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).

• ARCH(p) model:

Mean Equation: yt = a + t or yt = a + bXt + t

ARCH(1): t2 = + 2

t-1 + t > 0, >0

t is i.i.d.

• GARCH(p,q) model:

GARCH (2,1): t2 = + 12

t-1 + 22t-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.

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) = h2

t = 0 + 1t2 + h2

t-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 = + 2

t-1 + It-12t-1 + 2

t-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.

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 + 1,12

1,t-1 + 1,222,t-1 + 1,31,t-12,t-1 + 1,12

1,t-1

+ 1,222,t-1 + 1,3Cov1,2,t-1 + 1,t

2,t2 = 2 + 2,12

1,t-1 + 2,222,t-1 + 2,31,t-12,t-1 + 2,12

1,t-1

+ 2,222,t-1 + 2,3Cov1,2,t-1 + 2,t

Cov1,2,t = 3 + 3,121,t-1 + 3,22

2,t-1 + 3,31,t-12,t-1 + 3,12

1,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 + 1,12

1,t-1 + 1,121,t-1 + 1,t

2,t2 = 2 + 2,22

2,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.

2,2

2,1 tt

BEKK Model: guarantees the positive definiteness

MGARCH(1,1)

1,t2 = 1 + 2

1,121,t-1 + 21,12,11,t-12,t-1 + 2

2,122,t-1 + 2

1,121,t-1 +

21,12,1Cov1,2,t-1 + 22,12

2,t-1 + 1,t

2,t2 = 2 + 2

1,221,t-1 + 21,22,21,t-12,t-1 + 2

2,222,t-1 + 2

1,221,t-1 +

21,22,2Cov1,2,t-1 + 22,22

2,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 + 1,11,22

1,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)