Statistical Distributions BYU James B. McDonald. Statistical Distributions James B. McDonald Brigham Young University May 2013 The research assistance.

Statistical DistributionsBYU

James B. McDonald

Statistical Distributions

James B. McDonald

Brigham Young University

May 2013

The research assistance of Brad Larsen, Patrick Turley, and Sean Kerman is gratefully acknowledged as are comments from

Richard Michelfelder and Panayiotis Theodossiou.


1. Introduction

2. Some families of statistical distributions

3. Regression applications

4. Censored regression

5. Qualitative response models

6. Option pricing

7. VaR (value at risk)

8. Conclusion


1. Introduction 2. Some families of statistical distributions

a. Families

3. Regression applications4. Censored regression5. Qualitative response models6. Option pricing 7. VaR (value at risk)8. Conclusion

Some families of statistical distributions

a. Families f(y;θ), θ = vector of parameters i. GB: GB1, GB2, GG (0<Y)

GB distribution tree

Probability Density Functions

11 1 1 /

; , , , , , 0 / 1, 1 /

qaap

a ap qaap

a y c y bGB y a b c p q y b c

b B p q c y b

11 1 /

1 ; , , , ; , , 0, ,,

qaap

ap

a y y bGB y a b p q GB y a b c p q

b B p q


1

2 ; , , , ; , , 1, ,, 1 /

ap

p qaap

a yGB y a b p q GB y a b c p q

b B p q y b

/1

; , ,

ayap

ap

a y eGG y a p

p

0 / 1

,

a a

a controls peakedness

b is a scale parameter

c domain y b c

p q shape parameters

Probability Density FunctionsGB2 PDF evaluated at different parameter values:


a. Families i. GB: GB1, GB2, GG

ii. EGB: EGB1, EGB2, EGG (Y is real valued)

EGB distribution tree


1/ /

/

1 1; , , , ,

, 1

qp y m y m

p qy m

e c eEGB y m c p q

B p q ce

- 1 - <

1

y mfor n

c

1/ /1

1 ; , , ,,

qp y m y me e

EGB y m p qB p q


/

/2 ; , , ,

, 1

p y m

p qy m

eEGB y m p q

B p q e

//

; , ,y mp y m ee e

EGG y m pp

,

m controls location

is a scale parameter

c defines the domain

p q are shape parameters

Probability Density FunctionsEGB2 PDF evaluated at different parameter values:



ii. EGB: EGB1, EGB2, EGG

iii. SGT (Skewed generalized t): SGED, GT, ST, t, normal (Y is real valued)

SGT distribution tree

SGT5 parameter

SGED GT

SLaplace SNormal t SCauchy

Laplace Uniform Normal Cauchy

4 parameter

3 parameter

2 parameter

λ=0 p=2q→∞

ST

GED

λ=0λ=0

λ=0 λ=0λ=0

p=2 p=2

p=2

p=1

p=1

q→∞ q→∞

q=1/2

q→∞ q=1/2p→∞


; , , , ,SGT y m p q

1/

1/

2 1/ , 11

p

p

p p

q p

p

y mq B p q

sign y m q

/ 1

; , , ,2 1/

ppy m sign y m

peSGED y m p

p

= ( )

=

1 , -1 < < 1

2

, ,

m mode location parameter

scale

skewness area to left of m

p q shape parameters tail thickness moments of order pq df

Probability Density FunctionsSGT PDF evaluated at different parameter values:




iii. SGT (Skewed generalized t): SGED, GT, ST, t, normal

iv. IHS


sinh 0,1 /Y a b N k

22 22 2ln / / ln

2

22 2 2; , , ,

2 /

ky y

keIHS y k

y

2 2 2 2.5 .5.5 2 21/ , / , .5 , and .5 2 1k k k k

w w w w we e e e e e

2

k

mean

variance

skewness parameter

tail thickness

; , lim ; , , , 0kN y IHS y k

where

IHS

Probability Density FunctionsIHS PDF evaluated at different parameter values:





iv. IHS

v. g-and-h distribution (Y is real valued)

g-and-h distribution

Definition:

where Z ~ N[0,1]

2 / 2,

1gZhZ

g h

eY Z a b e

g

h>0 h<0

g-and-h distribution

2 20,0 ~ ,Y Z a bZ N a b

, 0

1gZ

g h

eY Z a b

g

2 / 20,

gZg hY Z a bZe

Is known as the g distribution where the parameter g allows for skewness.

Is known as the h distribution

• Symmetric

• Allows for thick tails

Probability Density Functionsg-and-h PDF evaluated at different parameter values with h>0:

Probability Density Functionsg-and-h PDF evaluated at different parameter values with h<0:


a. Families f(y;θ)i. GB: GB1, GB2, GG



iv. IHS

v. g-and-h distribution

vi. Other distributions: extreme value, Pearson family, …


a. Families f(y;θ)i. GB: GB1, GB2, GGii. EGB: EGB1, EGB2, EGG iii. SGT (Skewed generalized t): SGED, GT, ST, t,

normaliv. IHSv. g- and h-distributionvi. Other distributions: extreme value, Pearson

family, …vii. Extensions: 1. x , 2. Multivariate



a. Families b. Properties



b. Propertiesi. Moments

1. GB family

2 1

/ , / ; / , F

/ ;,

hh

GB

p h a h a cb B p h a qE Y

p q h aB p q

for h < aq with c=1



1. GB family

a. GB1

1

/ ,

,

hh

GB

b B p h a qE Y

B p q



1. GB family

a. GB1

b. GB2

2

/ , / - /

,

hh

GB

b B p h a q h aE Y p h a q

B p q



1. GB family

a. GB1

b. GB2

c. GG

/ /

hh

GG

p h aE Y for h a p

p



1. GB family

2. EGB family

2 1

, ; c,

p+q+t,

tty

EGB

p t te B p t qM t E e F

B p q

/ σ with 1for t q c

EGB moments

p p p q p q

2 ' p 2 ' 'p p q 2 ' 'p q

3 '' p 3 '' ''p p q 3 '' ''p q

4 ''' p 4 ''' '''p p q 4 ''' '''p q

EGG EGB1 EGB2

Mean

Variance

Skewness

Excess kurtosis

d n ss

ds

EGB2 moment space



1. GB family

2. EGB family

3. SGT family

SGT family

/

1 1

1,

1 1 12 1

,

h p

hh h h h

SGT

h hq B q

p pE y m

B qp

1 1

1

1 1 12 1

hh h h h

SGED

hp

E y m

p

for h < pq=d.f.

SGT moment space

SGT family moment space


a. Families


1. GB family

2. EGB family

3. SGT family

4. IHS

IHS moment space


a. Families


1. GB family

2. EGB family

3. SGT family

4. IHS

5. g-and-h family

g- and h-family

2

2 1

0,

1

1

i j gi ihj

njn n i i

g h ii

ie

n jE X a b

i g ih

Moments exist up to order 1/h (0<h)

g-and-h moment space (h>0)(visually equivalent to the IHS)

Moment space for g-and-h (h>0) and g-and-h (h real)

Moment space of SGT, EGB2, IHS, and g-and-h



ii. Cumulative distribution functions (see appendix)

• Involve the incomplete gamma and beta functions



ii. Cumulative distribution functions (see appendix)• Involve the incomplete gamma and beta functions

iii. Gini coefficients (G)

Gini Coefficients (G)

Definition:

0 0

1: :

2G x y f x f y dxdy

2

0

0

11

1

F y dy

F y dy

G

(Dorfman, 1979, RESTAT)

Gini Coefficients

Interpretation:

G = 2A

Gini Coefficients

Application:

Stochastic Dominance

Measures of income and wealth inequality





iv. Incomplete moments

Incomplete moments

Definition:

;

yh

h

s f s ds

y hE Y

Applications:

Option pricing formulas

Lorenz Curves

Incomplete moments

Convenient theoretical results:

;y h

2 2; ,LN y h

; , , /GG y a p h a

2 ; , , / , /GB y a b p h a q h a

Distribution

LN

GG

GB2






v. Mixture models

Mixture Models

Let denote a structural or conditional density of the random variable Y where and denote vectors of distributional parameters. Let the density of be given by the mixing distribution . The observed or mixed distribution can be written as

; ,f y

;g

; , ; , ;h y f y g d

Mixture Models

Observed model Structural model

Mixing distribution

; , , , ,SGT y m p q

; , ,GT y p q

2 ; , , ,EGB y p q

2 ; , , ,GB y a b p q

; , ,LT y q

; ,t y q

; , , ,SGED y m s p

; ,GED y s p

; , ln ,EGG y s p

; , ,GG y a s p

; ,LN y s

; ,N y s

1/; , ,pIGG s p q q

1/; , ,pIGG s p q q

1; , ,IGG s e q

; , ,IGG s a b q

1/ 2; 1,IGG s a q

1/ 2;IGA s q






v. Mixture models

vi. Hazard functions (Duration dependence)

Hazard functions

Definition:

Let denote the pdf of a spell (S) or duration of an event.

is the probability that that S>s.The corresponding hazard function is defined by

which can be thought of as representing the rate or likelihood that a spell will be completed after surviving s periods.

f s

1 F s

( )1

f sh s

F s

Hazard functions

Applications:

Does the probability of ending a strike, unemployment spell, expansion, or stock run depend on the length of the strike, unemployment spell, or of the run?

With unemployment, A job seeker might lower their reservation wage and become more likely to find a

job Increasing hazard function However, if being out of work is a signal of damaged goods, the longer they are

out of work might decrease employment opportunities Decreasing hazard function.

An alternative example might deal with attempts to model the time between stock trades. Engle and Russell (1998) Autoregressive conditional duration: a new model for

irregularly spaced transaction data. Econometrica 66: 1127-1162 Hazard function of time between trades is decreasing as t increases or the

longer the time between trades the less likely the next trade will occur.

Hazard functions

Applications:

Bubbles McQueen and Thorley (1994) Bubbles, stock returns, and duration dependence.

Journal of Financial and Quantitative Analysis, 29:379-401 Efficient markets hypothesis, stock runs should not exhibit duration dependence

(constant hazard function) McQueen and Thorley argue that asset prices may contain “bubbles” which grow

each period until they “burst” causing the stock market to crash. Hence, bubbles cause runs of positive stock returns to exhibit duration dependence—the longer the run the less likely it will end (decreasing hazard function), but runs of negative stock returns exhibit no duration dependence

Grimshaw, McDonald, McQueen, and Thorley. 2005, Communications in Statistics—Simulation and Computation, 34: 451-463.

What model should we use to characterize duration dependence? Exponential—constant Gamma—the hazard function can increase, decrease, or be constant Weibull—the hazard function can increase, decrease, or be constant Generalized Gamma: the hazard function can be increasing, decreasing, constant,

-shaped, or -shaped

Hazard functions

Possible shapes for the GG hazard functions



a. Families b. Propertiesc. Model selection


Some families of statistical distributionsc. Model selection

i. Goodness of fit statistics• Log-likelihood values

o for individual data

o for grouped data

Partition the data into g groups,

Empirical frequency:

Theoretical frequency:

1

:n

ii

n f y

1

!g

i i ii

n n n n p n n

1, , 1, 2,...,i i iI Y Y i g

1

/ , g

i i ii

p n n n n

;

i

i

I

p f y dy

Model Selection

i. Goodness of fit statistics• Log-likelihood values• Possible Measures

1

g

i ii

SAE p p

21

g

i ii

SSE p p

2

2 2

1

/ ~ # 1g

ii i

i

nn p p g parameters

n

Model Selection

i. Goodness of fit statistics• Log-likelihood values• Possible Measures• Akaike Information Criterion (AIC)

• A tool for model selection• Attaches a penalty to over-fitting a model

2AIC k

Model Selection

i. Goodness of fit statistics

ii. Testing nested models

Examples:1.

2.

: 0OH g

: : 0O OH SGT GT H

: : 2, 0, O OH SGT Normal H p and q

Testing nested models

Likelihood ratio tests

where r denotes the number

of independent restrictions

Wald test

22 * ~aLR r

21 2 * ~ 1a

SGT GTLR

22 2 * ~ 3a

SGT NormalLR

1 2' var ~a

MLE MLE MLEW g g g r

1

21

ˆ ˆ ˆ0 0 ~ 1aW Var

Statistical Distributions1. Introduction 2. Some families of statistical distributions

a. Families b. Propertiesc. Model selectiond. An example: the distribution of stock returns

3. Regression applications4. Qualitative response models5. Option pricing 6. VaR (value at risk)7. Conclusion

An example: the distribution of stock returns

1 11/ ~ 1t t t

t t tt t

P P Py n P P

P P

Daily, weekly, and monthly excess returns (1/2/2002 – 12/29/2006) from CRSP database (NYSE, AMEX, and NASDAQ)— 4547 companies

H0: skewness = 0

H0: excess kurtosis = 0

H0: returns ~ N(μ, σ2)

JB =

.95 2 6 / , 2 6 /CI n n

.95 2 24 / , 2 24 /CI n n

22

2.05

~ 2 5.99

6 24

excess kurtosisskewn

.95 0 5.99CI JB

An example: the distribution of stock returns (continued)

% of stocks for which excess returns statistics are in 95% C.I.

HO: Skewness=0 HO:Excess kurtosis=0 HO: Normal

Daily 16.38% 0.04% 0.09%

Weekly 30.61% 4.88% 4.75%

Monthly 66.79% 56.65% 53.77%


Daily excess returns plotted with admissible moment space of flexible distributions

-4 -3 -2 -1 0 1 2 3 40

10

20

30

40

50

60

Skewness

Kur

tosi

s

CRSP daily stocks--excess returns

CRSP stock

EGB2

SGTIHS

bound


Weekly excess returns plotted with admissible moment space of flexible distributions

-4 -3 -2 -1 0 1 2 3 40

10

20

30

40

50

60

Skewness

Kur

tosi

s

CRSP weekly stocks--excess returns

CRSP stock

EGB2

SGTIHS

bound


Monthly excess returns plotted with admissible moment space of flexible distributions

-4 -3 -2 -1 0 1 2 3 40

10

20

30

40

50

60

Skewness

Kur

tosi

s

CRSP monthly stocks--excess returns

CRSP stock

EGB2

SGTIHS

bound


Fraction of stocks in the admissible skewness-kurtosis space

daily weekly monthly

EGB2 15.48% 43.81% 50.80%

IHS 83.92% 84.39% 61.97%

SGT 87.62% 89.00% 95.10%

g-and-h 100.00% 99.98% 98.99%


Fitting a PDF to normal excess returns

Company Name Skew Kurtosis Jb Stat

US Steel 0.06 3.308 5.62

Estimated PDF logL SSE SAE Chi^2

Normal 2753.52 0.001 0.12 27.81

EGB2 2756.83 0.001 0.11 23.38

IHS 2756.76 0.001 0.11 23.46

SGT 2758.78 0.001 0.12 28.19

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

14

16

18

20

Excess returns

Estimated PDFs for US Steel daily excess returns

Returns

Normal

EGB2IHS

SGT


Company Name Skew Kurtosis Jb Stat

iShares -29.06 965.09 48733899.02

Fitting a PDF to leptokurtic excess returns

Estimated PDF logL SSE SAE Chi^2

Normal 2516.86 0.099 0.93 1433.33

EGB2 3713.99 0.002 0.13 43.47

IHS 3795.21 0.001 0.12 33.43

SGT 3810.07 0.003 0.21 79.35

-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.080

5

10

15

20

25

30

35

40

45

50

Excess returns

Estimated PDFs for iShares daily excess returns

Returns

Normal

EGB2IHS

SGT


1. Introduction 2. Some families of statistical distributions3. Regression applications4. Censored regression5. Qualitative response models6. Option pricing 7. VaR (value at risk)8. Conclusion


1. Introduction 2. Some families of statistical distributions3. Regression applications

a. Background

4. Censored regression models5. Qualitative response models6. Option pricing 7. VaR (value at risk)8. Conclusion

Regression applications--background

Model:

1xK vector of observations on the explanatory variables

Kx1 vector of unknown coefficients

independently and identically distributed random disturbances with pdf

t t tY X

tX

t ;f

Regression applications--background

If the errors are normally distributed OLS will be unbiased and minimum variance

However, if the errors are not normally distributed OLS will still be BLUE There may be more efficient nonlinear estimators



a. Backgroundb. Alternative estimators

4. Censored regression5. Qualitative response models6. Option pricing 7. VaR (value at risk)8. Conclusion

Alternative Estimators

i. Estimation

OLS

LAD

Lp

21

arg minn

OLS t tt

Y X

1

arg minn

LAD t tt

Y X

1

arg minp

pn

L t tt

Y X

Alternative Estimators (continued)

i. Estimation (continued)M-estimators:

Includes OLS, LAD, and Lp as special cases Includes MLE (QMLE or partially adaptive estimators) as a

special case where

SGT SGED EGB2 IHS

; ;n f

,1

arg min ;n

MLE t tt

Y X

1

arg minn

M t tt

Y X


i. Estimation

ii. Influence functions: ( ) '( )

OLS LADRedescending

influence function


i. Estimation

ii. Influence functions

iii. Asymptotic distribution of extremum

estimators

where

min H

1 1ˆ ~ ;asandwichN A BA

2

and ' '

d H dH dHA E B E

d d d d


i. Estimation ii. Influence functionsiii. Asymptotic distribution of extremum estimatorsiv. Other estimators

Semiparametric (Kernel estimator, Adaptive MLE)

where

denotes a kernel, and h is the window width

1

arg min n

SP K t tt

n f Y X

1

1 ni

Ki

ef K

nh h

i i i OLSe Y X

K

Regression applications (continued)

iv. Other estimators (continued) Generalized Method of Moments (GMM)

where

Z denotes a vector of instruments (can be X)

Q is a positive definite matrix

arg min 'GMM g Qg

1

n

i i i ii

g Z h Y X

1( )Q Var g



a. Backgroundb. Alternative estimatorsc. A Monte Carlo comparison of alternative estimators

4. Censored regression models5. Qualitative response models6. Option pricing 7. VaR (value at risk)8. Conclusion

A Monte Carlo comparison of alternative estimatorsc. A Monte Carlo comparison of alternative estimators

Model:

Error distributions: (zero mean and unitary variance)

Normal:

Mixture:

Skewness =0Kurtosis =24.3

Skewed:

Skewness=6.18Kurtosis=113.9

0;1N

.9* 0,1/ 9 .1* 0,9N N

.50,1 / 1LN e e e

1t t ty X

A Monte Carlo comparison of alternative estimators

Skewness

Kurtosis

Skewed

Mixture

Normal

A Monte Carlo comparison of alternative estimators

Estimators Normal Mixture-thick tails Skewed

OLS .275 .287 .280

LAD .332 .122 .159

SGED .335 .128 .060

ST .293 .112 .054

GT .314 .133 .135

SGT .335 .125 .073

EGB2 .287 .125 .049

IHS .285 .119 .054

SP = AML .285 .114 .128

GMM .319 .115 .088

Sample size = 50, T=1000 replicationsRMSE for slope estimators



a. Backgroundb. Alternative estimatorsc. A Monte Carlo comparison of alternative estimatorsd. An application: CAPM

i. Error distribution effectsii. ARCH effects

4. Censored regression5. Qualitative response models6. Option pricing 7. VaR (value at risk)8. Conclusion

An application: CAPM

i. CAPM and the error distribution


HO: Skewness=0 HO:Excess kurtosis=0 HO: Normal (JB)

Daily 14.14% 0.02% 0%

Weekly 28.13% 3.91% 3.43%

Monthly 67.56% 57.14% 54.76%

Percent of stocks for which OLS residual statistics are in 95% C.I.

An application: CAPM with and without ARCH effects (ST)




Daily 14.05% 0.02% 0%

Weekly 28.82% 3.83% 3.39%

Monthly 64.04% 54.72% 51.48%

Percent of stocks for which ST residual statistics are in 95% C.I.

An application: CAPM with and without ARCH effects (IHS)




Daily 13.99% 0.02% 0%

Weekly 27.89% 3.83% 3.36%

Monthly 65.54% 55.71% 52.32%

Percent of stocks for which IHS residual statistics are in 95% C.I.

An application: CAPM with alternative error distributions

Company Name Skewness Kurtosis JB stat

UNITED NATURAL FOODS INC -0.074 2.8004 0.1543

99 CENTS ONLY STORES 1.7541 7.6594 85.0456

Statistics of OLS residuals

Company Name OLS T GT SGED EGB2 IHS ST SGT

UNITED NATURAL FOODS INC 0.313 0.313 0.335 0.334 0.303 0.302 0.314 0.335

99 CENTS ONLY STORES 0.184 0.125 0.125 0.110 0.109 0.106 0.110 0.110

Estimated Betas

An application: CAPM with and without ARCH effects


ii. CAPM: how about ARCH effects? Review:

If errors are normal and no ARCH effects, OLS is MLE If errors are not normal and no ARCH effects OLS is

BLUE, but not MLE nor efficient If errors are normal and have ARCH effects OLS is

BLUE, but not efficient If errors are not normal and have ARCH effects OLS is

BLUE,but not efficient


ii. CAPM: ARCH effects (continued)

Model:

Percent of stocks exhibiting ARCH(1) effects (OLS) (% rejecting )1: 0OH

0.10 level 0.05 level

Daily 63.2% 60.0%

Weekly 29.2% 24.1%

Monthly 18.7% 13.7%

t t tY X .52

0 1 1t t tu


Percent of stocks exhibiting ARCH(1) effects (ST) (% rejecting )

Percent of stocks exhibiting ARCH(1) effects (IHS) (% rejecting )


Daily 63.2% 59.9%

Weekly 29.1% 23.9%

Monthly 16.9% 12.3%

1: 0OH


Daily 63.3% 60.0%

Weekly 29.3% 24.1%

Monthly 18.9% 13.9%

1: 0OH


ii. CAPM: ARCH effects (continued) ARCH Simulations

, t= 1, …, 60

X monthly excess market returns, 1/2002 to 12/31/2006

Error distributions

0 .9 t t ty X excess market return

2~ 0,t N

1

.520 11 : where ~ 0,1

tN t t tARCH u u N

1

.520 11 : where ~ (5)

tt t t tARCH u u t


ARCH Simulations (continued)

Errors

Estimation Non-ARCH ARCH Non-ARCH ARCH Non-ARCH ARCH

OLS/Normal 0.352 0.356 0.347 0.291 0.353 0.300

LAD 0.444 0.446 0.397 0.369 0.315 0.297

T 0.358 0.363 0.338 0.293 0.283 0.265

GED 0.381 0.389 0.357 0.318 0.306 0.285

GT 0.387 0.396 0.362 0.322 0.306 0.286

SGED 0.406 0.417 0.374 0.341 0.318 0.297

EGB2 0.371 0.376 0.352 0.312 0.300 0.281

IHS 0.368 0.377 0.348 0.319 0.291 0.275

ST 0.375 0.382 0.350 0.310 0.293 0.277

SGT 0.409 0.420 0.376 0.344 0.316 0.297

Root Mean Square Error (RMSE) for 10,000 replications

N(0,σ 2) N(0,1), Arch(1) t(5), Arch(1)


1. Introduction 2. Some families of statistical distributions3. Regression applications4. Censored regression models

a. Basic frameworkb. Simulation study

5. Qualitative response models6. Option pricing 7, VaR (value at risk)8. Conclusion

*i i iY X

Censored Regression a. Basic Framework

Model:

Log-likelihood function:

*i i iy X

* *

*

if y 0

= 0 if y < 0

i i i

i

y y

* 00

, ; ;ii

i i iyy

n f y X n F X

b. Censored regression: non-normality and heteroskedasticity

Qualitative Response Models


1. Introduction 2. Some families of statistical distributions3. Regression applications4. Censored Regression models5. Qualitative response models

a. Basic framework

6. Option pricing 7. VaR (value at risk)8. Conclusion

Qualitative Response—Basic Framework

Model:

if and 0 otherwise

Log-likelihood function:

*i i iy X

1 iy * 0iy

*Pr 1 Pri i i i i iy X y X X

Pr ; ;iX

i i iX f s ds F X

1

, ; 1 1 ;n

i i i ii

y n F X y n F X

Qualitative Response—Basic Framework (continued)

MLE of will be consistent and asymptotically distributed as

if the model is correctly specified.

Probit and logit estimators will be inconsistent if The error distribution is incorrectly specified heteroskedasticity exists, e.g. unmeasured heterogeneity is

present relevant variables have been omitted The index appears in a nonlinear form

Similar results are associated with Censored & Truncated regression models

12

ˆ ~ ;'

a dN E

d d


1. Introduction 2. Some families of statistical distributions3. Regression applications4. Censored regression 5. Qualitative response models

a. Basic frameworkb. An application: fraud detection


Prediction of corporate fraud (Y=1 fraud) Compare financial ratios of companies with averages

of five largest companies (“virtual” firm) 228 companies (114 fraud and 114 non-fraud) Variables: accruals to assets, asset quality, asset

turnover, days sales in receivables, deferred charges to assets, depreciation, gross margin, increase in intangibles, inventory growth, leverage, operating performance margin, percent uncollectables, receivables growth, sales growth, working capital turnover.

SGT, EGB2, & IHS formulations improve predictions

Qualitative response—An application: fraud detection


1. Introduction 2. Some families of statistical distributions3. Regression applications4. Censored regression5. Qualitative response models

a. Basic frameworkb. An applicationc. Some related issues


Qualitative response—Some related issues

Cost of misclassification Choice-based sampling Heterogeneity Semi-parametric estimation procedures


1. Introduction 2. Some families of statistical distributions3. Regression applications4. Censored regression5. Qualitative response models6. Option pricing: European call option 7. VaR (value at risk)8. Conclusion


1. Introduction 2. Some families of statistical distributions3. Regression applications4. Qualitative response models5. Option pricing: European call option

a. The Black-Scholes option pricing formula

6. VaR (value at risk)7. Conclusion

Option pricing—Black-Scholes

a. The Black Scholes option pricing formula

The equilibrium price of a European call option is equal to the present value of its expected return at expiration:

where involve “normalized

incomplete” moments

0, , ,0 ,

;1 ;0

rT rtf T T

X

rTT

T t

C S T X e E C S e S X f S S T dS

X XS e X

S S

; 1

yhh

y

h h

s f s dss f s ds

y hE y E y

; 1 ;y h y h

.



a. The Black-Scholes option pricing formulab. Some background and alternative formulations


Option pricing– Some background and alternative formulations

The Black Scholes (1973) option pricing formula corresponds to being the lognormal

, the cdf for the lognormal

The Black Scholes formula (Bookstaber and McDonald, 1991) corresponding to the Generalized Gamma is obtained from

, the cdf for the GG

The Black Scholes formula ( Bookstaber and McDonald, 1991) corresponding to the GB2 is obtained from

, the cdf for the GB2

Rebonato (1999) applied to the Deutschemark

f s

2 2; ; ,LN y h LN y h

; ; , ,GG

hy h GG y a p

a

2 ; 2 ; , , ,GB

h hy h GB y a b p q

a a

2 , ,GB TC S T X

Option pricing– Some background and alternative formulations

Sherrick, Garcia, and Tirupattur (1996) used to price soybean futures.

Theodosiou (2000) developed the

Savickas (2001) explored the use of

Dutta and Babbel (2005) explore the g- and h- family (4-parameter) of option pricing formulas, , based on Tukey’s nonlinear transformation of a standard normal.

Applied the g-and-h to pricing 1-month and 3-month London Inter Bank Offer Rates (LIBOR)

g- and- h distribution and GB2 perform much better (errors fairly highly correlated) than the Lognormal, Burr 3, and Weibull distributions

& , ,g h TC S T X

, ,SGED TC S T X

, ,Weibull TC S T X

3 , ,Burr TC S T X



a. The Black-Scholes option pricing formulab. Some background and alternative formulationsc. A comparison of pricing behavior


A comparison of pricing behaviorc. A comparison of pricing behavior (Dutta and Babbel, Journal of

Business, 2005) Calculates the difference between the market price and predicted price

for the g-and-h, GB2, lognormal, Burr3, and Weibull distributions

Option Pricing


1. Introduction 2. Some families of statistical distributions3. Regression applications4. Qualitative response models5. Option pricing: European call option 6. VaR (value at risk)7. Conclusion


1. Introduction 2. Some families of statistical distributions3. Regression applications4. Censored regression5. Qualitative response models6. Option pricing: European call option 7. VaR (value at risk)

a. Background and definitions

8. Conclusion

VaR—Background and definitions

i. Value at risk (VaR) is the maximum expected loss on a portfolio of assets over a certain time period for a given probability level.

R is the return on the asset

θ denotes the distributional parameters

α is the predetermined confidence level or coverage probability

is the corresponding maximum expected loss or conditional threshold

;R

f R dR

1 :RR F

R


R z

1 :; ,

R

Z Z

ZFf z dz Z

1 :ZR F

ii. Standardized returns


iii. Unconditional VaR formulation

Estimate f(R;θ)


iv. Conditional VaR formulation (AR(1) ABS-GARCH(1,1))

0 1 1t t t t t t tR R Z z

0 1 1 1 2 1t t t tz 1 :t t t ZR F


1. Introduction 2. Some families of statistical distributions3. Regression applications4. Censored regression5. Qualitative response models6. Option pricing: European call option 7. VaR (value at risk)

a. Background and definitionsb. Models and applications

8. Conclusion

VaR—Models and applications

i. Unconditional VaR formulation Exponential: (Hogg, R. V. and S. A. Klugman (1983)) Gamma: (Cummins, et al. 1990) Log-gamma: (Ramlau-Hansen (1988)), (Hogg, R. V. and

S. A. Klugman (1983)) Lognormal: (Ramlau-Hansen (1988)) Stable: (Paulson and Faris (1985) Pareto: (Hogg, R. V. and S. A. Klugman (1983)) Log-t: (Hogg, R. V. and S. A. Klugman (1983)) Weibull: (Cummins et al. (1990))


i. Unconditional VaR formulation (continued) Burr: (Hogg, R. V. and S. A. Klugman (1983)) Generalized Pareto: (Hogg, R. V. and S. A.

Klugman (1983)) GB2: (Cummins (1990, 1999, 2007) Pearson family: Aiuppa (1988) Extreme value distribution: Bali (2003), Bali and

Theodossiou (2008) IHS: Bali and Theodossiou (2008)


ii. Conditional VaR formulations (Bali and Theodossiou, JRI, 2008)

Data: S&P500 composite index, 1/4/50 – 12/29/2000 (n=12,832) Daily percentage log-returns: (Sample mean = .0341,

maximum=8.71, minimum=-22.90 standard deviation = .874 skewness =1.622 kurtosis=45.52


ii. Conditional distributions (Bali and Theodossiou, JRI, 2008) (continued)

Models Generalized extreme value EGB2 SGT IHS

Findings Out of sample VaR estimates are rejected for most unconditional

specifications Thresholds exhibit time varying behavior Out of sample VaR estimates for the conditional specifications

corresponding to the SGT, IHS, and EGB2 perform better than the extreme value distributions

Selected references for option pricing and VaR

Aiuppa, T. A. 1988. “Evaluation of Pearson curves as an approximation of the maximum probable annual aggregate loss.” Journal of Risk and Insurance 55, 425-441

Bali, T. G., 2003. “An Extreme Value Approach to Estimating Volatility and Value at Risk,” Journal of Business, 76:83-108 Bali, T. G. and P. Theodossiou, 2007. “A Conditional-SGT-VaR Approach with Alternative GARCH Models,” Annals of

Operations Research, 151: 241-267. Bali, T. G. and P. Theodossiou, 2008. “Risk Measurement Performance of Alternaitve Distribution Functions,” Journal of Risk

and Insurance, 75: 411-437. Black, F (1976). The Pricing of Commodity Contracts. Journal of Financial Economics 3:169-179. Cummins, J. D., G. Dionne, J. B. McDonald, and B. M. Pritchett 1990. “Applications of the GB2 family of distributions in

modeling insurance loss processes.” Insurance: Mathematics and Economics 9, 257-272. Cummins, J. D., C. Merrill, and J. B. McDonald, 2007. “Risky Loss Distributions and Modeling the Loss Reserve Pay-out Tail,”

Review of Applied Economics 3. Cummins, J. D., R. D. Phillips, and S. D. Smith 2001. “Pricing Excess of Loss Reinsurance Contracts against catastrophic

loss.” In Kenneth Froot, ed., The Financing of Catastrophe Risk (Chicago: University of Chicago Press) Dutta, K. K. and D. F. Babbel 2005. “Extracting Probabilistic Information from the Prices of Interest Rate Options: Tests of

Distributional Assumptions.” Journal of Business 78:841-870 Hogg, R. V. and S. A. Klugman, 1983. “On the Estimation of Long Tailed Skewed Distributions with Actuarial Applications.”

Journal of Econometrics 23, 91-102. McDonald, J. B. and R. M. Bookstaber (1991). “Option Pricing for Generalized Distributions.” Communications in Statistics:

Theory and Methods, 20(12), 4053-4068. Rebonato, R. (1999). Volatility and correlations in the pricing of equity. FX and interest-rate options. New York: John Wiley. Paulson, A. S. and N. J. Faris (1985). “A Practical Approach to Measuring the Distribuiton of Total Annual Claims.” In J. D.

Cummins, ed., Strategic Planning and Modeling in Property-Liability Insurance. Norwell, MA: Kluwer Academic Publishers. Ramlau-Hansen, H. (1988). “A Solvency Study in Non-life Insurance. Part 1. Analysis of Fire, Windstorm, and Glass Claims.”

Scandinavian Actuarial Journal, pp. 3-34. Rebonato, R. 1999. Volatility and correlations in the pricing of equity, FX and interest-rate options. New York: John Wiley. Reid, D. H. (1978). “Claim Reserves in General Insurance,” Journal of the Institute of Actuaries 105: 211-296 Savickas, R. (2001). A Simple option-pricing formula. Working paper, Department of Finance, George Washington University,

Washington, DC. Sherrick, B. J., P. Garcia, and V. Tirupattur (1996). Recovering probabilistic information for options markets: Tests of

distributional assumptions. Journal of Futures Markets 16:545-560. Theodossiou, Panayiotis, “Skewed Generalized Error Distribution of Financial Assets and Option Pricing,”


1. Introduction 2. Some families of statistical distributions3. Regression applications4. Censored regression5. Qualitative response models6. Option pricing: European call option 7. VaR (value at risk)8. Conclusion

Conclusion

END OF PRESENTATION

Appendices

Cumulative distribution functions1. GB, GB1, GB2, GG2. EGB23. SGT4. SGED5. IHS6. g-and-h distribution

Option pricing basics VaR—Models and applications discussion

Appendices—Cumulative distribution functions

1. GB, GB1, GB2, and GG

where and

denotes the incomplete beta function

2 1 ,1 ; 1;1 ; , , ,

,

,

p

z

z F p q p zGB y a b p q

pB p q

B p q

/a

z y b

11

0

1

,,

zqp

z

s s ds

B p qB p q


1. GB, GB1, GB2, and GG (continued)

where

2 1 ,1 ; 1;2 ; , , ,

,

,

p

z

z F p q p zGB y a b p q

pB p q

B p q

/

1 /

a

a

y bz

y b


1. GB, GB1, GB2, and GG (continued)

where

and

denotes the incomplete gamma function

Abramowitz and Stegun (1970, p. 932), McDonald (1984), and Rainville (1960,p. 60 and 125)

/

1 1

/; , , 1; 1; /

1

a apyae y

GG y a b p F p yp

z p

/a

z y

1

0

zp s

z

s e ds

pp


2. EGB2

where

3. SGT

where

2 ; , , , ,zEGB y m p q B p q

/

/1

y m

y m

ez

e

11

; , , , , 1/ ,2 2 z

sign y mSGT y m p q sign y m B p q

1

p

pp p

y mz

y m q sign y m

Appendix—Cumulative distribution functions

4. SGED

where

11; , , , 1/

2 2 z

sign y mSGED y m p sign y m p

1

p

pp

y mz

sign y m


5. IHS

where

; , , , Pr PrIHS y k Y y Z z

2; 0, 1 PrN z Z z 2

1 1

1 1 3 ; ;

2 2 2 22

z zF

2 / 2

1 1

2 2 2z

sign z

2

1y a y a

z k n kb b

/ wb 2 2 2.5 .52 2/ .5 2 1k k ke e e

2.5.5 kwa b b e e e

and with


6. g- and h-distribution Numeric procedures, based on the use of order statistics as

outlined in Exploring Data Tables, Trends, and Shapes by Hoaglin,, Mosteller, and Tukey (1985), Wiley.

For h > 0, the transformation

is one-to-one, (Martinez, J. and B. Iglewicz . 1984. “Some Properties of Tukey g and h family of distributions,” Communications in Statistics—Theory and Methods 13, 353-369). Even without an explicit functional form for the inverse, numerical “MLE” estimates” can be obtained.

2 / 2,

1gZhZ

g h

eY Z a b e

g

Appendices

Cumulative distribution functions Option pricing basics

1. European call option

2. Put option

3. Definitions of terms

4. Assumptions

5. Volatility

6. The Greeks

VaR—Models and applications discussion

Appendices—Option pricing basics

1. European call option

2. Put option

0

T 1 2

, , , ,0, , ,

;1 ;0 BS: S d

rT rtf T T

X

rT rTT

T t

C S T X r e E C S X r e S X f S S T dS

X XS e X e X d

S S

2 1 : -rTTBS Put formula e X d S d


3. Definitions of terms: T = time to expiration ST = Current market price r = interest rate (risk free rate) X = strike price (or exercise price)

call options: price at which the instrument can be purchased up to expiration profit per share gained upon exercising or selling the option >0 in the money <0 out of the money

put options: price at which the instrument can be sold up to expiration

TS X

TS X

TS X


4. Assumptions: Can short sell the underlying instrument No arbitrage opportunities Continuous trading in the instrument No taxes or transaction costs Securities are perfectly divisible Can borrow or lend at a constant risk free rate The instrument does not pay a dividend

5. Volatility (in the BS option pricing formula—based on the LN)

Appendices—Option pricing basics6. The Greeks:

(delta) measures the change in value of the instrument to a change in the current market price

(kappa or vega) measures the responsiveness of the value of the instrument in response to a change in volatility

(theta) responsiveness of the value of the instrument to T (time to expiration)

(rho) responsiveness to changes in the risk free rate

, , ,;1

f T

T T

C S T X r X

S S

, , ,

( )f TC S T X r

volatility

, , ,f TC S T X r

T

, , ,f TC S T X r

r

Appendices

Cumulative distribution functions Option pricing basics VaR—Models and applications

discussion

Appendices—VaR: Models and applications discussion

Paulson and Faris (1985) used the stable family and Aiuppa (1988) used the Pearson family to model insurance losses

Ramlau-Hansen (1988) modeled fire, windstorm, and glass claims using the log-gamma and lognormal

Cummins, et al. (1990) modeled fire losses using the GB2 Cummins, Lewis, and Phillips (1999) used the LN, Burr 12, and GB2 to

model hurricane and earthquake losses. Hogg, R. V. and S. A. Klugman, 1983. “On the Estimation of Long Tailed

Skewed Distributions with Actuarial Applications.” Journal of Econometrics 23, 91-102

Models loss distributions (a. Hurricaines (1949-1980), b. malpractice claims paid for insured hospitals in 1975)

Considers exponential, pareto (mixture of an exponential and inverse gamma), generalized pareto (mixture of gamma and inverse gamma), Burr distribution (mixture of a Weibull and inverse gamma), log-t (mixture of a lognormal and inverse gamma) and a log-gamma.

Consider alternative estimation procedures: maximum likelihood and minimum distance estimators

Many loss distributions are characterized by skewness and long tails such as associated with the flexible distributions coming from mixtures.


Cummins, J. D., G. Dionne, J. B. McDonald, and B. M. Pritchett, 1990. “Applications of the GB2 family of distributions in modeling insurance loss processes.” Insurance: Mathematics and Economics 9, 257-272. Models fire losses Considers the GB2 and special cases GG, BR3, BR12, LN, W, and

GA to model the fire loss data. MLE estimates of distributional parameters and Maximum Probably Yearly Aggregate Loss (MPY) were obtained at the .01 level.

Important to use distributions which permit thick tails Bali, T. G., 2003. “An Extreme Value Approach to Estimating

Volatility and Value at Risk,” Journal of Business, 76:83-108


Cummins, J. D., C. Merrill, and J. B. McDonald, 2007. “Risky Loss Distributions and Modeling the Loss Reserve Pay-out Tail,” Review of Applied Economics 3. Estimate aggregate loss distribution associated with claims incurred

in a given year, but settled in different years Data: U.S. products liability insurance paid claims (Insurance Services

Office (ISO)) Mixture model:

Consider different GB2 distributions for each cell (year) Multinomial distribution for fraction of claims settled at different lags

Single aggregate GB2 distribution for each year GB2 provides a significantly better fit to severity data than the LN, gamma, Weibull, Burr12, or generalized gamma

The Aggregate GB2 distribution has a thicker tail than does the mixture distribution


Bali, T. G. and P. Theodossiou, 2008. “Risk Measurement Performance of Alternative Distribution Functions,” Journal of Risk and Insurance, 75: 411-437. Models: Unconditional formulations

Generalized Pareto Generalized extreme value Box-Cox extreme value SGED SGT EGB2 IHS

Models: Conditional formulations (model time-varying VaR thresholds)

0 1 1t t t t t t tR R z z 0 1 1 1 2 1t t t tz

tL


Bali, T. G. and P. Theodossiou, 2008. “Risk Measurement Performance of Alternative Distribution Functions,” Journal of Risk and Insurance, 75: 411-437. (continued) Data

S&P500 composite index (1/4/1950 to 12/29/2000) Daily percentage log-returns: (n=12,832 maximum=8.71 minimum=-22.90 skewness =1.622 kurtosis=45.52

Findings Out of sample VaR estimates are rejected for most unconditional

specifications Thresholds exhibit time varying behavior Out of sample VaR estimates for the conditional specifications

corresponding to the SGT, IHS, and EGB2 perform better than the extreme value distributions

END OF APPENDICES

Statistical Distributions BYU James B. McDonald. Statistical Distributions James B. McDonald Brigham Young University May 2013 The research assistance.

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