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

Article abstract of DOI:10.1002/cctc.201900845

Au?Pd bimetallic catalyst has attracted significant research interests due to its fascinating properties. Herein, Au?Pd nanoparticles (NPs) supported on titanium silicalite-1 (denoted as Au?Pd/TS-1) were prepared by the alcohol reduction method. The Au?Pd alloy structure was proven via multiple characterization. This Au?Pd/TS-1 bimetallic catalyst showed improved activity compared with monometallic catalyst for propylene epoxidation with H₂ and O₂. The synergistic enhancement over Au?Pd/TS-1 could be elaborated by H-spillover process, which reduced the apparent activation energy significantly. The insights in this work are of referential importance to understanding the effect of interaction between bimetallic components on reaction system.

Full text of DOI:10.1002/cctc.201900845

∗
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}
|{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 , {Ft}t≥0, P(0) with the Grigelionis representation
(
0)
(0)
Z
Z
ꢀ
ꢁ
ꢀ
ꢁ
t
t
X = X +
bsds +
σ dW + ϑ1
? (p − q) + ϑ1
? pt,
(2)
t
0
s
s
{|ϑ|≤1}
{|ϑ|>1}
t
0
0
where W, p are a Wiener process and a Poisson random measure on R × E respectively. Here,
+
ꢀ
ꢁ
(
0)
(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
n
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
n
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
n
served transaction times with 0 = t < t < . . . , where N is the set of nonnegative integers.
+
0
1
We denote
n
n
nt =
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
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
t
i
i
i
n
n
discrete process, where Z is the set of integers, we denote V = V , 0 ≤ i ≤ n or V = V , 0 ≤ i ≤ n .
i
i
t
i
i
Let {δ } be a positive sequence of real numbers satisfying δ → 0 as n → ∞. Then, δ can
n
n
n
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
α
ꢂ
ꢂ
3
2
+
κ
n
ꢂ
ꢂ
α
δ(n, i) − δn ≤ Cδn
(4)
i−1
for some κ > 0.
Under the above assumptions, it follows that as n → ∞,
Z
t
δ n → A :=
αsds.
(5)
n
t
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
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
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
h
k+h
The sequence {χi}i∈Z is ρ mixing if ρ → 0 as k → ∞.
k
The following assumption characterises the degree of serial dependence of {χi}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
ρk ≤ kv ∀ k ∈ N+.
(7)
An immediate consequence of this assumption is that the autocovariance function of {χi}i
is decaying at a polynomial rate, i.e.,
C
k
|
Cov(χ , χi+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
· χi,
(9)
i
t
where γ is a (deterministic) positive Lipschitz continuous function of u. We further assume that
u
{
χi}i∈Z is 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
n
n
Y = X + ε .
(10)
i
i
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 , εi+`). The natural estimator is the sample analogue
i
`
i
n−`
1
n
n
n n
br :=
ε ε
,
i+`
`
∑
i
i=0
which is consistent and asymptotically normal under very mild conditions.
n
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
n
n
n
n
er :=
ε − ε
i−k0n
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
n
n
tor er follows the ReMeDI design and it provides another consistent estimator of r , provided:
`
`
0
0
n
k → ∞, k → ∞, k /n → 0, and k /n → 0. The intuition of the consistency becomes immedi-
n
n
n
n
ate if one rewrites er as
`
n−`−kn
n−`−kn
n−`−kn
n−`−kn
1
n
1
n
1
n
1
n
n
n n
n n
n
n
n
ni +`+kn .
er =
ε ε
−
ε ε
−
∑0 i−k0n i+`
ε
ε
+
∑0 i−k0n
i=k
n
ε
ε
(11)
`
∑0
i
i+`
∑0
i
i+`+kn
i=k
i=k
i=k
n
n
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`+kn , r`+k
0
, and r`+kn+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
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
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
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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
p
j ∈ J. The j-moments of χ, the stationary component of microstructure noise, are given by
!
q
r(j) = E
χ
.
(12)
∏
jp
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
n
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
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
2
tensen and Linton (2006) show how to use the moments E χ χ
to estimate θ.
i
i−j
1
1
Electronic copy available at: https://ssrn.com/abstract=3423607

n
n
Y
n
n
n
n
Y
i+j −2k
Y
Y
Y
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
k
n
n
n
n
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
n
n
n
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
n
E ∆ (ε) ≈ E
ε
.
j
i
∏
i+jp
p=1
ꢂ
ꢂ
ꢂ
ꢂ
If kp,n is 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
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
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
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Theorem 4.1. Let Assumptions (H, O, N) hold, assume v > 1 and k satisfies
n



−k1,n → ∞, k → ∞, ∀ p ≥ 2,


p,n
ꢂ
ꢂ
ꢂ
ꢂ
sup δ k
→ 0, ∀ p ≥ 1,
(15)
(16)
n
n,p
p




kp+1,n − kp,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
nt
At
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 n
n
follows:


−
kn
if p = 1,
kp,n
Then k satisfies the conditions in (15).
=

(
p − 1)kn if 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 n
2
kn → ∞, k /` → 0, ` δ = O(1).
n
n
n 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
n
n
`
0
n t
0
n t
0
n
δn`n
n
t− αt
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
0
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
0
0
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
jQq = (j : p ∈ Q ) for Q ⊂ {1, 2, . . . , q}.
p
q
q
1
3
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For each Q ⊂ {1, 2, . . . , q}, there is an associated (unique) pair of subsets:
q
c
0
Q := {1, 2, . . . , q}\Q , Qq
0
:= Q ∪ {q + 1, . . . , q }.
(17)
q
q
q
9
We denote for each k ∈ Z the following moments
ꢃ
ꢃ0
ꢄꢄ
ꢃ ꢄ
0
0
s (j, j ; k) := r j ⊕ j (+k) − r (j) r j ;
0
ꢀ
ꢀ
ꢁꢁ
0
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
0
0
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
−p
jp∈j,j ∈j0
p=p
0
jp∈j
0
p
0
6
−
r({j0
0
+ k} ⊕ j)r(j0 ).
−p0
∑
p
0
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
n


−
2
kn
p−1
if p = 1,
kp,n
=

kn if p ≥ 2,
(18)
ꢇ
ꢈ
ꢇ
ꢈ
−
$
1
1
1
v > 3, k ꢀ δ , $ ∈
, δ , where δ ∈
,
.
n
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
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) := nt
− R(j)t
.
t
nt
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
t
t
t
where the limit is centred Gaussian with (co)variances:
R
0
t
0
q+q
ꢃ
ꢄ
γ
dAs
s
0
0
t
0
σ(j, j ) := E Z(j) Z(j ) = S(j, j )
,
t
t
At
9By convention we let r(∅) = 1.
1
4
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and
2
0
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 nt. 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
nt
1
nt
ꢃ
ꢄꢃ
ꢄ
n
n
n
n
n
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
nt R − R
−L→ N 0, S`
s
,
n
b
t,`
t,`
At
where
R
t
0
2
∞
γ dA
ꢃ
ꢄ
s
s
2
Rt,` := r`
,
S :=
`
E((χ χ − r )(χ χ
− r )) + 3r .
(23)
∑
0
`
`
k
k+`
`
k
At
k=−∞
ꢃ
ꢄ
In the special case of i.i.d. noise, the asymptotic variance reduces to E χ4 + 2r up to
2
0
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
0
estimators. It has several constituents: ∑
s (j, j ; k) is the asymptotic variance of the sample
k∈Z
0
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
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
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
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)
+ un
n
b
Let Σ be a 2 × 2 matrix with components
t
n
n
n
n
0
n
n
0
0 n
t
b
b
b
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
t
t
t
t
t
where
!
nt−v0
n
in
ꢀ
ꢁ
1
0
n
n
n
n
n
n
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
nt i=2
kn
0
q −1
k=1
(26)
n
n
−1
n
n
n
−1
0 n
∆
(Y) := ∆ (Y) − v ReMeDIv (Y; j) ; ∆j
0
(Y) := ∆
0
(Y) − v ReMeDIv (Y; j ) ;
j
i
j
i
n
n
i
i
j
i
n
n
i
0
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 n
−
δ
−φ
i ꢀ δ , v ꢀ δn
,
2δ < φ < 1/2.
(27)
n
n
n
1
6
Electronic copy available at: https://ssrn.com/abstract=3423607

We have
!
Z(j)nt
L
n
t
L
−→ N (0, I ) ,
(28)
2
0
n
Z(j
t
ꢀ
ꢁ−
1
n
n
n>
n
t
b
where L is a matrix satisfying L L = Σ
.
t
t
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,`
t,`
σb(Y; j , j )
`
` t
!
n
b
R
n
−L→ N (0, 1) ,
t
t,`
− `
r
bn
bn
S
R
t,`
t,0
where






>
n
n
b
b
R
R
n

1
t,`
n
 1
t,`
b
b
S
:= 
, −
Σ 
, −
 ,
ꢀ
ꢁ
t
ꢀ
ꢁ
t,`
2
2
bn
bn
R
n
R
n
t,0
Rb
t,0
Rb
t,0
t,0
n
n
n
n
n
n
n
b
b
b
b
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
t
0
0 t
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
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
in
ꢀ
ꢁ
1
1
1
nt
0
n
n
n
n
n
n
n
e
e
e
e
e
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
n
n
n
t
e
e
∆
(Y) := ∆ (Y) −
;
∆
0
(Y) := ∆
0
(Y) −
.
j
i
j
i
j
i
j
i
nt
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
m
1
1
2
. For t < τ we have
≤ αt− ≤ κm,1 and α ≤ κm,1
.
m
t−
κm,1
n
. Let (F )
be the smallest filtration satisfying
t
t≥0
n
(
a) F ⊂ F ,
t
t
n
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
∞
t≥0
3
. With the restriction {tn < τ }, and for all p > 0,
m
i−1
ꢂ
ꢂ
ꢂ
ꢂ
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)
(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)
nt
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
i
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
i
i
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
u
t
nt−1
1
nt
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
χi,
t
i
where {χ } is a stationary mixing sequence, which could be serially dependent. The serial
i
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 Rt,0 is defined in (23). If ε = Sq /2, and {q } are uncorrelated with P(q = 1) =
i
i
i
i
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
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
i
q
ꢃ
ꢄ
n
P
t
q
measure. Let
q
n
t
n
2
n
n
b
b
AdjRoll := 2 (Roll /2) + 2R − R .
(36)
t
t,1
t,2
n
n
t
P
The adjusted Roll measure has the same probability limit as the IBAS : AdjRoll → IBAS ,
t
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
n
Empirically, we observe almost perfect matches of AdjRoll and IBAS , see Figure C.2 in our supplementary
t
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
`,κn
defined in (25)):
s
κn
n
n
n
b
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
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
i
i=1
(
buy) z (−z ) shares. The payoff is realized at time τ + 1 with price equal to the fundamental
i
i
value X , which evolves stochastically as
τ+1
Xτ+1 = X + θ ,
τ
τ
2
where θ has conditional mean 0 and variance Var (θτ) = σ . At time τ + 1, the intermediary
τ
τ
τ
has cash
n
cτ+1 = c + Yτ
zi,
τ
∑
i=1
and his aggregate wealth becomes
n
n
wτ+1 = −Xτ+1
zi + cτ+1 = (Y − (X + θ ))
z + c .
i
∑
τ
τ
τ
∑
τ
i=1
i=1
n
Thus, the conditional mean of wτ+1 is E (w ) = (Y − X ) ∑ z + c and the conditional
τ
τ+1
τ
τ
i=1
i
τ
2
0
Electronic copy available at: https://ssrn.com/abstract=3423607

variance is Var (w ) = (∑n z ) σ . The intermediary has mean-variance utility:
2
2
τ
τ+1
i=1
i
τ
ρ
a
U (w) = E (w) − Var (w) ,
τ
τ
τ
2
where ρ is the risk-aversion coefficient. To maximize his utility, the intermediary solves
a
max U (wτ+1).
τ
n
{
z }
i
i=1
At the optimum, the first order condition says that
n
Y − X
τ
τ
zi =
.
∑
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
i=1
i
ρaσ
that
n
2
Y = X + ρ σ
qi.
(39)
τ
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
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
τ
a
τ
2
2
Y = X + ρ σ ; thus the instantaneous spread S is given by S = 2ρ σ . Similar to the Roll’s
τ
τ
a
τ
1
1
a
τ
p
2
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
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 ∑ qi, i.e.,
a
τ
i=1
v
!
u
u
t
n
1
n
Sn := 2
Varτ ρ στ2
q .
i
a
∑
i=1
We denote S := lim
S ; S is the long run average measure. Let r = Cov(q , qi+k), and
∞
n→∞
n
∞
k
i
∞
assume that {q } are independent of σ and ∑
|r | < ∞. We have
i
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
i
q
q
S = S (1 + $ )/(1 − $ ). When the order flow is positively autocorrelated, i.e., $ ∈ (0, 1),
∞
1
q
q
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
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
κn ꢀ δn−ψ
1
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) ,
−L→ N (0, 1) ,
t
t
t
q
(41)
(42)
n,κn
(0, 0)/IBASnt
b
Σ
t
√
n
n (ABAS − ABAS )
t
t
t
q
n,κn
/ABASnt
b
2
Σ
t
where
and
κn
κn
n,κn
t
n,κn
n,κn
t
n,κn
t
0
b
b
b
b
Σ
:= Σt (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
n
n
n
n
n
∑
q
∆j(Y) ∆j0 (Y) + ∑
νk,in ∆j(Y) ∆j0 (Y)
+ ∆j0 (Y) ∆j(Y)
i=n +2 kn
i
i
i
i+k
i
i+k
n
T1
σb(Y; j , j )
:=
.
0
0 T ,T
1
2
n − n
T
2
T1
2
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
0
T
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
R
R
R
T2
T1
T4
T3
T2
T1
T4
2
2
2
2
γ dA
γ dA
γ dA
γ dA
T3
s
s
s
s
s
s
s
s
H0 :
=
;
H :
6
=
.
3
(44)
(45)
A
A − A
A − A
A − 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
ꢄ
ReMeDI(Y;j0)n
ReMeDI(Y;j0)n
T1,T2
T3,T4
n − n
n − n
−
T
2
T1
T
4
T3
nT −n
nT −n
2
T1
4
T3
T(Y; T , T , T , T ) :=
q
.
1
2
3
4 n
ꢃ
ꢃ
ꢄ
n
n
n − n σb(Y; j , j )
+ n − n σb(Y; j , j )
T
4
T
0
0
T
2
T1
0
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 HA.
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
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
n,q
t
ReMeDI(Y;j )
`
b
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
b
b
T
:= nt
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
n
n
n
n
b
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
n
n
n
b
Σ
(1, 1) := 4
ν
ν
0
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
n
t
VR
VR
b
b
b
b
b
S := R (0) + 2
ν R (`);
`,q
Σ
:=
+
Σ
(2, 2) −
.
3
∑
t
VR
ꢀ
ꢁ
ꢀ
ꢁ
VR
ꢀ
ꢁ
2
4
n,q
t
n,q
n,q
`
=1
b
b
b
R (0)
R (0)
R (0)
t
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
nt
T
− 1
n,q
VR,t
T
:=
q
.
bn
Σ
VR
We consider the following pairs of hypotheses:
q
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
q
q
q
A
0
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
n
ꢉ ꢇ
ꢈ
ꢊ
qn ꢀ δn−
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
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
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7
.1 The Model
Let {Yi}i∈N+ be an observed time series that can be decomposed into
Y = X + ε ,
(53)
i
i
i
where X and ε satisfy
Assumption 7.1.
1. The process {X } satisfies X
= µ + X + η , where {η } are centered
i
i
i+1
i
i
i i
ꢃ ꢄ
2
independent random variables with bounded second moments, i.e., sup E η ≤ C < ∞.
i
i
n
2
. The process {ε } is stationary, i.e., ε = C χ , where the process {χ } satisfies Assumption 2.1
i
i
i
γ
i
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
n
`
n−kn−`
1
n
n
kn
`
n
b
R :=
∆ (Y) .
`
∑
j
i
i=kn
ꢃ
ꢄ
u
1
1
3
Theorem 7.1. Let v > 0 and k satisfies k ꢀ n , u ∈
,
. Under Assumption 7.1, we have
n
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
n
P
2
b
b
σb :=
− R + R → σ .
(55)
X
0
1
X
n
2
5
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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 ϕχi
−
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
1
n
n
X
U((0, `))n
X
−1
˜
n
U((0, `)) →
,
E
=
+ O(k ),
t
t
˜
nk
2
2
n
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.12 Thus, 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
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−
10
8
·10−7
·
2
2
.4
.2
2
3.5
3
ReMeDI
RVs
ReMeDI
LAs
2
1
0
.5
2
1
1
1
1
.8
.6
.4
.2
1
.5
1
0
0
0
.8
.6
.4
.5
0
0
1
2
3
4
0
1
2
3
4
Log Ticks
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.13 Thus, 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
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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
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
0
u
2
u
2
R + n σ
1 + n σ /R
0
n
n
0
2
R = O (1) in either asymptotic framework; under infill asymptotics, σ = O (δ ), thus for
0
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
n
n
the following asymptotic variance estimator15
3
ꢀ
ꢁ
n
2
2
3
k
π σb(Y; j , j )
2
`
n
2,n
X
n
`
` t
σb
:=
σb
+
.
,robust
n
n
t
0
For u close to 1, we have

ꢀ
ꢁ
3
2
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
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
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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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