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Unformatted text preview: cont’d) • But in general a GARCH(1,1) model will be sufficient to capture the
volatility clustering in the data. • Why is GARCH Better than ARCH?
 more parsimonious  avoids overfitting
 less likely to breech nonnegativity constraints 16 The Unconditional Variance under the GARCH Specification • The unconditional variance of ut is given by when
• is termed “nonstationarity” in variance • is termed intergrated GARCH • For nonstationarity in variance, the conditional variance forecasts will
not converge on their unconditional value as the horizon increases. 17 Estimation of ARCH / GARCH Models • Since the model is no longer of the usual linear form, we cannot use
OLS. • We use another technique known as maximum likelihood. • The method works by finding the most likely values of the parameters
given the actual data. • More specifically, we form a loglikelihood function and maximise it. 18 Estimation of ARCH / GARCH Models (cont’d) • The steps involved in actually estimating an ARCH or GARCH model
are as follows 1. Specify the appropriate equations for the mean and the variance  e.g. an
AR(1) GARCH(1,1) model: 2. Specify the loglikelihood function to maximise: 3. The computer will maximise the function and give parameter values and
their standard errors
19 Parameter Estimation using Maximum Likelihood • Consider the bivariate regression case with homoscedastic errors for
simplicity: • , σ2) so that the
Assuming that ut ∼ N(0,σ2), then yt ∼ N(
probability density function for a normally distributed random variable
with this mean and variance is given by
(1) • Successive values of yt would trace out the familiar bellshaped curve. • Assuming that ut are iid, then yt will also be iid. 20 Parameter Estimation using Maximum Likelihood (cont’d) • Then the joint pdf for all the y’s can be expressed as a product of the
individual density functions (2)
for t = 1,...,T • Substituting into equation (2) for every yt from equation (1),
(3)
21 Parameter Estimation using Maximum Likelihood (cont’d)
• The typical situation we have is that the xt and yt are given and we want to
estimate β1, β2, σ2. If this is the case, then f(•) is known as the a likelihood
function, denoted LF(β1, β2, σ2), so we write (4) • Maximum likelihood estimation involves choosing parameter values (β1,
β2,σ2) that maximise this function. • We want to differentiate (4) w.r.t. β1, β2,σ2, but (4) is a product containing
T terms. 22 Parameter Estimation using Maximum Likelihood (cont’d)
• Since , we can take logs of (4). • Then, using the various laws for transforming functions containing
logarithms, we obtain the loglikelihood function, LLF: • which is equivalent to
(5) • Differentiating (5) w.r.t. β1, β2,σ2, we obtain
(6)
23 Parameter Estimation using Maximum Likelihood (cont’d)
(7) • (8)
Setting (6)(8) to zero to minimise the functions, and putting hats above
the parameters to denote the maximum likeli...
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This note was uploaded on 09/20/2013 for the course FINA 5170 taught by Professor Janebargers during the Summer '13 term at Greenwich School of Management.
 Summer '13
 JaneBargers
 Financial Markets

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