Nonparametric Inference on Stochastic Dominance …...Nonparametric Inference on Stochastic...

Nonparametric Inference on Stochastic Dominance andRelated Concepts

Yoon-Jae Whang

Seoul National University

Feb. 20, 2017Yale Econometrics Lunch

Y.J. Whang (SNU) Nonparametric Tests of SD Feb. 20, 2017 1 / 87

Why interested in Stochastic Dominance?

Stochastic Dominance : A popular (partial) ordering rule of variousdistribution functions. Useful in

Ranking portfolio investment strategiesComparing income distributions or poverty levelsDistributional treatment effectsEvaluation of forecasting models, etc.

Attractive in that it does not require restrictive assumptions on thedistributions of choice alternatives and preference structure ofeconomic agents or policy makers.

The stochastic dominance relation corresponds to an inequalityrestriction between nonparametric functions and its statisticalinference can be complicated.

This talk gives a brief overview of the literature and introducessome further developments.


Outline of the talk

1 Introduction

1 Concepts of Stochastic Dominance2 Existing SD Tests3 Hypotheses of Interest4 Extensions of SD Tests

2 Some Further Developments

1 Testing for Conditional SD: Functional Inequality Approach2 A Nonparametric Test of a Strong Leverage Hypothesis3 Distributional Tests under Measurement Errors

3 Conclusions


First Order Stochastic Dominance

Let

X & Y : rv’s with cdf FX(·) & FY (·)U1 = {u(·) : u′ ≥ 0} : Utility (Social Welfare) functions

X First Order Stochastic Dominates (FSD) Y (or FX � FY ) iff

(a) FX(x) ≤ FY (x) ∀x ∈ X or(b) E[u(X)] ≥ E[u(Y )] ∀u ∈ U1 or(c) QX(τ) ≥ QY (τ) ∀τ ∈ T , where QD(τ) is the τ -th quantile of

D = X,Y.


First Order Stochastic Dominance

-8 -6 -4 -2 0 2 4 6 8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CDF of X CDF of Y


Second Order Stochastic Dominance

LetU2 = {u(·) : u′ ≥ 0, u′′ ≤ 0}.

X Second Order Stochastic Dominates (SSD) Y iff

(a)∫ x−∞ FX(t)dt ≤

∫ x−∞ FY (t)dt ∀x ∈ X or

(b) E[u(X)] ≥ E[u(Y )] ∀u ∈ U2 or(c)

∫ τ0QX(t)dt ≥

∫ τ0QY (t)dt ∀τ ∈ T .


Second Order Stochastic Dominance

-8 -6 -4 -2 0 2 4 6 8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CDF of X CDF of Y

-8 -6 -4 -2 0 2 4 6 8

0

1

2

3

4

5

6

7

8

∫x

−∞FX(t)dt

∫x

−∞FY (t)dt


Higher Order Stochastic Dominance

For F = FX or FY ,

F (s)(x) =

{F (x) for s = 1∫ x−∞ F

(s−1)(t)dt for s ≥ 2.

Us = {u(·) : u′ ≥ 0, u′′ ≤ 0, ...(−1)s+1u(s) ≥ 0}.

X Stochastic Dominates Y at Order s iff

(a) F(s)X (x) ≤ F (s)

Y (x) ∀x ∈ X or(b) E[u(X)] ≥ E[u(Y )] ∀u ∈ Us or

(c) Q(s)X (t)dt ≥ Q(s)

Y (t)dt ∀τ ∈ T .


Existing SD Tests

Comparison of the distributions at finite number of grids:

Beach and Davidson (1983)Bishop, Fomby and Thistle (1992)Anderson (1996)Dardanoni and Forcina (1999)Davidson and Duclos (2000), etc.

Comparison of the distributions over the whole support:

McFadden (1989), Klecan, McFadden and McFadden (1991)Kaur, Rao and Singh (1994)Barrett and Donald (2003)Linton, Maassoumi and Whang (2005)Horvath, Kokoszka and Zitikis (2006)Linton, Song and Whang (2010)Donald and Hsu (2013)


Hypotheses of Interest

Three types of Hypothesis:

(1) H0 : FX = FY vs. H1 : FX � FY(2) H0 : FX � FY vs. H1 : FX � FY

(3) H0 : FX � FY vs. H1 : FX � FY

(1) is the usual ”one-sided” test. Assumes that SD holds and tries tofind a statistical evidence for strict SD. Both H0 and H1 can be falsefor arbitrary FX and FY .(2) looks for evidence against SD as a minimal requirement to pursuean analysis under SD (cf. Goodness of fit test problem). Non-rejectionof H0 may not imply existence of SD.(3) is attractive to assess the existence of SD. Need to restrict thedomain of interest by excluding the tails of the distributions.


Extensions of SD Tests

SD Efficiency: Post (2003), Scaillet and Topaloglou (2010), Linton,Post and Whang (2014)

Portfolio Returns:

Y : Evaluated portfolio (with cdf FY )X>λ : Portfolio (with cdf Fλ) for λ ∈ Λ0 ⊂ RK , whereX = (X1, . . . , XK)> are K - benchmark assets

We say that Y is s-th order SD efficient in Λ0 iff there does not exist

any portfolio in {X>λ : λ ∈ Λ0} that s-th order dominates it.

H0 : Y is s-th order SD efficient in Λ0.

SD efficiency criteria is useful as a portfolio screening/building device.

The test requires comparison among infinite number of distributionfunctions and raises some computational issues.


Extensions of SD Tests (Cont’d)

Stochastic Monotonicity : Lee, Linton and Whang (2007),Delgado and Escanciano (2012)

H0 : FY |X(y|x) ≤ FY |X(y|x′) for all y and x ≥ x′.

If X is some policy variable, it amounts to testing whether its effect onthe distribution of Y is increasing, e.g. Son’s income vs. Father’sincome), Dynamic macro models (Lucas and Stokey (1989), IO models(Olley and Pakes (1996)).

The test can be viewed as a continuum version of the standard SDtest.

LLW(2007) consider a test based on a rescaled U-process and proposea higher-order approximation to the limiting null distribution.



Conditional SD : Lee and Whang (2009), Gonzalo and Olmo(2014), Delgado and Escanciano (2013), Andrews and Shi (2015)

H0 : FY1|X(y|x) ≤ FY2|X(y|x) for all y and x.

Conditional SD implies that the SD relationship holds for everysubgroup of the population defined by X. Useful to control forsubgroup or individual heterogeneity in comparing distributions.

Asymptotics can be carried out using Poissonization (integral-typetest), strong approximation (sup-type test), or by the unconditionalmoment representation of conditional cdf’s.



Lorenz Dominance : Beach and Davidson (1983), Barrett, Donaldand Bhattacharya (2014)

H0 : L2(p) ≤ L1(p) for all p ∈ [0, 1],

where

Lk(p) =

∫ p0 F

−1k (t)dt

µk, k = 1, 2.

Lorenz curve(LC) plots the percentage of total income earned byvarious portions of the population when the population is ordered bythe size of their incomes, i.e. from poorest to richest. Fundamentaltool for the analysis of economic inequality.



Density (or Likelihood) Ratio Ordering : Dykstra et. al. (1995),Beare and Moon (2015), Beare and Shi (2016)

We say that X Density Ratio Dominates (DRD) Y iff

H0 : dFY (x)/dFX(x) is nonincreasing in x.

Density ratio ordering is a stronger property than FSD, i.e. if X DRDY, then X FSD Y. Various applications in finance, insurance,mechanism design and auction theory.

Equivalent to the hypothesis that the ordinal dominance curveR = FY ◦ F−1X is concave. Test based on the empirical analogue Rn ofR and its least concave majorant.



Positive Quadrant Dependence : Denuit and Scaillet (2004),Scaillet (2005)

We say that X is positive quadrant dependent on Y, or PQD(X|Y ), if

P (X ≤ x, Y ≤ y) ≥ P (X ≤ x)P (Y ≤ y) ∀(x, y) ∈ R2.

PQD is the same as saying that FX (·) first order stochastic dominatesFX (·|Y ≤ y) ∀y ∈ R.Applications (in finance, insurance and risk management): Dhaene andGoovaerts (1996), Denuit, Dhane and Ribas (2001), Embrechts,McNeil and Straumann (2000), Denuit and Scaillet (2004), Levy(1992), Shaked and Shanthikumar (1994), Drouet-Mari and Kotz(2001) and Scaillet (2005).



Expectation Dominance : Zhu, Guo, Lin and Zhu (2014), Linton,Whang and Yen (2016)

We say that Y positive expectation dependent (PED) on X if

E (Y ) ≥ E (Y | X ≤ x) ∀x.

Classical portfolio choice problem (Wright (1987)):

maxλ∈[0,1]

EU(λY + (1− λ)X).

When EY = EX, then the necessary and sufficient condition for theoptimal λ∗ ∈ (0, 1) (diversification) are that:

E(Y −X|X ≤ x) ≥ 0 ; E(X − Y |Y ≤ y) ≥ 0 ∀x, y.

PED is equivalent to

Cov (Yt, γ(X)) ≥ 0 for every increasing function γ.



Spatial Dominance : Park (2008)

Let {Xt : t ∈ [0, T ]} be a (possibly nonstationary) time series. Define

l(T, x) = limε→0

1

2ε

∫ T

0

1 {|Xt − x| < ε} dt,

Λ(T, x) =

∫ x

−∞

∫ T

0

e−rtEl(dt, x) =

∫ T

0

e−rtP {Xt ≤ x} dt,

to be the local time (the frequency at which Xt visits the spatial pointx up to time T ) and the spatial distribution of Xt., respectively.We say that Xt first-order spatially dominates Yt if

H0 : ΛX(T, x) ≤ ΛY (T, x) ∀ x ∈ R.This holds iff∫ T

0

e−rtEu(Xt)dt ≤∫ T

0

e−rtEu(Yt)dt ∀u ∈ U1.

Useful to compare the performance of two assets over a given timeinterval, because optimal investment strategies might be horizondependent. (Goyal and Welch (2008))



Almost Stochastic Dominance (Leshno and Levy (2002)) : Whichprospect to choose?

(X,PX) (Y, PY )

($2, 1/2) ($1, 1/2)($3, 1/2) ($1000, 1/2)

No FSD or SSD. Almost all ”reasonable” investor would choose ??.For ε ∈ (0, 1), FX dominates FY by ε−ASD if∫

[FX(x)− FY (x)]+ dx ≤ ε∫|FX(x)− FY (x)| dx

This holds iff

Eu(X) ≤ Eu(Y ) ∀u ∈ U1 (ε) ⊂ U1.

The ratio of the area between FX and FY for whichFX ≥ FY (violation area) to the total area between FX and FY is lessthan or equal to ε. (The limiting case ε = 0 corresponds to FSD.)



Utopia index : Anderson, Post and Whang (2017)

Let F = {F1, F2, ..., FM} be a class of CDFs for M ≥ 2. Define

Is,1 = 1− As,1Ts

,

where

As,1 =

∫X

[F

(s)1 (x)− min

F∈F{F (s)(x)}

]dx

Ts =

∫X

[maxF∈F{F (s)(x)} − min

F∈F{F (s)(x)}

]dx.

This is a comparison measure of M (integrated) distribution functions.Useful to rank incomparable prospects.APW develops the distribution theory under a time series setting.


2.1. Testing for Conditional SD: FunctionalInequality Approach

(with Sokbae Lee and Kyungchul Song)


Hypothesis of Functional Inequalities

Consider a set of nonparametric functions

vτ ,1(·), · · ·, vτ ,J(·)

for each index τ in a finite dimensional space T , e.g.vτ ,j(x) = E(Yj |X = x), Fj(τ |x), qj(τ |x) or some functions of them.

We are interesting in testing:

H0 : vτ ,j(x) ≤ 0 for all (x, τ , j) ∈ X × T × NJH1 : vτ ,j(x) > 0 for some (x, τ , j) ∈ X × T × NJ ,

where X × T is the domain of interest and NJ = {1, · · · , J}.


Motivating Example: Auction Models

Consider the following auction environment studied by Guerre,Perrigne, and Vuong (2009) (GPV hereafter).

Let I2 > I1 ≥ 2 be two different numbers of bidders, and for each τsuch that 0 < τ < 1, let qk(τ) denote the τ -th quantile of theobserved equilibrium bid distribution, say Gk(·), when the number ofbidders is Ik, where k = 1, 2.

In GPV, the support of Gk(·) is [b, bj ] ⊂ [0,∞) with b < bj . (Notethat b is common across j’s, while bj ’s are not.)


Auction Models (Cont’d)

Equation (5) of GPV provides the following testable restrictions:

I1 − 1

I2 − 1q2(τ) +

I2 − I1I2 − 1

b ≤ q1(τ) (1)

≤ q2(τ) ≤ I2 − 1

I1 − 1q1(τ) +

I1 − I2I1 − 1

b

for any τ ∈ (0, 1].



Suppose that I1 = 2 and I2 = 3. Then (1) is written as

1

2q2(τ) +

1

2b ≤ q1(τ) ≤ q2(τ) ≤ 2q1(τ)− b

for any τ ∈ (0, 1].

Arranging inequalities in (1) shows that there exist only two distinctrelations:

q1(τ)− q2(τ) ≤ 0,

b− 2q1(τ) + q2(τ) ≤ 0(2)

for any τ ∈ (0, 1].

The first equation of (2) corresponds to the hypothesis of stochasticdominance.



The restrictions in (1) are based on exogenous participation for whichthe latent private value distribution is independent of the number ofbidders.

If the auctions are heterogeneous so that the private values areaffected by observed characteristics, we may consider conditionallyexogenous participation with a conditional version of (2).



Consider a conditional version of (2):

vτ ,1(x) ≡ q1(τ |x)− q2(τ |x) ≤ 0,

vτ ,2(x) ≡ b− 2q1(τ |x) + q2(τ |x) ≤ 0(3)

for any τ ∈ (0, 1] and for any x ∈ X , where X is the (common)support of X, and X denotes the observed variables that characterizeauction heterogeneity, and qk(τ |x) denotes the τ -th conditionalquantile (given X = x) of the observed equilibrium bid distributionwhen the number of bidders is Ik.

Note that the first inequality of (3) corresponds to the hypothesis ofconditional stochastic dominance.


Timber Auction Data


Conditional Quantile Functions


Estimates of vτ ,1(x)

The figure shows that 20 estimated curves of vτ ,1(x) ≡ q1(τ |x)− q2(τ |x)at different quantiles, ranging from the 10th percentile to the 90thpercentile.


Estimates of vτ ,2(x)

The figure shows that 20 estimated curves ofvτ ,2(x) ≡ b− 2q1(τ |x) + q2(τ |x) at different quantiles, ranging from the10th percentile to the 90th percentile.


Test Statistic

Let qj(τ |x) be the local polynomial estimator of qj(τ |x) and definevτ ,j(x) to be vτ ,j(x) with qj(τ |x) replaced by qj(τ |x) for j = 1, 2.

(Joint) Test statistic:

θJoint =

2∑j=1

∫X×T

[√nhdvτ ,j(x)

]p+dQ(x, τ)

where p = 1 or 2. The conditional SD hypothesis can be tested usingthe first summand of the joint test statistic.

Let cα > 0 be a critical value. We

reject H0 if and only if θ > cα.

We suggest cα such that the rejection probability is not greater thanα under the null hypothesis, and yet leads to a test with good powerproperties.


Bootstrap Test Statistic

Let cn →∞ (at a suitable rate). Define

B(cn) ≡{

(x, τ) ∈ X × T : |√nhdvτ (x)| ≤ cn

}.

Our Bootstrap Test Statistic:

θ∗

=

∫B(cn)

[Z∗n(x, τ)]+ dQ(x, τ),

where B(cn) is a sample version of B(cn).


Contact Set (J=1)


Testing Procedure

Let c∗α be the (1− α)-th quantile from the bootstrap distribution of

θ∗

and takec∗α,η = max{c∗α, hd/2η + a∗}

as our critical value, where η > 0 is a small fixed number.

Then,Reject H0 if and only if θ > c∗α,η.


Asymptotic Validity

Theorem (1)

Suppose that Assumptions A1-A6 and B1-B4 hold. Then,

limsupn→∞

supP∈P0

P{θ > c∗α,η} ≤ α.

Theorem (3)

Suppose that Assumptions A1-A3 and B1-B4, and C1 hold and that weare under the fixed alternative H1,

P{θ > c∗α,η} → 1.


Test Results for Auction Data

We used the timber auction data used in Lu and Perrigne (2008).

In our illustration, we use the appraisal value as X.

Table: Summary Statistics

2 bidders 3 bidders(Sample size = 107) (Sample size = 108)

Standard StandardMean Deviation Mean Deviation

Appraisal Value 66.0 47.7 53.3 41.4Highest bid 96.1 55.6 100.8 56.7Second highest bid 80.9 49.2 83.1 51.5Third highest bid 69.4 44.6

Notes: Bids and appraisal values are given in dollars per thousand board-feet (MBF).

Looking at the data, it seems that average bids become higher as thenumber of bidders increases from 2 to 3, conditional on appraisalvalues.


Test Results for Auction Data (Cont’d)

The contact set was estimated with cn = Ccs log(n). We checked thesensitivity to the tuning parameters with Ccs ∈ {0.5, 1, 1.5} andh ∈ {0.3, 0.6, 0.9}.

All cases resulted in bootstrap p-values close to 1, thereby suggestingthat positive values of vτ ,1(x) at lowest appraisal values cannot beinterpreted as evidence against the null hypothesis beyond randomsampling errors.

Therefore, we have not found any evidence against the economicimplications imposed by auction theory.


2.2. A Nonparametric Test of a Strong LeverageHypothesis

(with Oliver Linton and Yu-Min Yen)


Motivation

Leverage Hypothesis: Negative shocks to stock prices affect theirvolatility more than equal magnitude positive shocks.

Whether this is attributable to changing financial leverage is stillsubject to dispute, but the terminology is in wide use.Black (1976), Christie (1982): A drop in the value of stock (negativereturn) increases the financial leverage (debt-to-equity ratio), whichmakes the stock riskier and increases volatility.

Most of the existing tests typically involve fitting of a generalparametric or semiparametric model to conditional volatility.

We propose a way of testing the leverage hypothesisnonparametrically without requiring a specific parametric orsemiparametric model.


Motivation (Cont’d)

Our null hypothesis is that the conditional distribution of volatilitygiven negative returns and past volatilities stochastically dominatesthe distribution of volatility given positive returns and past volatilities.

⇒ Hypothesis of ”Conditional Stochastic Dominance”

This hypothesis is in some sense stronger than those consideredpreviously, since we refer to the distribution of outcomes rather thanjust correlations.

If our null hypothesis is satisfied, then any investor who valuesvolatility negatively would prefer the distribution of volatility thatarises after positive shocks to returns to the distribution that arisesafter negative shocks.


Main Results

1 Extend the unconditional dominance test of Linton, Maasoumi andWhang (2005) to conditional stochastic dominance to test theleverage hypothesis.

2 Propose an inference method based on subsampling. The test isconsistent against a general class of alternatives.

3 Use Realized Volatility as a direct nonparametric measure ofvolatility, which allows us to avoid specifying a model for volatilityand makes our test model free.

4 Double asymptotic framework (n→∞ and T →∞), where n andT denote the high-frequency and low-frequency time periods. Biascorrection method under weaker conditions on n and T.

5 We find empirical evidences in favor of the strong leveragehypothesis.



Suppose that we observe a process {yt, xt, rt}Tt=1, where xt ∈ Rdx forsome dx. Let

F+(y|x) = Pr (yt ≤ y | rt−1 ≥ 0, xt = x)

F−(y|x) = Pr (yt ≤ y | rt−1 < 0, xt = x) .

We consider the hypothesis

H0 : F−(y|x) ≤ F+(y|x) a.s. for all (y, x) ∈ Xy ×XxH1 : F−(y|x) > F+(y|x) for some (y, x) ∈ Xy ×Xx,

where Xy ⊂ R denotes the support of yt and Xx ⊂ Rdx denotes thesupport of xt.

A leading example would be to take yt = σ2t and xt = σ2t−1. We mayalso take xt = h(σ2t−1, . . . , σ

2t−p), where h : Rp → Rdx for dx < p is a

measurable function.


Parametric vs. Our Approach

Parametric Approach: Suppose that σ2t was generated from a GJRGARCH(1,1) process, i.e.,

σ2t = ω + βσ2t−1 + γ+r2t−11(rt−1 > 0) + γ−r

2t−11(rt−1 ≤ 0).

The case where γ− > γ+ corresponds to the presence of a leverageeffect.

Our Approach: The null hypothesis is effectively that bad news onthe current returns leads to a bigger effect on the conditionaldistribution of future volatility than good news whatever the currentlevel of volatility.


Moment Inequality Representation

Let

π+0 (x) = Pr(rt−1 ≥ 0|xt = x),

π−0 (x) = Pr(rt−1 < 0|xt = x).

We can write the null hypothesis by the conditional momentinequality:

H0 : E

[1(yt ≤ y)

(1(rt−1 < 0)

π−0 (xt)− 1(rt−1 ≥ 0)

π+0 (xt)

)∣∣∣∣xt = x

]≤ 0

for all (y, x) ∈ Xy ×Xx.Equivalently, it can be written as:

H0 : E[1(yt ≤ y)

{π+0 (xt)− 1(rt−1 ≥ 0)

}∣∣xt = x]≤ 0

for all (y, x) ∈ Xy ×Xx, assuming π+0 (x) = 1− π−0 (x) > 0 for allx ∈ Xx.Y.J. Whang (SNU) Nonparametric Tests of SD Feb. 20, 2017 45 / 87

Moment Inequality Representation (Cont’d)

The hypothesis H0 can be equivalently stated using the unconditionalinequality:

H0 : E[1(yt ≤ y)g (xt)

{π+0 (xt)− 1(rt−1 ≥ 0)

}]≤ 0

for all (y, g) ∈ Xy × G, where g is an instrument that depends on theconditioning variable xt and G is the collection of instruments, e.g.

G = {ga,b : ga,b(x) = 1 (a < x ≤ b) for some a, b ∈ Xx} .

(cf.) Stinchcombe and White (1998), Andrews and Shi (2013).


The Test Statistic

In practice, the volatility σ2t is unobserved and needs to be estimatedusing high frequency data. Let yt and xt be estimators of yt and xt,respectively, based on the estimated volatility σ2t ,

Let π+ be nonparametric kernel estimators of π+0 , i.e.,

π+(x) =

∑Tt=2 1(rt−1 ≥ 0)Kh (x− xt)∑T

t=2Kh (x− xt).

The hypothesis can be tested based on the following statistic

ˆmT (y, g, π) =1

T

T∑t=2

1(yt ≤ y)g (xt) {π(xt)− 1(rt−1 ≥ 0)} .

Test statistic:

ST = sup(y,g)∈Y×G

√T ˆmT (y, g, π+).


Bias Correction

We consider a bias correction method that is designed to capture theleading consequence of estimating σ2t from the high frequency returnsdata.Bias correction term:

∆T (y, g) =

(κ− 1

κT

T∑t=2

RQtnt

)× 1

T

T∑t=1

{[F ′′(y|xt)− F+′′(y|xt)

]+[F (y|xt)− F+(y|xt)

] f ′′(xt)f(xt)

}g (xt) π

+0 (xt),

where

RQt = nt

nt∑j=1

r4tj , κ = E[η4tj

].

Bias corrected test statistic:

SbcT = sup(y,g)∈Y×G

√T{

ˆmT (y, g, π+) − ∆T (y, g)}.


Asymptotic Null Distribution

Theorem (2)

Suppose that Assumptions A and B1-B5 hold and nx/T →∞, wherex > 1/γ and γ > max{2ε+ 1

3 ,1+ε1+2λ}. Then, under the null hypothesis H0,

SbcT ⇒{

sup(y,g)∈B [ν(y, g)] if B 6= ∅−∞ if B = ∅ ,

whereB = {(y, g) ∈ Xy × G : E

[1(yt ≤ y)g (xt)

{π+0 (xt)− 1(rt−1 ≥ 0)

}]= 0}.


Asymptotic Validity

Theorem (3)

Suppose Assumptions A, B and C hold. Then, under the null hypothesisH0,

limT→∞

Pr[SbcT > gT,bT (1− α)] ≤ α,

with equality holding if B 6= ∅, where B is defined in Theorem 2 andgT,bT (1− α) denotes the subsample critical value.

Theorem (4)

Suppose that Assumptions A, B and C hold. Then, under the alternativehypothesis H1,

limT→∞

Pr[SbcT > gT,bT (1− α)] = 1.


Empirical Results

We focus on whether there is a leverage effect between dailyconditional variances and daily lagged returns in S&P500 (cash) indexand individual stocks.

Five constituents of Dow Jones Industrial Average (DJIA): Microsoft(MSFT), IBM (IBM), General Electronic (GE), Procter& Gamble(PG) and 3M (MMM).

The samples used for the test span from Jan-04-1993 to Dec-31-2009(4,283 trading days).


Estimators of Daily Conditional Variance

1. Realized Variance Estimator:

RVt =

nt∑j=1

r2tj ,

where

rtj = logP

(t− 1 +

j

nt

)− logP

(t− 1 +

j − 1

nt

)is the jth intraday log return on day t, nt is the total number ofintraday log return observations on day t, and P (t− 1 + j/M) is theintraday asset price at time stamp t− 1 + j/nt.


Estimators of Daily Conditional Variance (Cont’d)

2. Squared Intraday Range Estimator (Garman and Klass, 1980;Parkinson, 1980):

RG2t =

IG2t

4 log 2,

whereIGt = max

t−1≤τ<tlogP (τ)− min

t−1≤τ<tlogP (τ) ,

and P (τ) is the intraday asset price at time stamp τ on day t,t− 1 ≤ τ < t. Under mild regularity conditions, RG2

t is aconditionally unbiased estimator for σ2t .


Figure 1: MSFT and IBM5

10

15

20

sqrt

(RV

_5

min

)

05

10

15

20

25

RG

−0.1

5−

0.0

50.0

50.1

5

R

1995 2000 2005 2010

MSFT

05

10

15

20

sqrt

(RV

_5

min

)

05

10

15

20

25

RG

−0.1

5−

0.0

50.0

5

R

1995 2000 2005 2010

IBM

Figure: Time series plots of daily√RV 5min

t , RGt and rdailyt for the S&P500index and five constituents from Dow Jones Industrial Averages. The quantitiesof√RV 5min

t and RGt shown here are scaled by 252 (annualized). The sampleperiod is from Jan-04-1993 to Dec-31-2009 (4,283 trading days).


Figure 2: Plot of Surfaces

Sigma_upper

−2

−1

0

1

2

3

4

y

−2

−1

0

1

2

3

4

m_T

=ta

u_1−

tau_2

−0.010

−0.005

0.000

Figure: MSFT with j=1. Plot of ˆmTj

(y, g, π+

). Here RV 5min

t is the estimateddaily volatility.


Figure 4: Empirical P-Values (MSFT)

0.0

0.2

0.4

0.6

0.8

1.0

Subsample size

Em

pir

ical p−

valu

e

100 300 500 700 1000 1200 1400 1600 1800 2000

j=1j=5

Figure: The subsample empirical p-values for STjof MSFT. Here RV 5min

t is theestimated daily volatility.


Robust Checks

Time Periods: Two subperiods (1/4/1993-11/30/2001, 12/03/2001-12/31/2009) based on the U.S. recessions identified by the NBER.

Conditioning Variables: EMA(Exponential Moving Average),HAR-RV(Heterogeneous Autoregressive Realized Variance)

More Sophisticated Volatility Estimator: Pre-averaging basedestimator of quadratic variation (Jacod et. al. (2009), Hautsch andPodolskij (2013)), etc.

Alternative Methods: HAR-RV Type Models with the LeverageEffect, Wang and Mykland (2013), Aıt-Sahalia, Fan and Li (2013)


Remarks

We have found there is almost no evidence against the strongleverage effect in daily stock returns both at the individual stocklevel and the index level.

The null hypothesis we consider is quite strong, namely first orderdistributional dominance. Therefore, it is quite powerful that the datado not reject this hypothesis.

Our empirical evidence is robust along a number of directions


2.3. Distributional Tests under MeasurementErrors

(with Karun Adusumilli and Taisuke Otsu)


Motivation

Measurement errors are quite common in economic and statisticaldata.

Examples include income data and household surveys, e.g. CPS.Imprecise memory, Different recorder, Tax evasion, etc...

The traditional approach often assumes away the existence ofmeasurement errors. However, sometimes this practice may lead tomisleading conclusions on the underlying true distributions.

In this paper, we develop deconvolution methods to do inference onthe latent true cumulative distribution functions.


Motivation (Cont.)

Classical measurement error model:

X = X∗ + ε,

X∗ is not observable

Only the mis-measured variable X is observableε is a measurement error satisfying ε ⊥ X∗.

Rich literature for nonparametric methods on pdf fX∗ bydeconvolution technique, but relatively thin literature for inferencemethods on cdf FX∗ .


This Paper

We show validity of asymptotic and bootstrap approximations forthe distribution of

Tn = supt∈T|FX∗(t)− FX∗(t)|, (4)

where FX∗(t) is the deconvolution cdf estimator.

We allow fε to be ordinary or super smooth, or to be unknown andestimated by repeated measurements.

We show that the convergence rate of the bootstrap approximationerror is of polynomial order under ordinary smooth errors andlogarithmic order under super smooth errors.


This Paper (Cont.)

Applications

Uniform confidence bands for FX∗ and its quantilesCdf-based tests for goodness-of-fit of parametric models of FX∗Homogeneity tests for two distributionsStochastic dominance tests between true distributions.


Main Results

In my presentation, I will focus on the Kolmogorov-Smirnov type testfor stochastic dominance under measurement errors.

We consider a test based on the deconvolution distribution functionestimator of Hall and Lahiri (2008).

We propose a bootstrap test of stochastic dominance and verify itsvalidity.


Density Estimator

Assume that X∗ ⊥ ε and fε is known. Then, the Fourier transform(FT) of fX is given by

f ftX(t) =

∫exp(itx)fX(x)dx = f ftX∗(t) · f ftε (t) .

Naive Estimator of fX∗(t) : Let

Φn(t) =1

n

n∑j=1

exp(itXj)

f ftX∗(t) = Φn(t)/ f ftε (t).

If we can take inverse FT of f ftX∗(t), then we may get an estimatorfX∗ . However, the estimator is not well-defined as f ftX(t) is notintegrable. We need to regularize it!


Density Estimator (Cont’d)

Deconvolution Kernel Density Estimator: Define

fX∗(x) =1

2π

∫ 1

−1exp(−itx)

Φn(t)

f ftε (t)K ft(th)dt

=1

nh

n∑j=1

KX,n(x−Xi

h

),

where

KX,n(u) =1

2π

∫ 1

−1exp(−itu)

K ft(t)

f ftε (t/h)dt

and K is a kernel function with K ft bounded and supported on[−1, 1].


CDF Estimator

Deconvolution Kernel CDF Estimator for FX∗ under symmetric fε(cf. Hall and Lahiri (2008)):

FX∗(t) ≡∫ t

−∞fX∗(x)dx =

1

2+

1

n

n∑i=1

LX,n(t−Xi

h

),

where

LX,n(u) =1

2π

∫ 1

−1

sin(ωu)

ω

K ft(ω)

f ftε (ω/h)dω

Dattner, Goldenshluger and Juditsky (2011): General case of possiblyasymmetric fε.


Asymptotic Gumbel Approximation for Ordinary SmoothCase

Theorem (1)

Suppose that Assumptions C,OS, and G hold, and (nh)−1(log n)3 → 0 asn→∞. Then,

P{

(−2 log h)1/2(B−1/2tn − bn) ≤ c}→ exp(−2 exp(−c))

for all c ∈ R, where the constant B and sequence bn is defined in (42),where

tn = supt∈T|fX(t)−1/2{FX∗(t)− FX∗(t)}|.


Remarks

1. Theorem 1 implies that the asymptotic confidence band at level α forFX∗ is given by

CGn (t) = [FX∗(t)±B1/2fX(t)1/2{cGα (−2 log h)−1/2 + bn}],

for t ∈ T , where cGα solves exp(−2 exp(−cGα )) = α.

2. No asymptotic approximation available yet for the super-smooth case.Main difficulty is that the limiting form of the deconvolution kerneldoes not exist.

3. We consider a bootstrap approximation that does not requireexistence of the asymptotic distribution and has better finite sampleperformance.


SD Test

Model: Suppose we observe i.i.d. samples {Xi}ni=1 and {Yi}mi=1 of Xand Y , respectively, generated from

X = X∗ + ε

Y = Y ∗ + δ, (5)

where (ε, δ) are measurement errors and (X∗, Y ∗) are unobservableerror-free variables of interest.

Null Hypothesis of Interest:

H0 : FX∗(t) ≤ FY ∗(t) for all t ∈ T ,

where T is a compact interval of interest specified by the researcher.

Test Statistic:

Dn,m = supt∈T{FX∗(t)− FY ∗(t)}.


Motivation: Barrett and Donald (2003) Test

Design

X = X∗ + ε, Y = Y ∗ + δ, where X∗ ∼ N(0, 3.5), Y ∗ ∼ N(0, 3.5),ε ∼Laplace(0, σ2

ε), and δ ∼Laplace(0, 0.5).

Rejection Probability of BD Test for H0 : X FSD Y

n \ σ2ε 0.5 1.0 2.0 3.0

100 0.072 0.050 0.078 0.088

250 0.049 0.054 0.082 0.148

500 0.053 0.055 0.152 0.334

1000 0.051 0.098 0.275 0.667

⇒ BD Test rejects too often the hypothesis H0 : X∗ FSD Y ∗.


Bootstrap Procedure for SD Test

1 Resample {X#i }ni=1 and {Y #

i }mi=1 from {Xi}ni=1 and {Yi}mi=1,respectively.

2 Compute the bootstrap counterpart

D#n,m = sup

t∈T

{(F#X∗(t)− F

#Y ∗(t)

)−(FX∗(t)− FY ∗(t)

)},

where

F#X∗(t) =

1

2+

1

n

n∑i=1

LX,n

(t−X#

i

h

),

F#Y ∗(t) =

1

2+

1

m

m∑i=1

LY,m

(t− Y #

i

h

).

3 Let cα denote the (1− α)-th quantile of the bootstrap statistic D#n,m.

We reject H0 if Dn,m > cα.


Asymptotic Validity

Theorem (7)

Suppose that Assumption C holds true for both X = X∗ + ε andY = Y ∗ + δ, and that n/(n+m)→ τ ∈ (0, 1) as n,m→∞.

(i) Under H0,P{Dn,m > cDα } ≤ α+ %n,m,

for some positive sequence %n,m = O(n−c) (under Assumption OS forboth ε and δ) or %n,m = O((log n)−c) (under Assumption SS for bothε and δ) with c > 0.


Asymptotic Validity (cont’d)

Theorem (7)

(ii) Let P0 be the set of probability measures of (X,Y ) satisfying H0

(but fε and fδ are fixed) and

0 < cX ≤ inft∈T

fX(t) ≤ supt∈T

fX(t) ≤ CX <∞,

supω∈R{(1 + |ω|)γX |f ftX∗(ω)|} ≤MX <∞,

and likewise for fY and f ftY ∗ for some cX , γX , CX ,MX > 0 that areindependent of (fX , fY ). Then

supP∈P0

P{Dn,m > cDα } ≤ α+ %n,m,

for some positive sequence %n,m = O(n−c) (under Assumption OS) or%n,m = O((log n)−c) (under Assumption SS) with c > 0.


Asymptotic Validity (cont’d)

Theorem (7)

(iii) Under the alternative H1 (i.e., H0 is false) and Assumption OS or SS,

P{Dn,m > cDα } → 1.


Remarks

The proof of Theorem 7 does not require the existence of limitdistribution of Dn,m .

We verify Theorem 7 using uniform strong approximation of the teststatistics and the anti-concentration property of the supremum of theapproximating Gaussian process.(cf.) Chernozhukov, Chetverikov and Kato (2014, AoS).

Even if the error distributions are not known, we can perform a validinference if repeated measurements on the data are available.(cf.) Delaigle, Hall and Meister (2008, AoS).


Empirical Application: Motivation

Korea has become an aging society quite rapidly, but it is worried thatthe welfare status of the retirement age group is worseningsignificantly.

Among the OECD countries, Korea has the most significant variationsin within-age group income inequality.

We apply our deconvolution-based SD test to compare the incomedistributions of Korea between 2006 and 2012 across various agegroups.

We use household-based survey data, which are inherently affected byvarious sources of measurement errors. (see Bound, Brown, andMathiowetz(2000))


Gini Coefficients by Age Groups, Year 2005


Data Description

We use Korea Household Income and Expenditure Survey data,which contains incomes from various sources and consumption ofgoods and services for each household.

We process the data as follows.

1 Adjust the data for inflation using 2010 Consumer Price Index.2 Obtain the real household disposable income by adding all incomes,

public pension, social benefits and transfers, and subtracting publicpension premium and social security fees.

3 Compute the individual income by adjusting the total householdincome using the square-root equivalization scale, which is a commonpractice to approximate the individual welfare.



We considered two different null hypotheses for five different agegroups:

1 The 2006 income distribution stochastically dominates 2012 incomedistribution (abbreviated to 06SD12)

2 The 2012 income distribution stochastically dominates 2006 incomedistribution (abbreviated to 12SD06)

As a benchmark test, we consider the Barrett and Donald (2003,BD)’s test which does not take into account the presence ofmeasurement errors.


Observed Income Distributions

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

×106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CDF (Age:45-65)

2006

2012

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

×106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CDF (Age:60+)

2006

2012

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

×106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CDF (Age:65+)

2006

2012

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

×106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CDF (Age:70+)

2006

2012


Deconvolution CDF (Laplace Error)

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

×106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Deconvolution CDF (Age:45-65, Error: Laplace, SNR=3)

2006

2012

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

×106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Deconvolution CDF (Age:60, Error: Laplace, SNR=3)

2006

2012

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

×106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


2006

2012

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

×106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


2006

2012


Deconvolution CDF (Normal Error)

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

×106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Deconvolution CDF (Age:45-65, Error: Normal, SNR=3)

2006

2012

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

×106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Deconvolution CDF (Age:60, Error: Normal, SNR=3)

2006

2012

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

×106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


2006

2012

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

×106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


2006

2012


Bootstrap P-values(FSD)

AgeNullHyp.

BDLaplace Error Normal Error

SNR2 SNR3 SNR4 SNR2 SNR3 SNR4

25-4512SD06 1.000 0.998 1.000 1.000 1.000 1.000 1.00006SD12 0.000 0.000 0.000 0.000 0.000 0.000 0.000

45-6512SD06 1.000 1.000 1.000 1.000 1.000 1.000 1.00006SD12 0.000 0.000 0.000 0.000 0.000 0.000 0.000

60+12SD06 0.039 0.000 0.000 0.000 0.305 0.234 0.05406SD12 0.000 0.037 0.023 0.013 0.000 0.000 0.000

65+12SD06 0.000 0.001 0.000 0.000 0.027 0.013 0.00306SD12 0.353 0.400 0.652 0.704 0.143 0.240 0.189

70+12SD06 0.000 0.000 0.000 0.000 0.000 0.000 0.00106SD12 0.928 0.501 0.934 0.988 0.664 0.698 0.715


Empirical Results

For age groups, 25-45 and 45-65, the 2012 income (first and secondorder) dominates the 2006 income. However, for age group 70+, the2006 income (first and second order) dominates the 2012 income.This is consistent with the empirical findings that the welfare statusof the retirement age group is worsening.

The results of the BD test and our test are similar, except in the caseof FSD (for the age group 60+) and SSD (for the age group 65+).This implies that we may need to consider explicitly the presence ofmeasurement errors to draw sensible conclusions on the trueunderlying distributions.


3. Conclusions


Conclusions

Stochastic dominance is a useful concept in various areas ofeconomics.

Statistical inference on stochastic dominance and related concepts iscomplicated and construction of valid testing procedures in differentcontexts requires a careful study.

We review some of the existing tests of the SD hypothesis and theirextensions. There should be many other interesting hypotheses thatdeserve attention.


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