Chapter 3 - Tsay Flashcards
(31 cards)
what is conditional heteroscedasticity about
finding the conditional volatility.
This is the volatility that appears from the result of haivng prior infromation. It is therefore different from the unconditional volatility.
what can we say about IV vs that from GARCH
IV tend to be greater
elaborate on the characteristics of volatility
1) Not directly observable
2) Tendency to cluster
3) Continuous, jumps are very rare
4) does not diverge to infniity, it is mean reverting
5) Leverage effect (react differnetly to negative shocks than positive shocks)
Why is the point of these characteristics of volatility?
we want our model to be able to represent them
what is the foundation of conditional heteroskedastic models
Especially in the context of asset return series, the idea is that the series r_t is uncorrelated or correlted serially with just minor correlations, however r_t is what we call a “dependent” series.
The dependency is not linear.
It appears that the dependency is is in the magnitude, or squares of the returns.
what is the important “conditonal volatility” result in this early part of the chapter?
Var[r_t | prior info] == Var[a_t | prior info]
sigma_t^2 = Var[a_t | prior info]
This say that all the uncertainty is in the error term. This assumes that we are able to remove all other kinds of dependencies from the resduals I believe.
what is the model building procedure
1) Specify a mean equation, liek ARIMA. The goal is to remove linear dependencies
2) Use reisudals of the mean equaiton to check for ARCH effects.
3) Specify a volatility model if ARCH is present
4) Check the model for adequacy
elaborate on specifying the mean equation
The general idea is that it will usually be very simple. In some cases, AR is needed, in other cases we need ARIMAX.
In the simplest cases with no serial dependence, simply using the unconditional mean might be enough.
elaborate on testing for ARCH effects
There are 2 ways to do this:
1) Ljung-Box on the {a_t^2} series
2) ARCH-LM
elaborate on ARCH-LM
this is a regular F-test where we have a restricted-unrestricted divided by unrestricted regression with a certain set of degrees of freedom.
The unrestricted regression is the regular regression of squared residuals where the variables are lag variables. We include some arbitrary amount.
The restricted regression is under the null that these coeffs should be 0, so it basically only is the constant/mean.
If the difference, the ratio, is significantly large, it means that the restricted regression performs much worse than the unrestricted, which means that the unrestricted regressions has something to it. this means that there is a non-linear, a squared, relationship in th eresiduals.
how do we find the reisudal series?
We need some mean equation that removes the conditional mean from the time series.
a_t = r_t - mu_t
elaborate on ARCH model and what they consider to be factual
We assume that the condiitonal volatility sigma_t^2 is a funciton of the past residuals.
sigma_t^2 = alpha_0 + alpha_1 a_{t-1}^2 + … +
ALSO:
alpha_t = sigma_t x e_t.
It is important to understand that ARCH models view the CURRENT shock a_t as a consequence of the current level of volatility. The current level of volatility is given as a dependency to earlier values, and the volatility then influence the shock.
ARCH models view the current shock to be heavily influenced by the current volatility. while ARIMA models view the shock as “completely random” this is a result of their limit of being only linear. ARCH models take the current shock and view it differently. They view it as a result of some other random event, scaled by the current level of volatility.
give the uncodnitional mean of the shocks under ARCH
E[a_t] = E[sigma_t e_t] = E[sigma_t] E[e_t] = E[sigma_t] * 0 = 0
Var(a_t) = E[a_t^2] = E[sigma_t^2 e_t^2] = sigma_t^2 E[e_t^2]
the variance of e_t is 1
Var(a_t) = sigma_t^2
What is wrong?
We cannot pull sigma_t^2 out of the expression because it is not a constant.
give the unconditional variance of the shocks under ARCH
Var(a_t) = E[a_t^2] = E[E[a_t^2 | prior]]
Now we must assume a model shape. We consider a simple ARCH(1) model
From the definition of an ARCH model, we have that E[a_t^2 | prior] = Var(a_t | prior) = sigma_t^2 = alpha_0 + alpha_1 a_{t-1}^2
therefore we get:
= E[alpha_0 + alpha_1 a_{t-1}^2] = alpha_0 + alpha_1 E[a_{t-1}^2]
Now, in order to have it stationary, we require that the variance is constant.
Var(a_t ) = alpha_0 + alph_1 Var(a_{t-1})
Var(a_t) (1 - alpha_1) = alpha_0
Var(a_t) = alpha_0 / (1 - alpha_1)
we require that alpha_1 is not 1.
We also require that alpha_1 is so that the denominator is positive, because otherwise we get a ngative variance estimate.
why would we care about the fourth moment of a_t under ARCH
tells us about the tail characteristics. We want our model to create bigger chance for outliers. therefore we reuqire kurtosis to be larger than 3, as 3 is the normal distribution.
elaborate on the key resutls here
we require the fourht moment to exist, and therefore have some conditions on the denominator.
We also see that if we take the unconditional kurtosis, which this reuslt is, we have that it is always larger than 3. This means that the ARCH(1) model produce larger chance of outliers than the nromal distribution, which follow with the idea of volatility clustering.
elaborate on finding the fourth moment conditions
E[a_t^4] = ?
E[E[a_t^4 | prior]] = ?
E[a_t^4 | prior] = ?
= E[sigma_t^4 e_t^4 | prior]
Given prior informaiton, we know that our ARCH model specify the current volatility, sigma_t. Therefore this behaves as a constant, and we can pull it out
= sigma_t^4 E[e_t^4 | prior]
The remaining expectation is the conditional fourth moment of hte standard nromal distrbiution. It depend on nothing earlier, so the prior has nothintg ot say
= sigma_t^4 E[e_t^4]
The fourth moment of a stnadard normal distribuiton is given as 3.
therefore, we get:
= 3sigma_t^4
we can use our model instead. sigma_t^2 = alpha_0 + alpha_1 a_{t-1}^2
sigma_t^4 = (alpha_0 + alpha_1 a_{t-1}^2)^2
Thus we get:
E[a_t^4 | prior] = 3 (alpha_0 + alpha_1 a_{t-1}^2)^2
Now we can continue wiht the unconditional fourth moment:
E[E[a_t^4 | prio]] = E[3 (alpha_0 + alpha_1 a_{t-1}^2)^2]
= 3 E[(alpha_0 + alpha_1 a_{t-1}^2)^2]
= 3 E[ß_0^2 + 2ß_0ß_1 a_{t-1}^2 + ß_1^2 a_{t-1}^4]
= 3[ß_0^2 + 2ß_0ß_1 E[a_{t-1}^2] + ß_1^2 E[a_{t-1}^4]]
= 3[ß_0^2 + 2ß_0ß_1 Var(a_t) + ß_1^2 m_4]] where m_4 is the fourth moment.
We actually have m_4 at LHS and RHS now:
m_4 = 3[ß_0^2 + 2ß_0ß_1 Var(a_t) + ß_1^2 m_4]]
if we use the earlier expression for uncodnitional variance, we get the result on the image.
elaborate on weaknesses of ARCH mdoels
1) Assumes that positive and negative shocks have the same effect on volatility. We know that this does not align with empirical findings
2) the parameters are actually kind of restricted
3) It give no insight on CAUSE/SOURCE of variations. it only describe the mechanical system.
how to specify ARCH order
We use PACF of the squared shock series.
how to estimate paramters in ARCH
MLE
Elaborate on model checking the ARCH
the standardized residuals, a_t_bar = a_t/sigma_t SHOULD produce an iid sequence. Therefore we check adequacy by using the standardized series with something like the ljung box.
We can use a_t_bar with Ljung Box to check the adequanct of the mean equaiton, and a_t_bar^2 with Ljung box to check the volatility equation.
name an issue with ARCH models
Usually need a lot of parameters
how to make ARCH better
GARCH, generalized ARCH.
Includes past volatility values as well.
