1. 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).

2. 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.

3. 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.

4. 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.

5. 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.

6. 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

7. 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.

8. 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, (shock spillovers)
4 GARCH terms: 1,1 , 2,1 , 1,2 , 2, (volatility spillovers)