Autoregression and Cointegration
Probably the most commonly applied time series model structure in any field is the autoregressive model. Future values of a series are projected as weighted averages of recently exhibited values in that series. Examples already seen include the moving average and the exponentially weighted moving average. The autoregressive model of order p relates the series outcome at time t to a linear combination of the p immediately preceding outcomes yt i yt-i ----- t - pyt - p t The coefficients, 5 , or...
E[Pf PtiPti PtPtm x 1 E[Pt1 PtPt1 PtPt m x j
The two pieces correspond to a downward moves from a price above the median and b upward moves from a price below the median. It is important to keep the two parts separate because the expected values of each are generally different only for symmetric distributions are they equal. Consider the first term only. From Figure 7.1, the cases of interest define the conditional distribution represented by the shaded region. For any particular price Pt pt gt m, the expected amount FIGURE 7.1 Generic...
A Factor Model
The modeling discussion thus far has focused on spreads between pairs of stocks, the domain where statistical arbitrage, as pairs trading, had its genesis. Now we will discuss some modeling ideas applied to individual stock price series analyzed as a collection. The notion of common risk factors, familiar from Barra-type models, lies at the heart of so-called factor models for stock returns The basic idea is that returns on a stock can be decomposed into a part that is determined by one or more...
Statistical Arbitrage
Much of what happens can conveniently be thought of as random variation, but sometimes hidden within the variation are important signals that could warn us of problems or alert us to opportunities. Box on Quality and Discovery, G.E.P. Box The pair trading scheme was elaborated in several directions beginning with research pursued in Tartaglia's group. As the analysis techniques used became more sophisticated and the models deployed more technical, so the sobriquet by which the discipline became...
REVERSION IN fl STATIONARY RANDOM PROCESS
We begin the study with consideration of the simplest stochastic system, a stationary random process. Prices are supposed to be generated independently each day from the same probability distribution, that distribution being characterized by unchanging parameters. We shall assume a continuous distribution. Price on day t will be denoted by Pt, lowercase being reserved for particular values such as a realized price . Some considerations that immediately suggest themselves are 1. If Pt lies in...
Factor Analysis
A factor analysis of a multivariate data set seeks to estimate a statistical model in which the data are ''explained'' by regression on a set of m factors, each factor being itself a linear combination weighted average of the observables. Factor analysis has much in common with principal component analysis PCA which, since it is more familiar, is good for comparison. Factor analysis is a model based procedure whereas principal component analysis is not. PCA looks at a set of data and finds...
Defactored Returns
Another successful model based on factor analysis reversed the usual thinking Take out the market and sector movements from stock returns before building a forecast model. The rationale is this To the extent that market factors are unpredictable but sentiment about the relative position of individual stocks is stable over several days, such filtered returns should exhibit more predictable structure. Let's look in a little more detail at this interesting idea. Residuals from the fitted...
Preface
Acknowledgments xxiii CHAPTER 1 Spread Margins for Trade Rules 16 Correlation Search in the Twenty-First Century 26 Portfolio Configuration and Risk Control 26 Risk Control Using Event Correlations 31 Evolutionary Operation Single Parameter Illustration 34 Exponentially Weighted Moving Average 40 Classical Time Series Models 47 Autoregression and Cointegration 47 Doubling A Deeper Perspective 61 Prediction Model for Defactored Returns 65 Proof of the 75 percent Rule 69 Analytic Proof of the 75...
Factor Analysis Primer
The following material is based on the description of factor analysis in The Advanced Theory of Statistics, Volume 3, Chapter 43, by Sir Maurice Kendall, Alan Stuart, and Keith Ord now called Kendall's Advanced Theory of Statistics KS . The notation is modified from KS so that matrices are represented by capital letters. Thus, y in KS is r here. This makes usage consistent throughout. Suppose there are p stocks, returns for which are determined linearly from values of m lt p unobservable...
The 75 Percent Rule
The model just described is formalized as a probability model as follows. Define a sequence of identically distributed, independent continuous random variables Pt, t 1,2, with support on the nonnegative real line and median m. Then Pr Pt gt Pt-1 n Pt-1 lt m U Pt lt Pt-1 n Pt-1 gt m 0.75 In the language of the motivating spread problem, the random quantity Pt is the spread on day t a nonnegative value , and days are considered to be independent. The two compound events comprising the probability...
Proof of the 75 Percent Rule
The proof of the result uses a geometric argument to promote visualization of the problem structure. An added bonus is that one can see that certain structural assumptions made in the theorem may be relaxed. These relaxations are discussed following the proof of the basic result. Consider the joint distribution of two consecutive terms of the sequence, Pt-1 and Pt. Assuming independence, the contours of this joint distribution are symmetric about the line Pt Pt-1 regardless of the precise form...