Estimating the Distribution of the Market Invariants

In this chapter we discuss how to estimate the distribution of the market invariants from empirical observations. In Section 4.1 we define the concept of estimator, which is simply a function of current information that yields a number, the estimate. Such a general definition includes estimators that perform poorly, i.e. functions that yield an estimate which has little in common with the real distribution of the market invariants. Therefore we discuss optimality criteria to evaluate an estimator. After defining estimators and how to evaluate them, we need to actually construct estimators for the market invariants. Nevertheless, constructing estimators by maximizing the above optimality criteria is not possible. First of all, the search of the best estimator among all possible functions of current information is not feasible. Secondly the optimality criteria rarely yield a univocal answer. In other words, an estimator might perform better than another one in given circumstances, and worse in different circumstances. Therefore, we construct estimators from general intuitive principles, making sure later that their performance is acceptable, and possibly improving them with marginal corrections. In this spirit, we proceed as follows. In Section 4.2 we introduce nonparametric estimators. These estimators are based on the law of large numbers. Therefore, they perform well when the number of empirical observations in the time series of the market invariants is large, see Figure 4.1. When this is the case, nonparametric estimators are very flexible, in that they yield sensible estimates no matter the underlying true distribution of the market invariants. In particular we discuss the sample quantile, the sample mean, the sample covariance and the ordinary least square estimate of the regression factor loadings in an explicit factor model, stressing the geometrical properties of these estimators. We conclude with an overview of kernel estimators. When the number of observations is not very large, nonparametric estimators are no longer suitable. Therefore we take a parametric approach, by assuming that the true distribution of the market invariants belongs to a restricted class of potential distributions. In Section 4.3 we discuss maximum symmys.com