Portmanteau test of indepen- dence Stochastic Modelling in Economics and Finance 1 Petr Jon´ aˇ s 6 th January 2014
Portmanteau test of indepen-dence
Stochastic Modelling in Economics and Finance 1
Petr Jonas
6th January 2014
Contents
1 Introduction
2 Test procedure
3 Finite sample performance
4 Application to credit card transaction and geomagneticvariation
Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
Introduction
Most tools of FDA rely on the assumption of iid functionalobservations⇒ portmanteau test of independence for functionalobservationsThe functional observations Xn(t), n = 1, 2, . . . ,N areapproximated by the first p terms of the principal componentexpansion (Xkn are the scores).
Xn(t) ≈p∑
k=1
Xknvk(t), n = 1, 2, . . . ,N (1)
If the populations FPC’s vk(t) are known ⇒ Testing the iidassumption for the curves Xn(t) reduces to testing thisassumption for the random vectors [X1n, . . . ,Xpn]T .We find multivariate analogs of correlations and an analog ofthe sum of squares which has a χ2 asymptotic distribution.
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
We observe zero mean random function{Xn(t), t ∈ [0, 1] , n = 1, 2, . . . ,N} and we want to test
- H0 : the Xn(t) are independent and identically distributed(iid)
- HA : H0 does not hold
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
We approximate the Xn(t) by EFPCs
Xn(t) =
p∑k=1
Xknvk(t)
where
Xkn =
∫ 1
0Xn(t)vk(t)dt =
∫ 1
0Xknvk(t)dt (2)
we take p components explains so that large fraction of samplevariance.
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
To establish the null hypothesis distribution we need followingassumptionASSUMPTION 1The observations X1, . . . ,XN are iid in L2, have mean zero andsatisfy
E ‖Xn‖4 = E
[∫ 1
0X 2n (t)dt
]2<∞ (3)
The eigenvalues of the (population) covariance operator Csatisfy
λ1 > λ2 > · · · > λp (4)
whereC = E [〈(X − µ), .〉 (X − µ)]
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
We will work with the following notation. Denote
Xn =[X1n, . . . , Xpn
]T(5)
andXn = [X1n, . . . ,Xpn]T (6)
where
Xkn =
∫ 1
0Xn(t)vk(t)dt (7)
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
under H0, the Xn are iid mean zero random vectors in Rp forwhich we denote
v(i , j) = E [XinXjn], V = [v(i , j)]i ,j=1,...,p
The matrix V is p × p covariance matrix of the Xn. Let usdenote the autocovariance matrix Ch with elements
ch(k, l) =1
N
N−h∑n=1
XknXl ,n+h 0 ≤ h < N
Definerf ,h(i , j) =
(C−10 Ch
)(i ,j)
rb,h(i , j) =(ChC
−10
)(i ,j)
and denote
QN = NH∑
h=1
p∑i ,j=1
rf ,h(i , j)rb,h(i , j) (8)
analogously we can construct QN from the vectors X6
Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
Theorem 7.1
If Assumption 1 holds, then
QND→ χ2
p2H (9)
(Chi-square distribution with p2H degrees of freedom)
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
Let us consider a special case of functional AR(1) model
Xn+1 = Ψ(Xn) + εn+1 (10)
with iid zero mean innovations εn ∈ L2 and Ψ(x) the anintegral Hilbert-Schmidt operator. We assume that {Xn} isstationary solution to equation (10) (see chapter 13). Let usintroduce the p × p matrix Ψ with elements
ψlk = 〈vl ,Ψ(vk)〉 , l , k = 1, 2, . . . , p
where vk are the eigenfunctions of the covariance operator ofX1. If Ψ is not zero then Ψ is not zero for sufficiently large p
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
Theorem 7.2
Suppose the functional observations Xn follow a stationarysolution to equation (10), assumption 1 hold and p is so largethat p × p matrix Ψ is not zero. Then
QNP→ +∞
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
Finite sample performance
The simulation study focused on the power of the test forAR(1) model (10) which can be written as
Xn(t) =
∫ 1
0ψ(t, s)Xn−1(s)ds+εn(t), t ∈ [0, 1], n = 1, 2, . . . ,N
(11)The sufficient condition for the assumptions of Theorem 7.2 tohold is
‖Ψ‖S =
∫ 1
0
∫ 1
0ψ2(t, s)dtds < 1 (12)
In the study εn are Brownian Motions (BM) or BrownianBridges (BB)
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
Two kernels are used.Gaussian kernel
ψ(t, s) = C exp(t2 + s2
2), t, s ∈ [0, 1]
and Wiener kernel
ψ(t, s) = C min(s, t), t, s ∈ [0, 1]
The constant C is chosen so that ‖Ψ‖S = 0.3, 0.5, 0.7
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
- The power increases with N
- The power increases with ‖Ψ‖S- The power is highest for lag H = 1 because for AR(1) the
”correlation” between Xn and Xn−1 is largest at this lag
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
72.png
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
The BM and BB have a different covariance structure andtherefore the power of the test for BB is higher.
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
The portmanteau test will be applied to the date from VilniusBank about transactions and the geomagnetic data
Bank dataSuppose Dn(ti ) is the number of credit card transactions in dayn, n = 1, 2, . . . , 200 between times ti−1 and ti , where ti − ti−1= 8 min (i = 1, . . . , 128).To remove weekly periodicity, the study works withXn(ti ) = Dn(ti )− Dn−7(ti ), n = 1, 2, . . . , 193.
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation Example of first 3 weeks of the data
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
The test rejects H0 at 1% level for smooth and lagged values1 ≤ H ≤ 5 and the number of principal components equal to4, 5, 10 and 20.Study also tried to apply the test to the residualsεn = Xn − Ψ(Xn−1).Next table displays the p−values for the sequence of residuals.
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
Ground based magnetometer records
The study focus on horizontal (H) component measured atHonolulu in 2001. In the first two weeks the data are :
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
After subtracting linear change over the day (the linear lineconnecting first and last point of the day), we obtain.
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
Testing one year magnetometer data with lags H = 1, 2, 3 anddifferent numbers of principal components p = 3, 4, 5 yieldsp−values very close to zero. The result of the test applied onsmaller subset gives
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
It advocated that the transformed data can to a reasonableapproximation be viewed as a functional simple random sample,at least with respect to the second order properties.The two examples discussed in this section show that our testcan detect departures from the assumption of independence oridentical distribution (magnetometer data), and confirm bothassumptions when they are expected to hold.
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Portmanteautest
Introduction
Test procedure
Finite sampleperformance
Application tocredit cardtransactionandgeomagneticvariation
Thank You For Your Attention!
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