Extreme Value Theory
2. Caveats
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In practice life is not as simple as is suggested in the Introduction. A
particularly important issue is that extrapolation into the tail of a
probability distribution isn’t challenging because it is difficult to identify
possible probability distributions that might fit the observed data. Instead
the challenge is that the range of answers that can plausibly be obtained can
be very wide. Extrapolation of any sort (including, as here, extrapolation into
the tail of a distribution) is an intrinsically uncertain exercise, much less
reliable than interpolation, as is explained in Press et al
(2007).
Three other important caveats are relevant when EVT is
applied to financial data:
(a) EVT
relies on the tail of the distribution in question actually converging in some
suitable sense. This generally occurs for smooth distributions commonly used by
statisticians like the normal distribution, the Student’s t
distribution, the Pareto distribution etc. However, these sorts of
distributions are very ‘well behaved’ in a mathematical sense and also form an
infinitesimal proportion of the totality of possible distributions that might
apply. So it is by no means obvious that convergence of the sort required for
EVT to apply will actually take place in practice. It is relatively
straightforward to construct distributions where convergence doesn’t occur,
although whether they are plausible is again a matter of opinion.
Fundamentally, extrapolation involves exercise of judgement, and what one
person thinks is reasonable someone else may think is not.
(b) EVT
is usually developed from a univariate, i.e. single series, perspective. Some
important financial problems, in particular portfolio construction, are
intrinsically multivariate in nature. For example, most practical portfolio
construction problems require selection between asset categories, so require an
understanding of the joint behaviour of different return series. It is possible
to develop a multivariate extreme value theory (including results for
multivariate maxima and multivariate threshold exceedances), but the
mathematics is quite complicated, perhaps best analysed using copulas, and is
not very easily aligned to the portfolio construction problem.
(c) The
conceptual appeal of EVT may encourage researchers to leap in with the
technique without first trying to understand what might be causing the observed
tail behaviour. An important point here is that financial data often exhibits
time-varying volatility (also known as volatility clustering). If this point is
given insufficient weight then EVT results, even if theoretically applicable,
may be easily misinterpreted or misapplied.
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