MODELLING ASSET RETURN VOLATILITY
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1. Introduction
We introduce the ARCH\GARCH class of models which were developed to
account for the persistence in squared returns which, as we have seen, is a typical feature
of asset return data. ARCH and GARCH refer, respectively, to an Autoregressive
Conditional Heteroscedastic and a Generalized Autoregressive Conditional
Heteroscedastic model. These models are heteroscedatic and autoregressive because they
model time-varying volatility in returns and do that by taking account of past volatility.
2. ARCH Processes
Consider a very simple model for returns { , which in practice is not an
unreasonable characterization for mean returns, namely,
ty c tε= + (1)
where 2(0, )t WNε σ∼ and c is the intercept. This assumption means that the
unconditional mean and variance of the innovation tε are constant and finite (equal to
zero and 2σ , respectively) and that the innovations are serially uncorrelated at all leads
and lags. In topic 3, we made the further assumption that the innovations were
independently and identically distributed so that the innovations were characterized as a
strict white noise process, that is, 2)(0,t iidWNε σ∼ . The assumption of independence
means that the conditional probability density function of 1|t tε −Ω where
1 1 2{ , , }t t tε ε− − − …Ω = is the same as the unconditional probability density function of tε .
This has the implication that the conditional variance of the innovation, namely,
depends only on j and not on the conditioning information set . Thus,
for example, when j=1
1 1var( | ) var( | )t t t tε ε− +Ω = Ω
The arrival of “news” at t , namely tε , does not affect the value of conditional variance
of the innovation one period ahead, namely, at t+1. In other words, the conditional
variance does not adapt to the arrival of new conditioning information. In financial
markets, however, return volatility responds to the arrival of “news” and the present
specification cannot capture this feature of financial data.
To do so, we will continue to assume that the innovations are white noise but
that they are not strictly white noise by dropping the assumption of independence. In
particular, we will assume a particular dependence structure in the conditional variance of
the innovation.
The Autoregressive Conditional Heteroscedastic Model of order one (ARCH(1))
for the innovation tε is
21| (0,t t tN )ε σ−Ω ∼
2 20 1 1 0 10, 0 1t tσ α α ε α α−= + > ≤ < (2)
where the
1)t t t−Ω , conditional variance of t
2 var( |σ ε= ε based on the informatio
1 1 2{ , , }t t t
ε ε− − −Ω = … . The 't sε are serially uncorrelated b t they are not serially
independent because the current conditional variance of t
ε depends on the one period
lagged value of 2tε . The conditional distribution is assumed normal so that equations (1)
and (2) can be estimated jointly by maximum likelihood estimation.
It is important to stress that tε is a weak white noise process and is covariance
stationary. The unconditional mean and the unconditional variance of ε are constant and
nite and are, respectively,
var( ) ( ( ))
The restrictions 0 10, 0 1α α> ≤ < , ensure that the unconditional variance is positive
and finite and, also, that the conditional variance is always positive. The important po
is that the unconditional mean and variance are finite and constant as required for
covariance stationarity. Also, as sta
d earlier the te 't sε are serially uncorrelated. The
onditional mean and variance are
var( | ) [( ( | )) | ]
t t t t t t
that the conditional variance is time-varying and thus
teroscedastic. Note that
The important point here is
20var( | )t1tε α α εΩ = + +
and it follows that 1 1var( | ) var( | )t t t tε ε− +Ω ≠ Ω . The arrival of news, namely, tε , wi
affect the conditional variance for next period (at time t+1). This is the key feature of
ARCH type models: the conditional variance adapts to the conditioning information set.
Clearly, if 1 0
α = , there are no conditional variance dynamics and we are back to
independence where 0(0, )t iidWε α∼ from which it follows, for example, that
1 1| ) var( | )t t t tvar(ε ε− +Ω = Ω .
The ARCH model is capable of capturing the feature of volatility clustering that is
erved in financial data where large changes in returns tend to be followed by large
changes, and small changes in returns by small changes, of either sign. A large observe
ε − produces a large conditional variance at time t, that is, a large
tσ , thereby increasing
the likelihood of a large 2tε at time t. The ARCH process models conditional variance
dynamics in an autoregressive fashion.
Clearly, in the case of equation (1), the unconditional and conditional mean of
is just . The conditional variance of is time-varying and equal to:
or each parameter, namely, for c
var( | ) [( ( | )) | ]
t t t t t t
Equation (1) is referred to as the mean equation and equation (2) a he var nc
equation. In practice, both equations are estimated jointly using maximum likelihood
techniques and an estimate is obtained f , 0α and 1α .
here is no need for the mean equation to take the particularly simple form of equation
). The mean equation could easy be
0 1 1, 2 2,t ty X Xβ β β= + + + ε with 1, 2,( | , ) 0t t tE X Xε =
1t ty c ε θε −= + +
1t ty c yρ ε+ −= +
where, in each case, 2(0, )t WNε σ∼ . Similarly, there is no need to restrict the variance
quation to an ARCH specification with one autoregressive lag. The ARCH(q)
ecification for the variance equation is:
21| (0,t t tNε σ−Ω ∼
0 1 1 2 2t t t q t q
σ α α ε α ε α ε− − −= + + + +…
0, 0 for all 1 , 1i iii to qα α α=> ≥ = <∑
The stated restrictions are sufficient to ensure that the conditional and unconditional
variances are positive and finite and that tε is covariance stationary. Note that for the
mean, variance and covariance are, ARCH(q) process, the unconditional
cov( , ) 0 0.
hile the conditional mean and variance are, w
t t t t q t q
ε α α ε α ε α ε
Ω = + + + +
Consider now the process. It is important to realize that no matter the specific form of
e mean equation for , it will always be the case that
var( | )t t t tvar( | )
For example, suppose follows the AR(1) process given above. Then the conditional
ean of is
1( | )t t tc y E
( | ) [( ) |t t t t t
ρ ε− −= + Ω
which is the forecast value of
ty on the basis of time t-1 information.
var( | ) [( ( | )) | ]
[( ( )) | ]
t t t t t t
E c y c yρ ε ρ
= + + − + Ω
T nhe two-standard error co fidence interval for the one-step ahead forecast of , based
n information available at time t-1 (i.e.
o ) for the AR(1) model is: 1t−Ω
1 2t tc yρ σ−+ ±
the case of an ARCH(1) process it is In
1 0 1 12 t tc yρ α α ε− −+ ± +
On the basis of time t information, the two-standard error confidence interval for the one-
ep ahead forecast of , based on information available at time t (i.e. ) is:
st 1ty + tΩ
12t tc yρ σ ++ ±
the case of an ARCH(1) process it is In
0 12 t tc yρ α α ε+ ± +
Notice that not only is the forecasted value of y updated, as it now depends on , but
f the forecast is updated as it depends on
tε . Thus, the arrival of also the standard error o
new information at time t influences not only the forecast but the confidence interval
. GARCH Processes
The generalized ARCH or GARCH(1,1) model is:
associated with the forecast.
σ α α ε β σ
Here the conditional variance at time t 2( )tσ depends not only on last period’s squared
innovation 2 1( )tε − but also on the conditional variance last period
1( )tσ − . The parame
restrictions ensure that the unconditional variance and the conditional variance are
positive and finite and that ty is covariance stationary. Equation (3) is the mean equation
n this case, it
ery simple but it ca e any form, for example, a regression
quation o A or AR ess. E tion (4) is the variance equation. The
unconditional mean and the unconditional variance of
ε are constant and finite and are,
spectively,
Note that the unconditional covariance is
cov( , ) 0t t jε ε − = for j>0. The conditional mean
nd variance are
var( | ) [( ( | )) | ]
t t t t t t
he important point is that the unconditional variance is constant, as must be the case
nder covariance stationa ty, whereas the conditional variance is time-varying.
The two-standard error confidence interval for the one-step ahead forecast of ,
based on information available at time t-1 1t−Ω ) for the AR(1) model is:
1 2t tc yρ σ−+ ±
In the case of a GARCH(1,1) process it is
1 0 1 1 12 t tc yρ α α ε β σ− −+ ± + + 1t−
n the basis of time t information, the two-standard error confidence interval for the one-
ion available at time t (i.e. ) is:
step ahead forecast of ty , based on informat tΩ
12t tc yρ σ ++ ±
In the case of an GARCH(1,1) process it is
0 1 12 t t tc yρ α α ε β σ+ ± + +
Notice that not only is the forecasted value of y updated, as it now depends on ty , but
also the standard error of the forecast is updated as it depends on 2tε and
tσ . Thus, the
arrival of new information at time t influences not only the forecast but the confidence
interval associated with the forecast.
In practice, it was found that to adequately model volatility, the ARCH(q)
specification required a very long number of lags, that is, q was found to be very large.
Not only do a large number of lags use up degrees of freedom but with so many
parameters to estimate it is more probable that one or more of the estimated parameters
ay be negative, violating the sign restrictions on the parameters. This led to the
evelopment of the GARCH model. We will now show that the GARCH(1,1) model is
res RCH model of infinite order. Now
equivalent to a tricted A
2 20 1 1 1t tσ α α ε β σ− −= + +
Substitute for 2 1tσ − to get
0 1 1t tσ α α ε β
1 0 1 2 1 2
0 1 1 1 1 2 1
α β α ε β ε β σ
tinue i his way to get,
2 1 2 2 1 20 1 1 1 1 1 2 1 1(1 ) ( )
t t t t j t jσ α β β α ε β ε β ε β σ
− − −= + + + + + + + +… … −
2 2 2 1 20 1 1 1 2 1
σ α ε β ε β ε
− − −= + + + + +−
This follows because 1 1β < , given the sign restrictions of the GARCH(1,1) model. Thus, the GARCH(1,1) model is equivalent to an ARCH(∞) where the coefficients on the lagged squared innovations decline geometrically. The GARCH(1,1) is a parsimonious model since there are only three parameters to estimate: 0 1 1, ,α α β . In many finance applications, particularly in Value-at-Risk calculations and the ricing of options, foreca s of the conditional variance are required. We will now derive n expression for the h-step ahead forecast of the conditional variance for the he optimal forecast of GARCH(1,1) specification. By def 1( |t t tEσ ε −= Ω Updating by h periods, we obtain 2σ = 2 1( |t h t h t hE ε+ + +Ω T 2t hσ + based on information available at time t is ) . By construction, 2( |t h tE σ + Ω 1[ | ] [ ( | ) | ]t h t t h t h tE E Eσ ε+ + + −Ω = Ω Ω 2( | ) (5t h tE ε += Ω where the last equality follows from the law of iterative expectations. The GARCH(1,1) nditional expectation on the basis of the information set 0 1 1 1t h t h t hσ α α ε β σ+ + −= + + Take the co tΩ to get where the last equality follows from equation (5). By recursive substitution, we can write quation (6) as where the la 2 2 2( | ) ( | ) ( | )t hE E Eσ α α ε β σ+ Ω = + Ω + Ω0 1 1 1 1 0 1 1 1( ) ( | ) t t h t t h t t h tEα α β σ + −= + + Ω 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 ( | ) [1 ( ) ( ) ( ) ] ( ) ( | ) [1 ( ) ( ) ( ) ] ( ) E Eσ α α β α β α β α β σ α α β α β α β α β σ Ω = + + + + + + + + + Ω = + + + + + + + + + st step follows because 2 1tσ + is known at time t. This formula can also be use to calculate a two standard error confidence band around the optimal h-step ahead recast of y. Fortunately, EViews automatically calculates forecasts of the h-step ahead onditional variance using this formula for the case of a GARCH(1,1) process. Provided 1 1 1α β+ < ariance of , the forecast of the cond al variance will converge to the unconditional ition as . That is h → ∞v tε 2 0lim ( | )t h tE if tε follows a GARCH(1,1) process, then it can be shown that tε has an ARMA(1,1) representation, namely, 0 1 1 1 1 1( )t t t tv vε α α β ε β− −= + + − + where 2 2t t tv ε σ= − is the difference between the squared innovation and the condition variance at time t. (You will be asked to show this result in a tutorial exercise). Th ε is a noisy proxy for the conditional variance 2tσ : tε is an unbiased predictor of but it is more volatile. Also, recall that real world financial asset returns are typically unconditionally symmetric but leptokurtic (that is, more peaked in the centre and with fatter tails than a normal distribution). It turns out that the implied unconditional distribution of the conditionally normal GARCH process is also symmetric and leptokurtic. Finally, the GARCH(1,1) can be extended to a GARCH(p,q) process given by 0, 0, 0, 1 t i t i i t i σ α α ε β σ As before, the stated conditions ensure that tε is covariance stationary and that the conditional variance is positive. The unconditional mean and the unconditional variance of ε are constant and finite and are, respectively, var( ) ( ( )) The conditional mean and variance are var( | ) [( ( | )) | ] t t t t t t i t i i t i Also, as before, no matter the specific form of the mean equation for , it will always be the case that var( | ) var( | )t t t t 3. Maximum Likelihood Estimation of the GARCH(1,1) Process In lecture note 2, we discussed maximum likelihood estimation of the simple linear regression model 0 1 where ~ (0, )t t t tY X u u iid Nβ β σ= + + . The log-likelihood function for this model is ( , , | , 1, , ) ln(2 ) ln ( ) l Y X t T Yβ β σ π σ β β = = − − − − −∑… X Now lets assume the conditional mean equation is 0 1 t tY X tδ δ ε= + + and the conditional variance equation is 0, 0, 0, 1 σ α α ε β σ The log-likelihood function for this model is 0 1 1 1 1 2 2 1 1 0 1 1 1 1 ( , , , , ) ln(2 ) ln( ) π α α ε β σ α α ε β σ− −= = − − = − − + + − Once we substitute and 2 1 1 0 1 1(t t tY Xε δ δ− − −= − − in the log-likelihood function, it is then possible to maximize the log-likelihood with respect to each of the parameters ( 0 1 0 1 1, , , ,δ δ α α β ). The values of 0σ have to be set and they are usually set equal to the unconditional variance of the 'sε from the estimated mean equation, which is where the e’s are the OLS residuals. EViews actually uses a more sophisticated approach. Because the first-order conditions are nonlinear, they are solved using an iterative search procedure. In order to implement such a procedure, starting values for the parameters need to be specified. The OLS estimates of 0δ and 1δ from the mean equation are used as the starting values for these parameters, respectively. In the absence of GARCH effects, 0α can be interpreted as the unconditional variance of the 'sε .Thus, an appropriate starting value of 0α is ôlsσ . For 1α and 1β , arbitrarily select a small number (say, 0.05) for both as a starting value. The search procedure will iterate from these starting values and continue until a convergence criterion is satisfied. Hopefully, the resulting estimates will correspond to a global and not just a local maximum of the log-likelihood function. 4. Tests for ARCH\GARCH Effects To see whether there are ARCH\GARCH effects evident in the data, first estimate the mean equation by OLS and save the residuals. Denote the OLS residuals from the mean equation as . Consider the autocorrelations of the squared OLS residuals, that is, of . If the autocorrelations for the squared residuals are large and exceed the Bartlet bands, that is an indication of the presence of ARCH\GARCH effects in the data. Alternatively, one may use an LM test for ARCH\GARCH. Estimate the auxiliary regression 2 2 20 1 1t t q t qe e eγ γ γ− −= + + + +… tv where the e’s are the OLS residuals from the mean equation and is an error term. Obtain the from this regression. The LM test statistic is where T is number of OLS residuals. The null and alternative hypotheses are : 0 for all 1,2, , . : 0 for at least one 1,2, , . Large values of lead to a rejection of the null hypothesis of no ARCH\GARCH effects. Alternatively, an F-test for 0 for all 1,2, ,i i qγ = = … could also be used. 5. Tests of Model Adequacy It can be shown that if 21| (0,t t tN )ε σ−Ω ∼ then This provides a means to check on the adequacy of the estimated ARCH\GARCH model. Standardize the residuals by the conditional standard deviation from the fitted ARCH\GARCH model 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com