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Appendix

C

inferential Statistics

i

n Appendix A, we provided the basics of descriptive statistics. Our focus in
this appendix is on inferential statistics, covering the three major topics of
point estimators, confidence intervals, and hypothesis testing.

Point Estimators

Since it is generally infeasible or simply too involved to analyze an entire
population in order to obtain full certainty as to the true environment, we
need to rely on a small sample to retrieve information about the population
parameters. To obtain insight about the true but unknown parameter value,
we draw a sample from which we compute statistics or estimates for the
parameter.
In this section, we will learn about samples, statistics, and estimators.
In particular, we present the linear estimator, explain quality criteria (such
as the bias, mean squared error, and standard error) and the large-sample
criteria. In the context of large-sample criteria, we present the idea behind
consistency, for which we need the definition of convergence in probability
and the law of large numbers. As another large-sample criterion, we intro-
duce the unbiased efficiency, explaining the best linear unbiased estimator or,
alternatively, the minimum-variance linear unbiased estimator.

Sample, Statistic, and estimator

The probability distributions typically used in financial econometrics depend
on one or more parameters. Here we will refer to simply the parameter θ,
which will have one or several components, such as the parameters for the
mean and variance. The set of parameters is given by Θ, which will be called
the parameter space.