### Chapter 7

#### Inference

# 7.1 Motivation

In Section 1.7.7, we described three components of any value-at-risk measure:

- an inference procedure,
- a mapping procedure, and
- a transformation procedure.

In this chapter, we discuss inference procedures. Unfortunately, the discussion will be somewhat tentative. Whereas many sophisticated techniques are available to support mapping and transformation procedures, techniques for inference procedures are less developed. Researchers are studying ways to extend traditional methods of time series analysis to the needs of value-at-risk measures, but techniques currently used are largely ad hoc.

The purpose of an inference procedure is to characterize a distribution for key factors ^{1}** R** conditional on information available at time 0. Although practice varies, the characterization typically takes one of three forms:

- a fully specified conditional distribution;
- a conditional mean
^{1|0}**μ**and conditional covariance matrix^{1|0}**Σ**; - a realization {
^{1}*r*^{[1]},^{1}*r*^{[2]}, … ,^{ 1}*r*^{[m]}} of a sample for the conditional distribution.

In the following section, we discuss how to select key factors that will facilitate an inference procedure. We then describe current practice for constructing the first or second of the above three forms of characterizations. Next, we critique current practice and identify avenues for further research. We close with a discussion of the third of the three forms just listed.