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ARCH (Auto-Regressive Conditional Heteroscedasticity). An approach to modelling time-varying variance of a time series. ( t 2 : conditional variance )
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ARCH(Auto-Regressive Conditional Heteroscedasticity) • An approach to modelling time-varying variance of a time series. (t2 : conditional variance) • Mostly financial market applications: the risk premium defined as a function of time-varying volatility (GARCH-in-mean); option pricing; leptokurtosis, volatility clustering. More efficient estimators can be obtained if heteroscedasticity in error terms is handled properly. ARCH: Engle (1982), GARCH: Bollerslev (1986), Taylor (1986).
ARCH(p) model: Mean Equation: yt = a + toryt = a + bXt + t ARCH(1): t2 = + 2t-1 + t > 0, >0 tis i.i.d. • GARCH(p,q) model: GARCH (2,1): t2 = + 12t-1 + 22t-2 + 2t-1 +t > 0, >0, >0 Exogenous or predetermined regressors can be added to the ARCH equations. The unconditional variance from a GARCH (1,1) model: 2 = / [1-(+)] + < 1, otherwise nonstationary variance, which requires IGARCH.
Use of Univariate GARCH models in Finance Step 1: Estimate the appropriate GARCH specification Step 2: Using the estimated GARCH model, forecast one-step ahead variance. Then, use the forecast variance in option pricing, risk management, etc.
Use of ARCH models in Econometrics • Step 1. ARCH tests(H0: homoscedasticity) Heteroscedasticity tests: White test, Breusch-Pagan test (identifies changing variance due to regressors) ARCH-LM test: identifies only ARCH-type (auto-regressive conditional) heteroscedasticity. H0: no ARCH-type het. • Step 2. Estimate a GARCH model (embedded in the mean equation) Yt = 0 + 1Xt+ t and Var(t) = h2t = 0 + 1t2 + h2t-1 + vtwhere vt is i.i.d. Now, the t-values are corrected for ARCH-type heteroscedasticity.
Asymmetric GARCH (TARCH or GJR Model) Leverage Effect: In stock markets, the volatility tends to increase when the market is falling, and decrease when it is rising. To model asymmetric effects on the volatility: t2 = + 2t-1 + It-12t-1 + 2t-1 +t It-1 = { 1 if t-1 < 0, 0 if t-1 > 0 } If issignificant, then we have asymmetric volatility effects. If is significantly positive, it provides evidence for the leverage effect.
Multivariate GARCH If the variance ofa variable is affected by the past shocks to the variance of another variable, then a univariate GARCH specification suffers from an omitted variable bias. VECH Model: (describes the variance and covariance as a function of past squared error terms, cross-product error terms, past variances and past covariances.) MGARCH(1,1) Full VECH Model 1,t2 = 1 + 1,121,t-1 + 1,222,t-1 + 1,31,t-12,t-1 + 1,121,t-1 + 1,222,t-1+1,3Cov1,2,t-1 +1,t 2,t2 = 2 + 2,121,t-1 + 2,222,t-1 + 2,31,t-12,t-1 + 2,121,t-1+ 2,222,t-1 +2,3Cov1,2,t-1 +2,t Cov1,2,t = 3 + 3,121,t-1 + 3,222,t-1 + 3,31,t-12,t-1 + 3,121,t-1 + 3,222,t-1 +3,3Cov1,2,t-1 +3,t Two key terms: Shock spillover, Volatility spillover
Diagonal VECH Model: (describes the variance as a function of past squared error term and variance; and describes the covariance as a function of past cross-product error terms and past covariance.) MGARCH (1,1) Diagonal VECH 1,t2 = 1 + 1,121,t-1 + 1,121,t-1 + 1,t 2,t2 = 2 + 2,222,t-1 + 2,222,t-1 +2,t Cov1,2,t = 3 + 3,31,t-12,t-1 + 3,3Cov1,2,t-1 +3,t This one is less computationally-demanding, but still cannot guarantee positive semi-definite covariance matrix. Constant Correlation Model: to economize on parameters Cov1,2,t = Cor1,2 however, this assumption may be unrealistic.
BEKK Model: guarantees the positive definiteness MGARCH(1,1) 1,t2 = 1 + 21,121,t-1 + 21,12,11,t-12,t-1 + 22,122,t-1 + 21,121,t-1 +21,12,1Cov1,2,t-1 + 22,122,t-1 + 1,t 2,t2 = 2 + 21,221,t-1 + 21,22,21,t-12,t-1 + 22,222,t-1 + 21,221,t-1 +21,22,2Cov1,2,t-1 + 22,222,t-1 + 2,t Cov1,2,t = 1,2 + 1,1 1,2 21,t-1+(2,11,2+ 1,12,2)1,t-12,t-1 + 2,1 2,2 22,t-1 + 1,11,221,t-1 +(2,1 1,2+ 1,12,2)Cov1,2,t-1+ 2,12,222,t-1 + 3,t Interpreting BEKK Model Results: You will get: 3 constant terms: 1 , 2 , 1,2 4 ARCH terms:1,1 , 2,1 ,1,2 , 2,2 (shock spillovers) 4 GARCH terms: 1,1 , 2,1 , 1,2 , 2,2 (volatility spillovers)