dynamic negative binomial di erence model for high...

Dynamic Negative Binomial Di�erence Model

for High Frequency Returns ∗

PRELIMINARY AND INCOMPLETE

István Barra(a,b) and Siem Jan Koopman (a,c)

(a) VU University Amsterdam and Tinbergen Institute

(b) Duisenberg School of Finance

(c) CREATES, Aarhus University

January 19, 2015

Abstract

We introduce the dynamic ∆NB model for �nancial tick by tick data. Our model explicitly

takes into account the discreteness of the observed prices, fat tails of tick returns and intraday

patterns of volatility. We propose a Markov chain Monte Carlo estimation method, which

takes advantage of an auxiliary mixture representation of the ∆NB distribution. We illustrate

our methodology using tick by tick data of several stocks from the NYSE in di�erent periods.

Using predictive likelihoods we �nd evidence in favour of the dynamic ∆NB model.

Keywords: high-frequency econometrics, Bayesian inference, Markov-Chain Monte Carlo,

discrete distributions

1 Introduction

Stock prices are not continuous variables, as they are multiple of the so called ticksize, which

is the smallest possible price di�erence. For example, on US exchanges the tick size is set to

not smaller than 0.01$ for stock with a price greater than 1$ by the Security and Exchange

Commission in Rule 612 of the Regulation National Market System. This has a serious impact

on the distribution of the trade by trade log returns, resulting in multi modality and discontinuity

in the distribution. Alzaid and Omair (2010) and Barndor�-Nielsen et al. (2012) suggest to model

tick returns (price di�erences expressed in number of ticks) with integer valued distributions as

the distribution of these tick returns is more tractable.

In this paper we propose a model in which tick returns have ∆Negative Binomial (∆NB)

distribution condition on a Gaussian latent state. Our model provides a �exible framework to �t

∗Author information: István Barra, Email: [email protected]. Address: VU University Amsterdam, De Boelelaan1105, 1081 HV Amsterdam, The Netherlands. István Barra thanks for the Dutch National Science Foundation(NWO) for �nancial support.

1

the empirically observed fat tails of the tick returns and other stylized facts of trade by trade

returns, hence it is an attractive alternative of previously suggested models in the literature

(see Koopman et al. (2014)). Moreover our structural model allows us to decompose the log

volatility into a periodic and a transient volatility component. We propose a Bayesian estimation

procedure using Gibbs sampling. Our procedure is based on data augmentation and auxiliary

mixtures and it extends the auxiliary mixture sampling proposed by Frühwirth-Schnatter and

Wagner (2006) and Frühwirth-Schnatter et al. (2009). In the empirical part of our paper we

illustrate our methodology on six stocks from the NYSE in a volatile week of October 2008 and

a calmer week from April 2010. We compare the in-sample and out-of-sample �t of the dynamic

Skellam and dynamic ∆NB model using Bayesian information criteria and predictive likelihoods.

We �nd that the ∆NB model is favoured for stock with low relative tick size and in more volatile

periods.

Our paper is related to several strands of literature. Modelling discrete price changes with

Skellam and ∆NB distributions was introduced by Alzaid and Omair (2010) and Barndor�-

Nielsen et al. (2012), while the dynamic Skellam model was introduced by Koopman et al. (2014).

Our paper is related to stochastic volatility models see for example Chib et al. (2002) Kim et al.

(1998) Omori et al. (2007) and recently Stroud and Johannes (2014). We extend the literature on

trade by trade returns by explicitly account for prices discreteness and fat tails of the tick return

distribution (see Engle (2000), Czado and Haug (2010), Dahlhaus and Neddermeyer (2014) and

Rydberg and Shephard (2003)).

The rest of the paper is organized as follows. I Section 2 we discuss the issue with trade-

by-trade log return and we describe the Skellam and ∆NB distributions. Section 3 introduces

the dynamic ∆NB distribution while Section 4 explains our Bayesian estimation procedure. In

Section 5 we describe our dataset and cleaning procedure and Section 6 presents the empirical

�ndings.

2 Tick returns and integer valued distributions

Stock prices can be quoted as a multiple of the tick size. As a consequence prices are de�ned on

a discrete grid, where the grid points are a tick size distance away from each other. We can write

the prices at time tj as

p(tj) = n(tj)g (1)

where g is the tick size which can be the function of the price on some exchanges and n(tj) is a

natural number, denoting the location of the price on the grid. Modelling trade by trade returns

can pose di�culty as the e�ect of price discreteness on a few seconds frequency is pronounced

compared to lower frequencies such as one hour or one day. As described in Münnix et al.

(2010), the problem is that the return distribution is a mixture of return distributions ri, which

correspond to �x price changes ig

ri =

{p(tj)− p(tj−1)

p(tj−1)

∣∣∣∣ p(tj)− p(tj−1) =[n(tj)− n(tj−1)

]g = i(tj)g = ig

}(2)

2

where i(tj) is an integer, which express the price change in terms of ticks. The ri distributions

are on the intervals (for positive i) [ig

max pi,

ig

min pi

], (3)

where

pi =

{p(tj−1)

∣∣∣∣ p(tj)− p(tj−1) =[n(tj)− n(tj−1)

]g = i(tj)g = ig

}(4)

These interval and the center of the intervals ci can be approximated by[ig

p,ig

¯p

], (5)

and

ci ≈ig

2

(1

¯p− 1

p

)(6)

as max pi ≈ p and min pi ≈¯p for i close to 0.

First, note that the intervals corresponding to zero price change and one tick changes are

always non-overlapping. Secondly, the center of the intervals are approximately equally spaced,

however the intervals for higher absolute value changes are wider, which means that the intervals

are getting more and more overlapping as |i| is increasing. Thirdly, the intervals are less overlap-ping when the price is lower, the volatility is higher or the tick size is bigger. Figure 1 shows the

empirical trade by trade return distribution of several stocks from the New York Stock Exchange

(NYSE).

[ insert Figure 1 here ]

Modelling this special feature of the trade by trade return distribution is di�cult and often

neglected in the literature (see e.g., Czado and Haug (2010) and Banulescu et al. (2013)). We

use an alternative modelling framework. Following Alzaid and Omair (2010), Barndor�-Nielsen

et al. (2012), and Koopman et al. (2014) we can de�ne tick returns as

r(tj) =p(tj)− p(tj−1)

g= n(tj)− n(tj−1) = i(tj) (7)

which is obviously an integer. The advantage of this approach is that we can directly model the

price changes expressed in terms of ticks. Although the distribution of tick returns is integer

valued, it is still easier to model this distribution than the speci�c features of the log returns.

On issue with trick returns is that they are not properly scaled, hence we expect higher variance

at higher prices, however on shorter time intervals this e�ect should be small, moreover we can

count for this our model by introducing a time varying unconditional mean in the volatility

equation. Figure 2 show the empirical distribution of �ve stock from the NYSE along with �tted

Skellam densities.


3

In order to model integer returns we need a discrete distribution de�ned on integers. One of

the potential distributions is the Skellam distribution, which was suggested by Alzaid and Omair

(2010). The Skellam distribution is de�ned as the di�erence of two Poisson distributed random

variables. If P+ and P− are Poisson random variables with intensity λ+ and λ−, then

R = P+ − P−, R ∼ Skellam(λ+, λ−) (8)

with probability mass function given by

f(r;λ+, λ−) = exp(−λ+ − λ−)

(λ+

λ−

) r2

I|r|(2√λ+λ−) (9)

where Ir is the modi�ed Bessel function of the �rst kind. The Skellam distirbution has the

following �rst and second moments

E(R) = λ+ − λ− (10)

Var(R) = λ+ + λ− (11)

An important special case is the zero mean Skellam distribution is when λ = λ+ = λ−. In this

case

E(R) = 0 (12)

Var(R) = 2λ (13)


One issue with the Skellam distribution is that it has exponentially decaying tails and it

implies approximately normally distributed tick returns as the tick size goes to zero. Keeping

the variance of the tick returns �xed at σ2

limg→0

p(tj)− p(tj−1)

g= lim

g→0g

[n(tj)

g− n(tj−1)

g

]≈ g

[n∗(tj)− n∗(tj−1)

]= N(0, σ2), (14)

where

n(tj)d=n(tj−1) ∼ Poi

(σ2

2

)and n∗(tj)

d=n∗(tj−1) ∼ N

(σ2

2g,σ2

2g

),

moreover we used the fact that a Poisson distribution with intensity λ can be approximated with

a normal distribution with mean and variance equal to λ. The thin tailed trade by trade return

assumption might be implausible as we know there is some evidence for jumps in stock prices.


The ∆ NB distribution is an alternative integer valued distribution which was proposed by

Barndor�-Nielsen et al. (2012). ∆ NB distribution is de�ned as the di�erence of two negative

binomial random variables NB+ and NB− with number of failures λ+ and λ− and failure rates

ν+ and ν−

R = NB+ −NB−. (15)

4

Then R is distributed as

R ∼ ∆NB(λ+, ν+, λ−, ν−) (16)

with probability mass function given by

f∆NB(r;λ+, ν+, λ−, ν−) =

(ν+)ν+ (

ν−)ν− (

λ+)r

(ν+)rr! F

(ν+ + r, ν−, r + 1; λ+λ−

)if r ≥ 0(

ν+)ν+ (

ν−)ν− (

λ−)r

(ν−)rr! F

(ν+, ν− − r,−r + 1; λ+λ−

)if r < 0

where

ν+ =ν+

λ+ + ν+

ν− =ν−

λ− + ν−

λ+ =λ+

λ+ + ν+

λ− =λ−

λ− + ν−(17)

and

F (α, β; γ; z) =

∞∑n=0

(α)n(β)n(γ)n

zn

n!(18)

is the hypergeometric function with (x)n denoting the Pochhammer symbol of falling factorial

(x)n = x(x− 1)(x− 2) · · · (x− n+ 1) =Γ(x+ 1)

Γ(x− n+ 1). (19)

The ∆NB distribution has the following �rst and second moments

E(R) = λ+ − λ− (20)

Var(R) = λ+

(1 +

λ+

ν+

)+ λ−

(1 +

λ−

ν−

)(21)

An important special case is the zero mean ∆ NB distribution is when λ = λ+ = λ− and

ν = ν+ = ν−.

f(r;λ, ν) =

(ν

λ+ ν

)2ν ( λ

λ+ ν

)|r| Γ(ν + |r|)Γ(ν)Γ(|r|+ 1)

F

(ν + |r|, ν, |r|+ 1;

(λ

λ+ ν

)2)

(22)

In this case

E(R) = 0 (23)

Var(R) = 2λ

(1 +

λ

ν

)(24)

We can think about the zero mean ∆NB(λ, ν) distribution as the realization of a compound

5

Poisson process

R =

N∑i=1

Mi, (25)

where N is a Poisson distribution with intensity

λ(z1 + z2), z1, z2 ∼ Ga(ν, ν) (26)

and

Mi =

1, with P (Mi = 1) = z1z1+z2

−1, with P (Mi = −1) = z2z1+z2

(27)

This representation will be useful later on.

3 Dynamic ∆NB model

In order to build a sensible model of trade by trade tick returns we have to account for several

stylized facts. First of all, we model the tick returns with an integer valued distribution. We

propose to use the ∆NB distribution to account for the potential fat tails of the distribution. In

addition we use the zero in�ated version of the ∆NB distribution, because there a huge chunk

of the trade by trade returns are zeros. The number of zero trade by trade returns are higher for

more liquid stock as the available volumes on best bid and ask prices are higher in consequence the

price impact of one trade is lower. Taking the above considerations into account, our observation

density can be written as

yt =

rt with (1− γ)f∆NB(rt;λt, ν)

0 with γ + (1− γ)f∆NB(0;λt, ν), (28)

where γ is the zero in�ation parameter, λt is the volatility parameter at time t and ν is the

degree of freedom parameter of the ∆NB distribution, which determines the thickness of the tail

of the distribution.

Besides explicitly accounting for the discreteness of prices in our model, we also model the

daily volatility pattern and volatility clustering. We introduce these features into our model by

specifying the logarithm of the volatility as

log λt = µθ + st + xt (29)

where µθ is the unconditional mean of the log intensity, st is a spline which is standardized

such that it has mean zero and xt is a zero mean AR(1) process. This speci�cation implies a

decomposition of the volatility into a deterministic daily pattern and a stochastic time varying

component. The daily pattern of volatility usually associated to frequent trading during the

beginning of the day and lower activity during lunch. The xt process captures changes in volatility

due to new �rm speci�c or market information experienced during the day. We model the daily

patter with a periodic spline which has a daily periodicity. (See e.g., Bos (2008), Stroud and

6

Johannes (2014) andWeinberg et al. (2007)) The spline function is a continuous function built

up from piecewise polynomials. Using the results of Poirier (1973) we can write a cubic spline st

with K knots as a regression

st = wtβ (30)

where wt is a 1×K vector and β is K × 1 vector. Details about the spline are in Appendix C.

The latent state in our model is xt an AR(1) process, which accounts for the variation in

volatility on top of the daily variation. Because of identi�cations reasons we restrict the AR(1)

process to have zero mean, which yields to the following transition density

xt = φxt−1 + ηt, ηt ∼ N(0, σ2η), (31)

where φ is the persistence parameter and σ2η is the variance of the error term.

The full model would be

Tick return : yt =

rt with (1− γ)f∆NB(rt;λt, ν)

0 with γ + (1− γ)f∆NB(0;λt, ν)

Total log volatility : log λt = µθ + st + xt

Daily volatility pattern : st = wtβ

Transient volatility : xt = φxt−1 + σηηt, ηt ∼ N(0, σ2η)

4 Estimation

Our proposed estimation procedure relies on data augmentation and auxiliary mixture sampling

of Frühwirth-Schnatter and Wagner (2006) and Frühwirth-Schnatter et al. (2009). First for each

observation yt we introduce the variable Nt which is equal to the sum of NB+ and NB−.

Condition on the gamma mixing variables z1t and z2t and the intensity λt we can think about

Nt as a realization of a Poisson process on [0, 1] with intensity (z1t+ z2t)λt for every t = 1, . . . , T

and we can introduce the latent arrival time of the Nt-th jump of the Poisson process τt2 and the

arrival time between the Nt-th and Nt + 1-th jump of the process τt1. Obviously the interarrival

time τt1 has exponential distribution with intensity (z1t + z2t)λt while the Ntth arrival time has

a Ga(Nt, (z1t + z2t)λt) distribution, hence we can write

τt1 =ξt1

(z1t + z2t)λt, ξt1 ∼ Exp(1) (32)

τt2 =ξt2

(z1t + z2t)λt, ξt2 ∼ Ga(Nt, 1). (33)

By taking the logarithm of the equations we can rewrite them as

− log τt1 = log(z1t + z2t) + log λt + ξ∗t1, ξ∗t1 = − log ξt1 (34)

− log τt2 = log(z1t + z2t) + log λt + ξ∗t2, ξ∗t2 = − log ξt2. (35)

These equations are linear in the state, which would facilitate the use of Kalman �ltering,

7

however the error terms ξ∗t1 and ξ∗t2 are non normal. We can use the result of Frühwirth-Schnatter

and Wagner (2006) and Frühwirth-Schnatter et al. (2009) to come up with a normal mixture

approximation of these distributions

fξ∗(x;Nt) ≈R(Nt)∑r=1

ωr(Nt)φ(x,mr(Nt), vr(Nt)

). (36)

Using the mixture of normal approximation of ξ∗t1 and ξ∗t2, allows us to build an e�cient Gibbs

sampling procedure where we can sample the latent state paths in one block, e�ciently using

Kalman �ltering and smoothing techniques. This is crucial as in our application the number of

observation is large and updating the state time period by time period would make our estimation

slow and ine�cient.

The MCMC algorithm

1. Initialize µθ, φ, σ2η, γ, ν, R , τ , N , z1, z2, s and x

2. Generate φ,σ2η, µθ, s and x from p(φ, σ2

η, µθ, s, x|γ, ν,R, τ,N, z1, z2, s, y)

(a) Draw φ, σ2η from p(φ, σ2

η|γ, ν,R, τ,N, z1, z2, s, y)

(b) Draw µθ, s and x from p(µθ, s, x|φ, σ2η, γ, ν, R, τ,N, z1, z2, s, y)

3. Generate γ from p(γ|ν, µθ, φ, σ2η, x,R, τ,N, z1, z2, s, y)

4. Generate R, τ,N, z1, z2, ν from p(R, τ,N, z1, z2, ν|γ, µθ, φ, σ2η, x, s, y)

(a) Draw ν from p(ν|γ, µθ, φ, σ2η, x, s, y)

(b) Draw z1, z2 from p(z1, z2|ν, γ, µθ, φ, σ2η, x, s, y)

(c) Draw N from p(N |z1, z2, ν, γ, µθ, φ, σ2η, x, s, y)

(d) Draw τ from p(τ |N, z1, z2, ν, γ, µθ, φ, σ2η, x, s, y)

(e) Draw R from p(R|τ,N, z1, z2, ν, γ, µθ, φ, σ2η, x, s, y)

5. Go to 2

The detailed MCMC steps can be found in Appendix D.

4.1 Simulation exercise

To check our estimation procedure for the dynamic Skellam and ∆ NB models we simulate 20

000 observation and run 100 000 iterations of our MCMC procedure. We set µ = −1.7, φ = 0.97

, σ = 0.02, γ = 0.001 and ν = 15 which sensible parameters based on the estimates on real data.

Table 1 summarizes the results.

We �nd that the algorithm successfully estimate the parameters as the true parameters are

in the HPD regions. Based on the simulations the AR(1) coe�cient and volatility in the state

are the hardest parameters to estimate.

[ insert Table 1 here ]

8



5 Data

We have quotes and trades data from the Thomson Reuters Sirca dataset. In this data set we have

all the trades and quotes with millisecond time stamps for stocks from NYSE In our analysis we

use Alcoa (AA), Coca-Cola (KO) International Business Machines (IBM), J.P. Morgan (JPM),

Ford (F), Xerox (XRX), which di�er in liquidity and their price magnitude. In the paper we

concentrate on two months, namely October 2008 and April 2010. These months exhibit di�erent

market sentiments and volatility characteristics, as October 2008 is in the middle of the 2008

�nancial crises with record high realized volatility and some of the markets experienced their

worst weeks in October 2008 since 1929, while April 2010 is a calmer month with lower realized

volatility. We carry out the following �ltering steps to clean the data following a similar procedure

to described in Boudt et al. (2012), Barndor�-Nielsen et al. (2008) and Brownlees and Gallo

(2006).

First we �lter the trades from the the trades and quotes data set by selecting rows where

the 'Type' column equals 'Trade'. Large portion of the data is in fact consists of quotes, hence

by excluding the quotes we loose around 70-90 % of the data set. In the next step we we delete

the trades with missing or zero prices or volumes. Further more we restrict our analysis to the

trading period. The fourth step is to aggregate the trades which have the same time stamps. We

decided to use the trades with last sequence number when there are multiple trades at the same

millisecond. This choice is motivated by the fact that, that is the price which we can observe

with a millisecond �ne resolution. Finally we �lter the outliers using the the rule suggested by

Barndor�-Nielsen et al. (2008). We delete trades with price smaller then the bid minus the bid

ask spread and higher than the ask plus the bid ask spread. Table 2 and Table 3 summarizes

some descriptive statistics for the data from 3rd to 10th October 2008 and from 23rd to 30th

April 2010 respectively. Detailed summary of the cleaning can be found in Table 6 and 7.



6 Empirical results

We estimate the dynamic Skellam and ∆NB models for two di�erent stocks Alcoa (AA), Coca-

Cola (KO), International Business Machines (IBM), J.P. Morgan (JPM), Ford (F), Xerox (XRX),

in the period of 3rd to 9th October 2008 and 23rd to 29th April 2010. The results on the data

from 2008 are reported in Table 4 while Table 5 shows the parameter estimates on the data from

April 2010.

The unconditional mean volatility di�er across stocks and time periods. The unconditional

mean volatility is higher for stocks with higher price and it is higher in the more volatile period

9

in 2008 which is in line with our intuition. The AR(1) coe�cients are in the range of 0.88-0.99.

This suggest persistence in the volatility even after accounting for the daily volatility pattern,

however the transient volatility is less persistent in the more volatile crises period. The volatility

parameter of the AR(1) process is higher during the 2008 �nancial crisis period. We only used the

zero in�ation parameter when some additional �exibility was needed in the observation density.

This was the case in for stocks with higher price and the more volatile periods. In case of the

April 2010 period we used the zero in�ation only for IBM, while in the October 2008 period we

included the zero in�ation for all stock expect for the two lowest price stock F and XRX. The

tail parameter of the ∆NB distribution is higher during the calmer 2010 period which suggest

that the distribution of the tick return is closer to a thin tailed distribution in that period. In

addition the tail parameter is lower for stock with higher average price.



Using the output of our estimation procedure we can decompose the logarithm of the volatility

by

E(log λt) = E(µθ) + E(st) + E(xt) (37)

Figure 7 depicts the decomposition of the logarithm of the volatility from the Skellam model

estimated on IBM tick returns from 23rd to 29th April 2010.


6.1 In-sample comparison

The computation of Bayes factors is infeasible in this setup as it requires sequential parameter

estimation, which is computationally prohibitive with large time dimension of our model. Instead

we follow Stroud and Johannes (2014) and we calculate Bayesian Information Criteria (BIC)

BICT (M) = −2

T∑t=1

log p(yt|θ,M) + di log T (38)

where p(yt|θ,M) can be calculated with a particle �lter and θ is the posterior mean of the

parameters. The BIC gives an asymptotic approximation to the Bayes factor by BICT (Mi) −BICT (Mj) ≈ −2 logBFi,j . This approximation can be used for sequential model comparison.

Figure 8 and Figure 10 shows the in Bayes factors on the sample from 23rd to 29th October

2008 and 3rd to 9th April 2010. The pictures indicate that in 2008 there is evidence in favour of

the ∆NB model in case of AA, F and XRX, while in 2010 on IBM favours the ∆NB distribution.

This result is consistent with our prior conjecture that in the crisis period there are more jumps.

Based on the sequential Bayes factors, the ∆NB model tends to be favoured in case of sudden

big jumps in the data. Realization of returns from the tail suggest the need of ∆NB only in

cases where the volatility high. This in line with the intuition that in models with time varying

volatility identi�cation of the tails comes from observation of extreme realizations coupled with

low volatility.

10

6.2 Out-of-sample comparison

In order to compare the dynamic Skellam and ∆NB models we can use predictive likelihoods.

The one-step-ahead predictive likelihood for modelM is de�ned as

p(yt+1|y1:t,M) =

∫ ∫p(yt+1|y1:t, xt+1, θ,M)p(xt+1, θ|y1:t,M)dxt+1dθ

=

∫ ∫p(yt+1|y1:t, xt+1, θ,M)p(xt+1|θ, y1:t,M)p(θ|y1:t,M)dxt+1dθ. (39)

Noticed that the h-step-ahead predictive likelihood can be decompose to the sum of one-step-

ahead predictive likelihoods

p(yt+1:t+h|y1:t,M) =

h∏i=1

p(yt+i|y1:t+i−1,M) =

h∏i=1

∫ ∫p(yt+i|y1:t+i−1, xt+i, θ,M)

× p(xt+i|θ, y1:t+i−1,M)p(θ|y1:t+i−1,M)dxt+i.dθ (40)

The above formula suggests that we have to calculate p(θ|y1:t+i−1,m), the posterior of the

parameters using sequentially increasing data samples. This means that we have to run our

MCMC procedure as many times as many out of sample observations we have. Unfortunately in

our application this means several thousands of runs in case we would like to check the predictive

likelihood on an out of sample day, which is computationally not practical or even infeasible.

However we can use the vast amount of available data by using the following approximation

p(yt+1:t+h|y1:t,M) ≈h∏i=1

∫ ∫p(yt+i|y1:t+i−1, xt+i, θ,M)

× p(xt+i|θ, y1:t+i−1,M)p(θ|y1:t,M)dxt+idθ. (41)

This approximation can be motivated by the fact that, after observing a considerable amount

of data, which me means that t is su�ciently large, the posterior distribution of the static

parameters should not change that much, hence p(θ|y1:t+i−1,M) ≈ p(θ|y1:t,M). We carry out

the following exercise. We thin our MCMC output to get a sample from the posterior distribution

based on our in-sample observations. Then for each parameter draw we estimate the likelihood

by running a particle �lter through the out-of-sample period.

Figure 9 and Figure 11 shows the out of sample sequential predictive Bayes factors for 10th

October 2008 and 30th April 2010 respectively. Based on the results we can say that in the more

volatile period the ∆NB model is doing better in term of Bayes factor. On 10th October AA,

KO,XRX show evidence in favour the ∆NB model in the out-of-sample comparison, while on

30th October 2008 the dynamic Skellam model �ts the data better out-of sample, except for

IBM.



11



7 Conclusion

In this paper we introduced the dynamic ∆NB model for modelling trade by trade returns. We

developed a Gibbs type MCMC procedure for the Bayesian estimation of the dynamic Skellam

and ∆NB model. Moreover we showed some empirical evidence in favour of the ∆NB model

using di�erent stock and periods from the NYSE.

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13

A Numerical issues with the Skellam distribution

In general it is good to use the scaled version of the modi�ed Bessel function of the �rst kind

exp(−z)In(z). (A1)

For the special case of the Skellam

exp(−2λ)I|k|(2λ) (A2)

for small and large 2λ this still can be unstable, but we can use the following approximations

for small 2λ

exp(−2λ)I|k|(2λ) ≈ exp(−2λ)1

Γ(|k|+ 1)

(2λ

2

)|k|≈ 1× λ|k|

Γ(|k|+ 1)(A3)

While for large 2λ we can use

exp(−2λ)I|k|(2λ) ≈ exp(−2λ)exp(2λ)√

2π2λ=

1

2√πλ

(A4)

B NB distribution

Di�erent parametrization of the NB distribution

f(k; ν, p) =Γ(ν + k)

Γ(ν)Γ(k + 1)pk(1− p)ν (A5)

Using

λ = νp

1− p⇒ p =

λ

λ+ ν(A6)

f(k;λ, ν) =Γ(ν + k)

Γ(ν)Γ(k + 1)

(λ

ν + λ

)k ( ν

ν + λ

)ν(A7)

Mean

µ = λ (A8)

Variance

σ2 = λ

(1 +

λ

ν

)(A9)

Dispersion indexσ2

µ

(1 +

λ

ν

)(A10)

The NB distribution is over dispersed and which means that there are more intervals with low

counts and more intervals with high counts, compared to a Poisson distribution. As we increase

ν we get back to the Poission case.

14

The Poisson distribution can be obtained from the NB distribution as follows

limν→∞

f(k;λ, ν) =λk

k!limν→∞

Γ(ν + k)

Γ(ν)(ν + λ)k1(

1 + λν

)ν =λk

k!limν→∞

(ν + k − 1) . . . ν

(ν + λ)k1(

1 + λν

)ν=

λk

k!· 1 · 1

eλ= Poi(λ) (A11)

The NB distribution Y ∼ NB(λ, ν) can be written as a Poisson-Gamma mixture or Poisson

distribution with Gamma heterogeneity where the Gamma heterogeneity has mean 1.

Y ∼ Poi(λU) where U ∼ Ga(ν, ν), (A12)

where we use the Ga(α, β) is given by

f(x;α, β) =βαxα−1e−βx

Γ(α)(A13)

f(k;λ, ν) =

∞∫0

fPoisson(k;λu)fGamma(u; ν, ν)du

=

∞∫0

(λu)ke−λu

k!

ννuνe−νu

Γ(ν)du

=λkνν

k!Γ(ν)

∞∫0

e−(λ+ν)uuk+ν−1du

Substituting (λ+ ν)u = s we get

=λkνν

k!Γ(ν)

∞∫0

e−ssk+ν−1

(λ+ ν)k+ν−1

1

(λ+ ν)ds

=λkνν

k!Γ(ν)

1

(λ+ ν)k+ν

∞∫0

e−ssk+ν−1ds

=λkνν

k!Γ(ν)

Γ(k + ν)

(λ+ ν)k+ν

=Γ(ν + k)

Γ(ν)Γ(k + 1)

(λ

ν + λ

)k ( ν

ν + λ

)ν(A14)

C Daily volatility patterns

We want to approximate the function f : R → R with a continuous function which is built up

from piecewise polynomials of degree at most three. Let the set ∆ = {k0, . . . , kK} denote the

set of of knots kj j = 0, . . . ,K. ∆ is some times called a mesh on [k0, kK ]. Let y = {y0, . . . , yK}

15

where yj = f(xj). We denote a cubic spline on ∆ interpolating to y as S∆(x). S∆(x) has to

satisfy

1. S∆(x) ∈ C2 [k0, kK ]

2. S∆(x) coincides with a polynomial of degree at most three on the intervals[kj−1, kj

]for

j = 0, . . . ,K.

3. S∆(x) = yj for j = 0, . . . ,K.

Using the 2 we know that the S′′∆(x) is a linear function on

[kj−1, kj

]which means that we can

write S′′∆(x) as

S′′∆(x) =

[kj − xhj

]Mj−1 +

[x− kj−1

hj

]Mj for x ∈

[kj−1, kj

](A15)

where Mj = S′′∆(kj) and hj = kj − kj−1. Integrating S

′′∆(x) and solving the integrating for the

two integrating constants (using S∆(x) = yj) Poirier (1973) shows that we get

S′∆(x) =

[hj6− (kj − x)2

2hj

]Mj−1 +

[(x− kj−1)2

2hj− hj

6

]Mj +

yj − yj−1

hjfor x ∈

[kj−1, kj

](A16)

S∆(x) =kj − x

6hj

[(kj − x)2 − h2

j

]Mj−1 +

x− kj−1

6hj

[(x− kj−1)2 − h2

j

]Mj

+

[kj − xhj

]yj−1 +

[x− kj−1

hj

]yj for x ∈

[kj−1, kj

](A17)

In the above expression only Mj for j = 0, . . . ,K are unknown. We can use the continuity

restrictions which enforce continuity at the knots by requiring that the derivatives are equal at

the knots kj for j = 1, . . . ,K − 1

S′∆(k−j ) = hjMj−1/6 + hjMj/3 + (yj − yj−1)/hj (A18)

S′∆(k+

j ) = −hj+1Mj/3− hj+1Mj+1/6 + (yj+1 − yj)/hj+1 (A19)

which yields K − 1 conditions

(1− λj)Mj−1 + 2Mj + λjMj+1 =6yj−1

hj(hj + hj+1)− 6yjhjhj+1

+6yj+1

hj+1(hj + hj+1)(A20)

where

λj =hj+1

hj + hj+1(A21)

Using two end conditions we have K + 1 unknowns and K + 1 equations and we can solve the

linear equation system for Mj . Using the M0 = π0M1 and MK = πKMK−1 end conditions we

16

can write

Λ︸︷︷︸(K+1)×(K+1)

=

2 -2 π0 0 . . . 0 0 0

1-λ1 2 λ1 . . . 0 0 0

0 1-λ2 2 . . . 0 0 0...

......

......

...

0 0 0 . . . 2 λK−2 0

0 0 0 . . . 1-λK−1 2 λK−1

0 0 0 . . . 0 -2 πK 2

(A22)

Θ︸︷︷︸(K+1)×(K+1)

=

0 0 0 . . . 0 0 06

h1(h1+h2) - 6h1h2

6h2(h1+h2) . . . 0 0 0

0 6h2(h2+h3) - 6

h2h3. . . 0 0 0

......

......

......

0 0 0 . . . - 6hK−2hK−1

6hK−1(hK−2+hK−1) 0

0 0 0 . . . 6hK−1(hK−1+hK) - 6

hK−1hK6

hK(hK−1+hK)

0 0 0 . . . 0 0 0

(A23)

m︸︷︷︸(K+1)×1

=

M0

M1

...

MK−1

MK

(A24)

y︸︷︷︸(K+1)×1

=

y0

y1

...

yK−1

yK

(A25)

The linear equation system is given by

Λm = θy (A26)

and the solution is

m = Λ−1Θy (A27)

17

Using this result and equation (A17) we can calculate

S∆(ξ)︸︷︷︸N×1

=

S∆(ξ1)

S∆(ξ2)...

S∆(ξN−1)

S∆(ξN )

(A28)

Lets denote P the N × (K + 1) matrix where ith row i = 1, . . . , N1 given that kj−1 ≤ ξ ≤ kj

can be written as

pi︸︷︷︸1×(K+1)

=

0, . . . , 0︸︷︷︸�rst j − 2

,kj − ξi

6hj

[(kj − ξi)2 − h2

j

],ξi − kj−1

6hj

[(ξi − kj−1)2 − h2

j

], 0, . . . , 0︸︷︷︸last K + 1− j

(A29)

Moreover denote Q the N×(K+1) matrix where ith row i = 1, . . . , N1 given that kj−1 ≤ ξ ≤ kjcan be written as

qi︸︷︷︸1×(K+1)

=

0, . . . , 0︸︷︷︸�rst j − 2

,kj − ξihj

,ξi − kj−1

hj, 0, . . . , 0︸︷︷︸last K + 1− j

(A30)

Now using (A17) and (A27) we get

S∆(ξ) = Pm+Qy = PΛ−1Θy +Qy = (PΛ−1Θ +Q)y = W︸︷︷︸N×(K+1)

y︸︷︷︸(K+1)×1

(A31)

where

W = PΛ−1Θ +Q (A32)

In practical situations we might only know the knots but we don't know we observe the spline

values with error. In this case we have

s = S∆(ξ) + ε = Wy + ε, (A33)

where

s︸︷︷︸N×1

=

s1

s2

...

sN−1

sN

(A34)

18

and

ε︸︷︷︸N×1

=

ε1

ε2

...

εN−1

εN

(A35)

with

E(ε) = 0 and E(εε′) = σ2I (A36)

Notice that after �xing the knots we only have to estimate the value of the spline at he knots

and this determines the whole shape of the spline. We cab do this by simple OLS

y = (W>W )−1W>s (A37)

For identi�cation reasons we want∑j:uniqueξj

S∆(ξj) =∑

j:uniqueξj

wjy = w∗y = 0 (A38)

where wi is the ith row of W and

w∗︸︷︷︸1×(K+1)

=∑

j:uniqueξj

wj (A39)

The restriction can be enforced by one of the elements of y. This ensures that E(st) = 0 so st

and µθ can be identi�ed. If we drop yK we can substitute

yK = −K−1∑i=0

(w∗i /w∗K)yi (A40)

where w∗i is the ith element of w∗. Substituting this into

∑j:uniqueξj

S∆(ξj) =∑

j:uniqueξj

wjy =∑

j:uniqueξj

K∑i=0

wjiyi =∑

j:uniqueξj

K−1∑i=0

wjiyi − wjKK−1∑i=0

(w∗i /w∗K)yi

=∑

j:uniqueξj

K−1∑i=0

(wji − wjKw∗i /w∗K)yi =K−1∑i=0

∑j:uniqueξj

(wji − wjKw∗i /w∗K)yi

=

K−1∑i=0

(w∗i − w∗Kw∗i /w∗K)yi =

K−1∑i=0

(w∗i − w∗i )yi = 0 (A41)

Lets partition W in the following way

W︸︷︷︸N×(K+1)

= [W−K︸︷︷︸N×K

: WK︸︷︷︸N×1

] (A42)

19

where W−K is equal to the �rst K columns of W and WK is the Kth column of W . Moreover

w∗︸︷︷︸1×(K+1)

= [w∗−K︸︷︷︸1×K

: w∗K︸︷︷︸1×1

] (A43)

We can de�ne

W︸︷︷︸N×K

= W−K︸︷︷︸N×K

− 1

w∗KWK︸︷︷︸N×1

w∗−K︸︷︷︸1×K

(A44)

and we have

s = S∆(ξ) + ε = W︸︷︷︸N×K

y︸︷︷︸K×1

+ε. (A45)

D MCMC estimation of the dynamic ∆NB model

D.1 Generating the parameters x, µθ, φ, σ2η (Step 2)

Notice that conditional on R ={rtj , t = 1, . . . , T, j = 1, . . . ,min(Nt + 1, 2)

}, τ , N ,γ and s we

have

− log τt1 = log(z1t + z2t) + µθ + st + xt +mrt1(1) + εt1, εt1 ∼ N(0, v2rt1(1)) (A46)

and

− log τt2 = log(z1t + z2t) + µθ + st + xt +mrt2(Nt) + εt2, εt2 ∼ N(0, v2rt2(Nt)) (A47)

which implies the following following state space form

yt︸︷︷︸min(Nt+1,2)×1

=

1 wt 1

1 wt 1

︸︷︷︸

min(Nt+1,2)×(K+2)

µθ

β

xt

︸︷︷︸(K+2)×1

+ εt︸︷︷︸min(Nt+1,2)×1

, εt ∼ N(0,Ht) (A48)

αt+1 =

µθ

β

xt+1

︸︷︷︸

(K+2)×1

=

1 0 0

0 IK 0

0 0 φ

︸︷︷︸

(K+2)×(K+2)

µθ

β

xt

︸︷︷︸(K+2)×1

+

0

0

ηt+1

︸︷︷︸

(K+2)×1

, ηt+1 ∼ N(0, σ2η)(A49)

(A50)

where

µθ

β

x1

︸︷︷︸(K+2)×1

∼ N

µ0

β0

0

︸︷︷︸(K+2)×1

,

σ2µ 0 0

0 σ2βIK 0

0 0 σ2η/(1− φ2)

︸︷︷︸

(K+2)×(K+2)

(A51)

20

Ht = diag(v2rt1(1), v2

rt,2(Nt)) and

yt︸︷︷︸min(Nt+1,2)×1

=

− log τt1 −mrt1(1)− log(z1t + z2t)

− log τt2 −mrt2(Nt)− log(z1t + z2t)

(A52)

D.2 Generating γ (Step 3)

p(γ|ν, µθ, φ, σ2η, x,R, s, τ,N, z1, z2, y) = p(γ|ν, µθ, s, x, y) (A53)

because given ν, λ and y, the variables R, τ,N, z1, z2 are redundant.

p(γ|ν, µθ, s, x, y) ∝ p(y|γ, ν, µθ, s, x)p(γ|ν, µθ, s, x) = p(y|γ, ν, µθ, s, x)p(γ) (A54)

as γ is independent from ν and λt = exp(µθ + st + xt).

p(y|γ, ν, µθ, x)p(γ) =T∏t=1

[γ1{yt=0} + (1− γ)

(ν

λt + ν

)2ν ( λtλt + ν

)|yt| Γ(ν + |yt|)Γ(ν)Γ(|yt|)

× F

(ν + yt, ν, yt + 1;

(λt

λt + ν

)2) γa−1(1− γ)b−1

B(a, b)

∝T∏t=1

[γa(1− γ)b−1

1{yt=0} + γa−1(1− γ)b(

ν

λt + ν

)2ν ( λtλt + ν

)|yt| Γ(ν + |yt|)Γ(ν)Γ(|yt|)

× F

(ν + yt, ν, yt + 1;

(λt

λt + ν

)2)

We can carry out an independent MH step to sample from this density using a truncated normal

or normal density with mean equal to the mode of this above distribution and variance equal to

the Hessian at the mode.

D.3 Generating the auxiliary variables R, τ,N, z1, z2, ν (Step 4)

p(R, τ,N, z1, z2, ν|γ, µθ, φ, σ2η, s, x, y) = p(R|τ,N, z1, z2γ, p, µθ, φ, σ

2η, s, x, y)

× p(τ |N, z1, z2γ, ν, µθ, φ, σ2η, s, x, y)

× p(N |z1, z2γ, ν, µθ, φ, σ2η, s, x, y)

× p(z1, z2|γ, ν, µθ, φ, σ2η, s, x, y)

× p(ν|γ, µθ, φ, σ2η, s, x, y) (A55)

21

Generating ν (Step 4a)

Note that

p(ν|γ, µθ, φ, σ2η, s, x, y) = p(ν|γ, λ, y)

∝ p(ν, γ, λ, y)

= p(y|γ, λ, ν)p(λ|γ, ν)p(γ|ν)p(ν)

= p(y|γ, λ, ν)p(λ)p(γ)p(ν)

∝ p(y|γ, λ, ν)p(ν) (A56)

where p(y|γ, λ, ν) is a product of zero in�ated ∆NB probability mass functions. We can draw ν

using a Laplace approximation or an adaptive random walk Metropolis-Hasting procedure.

An alternative way of drawing ν is using a discrete uniform prior ν ∼ DU(2, 128) and a

random walk proposal in the following fashion as suggested by Stroud and Johannes (2014)

for degree of freedom parameter of a t density. We can write the posterior as a multinomial

distribution p(ν|µθ, x, z1, z2) ∼M(π∗2, . . . , π∗128) with probabilities

π∗ν ∝T∏t=1

[γI{yt=0} + (1− γ)f∆NB(yt;λt, ν)

]=

T∏t=1

gν(yt) (A57)

To avoid the computationally intense evaluation of these probabilities we can use a Metropolis-

Hastings update. We can draw the proposal ν∗ from the neighbourhood of the current value ν(i)

using a discrete uniform distribution ν∗ ∼ DU(ν(i) − δ, ν(i) + δ) and accept with probability

min

{1,

∏Tt=1 gν∗(yt)∏Tt=1 gν(i)(yt)

}(A58)

δ is chosen such that the acceptance rate is reasonable.

Generating z1, z2 (Step 4b)

Notice that z1, z2 are independent given γ, µθ, s, x, y.

p(z1, z2|γ, ν, µθ, φ, σ2η, s, x, y) =

T∏t=1

p(z1t, z2t|γ, ν, µθ, φ, σ2η, st, xt, yt) (A59)

p(z1t, z2t|γ, ν, µθ, φ, σ2η, st, xt, yt) ∝ p(z1t, z2t, γ, ν, µθ, φ, σ

2η, st, xt, yt)

= p(yt|z1t, z2t, γ, ν, µθ, φ, σ2η, st, xt)

× p(z1t, z2t|γ, ν, µθ, φ, σ2η, st, xt) (A60)

p(z1t, z2t|γ, ν, µθ, φ, σ2η, st, xt, yt) ∝ g(z1t, z2t)

ννzν1te−νz1t

Γ(ν)

ννzν2te−νz2t

Γ(ν)(A61)

22

where

g(z1t, z2t) =

γ1{yt=0} + (1− γ) exp[−λt(z1t + z2t)

](z1t

z2t

) yt2

I|yt|(2λt√z1tz2t)

(A62)

with λt = exp(µθ + st +xt). We can carry out an independent MH step by sampling z∗1t, z∗2t from

Ga(λt, ν) and accept it with probability

min

{g(z∗1t, z

∗2t)

g(z1t, z2t), 1

}(A63)

Generating N (Step 4c)

The number of jumps are independent given γ, µθ, φ, σ2η, s, x, z1, z2, y which means

p(N |γ, ν, µθ, φ, σ2η, s, x, z1, z2, y) =

T∏t=1

p(Nt|γ, ν, µθ, φ, σ2η, st, xt, z1t, z2t, yt). (A64)

For a given t we can draw Nt from a discrete distribution with

p(Nt = n|γ, ν, µθ, φ, σ2η, st, xt, z1t, z2t, yt) =

p(Nt = n, yt = k|γ, ν, µθ, φ, σ2η, st, xt, z1t, z2t)

p(yt = k|γ, p, µθ, φ, σ2η, st, xt, z1t, z2t)

= p(yt = k|Nt = n, γ, ν, µθ, φ, σ2η, st, xt, z1t, z2t)

×p(Nt = n|γ, ν, µθ, φ, σ2

η, st, xt, z1t, z2t, )

p(yt = k|γ, ν, µθ, φ, σ2η, st, xt, z1t, z2t)

=

γ1{k=0} + (1− γ)p

n∑i=1

Mi = k|Nt = n, , z1t, z2t

×p(Nt = n|γ, µθ, φ, σ2

η, st, xt, z1t, z2t)

p(yt = k|γ, µθ, φ, σ2η, st, xt, z1t, z2t)

(A65)

The denominator is easy to evaluate it is a Skellam distribution at k with intensity λtz1t andλtz2t

. The probability

p

n∑i=1

Mi = k|Nt = n, z1t, z2t

(A66)

is not standard. condition on z1 and z2, yt = has a Skellam distribution, hence

Mi =

1, with P (Mi = 1) = z1tz1t+z2t

−1, with P (Mi = −1) = z2tz1t+z2t

(A67)

which implies we can representn∑i=1

Mi with a tree structure and the binomial distribution.n∑i=1

Mi

has a binomial distribution with n trails, (n+ k)/2 successes and p = 0.5 success rate. Note that

even k can only happen in even number of trails and odd k can only happen in odd number of

23

trails.

p

n∑i=1

Mi = k|Nt = n, z1t, z2t

=

0, if k > n or |k mod 2| 6= |n mod 2|(nn+k

2

)(z1t

z1t + z2t

)n+k2(

z2t

z1t + z2t

)n−k2

, otherwise

(A68)

The probability p(Nt = n|γ, µθ, φ, σ2η, st, xt, z1t, z2t) is equal to p(Nt = n|µθ, st, xt) and it is

a Poission random variable with intensity equal to λt(z1t + z2t). In general we have to following

expression for p(N |γ, ν, µθ, φ, σ2η, s, x, z1, z2, y) when |yt| ≤ n[

I{yt=0}γ[λt(z1t + z2t)

]nexp

[−λt(z1t + z2t)

]Γ(n+ 1)

+ I{|yt| mod 2=n mod 2}(1− γ)

[λt(z1t + z2t)

]nexp

[−λt(z1t + z2t)

]Γ(n+yt

2 + 1)Γ(n−yt2 + 1)

(z1t

z1t + z2t

)n+k2 (

z2t

z1t + z2t

)n−k2

× 1[

γ1{yt=0} + (1− γ) exp[−λt(z1t + z2t)

] (z1tz2t

) yt2I|yt|(2λt

√z1tz2t)

] (A69)

otherwise it is zero.

We can draw Nt parallel over t = 1, . . . , T by drawing a uniform random variable ut ∼ U [0, 1]

and

Nt = min

n : ut ≤n∑i=0

p(Nt = i|γ, µθ, φ, σ2η, st, xt, yt, z1t, z2t)

(A70)

Generating τ (Step 4d)

Notice that p(τ |N, z1, z2, γ, ν, µθ, φ, σ2η, x, y) = p(τ |N,µθ, z1, z2, s, x). Moreover

p(τ |µθ, z1, z2, s, x) =

T∏t=1

p(τ1t, τ2t|Nt, µθ, z1t, z2t, st, xt)

=

T∏t=1

p(τ1t|τ2t, Nt, µθ, z1t, z2t, st, xt)p(τ2t|Nt, µθ, z1t, z2t, st, xt) (A71)

where we can sample from p(τ2t|Nt, µθ, z1t, z2t, st, xt) using the fact that conditionally on Nt

the arrival time τ2t of the Ntth jump is the maximum of Nt uniform random variables and it

has a Beta(Nt, 1) distribution. The arrival time of the (Nt + 1)th jump after 1 is exponentially

distributed with intensity λt(z1t + z2t), hence

τ1t = 1 + ξt − τ2t ξt ∼ Exp(λt(z1t + z2t)) (A72)

24

Generating R (Step 4e)

Notice that

p(R|τ,N, z1, z2, γ, ν, µθ, φ, σ2η, s, x, y) = p(R|τ,N, z1, z2, ν, s, x) (A73)

Moreover

p(R|τ,N, z1, z2, ν, s, x) =T∏t=1

min(Nt+1,2)∏j=1

p(rtj |τt, Nt, µθ, z1t, z2t, st, xt) (A74)

Sample rt1 from the following discrete distribution

p(rt1|τt, Nt, µθ, z1t, z2t, st, xt) ∝ wk(1)φ(− log τ1t − log[λt(z1t + z2t)],mk(1), v2k(1)) (A75)

where k = 1, . . . , R(1) If Nt > 0 then draw rt2 from the discrete distribution

p(rt2|τt, Nt, µθ, z1t, z2t, st, xt) ∝ wk(Nt)φ(− log τ1t − log[λt(z1t + z2t)],mk(Nt), v2k(Nt)) (A76)

for k = 1, . . . , R(Nt)

Tables and Figures

Table 1: Estimation results from a dynamic Skellam and ∆NB model based on 20000 observations and 100000 iterationsfrom which 20000 used as a burn in sample.The 95 % HPD regions are in brackets The true parameters are µ = −1.7,φ = 0.97 , σ = 0.02, γ = 0.001 and ν = 15

Skellam ∆NB

µ -1.72 -1.726

[-1.797,-1.642] [-1.804,-1.651]

φ 0.973 0.975

[0.965,0.979] [0.969,0.981]

σ2 0.018 0.015

[0.013,0.023] [0.011,0.02]

γ 0.005 0.003

[0,0.017] [0,0.01]

β1 1.139 1.128

[0.884,1.392] [0.875,1.38]

β2 -0.306 -0.297

[-0.453,-0.158] [-0.448,-0.151]

β3 -0.801 -0.793

[-0.943,-0.657] [-0.933,-0.65]

β4 0.091 0.099

[-0.052,0.23] [-0.04,0.24]

ν 12.191

[8,16.4]

25

Table 2: Descriptive statistics of the data from 3rd to 10th October 2008.

AA F IBM

In Out In Out In Out

Num. obs 64 807 14 385 32 756 14 313 68 002 20 800

Avg. price 16.75 11.574 3.077 2.112 96.796 87.583

Mean -0.007 -0.004 -0.007 0 -0.02 -0.004

Std 1.63 2.126 0.745 0.601 6.831 7.09

Min -33 -51 -18 -10 - 197 - 105

Max 38 39 21 9 186 140

% Zeros 48.76 48.76 77.08 77.08 39.9 39.9

JPM KO XRX

Out In Out In Out In

Num. obs 142 867 43 230 70 356 25 036 26 020 8 623

Avg. price 42.773 38.889 49.203 41.875 9.049 7.768

Mean -0.009 0.012 -0.012 0.005 -0.006 0.004

Std 2.368 2.779 1.758 2.734 0.816 1.285

Min -48 -40 -33 -50 -17 -17

Max 74 55 30 63 19 12

% Zeros 43.78 43.78 34.39 34.39 54.98 54.98

Table 3: Descriptive statistics of the data from 23rd to 30th April 2010.

AA F IBM

In Out In Out In Out

Num. obs 27 550 4 883 63 241 9 894 43 606 8 587

Avg. price 13.749 13.519 13.734 13.231 130.176 129.575

Mean -0.001 -0.006 -0.001 -0.006 0.001 -0.019

Std 0.468 0.502 0.448 0.454 1.424 1.371

Min -3 -2 -5 -2 -22 -15

Max 3 2 4 3 24 9

% Zeros 75.02 75.02 79.73 79.73 51.93 51.93

JPM KO XRX

Out In Out In Out In

Num. obs 101 045 21 443 34 469 6 073 36 332 4 326

Avg. price 43.702 42.854 53.628 53.732 11.164 11.025

Mean -0.001 -0.007 -0.003 -0.006 0 -0.007

Std 0.615 0.638 0.647 0.696 0.494 0.459

Min -5 -10 -9 -5 -9 -2

Max 5 5 7 5 7 3

% Zeros 68.73 68.73 65.09 65.09 79.29 79.29

26

Table 4: Estimation results from a dynamic Skellam and ∆ NB model during the period from 3rd to 9th October 2008.The posterior mean estimates are based on 100000 iterations from which 20000 used as a burn in sample.The 95 % HPDregions are in brackets

AA F IBM

Skellam ∆NB Skellam ∆NB Skellam ∆NB

µ -0.174 -0.262 -1.873 -1.861 1.935 1.246

[-0.236,-0.112] [-0.321,-0.204] [-1.942,-1.803] [-1.932,-1.791] [1.865,2.008] [1.198,1.294]

φ 0.929 0.941 0.939 0.945 0.881 0.935

[0.921,0.936] [0.935,0.946] [0.931,0.95] [0.934,0.955] [0.873,0.888] [0.93,0.94]

σ2 0.207 0.126 0.112 0.093 0.86 0.124

[0.184,0.235] [0.107,0.141] [0.095,0.132] [0.077,0.114] [0.796,0.926] [0.112,0.133]

γ 0.248 0.243 0.279 0.299

[0.24,0.258] [0.234,0.251] [0.274,0.285] [0.294,0.304]

β1 0.436 0.374 0.297 0.288 0.282 0.206

[0.325,0.544] [0.272,0.477] [0.156,0.433] [0.149,0.429] [0.153,0.406] [0.118,0.293]

β2 -0.185 -0.151 -0.117 -0.114 0.076 0.023

[-0.274,-0.097] [-0.234,-0.068] [-0.224,-0.01] [-0.223,-0.007] [-0.034,0.186] [-0.053,0.098]

ν 8.701 14.315 2

[6.6,11] [10.4,18.2] [2,2]

JPM KO XRX


µ 0.229 0.239 0.18 0.148 -1.418 -1.417

[0.188,0.272] [0.193,0.284] [0.138,0.222] [0.107,0.189] [-1.474,-1.363] [-1.473,-1.361]

φ 0.897 0.905 0.937 0.943 0.928 0.941

[0.893,0.902] [0.901,0.911] [0.932,0.943] [0.937,0.948] [0.912,0.944] [0.928,0.954]

σ2 0.459 0.378 0.083 0.067 0.071 0.048

[0.444,0.476] [0.336,0.401] [0.076,0.091] [0.059,0.075] [0.053,0.089] [0.035,0.061]

γ 0.197 0.205 0.103 0.103

[0.192,0.203] [0.199,0.213] [0.096,0.11] [0.096,0.109]

β1 0.358 0.343 0.569 0.543 0.564 0.536

[0.285,0.431] [0.272,0.416] [0.502,0.64] [0.476,0.611] [0.448,0.677] [0.423,0.654]

β2 0.011 0.015 -0.209 -0.196 -0.142 -0.132

[-0.056,0.077] [-0.05,0.081] [-0.277,-0.14] [-0.262,-0.129] [-0.23,-0.052] [-0.22,-0.042]

ν 87.27 34.756 8.697

[75.2,98.8] [28.4,41.6] [5.8,11.6]

27

Table 5: Estimation results from a dynamic Skellam and ∆ NB model during the period from 23rd to 29th April 2010.The posterior mean estimates are based on 100000 iterations from which 20000 used as a burn in sample.The 95 % HPDregions are in brackets

AA F IBM


µ -2.23 -2.227 -2.397 -2.393 -0.083 -0.224

[-2.29,-2.17] [-2.288,-2.167] [-2.442,-2.351] [-2.436,-2.348] [-0.154,-0.008] [-0.299,-0.146]

φ 0.956 0.958 0.942 0.944 0.975 0.983

[0.944,0.968] [0.947,0.971] [0.933,0.951] [0.936,0.953] [0.968,0.981] [0.976,0.988]

σ2 0.029 0.027 0.061 0.057 0.025 0.011

[0.02,0.04] [0.018,0.039] [0.051,0.078] [0.046,0.068] [0.018,0.033] [0.007,0.017]

γ 0.287 0.267

[0.278,0.297] [0.256,0.279]

β1 0.037 0.037 0.148 0.149 0.476 0.421

[-0.052,0.13] [-0.056,0.13] [0.089,0.207] [0.09,0.206] [0.359,0.6] [0.306,0.536]

β2 -0.041 -0.041 -0.188 -0.188 0.204 0.181

[-0.138,0.057] [-0.137,0.061] [-0.259,-0.115] [-0.26,-0.115] [0.082,0.329] [0.061,0.3]

ν 20.367 27.436 6.101

[15,25.8] [21.4,33.8] [4.2,7.8]

JPM KO XRX


µ -1.674 -1.673 -1.636 -1.637 -2.334 -2.328

[-1.716,-1.632] [-1.716,-1.631] [-1.693,-1.581] [-1.693,-1.581] [-2.393,-2.275] [-2.387,-2.271]

φ 0.992 0.993 0.98 0.98 0.943 0.947

[0.99,0.994] [0.991,0.994] [0.973,0.987] [0.973,0.987] [0.929,0.959] [0.934,0.959]

σ2 0.002 0.002 0.007 0.007 0.059 0.052

[0.002,0.003] [0.002,0.003] [0.004,0.01] [0.004,0.01] [0.037,0.076] [0.038,0.068]

γ

β1 0.195 0.195 0.355 0.351 0.647 0.641

[0.124,0.266] [0.121,0.266] [0.268,0.443] [0.262,0.439] [0.553,0.739] [0.548,0.733]

β2 0.029 0.029 0.067 0.069 -0.457 -0.455

[-0.039,0.1] [-0.043,0.098] [-0.032,0.164] [-0.031,0.166] [-0.545,-0.367] [-0.544,-0.368]

ν 36.288 22.356 17.029

[29.6,43.8] [16.6,28] [12.4,22.4]

28

Table

6:Summary

ofthecleaningandaggregationprocedure

onthedata

from

October

2008forAlcoa(A

A),Coca-Cola(K

O)InternationalBusinessMachines

(IBM),J.P.Morgan(JPM),

Ford

(F),Xerox

(XRXfrom

theNYSE.

AA

FIBM

JPM

KO

XRX

#%

dropped

#%

dropped

#%

dropped

#%

dropped

#%

dropped

#%

dropped

Raw

quotesandtrades

511185

311914

688805

984526

541616

371065

Trades

107448

78.98

59749

80.84

128589

81.33

298773

69.65

126509

76.64

40846

88.99

Nonmissingprice

andvolume

107434

0.01

59737

0.02

128575

0.01

298761

0126497

0.01

40834

0.03

Trades

between9:35and15:55

107421

0.01

59724

0.02

128561

0.01

298744

0.01

126484

0.01

40820

0.03

Aggrageted

trades

79623

25.88

47146

21.06

89517

30.37

188469

36.91

96482

23.72

34722

14.94

Withoutoutliers

79198

0.53

47075

0.15

88808

0.79

186103

1.26

95398

1.12

34649

0.21

Withoutopeningtrades

79192

0.01

47069

0.01

88802

0.01

186097

095392

0.01

34643

0.02

Table

7:Summary

ofthecleaningandaggregationprocedure

onthedata

from

April2010forAlcoa(A

A),Coca-Cola

(KO)InternationalBusinessMachines

(IBM),J.P.Morgan(JPM),

Ford

(F),Xerox

(XRXfrom

theNYSE.

AA

FIBM

JPM

KO

XRX

#%

dropped

#%

dropped

#%

dropped

#%

dropped

#%

dropped

#%

dropped

Raw

quotesandtrades

1487382

2737300

803648

2109770

692657

1038502

Trades

33684

97.74

77778

97.16

53346

93.36

126153

94.02

41184

94.05

43170

95.84

Nonmissingprice

andvolume

33675

0.03

77765

0.02

53332

0.03

126142

0.01

41173

0.03

43155

0.03

Trades

between9:30and16:00

33666

0.03

77757

0.01

53324

0.02

126136

041164

0.02

43149

0.01

Aggrageted

trades

32446

3.62

73160

5.91

52406

1.72

122579

2.82

40573

1.44

40673

5.74

Withoutoutliers

32439

0.02

73141

0.03

52199

0.39

122494

0.07

40548

0.06

40664

0.02

Withoutopeningtrades

32433

0.02

73135

0.01

52193

0.01

122488

040542

0.01

40658

0.01

29

0.0015 0.0010 0.0005 0.0000 0.0005 0.0010 0.00150

1000

2000

3000

4000

5000

6000

7000AA

0.004 0.002 0.000 0.002 0.0040

500

1000

1500

2000

2500F

0.0015 0.0010 0.0005 0.0000 0.0005 0.0010 0.00150

1000

2000

3000

4000

5000IBM

0.0015 0.0010 0.0005 0.0000 0.0005 0.0010 0.00150

10002000300040005000600070008000

JPM

0.0015 0.0010 0.0005 0.0000 0.0005 0.0010 0.00150

1000

2000

3000

4000

5000

6000KO

0.0015 0.0010 0.0005 0.0000 0.0005 0.0010 0.00150

1000

2000

3000

4000

5000

6000XRX

Empirical distribution of the log returns in 10/2008

Figure 1: Empirical distribution of the tick by tick log returns during October 2008 for Alcoa (AA), Ford (F), InternationalBusiness Machines (IBM),JP Morgan (JPM), Coca-Cola (KO) and Xerox (X)

30

50 40 30 20 10 0 10 20 30 40

10-5

10-4

10-3

10-2

10-1

log

dens

ity

AA

empiricalskellam

20 15 10 5 0 5 10 15 20 2510-5

10-4

10-3

10-2

10-1

log

dens

ity

F

empiricalskellam

150 100 50 0 50 100 150

10-5

10-4

10-3

10-2

10-1

log

dens

ity

IBM

empiricalskellam

60 40 20 0 20 40 60 80

10-5

10-4

10-3

10-2

10-1

log

dens

ity

JPM

empiricalskellam

60 40 20 0 20 40 60 80

10-5

10-4

10-3

10-2

10-1

log

dens

ity

KO

empiricalskellam

20 15 10 5 0 5 10 15 20

10-4

10-3

10-2

10-1

log

dens

ityXRX

empiricalskellam

Empirical distribution of the tick returns in 10/2008

Figure 2: Empirical distribution of the tick returns along with �tted Skellam density during October 2008 for Alcoa (AA),Ford (F), International Business Machines (IBM),JP Morgan (JPM), Coca-Cola (KO) and Xerox (X)

31

10 5 0 5 100.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35Zero mean Skellam distribution

Skellam γ=0 and λ=1Skellam γ=0 and λ=2Skellam γ=0.1 and λ=2

Figure 3: The picture shows the Skellam distribution with di�erent parameters

32

10 5 0 5 100.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35Zero Mean ∆NB Distribution

∆NB γ=0, λ=1 and ν=1∆NB γ=0, λ=1 and ν=10 ∆NB γ=0, λ=5 and ν=1∆NB γ=0, λ=5 and ν=10∆NB γ=0.1, λ=5 and ν=10

Figure 4: The picture shows the ∆NB distribution with di�erent parameters

33

1.90 1.85 1.80 1.75 1.70 1.65 1.60 1.55µ

02468

1012

0.955 0.960 0.965 0.970 0.975 0.980 0.985 0.990φ

020406080

100120

0.010 0.015 0.020 0.025 0.030

σ2

020406080

100120140160

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045γ

0

50

100

150

200

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8β1

0.00.51.01.52.02.53.03.5

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1β2

0123456

1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5β3

0123456

0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5β4

0123456

Posterior densities of the parameters

Figure 5: The posterior distribution of the parameters from a dynamic Skellam model based on 20000 observations and100000 iterations from which 20000 used as a burn in sample. Each picture shows the histogram of the posterior drawsthe kernel density estimate of the posterior distribution, the HPD region and the posterior mean. The true parameters areµ = −1.7, φ = 0.97 , σ = 0.02, γ = 0.001

34

1.95 1.90 1.85 1.80 1.75 1.70 1.65 1.60 1.55 1.50µ

02468

1012

0.960 0.965 0.970 0.975 0.980 0.985φ

020406080

100120140

0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.022 0.024 0.026

σ2

050

100150200

0.00 0.01 0.02 0.03 0.04 0.05γ

050100150200250300350400

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8β1

0.00.51.01.52.02.53.03.5

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1β2

0123456

1.1 1.0 0.9 0.8 0.7 0.6 0.5β3

0123456

0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5β4

0123456

5 10 15 20 25ν

0.000.050.100.150.20

Posterior densities of the parameters

Figure 6: The posterior distribution of the parameters from a dynamic ∆ NB model based on 20000 observations and100000 iterations from which 20000 used as a burn in sample. Each picture shows the histogram of the posterior drawsthe kernel density estimate of the posterior distribution, the HPD region and the posterior mean. The true parameters areµ = −1.7, φ = 0.97 , σ = 0.02, γ = 0.001 and ν = 15

35

26/04 27/04 28/04 29/0430

20

10

0

10

20

30tick retruns

26/04 27/04 28/04 29/0432101234

xt

26/04 27/04 28/04 29/040.80.60.40.20.00.20.40.6

st

26/04 27/04 28/04 29/0432101234

logλt

Volatility decompostion of IBM tick returns from 23rd to 29th April 2010

Figure 7: Decompostion of log volatility of IBM

40

30

20

10

0

10

20

30

40

Tick

ret

urns

AA

2015105

0510152025

Tick

ret

urns

F

100

50

0

50

100

Tick

ret

urns

IBM

60

40

20

0

20

40

60

80

Tick

ret

urns

JPM

40

30

20

10

0

10

20

30

Tick

ret

urns

KO

20

15

10

5

0

5

10

15

20

Tick

ret

urns

XRX

03/10 06/10 07/10 08/10 09/1050

40

30

20

10

0

10

20

2 lo

g B

F

03/10 06/10 07/10 08/10 09/1040

20

0

20

40

60

80

2 lo

g B

F

03/10 06/10 07/10 08/10 09/10800700600500400300200100

0100

2 lo

g B

F

03/10 06/10 07/10 08/10 09/10200

150

100

50

0

50

2 lo

g B

F

03/10 06/10 07/10 08/10 09/1060

50

40

30

20

10

0

10

2 lo

g B

F

03/10 06/10 07/10 08/10 09/1025

20

15

10

5

0

5

10

2 lo

g B

F

In-sample BIC comparison in October 2008

Figure 8: Sequential Bayes factors approximation based on BIC on data from 3rd to 9th October 2008.

36

60

40

20

0

20

40

Tick

ret

urns

AA

10

5

0

5

10

Tick

ret

urns

F

150

100

50

0

50

100

150

Tick

ret

urns

IBM

40

20

0

20

40

60

Tick

ret

urns

JPM

60

40

20

0

20

40

60

80Ti

ck r

etur

ns

KO

20

15

10

5

0

5

10

15

Tick

ret

urns

XRX

10:00 10/10/08

11:00 10/10/08

12:00 10/10/08

13:00 10/10/08

14:00 10/10/08

15:00 10/10/08

202468

10121416

2 lo

g B

F

10:00 10/10/08

11:00 10/10/08

12:00 10/10/08

13:00 10/10/08

14:00 10/10/08

15:00 10/10/08

20

15

10

5

0

5

10

2 lo

g B

F

10:00 10/10/08

11:00 10/10/08

12:00 10/10/08

13:00 10/10/08

14:00 10/10/08

15:00 10/10/08

300

250

200

150

100

50

0

50

2 lo

g B

F

10:00 10/10/08

11:00 10/10/08

12:00 10/10/08

13:00 10/10/08

14:00 10/10/08

15:00 10/10/08

100

80

60

40

20

0

20

2 lo

g B

F

10:00 10/10/08

11:00 10/10/08

12:00 10/10/08

13:00 10/10/08

14:00 10/10/08

15:00 10/10/08

20

0

20

40

60

80

100

120

2 lo

g B

F

10:00 10/10/08

11:00 10/10/08

12:00 10/10/08

13:00 10/10/08

14:00 10/10/08

15:00 10/10/08

20

0

20

40

60

80

100

120

2 lo

g B

F

Out of sample predicitve likelihood comparison in October 2008

Figure 9: Sequential predictive Bayes factors on 10th October 2008.

3

2

1

0

1

2

3

Tick

ret

urns

AA

54321

01234

Tick

ret

urns

F

30

20

10

0

10

20

30

Tick

ret

urns

IBM

6

4

2

0

2

4

6

Tick

ret

urns

JPM

108642

02468

Tick

ret

urns

KO

108642

02468

Tick

ret

urns

XRX

23/04 26/04 27/04 28/04 29/0460

50

40

30

20

10

0

10

2 lo

g B

F

23/04 26/04 27/04 28/04 29/0470

60

50

40

30

20

10

0

10

2 lo

g B

F

23/04 26/04 27/04 28/04 29/0420

10

0

10

20

30

40

50

2 lo

g B

F

23/04 26/04 27/04 28/04 29/04

80

60

40

20

0

2 lo

g B

F

23/04 26/04 27/04 28/04 29/0460

50

40

30

20

10

0

10

2 lo

g B

F

23/04 26/04 27/04 28/04 29/0430

25

20

15

10

5

0

5

2 lo

g B

F

In-sample BIC comparison in April 2010

Figure 10: Sequential Bayes factors approximation based on BIC on data from 23rd to 29th April 2010.

37

2.01.51.00.5

0.00.51.01.52.0

Tick

ret

urns

AA

2

1

0

1

2

3

Tick

ret

urns

F

15

10

5

0

5

10

Tick

ret

urns

IBM

108642

0246

Tick

ret

urns

JPM

6

4

2

0

2

4

6Ti

ck r

etur

ns

KO

2

1

0

1

2

3

Tick

ret

urns

XRX

10:00 30/04/10

11:00 30/04/10

12:00 30/04/10

13:00 30/04/10

14:00 30/04/10

15:00 30/04/10

10

8

6

4

2

0

2

2 lo

g B

F

10:00 30/04/10

11:00 30/04/10

12:00 30/04/10

13:00 30/04/10

14:00 30/04/10

15:00 30/04/10

12

10

8

6

4

2

0

2

2 lo

g B

F

10:00 30/04/10

11:00 30/04/10

12:00 30/04/10

13:00 30/04/10

14:00 30/04/10

15:00 30/04/10

5

0

5

10

15

20

25

30

2 lo

g B

F

10:00 30/04/10

11:00 30/04/10

12:00 30/04/10

13:00 30/04/10

14:00 30/04/10

15:00 30/04/10

15

10

5

0

5

10

2 lo

g B

F

10:00 30/04/10

11:00 30/04/10

12:00 30/04/10

13:00 30/04/10

14:00 30/04/10

15:00 30/04/10

10

8

6

4

2

0

2

2 lo

g B

F

10:00 30/04/10

11:00 30/04/10

12:00 30/04/10

13:00 30/04/10

14:00 30/04/10

15:00 30/04/10

7654321012

2 lo

g B

F

Out of sample predicitve likelihood comparison in April 2010

Figure 11: Sequential predictive Bayes factors on 30th April 2010.

38

dynamic negative binomial di erence model for high...

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