Chapter 3 - Conditional heteroskadastic models Flashcards
What is the objective in this chapter?
The objective is to understand methods used to estimate and model volatility of returns on assets. Such models are known as heteroskedastic models.
What do we mea nby volatility?
Conditional standard deviation of stock returns.
most important feature of volatility
It is not observable. Only estimatable
what can we say about IV compared to other types?
Tend to be larger than those produced by for instance GARCH models.
IV
Volatility is not observable, but it has some known characetistics. Elaborate on them
1) Volatility tends to cluster
2) Volatility operate witihn a fixed range
3) Tend to not be jumps, more like continuous manner
4) Leverage effect: Volatility tend to act differnetly to price increases and to price drops
elaborate on on this image. What are we seeing here? What is the conclusions?
We see autocorrelation function of the asset returns. They show very little sign of being autocorrelated.
However, we see that their magnitudes are correlated. This means that large levels of volatility follow from large levels of volatility, and the same for low levels.
This means that when stock prices move a lot, they will likely see large movements both up and down. And when they dont move a lot, they remain very flat.
This is an empirical result.
The conclusion is that log returns are not serially correlated, but they are dependent. Volatility models attempt to capture this-
what do we mean by conditional mean
expected value of some variable, for instance the stock return or log stock return, given all the infromation prior to that point in time.
elaborate on this so-called model
the assumption is that log returns have very little autocorrelation that can be captured anyways, and therefore a simple model is probably the best. Therefore we use an ARMA model to represent how r_t might be “generated” from its earlier lags.
THen we say move away from autocorrelation and say that the stock returns also depend on other things, which is why we add the regular linear regression part to it. So we end up with 1 part autocorrelation time series, and 1 part regular regression. They work together to produce the log return for the next period.
Then we make use of the fact that r_t = mu_t + a_t.
mu_t is the “conditional mean”, which is the expected log return given all prior information.
a_t represent the unpredicatable part. Therefore, we have that the log return consist of whatever we have to be the expected log return conditoonal on the prior infromation, and some other part that we dont know much about. Basically, we say “log return at time t is equal to our expectation of what it should be, plus some uncertainty that we cant account for”.
now we have a series like this:
r_t = ø_0 + ∑bi xit + ∑cj r_{t-j} + ∑d_k a_{t-1} + a_t
Then we use the fact that mu_t = r_t - a_t, and we get
mu_t = ø_0 + ∑bi xit + ∑cj r_{t-j} + ∑d_k a_{t-1}
Aftert this we do some assumptions and some more shit and I odnt know what we end up with
how can we remove linear dependence from a time series
First we need to understand what linear dependence is.
linear dependence is all kinds of dependencies that affect the dependent variable in some way. It is a broad categroy, and includes both the effect of autocorrelation AND the effect of explanatory variables.
We can remove the effect of autocorrelation by fitting an ARMA model. The residuals of the model is then generally assmed to be free of auto correlation. Thus, the variation that remains is not due to autocorrelation.
We can extend this, and remove teh effect of explanatory variables. This is done by adding those terms (regular linear regression) to the already existing ARMA model. The residuals of such a model will now contain neither autocorrelation nor other types of linear dependence.
The result is residuals that are purely random variation. This is the metric we are interested in when modeling volatility.
The key point is that while the residuals are now random in their mean, they can still exhibit dependent behavior, like clusteirng.
Take a second to consider what we have here.
We have residuals. Residuals are the variation we did nto explain wit hthe model. Their values are random in their mean, which means that we cannot hope to predict them. However, the variance of the residuals may still have patterns. ANd it is this pattern that we are intereseted in.
elaborate on building volatility models for asset return series
1) specify a mean equation and removing linear dependence
2) Use the residuals of the mean equation to test for ARCH effects
3) if arch effects are statistically significant, specify a volatility model.
4) check the fitted model, and refine if needed
what can we say about the variance of white noise?
homoskedastic.
This of course means that if we find that residuals have tendencies of volatility clusteirng, its’ variance is heteroskedastic which means that it is not white noise.
