Lecture 12 Heteroscedasticity 3

ARCH Models: Stylized Facts of Asset Returns - Expected events – Cornell (1978), Patell and Wolfson (1979), etc: Volatility is high at regular times such as news announcements or other expected events, or even at certain times of day –for example, less volatile in the early afternoon. - Volatility and serial correlation – LeBaron (1992): Inverse relationship between the two. - Co-movements in volatility – Ramchand and Susmel (1998): Volatility is positively correlated across markets/assets. • We need a model that accommodates all these facts. ARCH Models: Stylized Facts of Asset Returns • Easy to check leptokurtosis (Stylized Fact #1) Figure: Descriptive Statistics and Distribution for Monthly S&P500 Returns Statistic Mean (%) Standard Dev (%) Skewness Excess Kurtosis Jarque-Bera AR(1) 0.626332 (p-value: 0.0004) 4.37721 -0.43764 2.29395 145.72 (p-value: <0.0001) 0.0258 (p-value: 0.5249) 25 ARCH Models: Stylized Facts of Asset Returns • Easy to check Volatility Clustering (Stylized Fact #2) ARCH Models: Engle (1982) • We start with assumptions (A1) to (A5), but with a specific (A3’): 𝑦 (A3’) 𝜎 𝛾𝑋 𝜀, 𝑉𝑎𝑟 𝜀 𝜀 ~ 𝑁 0, 𝜎 𝐸 𝜀 𝜔 𝛼 𝜀 ⋯ 𝛼 𝜀 , which we can write, using the L operator, as: ∑ 𝜎 𝛼𝜀 𝜔 𝛼 𝐿 𝜀 • We can write the model in terms of an AR(q) for 𝜀 . Define 𝜈 ≡𝜀 𝜎 , -an error term for the variance. Then, 𝜀 𝜔 𝛼 𝐿 𝜀 𝜈 • Correlated 𝜀 ’s: High (low) past 𝜀 ’s produce a high (low) 𝜀 today. 52 26 ARCH Models: Engle (1982) • The model 𝜎 ∑ 𝛼𝜀 𝜔 𝛼 𝐿 𝜀 is an AR(q) model for squared innovations, 𝜀 . We have the ARCH model: Auto-Regressive Conditional Heteroskedasticity. • The ARCH(q) model estimates the unobservable (latent) variance. • Non-negative constraints: Since we are dealing with a variance, we usually impose 𝜔 > 0 and 𝛼 > 0 for all 𝑖. 𝜀 Notation: 𝐸 =𝐸 𝜀 |I Useful result: Since E[𝜀 ] = 0, then 𝐸 [𝜀 ] = 𝜎 53 Robert F. Engle (1942, USA) ARCH Models: Unconditional Variance • The unconditional variance is determined by: 𝜎 𝐸𝜎 𝜔 𝛼𝐸𝜀 𝜔 𝛼𝜎 That is, 𝜎 ∑ To obtain a positive σ2, we impose another restriction: (1 ∑ 𝛼)>0 Example: ARCH(1) 𝑌 𝜎 𝛽𝑋 𝜔 𝜀, 𝛼 𝜀 𝜀 ~ 𝑁 0, 𝜎 𝜎 • We need to impose restrictions: 𝜔 > 0, 𝛼 > 0, & (1 – 𝛼 ) > 0. 54 27 ARCH Models: Leptokurtosis • Errors may be serially uncorrelated, but they are not independent: There will be volatility clustering, which produces fat tails. We want to calculate the kurtosis of the errors: 𝜅 𝜀 𝐸 𝜀 /𝐸 𝜀 𝑧 • We define standardized errors: They have conditional mean zero and a time invariant conditional variance equal to 1. That is, 𝑧 ~ N(0, 1). • From the definition of 𝑧 we have: 𝜀 = 𝑧 𝜎 Now, we compute the fourth (also central, since E[𝜀 ]=0) moment: 𝐸𝜀 𝐸𝑧 𝐸𝜎 55 ARCH Models: Leptokurtosis • Then, using Jensen’s inequality: 𝐸𝜀 𝐸𝑧 𝐸𝜎 𝜅 𝜀 𝐸𝑧 𝐸𝜎 3𝐸 𝜀 𝐸 𝜀 /𝐸 𝜀 𝐸𝑧 𝐸𝜀 3. where we have used the fact that since E[𝜀 ] = 0, then E[𝜀 ] = E[𝜎 ] Technical point: It can be shown that for an ARCH(1), the 4th moment for an ARCH(1): 𝜅 𝜀 3 1 𝛼 1 3𝛼 if 3𝛼 1. 56 28 ARCH Models: Alternative Representation • More convenient, but less intuitive, presentation of the ARCH(1) model: 𝜀 𝜎 𝑧 where 𝜐 is i.i.d. with mean 0, and Var[𝑧 ]=1. Since 𝑧 is i.i.d., then: 𝐸 𝜀 𝐸 𝜎 𝑧 𝐸 𝜎 𝐸 𝑧 𝜔 𝛼 𝜀 • It turns out that 𝜎 is a very persistent process. Such a process can be captured with an ARCH(𝑞), where 𝑞 is large. This is not efficient. • In practice, 𝑞 is often large. A more parsimonious representation is the Generalized ARCH model or GARCH(𝑞, 𝑝): 𝜎 𝜔 𝜔 ∑ 𝛼𝜀 ∑ 𝛼 𝐿 𝜀 𝛽 𝐿 𝜎 𝛽𝜎 GARCH: Bollerslev (1986) • A more parsimonious representation is the GARCH(q, p): ∑ 𝛽𝜎 𝜔 ∑ 𝛼𝜀 𝜎 which is an ARMA(max(𝑝, 𝑞), 𝑝) model for the squared innovations. • Popular GARCH model: GARCH(1,1): 𝜎 𝜔 𝛼 𝜀 𝛽𝜎 with an unconditional variance: Var[𝜀 ] = σ =  Restrictions: 𝜔 > 0, 𝛼 > 0, 𝛽 > 0; (1 𝛼 𝛽 ) > 0. • Technical details: This is covariance stationary if all the roots of 𝛼(L) + 𝛽(L) = 1 lie outside the unit circle. For the GARCH(1,1) this amounts to 𝛼 + 𝛽 < 1. 29 GARCH: Bollerslev (1986) • Bollerslev (1986) showed that if 3α12 + 2α1β1 + β12 < 1, the second and 4th (unconditional) moments of εt exist: E[ t2 ]   (1  1  1 ) E[ t4 ]  3 2 (1  1  1 ) 2 2 (1  1  1 )(1  1  211  31 ) if (1  1  211  31 )  0 2 2 GARCH-X • In the GARCH-X model, exogenous variables are added to the conditional variance equation. Consider the GARCH(1,1)-X model: 𝜎 𝜔 𝛼 𝜀 𝛽 𝜎 +𝛿𝑓 𝑋 where f(𝑋 ,𝜃) is strictly positive for all 𝑡. Usually, 𝑋 , is an observed economic variable or indicator, say liquidity index, and f(.) is a nonlinear transformation, which is non-negative. Examples: Glosten et al. (1993) and Engle and Patton (2001) use 3mo T-bill rates for modeling stock return volatility. Hagiwara and Herce (1999) use interest rate differentials between countries to model FX return volatility. The US congressional budget office uses inflation in an ARCH(1) model for interest rate spreads. 30 IGARCH • Recall the technical detail: The standard GARCH model: 𝜎 𝜔 𝛼 𝐿 𝜀 𝛽 𝐿 𝜎 is covariance stationary if 𝛼 1 + 𝛽 1 < 1 • But strict stationarity does not require such a stringent restriction (That is, that the unconditional variance does not depend on t). In the GARCH(1,1) model, if 𝛼 + 𝛽 = 1, we have the Integrated GARCH (IGARCH) model. In practice, in many applications, 𝛼 + 𝛽 are close to one. • Implication: Shocks to the conditional variance are “persistent.” This can be seen in the variance forecasts: 𝐸 𝜎 𝜎  Today’s variance remains important for all future forecasts 𝑗𝜔 IGARCH • In practice, it is often found that 𝛼 𝛽 are close to 1. • It is often argued that IGARCH is a product of omitted variables; for example, structural breaks. 31 GARCH: Variations – GARCH-in-mean • The time-varying variance affects mean returns: Mean equation: 𝑦 𝑋𝛾 𝛿𝜎 𝜀, Variance equation: 𝜎 𝜔 𝛼 𝜀 𝛽𝜎 𝜀 ~ N(0, 𝜎 ) • We have a dynamic mean-variance relations. It describes a specific form of the risk-return trade-off. • Finance intuition says that 𝛿 has to be positive and significant. However, in empirical work, it does not work well: 𝛿 is not significant or negative. 63 GARCH: Variations – Asymmetric GJR • GJR-GARCH model – Glosten, Jagannathan & Runkle (JF, 1993): 𝜔 𝜎 where 𝐼 𝛼𝜀 𝛾𝜀 ∗𝐼 𝛽𝜎 if 𝜀 < 0; otherwise. =1 =0 • Using the indicator variable 𝐼 , this model captures sign (asymmetric) effects in volatility: Negative news (εt-i < 0) increase the conditional volatility (leverage effect). • The GARCH(1,1) version: 𝜎 where 𝐼 =1 =0 𝜔 𝛼 𝜀 if 𝜀 < 0; otherwise. 𝛾 𝜀 𝐼 𝛽𝜎 64 32 GARCH: Variations – Asymmetric GJR • The GARCH(1,1) version: 𝜎 When ε ε 𝜔 𝛼 𝜀 <0  𝜎 >0  𝜎 𝛾 𝜀 𝜔 𝜔 𝐼 𝛼 𝛾 𝛼 𝜀 𝛽𝜎 𝜀 𝛽𝜎 𝛽𝜎 • This is a very popular variation of the GARCH models. The leverage effect is significant. • There is another variation, the Exponential GARCH, or EGARCH, that also captures the asymmetric effect of negative news on the conditional variance. 65 GARCH: Variations – EGARCH • EGARCH model – Nelson (Econometrica, 1991). It models an exponential function for the time-varying variance: q p i 1 j 1 log( t2 )      i ( zt i   (| zt i |  E | zt i |))    j log( t2 j ) where z is a standardized i.i.d. D(0, 1) innovation. • By design, we have the variance follows an exponential function. Thus, no non-negative restrictions on the parameters are imposed. • Negative news (𝑧 < 0) increase 𝜎 (leverage effect). Note: Nelson provides formulas of the unconditional moments, under the GED. But, under leptokurtic distributions such as the Student-t the unconditional variance does not exist. (Intuition: we have an exponential formulation, with a large shock it can explode.) 33 GARCH: Variations – NARCH • Non-linear ARCH model NARCH – Higgins and Bera (1992) and Hentschel (1995). These models apply the Box-Cox-type transformation to the conditional variance: 𝜎 𝜔 𝛼 |𝜀 𝜅| 𝛽𝜎 Special case: γ = 2 (standard GARCH model). Note: The variance depends on both the size and the sign of the variance which helps to capture leverage type (asymmetric) effects. 67 GARCH: Variations – TARCH • Threshold ARCH (TARCH) – Rabemananjara & Zakoian (1993) Large events –i.e., large errors- have a different effect from small events. We use 2 indicator variables, 𝐼 𝜀 𝜅 &𝐼 𝜀 𝜅 : one for “large events,” 𝜀 𝜅 , & one for “small events,” 𝜀 𝜅 : 𝜎 𝜔 𝛼 𝐼 𝜀 𝜅 𝛼 𝐼 𝜀 𝜅 𝜀 𝛽𝜎 There are two variances: 𝜎 𝜔 𝛼 𝜀 𝛽𝜎 , if 𝜀 𝜅 𝜎 𝜔 𝛼 𝜀 𝛽𝜎 , if 𝜀 𝜅 • We can modify the model in many ways. For example, we can allow 68 for the asymmetric effects of negative news. 34 GARCH: Variations – SWARCH • Switching ARCH (SWARCH) – Hamilton and Susmel (JE, 1994). Intuition: σt2 depends on the state of the economy –regime. It’s based on Hamilton’s (1989) time series models with changes of regime:  t2   s , s t t 1  q  i 1 i , st , st 1  t21 The key is to select a parsimonious representation:  2t   s t q  i 1 i  t2 i s ti For a SWARCH(1) with 2 states (1 and 2) we have 4 possible σt2:  2 t    2 t    2 t    2 t   1   1  t2 1  1 /  1 , s t  1, s t  1  1 1   1 2 t 1  / 2, s t  1, s t  1  2 2   1 2 t 1  / , s t  2 , s t 1  1 2   1 2 t 1  / , s t  2 , s t 1  2 1 2 1 2 2 GARCH: Forecasting and Persistence • Consider the forecast in a GARCH(1,1) model 𝜎 𝜔 𝛼 𝜀 𝛽 𝜎 =𝜔 𝜎 𝛼 𝑧 𝛽 Taking expectation at time 𝑡 𝐸 𝜎 𝜔 𝜎 𝛼 1 𝜀 𝜎 𝑧 𝛽 Then, by repeated substitutions: 𝐸 𝜎 𝜔 𝜎 𝜔 1 𝛼 𝛼 𝛽 =𝜔 𝛽 𝜎 𝜔∑ 𝛼 𝛽 𝜔 𝛼 𝜎 𝛼 𝜎 𝛽 𝛼 𝛽 𝛼 𝛽 𝛽 Assuming (𝛼 𝛽 ) < 1, as 𝑗 → ∞, the forecast reverts to the unconditional variance: σ = 35 GARCH: Forecasting and Persistence • Today’s forecast in a GARCH(1,1) model for horizon 𝑗 : 𝜔∑ 𝐸 𝜎 When 𝛼 horizons: 𝛽 𝛼 𝛽 𝜎 𝛼 𝛽 1, today’s volatility affect future forecasts for all 𝐸 𝜎 𝜎 𝑗𝜔 A shock to 𝜎 stays forever in the forecast! ARCH Estimation: MLE • All of these models can be estimated by maximum likelihood. First we need to construct the sample likelihood. • Since we are dealing with dependent variables, we use the conditioning trick to get the joint distribution: 𝑓 𝑦 ,𝑦 ,…,𝑦 ;𝛉 𝑓 𝑦 𝐱 𝟏 ; 𝛉 𝑓 𝑦 𝑦 , 𝐱 , 𝐱 ; 𝛉 𝑓 𝑦 𝑦 , 𝑦 , 𝐱 , 𝐱 , 𝐱 ; 𝛉 ... . . . 𝑓 𝑦 |𝑦 , … , 𝑦 , 𝐱 ,…,𝐱 ;𝛉 . Taking logs: 𝐿 log 𝑓 𝑦 , 𝑦 , . . . , 𝑦 ; 𝛉 … log 𝑓 𝑦 |𝐱 𝟏 ; 𝛉 log 𝑓 𝑦 |𝑦 log 𝑓 𝑦 |𝑌 log 𝑓 𝑦 |𝑦 , 𝐱 , 𝐱 ; 𝛉 ,…,𝑦 ,𝐱 ,…,𝐱 ;𝛉 ,𝑋 ;𝛉 • We maximize this function with respect to the 𝑘 mean parameters (γ) and the 𝑚 variance parameters (𝜔, 𝛼 , 𝛽 ). Let 𝛉 =(γ, 𝜔, 𝛼 , 𝛽 ). 72 36