In financial econometrics, a stylized fact is a structural observation that is believed to hold for a diverse collection of instruments, markets, and time periods. Surveys of stylized facts provide a good hunting ground for the final projects for 956.
We recently ran into the issue of testing multiple hypothese when we wanted to know about the significance of the transition probablities of an estimated Markov matrix. To give some deeper background on this I have collected some resources on multiple test procedures, such as the Holm procedure that we discussed in class.
Hidden Markov Models are formally state-space models but they have a character of their own. We will use HMMs to develop our own view of the "variability of volatility." The methods used to estimate HMMs are closely linked with the famous EM Algorithm which we will also explore in due course. Graphical Models and Bayesian Networks are probabilistic structures that contain the HMMs as a special case. They are not directly part of our course, but they are definitely worth learning about.
State space models and the Kalman filter. This is a massive area which sustains departments of engineering all over the world, both in academia and in industry. Oddly enough, its shear girth may cause it to be under acknowledged outside of engineering. A great range of statistical models can be put into a state space framework, even including models that lack an explicit "time" element. The technology for dealing with state space models has been so intensively developed, one almost always has the opportunity to profit from this translation. The only bad news is that you may well find that your great new idea is actually old hat.
There is also a rapidly developing technology for dealing with state space models that are non-linear and/or non-Gaussian. One of the most promising ideas here is that of a particle filter.
Many financial time series suggest that the underlying "volatility" varies over time. There is now a giant literature that engages this possibility, and, even though we will spend perhaps a quarter of our course on the relevant models. This page provides pointers to resources that go from classics, to tutorials, to new topics for research. Despite an inevitable overlap, the links are split into two subtopics: ARCH-GARCH and Stochastic Volatility Models.
Yule-Walker, Durbin-Levinson, Burg, and such. This brief collection of links is largely self-explanitory. A few of the links will be discussed in class.
Is it noise? Or, can you detect evidence of non-independence or evidence of forecastability? The links reviewed here all address this issue. The collection includes a link to the original 1976 article of Ljung and Box.
Are they normal? Many of the procedures used in the analysis of time series depend to a lesser or greater extent on the assumption that the "error term" is normally distributed. This page collects current and classic resources that speak to this question.
Forecasting! There are many important applications of time series that have little to do with forecasting, yet forecasting has an inevitable appeal to people who are interested in financial time series. Even when forecasting can't be done very well, it is important to know what can be done. In many ways, forecasts are "the stuff that dreams are made of."
Spinning Coins? It has long been known that if you spin a US one cent piece on a smooth surface, the probability that it comes up heads is FAR from 1/2. In 2002 there was a rash of stories about a very small experiment with the spinning of a Belgian one Euro coin. This page collects some of the relevant links, and, along the way, has some basic information on Bayesian estimation.
Nutty Models? Not all statistical analyses meet the highest standards. In fact, some are nutty. The links offered here suggest where one might look when in search of "instructive examples."
Connections to physics, sound, speech, hearing, and other parts of physical reality? We can't do much about these in class, but some of the connections are fascinating.
We will soon start discussing the methods by which statistical insights can be turned into trading strategies. One part of the picture is given by the Kelly-Breiman theory of bet sizing. I will also collect some background material on hedge funds and their strategies.
Andrew Moore's Data Mining Tutorials. This is a lovely collection of PDFed-PowerPoint presentations. I found it when looking for material on cross-validation, but I ended up digging through a slew of the tutorials. In each case, the tutorial gets quickly to the heart of the issue. They are also littered with (mostly amusing) jokes. For our class, the most relevant tutorials are those on the Gaussian Distribution and on Hidden Markov Models. We may go over Moore's HMM tutorial in class.