Discussion Papers In Economics
And Business
Bayesian inference for time varying partial
adjustment model withapplication to intraday price discovery
Kenji Hatakenaka
Kosuke Oya

Discussion Paper 21-19

November 2021

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANBayesian inference for time varying partial adjustment model withapplication to intraday price discovery1
Kenji Hatakenaka2 , Kosuke Oya3

Abstract
Price discovery is an important built-in function of financial markets and the central issue
in the market microstructure research. Market participants need to know whether the pricediscovery has been achieved or how much progress has been made in order to trade at an
appropriate price they consider. Since various economic events such as earnings announcement affect the price discovery, the intraday transition of price discovery varies date-by-date.
In this study, we propose a statistical method to see when and how fast the intraday pricediscovery progresses using the high frequency price series on a daily basis. The proposed
method consists of estimating three candidate models which gauge the different types of
price discovery progress, i.e. no progress, smooth progress and abrupt progress, and selecting the most appropriate model based on Bayesian approach. We conduct simulation
analysis to assess the performance of our proposed method and confirm that the method
depicts the state of price discovery appropriately. The empirical study using the Japanesestock market index shows that the proposed method well categorizes the intraday pricediscovery progresses on a daily basis.
Keywords: pre-opening period, market microstructure, partial adjustment model
JEL: C11, G14

1

We would like to thank Yasuhiro Omori and Tsunehiro Ishihara for their valuable advice around MCMC.
We also would like to thank the participants of the 2021 international conference on Econometrics andStatistics, especially to Toshiaki Watanabe and Mike, So for helpful comments. This work is supported by
JSPS KAKENHI Grant Number JP16H03605, JP17H02513.
2
Graduate School of Economics, Osaka University Email:k.hatakenaka.o@icloud.com
3
Graduate School of Economics, Osaka University Email:oya@econ.osaka-u.ac.jp

1. Introduction
Price discovery is an important built-in function of financial markets and the central issue
in the market microstructure research. More specifically, the price discovery brings pricesof assets traded on markets closer to their fundamental values by reflecting all publicly
available information. Market participants need to know whether the price discovery has
been achieved or how much progress has been made in order to trade at an appropriate pricethey consider. In this paper, we propose a statistical method to see when and how fast pricediscovery progresses.
Biais et al. (1999) is well known previous research about price discovery during the
preopening period which is the period before the regular market session. They gauge thestate of price discovery at specific time by estimating the slope coefficient of unbiasedness
regression in which they regress the close-to-close return onto the return from the closing
price of previous trading day to the price at the specific time. In their analysis, the slope
coefficients at specific time t is obtained by one unbiasedness regression using the 19 trading
days of French stock market data between October 29 and November 26, 1991. They figureout the path of slope coefficients for each minute during the period from 9:30 to 12:00 which
includes the opening at 10:00 am in the Paris Bourse. However, not only does their approach
ignore the time series properties of the market data, but they also cannot have sufficient
sample size when our interest focuses on the economic event such as quarterly earnings
announcement since there are only four announcements in a year.
In a different way than Biais et al. (1999), we adopt a partial price adjustment model
which focuses on the difference between the fundamental value and the observed market
price of asset. The partial price adjustment model is often used in the research field of
market microstructure. For example, Amihud and Mendelson (1987) conduct the empiricalcomparison of price behavior under the different two trading mechanisms by using the partial
price adjustment model. It is worth noting that the magnitude of adjustment coefficient of1

the model represents the state of price discovery. We extend the partial adjustment model tomodels that allow time-varying adjustment coefficient using the mechanisms of the smoothtransition as in Teräsvirta (1994) and the threshold coefficient as in Tong (1978) to capture
how the price discovery progress.
In this paper, we consider the partial price adjustment models whose adjustment coefficient is constant, smoothly time varying, and switching abruptly. The estimation of unknown
parameters is not easy due to the identification problems inherent in the smooth transition
model and the threshold model. Following the previous study by Gerlach and Chen (2008),
Bayesian approach is used to estimate the proposed models in this paper. For the model
selection, we employ the deviance information criterion of Spiegelhalter et al. (2002) andthe marginal likelihood method of Chib (1995) and Chib and Jeliazkov (2001), and evaluate
these selection methods through the simulation studies as in So and Chan (2014).
The rest of this paper is organized as follows. We describe three types of partial adjustment model with an adjustment coefficient which is constant, smoothly time varying, andswitching abruptly in Section 2. Section 3 provides the estimation method of three candidate models and the model selection criteria. After that, we conduct simulation analysis in
Section 4 to assess the performance of our proposed method and give an empirical study
using actual observed financial data in Section 5. We offer concluding remarks in Section 6.

2. Model description of partial adjustment models
We see how price discovery progresses by estimating three candidate models and selecting
most appropriate model among them. In this section, we introduce three candidate models
and describe the identification problems related to them.

22.1. Constant adjustment modelFirst, we introduce a partial adjustment model used in Amihud and Mendelson (1987).
They analyze the price behaviors of the stocks on New York Stock Exchange with followings∆pt = g(mt − pt−1 ) + ut ,(1)

mt = mt−1 + et(2)

where pt is the logarithmic market price of asset at time t and ∆pt ≡ pt − pt−1 stands forthe return. Here, mt is the logarithmic efficient price which is given as the expectationof the fundamental value of the asset conditional on all publicly available information up
to time t. It is noted that mt is unobservable and the specification of which the efficient
price follows the random walk according to the efficient market hypothesis is conventional inmarket microstructure analysis (see Amihud and Mendelson (1987) and Hasbrouck (2007)).
The market return noise ut and the innovation of efficient price et are i.i.d. random variables
with mean zero, constant variances, and zero covariance. Further we impose the normality
assumption for ut and et for simplicity. Combining (1) and (2) gives the following equation
for the market price pt

pt = (1 − g)pt−1 + gmt−1 + ut + get .

The expectation of pt conditional on the information up to time t − 1 implies that thecurrent market price is determined between the previous market price and the efficient priceon average, i.e. the market price is partially adjusted to the efficient price. The magnitudeof adjustment coefficient g represents the state of price discovery. For the case of g = 1, thecurrent market price pt is appropriately adjusted to the efficient price on average. g < 1(> 1) represents the under (over) reaction of traders to new information. In addition, we
3

suppose g ∈ (0, 2) to ensure the stationarity of the return process. Hereafter, we call thismodel as the constant partial adjustment model.
2.2. Time varying adjustment model
As in Biais et al. (1999), it is interesting to assess the change of state of price discovery
through various economic events, e.g. before and after the opening period and earnings
announcement. In the following, we extend the constant partial adjustment model to thetwo types of the time-varying partial adjustment models, which are based on the smoothtransition autoregressive model proposed in Teräsvirta (1994) and the threshold autoregressive model proposed in Tong (1978). The smooth transition partial adjustment model isintroduced to depict the gradual change of state of price discovery as follows∆pt = st (mt − pt−1 ) + ut ,(3)

mt = mt−1 + et ,−1τt − cst = a1 + a2 1 + exp −γ
στ

(4)
(5)

where τt = t/T (t = 2, . . . , T ) and στ is the standard deviation of τt which is used tostandardization. Eq. (5) contains the logistic function of τt with the speed parameter γ > 0
and the location parameter 0 < c < 1. The adjustment coefficient st transits smoothly
from a1 to a1 + a2 where the parameter restrictions a1 , a1 + a2 ∈ (0, 2) are imposed for
non-explosive condition of ∆pt . As depicted in Figure 1, the adjustment coefficient stays
at a1 + a2 /2 when γ = 0, and instantaneously changes from a1 to a1 + a2 at τt = c whenγ = ∞. These are the special case of the smooth transition partial adjustment model.
We adopt the Markov chain Monte Carlo (MCMC) method for the estimation of thesmooth transition model because the maximization of the log likelihood without any restrictions is difficult, specially with respect to γ. As shown in Ekner and Nejstgaard (2013), the
4

γ=0
γ=1
γ=3
γ = 201.0

0.8

switching

0.6

0.4

0.2

0.0
0.0

0.2

0.4

0.6

0.8

1.0

Figure 1: Shape of adjustment coefficients st = (1 + exp{−γ(τt − 0.5)/στ })−1 with each speed parameter(γ = 0, 1, 3, 20) and switching adjustment coefficient with threshold c = 0.5. στ is standard
deviation of τt = t/T .

first partial derivative of the log likelihood function with respect to γ takes 0 at γ = ∞.
This implies that the likelihood becomes at least locally maximum at γ = ∞ even if thetrue γ is not ∞. Moreover, Figure 1 shows that the γ in (5) which takes the value over
20 makes the path of adjustment coefficient by the smooth transition model virtually indistinguishable from that by the threshold model which will be given below, see also Gerlach
and Chen (2008). Depending on the purpose of analysis, the accuracy of distinguishing thedifference between the paths from the two different models will differ. For our purpose of
accurately capturing the state of price discovery, it is not necessary to distinguish between
the smooth transition model with γ of 20 or more and the threshold model. Instead of thesmooth transition model with the large value of γ, we provide the following threshold model

5

for the case of abruptly changing st as an alternative.

∆pt = st (mt − pt−1 ) + ut ,(6)

mt = mt−1 + et ,(7)

st = a1 + a2 Iτt ≥c , c ∈ {2/T, 3/T, . . . , 1}.

(8)Eq. (8) makes st switching abruptly from a1 to a1 + a2 instead of smooth transition with (5).
As described above, for the case of γ = ∞, the smooth transition partial adjustment model
collapses to the threshold partial adjustment model. For the threshold partial adjustment
model (6)–(8), the non-explosive condition (a1 , a1 + a2 ∈ (0, 2)) is also necessary. Hereafter
we refer (1)–(2), (3)–(5), and (6)–(8) as models C, S, and T for brevity.
2.3. Restrictions for parameters
We have to consider an identification problem for models S and T . It is obvious that theidentification problem for the parameters γ and c is caused by imposing a2 = 0 on model S
for the expression of constant adjustment coefficient and the same problem occurs for theparameter c in model T . Therefore, we impose the restriction a2 6= 0 for the exclusion of
constant adjustment coefficient case from the scope of models S and T .
By restricting parameters in each model and selecting appropriate model from the candidates, we can avoid the identification problems and categorize the path of adjustment
coefficient into one of three different types: constant, smoothly time-varying, and switchingabruptly, with respective corresponding models.
3. Bayesian inference
We estimate all three candidate models and carry out a model selection, using Bayesian
approach. The inference based on the posterior distribution through Markov chain Monte
Carlo (MCMC) is convenient to incorporate parameter restrictions described in the previous
6section. Further, the Bayesian approach adopted in this paper does not involve the numerical
optimization unlike the maximum likelihood method which is often associated with somedifficulties in maximizing the log likelihood and obtaining the standard error of estimators
for the inference. In following, we provide the Bayesian estimation and model selection indetail.
3.1. Prior distributions
We first introduce the prior distributions for the speed parameter γ and the location
parameter c of model S. For the reason mentioned in subsection 2.2, we adopt the truncatednormal prior for log γ, whose mean µγ , variance σγ2 , and the lower and upper bounds log `γ
and log uγ , that is, log γ ∼ T N (µγ , σγ2 )(log `γ ,log uγ ) to distinguish between model S and model
T . For the location parameter c in model S, we set c ∼ T N (µc , σc2 )(`c ,uc ) as the priordistribution. For the case of model T , we use the discrete uniform prior for c with thesame lower and upper bounds `c and uc as used for model S, i.e. `c < uc and `c , uc ∈{2/T, 3/T, . . . , 1}. In addition, as we mentioned earlier, we need to restrict the range ofadjustment coefficient for the non-explosiveness of return process and identification problems
in models S, T . So we set the following truncated normal prior with the positive lower bound:a1 ∼ T N (µa1 , σa21 )(`a1 ,2) . a2 needs to satisfy a2 ∈ (−a1 , 2−a1 ) for the non-explosive condition
and a2 6= 0 for the identification problem. Because we focus on how the price discovery
progress, i.e. the non-decreasing adjustment coefficient case in this paper, we assume theconditional truncated normal prior for a2 as follow a2 |a1 ∼ T N (1 − a1 , σa22 )(`a2 ,2−a1 ) where`a2 > 0. For g which represents the state of price discovery in model C, similar to a1 anda2 , we adopt the same prior as a1 . For the noise variance σu2 , we use the inverse gamma
distribution with shape parameter αu and rate parameter βu , that is IG(αu , βu ), as the
conjugate prior in the simulation study and the uniform prior U (`u , uu ) in the empirical2
study. For the sampling procedure of σmwhich is the innovation variance of unobservableefficient price mt , we conduct ancillarity-sufficiency interweaving strategy (ASIS) proposed7

in Yu and Meng (2011). We will give the prior distribution and the sampling procedure2of σmin the following subsection. Overall, we assume that the prior distributions for eachparameter are independent each other except for a1 and a2 . The detail of sampling procedure
is described in following.
3.2. Sampling procedure
Firstly, we describe the sampling procedure for model S. Denote Θ the parameter set of2the target model, here Θ = (γ, c, a1 , a2 , σm
, σu2 ).

step 0. Determine Θ(0) and set i = 1.
(i)

(i)

(i)

(i)

step 1. Draw the state variables me T = (m2 , m3 , · · · , mT ).
step 2. Draw individually each parameter in Θ(i) .
step 3. Repeat step1 and step2 for i = 1, . . . , N .
In estimating the other candidate models, we similarly use the same procedure for the otherparameter set in those models.
As for step 1, by combining with the Carter and Kohn (1994) algorithm, Gibbs samplingis possible to obtain a draw for the unobservable efficient price meT.
As for the sampling method of γ and c in model S, we implement the MetropolisHastings (MH) algorithm. In drawing proposals of γ and c, we adopt the random walk MH
algorithm during burn-in period (first M iterations) and the adaptive MH algorithm afterburn-in period (another M ∗ iterations). The proposal mean and variance of the adaptive
MH algorithm are chosen to be the sample mean and variance for the last quarter of burn-in
period. As for the random walk MH algorithm, we use the following proposal distribution
for γ

γ|γ

i−1


∼G

∆γ
∆γ
, i−1 2
i−1
γ
(γ )
8


(9)
(`γ ,uγ )where G(α, β)(a,b) denotes a truncated gamma distribution with shape parameter α, rate
parameter β, lower bound a, and upper bound b, and ∆γ is the pre-determined constantfor variance of γ. For c of model S, we conduct following sampling method to restrict theparameter space of c. Firstly, we draw a random variable x∗ from N (µx , σx2 ) and performthe following transformation to generate c∗ ,c=exp(x)(uc − `c ) + `c , c ∈ (`c , uc ).
1 + exp(x)After the burn-in period, random samples of γ and x are obtained from the updated proposal
distribution.
2, and σu2 ), Gibbs
As for the sampling of the other parameters (c in model T , g, a1 , a2 , σmsampling is adopted. Using the conjugate prior distribution described in the previous section,
we can easily obtain samples of the parameters a1 , a2 , g, and σu2 by Gibbs sampling. The fullconditional posterior distributions of a1 , a2 , and g are the truncated normal distribution andthat of σu2 is the inverse gamma distribution. Note that the posterior distribution of σu2 is thetruncated inverse gamma distribution when we adopt the uniform prior as σu2 ∼ U (`u , uu ).
As for c in model T , we employ the method to generate a change point as in Carlin et al.
(1992).
2
For the estimation of σmin all candidate models, we conduct ASIS to improve thesampling efficiency. By setting mt = m∗t σm and et = e∗t σm , we can reparameterize models
C, S, and T . For example, models S and T can be rewritten as∆pt = st (m∗t σm − pt−1 ) + ut ,(10)m∗t = m∗t−1 + e∗t(11)where σm is shifted from the state equation to the observation equation. The model C
9in another parameterization is obtained by setting st = g in (10). The full conditional
posterior distribution of σm is a truncated normal by setting the prior density as σm ∼T N (µ0 , σ02 )(0,∞) . In the parameterization utilized in the previous section, the full conditional2
is an inverse gamma distribution when we use an inverse gamma
posterior distribution of σmdistribution as the prior whereas a truncated inverse gamma distribution using a uniform
2
prior. The ASIS we adopt for σmis the following
2
Step 1. Calculate me ∗T = me T /σm . Here we consider Θ contains σm instead of σm. Draw σm

from

σm |e
pT , me ∗T , Θ−σm ∼ T N (µ1 , σ12 )(0,∞) , (Ancillary)PT(st m∗ )(∆pt + st pt−1 )σ02 + µ0 σu2 2
σ02 σu2,σ=µ1 = t=2 PtT
P
1
T
∗ 2 2
∗ 2 2
2
2
t=2 (st mt ) σ0 + σut=2 (st mt ) σ0 + σu
where peT = (p1 , . . . , pT ), st is given by (5) for model S, (8) for model T and g formodel C, and µ0 , σ02 are the hyper-parameters of the prior distribution T N (µ0 , σ02 )(0,∞)
for σm .
2
fromStep 2. Calculate meT = me ∗T σm and draw σm22 ∼ IG(α1 , β1 ),σm|epT , me T , Θ−σm
(Sufficient)

T −1
α1 =
+ αm ,2T

1X
β1 = βm +(mt − mt−1 )2
2 t=3

where αm , βm are the hyper-parameters of the prior distribution IG(αm , βm ) for22σm. When we set the prior density as σm∼ U (`m , um ), draw from the truncatedinverse gamma posterior with shape parameter α1 and rate parameter β1 whereαm = −1, βm = 0.
For detail of ASIS, see Yu and Meng (2011) and Kastner and Frühwirth-Schnatter (2014).
In order to select the appropriate path type of adjustment coefficient, we conduct the10Bayesian model selection. The model selection criteria we adopt are Deviance Information
Criterion (DIC) of Spiegelhalter et al. (2002) and the marginal likelihood method of Chib
(1995) and Chib and Jeliazkov (2001). We will evaluate these selection criteria in thefollowing simulation analysis.

4. Simulation analysis
We are interested in whether the selection criteria provide an appropriate model withrespect to the transition type of adjustment coefficient and the signal to noise ratio for theobserved data. In this section, we conduct a simulation analysis to see how our proposed
estimation method and model selection criteria work.
4.1. Settings for estimation and simulation
For the estimation of three candidate models, we apply the sampling scheme describedin the previous section. We employ M = 10000 iterations as burn-in and another M ∗ =15000 iterations for posterior sampling of parameters and calculating DIC. Chib’s marginal
likelihood requires another 5500 iterations where we discard the first 500 iterations as burnin. In the calculation of Chib’s marginal likelihood of the model S, we conduct the same
adaptive sampling method as in the estimation of the model S.
The simulation data is generated from the data generating process (3) and (4) in various
2
settings for the transition speed γ and the signal to noise ratio ν ≡ σm/σu2 . The speed

parameter is set as γ = 0 (st is constant), γ = 5, 15 (st is smoothly time varying), and
γ = 150, ∞ (st switches abruptly). Because model S collapses to model T when γ = ∞,we generate the simulated data using model T in the case of γ = ∞. The signal to noise
ratio is set as ν = 5, 10 (efficient case, which is the case where the information for efficient
price predominates at the observed price) and ν = 1 (noisy case). We provide the simulated
data with 15 settings with γ = 0, 5, 15, 150, ∞ and ν = 1, 5, 10. We set other parameters as
11

Table 1: Model selection result using Bayes Factor and DIC
model type
C
S
S
T
T

setting of γ
0
5
15
150
∞

ν=1
BF DIC
98
49
80
75
30
52
96
93
97
92

ν=5
BF DIC
99
37
100 100
89
93
98
96
100 100

ν = 10
BF DIC
99
44
100 100
97
99
98
97
100 100

Figures in table are the proportion (in %) of correct selection. 100 replications are
conducted for each setting. BF in this table represents Bayes Factor calculated by
Chib’s method.

c = 0.5, a1 = 0.1, a2 = 0.9, σu2 = 0.001, and initial value of log price p1 = log 5000 and log
efficient price m1 = log 5050. The sample size of simulated data is set to 1800, assuming 30
minutes observations in seconds and the log price series multiplied by 100 is used to display
the return in percent. We generate 100 replications for all settings of γ and ν.
For the settings of boundary, we set `γ = 1, uγ = 20, `c = 0.1, uc = 0.9, `g =`a1 = 0.01, `a2 = 0.05. For the prior distributions, we set log γ ∼ T N (10, 32 )(0,log 20) ,c ∼ T N (0.45, 0.52 )(0.1,0.9) (for the model S), g ∼ T N (0.4, 52 )(0.01,2) , a1 ∼ T N (0.4, 52 )(0.01,2) ,2
a2 |a1 ∼ T N (1 − a1 , 52 )(0.05,2−a1 ) , σu2 , σm
∼ IG(10−3 , 10−3 ), σm ∼ T N (0, 32 )(0,∞) . For thehyper-parameters of the proposal distributions, we set ∆γ = 2.52 , σx2 = 0.032 . In the following, we present the results of model selection and estimation for each setting of simulation.
4.2. Model selection result
First of all, we define the correct model selection in this simulation study as follows: theselection is correct when the model T is selected in the case of γ = 150, ∞, the model S is
selected in the case of γ = 5, 15, and the model C is selected in the case of γ = 0. In thefollowing, we will compare the model selection criteria in terms of the correct selection ratio.
To assess the statistical inference proposed in this paper, we conduct the estimation andmodel selection iteratively 100 times with the simulated data for each parameter setting.
Table 1 shows the ratio of correct selection out of 100 replications. In the efficient case(ν = 5, 10), Chib’s method performs well for all settings of γ while DIC does not for the case12

of γ = 0 (the model C is correct). In the noisy case (ν = 1), Chib’s method is less accurate
for the case of γ = 5, 15 (the model S is correct) while DIC is also less accurate for the caseof γ = 0, 5, 15. We will see the case of (γ, ν) = (15, 1) in detail below.
For convenience sake, we denote Bayes factor in favor of model M0 and against modelM1 as BF(M0 /M1 ). We select the model M0 when log BF(M0 /M1 ) > 0. The first andsecond rows of Figure 2 depict boxplots of log BF(correct model/incorrect model). Boxplotsof log BF(C/S) and log BF(C/T ) are omitted except for the case of γ = 0 because none oftwo criteria select the model C when γ > 0. The boxplot of log BF(S/T ) for γ = 15 indicatesthat model T tends to be incorrectly selected instead of model S. In general, model S andmodel T become indistinguishable as the observed price data become noisy.
The third and fourth rows of Figure 2 provide the boxplots of difference of DICs forthe correct model and the incorrect model, that is DICcorrect − DICincorrect . We select modelM0 against model M1 when DICM0 − DICM1 < 0. The boxplots for DIC show the sametendency as the selection using the Bayes factor for the case of less informative observed
price data. Moreover, selection using DIC differences does not clearly distinguish between
model C and other model types when the model C is correct whereas the selection based on
Bayes factor does.
Regarding DIC vs Chib’s marginal likelihood approach, the performances of both criteria
are almost the same or Chib’s marginal likelihood approach is superior to DIC except forthe case of (γ, ν) = (15, 1). Although neither method makes a clear distinction between
model S and model T when we use the noisy data, Chib’s method can distinguish between
model C and other model types more accurately than DIC does. From those results, Chib’s
method is suitable for our purpose. We will use Chib’s method to select appropriate modelfor empirical study in the next section.

13γ = ∞: logBF(T/S)γ = 150: logBF(T/S)

30

20

20

20

10

10

10

0

0

0

ν=5

ν = 10ν=1

γ = 5: logBF(S/T)ν=5ν = 10

40

10

20

5

0

0

ν=1

γ = 0: logBF(C/S)

1560

γ = 15: logBF(S/T)

30

ν = 10

ν=5

ν=1

ν = 10

ν=5

ν=1

ν = 10

ν=5

ν=1

ν = 10ν=5

ν=1

γ = 0: logBF(C/T)

8
6
4
2
0

ν = 10

ν=5

ν=1

ν=5

ν = 10γ = ∞: DICT − DICS

ν=1

−2

γ = 150: DICT − DICS

γ = 15: DICS − DICT

0
0

0

−20
−20

−20
−40

−40
−40

−60

−60

ν = 10

ν=5

ν=1

ν = 10γ = 5: DICS − DICT

ν=5

ν=1

γ = 0: DICC − DICS0

γ = 0: DICC − DICT

15

5
−25

10
0

−50

5

−75
−5

0

−100
−125

ν = 10

ν=5

ν=1

−10

−5

ν = 10

ν=5

ν=1Figure 2: Boxplots of log of the Bayes factor calculated by Chib’s method and the difference of DIC foreach setting of γ and ν. The lower whisker is at the lowest data above Q1 − 1.5(Q3 − Q1 ) andthe higher whisker is at the highest data below Q3 + 1.5(Q3 − Q1 ) where Qi is i-th quartile. The
points which are outside the whiskers are defined as outliers and they are plotted as individual
points.

4.3. Estimation result of the adjustment coefficient and parameters
We also report the estimation result of st and all parameters. For brevity, ŝt denotes theestimate of the adjustment coefficient, that is, ŝt = â1 + â2 (1 + exp{−γ̂(τt − ĉ)/στ })−1 in
14

DGP: γ = ∞, model: 

1.0

DGP: γ = 150, model: 

1.0
0.5

0.5
0.0

0.2

1.0

0.4

0.6

DGP: γ = 15, model: 

0.8

1.0

0.0

0.2

0.0

0.2

0.0

0.2

1.0

0.4

0.6

0.8

1.0

0.4

0.6

0.8

1.0

0.4

0.6

0.8

1.0

DGP: γ = 15, model: 

0.5

0.5
0.0

0.2

1.0

0.4

0.6

DGP: γ = 5, model: 

0.8

1.0
1.0

DGP: γ = 0, model: 

0.5

0.5
0.0

0.2

0.4

0.6

0.8

1.0

0.0

Figure 3: st = 0.1+0.9(1+exp{−γ(τt −0.5)/στ })−1 for the simulation data with γ = 0, 5, 15, 150, ∞ when
ν = 1 (noisiest case). For each figure, true st is plotted by solid line and 5 and 95 percentilesof ŝt in 100 replications are plotted by the upper and lower dotted lines.

model S , ŝt = â1 + â2 Iτt ≥ĉ in model T , and ŝt = ĝ in model C (θ̂ represents a posterior mean
estimate of θ). We have similar estimation results of st in all cases of ν. So we report only forthe noisiest case (ν = 1). Figure 3 represents the result of ŝt for each setting of γ. For eachsetting, we plot true st and 5 and 95 percentiles of ŝt obtained by 100 replications. In Figure
3, the most selected model in 100 replications is denoted as “selected model”. Only for thecase of γ = 15, the selected model differs from the model that provides the simulated data.
Overall, the interval between 5 and 95 percentiles of ŝt by the selected model is moderately
narrow and contains true st including the case of γ = 15.
Finally, we provide the summary statistics of posterior mean of parameters for 100 replications in Table 2. Similar to ŝt , we report the estimation results of models S and T only forthe case of (γ, ν) = (15, 1). For the remaining easily distinguishable cases (γ = 0, 5, 150, ∞),we report only the results of the model selected by Chib’s method, which is a correct model.
The selected model name is below the setting of γ in Table 2.
15

Table 2: Mean and standard deviation of the posterior mean for each parameter in 100 replications
γ=0model: C
Mean
Std.
γ
c
a1
a2
g
σu2
2
σm
ν

0.55030.00100.00101.02590.04280.00010.00010.1595γ=0model: C
Mean
Std.

16

γ
c
a1
a2
g
σu2
2
σm
ν

0.53600.00110.00505.22140.05880.00020.0004
1.2223γ=0model: C
Mean
Std.
γ
c
a1
a2
g
σu2
2
σm
ν

0.53270.00110.0102
10.5577

0.04640.00030.00082.8886γ=5model: S
Mean
Std.
7.7064 3.0786
0.5041 0.0241
0.1106 0.0304
0.8848 0.0629
0.00100.00101.01310.00010.00010.1537

ν=1
γ = 15model: S
Mean
Std.
14.0650 2.3390
0.5003 0.0118
0.1120 0.0261
0.8991 0.0490

γ = 15model: T
MeanStd.

γ = 150
model: T
Mean
Std.

γ=∞
model: T
Mean
Std.

0.5036
0.11480.86730.01720.0266
0.04620.50070.11450.89080.0046
0.03490.04260.50010.1066
0.89430.00440.02620.04610.0010 0.0001
0.0010 0.0001
1.0444 0.1188
ν=5
γ = 15model: S
Mean
Std.
14.9304 2.1573
0.5007 0.0063
0.1022 0.0222
0.9036 0.0346

0.00100.00101.00740.00010.00010.12060.00100.00101.01500.00010.00010.12950.00100.00101.03230.00010.00010.1198γ = 150
model: T
Mean
Std.

γ=∞
model: T
Mean
Std.

0.50000.10540.89820.00240.02020.03230.49990.10150.89740.00130.02220.03330.00100.00505.01040.00010.00040.49950.00100.00515.12900.00010.00040.4883γ=5model: S
Mean
Std.
5.2748 0.8084
0.5014 0.0110
0.1022 0.0212
0.9005 0.0371

0.0010 0.0001
0.0050 0.0004
5.0716 0.4996
ν = 10γ = 15model: S
Mean
Std.
15.1396 2.0692
0.5006 0.0050
0.1003 0.0155
0.9040 0.0314

0.00100.0101
10.17860.00100.010010.1080

γ=5model: S
Mean
Std.
5.5829 1.1481
0.5019 0.0135
0.1055 0.0273
0.8965 0.0417
0.00100.00505.09470.00010.0004
0.63740.00010.00091.24120.00010.00090.9981

γ = 150
model: T
Mean
Std.

γ=∞
model: T
Mean
Std.

0.49930.10320.89900.00200.01560.03050.50000.10090.89720.00100.01590.02940.00100.010010.00190.00010.00080.96540.00100.0102
10.26410.00010.00070.97462
ν = σm/σu2 and the true values of parameters are a1 = 0.1, a2 = 0.9, c = 0.5, g = 0.55, σu2 = 0.001.

For all settings of γ, true values of all parameters are included within two standard
deviations of their means. The posterior mean of γ tends to deviate slightly from the true
value. However, for our purpose, what is important is that the path of estimated ŝt is close
to the true st , not whether the estimate of γ is highly accurate.
In fact, we have selected the incorrect model T instead of the correct model S for thecase of (γ, ν) = (15, 1). But the implication from the estimated path of st is almost thesame as that from the true path of st for the following reasons. In this case, the true startand end points of change in the adjustment coefficient are very close, the estimated change
point ĉ by the incorrectly selected model T is in between and the parameters a1 and a2 arecorrectly estimated.

5. Empirical analysis
We apply our proposed method to the financial time series data observed in Tokyo Stock
Exchange (TSE) to examine how the intraday price adjustment during the preopening periodvaries on a daily basis. The data for this study is TOPIX Core30 which is a market index
composed of the 30 issues with the highest market capitalization and liquidity in the firstsection of TSE. The sampling frequency of the data we use is one second and the sample
period is 15 minutes before and after the opening auction at 9:00 am, i.e. from 8:45 to 9:15,
on the five trading days of the second week of June, 2018.
2
The same prior distributions are adopted as in subsection 4.1 except for σu2 and σm. We
2use a uniform prior U (0, 1) for σmand σu2 . The settings of iterations for the MCMC sampling

and obtaining the marginal likelihood for each model are also the same as in subsection 4.1.
Our proposed method clearly categorized the intraday price discovery progresses on adaily basis as shown in Table 3. Table 4 summarizes the posterior samples of parameters
for the selected model on each day. All the absolute values of CD are smaller than 2 except
a1 for June 4 and the inefficient factor ranges from 1.245 to 71.989, suggesting that the17Table 3: Result of model selection for TOPIX Core30

June 4, 2018
T is selected

Model
S
T
C

Log of marginal likelihood S.E.
5961.97
0.15356114.26
0.03445981.95
0.1901

June 5, 2018
T is selectedSTC

7135.727153.396989.66

0.05580.04400.1336

June 6, 2018
T is selectedSTC

6694.01
6992.506597.38

0.17730.03050.1610June 7, 2018
S is selectedS
TC

6913.056741.30
6712.25

0.04510.07180.1295June 8, 2018
C is selectedSTC

2662.852683.932705.770.08380.09330.0945

sampling algorithm adopted in this study works well. As a point to note, CD and Inef forJune 4, 5 and 6 for the change point c of model T were not available because the posterior
sample distributions were degenerated. At these change points, the observed data clearlychanged their distributions as depicted in Figure 4. As the model T , i.e. eqs. (6)–(8),suggests a high adjustment coefficient a1 + a2 contributes to increase the observed return
variance relative to the return variance when the adjustment coefficient is a1 .
The estimated paths of adjustment coefficient in Figure 5 show that how the price discovery works is also various for each day. We find that the improvements in the priceinformational efficiency were observed on June 4, 5, 6, and 7, and the progresses of the first
three were sharp and the last one was gradual. The adjustments started to change just before the market opening period of June 5, 6, and 7. In contrast, the information efficiency for
June 8, 2018 was low and constant even before the market opened, and remained constantafter the market opened. It is worth noting that all of the estimated price adjustments were18

Table 4: Summaries of posterior sample
Date
June 4, 2018
model T

June 5, 2018
model T

June 6, 2018
model T

June 7, 2018
model S

June 8, 2018
model C

c
a1
a2
σu2
2
σm
ν
c
a1
a2
σu2
2
σm
ν
c
a1
a2
σu2
2
σm
ν
γ
c
a1
a2
σu2
2
σm
ν
g
σu2
2
σm
ν

Mean
0.108
0.012
0.581
2.753E-05
7.082E-05
2.583
0.480
0.016
0.511
1.052E-05
5.838E-05
5.561
0.434
0.011
0.710
1.360E-05
2.572E-05
1.895
19.521
0.395
0.013
0.632
9.544E-06
7.360E-05
7.733
0.297
1.278E-03
9.917E-03
7.793

Std.
0.000
0.002
0.028
1.757E-06
5.905E-06
0.276
0.000
0.006
0.033
5.519E-07
6.448E-06
0.673
0.000
0.001
0.027
5.649E-07
2.162E-06
0.174
0.383
0.003
0.003
0.032
5.322E-07
6.985E-06
0.831
0.042
1.488E-04
1.446E-03
0.984

CI95 lower
0.108
0.010
0.527
2.428E-05
6.016E-05
2.087
0.480
0.010
0.446
9.490E-06
4.708E-05
4.379
0.434
0.010
0.658
1.253E-05
2.192E-05
1.583
18.547
0.388
0.010
0.571
8.576E-06
6.118E-05
6.240
0.223
9.780E-04
7.533E-03
6.048

Median
0.108
0.011
0.581
2.747E-05
7.052E-05
2.566
0.480
0.015
0.511
1.051E-05
5.795E-05
5.508
0.434
0.011
0.710
1.358E-05
2.560E-05
1.887
19.604
0.395
0.012
0.632
9.524E-06
7.314E-05
7.688
0.296
1.281E-03
9.733E-03
7.736

CI95 upper
0.108
0.016
0.636
3.112E-05
8.319E-05
3.170
0.480
0.030
0.577
1.165E-05
7.232E-05
7.032
0.434
0.013
0.762
1.475E-05
3.030E-05
2.262
19.981
0.401
0.021
0.695
1.066E-05
8.832E-05
9.513
0.387
1.560E-03
1.315E-02
9.859

CD
NA
2.908
0.247
-0.053-0.169-0.208NA
0.533-0.479-0.448
0.159
0.288
NA
0.882
-0.3051.019
0.138-0.106-0.346-1.4140.084-1.2541.989
0.733-0.0191.397-1.355-1.544-0.926

Inef.
NA
2.146
16.36010.068
11.6368.188
NA10.22221.8036.736
24.43218.627NA
1.245
10.5142.697
13.06910.098
4.716
5.009
3.151
26.2447.170
24.93215.93071.98956.89856.83916.567In this table, the mean, standard deviation (Std.), 95% credible interval (CI95 lower and upper),median, z-value of convergence diagnostic test (CD) by Geweke (1992), and inefficiency factor (Inef.)by Chib (2001) for each sample and parameter are given.

below unity, which implies the adjustments were the under reaction to new information,
especially that of June 8 was remarkably low.
6. Conclusion
This paper proposes a Bayesian approach to evaluate the intraday price discovery on adaily basis. Based on Amihud and Mendelson (1987), we introduce three candidate partial
adjustment models: the models S, T , and C whose adjustment coefficient is smoothly timevarying, switching abruptly, and being constant, respectively.
19





June 4, 2018:

June 5, 2018:

June 6, 2018:
10040

10000

10050

10020
10025

9980

10000

10000
8:30

9:00

8:30

9:00

8:30

9:00

8:30

9:00

8:30

9:00

0.00050.001

0.001
0.0000

0.000

0.000
−0.001
8:30

9:00

8:30

9:00

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0.0
8:30

9:00

0.0
8:30

9:00

Figure 4: Price (top), return (middle) and the estimated path of adjustment coefficient (bottom) for June4, 5 and 6, 2018, when the model T is selected.

June 4, 2018:

June 5, 2018:

June 6, 2018:

1.0

1.0

1.0

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.0

0.0
8:45

9:00

9:15

0.0
8:45



9:00

9:15

8:45

9:00

9:15



June 7, 2018:

June 8, 2018:

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0
8:45

9:00

9:15

8:45

9:00

9:15

Figure 5: The estimation example of ŝt for the real data from TOPIX Core 30 index

A simulation study shows that the proposed method aptly categorizes the type of pricediscovery. Though the model selection between model S and model T tends to be difficult
for the case of less informative observed price data, the adjustment coefficient, that is the20most important target, is correctly estimated even in such a case. The empirical study
illustrates our methods using intraday high frequency time series of TOPIX Core30 of Tokyo
Stock Exchange. We confirm that the proposed method gauge the price adjustment progress
properly and find that the adjustment already starts before market opening via the indicative
price information during preopening period.
We have considered the single transition scheme for the price adjustment in this paper.
For the analysis of price discovery during the preopening period, the single transition isappropriate because the information that flows into the market is accumulated and reflectedin the price as time passed whereas the analysis for the intraday period from open to closerequires more flexible transition scheme. Future work could consider a model with a generalpath of progress of price discovery by considering three or more states.

References
Amihud, Y. and H. Mendelson (1987): “Trading mechanisms and stock returns: an empirical investigation,” Journal of Finance, 42, 533–553.
Biais, B., P. Hillion, and C. Spatt (1999): “Price discovery and learning during the preopening period
in the Paris Bourse,” Journal of Political Economy, 107, 1218–1248.
Carlin, B., A. Gelfand, and A. Smith (1992): “Hierarchical Bayesian analysis of changepoint problems,” Applied Statistics, 41, 389–405.
Carter, C. and R. Kohn (1994): “On Gibbs sampling for state space models,” Biometrika, 81, 541–553.
Chib, S. (1995): “Marginal likelihood from the Gibbs output,” Journal of the American Statistical Association, 90, 1313–1321.
——— (2001): “Markov Chain Monte Carlo method: computation and inference,” in Handbook of Econometrics, ed. by J. Heckman and E. Leamer, North Holland, Amsterdam, vol. 5, 3569–3649.
Chib, S. and I. Jeliazkov (2001): “Marginal likelihood from the Metropolis-Hastings output,” Journalof the American Statistical Association, 96, 270–281.
Ekner, L. and E. Nejstgaard (2013): “Parameter identification in the logistic STAR model,” Discussion
paper, Department of Economics, University of Copenhagen.

21Gerlach, R. and C. Chen (2008): “Bayesian inference and model comparison for asymmetric smoothtransition heteroskedastic models,” Statistics and Computing, 18, 391–408.
Geweke, J. (1992): “Evaluating the accuracy of sampling-based approaches to the calculation of posterior
moments,” in Bayesian Statistics 4, ed. by J. Bernardo, J. Berger, A. Dawid, and A. Smith, Oxford:
Oxford University Press, 169–193.
Hasbrouck, J. (2007): Empirical Market Microstructure, Oxford University Press.
Kastner, G. and S. Frühwirth-Schnatter (2014): “Ancillarity-sufficiency interweaving strategy(ASIS) for boosting MCMC estimation of stochastic volatility models,” Computational Statistics andData Analysis, 76, 408–423.
So, M. and R. Chan (2014): “Bayesian analysis of tail asymmetry based on a threshold extreme value
model,” Computational Statistics and Data Analysis, 71, 568–587.
Spiegelhalter, D. J., N. G. Best, B. P. Carlin, and A. Van Der Linde (2002): “Bayesian measuresof model complexity and fit,” Journal of the Royal Statistical Society. Series B: Statistical Methodology,
64, 583–616.
Teräsvirta, T. (1994): “Specification, estimation, and evaluation of smooth transition autoregressivemodels,” Journal of the American Statistical Association, 89, 208–218.
Tong, H. (1978): “On a threshold model,” in Pattern recognition and signal processing, ed. by C. Chen,
Amsterdam: Sijhoff and Noordhoff, 575–586.
Yu, Y. and X. L. Meng (2011): “To center or not to center: That is not the question-an ancillaritysufficiency interweaving strategy (ASIS) for boosting MCMC efficiency,” Journal of Computational andGraphical Statistics, 20, 531–570.

22