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Unformatted text preview: astic model
for the variance of the errors:
= α0 + α1
• This is known as an ARCH(1) model.
8 Autoregressive Conditionally Heteroscedastic
(ARCH) Models (cont’d)
• • •
• The full model would be
yt = β1 + β2x2t + ... + βkxkt + ut , ut ∼ N(0, )
where
= α0 + α1
We can easily extend this to the general case where the error variance
depends on q lags of squared errors:
= α0 + α1
+α2
+...+αq
This is an ARCH(q) model.
Instead of calling the variance , in the literature it is usually called ht,
so the model is
yt = β1 + β2x2t + ... + βkxkt + ut , ut ∼ N(0,ht)
where ht = α0 + α1
+α2
+...+αq 9 Another Way of Writing ARCH Models • For illustration, consider an ARCH(1). Instead of the above, we can
write
yt = β1 + β2x2t + ... + βkxkt + ut , ut = vtσt
, • vt ∼ N(0,1) The two are different ways of expressing exactly the same model. The
first form is easier to understand while the second form is required for
simulating from an ARCH model, for example. 10 Testing for “ARCH Effects”
1. First, run any postulated linear regression of the form given in the equation
above, e.g.
yt = β1 + β2x2t + ... + βkxkt + ut
saving the residuals, .
2. Then square the residuals, and regress them on q own lags to test for ARCH
of order q, i.e. run the regression
where vt is iid.
Obtain R2 from this regression
3. The test statistic is defined as TR2 (the number of observations multiplied
by the coefficient of multiple correlation) from the last regression, and is
distributed as a χ2(q).
11 Testing for “ARCH Effects” (cont’d) 4. The null and alternative hypotheses are
H0 : γ1 = 0 and γ2 = 0 and γ3 = 0 and ... and γq = 0
H1 : γ1 ≠ 0 or γ2 ≠ 0 or γ3 ≠ 0 or ... or γq ≠ 0.
If the value of the test statistic is greater than the critical value from the
χ2 distribution, then reject the null hypothesis.
• Note that the ARCH test is also sometimes applied directly to returns
instead of the residuals from Stage 1 above. 12 Problems with ARCH(q) Models • How do we decide on q?
• The required value of q might be very large
• Nonnegativity constraints might be violated.
– When we estimate an ARCH model, we require αi >0 ∀ i=1,2,...,q
(since variance cannot be negative)
• A natural extension of an ARCH(q) model which gets around some of
these problems is a GARCH model. 13 Generalised ARCH (GARCH) Models • • Due to Bollerslev (1986). Allow the conditional variance to be dependent
upon previous own lags
The variance equation is now
(1)
This is a GARCH(1,1) model, which is like an ARMA(1,1) model for the
variance equation.
We could also write • Substituting into (1) for σt12 : •
• 14 Generalised ARCH (GARCH) Models (cont’d)
• Now substituting into (2) for σt22 • An infinite number of successive substitutions would yield • So the GARCH(1,1) model can be written as an infinite order ARCH model. • We can again extend the GARCH(1,1) model to a GARCH(p,q): 15 Generalised ARCH (GARCH) Models (...
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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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