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).

Mean Equation: yt = a + t or yt = a + bXt + t ARCH(p) model: Mean Equation: yt = a + t or yt = a + bXt + t ARCH(1): t2 =  + 2t-1 + t  > 0,  >0 t is i.i.d. GARCH(p,q) model: GARCH (2,1): t2 =  + 12t-1 + 22t-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 + 1t2 +  h2t-1 + vt where 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-12t-1 + 2t-1 + t It-1 = { 1 if t-1 < 0, 0 if t-1 > 0 } If  is significant, then we have asymmetric volatility effects. If  is significantly positive, it provides evidence for the leverage effect.

MGARCH(1,1) Full VECH Model Multivariate GARCH If the variance of a 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 + 1,121,t-1 + 1,222,t-1 + 1,31,t-12,t-1 + 1,121,t-1 + 1,222,t-1 + 1,3Cov1,2,t-1 + 1,t 2,t2 = 2 + 2,121,t-1 + 2,222,t-1 + 2,31,t-12,t-1 + 2,121,t-1 + 2,222,t-1 + 2,3Cov1,2,t-1 + 2,t Cov1,2,t = 3 + 3,121,t-1 + 3,222,t-1 + 3,31,t-12,t-1 + 3,121,t-1 + 3,222,t-1 + 3,3Cov1,2,t-1 + 3,t Two key terms: Shock spillover, Volatility spillover

MGARCH (1,1) Diagonal VECH 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 + 1,121,t-1 + 1,121,t-1 + 1,t 2,t2 = 2 + 2,222,t-1 + 2,222,t-1 + 2,t Cov1,2,t = 3 + 3,31,t-12,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 + 21,121,t-1 + 21,12,11,t-12,t-1 + 22,122,t-1 + 21,121,t-1 + 21,12,1Cov1,2,t-1 + 22,122,t-1 + 1,t 2,t2 = 2 + 21,221,t-1 + 21,22,21,t-12,t-1 + 22,222,t-1 + 21,221,t-1 + 21,22,2Cov1,2,t-1 + 22,222,t-1 + 2,t Cov1,2,t = 1,2 + 1,1 1,2 21,t-1+(2,11,2+ 1,12,2)1,t-12,t-1 + 2,1 2,2 22,t-1 + 1,11,221,t-1 +(2,1 1,2+ 1,12,2)Cov1,2,t-1+ 2,12,222,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)