Reliable Inference in Conditions of Extreme Events by Adriana Cornea University of Exeter Business School Department of Economics ExISta Early Career Event October 17, 2012
Outline of the talk Extreme events & fat-tails Bootstrap Two papers joint with K. M. Abadir (Imperial College): Approximating moments by nonlinear transformations, with an application to bootstrapping for fat-tails Bootstrapping with fat-tailed asymmetry
Black Monday: Extreme event Black Monday, 19 Oct. 1987: Fall of more than 20% Average daily % change, Jan. 1 to Oct. 19, 1987: µ = 0.055% Standard deviation, Jan. 1 to Oct. 19, 1987: σ = 0.95%
Assume normal distribution N(µ, σ 2 ) normal density f(x) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 µ = 0, σ 2 = 0.2 µ = 0, σ 2 = 1.0 µ = 0, σ 2 = 5.0 µ = 2, σ 2 = 0.5 0.2 0.1 0 5 4 3 2 1 0 1 2 3 4 5 x Normal densities
µ = 0.055%, σ = 0.95% Pr(x < 22) 2 10 126 How small is 2 10 126? (Stock and Watson (2007)) The world population is about 6 billion. Probability of winning a lottery among all living people is 1 6 10 9 2 10 10. The universe has existed for 15 billion years, or about 5 10 17 sec. Probability of choosing a particular second at random from all seconds since the beginning of time is 2 10 18. There are approximately 10 43 molecules of gas in the first kilometer above the earth s surface. Probability of choosing one is 10 43. It is extremely unlikely that the return distribution is normal. Tails of the return distribution: much fatter (heavier) than those of the normal distribution.
Normal distribution N(0, 1) is extremely light-tailed Pr( X > x) 1 2πxe x 2 /2 (1) Many economic & financial variables are fat-tailed power laws Pr( X > x) C x α (2) α: tail index (fat-tailedness measure) E X p exists for p < α E X p does not exist for p α Empirics: economics losses from earthquakes (α (0.6, 1.5)), income (α (1.5, 3)), wealth (α 1.5), returns on many stocks (α (2, 4) infinite fourth moment)
Light vs. fat tails 0.4 0.35 0.3 Normal N(25,1) Asymmetric α stable, α=1.2 Lévy, α = 0.5 density f(x) 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25 30 35 40 45 50 x Tails of asymmetric α-stable distributions with α = 1.2 < 2 are fatter than those of the normal distribution (for which α = 2). Tails of Lévy distribution are fatter than those of the asymmetric α-stable distribution with α = 1.2.
Bootstrap Pulling yourself by the bootstraps. Illustration in the Dec. 04 issue of Significance
More serious bootstrap Suppose we have an i.i.d. sample: x = (x 1,, x n ) from an unknown distribution F (θ) Compute ˆθ and a statistic of interest t(n, ˆθ) for testing H 1 : θ = θ 0 vs. H 1 : θ θ 0 A simple bootstrap draws randomly with replacement from the empirical distribution function of x say B times Bootstrap sample x j = (x n 1, x 1, x 5, x 10,, x 5 ), j = 1,, B For each x j compute ˆθ j and t j (n, ˆθ j) Reject H 0 if 1 ( N B j=1 I t j (n, ˆθ j) < t(n, ˆθ) ) is smaller than 0.05 or 0.10 Or build a 90% confidence interval for θ with limits ˆθ (ˆθ 0.95 ˆθ), ˆθ (ˆθ 0.05 ˆθ)
Different bootstraps Naive or nonparametric bootstrap: i.i.d. Wild bootstrap: heteroskedasticity Block bootstrap, subsampling: autocorrelation M out of n bootstrap, subsampling: non-smooth statistics (max(x)), x has an infinite variance distribution Rich literature: Efron (1979), Godfrey (2009), Shao & Tu (1995), Politis, Romano & Wolf (1999), Good (1994), Efron & Tibshirani (1993) and many more
Bootstrap and fat-tails Let s take θ = E(x) and assume x F unknown and var(x) = We want to build a 90% CI about E(x) Naive bootstrap not valid if var(x) = : Athreya (1989), Knight (1989) Previous work: m out of n bootstrap, subsampling: Politis, Romano & Wolf (1999) parametric bootstrap: Cornea & Davidson (2011) Nothing works if tails are fat and asymmetric
An idea Take any transformation of x, x = g(y) such that var(y) < For simplicity we take x = exp(y) Suppose x has a Pareto distribution, F x (u) = 1 u α, 1 < α < 2 Then, F y (w) = 1 e αw E(x) = α/(α 1), E(y) = 1/α, hence E(x) = 1/(1 E(y)) Upper/lower limits of 90% bootstrap CI for E(x) are x n ( x 0.95 x n ), x n ( x 0.05 x n ) ( ) ( ) x 0.95 := 1/ 1 ȳ(b+1)(0.95), x 0.05 := 1/ 1 ȳ(b+1)0.05
Some simulations to illustrate Naive bootstrap M out of n bootstrap 0.90 0.95 0.99 0.90 0.95 0.99 α = 1.1 0.20 0.21 0.23 0.27 0.28 0.28 α = 1.3 0.48 0.52 0.56 0.73 0.76 0.79 α = 1.5 0.63 0.66 0.63 0.89 0.91 0.95 Table 1: Bootstrap coverage probabilities for E(x) without transformation; B = 399; 10,000 replications
Naive bootstrap 0.90 0.95 0.99 α = 1.1 0.89 0.93 0.98 α = 1.3 0.89 0.94 0.98 α = 1.5 0.88 0.93 0.98 Table 2: Naive bootstrap coverage probabilities for E(x) using the transformation; B = 399; 10,000 replications
Relaxing the assumptions In reality we do not know F x and F y and the link between E(x) and E(y) We can use power series expansions of exp(y) Raw expansion Centered expansion x = k j=0 y j j! + R k (3) k x = e E y e y E y = e E y (y E y) j j! j=0 + R c k (4) Higher-order terms create the problems when x has a fat-tailed distribution Bounding R k by a low power term will (hopefully) solve the problem
A crazy expansion Let 1 i 2 and (y E y) /m ζ + 2πi y, where i y Z, m N, and ζ ( π, π] Then we have the expansion x = e E y e 2πmiy (exp ( )) ζ im e E y e ( ) 2πmiy im ξ i k + ϱ x,k ξk := k j=0 ζj /(i j j!) iy is random, but m is deterministic and to be chosen later Binomial expansion gives x = e E y e 2πmiy Re(ξ im k ) + Rc x,k
This expansion allows us to conclude that Rx,k c = O ( p ζ k+1 ) And find an accurate bound for Rθ,k c, θ = E(x) R c θ,k [ e E y E e ( )] 2πmiy ξ im ζ k+1 k H (k + 1)! ξ k Denote ψ = ζ k+1 (k+1)! ξ k Where 1 2e m sin 1 ψ cos (m log (1 ψ )) + e 2m sin 1 ψ H( ψ ) := 1 + e m sin 1 ψ 1 + e mπ [ ) 0, 1 e π/m [ for ψ 1 e π/m, 1 ] (1, ) respectively
Application: transformation-based bootstrap Letting y := log(x) and z := e E y e 2πmiy Re(ξ im k ), we have x = z + Rx,k c and the remainder has the bound Bc x,k By applying the triangle inequality twice, z B c x,k x z + Bc x,k which can be used to build conservative CIs for E(x) To do this, consider an i.i.d. sample x 1,, x n and compute x n := 1 n n x j, j=1 z + n := 1 n n z j, B c x := 1 n,k n j=1 n j=1 B c x j,k By the triangle inequality, t 1,n x n t 2,n, t 1,n := z + n B c x n,k, t 2,n := z + n +B c x n,k
We can bootstrap t 1,n and t 2,n instead of x n for appropriate choice of k and m If Bx,k c is too large then the CI is too conservative (we don t want that). If k or m, then Bx,k c vanishes and z coincides with x and we are back to the original invalid bootstrap. Thus k, m have to be finite and their value chosen depending on the thickness of the tail of x, α.
In practice, first estimate α (using for instance Hill (1975) method) Then, for extreme quantile (99%) take k = 1 and an estimate of m is given by the integer part of exp (1.44 37.90n 1/2 + 14.41α 1 15.42α 2)
n = 100 n = 1000 m 0.90 0.95 0.99 m 0.90 0.95 0.99 α = 1.1 1 0.98 0.98 0.99 1 1 1 1 2 0.58 0.58 0.60 2 0.99 0.99 0.99 3 0.48 0.49 0.54 3 0.67 0.71 0.75 4 0.51 0.53 0.57 4 0.71 0.74 0.79 α = 1.3 1 0.99 0.99 0.99 9 0.99 0.99 0.99 2 0.88 0.90 0.93 12 0.97 0.98 0.99 3 0.88 0.90 0.93 16 0.95 0.96 0.98 4 0.87 0.89 0.92 20 0.92 0.93 0.96 α = 1.5 1 0.99 0.99 0.99 16 0.98 0.99 0.99 2 0.96 0.97 0.98 20 0.97 0.98 0.99 3 0.93 0.95 0.97 25 0.95 0.96 0.98 4 0.86 0.88 0.91 30 0.93 0.94 0.97 Table 3: Transformation-based bootstrap coverage probabilities for E(x), k = 1, B = 399, 10,000 replications; data from Pareto(α)
For lower quantiles take k = 2, an estimate of m is given by the integer part of exp (3.22 66.36n 1/2 1.16α 2 + 102.58n 1 α 2) exp (40.36n 1/2 sign. level 1)
n = 100 n = 1000 m 0.90 0.95 0.99 m 0.90 0.95 0.99 α = 1.1 1 0.90 0.90 0.92 3 0.99 0.99 0.99 2 0.82 0.84 0.86 4 0.98 0.98 0.99 3 0.77 0.79 0.81 5 0.93 0.94 0.95 4 0.66 0.67 0.69 6 0.86 0.87 0.89 α = 1.3 1 0.98 0.99 0.99 4 0.99 0.99 0.99 2 0.96 0.97 0.98 5 0.99 0.99 0.99 3 0.90 0.91 0.93 6 0.96 0.96 0.98 4 0.82 0.83 0.86 7 0.92 0.94 0.95 α = 1.5 1 0.99 0.99 0.99 6 0.97 0.97 0.98 2 0.97 0.98 0.99 7 0.94 0.95 0.97 3 0.92 0.93 0.95 8 0.91 0.93 0.96 4 0.86 0.88 0.91 9 0.89 0.91 0.94 Table 4: Transformation-based bootstrap coverage probabilities for E(x), k = 2, B = 399, 10,000 replications; data from Pareto(α)