Question

Question

a) Explain the following characteristics of financial asset returns

i) Leptokurtosis

ii)Volatility Clustering

iii) Leverage effects

b)

i)Outline the main steps in testing for 'ARCH Effects"in financial Asset returns

ii) Explain three limitations of the ARCH (q) models

iii) State the stapes in estimating a GARCH Model

c) Distinguish between the following extensions to the GARCH Model

i) GARCH -M

ii) EGARCH

iii) GJR-GARCH

Answer 1

a)

i) Normal Distriby=utio: Kurtosis = 3 and Skewness = 0

Lepokurtosis: positive excess kurtosis, i.e.Kurtosis > 3

ii) Volatlity Clustering : Returns large changes tend to be followed by large changes...and small changes tend to be followed by small change.

iii) Leverage Effect: Negative return increases variance by more than a positive return of the same magnitude.

b)

i)ARCH- Autoregressive Conditional Heteroscedasticity-

The AR comes from the fact that these models are autoregressive models in squared returns, The conditional comes from the fact that in these models, next period’s volatility is conditional on information this period. In an ARCH(1) model, next period's variance only depends on last period's squared residual so a crisis that caused a large residual would not have the sort of persistence that we observe after actual crises.

Testing ARCH effects in the conditional variance of residuals

1. Consider the AR (1) model for the returns series of each individual index as:

                      R_t = ?₀ + ?₁ * R_t-1 +u_t

2- Use Linear Regression to calculate residuals (u_t)

2- Run Linear Regression on square of residuals on q lags.

The ARCH (q) specification for is denoted as:

Null Hypothesis: H0 = no Arch Effect:                 

Aletranate Hupothesis:

If the value of the Linear Model statistic is greater than the critical value from the

distribution, the coefficient of the lagged term is statistically significant, then the null hypothesis is rejected i.e. there is no ARCH effect in equation.

ii) Limitations: The model assumes that positive and negative shocks have the same effects on volatility.

iii) GARCH model assumes that eventually in the future variance will revert to the average value.

In this model, the residual term from white noise to an ARMA (p; q) process is expanded. In GARCH next period's variance depends on last period's squared residual and on the variance of last period or next period forecast of variance is a blend of our last period forecast and last period's squared return.