Some applications of functional data analysis to ...piotr/ucsd.pdfSome applications of functional data analysis to econometrics and ﬁnance Piotr Kokoszka Department of Statistics,

Some applications of functional data analysisto econometrics and finance

Piotr KokoszkaDepartment of Statistics, Colorado State University

Joint work with

Lajos HorvathDepartment of Mathematics, Utah State University

Hong MiaoDepartment of Finance, Colorado State University

Matthew ReimherrDepartment of Statistics, Penn State University

Piotr Kokoszka FDA in econometrics and finance

Outline

Functional time series

A factor model for cumulative intraday returns

Predictability of cumulative intraday returns

Stationarity and KPSS tests for functional time series

Aspects of asymptotic theory and some simulations


Cumulative intraday returns on SP500;

Lucca and Moench 2014, Journal of Finance


Microsoft stock prices in one-minute resolution

Clo

sin

g p

rice

23

.02

3.5

24

.02

4.5

M T W TH F M T W TH F


Centered Eurodollar futures curves over a 50 day period.

0 1140 2280 3420 4560 5700

−10


An example of a functional time series model

Functional autoregressive (FAR, ARH) process

Zi = Φ(Zi−1) + εi ,

Zi(t) =

∫φ(t, s)Zi−1(s)ds + εi (t), 0 ≤ t ≤ 1.

Zi =

p∑

j=1

Φj(Zi−j) + εi .


Some recent research

Optimal prediction using the FAR model(Eurodollar futures).

Order selection in the FAR(p) model.

General weakly dependent functional time series, includingARCH type models.

Two sample problems, change point analysis.

Limit theory for the mean function,estimation of the LRV.

Spectral analysis of functional time series(lagged regression, tests for periodicity)


A factor model for cumulative intraday returns


Cumulative intraday returns

Pn(t) - price of an asset on day n at time t within that day(trading hours).

Normalize the trading day to be the interval [0, 1].

Cumulative intraday return (CIDR) on day n:

rn(t) = 100(pn(t)− pn(0)), pn(t) = logPn(t). (1)

Compare: daily log return is pn(1) − pn−1(1).

Sincern(t) ≈ 100(Pn(t)− Pn(0))/Pn(0),

the curves rn(t) and Pn(t) have similar shapes but different scalesand origins.(Pn(0) constant for a given day n.)


CIDR’s vs. daily price curves

Top panel: XOM price curves on five consecutive days.Bottom panel: cumulative intraday returns on the same days.


Factor models for CIDR’s

Rn CIDR curve on day n (a single asset)Mn CIDR curve for a market index

Rn(t) = β0(t) + β1Mn(t) + εn(t).

Sn,Hn potential scalar risk factors

Rn(t) = β0(t) + β1Mn(t) + β2Sn + β3Hn + εn(t).

Cn CIDR curves for oil futures

Rn(t) = β0(t) + β1Mn(t) + β2Cn(t) + εn(t).

Combinations of scalar and curve factors are allowed.


The general model

Rn(t) = β0(t) +

p∑

j=1

βjFnj(t) + εn(t).

Question: are any of the scalar coefficients βi significant?

Statistical factor model of Hays, Shen and Huang (2012):

Xn(t) =K∑

k=1

γnkFk(t) + εn(t).

The factors Fk do not depend on n and are orthonormal functionsto be estimated.


Estimation and results

Functional β0(·), scalar βihybrid estimators: method of moments / least squares

Consistency and asymptotic normality established under a weakdependence condition.SE’s of the βi can be estimated using asymptotic variances.

Conclusions:

The coefficients of the market CIDR’s are significant.

Scalar factors are not significant (shape!).

Oil futures CIDR’s are sometimes significant(oil companies, IT companies).


A weak dependence condition

Bernoulli shifts

Fnj = fj(δn, δn−1, . . .), εn = e(ηn, ηn−1, . . . ).

Definition

A sequence {Xn} is called Lp–m–approximable if each Xn is aBernoulli shift Xn = f (un, un−1, . . .) and

∞∑

m=1

νp(Xn − X

(m)n

)< ∞,

where νp(X ) =(E‖X‖p

)1/p;

X(m)n = f (un, un−1, . . . , un−m+1, u

′n−m, u

′n−m−1, . . .).


Predictability of cumulative intraday returns


Predictability

Related to Efficient Market Hypothesis

Many forms:Introduction in Campbell, Lo, MacKinlay (1997)

Celebrated history:Bachelier, Working, Cowles, Granger, Fama, Samuelson andMandelbrot.

Still subject of research:If frictions and nonstationarities are eliminated,the direction of returns cannot be predicted.

Question: Can CIDR curves be predicted form the past CIDRcurves of the same asset?


Functional Principal Components

Normalize the trading day to be the interval [0, 1].Treat the CIDR’s as random elements of L2([0, 1]).Assume that the L2–valued sequence {rn} is stationary(under H0 to be specified in the following).

r – a random function with the same distribution as each rn.

r(t) = µ(t) +

∞∑

k=1

ξkvk(t), (2)

µ(t) = Er(t) – mean function,vk(·), k ≥ 1, – functional principal components(optimal orthonormal factors).Scores:

ξk = E

∫(r(t)− µ(t))vk(t)dt


µ(·) and vk(·), k ≥ 1, are population parameters.

They are estimated by µ(·) (sample mean)and the estimated FPC’s (EFPC’s) vk(·), 1 ≤ k ≤ p.

All procedures of FDA that use FPC depend on p.

There are several ways of selecting optimal p,but we feel most comfortable when conclusions do not dependstrongly on p in a reasonable range.

The estimated mean function µ(·) is very close to zero, for mostblue chip stocks statistically not significantly different from zero.


First four EFPC’s of XOM.


Idea of testing predictability of shapes

The shape of the curve rn observed on day n is quantified by thevector of scores

[ξ1n, ξ2n, . . . , ξpn]T , ξkn =

∫{rn(t)− µ(t)} vk(t)dt

These vectors describe the temporal evolution of the shapes.

Example of interpretation: The sequence of the scores ξ1n shows“how much” component v1 is present on day n.If the sign of ξ1n is positive, the rn is mostly increasing.If P(ξ1n > 0|ξ1,n−1 > 0) > 1/2, there is some predictability ofshapes: increasing curves tend to follow increasing curves.


Time series of scores for WMT: left ξ1n, right ξ2n, 1 ≤ n ≤ 2500.


Triangular array of score signs

Centered CIDR’s:Xn(t) = rn(t)− µ(t)

Define the triangular array

I(k)N,n = sign{〈Xn, vk〉}, 1 ≤ n ≤ N, 1 ≤ k ≤ p.

Unlike the sign of a scalar return, the sign

I(k)n = sign{〈Xn, vk〉}

is not observable.

Notice that I(k)n I

(k)n+1 is positive for a sequence and negative for a

reversal.


Null and alternative hypotheses

Null Hypothesis:The sequence {Xn} is conditionally symmetric:

L(Xn|Xn−1,Xn−2, . . . ) = L(−Xn|Xn−1,Xn−2, . . . ) a.s.

Alternative Hypotheses:

HA : null hypothesis does not hold.

H−A,j : E [I

(j)1 I

(j)2 ] < 0 or H+

A,j : E [I(j)1 I

(j)2 ] > 0.

1 under H+A,j , P(ξj ,n > 0|ξj ,n−1 > 0) > 1/2 and

P(ξj ,n < 0|ξj ,n−1 < 0) > 1/2, (sequences preferred),

2 under H−A,j , P(ξj ,n > 0|ξj ,n−1 > 0) < 1/2 and

P(ξj ,n < 0|ξj ,n−1 < 0) < 1/2 (reversals preferred).


The tests

To test against H+A,j and H−

A,j we use statistics Λ(j).

To test against the general HA we use a statistics Λp based on thefirst p EFPC’s vj .Λp weighs the Λ(j) according to their importance.Average percent of variance explained by the first four EFPC’s:85.0%, 7.0%, 2.5%, 1.3%

Last part of the talk:

Definitions of these statistics and their asymptotic theory.

Simulation study assessing the finite sample performance of thetests.


Empirical evidence

P–values for the test based on Λp (Dec 1999-Apr 2007).We denote with * the P–values under 10% favoring sequences.

p 1 2 3 4

BOA 0.255 0.243 0.232 0.240CITI 0.194 0.213 0.229 0.223COCA 0.053* 0.062* 0.063* 0.060*CVX 0.106 0.093 0.086 0.085DIS 0.181 0.226 0.204 0.193IBM 0.590 0.656 0.653 0.642MCD 0.239 0.218 0.212 0.209MSFT 0.952 0.841 0.887 0.896WMT 0.984 0.927 0.916 0.897XOM 0.016 0.015 0.015 0.015


P–values of the tests based on statistics Λ(j).We denote with * the P–values under 10% favoring sequences.

j 1 2 3 4

BOA 0.255 0.646 0.309 0.194CITI 0.194 0.618 0.223 0.290COCA 0.053* 0.413 0.857 0.168CVX 0.106 0.419 0.196 0.628DIS 0.181 0.098 0.040* 0.033*IBM 0.590 0.290 0.857 0.369MCD 0.239 0.391 0.536 0.536MSFT 0.952 0.069 0.040* 0.413WMT 0.984 0.194 0.646 0.115XOM 0.016 0.592 0.834 0.625


Summary

The statistic Λp yields consistent conclusions across various p.

The auxiliary statistics Λ(j) allow us to identify thecomponents with predicatability and its direction.

For most blue chip stocks (2000-2007) there is no evidence ofpredictability.

If overall predictability exists, it is in the the first FPC(increasing/decreasing shape).

These conclusions generally remain valid for the period of thefinancial cricis 2008/03/18 to 2009/03/31, except that

MCD becomes predictable (sequences). P-value drops from0.2 to 0.02.XOM becomes unpredictable. P-value increases to 0.6 from0.15.


Stationarity and KPSS tests


Null hypotheses

Level stationarity:

H0 : Xi(t) = µ(t) + ηi (t), 1 ≤ i ≤ N.

Trend stationarity:

H0 : Xi(t) = µ(t) + β(t)i + ηi (t), 1 ≤ i ≤ N.

The ηi are assumed to form L2+δ–m–approximable sequence(Bernoulli shifts are stationary and ergodic).

To avoid long formulas, we focus in the following on levelstationarity.


Alternatives to level stationarity

Change point alternative:

HA,1 : Xi(t) = µ(t) + δ(t)I{i > k∗}+ ηi (t), 1 ≤ i ≤ N,

for some 1 ≤ k∗ < N.The mean function µ(t), the size of the change δ(t), and the timeof the change, k∗, are all unknown parameters. We assume thatthe change occurs away from the end points, i.e.

k∗ = [Nτ ] with some 0 < τ < 1.


Alternatives to level stationarity

Integrated alternative:

HA,2 : Xi (t) = µ(t) +

i∑

ℓ=1

ηℓ(t).

Alternative to trend stationarity:

Xi(t) = µ(t) +i∑

ℓ=1

uℓ(t) + ηi (t).

Deterministic trend alternative:

HA,3 : Xi(t) = µ(t) + g(i/N)δ(t) + ηi (t),

where

g(·) is a piecewise Lipschitz continuous function on [0, 1].


General idea of tests

TN =

1∫

0

{∫Z 2N(x , t)dt

}dx ,

ZN(x , t) = SN(x , t)− xSN(1, t), 0 ≤ x ≤ 1,

SN(x , t) = N−1/2

⌊Nx⌋∑

i=1

Xi(t), 0 ≤ x ≤ 1.

Idea: Under H0, ZN(x , t) computed from the observations Xi is thesame as if it were computed from the unobservable errors ηi .Under HA, deterministic or random trends show up in ZN(x , t).The same idea as in the scalar case. Difficulty: what are the teststatistics and their limits for functions?


Long run covariance function

C (t, s) = Eη0(t)η0(s) +

∞∑

ℓ=1

Eη0(t)ηℓ(s) +

∞∑

ℓ=1

Eη0(s)ηℓ(t).

The function C is positive definite and symmetric.

λiϕi (t) =

∫C (t, s)ϕi (s)ds, 1 ≤ i < ∞.

Kernel estimators C can be constructed which give empiricalcounterparts of the eigenvalues λi and the eigenfunctions ϕi :

λi ϕi (t) =

∫CN(t, s)ϕi (s)ds, 1 ≤ i ≤ N.


Null distributions

Consider iid Brownian bridges Bi .

TND−→

∞∑

i=1

λi

∫B2i (x)dx .

The above result leads to a MC test.Approximate the null distribution of TN by that of

N∑

i=1

λi

∫B2i (x)dx

Note: For scalar series, λ1 is the usual long-run variance, there areno other λi .For functional series, the λi are long run variances in principaldirections of the curves in the function space.


Null distributions

T 0N(d) =

d∑

i=1

1

λi

∫〈ZN(x , ·), ϕi 〉2 dx D−→

d∑

i=1

B2i (x)dx .

Test based on T 0N(d) is the usual asymptotic test with parameter

free limit distribution, but it depends on d .

We considered four other tests, similar in spirit. We compared allsix tests theoretically and using simulations.Greatest difficulty: divergence rates under the alternatives.

Some empirical findings: CIDR are stationary. Yield curves aregenerally not even trend stationary. On some short time intervals,less than 100 days, trend stationarity of yield curves cannot berejected.


Some asymptotic theory and simulations


Key tools for stationarity and KPSS tests

Partial sum processes:

SN(x , t) =1√N

⌊Nx⌋∑

n=1

Xn(t), VN(x , t) =1√N

⌊Nx⌋∑

n=1

ηn(t), 0 ≤ x ≤ 1.

Weak approximation: (Berkes, Horvath, Rice, 2013) There is asequence of Gaussian processes ΓN(x , t) such that for every N

{ΓN(x , t), 0 ≤ x , t ≤ 1} D= {Γ(x , t), 0 ≤ x , t ≤ 1},

Γ(x , t) =

∞∑

i=1

λ1/2i Wi (x)φi (t),

hN = sup0≤x≤1

‖VN(x , ·) − ΓN(x , ·)‖ = op(1).

The Wi are independent standard Wiener processes.The norm is the L2–norm of functions.


Key tools for stationarity and KPSS tests

A general approximation result: (Billingsley, 1968)Suppose ZN , YN , Y are random variables taking values in a

separable metric space with the distance function ρ. If YND→ Y

and ρ(ZN ,YN)P→ 0, then ZN

D→ Y .

In our setting, we work in the metric space D([0, 1], L2) which isthe space of right-continuous functions with left limits takingvalues in L2. A generic element of D([0, 1], L2) is

z = {z(x , t), 0 ≤ x ≤ 1, t ∈ T }.

For each fixed x , z(x , ·) ∈ L2, so ‖z(x , ·)‖2 =∫z2(x , t)dt < ∞.

The uniform distance between z1, z2 ∈ D([0, 1], L2) is

ρ(z1, z2) = sup0≤x≤1

|z1(x , ·)−z2(x , ·)‖ = sup0≤x≤1

{∫(z1(x , t)−z2(x , t))

2dt}1/


Key tool for the predictability test

Difficulty: The usual bound

E

∫(vk(t)− vk(t))

2 dt = O(N−1

)

cannot be used when working with signs (not continuous).

We used a martingale limit theorem due to D. L. McLeish.(The Annals of Probability, 1974, 2, 620–628.)

Theorem

{Yn,N} – array of martingale differences such that(a) max1≤n≤N |Yn,N | is uniformly bounded in L2(P) norm,(b) max1≤n≤N |Yn,N | → 0, in probability,(c)

∑1≤n≤N Y 2

n,N → 1, in probability.

Then,∑

1≤n≤N Yn,ND→ N(0, 1).


Predictability test statistics

Verify that the assumptions of the theorem of McLeish hold with

Yn,N = N−1/2I(j)N,nI

(j)N,n+1.

Conclude that

Λ(j) := (N − 1)−1/2N−1∑

i=1

I(j)N,nI

(j)N,n+1

D→ N(0, 1);

(Λ(1), . . . ,Λ(p))TD→ N(0,Σ).

The matrix Σ can be estimated. (Under addition symmetryassumptions, it is a p × p identity matrix)

Λp =

p∑

j=1

λjΛ(j) D→ N(0, λTΣλ).


Documents

Some applications of functional data analysis to ...piotr/ucsd.pdfSome applications of functional data analysis to econometrics and ﬁnance Piotr Kokoszka Department of Statistics,