# Question: the probability density function of the normal distribution is given...

###### Question details

The probability density function of the normal distribution is given by

$f(x;\mu ,{\sigma}^{2})=\frac{1}{\sqrt{2\pi {\sigma}^{2}}}exp\left(-\frac{{(x-\mu )}^{2}}{2{\sigma}^{2}}\right)$

where µ is the mean and σ 2 is the variance of the distribution.

(a) [20 marks] Assuming that µ = 0, derive the maximum likelihood estimate of ${\sigma}^{2}$ given the sample of i.i.d data $({x}_{1},{x}_{2},...,{x}_{T}).$

(b) [20 marks] Now assume that ${x}_{t}$ is conditionally normally distributed as N(0, ${\sigma}_{t}^{2}$), where

${\sigma}_{t}^{2}=\omega +\beta {\sigma}_{t-1}^{2}+\alpha {x}_{t-1}^{2}$

Write down the likelihood function for this model given a sample of data$({x}_{1},{x}_{2},...,{x}_{T}).$

(c) [15 marks] Describe how we can obtain estimates for {ω, α, β} for the GARCH(1,1) model and discuss estimation difficulties.

(d) [20 marks] Describe in your own words what graphical method and formal tests you can use to detect volatility clustering.

(e) [25 marks] Describe the RiskMetrics exponential smoother model for multivariate volatility, and discuss the pros and cons of the constant conditional correlation model of Bollerslev (1990) versus the RiskMetrics approach.