Synergistic Enhancement over Au-Pd/TS-1 Bimetallic Catalysts for Propylene Epoxidation with H2 and O2

∗

A ReMeDI for Microstructure Noise

†

‡

Z. Merrick Li

Oliver Linton

July 15, 2019

Abstract

We introduce the Realized moMents of Disjoint Increments (ReMeDI) paradigm to mea-

sure microstructure noise (the deviation of the observed asset prices from the fundamental

values caused by market imperfections). We propose consistent estimators of arbitrary ﬁ-

nite moments of a microstructure noise process, which could be serially dependent and

nonstationary, based on high-frequency data. We characterize the limit distributions of the

proposed estimators and construct robust conﬁdence intervals under inﬁll asymptotics.

We further demonstrate that the ReMeDI approach also works on low-frequency, non-inﬁll

data. It thus can be applied to many asset pricing and macroeconomic models, in which

the time series have a permanent and a transitory component.

We propose two liquidity measures that gauge the instantaneous and average bid-ask

spread with potentially autocorrelated order ﬂows. They can be consistently estimated

within our framework. We provide an economic model to justify such measures as an inter-

mediary’s inventory risk measure when meeting serially dependent liquidity demand. Em-

pirically we ﬁnd our new liquidity measures are very effective to identify liquidity drains

during the Flash Crash, when the market experienced extreme selling pressures.

KEYWORDS: Microstructure noise, liquidity measures, inventory models, order ﬂows, in-

ﬁll asymptotics, permanent and transitory components, Flash Crash

1

Introduction

Economic time series are often modelled as the sum of a latent process obtained from an un-

derlying economic model and another additive term that is either an error reﬂecting a variety

∗

Part of the paper was circulated under various titles, including, “A simple measure of microstructure noise”and

“

Robust measures of microstructure noise”. We beneﬁt from discussions with Yacine Aït-Sahalia, Torben Andersen,

Peter Boswijk, Rob Engle, Christian Gouriéroux, Peter Reinhard Hansen, Jean Jacod, Ilze Kalnina, Frank Kleiber-

gen, Yingying Li, Albert Menkveld, Per Mykland, George Tauchen, Michel Vellekoop, Dacheng Xiu, Xiye Yang,

Xinghua Zheng, and seminar participants at the 2017 North American Summer Meeting (St. Louis, June, 2017),

2

017 European Meeting (Lisbon, August, 2017) and 2018 China Meeting (Shanghai, June, 2018) of the Econometric

Society, International Conference on Quantitative Finance and Financial Econometrics (Marseille, June, 2018), the

2th Annual SoFiE Conference (Shanghai, June, 2019), Hong Kong University of Science and Technology, Univer-

1

sity of Amsterdam and University of Cambridge.

†

Department of Economics, University of Cambridge, Austin Robinson Building, Sidgwick Avenue, Cambridge,

CB3 9DD, United Kingdom. Email: z.merrick.li@gmail.com

‡

Department of Economics, University of Cambridge, Austin Robinson Building, Sidgwick Avenue, Cambridge,

CB3 9DD, United Kingdom. Email: obl20@cam.ac.uk .

1

Electronic copy available at: https://ssrn.com/abstract=3423607

of adjustments to the frictionless theoretical model, or a term that represents departures from

the economic model, thus

Y

=

X

+

ε

.

(1)

|

{z}

|{z}

observed series

underlying process

deviation

The two processes X and ε are generated by different mechanisms, and can have quite dis-

tinct statistical properties and economic interpretations. Both quantities may be of interest as

they give interpretation to the underlying economic theory and its relevance for the observed

data, or distinguish classes of economic models. However, since only the sum process Y is

observable, this makes estimation and inference challenging.

We are concerned with applications of this framework in ﬁnancial markets where the ob-

1

served asset price (Y) contains market microstructure noise (ε) that blurs the efﬁcient price (or

fundamental value) (X). The fundamental theorem of asset pricing says that X should be a

semimartingale process (Delbaen and Schachermayer (1994)). In practice however, there are

many market frictions that may cause observed prices to deviate from this ideal price, such

as: transaction costs, price discreteness, inventory holdings, information asymmetry, measure-

ment errors. One may also allow for temporary misspricing (French and Roll (1986)) or fads

effects (Lehmann (1990)); see also O’Hara (1995), Hasbrouck (2007) and Foucault et al. (2013)

for insightful reviews. A lot of early work proceeded on the basis that the microstructure noise

process was i.i.d., but recently this assumption has been shown to be too strong; both theo-

retically and empirically the microstructure noise may exhibit rich dynamics depending on

its sources. If the microstructure effects are negligible, the observed price should be close to

the efﬁcient price and be unpredictable. Therefore, the dispersion and persistence of the mi-

crostructure noise serve as natural measures of market quality. Market quality is of concern to

regulators and practitioners as well as academics; proxies for market quality are widely used

in empirical analysis, see Linton and Mahmoodzadeh (2018).

We next outline our contribution. We introduce a general econometric approach to mea-

suring microstructure noise in a nonparametric setting. First, we propose a new estimator of

the moments of a general dependent noise process called the Realized moMents of Disjoint Incre-

ments (ReMeDI) based on observed noisy high-frequency transaction prices. We assume that

the underlying efﬁcient price follows a semimartingale, which may accommodate stochastic

volatility, jumps, etc. We allow the microstructure noise to be weakly dependent and to have

serially correlation (of unknown form) that may decay algebraically; this may capture, for in-

stance, the effects of clustered (or hidden) order ﬂows (Gerig (2008)), or herding (Park and

Sabourian (2011)). The microstructure noise is allowed to have time-varying heteroskedastic-

ity, which allows for intraday variaiton in the scale of the noise. We develop the joint estimators

of arbitrary moments of microstructure noise and derive the associated limit distributions. We

provide a consistent estimator of the asymptotic variance that is carefully designed to improve

its ﬁnite sample performance. Our setting also allows for certain degrees of irregularities in

¹By price it always means the logarithmic price in this paper unless stated otherwise.

2

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the observation schemes. Statistically, we provide a general method to separate a nonstation-

ary and serially dependent sequence from a semimartingale; from an economic point of view,

we identify the components of asset prices arising from market frictions.

A second contribution we make is to propose two new liquidity measures based on the

autocovariance function of the microstructure noise. The ﬁrst measure can be considered as a

generalization of the Roll effective spread measure that is robust to autocorrelated order ﬂow

patterns. It measures the spread associated with a single trade, whence we think of it as an

instantaneous measure. The second measure is designed to capture the average spread of multi-

ple trades, where patterns of the order ﬂow play a role. We develop a simple inventory model

to provide economic insights about the average measure: it can be interpreted as an interme-

diary’s inventory risk when he meets a sequence of possibly autocorrelated high-frequency

liquidity demands that could reﬂect order splitting behavior or herding behavior, see Toth

et al. (2015) and references therein. We also show how to estimate and conduct inference about

these liquidity measures using our ReMeDI procedures.

Our third contribution is to develop several hypotheses tests about the microstructure

noise. In particular, we test for the intraday patterns by comparing the magnitudes of noise

in two intervals. We also provide tests that the microstructure noise is not autocorrelated. All

tests are model-free, and can detect alternatives with rich dynamics and model speciﬁcations.

Our fourth contribution is that the ReMeDI approach is robust to data frequencies in a sense

that it remains valid when applied to low-frequency data under long-span asymptotics. We show

that the ReMeDI estimators are able to effectively separate a process with independent incre-

ments (X) and a stationary mixing sequence (ε). It is a surprising yet intuitive property of the

ReMeDI design: the increments of X over disjoint intervals (the efﬁcient returns) are uncorre-

lated, and what remains is attributed to ε. This property not only distinguishes the ReMeDI

approach from alternative high-frequency estimators that rely structurally on the inﬁll asymp-

totics, but also broadens the range of potential applications in other ﬁelds of economics and

ﬁnance that utilize lower frequency data. In this setting the large sample variance is domi-

nated by the variation of the efﬁcient price process; we obtain a CLT for the estimator but at a

slower rate than in the inﬁll case. Finally, we provide a novel implementation rule to calculate

the asymptotic variance of the ReMeDI estimators if one feels uncertain about the underlying

asymptotic framework of his dataset.

We apply our methods to data from the E-mini stock index futures contract during the

month of May 2010. There are several interesting empirical ﬁndings. First, we ﬁnd that our

liquidity measures are lower during the "European" segment of the trading day, and that these

measures are fairly stable except during the Flash Crash day, May 6th, when the liquidity

measures during the "American" segment rose dramatically. We ﬁnd positive and signiﬁcant

autocorrelations only for the ﬁrst few lags on the non Flash Crash days, much less dependence

than for the individual stock case we discuss below. However, during the Flash Crash day this

autocorrelation became much stronger and signiﬁcant for many lags. This is consistent with

for example the positive feedback loop hypothesis (Beddington et al. (2012)).

3

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We also apply our method to individual stock price data from 2016 (presented in the sup-

plementary materials Li and Linton (2019)). We ﬁnd that the microstructure noise exhibits

strong and positive autocorrelations in the ﬁrst several lags, and has minor negative autocor-

relations after many ticks. The strong and positive autocorrelations induce signiﬁcant and

persistent differences between the three liquidity measures: the average measure tends to be

larger than the instantaneous one, and the classic Roll measure is further downward biased

relative to our instantaneous measure. The magnitude of noise in the ﬁrst ﬁve minutes is sig-

niﬁcantly larger than during the regular trading hours (9:35 — 15:55), but the magnitude of

noise in the last ﬁve minutes is not so statistically different.

The rest of the paper is organized as follows. Section 1.1 reviews the related literature.

Section 2 sets the continuous-time framework for the efﬁcient price and microstructure noise.

Section 3 introduces the ReMeDI estimators and outlines the intuition of the design. Section 4

presents the asymptotic theory. Section 5 and Section 6 demonstrate the economic and sta-

tistical applications. Section 7 extends the ReMeDI approach to low-frequency settings. The

empirical studies are presented in Section 8, and Section 9 concludes. All mathematical proofs,

simulation studies, the procedures to select the tuning parameters and additional empirical

studies on an individual stock are collected in the supplementary materials Li and Linton

(2019).

1

.1 Related literature

1

.1.1 High-frequency econometrics

There are a number of methods for estimation of the moments of noise and the parameters

of the efﬁcient price. Speciﬁcally: the two-scale/multi-scale realized volatility by Zhang et al.

(

2005), Zhang (2006), Aït-Sahalia et al. (2011); the optimal-sampling realized variance by Bandi

and Russell (2006, 2008); the maximum likelihood estimators by Aït-Sahalia et al. (2005), Xiu

2010); the pre-averaging method developed in Podolskij and Vetter (2009), Jacod et al. (2009);

(

and the realized kernel by Hansen and Lunde (2006), Barndorff-Nielsen et al. (2008). Most of

this literature only considers i.i.d. microstructure noise.

Several recent papers explore richer microstructure models by considering autocorrelated

noise. The estimators of the second moments of noise in Da and Xiu (2019) and Li et al. (2019)

are by-products of the integrated volatility estimators in the presence of autocorrelated noise.

In a recent seminal work, Jacod et al. (2017) introduce the ﬁrst estimator, called local averaging

(LA) method to measure arbitrary moments of microstructure noise using high-frequency data.

They also introduce a general framework of stochastic observation scheme and microstructure

noise with a semimartingale “size process”. Our main results are derived under slightly less

general assumptions than theirs, although we generalize the consistency result to the precise

setting of Jacod et al. (2017). We differentiate our paper from Jacod et al. (2017) as follows.

First, the ReMeDI method is based on differencing, while the LA is based on deviations from

local averages, both ideas are widely used in other contexts such as panel data to eliminate nui-

sance parameters. Second, the ReMeDI approach works beyond the inﬁll framework. It can

4

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be applied on low-frequency data and it has a limit distribution under the long-span asymp-

totics. This statistical property has an economically intuitive interpretation. Thus, the ReMeDI

estimators can be used by asset pricers with daily or coarser returns, or researchers in mar-

ket microstructure working on millisecond prices. The LA method, however, is inconsistent

when applied to low-frequency data. Third, we propose new measures of market liquidity

and provide economic interpretations. Next, even within the high-frequency framework, the

ReMeDI estimators are very robust. The LA estimators are more sensitive to data frequencies.

As showed by Jacod et al. (2017) that the LA estimators have a ﬁnite sample bias, which is a

fraction of the a priori unknown integrated volatility of the efﬁcient price. The bias could dom-

2

inate the noise parameters in practical situations, and this might cause many issues in the

3

implementations with real data. The bias of the ReMeDI estimators by contrast only depends

on the slope of the microstructure autocovariance function. Last, the ReMeDI approach has

4

another two advantages in real implementations: it is computationally very efﬁcient, and it

has a clear rule to select the tuning parameters.⁵

Chen and Mykland (2017) propose several tests of the intraday pattern, or stationarity of

microstructure noise. The tests are formed by comparing the TSRV developed by Zhang et al.

(2005) and a modiﬁed TSRV introduced by Kalnina and Linton (2008). We distinguish our

test of the intraday pattern from two aspects. First, our test is designed to improve the ﬁnite

sample performance. It is well established (see Hansen and Lunde (2006)) that the realized

volatility type estimators of noise variance have a ﬁnite sample bias component that is a frac-

tion of the integrated volatility of the underlying efﬁcient price process. Such bias term, as

analysed by Li et al. (2019), could wipe out the moments of noise in practical circumstances.

Thus to investigate the intraday patterns of microstructure noise, it is essential to effectively

separate microstructure noise and volatility in a ﬁnite sample, as the latter also exhibits promi-

nent intraday patterns, see Andersen and Bollerslev (1997). Second, we explicitly incorporate

autocorrelated microstructure noise. The TSRV approach of volatility estimation allows for

certain forms of weakly dependent noise, see Aït-Sahalia et al. (2011). But the statistical tests

developed in Chen and Mykland (2017) may not be applicable then, as the asymptotic vari-

ances of the test statistics depend on other higher moments of noise.

²The detailed analysis by Li et al. (2019) reveals that the bias is determined by both the data frequency and the

noise-to-signal ratio (the ratio of the variance of noise and the integrated volatility). Empirically, they show that

even using tick by tick data without any ﬁltration, the bias remains signiﬁcant.

3

In the empirical analysis by Jacod et al. (2017), the authors are puzzled about the strong dependence in the

microstructure noise. As we will show in the simulation studies that the strong and positive dependence in noise

after many lags is largely due to the ﬁnite sample bias of the LA estimators. However, correcting their bias in

practice is not trivial as one needs a proxy of the integrated volatility. Li et al. (2019) encountered a similar situation

and they propose a two/multi-step approach to make the bias correction.

4

For example, the LA (ReMeDI) takes 99.77% (0.23%) of the CPU time to estimate the variance of noise using

noisy price from a random walk plus AR(1) noise model, base on 1,000 simulated samples of size 23,400.

5

Jacod et al. (2017) also propose a heuristic rule to select the tuning parameters for the LA method by comparing

the LA estimates to a variant of the realized variances. However, the variant of the realized variances needs an

estimate of the integrated volatility to make a bias correction. The estimation of the integrated volatility in the

presence of serially dependent microstructure noise is not trivial (Li et al. (2019)). Moreover, the LA estimates have

a ﬁnite sample bias as well. Thus comparing two sequences of statistics both coupled with signiﬁcant biases may

lack insights on the choice of the tuning parameters.

5

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1

.1.2 Market microstructure

Our paper is also related to the market microstructure literature that seek to measure market

liquidity and study the impacts of order ﬂow patterns on market liquidity, see Roll (1984), Has-

brouck (1993), Aït-Sahalia and Yu (2009), Corwin and Schultz (2012) and Abdi and Ranaldo

(2017). The patterns of order ﬂows affect the measurement of market liquidity. Speciﬁcally,

Roll (1984) assumes i.i.d. order ﬂows; Hasbrouck and Ho (1987), Choi et al. (1988) consider an

AR(1) model with positive coefﬁcient to model the continuation of transactions; AR(1) models

with negative coefﬁcient are studied to model the inventory costs, see Ho and Stoll (1981), Hen-

dershott and Menkveld (2014). Some recent studies on infrequent rebalancing (Bogousslavsky

(2016)), limit attention (Hendershott et al. (2018)), or the interaction of high-frequency traders

and low-frequency traders (Aït-Sahalia and Sa g˘ lam (2017a,b)) reveal that the microstructure

noise has more complicated patterns. Our setting is more general, and our empirical studies

suggest the presence of more complex dynamics that fall outside the above standard frame-

works .

Our paper introduces an econometric approach to richer microstructure models. It aims

to integrate the market microstructure and ﬁnancial econometrics literature. It is, however,

not the ﬁrst attempt to push towards the integration of the two ﬁelds. Diebold and Strasser

(2013) focus on the correlation of efﬁcient price and noise in several leading microstructure

models, and study the implications for integrated volatility estimation. Li et al. (2016) model

the microstructure noise as a parametric function of the trading information and develop an

efﬁcient volatility estimator, see also Chaker (2017) and Clinet and Potiron (2017) for similar

approaches. Bandi et al. (2017) develop a novel measure of the staleness of stock returns under

the inﬁll asymptotic framework. Bollerslev et al. (2018) study the relationship between trading

volume and return volatility around important public news announcements using intraday

high-frequency data. The study relies critically on high-frequency econometric techniques to

identify jumps. Da and Xiu (2019) advocate the quasi-maximum likelihood approach to esti-

mate both the volatility and the autocovariances of moving-average microstructure noise.

1

.1.3 Macroeconomics and asset pricing

The random walk plus stationary component decompositions are popular in macroeconomics

and ﬁnance. The random walk part carries the permanent variations of the observed time series

while the stationary component carries the temporary or transitory ﬂuctuation. Cochrane (1988)

measures the random walk component of GNP growth, see also Cochrane (1994), Campbell

and Mankiw (1987). Beveridge and Nelson (1981) introduce a general method to decompose

time series into permanent and transitory components, and the latter is used to date the busi-

ness cycles, see recent studies by Oh et al. (2008) and Sinclair (2009). Summers (1986) spec-

iﬁes a random walk plus AR(1) process as an alternative hypothesis to market efﬁciency, see

also Poterba and Summers (1988) and Fama and French (1988).

The time series models in the above papers are essentially discrete, and estimations are

based on low-frequency data. The discrete-time model we considered in this paper is very

6

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general (see Section 7), nesting the popular random walk plus autoregressive processes, and

the ReMeDI approach remains effective to separate the two components in such settings.

2

Continuous-Time Framework and Assumptions

In this section we state precisely our assumptions regarding the continuous-time efﬁcient price

process, the observation scheme, and the microstructure noise.

2

.1 Efﬁcient price process

We assume that the efﬁcient price process X is an Itô semimartingale deﬁned on a ﬁltered

ꢀ

ꢁ

probability space Ω , F , {F_t}_t_≥₀, P⁽⁰⁾with the Grigelionis representation

(

0)

(0)

Z

ꢀ

ꢁ

ꢀ

ꢁ

t

X = X +

b_sds +

σ dW + ϑ1

? (p − q) + ϑ1

? p_t,

(2)

t

0

s

{|ϑ|≤1}

{|ϑ|>1}

t

0

where W, p are a Wiener process and a Poisson random measure on R × E respectively. Here,

+

ꢀ

ꢁ

(

0)

(0)

(

E, E) is a measurable Polish space on Ω , F , {F } , P

and the predictable compen-

t

t≥0

sator of p is q(ds, dz) = ds ⊗ λ(dz) for some given σ-ﬁnite measure on (E, E), see also Aït-

Sahalia and Jacod (2014) and Jacod and Shiryaev (2003) for detailed introduction of the last

two integrals. We assume that X satisﬁes the following assumption:

Assumption (H). The process b is locally bounded, the process σ is càdlàg , there is a localizing se-

quence {τ } of stopping times and for each n a deterministic nonnegative function Γ on E satisfying

n

R

2

Γ (z)λ(dz) < ∞ such that |ϑ(ω, t, z)| ∧ 1 ≤ Γ (z) for all (ω, t, z) satisfying t ≤ τ (ω).

n

2

.2 Observation scheme

n

For each n, let {t : i ∈ N } be a sequence of strictly increasing ﬁnite and deterministic ob-

+

i

n

served transaction times with 0 = t < t < . . . , where N is the set of nonnegative integers.

+

0

1

We denote

n

n_t=

1

n

,

δ(n, i) = t − t

.

(3)

∑

{t ≤t}

i

i−1

i

i≥1

Here, n is the number of observations recorded on the interval [0, t] for t ∈ R , while δ(n, i)

t

+

th

is the i spacing of the transaction times.

n

Deﬁnition 2.1 (The notation V ). Since we consider both the inﬁll and long-span asymptotics, we

i

introduce a uniﬁed notation that works in both settings for any process V. First, we use n (inﬁll setting)

t

and n (long-span setting) to denote the sample size. If V is a continuous-time function or process, we

n

use V to denote its discretized observation upon time t, i.e., V = V

n

i

, i = 0, . . . , n ; if {V }

is a

i∈Z

t

i

n

discrete process, where Z is the set of integers, we denote V = V , 0 ≤ i ≤ n or V = V , 0 ≤ i ≤ n .

i

t

i

Let {δ } be a positive sequence of real numbers satisfying δ → 0 as n → ∞. Then, δ can

n

be considered as the time lag between successive observations when the transaction times are

7

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equally spaced. It serves as a benchmark to compare with the real observations schemes that

will be speciﬁed below. More general schemes are relegated to Section 4.5.

Assumption (O). Let α be a nonnegative deterministic càdlàg process satisfying 0 < c ≤ α ≤ C <

α

t

α

∞

∀t for some positive constants c , C . Further assume

α

ꢂ

3

2

+

κ

n

ꢂ

α

δ(n, i) − δ_n≤ Cδ_n

(4)

i−1

for some κ > 0.

Under the above assumptions, it follows that as n → ∞,

Z

t

δ n → A :=

α_sds.

(5)

n

t

0

Thus, α captures the “density” of observations. If the observation scheme is regular, i.e.,

δ(n, i) = δ for all i, thus n = [t/δ ], then α ≡ 1 for all t satisﬁes (4) and A = t.

n

t

n

t

2

.3 Microstructure noise

We allow the microstructure noise to be dependent according to the ρ-mixing property.⁶

Deﬁnition 2.2. Let {χ }

be a sequence of stationary random variables deﬁned on a probability space

i

i∈Z

(

1)

(1)

q

(

Ω , G, P ). The probability space has discrete ﬁltrations G := σ{χ : p ≥ k}, : G := σ{χ :

p

k

−

∞

q ≤ k} satisfying G

= G = G. For any k ∈ N , we deﬁne the following mixing coefﬁcients for

∞

+

k ∈ N :

+

n

o

k+h

ρ := sup |E(V V )| : E(V ) = E(V ) = 0, kV k ≤ 1, kV k ≤ 1, V ∈ G , V

∈ G

. (6)

k

h

k+h

k

k+h

h

2

k+h

2

h

k+h

The sequence {χ_i}_i_∈_Zis ρ mixing if ρ → 0 as k → ∞.

k

The following assumption characterises the degree of serial dependence of {χ_i}_i_∈_Z, which

is necessary to obtain limit results.

Assumption 2.1 (Polynomially mixing coefﬁcients). There is some C > 0, v > 0 such that

C

ρ_k≤ _k_v∀ k ∈ N₊.

(7)

An immediate consequence of this assumption is that the autocovariance function of {χ_i}_i

is decaying at a polynomial rate, i.e.,

C

k

|

Cov(χ , χ_i₊_k)| ≤

,

(8)

i

v

⁶The key results in this paper also hold under a strong mixing condition, which is weaker than ρ-mixing,

see Bradley (2005). However, the moment conditions and the restrictions on the mixing coefﬁcients will be more

involved. On the other hand, Jacod et al. (2017) employ ρ-mixing sequence to model microstructure noise. Thus it

will facilitate the comparison with their result if we stick to the same setting.

8

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where C > 0 is some positive constant. We shall suppose that v > 1 for consistency and

that v > 3 for the CLT, which allows for quite strong dependence close to the long memory

boundary.

There is a large literature seeking to characterize the economic mechanisms that govern the

dynamic properties of microstructure noise, for example, the continuation of order ﬂows mod-

eled by Hasbrouck and Ho (1987), Choi et al. (1988), reversal order ﬂows due to market maker’s

risk aversion by Grossman and Miller (1988) and Campbell et al. (1993) or inventory controls

by Ho and Stoll (1981), Hendershott and Menkveld (2014), and the presence of inattentive (or

infrequent) traders by Bogousslavsky (2016) and Hendershott et al. (2018). Econometric mod-

els of microstructure noise focus usually on the i.i.d. case, although there are some works on

the MA(q) process, see Hansen and Lunde (2006), Hansen et al. (2008), Hautsch and Podolskij

(2013) and the ARMA(p, q) case, see Barndorff-Nielsen et al. (2008), Hendershott et al. (2013).

Note that our setting incorporates all the models.

Empirical studies have documented overwhelming evidence of various intraday patterns

of microstructure noise, see, e.g., Chan and Lakonishok (1995), Hasbrouck (1993), Madhavan

et al. (1997), McInish and Wood (1992) and Wood et al. (1985). The magnitudes of microstruc-

ture noise typically exhibit a U-shape or reverse J-shape. To capture such patterns, we intro-

duce a “size function” γ to allow for time-varying microstructure noise.⁷

Assumption (N). Let {χ }

be a stationary ρ-mixing random sequence with mixing coefﬁcients

i∈Z

i

(

1)

(1)

n

{

ρ }

on some probability space (Ω , G, P ). At stage n, the noise at time t is given by

k

k∈N₊

i

n

ε = γ

n

i

· χ_i,

(9)

i

t

where γ is a (deterministic) positive Lipschitz continuous function of u. We further assume that

u

{

χ_i}_i_∈_Zis centred at 0 with variance 1 and ﬁnite moments of all orders, and G is independent of F⁽⁰⁾.

n

Finally, the noisy observed price Y is given by (for i = 1, . . . , n )

t

i

n

Y = X + ε .

(10)

i

n

The process ε is consistent with the model of Engle and Rangel (2008), which allows for

i

stochastic volatility of the GARCH type as well as slow variation coming from γ. We may also

n

allow ε to have a slowly changing non-zero mean, which may be motivated by the "Drift burst

i

hypothesis" (Christensen et al. (2018)), because our method will difference this out. The as-

sumption of independence between the efﬁcient price and the microstructure noise, however,

excludes many interesting microstructure models, in which the efﬁcient returns are usually

⁷Jacod et al. (2017) model γ as a general nonnegative semimartingale process such that γ could depend on X.

Our setting is simpler and is motivated by the empirical facts we ﬁnd that the magnitude of microstructure noise is

close to a constant during the trading day and some irregularity arises only in the beginning of the trading session.

If, however, the measurement of microstructure noise has a ﬁnite sample bias that depends on X, e.g., the inte-

grated volatility of X, one could conclude that the size of microstructure noise is dependent on X and the intraday

pattern he discovers may be the manifestation of the intraday pattern of X instead. Or equivalently, any estimators

designed to recover the patterns of noise should be able to distinguish the efﬁcient price and microstructure noise

effectively.

9

Electronic copy available at: https://ssrn.com/abstract=3423607

correlated with the microstructure noise to reﬂect informational effects, see, e.g., Hendershott

et al. (2013). To the best of our knowledge, we are not aware of any nonparametric method that

can separate X and ε without assuming uncorrelatedness (Kalnina and Linton (2008) allow for

a speciﬁc type of small dependence between the noise and the efﬁcient price, Jacod et al. (2017)

and Da and Xiu (2019) allow for high-order dependence, but maintain the assumption of un-

correlatedness). However, it seems that the ReMeDI estimators are quite robust to such mis-

speciﬁcation, as we show in our extensive simulation studies in the supplementary material Li

and Linton (2019). We also present a consistency result for our estimator under the weaker

conditions of Jacod et al. (2017).

3

The Design and the Intuition of the ReMeDI Estimators

3

.1 The estimator of the autocovariance function

The intuition of the ReMeDI design can be best explained by considering the estimation of the

autocovariances of a stationary time series. Let {ε }

be a mixing sequence with mean zero;

i∈Z

we would like to estimate r := Cov(ε , ε_i₊_`). The natural estimator is the sample analogue

i

`

i

n−`

1

n

n n

br :=

ε ε

,

i+`

`

∑

i

i=0

which is consistent and asymptotically normal under very mild conditions.

n

We consider instead an estimator that replaces the “observations” ε , ε

by the “differ-

i

i+`

ences”, i.e.,

n−`−kn

ꢀ

ꢁ

1

n

ꢃ

ꢄ

n

er :=

ε − ε

_i₋_k0_n

ε

− ε

,

`

∑₀

i

i+`

i+`+kn

i=k

n

0

where k , k are integers that will grow at certain rates as the sample size increases. The estima-

n

tor er follows the ReMeDI design and it provides another consistent estimator of r , provided:

`

0

n

k → ∞, k → ∞, k /n → 0, and k /n → 0. The intuition of the consistency becomes immedi-

n

ate if one rewrites er as

`

n−`−kn

1

n

1

n

1

n

1

n

n n

n

ⁿ_i₊_`₊_k_n.

er =

ε ε

−

ε ε

−

∑₀_i₋_k0_ni+`

ε

+

∑₀_i₋_k0_n

i=k

n

ε

(11)

`

∑₀

i

i+`

∑₀

i

i+`+kn

i=k

n

The ﬁrst average is (asymptotically) equivalent to the sample analogue, thus it converges in

probability to r ; the remaining three averages are centred at r_`₊_k_n, r_`₊_k

0

, and r_`₊_k_n₊_k, which

0

n

`

n

themselves converge to zero as n → ∞ at a rate depending on (8).

Taking differences seems redundant if the time series {ε } is observable. However, in our

i

framework ε is masked by the efﬁcient price X, and what is observable is Y = X + ε. Taking

differences removes the effect of the efﬁcient price. The intuition of such removal under inﬁll

n

asymptotics is that the differences of the efﬁcient prices, say, X − X

_i₋_k0

, are much smaller

i

n

than the difference of the noise as n increases. In the absence of inﬁll asymptotics, the method

1

0

Electronic copy available at: https://ssrn.com/abstract=3423607

works because of the zero autocorrelations of the efﬁcient returns. We provide more details on

this case in Section 7.

3

.2 The general ReMeDI design

We next generalize the above approach to formally deﬁne our ReMeDI (Realized moMents of

Disjoint Increments) estimator. First, we provide some notations that will be used below. Let

J be the set of all ﬁnite sequences of integers satisfying

ꢅ

ꢆ

J = j = (j , j , . . . , j ) : j ∈ Z, p = 1, 2, . . . , q; q ≥ 2 .

1

2

q

p

In the sequel, we will assume without loss of generality that j = max{j : j ∈ j} for any

1

p

j ∈ J. The j-moments of χ, the stationary component of microstructure noise, are given by

!

q

r(j) = E

χ

.

(12)

∏

j_p

p=1

We are interested in such moments as target values, both in their own right, but also as they

provide information about parameters of a model for χ. For example, Hua et al. (2019) consider

ꢃ

ꢄ

an AR(1) process for χ, in which case moments E χ χ

can be used to estimate the process’

i

i−j

8

parameters. We may also be interested in functionals of the nonparametric scale function γ .

u

n

Let k = (k , . . . , k ) be a q-tuple of integers. For any j ∈ J and any process V, let I(k, j) be

1

q

t

k

n

the set of observation indices on [0, t] for which the following multi-difference operator ∆ (·) is

j

i

well deﬁned:

q ꢀ

ꢁ

k

n

∆

(V) :=

V

− V

i+jp−kp

.

(13)

j

i

∏

i+jp

p=1

Then the ReMeDI estimator associated with (V, j, k) is deﬁned by

n

k

n

ReMeDI(V; j, k) :=

∆ (V) .

(14)

t

∑

j

i

n

i∈I(k,j)_t

n

Remark 3.1. Using the above notations, we rewrite the estimator er as follows

`

n−`−kn

1

n

0

n

−kn,kn

n

er =

∆

(ε) .

i

`

∑₀

0,`

i=k

n

The general ReMeDI approach inherits two salient features of this estimator determined by the choices of

k: 1) the ﬁrst entry of k will be negative whereas the remaining ones are positive, i.e., the ﬁrst difference

ꢂ ꢂ

is a forward difference and the remaining ones are backward differences; 2) ∀1 ≤ p ≤ q, k → ∞

p

as n → ∞, and we will often write k = (k , . . . , k ) in the sequel to reﬂect such dependence.

n

1,n

q,n

We discuss a little more the intuition of the general ReMeDI design under inﬁll asymp-

8

A slightly more complicated case arises if χ satisﬁed a GARCH process with parameters θ; in that case, Kris-

i

ꢀ

ꢁ

2

tensen and Linton (2006) show how to use the moments E χ χ

to estimate θ.

i

i−j

1

Electronic copy available at: https://ssrn.com/abstract=3423607

n

Y

n

Y

i+j −2k

Y

i+j1+kn

i+j3−4kn

2

n

i+j3 i+j2

i+j1

Figure 1: Illustration the ReMeDI estimator of j-moments with j = (j1, j2, j3) and kn = (−kn, 2kn, 4kn).

totics. For this purpose, suppose that the noise size process γ is constant and we are estimating

ꢀ

ꢁ

q

n

E ∏

ε

. Suppose k satisﬁes the two properties in Remark 3.1. Next, we explain how

n

p=1 i+jp

ꢀ

ꢁ

q

k

n

to connect E ∏

ε

and ∆ (Y) by ∆ (ε) .

j j

i i

p=1 i+jp

To see this, we ﬁrst note {i + j − k } are the “distant” indices of the intervals on which

p

p,n p

the backward and forward differences are taken. Figure 1 illustrates a simple example with

j = (j , j , j ), k = (−k , 2k , 4k ) for some k ∈ N . The forward difference starts at the

1

2

3

n

+

(

i + j )-th observation and ends at the (i + j + k )-th observation; for the remaining indices

1 1

n

in j, the associated differences start from i + j , i + j and end at i + j − 2k , i + j − 4k , re-

2

3

2

n

3

n

spectively. The intuition of the ReMeDI approach is that the “distant” noise terms are approx-

n

imately independent of each other, and are also independent of the “clustered” noise {ε

}_p

i+jp

(

recall a special case outlined in (11)), therefore

!

ꢀ

ꢁ

q

kn

n

E ∆ (ε) ≈ E

ε

.

j

i

∏

i+jp

p=1

ꢂ

If k_p_,_nis still relatively small such that sup δ k

→ 0, the differences/increments of the

n

n,p

p

efﬁcient price over the intervals are asymptotically negligible. That is,

kn

n

kn

n

∆_j(Y) ≈ ∆ (ε) .

i

j

i

ꢀ

ꢁ

kn

n

q

n

Thus the averages of ∆ (Y) will converge in probability to E ∏

ε

by the law of large

, the ﬁrst

j

i

p=1 i+jp

numbers. This explains the intuition of the identiﬁcation.

A salient feature of the ReMeDI design is that the noisy log-return Yⁿ− Yⁿ

factor of ∆ (Y) , is taken on an interval disjoint from others. As we will see in Section 7 that

i+j1

1 1,n

i+j +k

kn

n

j

i

such design achieves consistency out of the inﬁll asymptotics framework.

4

The ReMeDI Estimators under Inﬁll Asymptotics

4

.1 Consistency

We next give the large sample properties of the ReMeDI (for a given choice of k ) estimator in

n

ꢀ

ꢁ

q

n

our general setting. Given j, the noise moments E ∏

ε

= C r(j) if γ ≡ C ∀u. For

γ

u

γ

p=1 i+jp

a general γ function that satisﬁes Assumption (N), the “average size” of the noise moments

R

t

0

q

becomes

γ dA /A instead of C , and it appears in the probability limit of the ReMeDI

s

s t

γ

estimators.

1

2

Electronic copy available at: https://ssrn.com/abstract=3423607

Theorem 4.1. Let Assumptions (H, O, N) hold, assume v > 1 and k satisﬁes

n





−k₁_,_n→ ∞, k → ∞, ∀ p ≥ 2,





p,n

ꢂ

sup δ k

→ 0, ∀ p ≥ 1,

(15)

(16)

n

n,p

p





k_p₊₁_,_n− k_p_,_n→ ∞, ∀ p ≥ 2,

as n → ∞. For j ∈ J, we have the following convergence in probability:

R

t

0

q

ReMeDI(Y; j, k )ⁿ

γs dAs

n t

P

→

R(j) :=

r(j),

t

n_t

A_t

where r(j) is deﬁned in (12) and A in (5).

t

Remark 4.1. Let {k } be a sequence of integers satisfying k → ∞, k δ → 0. Let k be speciﬁed as

n

n n

n

follows:





−

k_n

if p = 1,

k_p_,_n

Then k satisﬁes the conditions in (15).

=



(

p − 1)k_nif p ≥ 2.

n

The microstructure noise speciﬁed in (9) has a multiplicative structure; the following result

states that we can separate the components of microstructure noise locally.

Theorem 4.2 (A local estimator of γ). Let Assumptions (H, O, N) hold and assume v > 1. Let

{

k } , {` } be two sequences of integers satisfying

n

n n

2

k_n→ ∞, k /` → 0, ` δ = O(1).

n

n n

n

Let j = (0, 0), k = (−k , k ). Let I (j , k ) := I(j , k ) \I(j , k )

. A ReMeDI estimator

0

n

`

0

n t

0

n t

0

n

δn`n

n

t− _α_t

of γ is given by

t

n

kn

0

n

P

γb :=

∆ (Y) /` → γ .

t

∑

j

i

n

t

n

i∈I_`_n(j0,kn)_t

One could use this estimator to re-estimate the noise moments using the rescaled data,

thereby separating the scaling factor from the noise. We use it below to obtain consistent

estimates of the asymptotic variance of the ReMeDI estimator.

4

.2 Limit distribution

0

In the sequel whenever we have two vectors j = (j , . . . , j ), j = (j , . . . , j

0

) ∈ J, we suppose

1

q

1

q

0

without loss of generality that q ≤ q . We denote

0

j ⊕ j = (j , j , . . . , j , j , j , . . . , j

0

),

j

= j\{j },

1

2

q

1

2

q

−p

p

j(+k) = (j + k, j + k, . . . , j + k), for k ∈ Z,

1

2

q

j_Q_q= (j : p ∈ Q ) for Q ⊂ {1, 2, . . . , q}.

p

q

1

3

Electronic copy available at: https://ssrn.com/abstract=3423607

For each Q ⊂ {1, 2, . . . , q}, there is an associated (unique) pair of subsets:

q

c

0

Q := {1, 2, . . . , q}\Q , Q_q

0

:= Q ∪ {q + 1, . . . , q }.

(17)

q

9

We denote for each k ∈ Z the following moments

ꢃ

^ꢃ0

ꢄꢄ

ꢃ ꢄ

0

s (j, j ; k) := r j ⊕ j (+k) − r (j) r j ;

0

ꢀ

ꢁꢁ

0

p

s (j, j ; k) :=

r j ⊕ j (+k)

r(j , j + k);

p

1

∑

Qq

Q

∏

q⁰

c

Qq({1,2,...,q}

p∈Qq

0

s (j, j ; k) :=

r(j , j

0

+ k)r(j )r(j ) −

₋_p0

r({j } ⊕ j (+k))r(j

)

2

∑

p

−p

∑

p

−p

jp∈j,j ∈j⁰

p=p

0

jp∈j

0

p

0

6

−

r({j⁰

0

+ k} ⊕ j)r(j⁰).

₋_p0

∑

p

0

j

∈j

0

p

Now we further specify the choice of k to derive the limit distributions of the ReMeDI esti-

n

mators. k will solely be determined by an integer k , which is related to v as follows

n





−

2

k_n

p−1

if p = 1,

k_p_,_n

=



k_nif p ≥ 2,

(18)

ꢇ

ꢈ

ꢇ

ꢈ

−

$

1

v > 3, k ꢀ δ , $ ∈

, δ , where δ ∈

,

.

n

2

v

v + 2 5

The lower bound of k is to guarantee the estimator converges faster enough to the true param-

n

eters than the convergence rate, while the upper bound is to satisfy a Lindeberg’s condition.

n

Remark 4.2. Note that (18) implies (15). In the sequel, we will omit k and simply write ∆ (Y) and

n

j

i

n

kn

n

ReMeDI(Y; j) instead of ∆ (Y) and ReMeDI(Y; j, k ) when k satisﬁes (18).

t

j

i

n t

n

Let

ꢇ

ꢈ

ReMeDI(Y; j)ⁿ

√

n

t

Z(j) := n_t

− R(j)_t

.

t

n_t

n

0 n

t

The following theorem presents the joint limit distribution of Z(j) , Z(j ) .

t

0

Theorem 4.3. Let Assumptions (H), (O) and (N) hold, and k , v satisfy (18). For any t > 0, j, j ∈ J,

n

we have the following convergence in law

ꢃ

ꢄ

ꢃ

ꢄ

L

n

0 n

0

Z(j) , Z(j ) −→ Z(j) , Z(j ) ,

(19)

(20)

t

where the limit is centred Gaussian with (co)variances:

R

0

t

0

q+q

ꢃ

ꢄ

γ

dA_s

s

0

t

0

σ(j, j ) := E Z(j) Z(j ) = S(j, j )

,

t

A_t

⁹By convention we let r(∅) = 1.

1

4

Electronic copy available at: https://ssrn.com/abstract=3423607

and

2

0

S(j, j ) :=

s (j, j ; k).

(21)

∑

m

k∈Z m=0

This limit theory makes use of the assumption that the noise process is bounded away from

zero and so relatively large on the small time scale. One can allow a more general setting such

as Kalnina and Linton (2008) wherein the noise process shrinks to zero with sample size but

√

at a rate slower than n_t. In that case, one can obtain a self-normalized CLT. We consider this

case further in Section 7.

4

.3 Estimating the autocovariances of microstructure noise

In this section we consider the special case concerning the estimation of the variance and au-

tocovariances of microstructure noise. Let j = (0, `), ` ∈ N .

`

+

nt−kn−`

1

n_t

1

n_t

ꢃ

ꢄꢃ

ꢄ

n

b

R

:= ReMeDI(Y; j ) =

Y

− Y

Y − Y

.

(22)

t,`

` t

∑

i+`

i+`+kn

i

i−2kn

i=2kn

The following corollary provides the limit distribution.

Corollary 4.1. Under the conditions of Theorem 4.3, we have

R

!

t

0

ꢀ

ꢁ

4

√

γ dAs

n_tR − R

−^L→ N 0, S_`

s

,

n

b

t,`

A_t

where

R

t

0

2

∞

γ dA

ꢃ

ꢄ

s

2

R_t_,_`:= r_`

,

S :=

`

E((χ χ − r )(χ χ

− r )) + 3r .

(23)

∑

0

`

k

k+`

`

k

A_t

k=−∞

ꢃ

ꢄ

In the special case of i.i.d. noise, the asymptotic variance reduces to E χ⁴+ 2r up to

2

0

a constant magnitude of noise. We detail the intuition of asymptotic variances in the next

section.

4

.4 The asymptotic variance and its estimation

Like the probability limit, the asymptotic variance of the ReMeDI estimators has two com-

0

ponents. Speciﬁcally, S(j, j ) is the asymptotic variance of the stationary part of the ReMeDI

0

estimators. It has several constituents: ∑

s (j, j ; k) is the asymptotic variance of the sample

k∈Z

0

analogue of r(j), whereas ∑

s (j, j ; k) + s (j, j ; k) are attributed to noise at the “distant”

k∈Z

1

2

indices at the ends of each intervals. Consider S deﬁned in (23) as an example. If {χ }

`

i

i∈N

is directly observable and we estimate r by the sample analogue ∑ χ χ /n , the asymptotic

`

i

i+`

t

variance will be ∑^∞

E((χ χ − r )(χ χ

− r )). The presence of the “distant” noise in-

k=−∞

0

`

k

k+`

2

`

creases the asymptotic variance by ∑

3r , which is the asymptotic variances of the last three

i

i∈Z

0

terms in (11) (with k = 2k ).

n

1

5

Electronic copy available at: https://ssrn.com/abstract=3423607

Compared to the ReMeDI estimators, the local averaging (LA) estimators proposed by Ja-

0

cod et al. (2017) are asymptotically more efﬁcient in our settings, as only ∑

s (j, j ; k) ap-

k∈Z

0

pears in the asymptotic variance of the stationary part (although one can improve the efﬁ-

ciency of ReMeDI by taking averages of estimators computed using different k ). However,

n

simulation studies show that the ReMeDI class works better in ﬁnite samples with realistic

sample sizes or equivalently, data frequency — it has smaller ﬁnite sample variance and is al-

most unbiased under various model speciﬁcations. Moreover, the loss of asymptotic efﬁciency

is compensated for by the greater computational efﬁciency of the ReMeDI approach, which

pays off when one is working with massive high-frequency datasets (recall Footnote 4).

We now turn to the inference question. One can just estimate term by term using ReMeDI

for diferent combinations ignoring asymptotic independence and using Parzen-Newey-West

weighting. However, we ﬁnd that this approach poorly represents the ﬁnite sample variance.

We propose instead to ﬁrst recenter with local ReMeDI estimator; it seems to work better in

practice.

For given integers i, v , we deﬁne a local ReMeDI statistic

n

i+vn

n

ReMeDI (Y; j) :=

∆ (Y) .

(24)

vn

i

∑

=i+1

j

`

−

1

n

so that v ReMeDI (Y; j) will be a local estimate of the j-moments of noise. We also intro-

n

vn

i

duce the Newey-West weights to improve the ﬁnite sample performance (see Newey and West

(

1987)): for any i, u ∈ N ,

n

+

i

ν

i,un

= 1 − ₁

.

(25)

+ u_n

n

b

Let Σ be a 2 × 2 matrix with components

t

n

0

n

0

0 n

t

b

Σ (1, 1) := σb(Y; j, j) , Σ (1, 2) = Σ (2, 1) := σb(Y; j , j) , Σ (2, 2) := σb(Y; j , j ) ,

t

where

!

nt−v⁰

n

i_n

ꢀ

ꢁ

1

0

n

σb(Y; j, j ) :=

∆ (Y) ∆

0

(Y) +

ν

∆ (Y) ∆

0

(Y) + ∆_j

0

(Y) ∆ (Y)

;

t

∑

j

i

j

i

∑

k,in

j

i

j

i+k

i

j

i+k

ⁿ^ti=2

kn

0

q −1

k=1

(26)

n

−1

n

−1

0 n

∆

(Y) := ∆ (Y) − v ReMeDIv (Y; j) ; ∆_j

0

(Y) := ∆

0

(Y) − v ReMeDIv (Y; j ) ;

j

i

j

i

n

i

j

i

n

i

0

1

v := vn + kn + in + j ∨ j .

n

1

Theorem 4.4. Assume all the conditions of Theorem 4.3 hold, let {i } , {v } satisfy

n

n n

−

δ

−φ

i ꢀ δ , v ꢀ δ_n

,

2δ < φ < 1/2.

(27)

n

1

6

Electronic copy available at: https://ssrn.com/abstract=3423607

We have

!

Z(j)ⁿ_t

L

n

t

L

−→ N (0, I ) ,

(28)

2

0

n

Z(j

t

ꢀ

ꢁ₋

1

n

n>

n

t

b

where L is a matrix satisfying L L = Σ

.

t

Corollary 4.2 (Estimation of the autocovariance and autocorrelation function). Let j = (0, `), ` ∈

`

N . Under the conditions of Theorem 4.4, we have

+

r

s

ꢀ

ꢁ

ⁿt

−^L→ N (0, 1) ,

n

b

R − R

(29)

(30)

n

t,`

σb(Y; j , j )

`

` t

!

n

b

R

n

−^L→ N (0, 1) ,

t

t,`

− _`

r

bn

S

R

t,`

t,0

where













>

n

b

R

n



1

t,`

n

 1

t,`

b

S

:= 

, −

Σ 

, −

 ,

ꢀ

ꢁ

t

ꢀ

ꢁ

t,`

2

bn

R

n

R

n

t,0

Rb

t,0

Rb

t,0

n

b

and Σ is a 2×2 matrix with Σ (1, 1) = σb(Y; j , j ) , Σ (2, 2) = σb(Y; j , j ) , Σ (1, 2) = Σ (2, 1) =

t

`

` t

t

0

0 t

t

n

σb(Y; j , j ) .

`

0 t

The local ReMeDI statistics (24) are introduced to capture the time-varying noise, and the

n

statistics can be replaced by a global one ReMeDI(Y; j) /n if stationarity holds; the global

t

estimator requires less tuning parameters and reduces the computational cost.

Corollary 4.3 (Asymptotic (co)variances estimators with stationary noise). Assume γ ≡ C > 0.

γ

Let

!

0

nt−kn−j ∨j −in

i_n

ꢀ

ꢁ

1

nt

0

n

e

σe(Y; j, j ) :=

∆ (Y) ∆

0

(Y) +

ν

∆ (Y) ∆

0

(Y) + ∆_j

0

(Y) ∆ (Y)

;

t

∑

j

i

j

i

∑

k,in

j

i

j

i+k

i

j

i+k

0

q −1

k=1

i=2

kn

n

t

0 n

ReMeDI(Y; j)

ReMeDI(Y; j )

n

t

e

∆

(Y) := ∆ (Y) −

;

∆

0

(Y) := ∆

0

(Y) −

.

j

i

j

i

j

i

j

i

nt

Then

0

q+q

P

0

n

0

σe(Y; j, j ) → C

S(j, j ).

t

γ

4

.5 Extensions to the model setting

In this section, we extend the consistency of the ReMeDI estimators under a broader setting

introduced by Jacod et al. (2017) in which the observation times and the size parameters of the

noise, γ, are stochastic and possibly related to the efﬁcient price process.

1

7

Electronic copy available at: https://ssrn.com/abstract=3423607

Assumption (E-O) (Assumption (O) in Jacod et al. (2017)). α, α are two Itô semimartingales deﬁned

ꢀ

ꢁ

on Ω , F , {F } , P⁽⁰⁾satisfying Assumption (H). We further assume there is a localizing

(

0)

(0)

t

t≥0

sequence {τ } of stopping times and positive constants κ

and κ such that:

m,p

m

1

2

. For t < τ we have

≤ α_t₋≤ κ_m_,₁and α ≤ κ_m_,₁

.

m

t−

κm,1

n

. Let (F )

be the smallest ﬁltration satisfying

t

t≥0

n

(

a) F ⊂ F ,

t

n

(

b) t is a {F }

stopping time for i = 0, 1, 2, . . . ,

t

t≥0

i

W

n

(

c) δ(n, i), conditional F , is independent of F :=

F for i = 0, 1, 2, . . .

t

∞

t≥0

3

. With the restriction {tⁿ< τ }, and for all p > 0,

m

i−1

ꢂ

3

2

δn ꢂ

+κ

n

ꢂE (δ(n, i) |F ) −

ꢂ≤ κm,1δn

,

t

n

ꢂ

α

ꢂ

i−1

ꢂ ꢀ

ꢁ

ꢂ

ꢃ

ꢄ

(31)

ꢂ

n

2

n

2

n ꢂ

2+κ

ꢂE δ(n, i)α − δn |F − δ α ꢂ≤ κm,2δ

,

i−1

t

n

i−1

n

p

n

p

E (δ(n, i) |F ) ≤ κ δ .

t

m,p n

Assumption (E-N). The process χ satisﬁes Assumption (N). Moreover, it is independent of F . As-

∞

ꢀ

ꢁ

(

0)

(0)

sume the process γ is a positive Itô semimartingale on Ω , F , (F ) , P

and its coefﬁcients

t

t≥0

satisfy Assumption (H).

Theorem 4.5. Assume Assumptions (H, E-O, E-N) hold, and k satisﬁes (15). For j ∈ J, we have the

n

following convergence in probability:

ReMeDI(Y; j, k )ⁿ

n t

P

→

R(j)_t.

(32)

n_t

5

Economic Applications: New Measures of the Bid-Ask Spread

In this section we introduce two liquidity measures and deﬁne ReMeDI estimators of them.

The measures are based on the rationale that the dispersion of microstructure noise, the devi-

ation of the observed prices from fundamental values, serves as a natural measure of market

quality, see Hasbrouck (1993) and Aït-Sahalia and Yu (2009), among others. The ﬁrst measure,

which we call the instantaneous bid-ask spread (IBAS), can be considered as a generalized and

robust Roll measure (Roll (1984)). The second liquidity measure gauges the average bid-ask

spread (ABAS). It can be interpreted as the inventory risk that an intermediary takes when he

meets a liquidity demand that may exhibit certain autocorrelation patterns.

5

.1 The Roll measure and the IBAS measure

Roll (1984) proposes a simple measure of the effective bid-ask spread. In Roll’s model, the

efﬁcient price follows a random walk, and the microstructure noise becomes the half-signed

1

8

Electronic copy available at: https://ssrn.com/abstract=3423607

bid-ask spread, i.e., ε = Sq /2, where q = 1 or −1 if the trade is initiated by a buyer or seller

i

and S is the spread. He further assumes that the efﬁcient price is independent of the order

ﬂow process {q } , which is serially uncorrelated with P(q = 1) = P(q = −1) = 0.5. The

i

Roll measure is based on the insight that the autocovariance of the adjacent observed returns

captures the bid-ask bounce. Using our notations, the Roll measure is

v

u

t

nt−1

1

n_t

n

−1,1

n

Roll := 2

∆

(Y) .

(33)

t

∑

0,0

i

i=1

In this case, the Roll measure provides a consistent estimator of the bid-ask spread, i.e.,

q

n

P

Roll → S = 2 Var(Sq /2).

t

i

n

We consider a general setting introduced in Section 2 for the half-spread, i.e., ε = γ

n

i

χ_i,

t

i

where {χ } is a stationary mixing sequence, which could be serially dependent. The serial

i

dependence in ε may reﬂect various order ﬂow patterns triggered by economic agents, and γ

can capture the time-varying spread. There is considerable evidence that order ﬂow is strongly

positively autocorrelated, see, for example, Gerig (2008).

We propose an alternative liquidity measure

p

IBAS := 2 R ,

(34)

t

t,0

where R_t_,₀is deﬁned in (23). If ε = Sq /2, and {q } are uncorrelated with P(q = 1) =

i

P(q = −1) = 0.5, we are back to the Roll’s setting and we have IBAS = S. A consistent

i

t

estimator of IBAS is given by

t

q

P

n

b

IBAS := 2

R

→ IBAS .

(35)

t

t,0

t

n

Compared to the Roll measure, IBAS is robust to autocorrelations in the half-spread ε . In

t

i

fact, it is well known, see, e.g., Choi et al. (1988), that the Roll measure becomes biased if

√

n

P

the order ﬂows are autocorrelated. One can show that Roll → S 1 − 2r + r , where r =

t

1

2

k

Cov(q , q ). In particular, if {q } is an AR(1) process with coefﬁcient 1 > $ > 0, we have

Roll → S 1 − $ < S. We may apply the ReMeDI estimators to correct the bias of the Roll

i

i+k

i

q

ꢃ

ꢄ

n

P

t

q

measure. Let

q

n

t

n

2

n

b

AdjRoll := 2 (Roll /2) + 2R − R .

(36)

t

t,1

t,2

n

t

P

The adjusted Roll measure has the same probability limit as the IBAS : AdjRoll → IBAS ,

t

which reduces to S if the spread is constant.¹⁰

IBAS is an instantaneous measure as it gauges the spread of a single trade. Next, we

t

propose an average measure that captures the spread of multiple trades.

1

0

n

Empirically, we observe almost perfect matches of AdjRoll and IBAS , see Figure C.2 in our supplementary

t

material Li and Linton (2019).

1

9

Electronic copy available at: https://ssrn.com/abstract=3423607

5

.2 The ABAS measure and its interpretation

We introduce a new liquidity measure that measures the average long run dispersion of the

spread,

s

∞

ABAS := 2 R + 2

R .

t,`

(37)

t

t,0

∑

`

=1

ꢃ

ꢄ

nt

n

∞

This makes intuitive sense since Var ∑ ε /n → R + 2 ∑

R . The ReMeDI estimator

`=1 t,`

t

t,0

i=1

i

of ABAS is given by (the choice of κ will be speciﬁed later, also recall the weights ν

are

n

`,κ_n

deﬁned in (25)):

s

κn

n

b

ABAS := 2 R + 2

R .

(38)

t

t,0

∑

t,`

`

=1

n

We show below that ABAS consistently estimates ABAS and we derive its limiting dis-

t

tribution.

In the next section we give a speciﬁc economic interpretation of the ABAS measure.

t

5

.2.1 A simple model of inventory risk when order ﬂows are autocorrelated

We introduce a simple model motivated by the pioneering work of Stoll (1978). The novelty

lies in the explicit modelling of the order ﬂows and a simple measure of the imbalances in

order ﬂows.

At time τ, a representative intermediary has cash c with a position 0 in a risky stock.

τ

n

There are n trades occurred at time {T } , with τ < T < · · · < T = τ + 1, which could be

i

1

n

i=1

clustered orders submitted by traders trading at a faster speed (Li (2017)), or orders received at

different exchanges (Menkveld (2008)). The intermediary sets a price Y for all trades between

τ

n

τ and τ + 1, and supplies {z } ; z > 0(z < 0) indicates the intermediary is willing to sell

i

i=1

(

buy) z (−z ) shares. The payoff is realized at time τ + 1 with price equal to the fundamental

i

value X , which evolves stochastically as

τ+1

X_τ₊₁= X + θ ,

τ

2

where θ has conditional mean 0 and variance Var (θτ) = σ . At time τ + 1, the intermediary

τ

has cash

n

c_τ₊₁= c + Y_τ

z_i,

τ

∑

i=1

and his aggregate wealth becomes

n

w_τ₊₁= −X_τ₊₁

z_i+ c_τ₊₁= (Y − (X + θ ))

z + c .

i

∑

τ

∑

τ

i=1

n

Thus, the conditional mean of w_τ₊₁is E (w ) = (Y − X ) ∑ z + c and the conditional

τ

τ+1

τ

i=1

i

τ

2

0

Electronic copy available at: https://ssrn.com/abstract=3423607

variance is Var (w ) = (∑ⁿz ) σ . The intermediary has mean-variance utility:

2

τ

τ+1

i=1

i

τ

ρ

a

U (w) = E (w) − Var (w) ,

τ

2

where ρ is the risk-aversion coefﬁcient. To maximize his utility, the intermediary solves

a

max U (w_τ₊₁).

τ

n

{

z }

i

i=1

At the optimum, the ﬁrst order condition says that

n

Y − X

τ

z_i=

.

∑

2

ρ σ

a

τ

i=1

n

In equilibrium, the representative intermediary meets the (stochastic) demand {q } , where

i

i=1

n

Yτ−Xτ

2

τ

we assume q = 1 indicating a buy and q = −1 a sell. Thus, ∑ q =

, which implies

i

i=1

i

ρaσ

that

n

2

Y = X + ρ σ

q_i.

(39)

τ

a

τ

∑

i=1

The deviation of the transaction price Y from the fundamental value reﬂects the intermedi-

τ

ary’s inventory risk to meet an incoming order ﬂow {q } . Therefore, the spread compensates

i

the intermediary to provide liquidity.

2

When there is only one trade, i.e., n = 1, then the bid and ask prices are Y = X − ρ σ and

τ

a

τ

2

Y = X + ρ σ ; thus the instantaneous spread S is given by S = 2ρ σ . Similar to the Roll’s

τ

a

τ

1

a

τ

p

2

model, S = 2 Var (ρ σ q ), twice the dispersion of the deviation ρ σ q , if P(q = 1) =

1

τ

a

τ

1

a

τ

1

P(q = −1) = 0.5.

1

When n is large, we need another measure to reﬂect the overall inventory risk the inter-

mediary takes. A natural measure of such risk is provided by twice the average dispersion of

2

n

ρ σ ∑ q_i, i.e.,

a

τ

i=1

v

!

u

t

n

1

n

S_n:= 2

Var_τρ στ²

q .

i

a

∑

i=1

We denote S := lim

S ; S is the long run average measure. Let r = Cov(q , q_i₊_k), and

∞

n→∞

n

∞

k

i

∞

assume that {q } are independent of σ and ∑

|r | < ∞. We have

i

τ

k=−∞

k

s

∞

2

S = 2ρ σ r₀+ 2

r .

k

∞

a

τ

∑

k=1

Consider the special case where {q } is an AR(1) process with coefﬁcient $ . Then, we have

i

q

S = S (1 + $ )/(1 − $ ). When the order ﬂow is positively autocorrelated, i.e., $ ∈ (0, 1),

∞

1

q

the average measure S is greater than the instantaneous measure S , as the intermediary is

∞

1

more likely to accumulate inventories on one side of the market, which entails a larger spread

to compensate his risk.

2

1

Electronic copy available at: https://ssrn.com/abstract=3423607

n

t

5

.3 The limit distribution of IBAS and ABAS

t

Theorem 5.1. Assume all conditions of Theorem 4.4 hold, and further assume

ꢉ

ꢊ

ꢉ ꢇ

ꢈ

ꢊ

2

1

κ_nꢀ δ_n⁻^ψ

1

2

φ

1

2

v > 1 + max

,

< ψ < min

− δ , − φ .

φ − 2δ 1 − 2φ

2v − 2

2

(40)

Then we have the following limit distribution

√

n

n (IBAS − IBAS )

−^L→ N (0, 1) ,

t

q

(41)

(42)

n,κn

(0, 0)/IBASⁿ_t

b

Σ

t

√

n

n (ABAS − ABAS )

t

q

n,κn

/ABASⁿ_t

b

2

Σ

t

where

and

κn

n,κn

t

n,κn

t

n,κn

t

0

b

Σ

:= Σ_t(0, 0) + 4

Σ

(`, 0) + 4

Σ

(`, ` ),

∑

0

`,` =1

`

=1

n,κn

0

n

b

Σ

(`, ` ) := σb(Y; j , j

0

) .

(43)

t

`

t

Remark 5.1. The asymptotic conditions in (40) seem quite complicated. But if the mixing coefﬁcients

(

recall ρ in Deﬁnition 2.2) are exponentially decaying, e.g., a stationary invertible ARMA(p, q) model

k

of microstructure noise, the conditions will hold.

6

Statistical Applications: Testing Statistical Properties of Microstruc-

ture Noise

6

.1 Testing for intraday patterns

It is well documented that microstructure noise exhibits intraday patterns. This section devel-

ops a formal test of the intraday patterns by examining the difference of the average magni-

n

tudes of noise over two intervals. Given s ≥ 0, let n = inf{i ∈ N : t ≥ s}, n = sup{i ∈ N :

i

s

i

s

n

t < s}. For any 0 ≤ T < T , let

1

2

nT −kn−j1

2

ReMeDI(Y; j₀)ⁿ:=

∆ (Y) ;

n

T₁,T₂

∑ _q

j

i

i=n +2 kn

T1

ꢀ

ꢁ

ꢃ

ꢄ

nT −vn

i

n

k=1

2

n

∑

q

∆j(Y) ∆j0 (Y) + ∑

^ν^k^,ⁱn ∆j(Y) ∆j0 (Y)

+ ∆j0 (Y) ∆j(Y)

i=n +2 kn

i

i+k

i

i+k

n

T1

σb(Y; j , j )

:=

.

0

0 T ,T

1

2

n − n

T

2

T₁

2

Electronic copy available at: https://ssrn.com/abstract=3423607

Thus the statistics ReMeDI(Y; j₀)ⁿ, σb(Y; j , j )

n

are the ReMeDI estimators based on obser-

0

T

1

,T

2

1

,T

2

vations between time T and T . The following result provides a device to compare the average

1

2

variances of microstructure noise on the intervals [T , T ] and [T , T ], corrected for the number

1

2

3

4

of observations.

Let 0 ≤ T < T ≤ T < T .

1

2

3

4

R

T2

T1

T4

T3

T2

T1

T4

2

γ dA

T3

s

H₀:

=

;

H :

6

=

.

3

(44)

(45)

A

A − A

T

2

T

1

T

4

T

3

T

2

T

1

T

4

T

Deﬁne the testing statistics

q

ꢀ

ꢁ

ꢃ

ꢄꢃ

ꢄ

_R_e_M_e_D_I₍_Y_;_j₀₎n

T1,T2

T3,T4

n − n

−

T

2

^T1

T

4

^T3

nT −n

2

T1

4

T3

T(Y; T , T , T , T ) :=

q

.

1

2

3

4 n

ꢃ

ꢄ

n

n − n σb(Y; j , j )

+ n − n σb(Y; j , j )

T

4

T

0

T

2

T₁

0

T

3

T

1

,T

2

3

,T

4

Theorem 6.1. Under the conditions of Theorem 4.4, we have



L



T(Y; T , T , T , T ) −→ N (0, 1) under H₀,

1

2

3

4 n

P



T(Y; T , T , T , T ) → ±∞

under H_A.

1

2

3

4 n

6

.2 Joint tests of zero autocorrelations

This section develops tests of the joint hypothesis of zero autocorrelations based on the dis-

n,q

b

tribution theory of the ReMeDI estimators. Recall that for any ` ∈ N , j = (0, `). Let Σ

+

`

t

n,q

0

n

0

b

be a q × q matrix with the (`, ` ) component Σ (`, ` ) = σb(Y; j , j

0

) for all 1 ≤ `, ` ≤ q. Let

t

`

t

n

t

n,q

t

ReMeDI(Y;j )

`

b

ꢀ

R

be a q-vector with the `-th element R (`) =

, 1 ≤ ` ≤ q. We ﬁrst introduce a

t

n

t

Box-Pierce (BP) statistic:

ꢁ ꢀ

ꢁ₋₁

ꢀ

ꢁ

>

n,q

BP,t

n,q

t

n,q

t

n,q

t

b

T

:= n_t

R

Σ

R

.

Next, we will introduce a variance-ratio (VR) type statistic (Lo and MacKinlay (1988), Hong

et al. (2017)). Let

q−1

n

b

Σ

(2, 2) := σb(Y; j , j ) ;

Σ

(1, 2) := 2

σb(Y; j , j

ν

σb(Y; j , j ) + σb(Y; j , j ) ;

VR

0

0 t

VR

∑

`,q−1

`

0 t

0

0 t

`

=1

q−1 q−1

q−1

∑

n

b

Σ

(1, 1) := 4

ν

0

) + 4

ν

σb(Y; j , j ) + σb(Y; j , j ) ;

VR

∑ ∑ `,q−1 ` ,q−1

`

t

`,q−1

`

0 t

0

0 t

0

`

=1 ` =1

`=1

ꢀ

ꢁ

2

q

q−1

n

b

q

n

b

S

bb

Σ

(1, 1)

t

2S Σ (1, 2)

q

t

n,q

t

n,q

n

t

VR

b

S := R (0) + 2

ν R (`);

`,q

Σ

:=

+

Σ

(2, 2) −

.

3

∑

t

VR

ꢀ

ꢁ

ꢀ

ꢁ

VR

ꢀ

ꢁ

2

4

n,q

t

n,q

`

=1

b

R (0)

t

The statistic is

q

b

S

n,q

t

T

:=

,

VR,t

n,q

b

R (0)

t

2

3

Electronic copy available at: https://ssrn.com/abstract=3423607

and the studentized version is given by

ꢀ

ꢁ

√

n,q

VR,t

n_t

T

− 1

n,q

VR,t

T

:=

q

.

bn

Σ

VR

We consider the following pairs of hypotheses:

q

A

H : r = · · · = r = 0;

H : ∃` such that 1 ≤ ` ≤ q, r

6

= 0.

(46)

(47)

1

q

`

0

∞

H : r = 0 : ∀` ≥ 1;

H : ∃` ≥ 1, r

6

= 0.

0

`

A

`

q

A

0

H :

ν

r = 0;

H

:

ν

r

6

= 0.

(48)

0

∑

`,q−1 `

∑

`=1

`,q−1 `

`

=1

These null hypotheses are implicit in some stylized models of market microstructure such as

the Roll model.

Theorem 6.2. Assume all the conditions of Theorem 4.4 hold, for a given q ∈ N , and a sequence

+

{

q } satisfying

n

ꢉ ꢇ

ꢈ

ꢊ

q_nꢀ δ_n⁻

β

,

0 < β < min

2

3

φ

− δ , (1 − 2φ) .

2

3

(49)

2

We have



n,q

BP,t

n,q

L

q





2

T

−→ χ under the H ;

q

0

(50)

(51)

(52)

P

q

T

→ ∞

under the H .

BP,t

A



q

ꢀ

ꢁ

n,qn

qn

T

qn

L



t

∞

−

1 −→ N (0, 1) under the H ;

2

0

n,qn

P



∞

under the H .

A

T

→ ∞

t





n,q

VR,t

n,q

L

q

0

T

−→ N (0, 1) under the H ;

0

P

q

A



0

→ ±∞

under the H .

VR,t

7

Beyond the Inﬁll Asymptotics and High-frequency Data

All the properties of the ReMeDI estimators introduced in previous sections are developed

under inﬁll asymptotics, i.e., the data frequencies increase without bound within a ﬁxed time

interval. Other approaches, e.g., the realized volatility estimators of the variance and covari-

ances of noise (see Bandi and Russell (2008), Hansen and Lunde (2006), Zhang et al. (2005), Li

et al. (2019)), the local averaging estimators proposed by Jacod et al. (2017), are also devel-

oped under the inﬁll asymptotics. This section demonstrates that the ReMeDI approach works

well beyond the inﬁll framework, and such robustness to data frequencies distinguishes the

ReMeDI estimators from the above alternatives. As a consequence, the two liquidity measures

developed in Section 5 can be applied to data at coarser grids, even daily data.

2

4

Electronic copy available at: https://ssrn.com/abstract=3423607

7

.1 The Model

Let {Y_i}_i_∈_N₊be an observed time series that can be decomposed into

Y = X + ε ,

(53)

i

where X and ε satisfy

Assumption 7.1.

1. The process {X } satisﬁes X

= µ + X + η , where {η } are centered

i

i+1

i

i i

ꢃ ꢄ

2

independent random variables with bounded second moments, i.e., sup E η ≤ C < ∞.

i

n

2

. The process {ε } is stationary, i.e., ε = C χ , where the process {χ } satisﬁes Assumption 2.1

i

γ

i

i=0

and is independent of X.

Model (53) and Assumption 7.1 characterize a large class of time series, including any time

series with stationary ﬁrst differences, see Cochrane (1988). The decomposition in (53) are

widely applied to model various economic phenomena. In ﬁnancial economics, X is the efﬁ-

cient price and ε is the pricing error (Hasbrouck (1993), Hendershott and Menkveld (2014)), or

transitory component in stock prices due to, e.g., noise trading (Poterba and Summers (1988)).

In macroeconomics, X is identiﬁed with the permanent component of the observed times series

while ε is the cyclical component (Cochrane (1994), Beveridge and Nelson (1981) and Campbell

and Mankiw (1987)). The random walk component X may also arise from a modiﬁcation of

existing model, say, by adding a random walk in the technology change (King et al. (1991)).

7

.2 The ReMeDI estimation

Now we show the ReMeDI approach can effectively separate the two components X and ε

under Assumption 7.1. In general, we may estimate the drift term µ root-n-consistently by

taking the sample average of the ﬁrst difference of the observed process. The effect of this

initial estimation may be shown to be of smaller order, and so for simplicity here suppose that

µ is known and without loss of generality is equal to zero.

Let k = (−k , k ), j = (0, `). Let

n

`

n−kn−`

1

n

kn

`

n

b

R :=

∆ (Y) .

`

∑

j

i

i=kn

ꢃ

ꢄ

u

1

3

Theorem 7.1. Let v > 0 and k satisﬁes k ꢀ n , u ∈

,

. Under Assumption 7.1, we have

n

3

+2v

P

n

2

b

R → R := C r .

(54)

`

γ `

ꢃ ꢄ

If X has stationary increments with E η²= σ , we have

2

i

X

n

1,1

n

i

∑

∆ (Y)

2

,n

i=1 0,0

n

P

2

b

σb :=

− R + R → σ .

(55)

X

0

1

X

n

2

5

Electronic copy available at: https://ssrn.com/abstract=3423607

b

Moreover, we have the following limit distribution for R :

`

s

ꢀ

ꢁ

1

3n

R − R −^L→ N (0, 1) .

n

b

(56)

`

2

,n

3

2

k

σb

n

X

The relative size of the differences of X and ε varies in empirical studies. In particu-

lar, Campbell and Mankiw (1987) ﬁnd large random walk component (X) while Clark (1987)

ﬁnd small permanent component (ε). Next, we present a result when the relative size of the

two components is asymptotically large or small.¹¹

Corollary 7.1 (Small and large ε). Assume all conditions in Assumption 7.1 hold except ε = n ^ϕχ_i

−

i

ꢀ

ꢁ

1

−4ϕ

1

u

with ϕ < 1/4. Let k ꢀ n , u ∈ 0,

∧ ₂. A consistent ReMeDI estimator of the autocorrelation

n

3

function is given by

n−kn−` kn

n

i

∑

∆ (Y)

j`

n,ϕ

`

i=kn

n−kn kn

P

br :=

→ r .

(57)

`

n

i

∑

∆ (Y)

j0

i=kn

We next discuss why the LA method is inconsistent under long-span asymptotics. Suppose

ꢃ ꢄ

that Assumption 7.1 holds and that E η²= σ . Then, one can show under some regularity

2

i

X

conditions that

ꢇ

ꢈ

P 3σ²

3knσ²

˜

1

n

X

U((0, `))ⁿ

X

−1

˜

n

U((0, `)) →

,

E

=

+ O(k ),

t

˜

nk

2

n

˜

n

where U((0, `)) /n is the LA estimator applied to the efﬁcient price and and k is the tuning

t

parameter of the LA method.¹²Thus, the LA method is inconsistent when applied to low-

frequency data under long span asymptotics, and this is in line with our observations in the

simulation studies. The ReMeDI estimator by contrast has a bias that only depends on the

microstructure noise, and so its magnitude is not affected by the sampling scheme.

7

.3 The intuition and consequence of the robustness to data frequency

The results presented in Theorem 7.1 have a very clear economic intuition, which is quite dif-

ferent from the identiﬁcation under inﬁll asymptotics illustrated in Section 3. The consistency,

without any restrictions on the data frequencies, relies on the zero autocorrelations of the efﬁ-

cient returns. Therefore, the ReMeDI estimators effectively “remove” the efﬁcient price as the

efﬁcient returns on non-overlapping intervals are uncorrelated. What remains is simply due to

the market microstructure, and by tuning the “distance” and “lengths” of the non-overlapping

intervals, we can freely estimate the targeted moments of noise.

¹¹See also the studies on “small noise” by Da and Xiu (2019) and Aït-Sahalia and Xiu (2019) on high-frequency

analysis.

1

2

The result is consistent with Equation (3.35) in Jacod et al. (2017), in which they show, under inﬁll asymptotics

3

QV

2

t

that the LA applied to the efﬁcient price has a bias (after proper scaling) equal to

variation of the efﬁcient price process.

, where QV is the quadratic

t

2

6

Electronic copy available at: https://ssrn.com/abstract=3423607

−

10

8

·10⁻⁷

·

2

.4

.2

2

3.5

3

ReMeDI

RVs

ReMeDI

LAs

2

1

0

.5

2

1

.8

.6

.4

.2

1

.5

1

0

.8

.6

.4

.5

0

1

2

3

4

0

1

2

3

4

Log Ticks

Figure 2: Estimates of the variance of noise using transaction prices of E-mini S&P 500 futures on May

3

, 2010 using samples at different frequencies.

Why is the robustness to data frequency crucial? First, such robustness greatly broadens the

applications of the ReMeDI estimators. They can be used by microstructure researchers using

tick by tick data, or by a macroeconomist using monthly or annual data. Second, the sampling

frequency matters in the estimation of the volatility of X, especially if one uses high-frequency

data which may be deemed “noisy” whence a de-noise procedure is essential (Jacod et al.

(2009), Zhang et al. (2005), Barndorff-Nielsen et al. (2008), Xiu (2010)). Such de-noise methods

rely on the estimation of the moments of noise, in particular, the variance of noise. Using real

transaction prices, Figure 2 compares the ReMeDI estimates of the variance of noise to two

alternatives: the realized volatility (RV) estimator and the local averaging (LA) estimators.

The estimation is performed using the transaction prices of the same stock but sampled at

different frequencies, from tick by tick data to 1 minute returns. Both the RV and LA estimators

suggest that the magnitude of noise is larger when the sampling grid is coarser. However, such

monotone correspondence is a consequence of the low-frequency bias.¹³Thus, any de-noise

method using the RV or LA estimators to correct the microstructure effects will return very

different estimates when the sampling frequency varies. The ReMeDI estimators, however,

are quite robust to the sampling frequencies. Third, an economic concern arises when one

treats the magnitude of noise as a proxy of liquidity, see Hasbrouck (1993), Aït-Sahalia and Yu

(2009) and Chen and Mykland (2017). Trading frequencies have increased enormously in the

¹³See the analysis in Hansen and Lunde (2006), Jacod et al. (2017) and Li et al. (2019).

2

7

Electronic copy available at: https://ssrn.com/abstract=3423607

past decade, in tandem with improvements in liquidity, see Hendershott et al. (2011). Figure 2

provides a warning to use such measures: the RV and LA measures suggest that the liquidity

improves as data frequency increases—this is not intuitive as the measures are obtained on

subsamples of the transaction prices generated by the same market mechanism.

7

.4 Implementation: Which asymptotics?

We have shown that the ReMeDI estimator is robust to the data frequency, however the lim-

iting distribution of the ReMeDI estimator does depend on the sampling scheme. In many

practical circumstances, it is hard to decide which asymptotic distribution provides a better

approximation to a given sample of transaction prices, e.g., 5-min returns over one trading

month. In this section we introduce a practical method to construct estimators of the asymp-

totic variance for the ReMeDI estimators that are also “robust” to data frequencies.

ꢃ

ꢄ

2

n

2

14

Let R be the variance of ε and σ = E (X − X ) . Deﬁne a signal-to-noise ratio

0

n

i

i−1

R

1

0

π :=

n

=

.

0

u

2

u

2

R + n σ

1 + n σ /R

0

n

0

2

R = O (1) in either asymptotic framework; under inﬁll asymptotics, σ = O (δ ), thus for

0

p

n

p

n

2

u ∈ (0, 1), π → 1; under long-span asymptotics σ is a constant, thus π → 0. Now we deﬁne

n

the following asymptotic variance estimator¹⁵

3

ꢀ

ꢁ

n

2

3

k

π σb(Y; j , j )

2

`

n

2,n

X

n

`

` t

σb

:=

σb

+

.

,robust

n

t

0

For u close to 1, we have



ꢀ

ꢁ

3

2

k

2,n



n

σb

+ o (1/n), long span;

p

2

3n

X

σb

=

`

,robust

n



σb(Y;j ,j )

`

t

3 2

+

O (k δ ),

inﬁll.

p

n n

nt

2

`

The designed estimator σb

will reduce to inﬁll estimator or long-span estimator when the

,robust

2

`

dataset tells that the noise is relatively large or small. Thus, σb

will automatically decide

,robust

which asymptotic framework is proper after collecting some information (signal-to-noise ratio)

from the dataset.

8

Empirical Studies

Our empirical study relies on the E-mini S&P 500 futures contracts from Chicago Mercantile

Exchange (CME) Globex, which are futures on the S&P 500 stock market index. We use the

1

4

2

To calculate πn, we need estimates of R , σ . R can be estimated by the ReMeDI estimator, and σ can be

0

n

0

n

2

estimated by the likelihood approach (Aït-Sahalia et al. (2005)). Note that accurate estimates of R , σ are not

0

n

required; only the asymptotic order of the ratio is needed.

¹⁵Note that we stick to the notations introduced in previous sections. It should be clear that both nt, n denote the

number of observations of a given sample. Similarly, 1/n and δn are of the same order.

2

8

Electronic copy available at: https://ssrn.com/abstract=3423607

transaction data for all 20 trading days in May, 2010, including May 6, 2010 — the day of the

Flash Crash” — when U.S. equity indices collapsed and rebounded abruptly after a rapid ex-

“

ecution of a large selling program. Thus, it is of particular interest to measure market liquidity

during period of market stress using our proposed method.

Throughout the empirical studies, we select various tuning parameters via the algorithm

we developed in the supplementary material, see Li and Linton (2019).¹⁶

8

.1 Liquidities in three trading zones

We focus on the trading session that spans from 18:00 of the prior day to 16:15 Eastern Standard

Time (ET). This session covers three trading zones: 18:00 to 3:00 ET, 3:00 to 9:30 ET and 9:30 to

1

6:15 ET are the periods of regular trading in Asia, Europe and the U.S., respectively. Table 1

17

reports some descriptive statistics of the trading activities for the three segments. It is evident

that trading in the U.S. segment is more active.

18

Table 1 also reports our spread measures , the IBAS and ABAS as measures of liquidity

costs of the E-mini futures market. Among the three trading zones, the European trading

segment has relatively smaller liquidity cost, as measured by both the IBAS and ABAS. The

relatively magnitudes of liquidity costs in the three trading zones are further compared in the

top panel of Figure 3 on a daily level. Figure 6 depicts pairwise comparisons of the variances of

microstructure noise in three trading sessions. We label the paired estimates that are statistically

different using our testing devices developed in Section 6. The top and the bottom panels

illustrate that the variance of noise in the European session is signiﬁcantly smaller than the

Asia (14 days out of 20) and the U.S. sessions (17 days out of 20). We also investigate other

statistics of the U.S. session that reﬂect the turbulent market conditions on 6 May. For example,

3

/2

we ﬁnd R(0 ) /R

equal to -0.5793 and 1.8531 for the non Flash Crash days and the Flash

3

t

t,0

2

Crash day, respectively; and the estimates of R(0 ) /R are 31.2912 and 159.5389, respectively.

4

t

t,0

All days in May, 2010 (exclude 6, May)

May 6, 2010

Europe

Asia

Europe

3.72 × 10

4.20 × 10

The U.S.

Asia

The U.S.

5

4

5

4

6

4

3

5

6

Volume

1.00 × 10

2.43 × 10

7.82 × 10

4.89 × 10

5.09 × 10

5

4

5

Transactions 1.36 × 10

1.90 × 10

7.97 × 10

4.77 × 10

3.83 × 10

IBAS

1.55 (0.013) 1.38 (0.0026) 1.57 (0.0070) 1.38 (0.013) 1.29 (0.0073) 2.78 (0.041)

ABAS

1.61 (0.18)

1.53 (0.026)

1.69 (0.053) 1.51 (0.043) 1.33 (0.021)

4.78 (0.49)

Table 1: Statistics of trading activities and liquidity costs of E-mini S&P 500 futures in May, 2010. In the

left panel, the trading volume and the number of transactions are averages over the 20 trading days.

The two liquidity measures IBAS and ABAS are estimates based on all observations in the 20 trading

days. The standard deviations are reported in parentheses. The right panel reports the statistics using

the transactions on May 6 only.

¹⁶Typical choices of kn on the Flash Crash day are around 8 or 9, depending on the parameters to execute the

algorithm; for the remaining days, kn would be 3 or 4.

¹⁷The statistics are computed after some minor data ﬁltration using the procedures proposed by Andersen et al.

(

2018).

¹⁸The results are stated in basis points. One basis point is equivalent to 0.01% or 0.0001 in decimal form.

2

9

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8

.2 Order ﬂow patterns and autocovariances of microstructure noise

As we observe from Table 1 that the ABAS measure is statistically quite large compared with

the IBAS in all segments. We have similar observations on a daily basis — as revealed in the

bottom panel of Figure 3 — the ABAS measures are persistently larger than the IBAS measures,

which are persistently larger than the Roll measures. The discrepancy, as suggested by earlier

analysis, reﬂects the order imbalances induced by the (positively) autocorrelated order ﬂows.

The autocovariances of microstructure noise provide an angle to look at the dynamics of

order ﬂow. We estimate the autocovariances of microstructure noise on each trading day for

the U.S. trading segment. The results are presented in Figure 4. The microstructure noise ex-

hibits positive albeit weak autocorrelation patterns on most trading days. The autocovariances

usually die out after three lags. However, the (1, 4)-subplot in Figure 4 reveals that the autoco-

variance function of noise becomes extremely strong on May 6, 2010 — when the U.S. market

experienced chaotic market conditions — an episode dubbed the Flash Crash.

8

.3 The intraday pattern of spread measures

To study the intraday pattern of the spread measures, we estimate IBAS and ABAS in each

5 minutes window. The top and bottom panels of Figure 5 report the estimates for the non

1

Flash Crash days and the Flash Crash day. In line with the statistics reported in Table 1, the

European session has smaller spreads. An interesting observation, as is evident in the top

panel of Figure 5 is that when switched from the Asia trading session to the European trading

session, both spreads exhibit signiﬁcant drops, and gradually revert to a level that is close to

the spreads in the U.S. trading zone. Moreover, the trading volumes have approximately a U-

pattern in each trading session. On the Flash Crash day, we observe the simultaneous abrupt

increases in spreads measures and trading volume.

8

.4 The Flash Crash: Liquidity evaporation under large selling pressure

The right panel of Table 1 suggests that May 6 is “an unusually turbulent day for the mar-

kets” CFTC-SEC (2010): trading activities and liquidity costs almost double. This is an unique

phenomenon only when the U.S. market is active, which “opened to unsettling political and

economic news from overseas concerning the European debt crisis” CFTC-SEC (2010). As the

negative market sentiment continued to grow in the afternoon, a large fundamental trader

initiated a sell program to sell rapidly a total of 75,000 E-Mini contracts at 2:32 p.m. (ET). It

triggered larger selling pressure and a positive feedback loop that drove prices down. The

positive feedback loop appears to be consistent with positively serially correlated market mi-

crostructure noise as we demonstrated in Figure 4. As a consequence, the E-mini price dropped

sharply in a short time period, and liquidity drained quickly.

Now we demonstrate how to apply our new liquidity measures to identify the liquidity

evaporation. The left bottom panel of Figure 3 shows that the three liquidity measures yield

very different estimates on the day of the Flash Crash. All three measures have peaks in impact

3

0

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