13) t 0 i t i i t i j (cid:0) j (cid:0) i=1 i=1;p X X Nest the standard GARCH model. If (cid:13) = 2, it is a GARCH; if (cid:13) = 1, it is an (cid:15) absolute value ARCH. 1.8 Quadratic ARCH (Sentana, 1991) (cid:27)2 = (cid:11) +(cid:12)y2 +(cid:14)y +(cid:11)h t 0 t 1 t 1 t 1 (cid:0) (cid:0) (cid:0) (cid:14) < 0 means positive returns increase volatility less than negative returns. (cid:15) 1.9 Switching ARCH (Cai, 1994; Hamilton and Susmel, 1994) y = (cid:27) " t t t (cid:27)2 = (cid:13)(S )+ (cid:12) y2 8 t t i=1;q i t i ; (cid:0) (cid:13)(S ) = (cid:13) +(cid:13) S > t 0 1 t >< P(S = i S =Pj) = p t t ij j > > There are two ARCH: models and the economy switches from one to another (cid:15) following a Markov chain. 9 1.10 ARCH in Mean (Engle, Lilien and Robin(1987)) Allow inter-temporal risk-return trade-o⁄s y = (cid:14)(cid:27)2 +(cid:27) " t t t t ; (cid:27)2 = (cid:11)+(cid:12) y2 + +(cid:12) y2 (cid:26) t 1 t 1 (cid:1)(cid:1)(cid:1) q t q (cid:0) (cid:0) This model is the extension of the Capital Asset Pricing Model (CAPM). (cid:15) 10

Econ 623 Econometrics II Topic 7: ARCH Models 1 Forms and Properties of ARCH#type Models

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