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

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The probability density function of the normal distribution is given by

f(x; µ, σ2 ) =  12πσ2  exp  (x  µ)2 2σ2 

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 σ2 given the sample of i.i.d data (x1, x2, . . . , xT ).

(b) [20 marks] Now assume that xt is conditionally normally distributed as N(0, σt2), where

σt2= ω + βσt-12 + αxt12 

Write down the likelihood function for this model given a sample of data(x1, x2, . . . , xT ). 

(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.

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