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Modeling, simulation and inference for multivariatetime series of counts using trawl processes
Almut E. D. VeraartImperial College London
2018 Workshop on Finance, Insurance, Probability and Statistics(FIPS 2018)
King’s College London, 10-11 September 2018
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IntroductionAim of the Project
ä Modelling multivariate time series of counts.
ä Count data: Non-negative and integer-valued, and often over-dispersed(i.e. variance > mean).
ä Various applications: Medical science, epidemiology, meteorology,network modelling, actuarial science, econometrics and finance.
Aim of the project
Develop continuous-time models for time series of counts that
à allows for a flexible autocorrelation structure;
à can deal with a variety of marginal distributions;
à allows for flexibility when modelling cross-correlations;
à is analytically tractable.
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IntroductionAim of the Project
ä Modelling multivariate time series of counts.
ä Count data: Non-negative and integer-valued, and often over-dispersed(i.e. variance > mean).
ä Various applications: Medical science, epidemiology, meteorology,network modelling, actuarial science, econometrics and finance.
Aim of the project
Develop continuous-time models for time series of counts that
à allows for a flexible autocorrelation structure;
à can deal with a variety of marginal distributions;
à allows for flexibility when modelling cross-correlations;
à is analytically tractable.
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IntroductionShort and Incomplete Review of the Literature
ä Overall, two predominant modelling approaches:
à Discrete autoregressive moving-average (DARMA) models introduced byJacobs & Lewis (1978a,b).
The advantage of such stationary processes is that their marginaldistribution can be of any kind. However, this comes at the cost that thedependence structure is generated by potentially long runs of constantvalues, which results in sample paths which are rather unrealistic in manyapplications (see McKenzie (2003)).
à Models obtained from thinning operations going back to the influential workof Steutel & van Harn (1979), e.g. INARMA. See also Zhu & Joe (2003) forrelated more recent work.
ä Key idea of this paper: Use trawling for modelling counts! – Nested withinthe framework of ambit fields (Barndorff-Nielsen & Schmiegel (2007)) andextends results by Barndorff-Nielsen, Pollard & Shephard (2012) andBarndorff-Nielsen, Lunde, Shephard & Veraart (2014).
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IntroductionShort and Incomplete Review of the Literature
ä Overall, two predominant modelling approaches:
à Discrete autoregressive moving-average (DARMA) models introduced byJacobs & Lewis (1978a,b).
The advantage of such stationary processes is that their marginaldistribution can be of any kind. However, this comes at the cost that thedependence structure is generated by potentially long runs of constantvalues, which results in sample paths which are rather unrealistic in manyapplications (see McKenzie (2003)).
à Models obtained from thinning operations going back to the influential workof Steutel & van Harn (1979), e.g. INARMA. See also Zhu & Joe (2003) forrelated more recent work.
ä Key idea of this paper: Use trawling for modelling counts! – Nested withinthe framework of ambit fields (Barndorff-Nielsen & Schmiegel (2007)) andextends results by Barndorff-Nielsen, Pollard & Shephard (2012) andBarndorff-Nielsen, Lunde, Shephard & Veraart (2014).
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IntroductionWhat is trawling...? A first ”definition”
“Trawling is a method of fishing that involves pulling a fishing net through thewater behind one or more boats. The net that is used for trawling is called atrawl.” (Wikipedia)
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Theoretical frameworkInteger-valued, homogeneous Levy bases
ä Let N be a homogeneous Poisson random measure on Rn ×R2 withcompensator
E(N(dy,dx ,dt)) = ν(dy)dxdt ,
where ν is a Levy measure satisfying∫ ∞−∞ min(1, ||y||)ν(dy) < ∞.
ä Assume that N is positive integer-valued, i.e. ν is concentrated on N.
ä Then we define an Nn-valued, homogeneous Levy basis on R2 in termsof the Poisson random measure as
L(dx ,dt) = (L(1)(dx ,ds), . . . ,L(n)(dx ,ds))′ =∫ ∞
−∞yN(dy,dx ,dt). (1)
ä The Levy basis L is infinitely divisible with cumulant function
CL(dx ,dt)(θ)..= log(E(exp(iθL(dx ,dt))) =
∫R
(eiθy − 1
)ν(dy)dxdt
= CL′(θ)dxdt , where L′ is the Levy seed.
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Theoretical frameworkInteger-valued, homogeneous Levy bases: The cross-correlation
ä From Feller (1968), we know that any non-degenerate distribution on Nn
is infinitely divisible if and only if it can be expressed as a discretecompound Poisson distribution.
ä A random vector with infinitely divisible distribution on Nn always hasnon-negatively correlated components.
ä We model the Levy seed by an n-dimensional compound Poissonprocess given by
L′t =Nt
∑j=1
Zj ,
where N = (Nt )t≥0 is an homogeneous Poisson process of rate v > 0and the (Zj )j∈N form a sequence of i.i.d. random variables independentof N and which have no atom in 0, i.e. not all components aresimultaneously equal to zero, more precisely, P(Zj = 0) = 0 for all j .
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Theoretical frameworkDefinition of a Trawl
Definition 1A trawl for the i th component is a Borel set A(i) ⊂ R× (−∞,0] such thatLeb(A(i)) < ∞. Then, we set
A(i)t = A(i) + (0, t), i ∈ {1, . . . ,n}.
ä Typically, we choose A(i) to be of the form
A(i) = {(x , s) : s ≤ 0, 0 ≤ x ≤ d (i)(s)}, (2)
where d (i) : (−∞,0] 7→ R is a cont. and Leb(A(i)) < ∞.
ä Then A(i)t = A(i) + (0, t) = {(x , s) : s ≤ t , 0 ≤ x ≤ d (i)(s− t)}.
ä If d (i) is also monotonically non-decreasing, then A(i) is a monotonictrawl.
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Theoretical frameworkDefinition of a Trawl
Definition 1A trawl for the i th component is a Borel set A(i) ⊂ R× (−∞,0] such thatLeb(A(i)) < ∞. Then, we set
A(i)t = A(i) + (0, t), i ∈ {1, . . . ,n}.
ä Typically, we choose A(i) to be of the form
A(i) = {(x , s) : s ≤ 0, 0 ≤ x ≤ d (i)(s)}, (2)
where d (i) : (−∞,0] 7→ R is a cont. and Leb(A(i)) < ∞.
ä Then A(i)t = A(i) + (0, t) = {(x , s) : s ≤ t , 0 ≤ x ≤ d (i)(s− t)}.
ä If d (i) is also monotonically non-decreasing, then A(i) is a monotonictrawl.
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Theoretical frameworkDefinition of a Trawl Process
Definition 2We define an n-dimensional stationary integer-valued trawl (IVT) process(Yt )t≥0 by Yt = (L(1)(A(1)
t ), . . . ,L(n)(A(n)t ))′,
à whereL(i)(A(i)
t ) =∫
R×RIA(i) (x , s− t)L(i)(dx ,ds), i ∈ {1, . . . ,n}.
à L is the n-dimensional integer-valued, homogeneous Levy basis on R2 (see (1)).
à A(i)t = A(i) + (0, t) with A(i) ⊂ R× (−∞,0] and Leb(A(i)) < ∞ is the trawl.
ä Wolpert & Taqqu (2005) study a subclass of general (univariate) trawl processes(not necessarily restricted to IV) under the name “up-stairs” representation,“random measure of a moving geometric figure in a higher-dimensional space”
ä Wolpert & Brown (2011) study so-called “random measure processes” which alsofall into the (univariate) trawling framework.
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Theoretical frameworkDefinition of a Trawl Process
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Theoretical frameworkDefinition of a Trawl Process
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Theoretical frameworkDefinition of a Trawl Process
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ExamplesNegative Binomial exponential-trawl process
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ExamplesNegative Binomial exponential-trawl process
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ExamplesNegative Binomial exponential-trawl process
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Some key properties of IVT processesCumulants
ä The IVT process is stationary and infinitely divisible.ä The IVT process is mixing⇒ weakly mixing⇒ ergodic.ä The cumulant function of a trawl process is given by
CY (i)
t(θ) = C
L(i)(A(i)t(θ) = Leb(A(i))CL′(i)(θ),
à I.e. to any infinitely divisible integer-valued law π, say, there exists astationary integer-valued trawl process having π as its one-dimensionalmarginal law.
ä The covariance between two (possibly shifted) components 1 ≤ i ≤ j ≤ nfor t ,h ≥ 0 is given by
Cov(
L(i)(A(i)t ),L(j)(A(j)
t+h))= Leb
(A(i) ∩ A(j)
h
)(∫R
∫R
yiyj ν(i,j)(dyi ,dyj )
),
Cor (L(i)(A(i)t ),L(i)(A(i)
t+h)) =Leb(A(i) ∩ A(i)
h )
Leb(A(i)).
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Multivariate law of the Levy seedPoisson mixtures
ä The law of L′ is of discrete compound Poisson type by construction.
ä Use Poisson mixtures based on random additive effect models, seeBarndorff-Nielsen et al. (1992).
ä Consider random variables X1, . . . ,Xn and Z1, . . . ,Zn, such that,conditionally on {Z1, . . . ,Zn}, the X1, . . . ,Xn are independent andPoisson distributed with means given by the {Z1, . . . ,Zn}.
ä Model the joint distribution of the {Z1, . . . ,Zn} by a so-called additiveeffect model as follows:
Zi = αiU + Vi , i = 1, . . . ,n,
where the random variables U,V1, . . . ,Vn are independent and theα1, . . . , αn are nonnegative parameters.
ä We have explicit formulas for the joint law of (X1, . . . ,Xn) and
E(Xi ) = αi E(U) + E(Vi ), i = 1, . . . ,n,Cov(Xi ,Xj ) = αi αj Var(U), if i 6= j .
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Theoretical resultsRepresentation as compound Poisson distribution and as a multivariatenegative binomial distribution
ä The Poisson mixture model of random-additive-effect type can berepresented as a compound Poisson distribution
ä If U and Vis follow suitable Gamma distributions, then negative binomialmarginal law can be achieved allowing for
à 1) independence,
à 2) complete dependence, or
à 3) dependence with additional independent factors between thecomponents.
ä Recall that the (multivariate) negative binomial distribution can berepresented as a compound Poisson distribution with (multivariate)logarithmic jump size distribution.
ä The representation result is used in the simulation algorithm for themultivariate trawl process.
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Key properties of IVT processesOverview
Flexible marginal distributions:
ä Poisson trawl process;
ä Negative binomial trawl process;
ä other compound Poisson distributions.
Various choices of the trawl function:
ä Superpositions of exponential trawls: d (i)(z) =∫ ∞
0 eλz π(i)(dλ), for z ≤ 0, for aprobability measure π(i) on (0,∞).
ä Possibility of allowing for long memory.
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Key properties of IVT processesOverview
Flexible marginal distributions:
ä Poisson trawl process;
ä Negative binomial trawl process;
ä other compound Poisson distributions.
Various choices of the trawl function:
ä Superpositions of exponential trawls: d (i)(z) =∫ ∞
0 eλz π(i)(dλ), for z ≤ 0, for aprobability measure π(i) on (0,∞).
ä Possibility of allowing for long memory.
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Inference(Generalised) Method of Moments
ä Use a (generalised) method of moments in a two-stageequation-by-equation approach to estimate the marginal parameters first,followed by the dependence parameters.
ä Step 1a) Use the acf r (i)(h) = Leb(A(i)∩A(i)h )
Leb(A(i))to infer the trawl parameters.
ä Step 1b) Use the cumulant function CY (i)
t(θ) = Leb(A(i))CL′(i)(θ) to infer
the marginal parameters of the Levy basis.
ä Step 2a) Compute Leb(A(i) ∩ A(j)) for i 6= j .
ä Step 2b) Use the cross-covariance function
Cov(
L(i)(A(i)t ),L(j)(A(j)
t ))= Leb
(A(i) ∩ A(j)
)(∫R
∫R
yiyj ν(i,j)(dyi ,dyj )
)to infer the dependence parameters.
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Empirical illustrationHigh frequency financial data from LOBSTER
ä Study high frequency limit order book data from LOBSTER.
ä We picked the Apple data for August 8, 2017: Start at 11:00am, end at12:00 (noon).
ä We analyse the joint behaviour of the number of newly submitted andfully deleted limit orders over 5s intervals (720 observations in total).
ä We fit a bivariate trawl model with double exponential trawl function andbivariate negative binomial law.
ä Summary statistic:
Min 1st Quartile Median Mean 3rd Quartile MaxNo. of submissions 0.00 19.00 44.00 53.46 74.25 290.00No. of cancellations 0.00 22.00 38.00 46.77 63.00 243.00
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Number of limit order submissions and deletionsTime series, acf and crosscorrelation
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Number of limit order submissions and deletionsHistograms: submissions (right), full cancellations (top)
TS
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Empirical illustrationFitted trawl (sum of two exponentials) to number of submissions and deletions
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Empirical illustrationFitted bivariate negative binomial marginal fit (to number of submissions anddeletions)
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Empirical illustrationFitted bivariate negative binomial: Bivariate histogram assessment
Cancellations
Sub
mis
sion
s
[0,5) [20,25) [40,45) [60,65) [80,85) [105,110) [135,140) [165,170) [195,200) [225,230) [255,260) [285,290)
[0,5
)[3
0,35
)[6
5,70
)[1
05,1
10)
[150
,155
)[1
95,2
00)
[240
,245
)[2
85,2
90)
diff
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Main contributions
ä New continuous-time framework for modelling multivariate stationary,serially correlated count data.
ä Two key components:à Integer-valued, homogeneous Levy basis: Generates random point pattern
and determines marginal distribution and cross-sectional dependence.à Trawl: Thins the point pattern and determines the autocorrelation structure.
ä Simulation algorithm & parameter estimation of IVT processes withmonotonic trawl.
ä Simulation studies reveal good performance of (generalised) method ofmoments and quasi-maximum-likelihood methods for parameterestimation.
ä Empirical application to high frequency financial data.
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You want to know more...?!
ä Preprint: Available on my website (article forthcoming in the Journal ofMultivariate Analysis).
ä New R package trawl available on CRAN.
ä Many more details in our new book:
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Bibliography I
Barndorff-Nielsen, O. E., Blæsild, P., & Seshadri, V. (1992). Multivariatedistributions with generalized inverse Gaussian marginals, and associatedPoisson mixtures. The Canadian Journal of Statistics. La RevueCanadienne de Statistique, 20(2), 109–120.
Barndorff-Nielsen, O. E., Lunde, A., Shephard, N., & Veraart, A. E. D. (2014).Integer-valued trawl processes: A class of stationary infinitely divisibleprocesses. Scandinavian Journal of Statistics, 41, 693–724.
Barndorff-Nielsen, O. E., Pollard, D. G., & Shephard, N. (2012).Integer-valued Levy processes and low latency financial econometrics.Quantitative Finance, 12(4), 587–605.
Barndorff-Nielsen, O. E. & Schmiegel, J. (2007). Ambit processes: withapplications to turbulence and cancer growth. In F. Benth, G. Di Nunno, T.Lindstrøm, B. Øksendal, & T. Zhang (Eds.), Stochastic Analysis andApplications: The Abel Symposium 2005 (pp. 93–124). Heidelberg:Springer.
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Bibliography II
Jacobs, P. A. & Lewis, P. A. W. (1978a). Discrete time series generated bymixtures. I. Correlational and runs properties. Journal of the RoyalStatistical Society. Series B. Methodological, 40(1), 94–105.
Jacobs, P. A. & Lewis, P. A. W. (1978b). Discrete time series generated bymixtures. II. Asymptotic properties. Journal of the Royal Statistical Society.Series B. Methodological, 40(2), 222–228.
McKenzie, E. (2003). Discrete variate time series. In Stochastic processes:modelling and simulation, volume 21 of Handbook of Statistics (pp.573–606). Amsterdam: North-Holland.
Steutel, F. W. & van Harn, K. (1979). Discrete analogues ofself-decomposability and stability. The Annals of Probability, 7(5), 893–899.
Veraart, A. E. D. (2015). Modelling multivariate serially correlated count datain continuous time. In preparation.
Wolpert, R. L. & Brown, L. D. (2011). Stationary infinitely-divisible Markovprocesses with non-negative integer values. Working paper, April 2011.
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Bibliography III
Wolpert, R. L. & Taqqu, M. S. (2005). Fractional Ornstein–Uhlenbeck Levyprocesses and the Telecom process: Upstairs and downstairs. SignalProcessing, 85, 1523–1545.
Zhu, R. & Joe, H. (2003). A new type of discrete self-decomposability and itsapplication to continuous-time Markov processes for modeling count datatime series. Stochastic Models, 19(2), 235–254.
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