Applications of bootstrap method to finance Chin-Ping King

Population distribution function F empirical distribution function(EDF) F n F (x 1, x 2,…, x n ) where x= (x 1, x 2,…, x n ) F n (x 1 *, x 2 *,…, x n * ) where x * = (x 1 *, x 2 *,…, x n * ) Probability of elements of population which occur in x : P Probability of elements of EDF which occur in x * : P n P n ~ Bi(P, (p*(1-p)/n)) By Weak Law of Large Number and Central Limit Theorem (n) 1/2 (F n - F) d N(0, p*(1-p))

Estimation of standard deviation and bias Estimator of θ : θ ’ θ ’ =s(x) Standard deviation: se={∑ n j=1 [θ ’ j - θ ’ (.)] 2 /(n-1)} se(.)= ∑ n j=1 θ ’ j /n Bias: bias=E[θ ’ ]- θ Root mean square error of an estimator θ ’ for θ: E[(θ ’ - θ) 2 ]= se 2 *{1+(1/2)*(bias/se)} 2

Nonparametric bootstrap The bootstrap algorithm for estimating standard errors(or bias) 1. Select B independent bootstrap samples x *1, x *2, …, x *B, each consisting of n data drawn with replacement from x. Total possible number of distinct bootstrap samples is C(2n-1,n). 2. Evaluate the bootstrap replication corresponding to each bootstrap sample θ ’* (b)=s(x *b ) b=1,2,…,B 3. Estimate the standard error (or bias) by the sample standard deviation (or bias) of the B replications: se ’ B = se={∑ B b=1 [θ ’* (b)- θ ’* (.)] 2 /(B-1)} θ ’* (.)= ∑ B b=1 θ ’* (b) /B Bias ’ B =θ ’* (.)- θ ’

A Schematic diagram of the nonparametric bootstrap Unknown Observed Random Empirical Bootstrap Population Sample Distribution Sample Distribution F x= (x 1, x 2,…, x n ) F n x * = (x 1 *, x 2 *,…, x n * ) θ ’ =s(x) θ ’* (b)=s(x *b ) Statistic of interest Bootstrap Replication

Parametric bootstrap Function form of population probability distribution F has been known, but parameters in population probability distribution F are not known Parametric estimate of population probability distribution : F par We draw B samples of size n from the parametric estimate of estimate of the population probability distribution F par : F par x * = (x 1 *, x 2 *,…, x n * )

Error in bootstrap estimates m i = the ith moment of the bootstrap distribution of θ ’ Var(se ’ B ) = Var(m 2 1/2 ) + E[(m 2 ( △ +2))/4B] △ = m 4 /m , the kurtosis of the bootstrap distribution of θ ’ Var(m 2 1/2 ):sample variation, it approaches zero as the sample size n approaches infinity E[(m 2 ( △ +2))/4B]:resampling variation, it approaches zero as B approaches infinity

Confidence intervals based on bootstrap percentiles (1-α) Percentile interval: [θ ’ % low, θ ’ % up ]= [θ ’*(α/2) B, θ ’*(1-(α/2)) B ] θ ’*(α/2) B : 100*(α/2)th empirical percentile, or B *(α/2)th value in the ordered list of the B replications of θ ’* θ ’*(1-(α/2)) B : 100*(1-(α/2))th empirical percentile, or B *(1-(α/2))th value in the ordered list of the B replications of θ ’*

Percentile interval lemma Suppose the transformation ψ ’ =t(θ ’ ) perfectly normalize the distribution of θ ’ : ψ ’ ~ N(ψ, c 2 ) For some standard deviation c. Then the percentile interval based on θ ’ equals [t -1 (ψ ’ -z (1-(α/2)) *c), t -1 (ψ ’ -z (α/2) *c)] Example: θ ’ =exp(x) x ~ N(0,1) ψ ’ =t(θ ’ )=logθ ’

Coverage performance Results of 300 confidence interval realizations for θ ’ =exp(x) Method % miss left % miss right Standard normal Interval Bootstrap percentile Interval miss left: left endpoint >1 Miss right: right endpoint <1

Transformation-respecting property The percentile interval for any (monotone) parameter transformation ψ ’ =t(θ ’ ) is simply the percentile interval for θ ’ mapped by t(θ ’ ) : [ψ ’ % low, ψ ’ % up ]= [t(θ ’ % low ), t(θ ’ % up ) ]

Better bootstrap confidence intervals (1-α) BC a interval: [θ ’ low, θ ’ up ]= [θ ’*(α1), θ ’*(α2) ] α1 and α2 are obtained by standard normal cumulative distribution function of some correction formulas for bootstrap replications. BC a interval is transformation respecting.

Accuracy of bootstrap confidence interval For (1- α )coverage, approximate confidence interval points θ ’ low and θ ’ up are called first order accurate if: Pr(θ ≦ θ ’ low )= (α/2 )+ O(n -1/2 ) Pr(θ ≧ θ ’ up )= (α /2)+ O(n -1/2 ) And second order accurate if Pr(θ ≦ θ ’ low )= (α/2 )+ O(n -1 ) Pr(θ ≧ θ ’ up )= (α/2 )+ O(n -1 ) Percentile interval : first order accurate. BC a interval : second order accurate.

Calibration of confidence interval points 1.Generate B bootstrap samples x *1, x *2, …, x *B. For each sample b=1,2,…,B: 1a) Compute a λ-level confidence interval point θ ’* λ (b) for a grid of values of λ. Where θ ’* λ (b) can be θ ’* (b)-z 1-λ *se ’* (b). 2. For each λ compute p ’ (λ)=#{θ ’ ≦ θ ’* λ (b) }/B. 3. Find the value of λ satisfying p ’ (λ)= α/2

Calibration of percentile interval and BC a interval Once calibration of percentile interval: second order accurate Pr(θ ≦ θ ’ low )= (α/2 )+ O(n -1 ) Pr(θ ≧ θ ’ up )= (α/2 )+ O(n -1 ) Once calibration of BC a interval: third order accurate Pr(θ ≦ θ ’ low )= (α/2 )+ O(n -3/2 ) Pr(θ ≧ θ ’ up )= (α /2)+ O(n -3/2 )

Computation of the bootstrap test statistics 1.Draw B samples of size n with replacement from x. 2.Evaluate ϕ(.) on each sample, ϕ(x *b ) where ϕ(.) is test statistics b=1,2,…,B 3. Approximate P-value by P-value=#{ϕ(x *b ) ≧ ϕ obs }/B or P-value=#{ϕ(x *b ) ≦ ϕ obs }/B Where ϕ obs = ϕ(x) the observed value of test statistics

Asymptotic refinement Asymptotically normal test statistics ϕ ϕ d N(0,σ 2 ) ϕ ~ G n (u,F) G n (u,F): exact cumulative distribution G n (u,F)=Pr(|ϕ| ≦ u|F) G n (u,F) φ(u) as n approaches infinity (assume σ=1) φ(u): standard normal cumulative distribution

An asymptotic test is based on φ(u) φ(u)- G n (u,F)=O(n -1 ) G * n (u): bootstrap cumulative distribution A bootstrap test is based on G * n (u) G * n (u)-G n (u,F)= O(n -3/2 )

Reality test for data snooping Forecasting model: l k Benchmark model: l 0 d k = l k - l 0 H 0 :max k=1,2,…,n E(d k ) ≦ 0 Data: 1000 daily closing stock prices of UMC Benchmark model: random walk with drift lnP t = a + lnP t-1 + ε t

Forecasting model : lnP t = a + ΔlnP t-1 + ε t where ΔlnP t = lnP t – lnP t-1 V=(1/B) ∑ B b=1 d 1 (b) Quantile of bootstrap distribution Statistics V for V Critical value * The difference is significant, so reject H 0 Forecasting model beat random walk model

Inference when a nuisance parameter is not identified the null hypothesis Threshold Autoregressive (TAR)model: α 10 + α 11 y t -1 + ε 1t y t -1 ≦ η y t = α 20 + α 21 y t -1 + ε 2t y t -1 > η η : threshold value H 0 : time series is linear H 1 : time series is TAR process

Data: monthly data of U.S. dollar/Sweden krona exchange rate from January 1974 to December 1998 U.S. dollar/Sweden krona Bootstrap P-value Reject H 0 U.S. dollar/Sweden krona exchange rates follow TAR process

Bootstrap percentile confidence interval

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