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ARCH (Auto-Regressive Conditional Heteroscedasticity) An approach to modelling time-varying variance of a time series. ( t 2 : 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).

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ARCH(p) model: Mean Equation: y t = a + t or y t = a + bX t + t ARCH(1): t 2 = + 2 t-1 + t > 0, >0 t is i.i.d. GARCH(p,q) model: GARCH (2,1): t 2 = + 1 2 t-1 + 2 2 t-2 + 2 t-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.

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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.

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Use of ARCH models in Econometrics Step 1. ARCH tests (H 0 : 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. H 0 : no ARCH-type het. Step 2. Estimate a GARCH model (embedded in the mean equation) Y t = 0 + 1 X t + t and Var( t ) = h 2 t = 0 + 1 t 2 + h 2 t-1 + v t where v t is i.i.d. Now, the t-values are corrected for ARCH-type heteroscedasticity.

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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: t 2 = + 2 t-1 + I t-1 2 t-1 + 2 t-1 + t I t-1 = { 1 if t-1 0 } If is significant, then we have asymmetric volatility effects. If is significantly positive, it provides evidence for the leverage effect.

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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,t 2 = 1 + 1,1 2 1,t-1 + 1,2 2 2,t-1 + 1,3 1,t-1 2,t-1 + 1,1 2 1,t-1 + 1,2 2 2,t-1 + 1,3 Cov 1,2,t-1 + 1,t 2,t 2 = 2 + 2,1 2 1,t-1 + 2,2 2 2,t-1 + 2,3 1,t-1 2,t-1 + 2,1 2 1,t-1 + 2,2 2 2,t-1 + 2,3 Cov 1,2,t-1 + 2,t Cov 1,2,t = 3 + 3,1 2 1,t-1 + 3,2 2 2,t-1 + 3,3 1,t-1 2,t-1 + 3,1 2 1,t-1 + 3,2 2 2,t-1 + 3,3 Cov 1,2,t-1 + 3,t Two key terms: Shock spillover, Volatility spillover

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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,t 2 = 1 + 1,1 2 1,t-1 + 1,1 2 1,t-1 + 1,t 2,t 2 = 2 + 2,2 2 2,t-1 + 2,2 2 2,t-1 + 2,t Cov 1,2,t = 3 + 3,3 1,t-1 2,t-1 + 3,3 Cov 1,2,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 Cov 1,2,t = Cor 1,2 however, this assumption may be unrealistic.

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BEKK Model: guarantees the positive definiteness MGARCH(1,1) 1,t 2 = 1 + 2 1,1 2 1,t-1 + 2 1,1 2,1 1,t-1 2,t-1 + 2 2,1 2 2,t-1 + 2 1,1 2 1,t-1 + 2 1,1 2,1 Cov 1,2,t-1 + 2 2,1 2 2,t-1 + 1,t 2,t 2 = 2 + 2 1,2 2 1,t-1 + 2 1,2 2,2 1,t-1 2,t-1 + 2 2,2 2 2,t-1 + 2 1,2 2 1,t-1 + 2 1,2 2,2 Cov 1,2,t-1 + 2 2,2 2 2,t-1 + 2,t Cov 1,2,t = 1,2 + 1,1 1,2 2 1,t-1 +( 2,1 1,2 + 1,1 2,2 ) 1,t-1 2,t-1 + 2,1 2,2 2 2,t-1 + 1,1 1,2 2 1,t-1 +( 2,1 1,2 + 1,1 2,2 )Cov 1,2,t-1 + 2,1 2,2 2 2,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)

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