Wednesday, January 23

Estimating GARCH models: when to use what?

For those who do not like mathematics, turn away now!

If you have been following the financial markets, you will see that the volatility has shot through the roof and most of the indices are deep in the red. One of the crucial ways of modelling the behaviour of financial markets is to use GARCH models (Generalized AutoRegressive Conditional Heteroscedastic) to model the time series with time varying volatility. For example, these models are particularly useful for modelling when the distribution has fat tails.

What are fat tails? Well, we are not talking about sheep here, it is just a statistical term for a bit of the probability of certain events happening. Confused? So am I. Let me try to explain, if you take your classmates and then arrange them in the following manner in a line. Plonk the average in height people in the middle and cluster them according to the number. So you have a big thick clump in the middle. On the right, you have very few tall people and on the left, you have very few short people. In general, take anything in nature, you would get this "normal" distribution of "stuff". It looks like the bell curve. Size of lemons. Amount of hair. So if you take a bunch of lemons or a bunch of guys, you will find that the majority will be around the mean, with progressively lesser numbers as you move away from the mean. At the far end, you will have one hirsute gorilla and on the other hand, you will have Apurva Agnihotri (chap looks like a boiled potato to me!). In other words, the tails, or ends of the distribution are for very rare occurrences.

So how does it matter? Well, if you were trying to figure out what is the size of next lemon you are going to pick up, then you need to have an idea about how does the distribution lie. If you know the distribution, then you can predict, with a fair degree of accuracy, as to what is the size of the next lemon you pick up. (Replace lemon with head hair, ladies, and you see the usefulness of this theory - helps you avoid lemons). Similarly, in the financial markets, knowing the distribution helps us to know how the markets will behave. Each market, product, currency, stock etc. has its own distribution and it changes in time.

And boyo, do we have fat tails or do we? For example, we saw events which were 25 standard deviation events last summer. This means that the last time this could have happened, we were covered in hair, used bone implements and grunted at each other. In other words, these events are frankly not predictable and if you did predict it using standard mathematics, these would only occur once every zillions of years. Instead of that, we have these events occuring every now and then. Putting it in another way, usually you would find only 1 out of 100 being totally bald if it was normal distribution, but if it was a fat tail, we would end up having say 10 in 100 men being bald (but never fear, everything evens up, if you have less hair to comb, you have more face to wash!)

Now this is indeed the problem, how on earth do you manage to model something like this? How unlikely an event can be before you manage it? You might as well as ask me (if I put on my mathematician or statistician hat) to be omnipotent. But we try.

I used these models in the dim and distant past in 1995 when I was comparing the usage of nonparametric sigmoid based models with GARCH models in a variety of asset classes. While you do know that GARCH is good, the next question which emerges is, which estimator to use and I had problems back then? It was interesting to see that we are still facing the problem and this new paper shows an interesting way of handling this problem and they use two financial time series to prove their hypothesis. Right now, I would bet that every rocket scientist in the city will be pouring over his models to figure out why they didnt predict the current splash.

But then, if you do manage to figure it out, you get to get quoted like this:

Their wealth is not exaggerated. Just days ago, one hedgie calculated the final numbers on what might be history’s biggest pay cheque: funds run by John Paulson are up £6 billion after doing nicely from the global credit crisis, and his share is estimated at a billion pounds or so. Indeed, several got close to a billion a year ago. There was John Arnold, then 33, who correctly judged that natural gas prices would go down, not up. And the mathematician Jim Simons, 69, whose fees, at 44% of investors’ profits, mean that he, too, was near nine zeroes sterling.

Estimating GARCH models: when to use what? Da Huang, Hansheng Wang, Qiwei Yao, The Econometrics Journal

Summary The class of generalized autoregressive conditional heteroscedastic (GARCH) models has proved particularly valuable in modelling time series with time varying volatility. These include financial data, which can be particularly heavy tailed. It is well understood now that the tail heaviness of the innovation distribution plays an important role in determining the relative performance of the two competing estimation methods, namely the maximum quasi-likelihood estimator based on a Gaussian likelihood (GMLE) and the log-transform-based least absolutely deviations estimator (LADE) (see Peng and Yao 2003 Biometrika, 90, 967–75). A practically relevant question is when to use what. We provide in this paper a solution to this question. By interpreting the LADE as a version of the maximum quasi-likelihood estimator under the likelihood derived from assuming hypothetically that the log-squared innovations obey a Laplace distribution, we outline a selection procedure based on some goodness-of-fit type statistics. The methods are illustrated with both simulated and real data sets. Although we deal with the estimation for GARCH models only, the basic idea may be applied to address the estimation procedure selection problem in a general regression setting.

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