Bootstrap Methods with application in

Econometrics and Finance*

WEI Zhen+

Department of Statistics and Probability

School of Mathematical Sciences

Peking University

People’s Republic of China

Email: dreamiwz@hotmail.com

Abstract

This paper studies both the traditional and the state of art aspects of Bootstrap

Resampling Procedures. The theoretical asymptotic performances of such procedures are

explored by an intuitive way of Edgeworth Expansion. Several refinements of standard

bootstrap methods are advanced by a brief introduction, with suggestions for further

research frontiers. Further more, empirical experiments are carried out in terms of

bootstrap confidence intervals and hypothesis tests to evaluation the managerial

performance of several representative funds in the financial market of People’s Republic

China. Another contribution of this paper is that it develops an integrated numerical

method to analyze the overall performance of each fund instead of relying on the traditional

four types of performance measures respectively. Concluding remarks are forwarded in an

attempt to explicit that any experimental analysis should be carried out with great caution

because a minor intrusion of theoretical assumptions may result in catastrophic failure in

understanding the intrinsic properties of the data.

Keywords bootstrap, resampling, jackknife, block bootstrap, circular bootstrap, wild

bootstrap, paired bootstrap, econometrics, financial time series, Edgeworth expansion,

cross score, performance measure, computer intensive methods

* This paper is presented as the undergraduate thesis work of Mr. WEI Zhen in the Department of Statistics and Probability, School of Mathematical Sciences, Peking University, PRC. + The author is grateful to Doctor QIN Huaizhen for his helpful suggestions and comments, to Doctor DAI Ming, HUANG Hai and professor YANG jingping for their gracious support in obtaining the financial data presented in this work.

Contents

1 Introduction 1

2 Bootstrap approach to Confidence Intervals and Hypothesis Tests 2 2.1 Bootstrap Confidence Intervals 3 2.2 Bootstrap Hypothesis Tests 4 2.3 Refinements of standard bootstrap techniques 5 2.3.1 Serial Correlation 5 2.3.2 Conditional Heteroskedasticity 6 2.3.3 Structural Equation Models 8 2.3.4 Non-stationary time series

3 The asymptotic theories of Bootstrap Statistics 9

4 Experimental Bootstrap with Financial Data 13 4.1 Bootstrapping the confidence intervals and hypothesis tests 13 4.2 Cross Scoring method for ranking the performance of funds 13 4.3 The Experimental Data 14 4.4 Experimental Results 15

5 Conclusion 17

Reference 25

Figures

Figure 1 Serial correlation … 6 Figure 2 Conditional Heteroskedasticity… 7 Figure 3 Co-integration of fund index… 9 Figure 4 Histograms and normal quantile-comparison plots… 18 Figure 5 Confidence ellipses of bootstrap samples… 19

Tables

Table 1 Ranking 33 representative close funds… 20 Table 2 Cross Scoring of 33 representative funds… 21 Table 3 Bootstrapping the 95% Confidence Intervals… 22 Table 4 The 95% Confidence Intervals of the transformed difference… 23 Table 5 Bootstrapping the 95% Confidence Intervals of the difference… 24

1. Introduction

The past several decades have experienced an explosive development in Computer Science

and Technology. One tremendous advance among them is the dramatic improvement in

Hardware capacity and Computational efficiency, which allows otherwise impossible access

into a new era of Statistical and Computational research frontier. That is what we often call

Computer Intensive Methods (CIM)1 with their wide applications in various disciplines and

areas such as Biological Computation, Signal Processing, Financial Analysis, Actuarial

Science etc. For recent references, see, for example, James MacKinnon (2002), Krzysztof

Ostaszewski and Grzegorz Rempala (2000) and Ansgar Heinrich Ludolf West (2000). Among

many versions of CIMs, the Bootstrap Methods receive particular recognition partly due to

its underlying seemingly simple but actually profound idea and its easy implementation

using computer simulation either through software packages that are readily available (e.g.

The boot library in R or S-plus) or through recoding according to user-defined algorithms

borrowing new features.

The term “Bootstrap” (or Bootstrapping) which is due to Efron (1979) is an allusion to

the expression "pulling oneself up by one's bootstraps" meaning doing the impossible [see

also, e.g. Efron and Tibshirani (1993)]. As it turns out, through a decade’s practices and

advances both in theoretical framework and simulation techniques, it does provide a

promising tool to practitioners to solve the traditionally unsolvable problems. On the other

hand, I should also note that there is no universal rule in the quantitative world which

confronts us in that its intrinsic complexity may well be far beyond what the data can tell us.

Then, it is natural to see that there are cases when standard bootstrap performs not as well

as the traditional asymptotic theories or simple Markov Chain Monte Carlo (MCMC)

Methods. However, to be generally speaking, as will be shown later in this paper, under

certain probabilistic conditions (which is often true in reality), bootstrap usually produces

more accurate and reliable results than traditional methods which are either based upon

the large number distribution properties of the observations [for example, see Pakes (1986),

Rust (1987) and Stern (1997)] or the prior beliefs of the authors [see, Albert and Chib (1993),

Geweke (1999) and Elerian, Chib, and Shephard (2001), among many others], especially for

inferences based upon small samples.

Furthermore, in cases when standard bootstrap procedures do not produce unbiased or

consistent estimate of statistics of interest, refinements are suggested to handle this

incompatibility. Specifically, Künsch (1989), Politis and Romano (1992), Liu (1988) and

Mammen (1993) and Freedman (1981) proposed different novel bootstrap methods to solve

problems in empirical econometrical and financial analysis. For a recent summarization of

these methods and their recent developments, see for example, Emmanuel Flachaire (2003)

and James MacKinnon (2002). For empirical study on the numerical performance of Fast

Bootstrap Procedures, see the work of Francois Lamarche (2003) as a good reference.

The remaining part of this paper is organized as follows. Section 2 introduces elements 1 Including various statistical and computational techniques such as Delta-methods, Cross validation, Jackknife, and AI Networks related algorithms.

1

of employing basic Bootstrap resampling methods to construct Confidence Intervals and

perform Hypothesis Tests, which ends with a discussion of some situations when standard

bootstrap resampling may not produce satisfying results and suggests several techniques

that are useful to overcome these difficulties. Asymptotic superiorities of general Bootstrap

methods over the traditional simulation methods are the main focus of Section 3, where a

fresh probabilistic interpretation of Edgeworth expansion is put forward to prove analytical

results. Several numerical experiments are carried out in Section 4 to give the reader an

intuitive understanding of the performance of Bootstrap Methods and its promising

application in real practices. Lastly, Section 5 supplements a short conclusion of this paper

and provides suggestions for further research directions.

2. Bootstrap approach to Confidence Intervals and Hypo- thesis Tests

There are two major resampling methods in the family of Bootstrap: parametric and

nonparametric. The former does assume any prior knowledge of the distribution of the

studied data set and samples directly from the original data (or the empirical distribution

function implied by the data) with replacement, hence is robust and more favored by

researchers (because therein lies its strength), while the latter makes assumptions about the

underlying distribution of the data, estimates its parameters (using maximum likelihood,

least square methods or other numerical procedures) and then resamples from the

estimated parametric distribution. Here, I focus on how to use nonparametric bootstrap to

construct confidence intervals and perform hypothesis test while assuming the reader can

easily carry out the parametric case following the steps described in this section.

Suppose we draw a sample { }1 2= , , ,… nX X XS from a population { }1 2= , , , Nx x xP … .

We proceed to draw a sample of size n from among the elements of , sampling with

replacement. The result is called bootstrap sample

S

{ }* * *1 2= , , , nX X X*S … . The key

bootstrap analogy is as follows:

The population is to the sample as the sample is to the bootstrap samples

That is

*v. s. v. s.∼P S S S

Following this conception, we could use the average of the bootstrapped statistics,

*** * 1( ) == = ∑

R

bbT

T E TR

(1)

to estimate the expectation of the bootstrapped statistics. Similarly, the estimated bootstrap

variance of *T

** 2* * 1

( )( )

1=

−=

−∑R

bbT T

V TR

(2)

2

estimates the sampling variance of T, and SE T V T=*( ) ( )* *

* is the bootstrap estimate of

the standard error of T.

2.1 Bootstrap Confidence Intervals

Standard (first suggested by Efron (1979) in his pioneering paper), studentized,

normal-theory, percentile, bias-corrected accelerated (or BCa) percentile and double

bootstrap confidence intervals are widely used. In Section 4 where numerical experiments

are carried out, I only stress on normal-theory, percentile and bias-corrected accelerated

CIs. Following is a brief introduction of how to construct these three types of confident

intervals. The theoretical deductions of these intervals are omitted because they can be

constructed through either simple statistical intuition or mathematical procedure that is

well described in various text books.

To construct a α( - )100 1 percent confidence interval, following the results in equations

(1) and (2), we have

1 2T B z SE T SE T V Tαθ −= − ± =* * * *

* */( ) ( ) where ( ) ( ) * (3)

* *Where B T T= − is the bootstrap estimate of the bias of the statistic T.

The so-called bootstrap percentile interval is to use the empirical quantiles of to

form a confidence interval for θ:

*bT

R RT Tα θ+ +< <* *[( ) / ] [( )( / )]1 2 1 1 2α− (4)

where are the ordered bootstrap replicates of the statistic, and the

operator nT T T…* * *

( ) ( ) ( ), , ,1 2

[ ]i indicates rounding to the nearest integer.

A more accurate approach which is recommended by many literatures is the so-called

bias-corrected, accelerated percentile (BCa) intervals. This type of interval can be calculated

from observations through the following steps:

1. Let correction factor 1

R

bb

I T Tz

R− =

⎡ ⎤≤⎢ ⎥

⎢ ⎥=+⎢ ⎥

⎢ ⎥⎣ ⎦

∑Φ

*( )1 1

1 (5)

where is the inverse of standard normal cumulative distribution function, is

an indicator function satisfying:

( )Φ i-1 ( )I i

( )I A⎧

= ⎨⎩

if A is true

if A is false

10

(6)

3

2. Let be the jackknife value( )iT −2 of T , and T be the average of . Then

correction factor 2 ( )iT −

( )( )( )( )

n

ii

n

ii

T Ta

T T

−=

−=

−=

⎡ ⎤−⎢ ⎥⎣ ⎦

∑

∑/

3

13 22

16

(7)

3. Having the two correction factors at hand, we compute

( )

( )

z za z

z z

z za z

z z

α

α

α

α

α

α

−

−

−

−

⎡ ⎤−⎢ ⎥= +

− −⎢ ⎥⎣ ⎦⎡ ⎤+⎢ ⎥= +

− −⎢ ⎥⎣ ⎦

Φ

Φ

/

/

/

/

1 21

1 2

1 22

1 2

1

1

(8)

where ( )Φ i is the standard normal cumulative distribution function.

4. The corrected percentile confidence interval is as follows

[ ] [ ]RaT Tθ< <*

1 2Ra* (9)

where the operator [ ]i indicates rounding to the nearest integer as before. It should be

noted that when , BCa confidence interval reduces to the percentile interval stated

previously.

a z= = 0

2.2 Bootstrap Hypothesis Tests

Although there is a natural correspondence between constructing confidence intervals and

performing hypothesis tests, it is rather important to stress out some important aspects of

bootstrap hypothesis tests for at least two reasons:

1. There so many ways to construct Bootstrap confidence intervals that we occasionally get

contradictory or inconsistent results. Bootstrap Hypothesis Testing, at this time, may be

more informative and reliable than a set of confidence intervals

2. In the fields such as Financial Analysis, innovative ideas often allows us to perform

original hypothesis tests which are beyond the scope of traditional confidence intervals.

Standard bootstrap hypothesis test can be explained as follows. Suppose the statistics

of interest isθ and the null hypothesis we want to test is ( )θ θ θ θ≤ =0 0 for someθ0 , the

alternative hypothesis isθ θ> 0 . Then, a pivotal statisticsV which is a function ofθ andθ0 is

constructed (e.g. the t statistics usually used in regression model is ( ) ( )Varθ θ θ− 0 ).

From the sample data, we can compute the value of such statistics d. Further, we employ

the standard bootstrap resampling procedure to calculate the bootstrap sample value of , V

2 Jackknife value refers to the value calculated from the sample when the i-th observation is

removed. ( )iT−

4

denoted as . Thereafter, we calculate the frequency of the event that is

greater than :

* ( , , ,id i n= 1 2 … ) *id

d

(* *n

d ni

)P I d dn =

= ≤∑1

1

which is the bootstrap P-value of the original hypothesis tests. Suppose *dP α< at a given

significant levelα , we have sufficient reason to discredit the null hypothesis.

While the standard form of bootstrap hypothesis test is appealing as it stands, here, I

instead focus on a particular case where Bootstrap hypothesis test is applied to a problem in

Finance and a novel approach is advanced and further experiments and discussion will be

carried out in Section 4 of this paper.

The Sharpe (1966) and Treynor (1965) performance measures are widely accepted in

both academia and industry to assess the Risk-adjusted value of a particular portfolio. It

can be shown, after some mathematical treatment, that the Sharpe performance measure is

useful when the portfolio of interest represents all of the investor’s investment, while

Treynor’s measure is preferred when the portfolio under discussion is only a portion of the

whole investment package. Previous efforts of the estimation of the two measures are based

on the assumption of the asymptotic normality of the data. For example, see the pioneering

paper of Jobson and Korkie (1981), in which a Z statistics is constructed to facilitate

Hypothesis Testing and construct Confident Intervals. However, practice tells us that under

some circumstances, which are not rare, such assumption can be seriously cast into doubt.

Besides, the data transformation process used to construct the Z statistics is also

questionable because certain regularity condition in the transformation is not necessarily

satisfied. Vinod and Morey (1999) suggested a Bootstrapping Methodology to reconcile

these problems and to produce more reliable hypothesis tests. Another two main stream

performance measures are the Jensen’s measure and Appraisal Ratio, which are also

employed by financial and rating system. In section 4, I apply both the JK (Jobson and

Korkie) Method and the Bootstrap Method (suggested by Vinod and Morey) to the data of

several funds in China, and comment on their relative strength.

2.3 Refinements of standard Bootstrap techniques

As every coin has its two sides, there are also cases where standard bootstrap procedures

may fail to do the right job. Thus we should exercise extreme caution when applying

bootstrap methods to perform real analysis and draw a conclusion.

2.3.1 Serial Correlation

It is not difficult to show that the standard Bootstrap resampling procedure may fail to

generate unbiased and consistent estimation of a statistic when the observed data is serially

correlated. Unfortunately, in industrial and financial practices, the retrieved data may well

violate the independent assumption of many statistical frameworks. Time Series Analysis

mainly focuses on the short-term and long-term interdependence and correlation of the

5

data, in which AC (Auto-Correlation) and PAC (Partial Auto-Correlation) Statistics are used

to describe the serial dependence. One particular case can be shown by Figure 1, which

describes the AC and PAC of the time series of a GDP data set.

To solve this problem, many refinements are suggested by academia. Particularly,

Künsch (1989) considered a moving block bootstrap procedure which resamples a block of

observations each time instead of only one observation. This approach does not generate an

equal sampling because the middle region of the data seems to have a greater chance to be

sampled. Politis and Romano (1992) thus forwarded a circular bootstrap, which takes the

data series as a circle and resamples the moving block circularly from the data. Empirical

evidence shows that both block bootstrap and circular bootstrap can capture the short-term

correlation of the data and outperform the standard bootstrap resampling method.

2.3.2 Conditional Heteroskedasticity

Financial and Economic data often exhibits some form of unknown volatility that beyond

the explanation of traditional Econometric Models. For example, the GARCH-related

models (such as ARCH, TARCH, EGARCH, GJR-GARCH, etc.) are particularly designed to

solve the problem of conditional heteroskedasticity that is produced by the error term of

traditional least square regression. This phenomenon is often called Volatility Clustering of

financial and macroeconomic time series, which has brought recognition in industry long

Figure 1 Serial correlation of Gross Domestic Productivity from January 1952 to

March 1996. The data is from the example file of the software package

Eviews 4.1

6

before any theoretical model was established to give explanation and prediction. As an

illustration, I regression the daily return of Fund index by that of Market Index of the

Financial Market of China3, producing error terms whose variances are AR(1) correlated.

The result is demonstrated in Figure 2.Under such circumstances, standard bootstrap also

can not make satisfying results.

Basically, there are two ways to alleviate this drawback. They are namely, Wild

Bootstrap and Paired Bootstrap (or Pairwise Bootstrap). The wild bootstrap which was

suggested by Liu (1988) and Mammen (1993) considers a bootstrap data generation process

(DGP)

( )tt ty fβ ε= +X tv (10)

where β is the OLS estimates of the standard regression model, tε is the corresponding

residual term and ( )f i is a transformation probability function, usually takes on binary

distribution having equal probability on -1 and 1. Bose (1988), Kreiss (1997) and He and

Teräsvirta (1999) developed and experimented many new version of the Wild Bootstrap

application, such as Fixed-Design and Recursive-Design Wild Bootstrap Methods.

-2

-1

0

1

2

3

-4

-2

0

2

4

6

50 100 150 200 250

ErrorReturn of Fund IndexFitted value

Figure 2 Conditional Heteroskedasticity of Regression Model of Financial data

The Paired Bootstrap was first proposed by Freedman (1981), which resamples the

dependent and independent variables in pairs in the regression or auto-regression models.

Silvia Goncalves and Lutz Kilian (2002) shows that, if certain regularity conditions are

3 The data are daily index extracted from January, 3, 2003 to February, 12, 2004.

7

satisfied, for the fixed-design Wild Bootstrap and the Paired Bootstrap

( ) ( )*

P* * * *supx

P t x P t xθθ∈

≤ − ≤ ⎯⎯→R

0 (11)

where **tθ

and *tθ are the bootstrap-based and ordinary t statistics for the regression model,

respectively. Furthermore, if more constraints are added, it may be shown that equation

(11) also holds for Recursive-Design Wild Bootstrap.

2.3.3 Structural Equation Models

Macroeconomic models usually deals with various equilibrium of the general Market. A

classical Supply-Demand Model that can be found in many texts can be written as

St t t

Dt t t

S Dt t

Q a a P a P

Q b b P b P

Q Q

t

t

ε

ε−

−

⎧ = + + +⎪⎪ = + + +⎨⎪ =⎪⎩

0 1 2 1 1

0 1 2 1 2 (12)

where StQ and D

tQ denote the amount of supply and demand at time respectively. t

tP and ty represent the price and income at time t .

In such cases, we should also be cautious when applying bootstrap resampling to the

standard techniques in SEM, such as TSLS (Two-Stage Least Squares), SUR (Seemingly

Unrelated Regression) and 3SLS (Three Stage Least Squares). For detailed discussion of

such events, I recommend James MacKinnon (2002) for reference. To see more topics on

residual resampling dealing with more generalized cases, see, for example, Urban Hjorth

(1994).

2.3.4 Non-stationary time series

It also can be shown that for models dealing with times series that are unstationary,

standard bootstrap procedure also fails. In Financial practices, we often need to extract the

underlying stationary or equilibrium characteristics from several un-stationary time series.

This is often modeled by Co-integration, which claims that a certain linear combination of

I(di) process4 (where di are positive integers) may yield a stationary outcome. The result

provides evidence for long-term equilibrium of many macroeconomic factors which are

themselves exposed to uncertainties of unknown form. Seeking out an efficient way to

bootstrap such processes still remains one of the cutting edge frontiers of theoretical and

empirical Bootstrap researches.

An example of co-integration of two I(1) sequences are shown in Figure 3, where Fund

Index and Market Index of Chinese Market are both I(1) time series5 and the residual

produced by the regression of the two series is a stationary process, which reflects the

long-term equilibrium relationship between the two macroeconomic indicators.

4 I(d) process refers the a series whose difference of d-th order is stationary. 5 This can be validated through uniroot test by Augmented Dickey-Fuller Test.

8

Figu

3. T

Th

Edgew

of othe

Su

which

for som

order m

Fo

mome

where

-40

-20

0

20

40

60

880

920

960

1000

1040

1080

50 100 150 200 250

Error Fund Index Fitted value

re 3 Co-integration of Fund Index and Market Index in the financial market of

China. The data are the daily close index extracted from January 3, 2003 to

February 12, 2004.

he asymptotic theories of Bootstrap Statistics

e asymptotic behaviors of Bootstrapping Methods are essentially examined by

orth Expansion, although it is frequently addressed by researchers that the omission

r potential aspects of bootstrap methodologies may blind us from the large picture.

ppose random variable has finite moments of all orders, one particular case of

is that decreases in tails at least in a exponential rate, or

X

X

( ) ( )Pr expX x c c x> < −1 2 (13)

e positive real number c and . In the following paper, I denote the j-th centered

oment of by1 c2

X ju , i.e.

( )jju E X= (14)

llowing probability theory, there is a one to one relationship between and its

nt generating function

X

( )XM t , which is defined as

( ) ( ) ( )dtX txXM t E e e F x= = ∫ (15)

is the cumulative distribution function of . Furthermore, if follows standard ( )F i X X

9

normal distribution, we have

( ) ( )dtxXM t e x e= ∫ Φ 2

t

=2

(16)

Define the culmulant generating function ( )XK t as

( ) ( )log!

r

X Xr

tK t M t

rκ

∞

=

= =∑1

r (17)

where is called the r-th cumulant ofrκ X ( ), ,r = 1 2 …

Theorem 1 Under the condition that has finite moments of all orders and mild

regularity constraints, we have

X

( ) ( ) !!

!!

i

i

rsr

r pr s

k i ii

i

u rk

pr

κ∆

= =

=

⎛ ⎞= − − ⎜ ⎟

⎝ ⎠∑ ∑ ∏

∏1 1

1

1 1 (18)

where the sum extends over all possible combinations of positive integers

and , such that ∆Σ sp p< <1

, sr1 r

s

i ii

p r p=

=∑1

and s

ii

r r=

=∑1

Proof We expand ( )log XM t using Taylor’s formula round t = 0

( ) ( ) ( )dlog

!d

r rX

X X rr t

K t tK t M t

rt

∞

= =

= =∑1 0

(19)

Comparing (18) and (19) yields

( ) ( ) 'd d log d

d d d

r r rX X X

r r r rX tt t

K t M t M

Mt t tκ −

== =

⎛ ⎞= = = ⎜ ⎟

⎝ ⎠1

00 0

(20)

On the other hand, under mild regularity conditions

( ) ( ) ( ) (d dd d

d d

r rtx r tx r tX

Xr rM t e F x t e F x E X e

t t= = =∫ ∫ ) (21)

Then, it is easy to see

( ) ( )d

d

rr

Xr

t

M t E X ut

=

r= =0

(22)

From (20) and (22), through a brief mathematical deduction, we get (18). ■

From formula (18), under the condition that ( )u E X= =1 0 , we can easily calculate

10

that: for , and so on. r uκ = r 2, or r = 1 2 3 u uκ = − 24 4 3

Let be independently identically distributed variables satisfying

conditions described in Theorem 1. If we denote their population mean and standard

error by

, ,X X1 2 …

µ andσ respectively, from Lindeberg-Levy Theorem, we know that

( ) ( )/ dnZ n X Nµ σ= − ⎯⎯→ 0 1, (23)

when n approximates infinity, and ( ) t2

nZM t e→ 2 equivalently.

From theorem 1, it is easy to show that

( ) ( )/n

n

Z X uM t M t nσ−⎡ ⎤= ⎣ ⎦ (24)

It follows that

( ) ( )( ) ( )

log /

nZ X uM t nK t n

uutt t

nn

σ

σ

σσ

−=

−= + + +

4243 43

43

32 246

o n (25)

Then, we know that

( ) ( ) ( )t

nZM t e c t c t c t o nnn

⎡ ⎤= + + + +⎢

⎣ ⎦

22 3 4 6

1 2 31 11 ⎥ (26)

where ( )

, ,uu u

c c cσ

σ σ σ

−= = =

243 3

1 2 33 4

36 24 7 62

.

Theorem 2 ( ) ttX rrE H X e e t⎡ ⎤ =⎣ ⎦

22 , where conforms standard normal

distribution, and

X

( )rH i refers to the Hermitian polynomials function, which is

recursively defined as:

( ) ( ),H x H x= =0 11 x and ( ) ( ) ( ) ( )r r rH x xH x r H x− −= − −1 21

Proof It is easy to see that

( ) ( )

( ) ( )

d

d

x

t

tX txr r

r

E H X e H x e e x

e H t x x

π

+∞ −

−∞

+∞

−∞

⎡ ⎤ =⎣ ⎦

= +

∫

∫ Φ

22

22

12 (27)

Following the definition of ( )rH x , we can employ the routine of Mathematical Deduction

again to prove that

( ) ( )d rrH t x x t

+∞

−∞+ =∫ Φ (28)

Combining equations (27) and (28) justifies the claim of Theorem 2. ■

11

Particularly, substituting r with 0 yields ( ) t

XM t e=22 . So Theorem 2 can be seen as a

generalization of Moment of generation function.

From Theorem 2, it can be easily show that the probability density function of has

the form: nZ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )nZ

x xf x x c H x c H x c H x o n

nn

φ φφ −⎡ ⎤= + + + +⎣ ⎦

11 3 2 4 3 6 (29)

where ( )φ i is the density function of standard normal distribution. Moreover, from the

notion that ( ) ( ) ( )dr rH x x x H xφ −= −∫ 1 , we can integration equation (29) to get the form

of the cumulative distribution function of nZ

( ) ( ) ( ) ( ) ( ) ( ) ( ) (nZ

x x )F x x c H x c H x c H x o nnn

φ φ −⎡ ⎤= − − + +⎣ ⎦Φ 11 2 2 3 3 5 (30)

From Theorem 1 and Theorem 2, an intuitive understanding of the asymptotic

behavior of various bootstrap resampling procedures can be explained as follows. Suppose

we computed the bootstrap estimate of a statistics ( )Xθ as ( )* Xθ , if ( )Xθ is pivotal, then

its cumulative distribution function (CDF) of the traditional normal theory estimateθ can

be expanded as

( ) ( ) ( ) ( ) ( ) ( ) (x x )F x F x G x G x o nnn

θθ

φ φ −= − − + 11 2 (31)

where ( )F xθ is the CDF of the true parameterθ . Similarly, the CDF of ( )* Xθ can also be

expanded as

( ) ( ) ( ) ( ) ( ) ( ) (*

x x )F x F x G x G x o nnn

θθ

φ φ −= − − + 11 2 (32)

According to the bootstrapping estimate procedure, we know that can be seen as

an asymptotic approximation of

( )G1 i( )G ≡ 0i . Following the traditional asymptotic theory, we

get

( ) ( ) ( )/G o n o n−= + 1 2 1 21 0i ∼ /− (33)

Comparing equations (31)-(33), we can write

( ) ( ) ( )*F x F x o nθθ−= + 1 (34)

whereas ( ) ( ) ( /F x F x o nθθ−= + 1 2 ) , which consolidates our former claim of the asymptotic

superiority of bootstrap procedures over the traditional ones.

From the steps above, it is not difficult to show that when the statistic ( )Xθ is not

pivotal, bootstrap can also provide consistent estimates while traditional approach can not,

but the asymptotic order is reduced to ( )/o n−1 2 .

12

4. Experimenting Bootstrap with Financial Data

In this section, I employ the above-mentioned bootstrap methods to construct confidence

intervals and perform hypothesis tests on the data of several representative funds in the

financial market of China. In the context of hypothesis testing, I compare the result

provides by JK’s method and bootstrap methods. From the result of the experiments, it

seems evident that bootstrapping hypothesis tests generally provide more accurate results

and more convincing conclusion. Furthermore, in order to provide an objective and

integrated evaluation of the collective performance of each fund, I proposed a Cross

Scoring method to rank these funds. The ultimate rank is summarized in Table 2.

4.1 Bootstrapping the confidence intervals and hypothesis tests

In the paper of Jobson and Korkie (1981), they used the following transformation

( ) ( )

( )

kn n k k n

nm kmkn k n

m m

E G ShT T

E G Tr

σ µ σ µ

σ σµ µ

σ σ

⎛ ⎞⎡ ⎤ = − − +⎜ ⎟⎣ ⎦ ⎝ ⎠⎛ ⎞⎡ ⎤ = −⎜ ⎟⎣ ⎦ ⎝ ⎠

2

2 2

1 114 32

(35)

to perform hypothesis tests on the difference of the performances of two funds,

particularly in terms of Sharpe and Treynor performance measures, where is the

transformation function, n and k represent two arbitrarily selected funds and T is the

sample size at hand. Essentially, the subsequent Z statistic is constructed from equation

(35). JK (1981) found that the Z statistics of Sharpe measure performs well for small

sample size, but the Treynor measure behaves not as well in some cases.

( )G i

4.2 Cross Scoring method for ranking the performance of funds

Having all the point estimates of the four performance measures, I suggest employing the

integrated rule to rank the performance of the funds. I define the Cross Scoring (CS) as

the following form

( ) ( )ω signk ij i ii j k

Score Perf k Perf j= ≠

⎡ ⎤= −⎣ ⎦∑∑4

1 (36)

where i represents the four types of performance measures, N is the number of funds to be

ranked, ( )sign x is a function taking values in { }, ,−1 0 1 when , ,x < = > 0 respectively

representing individual penalty (award) in each performance measures and ijω is a

weighting parameter satisfying

ωiji j k= ≠

=∑∑4

11 (37)

In the experiment, I fix ijω to ( )N −1 4 1 while more realistic approach is to adjust the

13

distribution of ijω s according to the specific property of the investor’s portfolio. For

example, when assessing the portfolio which represents the entire investment of the

investor, we can justifiably define:

when ω

otherwiseij

iN

⎧ =⎪= −⎨⎪⎩

1 11

0 (38)

This corresponds to using Sharpe’s Measure alone.

One natural explanation of using formula (36) to evaluate the performance of any

fund particularly or any financial institution in general is that we can adjust the

parameters freely to fit the need of individual interest. As has been formerly stated, the

Sharpe, Treynor measures and Appraisal Ratio have their respective importance in the

evaluation. However, in the real financial practices, our portfolio at hand does not

necessarily satisfy the conditions when a specific type of performance measure works

extremely well. So, it is realistic and promising to employ this hybrid rule. For instance, if

the portfolio of interest consists mainly of one stock and a combined investment on

another stock and the fund index, none of the four performance measures will work well in

this case alone. Following our idea of integration of these measures, one possible solution

is to take the weighted average of Appraisal Ratio and Sharpe (Treynor) measure. The

weights are proportional to the net value of each type of investment respectively. This idea

can be applied to more complicated situations in more or less the same manner.

In the experiments, using the results of section 4.1, we can easily calculate the Cross

Scores of all the funds of interest from equation (36). Here, I use the equal weights to

produce the rank according to the cross score. The results are shown in Table 4.

4.3 The Experimental Data

Due to the huge amount of the experimental data that have to be employed in order to give

a comprehensive rank of all the representative funds in Chinese Market, I use SAS system

to manipulate the daily returns of all the funds. Of 54 funds6 that currently exist in Chinese

Market, only 33 funds functions smoothly more than 3 years. Since it is uninformative to

explore the less-than-one-year daily returns of the remaining 21 funds, I only ranked the

33 representative funds using the scoring method described above. The experimental data

then is reduced to the daily returns of these representative funds from Jan. 3, 2001 to Dec.

31, 2003 (717 observations).

Then, the experiments in this section can be described as the following steps:

1. Rank the selected funds using the traditional four types of performance measure

respectively.

2. Rank the selected funds using the cross scoring method while assuming the investor

6 We only discuss close fund here since the inaccuracy of the information gained from the daily prices of open funds in China lends little insights into what we are interested in.

14

takes equal weights on each measures.

3. Comparison of the results in steps 1 and 2 are stated, and the top 3 and bottom 3 funds

according to step 2 are selected to construct confidence intervals and perform

hypothesis tests following the way described below.

4. Bootstrapping the confidence intervals of all the performance measures of the selected

funds in step3 and compare the results with their point estimates in step 1.

5. Perform hypothesis tests using JK’s method and Bootstrap method to access the

difference of the performance measures of the top 2 and bottom 2 funds according to

step 2 and comparison is stated to evaluate the accuracy of these tests.

4.4 Experimental Results

Table 1 reports the rank of the 33 representative close funds in the financial market of

China. The result is ordered by the four types of performance measures: Sharpe’s measure,

Treynor’s measure, Jensen’s measure and Appraisal Ratio in each category. Table 2

represents the cross rank of the 33 representative close funds using the method describe in

Section 4.2. Comparing the two tables, we can see that they generally produce similar

results with only a little deviation from each other which reflects the intrinsic differences

in the nature of these measures. Table 2 generates an objective rank of the funds due to the

reason that has been formerly stated. However, if the investor has a particular taste of

portfolio management, for instance, if he or she would like to risk all his investment in an

individual fund, then it may not be wise to use the rank in Table 2. Under such

circumstances, it is more reasonable to refer to the rank of the Sharpe performance

measure in Table 1.

Table 3 summarizes the point estimates and 95% bootstrap confidence intervals

(lower and upper bounds) of four performance measures of the top 3 and bottom 3

representative funds in Chinese Market which are selected according to the cross rank in

produced in Table 2. It can be seen from the numerical results that the bootstrap estimates

of confidence intervals of the two groups of funds 1-3 and 31-33 exhibit a sort of in-group

similarity, producing nearly same interval width, but the confidence intervals between the

two groups are somewhat different from each other. Then, we may further ask that is there

significant evidence to discredit the hypothesis that all funds in the Chinese Market are

managed the same? If this is true, it is useless to discriminate which portfolio of funds is

the best choice, just throw your money into a random bundle of them. This question will

be answered through the hypothesis techniques using JK’s method (see Section 4.1) and

bootstrap hypothesis test.

In order to explicit that the bootstrap data generating process does conform to the

theoretical assumption of creating asymptotically normally distributed resamples, I plot

both the histograms and the quantile comparison graphs of the bootstrap distribution of

the samples (the sample size here is fixed at 3999) of four performance measures given in

Table 3. To see the interactive effects of the performances on these funds, I plot the 5%, 95% and 99% confidence ellipses of the bootstrap samples of these measures either from

15

the same fund or from different funds. From figure 5, the Sharpe and Treynor

performance measures are strictly positively correlated, as can be seen from (a1) where the

three confidence ellipses reflects the close positive correlation between the two

performance measures within a single fund. This phenomenon can be explained by the

similarity between the definitions of the two measures. From (a2), we can see that the

Sharpe measure and Jensen measure for a single fund are also positive correlated, but the

correlation is reduced and not so evident. Also, it is not difficult to show that Jensen

measure and the Appraisal Ratio are linearly correlated as can be seen from (a3). Further

more, if we want to look at the interactive relationship between the measures of different

funds, we could also follow this graphical way. For example, (b1) tells us that the Sharpe

measure of different funds are also positively correlated, but comparing with (a1), we

conclude intuitively that this relationship is not as strong as that of Sharpe and Treynor

measures within one fund. (b2) and (b3) also show some positive correlation between

different measures of different funds.

Economic instinct tells us that the performances of all funds are largely subjected to

the macroeconomic factors such as the functioning of the national economy, the influence

of the international politics and trading, the wholesomeness of the financial and

managerial system and so on. This partly explains the positive correlation of different

types of performance measures. It also provides evidence of the consistency of employing

these measures to evaluate the performance of individual fund or the fund index of a

regional or national economy.

Table 4 provides the results of hypothesis testing using JK’s method. According to the

correspondence of confidence interval and hypothesis testing, if the 95% confidence

interval of the transformed difference of performance measure does not contain 0, then we

have sufficient reason to belief that there is a significant difference between the

performances of the two funds. The same concept can also be applied to the result of Table

3, where bootstrapping confidence intervals of the original differences (not the

transformed ones) of the performance measures are calculated for each combination of

two funds. From the results we conclude that there is no significant difference in the

performance of the funds under investigation, confirming our former intuition about the

analogous bootstrap CIs of the performance measures of the selected funds.

Accordingly, Table 5 describes the 95% bootstrap Confidence Intervals of the

difference in the four performance measures of top 2 and bottom 2 funds selected from

Table 2. The Confidence Intervals are constructed through three popular bootstrapping

methods, namely, the normal-theory CI, the percentile CI and the BCa CIs, respectively.

The bolded values in the following table indicate that the specified confidence interval

does not contain 0, so we have sufficient evidence to believe that there is significant

difference between the corresponding measures. Comparing the result in Table 4 and

Table 5, we can see that there are several discrepancies as are indicated by the bold values

of the confidence intervals. For example, the strict positive values in the three types of

confidence intervals of the difference in the Sharpe measure of Fund 1 (184695) and Fund

33 (500015) indicate that we have sufficient information to believe that investing your total

16

portfolio in Fund 1 is more promising than in Fund 33. Other results can be explained in

the similar way. These results are quite different from those in Table 4, which states that

all the 33 funds are managed similarly (since the top and bottom funds have similar

managerial performances). Intuition and real evidence both discredit this allegation,

which in turn consolidates that accuracy of employing bootstrap hypothesis tests to

generate results and gain useful information for decision making.

5. Concluding Remarks

This paper studied the theoretical and numerical aspects of various bootstrap

methodologies, especially on how to employing bootstrap ideas to construct confidence

intervals and perform hypothesis tests. Theoretical findings based upon some mild

regularity conditions are proved in an attempt to provide a straightforward explanation of

the asymptotic superiority of bootstrap procedures over the traditional ones.

Numerical experiments are carried out to evaluate the managerial performances of 33

representative funds in the financial market of China. The result is forwarded in a

graphical and tabular way to give the reader an intuitive understanding of how bootstrap

works and excels. Furthermore, this paper also advanced a cross scoring methods to assess

the various measures in a synthesizing form. The resulting rank of the 33 representative

funds in the financial market of China provides a pioneering step to evaluate their overall

performance and suggests an objective standard for the investors to resort to. Further

more, the comparison between the accuracy of employing JK’s method and bootstrap

method to perform hypothesis test reveals the fact that the JK’s asymptotic statistics can

not provide a coherent way to test the managerial performance of financial products.

To conclude this paper, I should emphasize again that the utilization of any new

theoretical methodology should be exercised with great care. The real world is not as

perfect as that is assumed in theories, so it is natural to see that some constraints in our

theory are too demanding to be satisfied. However, every problem has its solution as long

as you can realize it. The various refinements of standard bootstrap procedures can be

seen as pioneering steps to reconcile theory and reality. As has been formerly stated, it is

unrealistic to make a tool omnipotent. So, there are still tasks left to us to finish, one of

which mentioned in Section 2 is to bootstrap the models involving unstationary process. In

summary, to unravel the mysteries of the universe is to experience a joyous trip of life and

to contribute what we can to the present world and the future.

17

(a)

Histogram of t

t*

Den

sity

-1.1 -1.0 -0.9 -0.8 -0.7 -0.6

02

46

-3 -2 -1 0 1 2 3

-1.0

-0.9

-0.8

-0.7

-0.6

Quantiles of Standard Normal

t*

(b)

Histogram of t

t*

Den

sity

-2.6 -2.4 -2.2 -2.0 -1.8 -1.6 -1.4

0.0

0.5

1.0

1.5

2.0

2.5

-3 -2 -1 0 1 2 3

-2.6

-2.4

-2.2

-2.0

-1.8

-1.6

Quantiles of Standard Normal

t*

(c)

Histogram of t

t*

Den

sity

-0.016 -0.015 -0.014 -0.013 -0.012

020

040

060

080

0

-3 -2 -1 0 1 2 3

-0.0

16-0

.015

-0.0

14-0

.013

-0.0

12

Quantiles of Standard Normal

t*

(d)

Histogram of t

t*

Den

sity

-1.4 -1.2 -1.0 -0.8

01

23

45

6

-3 -2 -1 0 1 2 3

-1.4

-1.3

-1.2

-1.1

-1.0

-0.9

-0.8

-0.7

Quantiles of Standard Normal

t*

Figure 4 Histograms and normal quantile-comparison plots for the bootstrap

replicates of performance measures of Fund JingBao (184695), where (a) Sharpe Measure,

(b) Treynor Measure, (c) Jensen Measure and (d) Appraisal Ratio. From the graphics, we

can see that the bootstrap samples of all measures are approximately normally distributed.

18

-1.0 -0.9 -0.8 -0.7 -0.6

-2.6

-2.4

-2.2

-2.0

-1.8

-1.6

Sharpe Measure 1

Trey

nor M

easu

re 1

(a1)

-1.0 -0.9 -0.8 -0.7 -0.6

-1.5

-1.4

-1.3

-1.2

-1.1

-1.0

-0.9

-0.8

Sharpe Measure 1

Sha

rpe

Mea

sure

2

(b1)

-1.0 -0.9 -0.8 -0.7 -0.6

-0.0

16-0

.015

-0.0

14-0

.013

-0.0

12

Sharpe Measure 1

Jens

en M

easu

re 1

(a2)

-1.0 -0.9 -0.8 -0.7 -0.6

-2.8

-2.6

-2.4

-2.2

-2.0

-1.8

Sharpe Measure 1

Trey

nor M

easu

re 2

(b2)

-1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7

-0.0

16-0

.015

-0.0

14-0

.013

-0.0

12

Appraisal Ratio 1

Jens

en M

easu

re 1

(a3)

-1.0 -0.9 -0.8 -0.7 -0.6

-0.0

155

-0.0

150

-0.0

145

-0.0

140

-0.0

135

-0.0

130

Sharpe Measure 1

Jens

en M

easu

re 2

(b3)

Figure 5 Confidence ellipses of bootstrap samples of performance measures representing the interactive dependence of individual funds and the measures used to evaluate them. (a1) Sharpe measure v.s. Treynor measure, (a2) Sharpe measure v.s. Jensen measure, (a3) Appraisal Ratio v.s. Jensen measure of the same fund. (b1) Sharpe measures of two funds, (b2) Sharpe measure v.s. Treynor measure of another fund and (b3) Sharpe measure v.s. Jensen measure of another fund.

19

Table 1 Ranking 33 representative close funds in the financial market of China. The

result is ordered by the four types of performance measures: Sharpe’s measure, Treynor’s

measure, Jensen’s measure and Appraisal Ratio in each category, using the daily returns of

these representative funds from Jan. 3, 2001 to Dec. 31, 2003.

Sharpe Measure Treynor Measure Jensen Measure Apprasial Ratio

Fund Code value Fund Code value Fund Code value Fund Code value hand 500025 -0.755 taih 500002 -1.862 xingh 500008 -0.014 hand 500025 -0.878 jingb 184695 -0.803 jind 500021 -1.877 xingk 184708 -0.014 jingb 184695 -0.957 tongy 184690 -0.856 jiny 500010 -1.883 taih 500002 -0.014 anx 500003 -0.966 kaiy 184688 -0.858 hand 500025 -1.928 xinghe 500018 -0.014 tongy 184690 -0.977 anx 500003 -0.867 jingb 184695 -1.933 yuz 184705 -0.014 kaiy 184688 -0.983 yuy 500006 -0.873 hanbo 500035 -1.934 jins 184703 -0.014 yuy 500006 -1.023 huip 184689 -0.888 yuz 184705 -1.954 xingan 184718 -0.014 huip 184689 -1.031 taih 500002 -0.897 xingan 184718 -1.964 yuyuan 500016 -0.014 jingy 500007 -1.061 jingy 500007 -0.922 jins 184703 -1.973 yuh 184696 -0.014 jint 500001 -1.093 jingh 184691 -0.925 longy 184710 -1.977 tiany 184698 -0.014 yuyuan 500016 -1.135 jint 500001 -0.950 jingf 184701 -2.070 jingy 500007 -0.014 xingh 500008 -1.139 yul 184692 -0.950 jingh 184691 -2.072 jingb 184695 -0.014 jingh 184691 -1.149 jiny 500010 -0.950 xingk 184708 -2.091 jingf 184701 -0.014 taih 500002 -1.169

yuyuan 500016 -0.960 hans 500005 -2.095 jiny 500010 -0.014 yul 184692 -1.183 xingh 500008 -0.960 yul 184692 -2.118 jinx 500011 -0.014 hans 500005 -1.222 hanbo 500035 -0.963 tiany 184698 -2.139 tongs 184699 -0.014 huif 184693 -1.240 hans 500005 -0.965 tongz 184702 -2.141 yul 184692 -0.014 ans 500009 -1.256 jind 500021 -0.978 hanx 500015 -2.197 jind 500021 -0.014 tiany 184698 -1.285 yuz 184705 -0.992 yuy 500006 -2.200 longy 184710 -0.014 xinghe 500018 -1.285 jins 184703 -1.000 jinx 500011 -2.253 ans 500009 -0.014 jiny 500010 -1.289

tiany 184698 -1.003 yuh 184696 -2.257 hanbo 500035 -0.014 hanbo 500035 -1.292 xingan 184718 -1.003 xinghe 500018 -2.272 hans 500005 -0.014 tongz 184702 -1.325

huif 184693 -1.011 kaiy 184688 -2.290 tongy 184690 -0.014 tongs 184699 -1.342 longy 184710 -1.017 huip 184689 -2.292 hanx 500015 -0.014 yuz 184705 -1.354 tongz 184702 -1.021 tongs 184699 -2.312 huif 184693 -0.014 jins 184703 -1.362

xinghe 500018 -1.026 tongy 184690 -2.313 tongz 184702 -0.014 jind 500021 -1.367 jingf 184701 -1.036 huif 184693 -2.316 huip 184689 -0.014 xingan 184718 -1.376 ans 500009 -1.048 xingh 500008 -2.352 yuy 500006 -0.014 jinx 500011 -1.391

xingk 184708 -1.061 yuyuan 500016 -2.374 jingh 184691 -0.014 jingf 184701 -1.396 tongs 184699 -1.061 jingy 500007 -2.446 kaiy 184688 -0.014 longy 184710 -1.405 jinx 500011 -1.072 jint 500001 -2.517 jint 500001 -0.014 xingk 184708 -1.448

hanx 500015 -1.122 ans 500009 -2.522 hand 500025 -0.015 yuh 184696 -1.539 yuh 184696 -1.135 anx 500003 -2.538 anx 500003 -0.015 hanx 500015 -1.542

20

Table 2 Cross Scoring of 33 representative funds in the financial market of China. The

result is calculated from the cross scoring algorithm using the daily returns of these

representative funds from Jan. 3, 2001 to Dec. 31, 2003. The rank corresponds to both the

name (abbreviation of Chinese character) and the trading code in the security exchange of

China.

Rank Fund Code Sharpe Treynor Jensen ApRatio Cross score

1 jingb 184695 -0.803 -1.933 -0.014 -0.957 0.734 2 taih 500002 -0.897 -1.862 -0.014 -1.169 0.672 3 hand 500025 -0.755 -1.928 -0.015 -0.878 0.469 4 jiny 500010 -0.950 -1.883 -0.014 -1.289 0.281 5 yuz 184705 -0.992 -1.954 -0.014 -1.354 0.203 6 xingh 500008 -0.960 -2.352 -0.014 -1.139 0.203 7 tongy 184690 -0.856 -2.313 -0.014 -0.977 0.188 8 yul 184692 -0.950 -2.118 -0.014 -1.183 0.156 9 jingy 500007 -0.922 -2.446 -0.014 -1.061 0.156

10 yuy 500006 -0.873 -2.200 -0.014 -1.023 0.141 11 jins 184703 -1.000 -1.973 -0.014 -1.362 0.125 12 yuyuan 500016 -0.960 -2.374 -0.014 -1.135 0.109 13 kaiy 184688 -0.858 -2.290 -0.014 -0.983 0.094 14 jingh 184691 -0.925 -2.072 -0.014 -1.149 0.078 15 xingan 184718 -1.003 -1.964 -0.014 -1.376 0.063 16 jind 500021 -0.978 -1.877 -0.014 -1.367 0.063 17 hanbo 500035 -0.963 -1.934 -0.014 -1.292 0.063 18 huip 184689 -0.888 -2.292 -0.014 -1.031 0.047 19 tiany 184698 -1.003 -2.139 -0.014 -1.285 0.047 20 hans 500005 -0.965 -2.095 -0.014 -1.222 0.000 21 xinghe 500018 -1.026 -2.272 -0.014 -1.285 -0.047 22 anx 500003 -0.867 -2.538 -0.015 -0.966 -0.094 23 xingk 184708 -1.061 -2.091 -0.014 -1.448 -0.109 24 jingf 184701 -1.036 -2.070 -0.014 -1.396 -0.188 25 jint 500001 -0.950 -2.517 -0.014 -1.093 -0.219 26 longy 184710 -1.017 -1.977 -0.014 -1.405 -0.234 27 tongz 184702 -1.021 -2.141 -0.014 -1.325 -0.344 28 huif 184693 -1.011 -2.316 -0.014 -1.240 -0.359 29 tongs 184699 -1.061 -2.312 -0.014 -1.342 -0.406 30 jinx 500011 -1.072 -2.253 -0.014 -1.391 -0.406 31 yuh 184696 -1.135 -2.257 -0.014 -1.539 -0.422 32 ans 500009 -1.048 -2.522 -0.014 -1.256 -0.453 33 hanx 500015 -1.122 -2.197 -0.014 -1.542 -0.609

21

Table 3 Bootstrapping the 95% Confidence Intervals of performance measures of

several funds selected from Chinese Market. The Confidence Intervals are constructed

through three popular bootstrapping methods, namely, the normal-theory CI, the

percentile CI and the BCa CIs, respectively. The funds are ranked according to the

performance measures in each category. The rank corresponds to both the name

(abbreviation of Chinese character) and the trading code in the security exchange of China.

Norm-theory

CIs Percentile CIs BCa CIs

Fund Name

Code Point

Estimate Lower Upper Lower Upper Lower Upper

hand 500025 -0.755 -0.876 -0.626 -0.894 -0.642 -0.881 -0.631 jingb 184695 -0.803 -0.914 -0.683 -0.928 -0.698 -0.915 -0.686 taih 500002 -0.897 -1.078 -0.676 -1.139 -0.738 -1.082 -0.695 ans 500009 -1.048 -1.314 -0.712 -1.400 -0.828 -1.317 -0.749 hanx 500015 -1.122 -1.307 -0.912 -1.348 -0.954 -1.307 -0.928

Sharpe

Measure

yuh 184696 -1.135 -1.297 -0.958 -1.319 -0.976 -1.287 -0.936 taih 500002 -1.862 -2.132 -1.563 -2.194 -1.619 -2.156 -1.596 hand 500025 -1.928 -2.274 -1.548 -2.359 -1.633 -2.319 -1.621 jingb 184695 -1.933 -2.228 -1.606 -2.303 -1.665 -2.272 -1.651 hanx 500015 -2.197 -2.501 -1.859 -2.564 -1.923 -2.502 -1.881 yuh 184696 -2.257 -2.564 -1.909 -2.623 -1.975 -2.558 -1.913

Treynor

Measure

ans 500009 -2.522 -2.929 -2.064 -3.002 -2.150 -2.933 -2.076 taih 500002 -0.014 -0.015 -0.013 -0.015 -0.013 -0.015 -0.013 yuh 184696 -0.014 -0.015 -0.013 -0.015 -0.013 -0.015 -0.013 jingb 184695 -0.014 -0.015 -0.013 -0.015 -0.013 -0.015 -0.013 ans 500009 -0.014 -0.015 -0.013 -0.015 -0.014 -0.015 -0.014 hanx 500015 -0.014 -0.015 -0.014 -0.015 -0.014 -0.015 -0.014

Jensen

Measure

hand 500025 -0.015 -0.016 -0.013 -0.016 -0.013 -0.016 -0.013 hand 500025 -0.878 -1.016 -0.724 -1.045 -0.750 -1.019 -0.734 jingb 184695 -0.957 -1.094 -0.805 -1.118 -0.830 -1.099 -0.812 taih 500002 -1.169 -1.488 -0.743 -1.629 -0.920 -1.548 -0.822 ans 500009 -1.256 -1.701 -0.629 -1.864 -0.923 -1.735 -0.794 yuh 184696 -1.539 -1.698 -1.363 -1.726 -1.394 -1.700 -1.371

Appraisal

Ratio

hanx 500015 -1.542 -1.829 -1.190 -1.866 -1.271 -1.792 -1.147

22

Table 4 The 95% Confidence Intervals of the transformed difference in four

performance measures of several funds selected from Chinese Market using JK’s method,

where the numbers in the column “Diff” represents the Cross Rank generated in Table 2.

Transformed Sharpe

Measure

Transformed Treynor

Measure Diff

Estiamte Lower Upper Estimate Lower Upper

1-2 2.70E-05 -5.15E-04 5.69E-04 -4.07E-06 -1.83E-03 1.83E-03 1-3 -1.67E-05 -4.87E-04 4.54E-04 -2.61E-07 -1.85E-03 1.85E-03

1-31 7.56E-05 -4.76E-04 6.27E-04 1.54E-05 -1.76E-03 1.79E-03 1-32 6.11E-05 -3.96E-04 5.18E-04 2.53E-05 -1.73E-03 1.78E-03 1-33 7.46E-05 -4.56E-04 6.05E-04 1.31E-05 -1.79E-03 1.81E-03 2-3 -4.42E-05 -5.33E-04 4.44E-04 3.90E-06 -1.85E-03 1.85E-03

2-31 4.80E-05 -5.20E-04 6.16E-04 1.93E-05 -1.76E-03 1.80E-03 2-32 3.34E-05 -4.87E-04 5.54E-04 2.91E-05 -1.73E-03 1.79E-03 2-33 4.66E-05 -5.32E-04 6.25E-04 1.71E-05 -1.78E-03 1.82E-03 3-31 9.40E-05 -4.65E-04 6.53E-04 1.60E-05 -1.78E-03 1.81E-03 3-32 7.93E-05 -3.84E-04 5.42E-04 2.62E-05 -1.75E-03 1.80E-03 3-33 9.32E-05 -4.35E-04 6.21E-04 1.37E-05 -1.81E-03 1.83E-03

31-32 -1.51E-05 -6.05E-04 5.74E-04 9.69E-06 -1.71E-03 1.73E-03 31-33 -2.20E-06 -6.26E-04 6.22E-04 -2.51E-06 -1.76E-03 1.75E-03

Jobson

Korkie

Method

32-33 1.31E-05 -6.32E-04 6.59E-04 -1.24E-05 -1.75E-03 1.72E-03

23

Table 5 Bootstrapping the 95% Confidence Intervals of the difference in four

performance measures of top 2 and bottom 2 funds selected from Chinese Market. The

Confidence Intervals are constructed through three popular bootstrapping methods,

namely, the normal-theory CI, the percentile CI and the BCa CIs, respectively. The bolded

values in the following table indicate that the specified confidence interval does not

contain 0, so we have sufficient evidence to believe that there is significant difference

between the corresponding measures.

Norm-theory CIs Percentile CIs BCa CIs

Measure Diff

Point Estimate Lower Upper Lower Upper Lower Upper

1-2 0.094 -0.108 0.265 -0.066 0.296 -0.119 0.255 1-32 0.246 -0.069 0.504 0.025 0.558 -0.063 0.505 1-33 0.319 0.154 0.471 0.172 0.488 0.154 0.472 2-32 0.152 -0.190 0.467 -0.146 0.486 -0.199 0.445 2-33 0.225 0.023 0.444 0.007 0.428 0.038 0.468

Sharpe

Measure

32-33 0.073 -0.103 0.293 -0.151 0.223 -0.116 0.254 1-2 -0.071 -0.321 0.173 -0.324 0.181 -0.325 0.180 1-32 0.589 0.219 0.933 0.266 0.975 0.253 0.959 1-33 0.264 -0.019 0.534 0.001 0.555 -0.031 0.536 2-32 0.660 0.321 0.980 0.373 1.033 0.372 1.032 2-33 0.335 0.093 0.570 0.109 0.591 0.116 0.599

Treynor

Measure

32-33 -0.325 -0.611 -0.027 -0.649 -0.064 -0.652 -0.064 1-2 0.000 -0.001 0.001 -0.001 0.001 -0.001 0.001 1-32 0.000 -0.001 0.001 -0.001 0.001 -0.001 0.001 1-33 0.000 -0.001 0.001 -0.001 0.001 -0.001 0.001 2-32 0.000 -0.001 0.001 -0.001 0.001 -0.001 0.001 2-33 0.000 -0.001 0.001 -0.001 0.001 -0.001 0.001

Jensen

Measure

32-33 0.000 -0.001 0.001 -0.001 0.001 -0.001 0.001 1-2 0.212 -0.230 0.562 -0.085 0.684 -0.186 0.598 1-32 0.299 -0.324 0.766 -0.056 0.908 -0.188 0.808 1-33 0.585 0.236 0.895 0.285 0.910 0.189 0.847 2-32 0.087 -0.605 0.715 -0.500 0.764 -0.567 0.712 2-33 0.373 -0.090 0.889 -0.164 0.789 -0.120 0.828

Appraisal

Ratio

32-33 0.286 0.047 0.643 -0.144 0.428 -0.069 0.465

24

References

Albert, J., and S. Chib (1993) ‘Bayes inference via Gibbs sampling of autoregressive time series subject to Markov mean and variance shifts,’ Journal of Business and Economic Statistics 11, 1–15

Berkowitz, J., and L. Kilian (2000) ‘Recent developments in bootstrapping time series,’ Econometric Reviews 19, 1–48

Beran, R. (1988) ‘Prepivoting test statistics: A bootstrap view of asymptotic refinements,’ Journal of the American Statistical Association 83, 687–97

Bühlmann, P. (1997) ‘Sieve bootstrap for time series,’ Bernoulli 3, 123–48B¨uhlmann, P. (1998) ‘Sieve bootstrap for smoothing nonstationary time series,’ Annals of Statistics 26, 48–83

Carlstein, E. (1986) ‘The use of subseries methods for estimating the variance of a general statistic from a stationary time series,’ Annals of Statistics 14, 1171–79

Chang, Y., and J. Y. Park (2002) ‘A sieve bootstrap for the test of a unit root,’ Journal of Time Series Analysis 23, forthcoming

Chesher, A. and I. Jewitt (1987) “The bias of a heteroskedasticity consistent covariance matrix estimator”. Econometrica 55, 1217–1222.

Chib S., Greenberg E. (1995) “Understanding the Metropolis–Hastings Algorithm”. The American Statistician, 49, pp. 327-335.

Davidson, R. and E. Flachaire (2001) “The wild bootstrap, tamed at last”. working paper IER#1000, Queen’s University.

Davidson, R. and J. G. MacKinnon (1985) “Heteroskedasticity-robust tests in regression directions”. Annales de l’INSEE 59/60, 183–218.

Davidson, R. and J. G. MacKinnon (1998) “Graphical methods for investigating the size and power of hypothesis tests”. The Manchester School 66(1), 1–26.

Davidson, R. and J. G. MacKinnon (1999) “The size distortion of bootstrap tests”. Econometric Theory 15, 361–376.

Davidson, R. and J. G. MacKinnon (2002) “The power of bootstrap and asymptotic tests”. unpublished paper, revised November.

Eicker, B. (1963) “Limit theorems for regression with unequal and dependant errors”. Annals of Mathematical Statistics 34, 447–456.

Flachaire, E. (1999) “A better way to bootstrap pairs”. Economics Letters 64, 257–262.

Freedman, D. A. (1981) “Bootstrapping regression models”. Annals of Statistics 9, 1218–1228.

25

Godfrey, L. G. and C. D. Orme (2001) “Significance levels of heteroskedasticity-robust tests for specification and misspecification: some results on the use of wild bootstraps”. paper presented at ESEM’2001, Lausanne.

Hall, P., Horowitz, J. L. and Jing, B.-Y. (1995) “On Blocking Rules for the Block Bootstrap and Dependent Data”, Biometrika, 82, 561-574.

Hjorth, J. S. U. (1994) “Computer Intensive statistical Methods - Validation model selection and bootstrap” , Chapman & Hall.

Kuensch, H. R. (1989) “The Jackknife and the Bootstrap for General Stationary Observations”, The Annals of Statistics, 17, 1217-1241.

Margasalia, G. (1984) “A Current View of Random Number Generators, The Proceedings of the 16th Symposium on the Interface”, Atlanta 1984. Elsevier Press.

Phillips, P. C. B., and J. Y. Park (1988) “On the formulation of Wald tests of nonlinear restrictions,” Econometrica, 56, 1065–83.

Prakasa-Rao B. L. S. (1983). “Nonparametric Functional Estimation”, Academic Press.

Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling (1992) Numerical Recipes in C, Cambridge University Press, Cambridge.

Shao J., Tu D. (1995) “The Jackknife and Bootstrap”, Springer.

Tong, H. and Lim, K. S. (1980) “Threshold Autoregression, Limit Cycles and Cyclical Data”, Journal of Royal Statistical Society, B 42, 245-292.

Tong, H. (1983) “Threshold Models in Non-Linear Time Series Analysis”, Lecture Notes in Statistics, Vol. 21, Springer-Verlag.

Tong, H. (1990) “Non-Linear Time: A dynamical System Approach”, Oxford University Press.

Tsay, R. S. (1989) “Testing and Modeling Threshold Autoregressive Processes”, Journal of American Statistical Association, 84, 231-240.

Wellner J.A. and Zhan, Y. (1997) “A hybrid algorithm for computation of the non-parametric maximum likelihood estimator from censored data”. J. Amer. Statist. Assoc. 92 945-959.

Weng, C.S. (1989) “On a second-order asymptotic property of the Bayesian bootstrap mean". Ann. Statist. 17, 705-710.

Wong, C. S. and Li, W. K. (1998) “A Note on the Corrected Akaike Information Criterion for Threshold Autoregressive Models”, Journal of Time Series Analysis 19, 113-124.

Wu CFJ (1986) “Jackknife, Bootstrap, and other Resampling Methods in Regression Analysis”, Annals of Statistics, 14: 1261-1295.

Rust, J. (1987) ‘Optimal replacement of GMC bus engines: An empirical model of

26

Harold Zurcher,’ Econometrica 55, 999-1033

Schwert, G. W. (1989) ‘Testing for unit roots: a Monte Carlo investigation,’ Journal of Business and Economic Statistics 7, 147–159.

Smith, R. J. (1987) “Alternative asymptotically optimal tests and their application to dynamic specification,” Review of Economic Studies, 54, 665–80.

Staiger, D., and J. H. Stock (1997) ‘Instrumental variables regressions with weak instruments,’ Econometrica 65, 557–86.

Stern, S. (1997) ‘Simulation-based estimation,’ Journal of Economic Literature 35, 2006–39.

27