#  $K^{*0}$ and $\Sigma^{*}$ production in Au+Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV and 62.4 GeV

Kai Zhang  Department of Physics, Qufu Normal University, Shandong 273165,
People’s Republic of China  Jun Song  Department of Physics, Jining
University, Shandong 273155, People’s Republic of China  Feng-lan Shao
<EMAIL_ADDRESS>Department of Physics, Qufu Normal University, Shandong
273165, People’s Republic of China

###### Abstract

Applying a quark combination model for the hadronization of Quark Gluon Plasma
(QGP) and A Relativistic Transport (ART) model for the subsequent hadronic
rescattering process, we investigate the production of $K^{*0}$ and
$\Sigma^{*}$ resonances in central Au+Au collisions at $\sqrt{s_{NN}}=$ 200
GeV and 62.4 GeV. The initial $K^{*0}$ produced via hadronization is higher
than the experimental data in the low $p_{T}$ region and is close to the data
at 2-3 GeV/c. We take into account the hadronic rescattering effects which
lead to a strong suppression of $K^{*0}$ with low $p_{T}$ , and find that the
$p_{T}$ spectrum of $K^{*0}$ can be well described. According to the
suppressed magnitude of $K^{*0}$ yield, the time span of hadronic rescattering
stage is estimated to be about 13 fm/c at 200 GeV and 5 fm/c at 62.4 GeV. The
$p_{T}$ spectrum of $\Sigma^{*}$ directly obtained by quark combination
hadronization in central Au+Au collisions at 200 GeV is in well agreement with
the experimental data, which shows a weak hadronic rescattering effects. The
elliptic flow v2 of $\Sigma^{*}$ in minimum bias Au+Au collisions at 200 GeV
and $p_{T}$ spectrum of $\Sigma^{*}$ at lower 62.4 GeV are predicted.

###### pacs:

25.75.Dw, 24.10.Lx, 25.75.Nq, 25.75.-q

##  I Introduction

The short-lived resonances are efficient tools of probing the properties of
the hot and dense medium produced in relativistic heavy ion collisions. At
RHIC energies, QGP with extremely high energy density is created in primordial
collision stage stock08RHICrev , and the system undergoes a long time to
expand and cool Kolb0305084nuth . The lifetime of the resonance is about a few
fm/c, which is less than (or roughly the order of) the lifetime of the system
formed in heavy ion collisions. After QGP hadronization, but before the
interactions of hadrons cease, the initially produced resonances and stable
hadrons will undergo a hadronic rescattering stage. The resonance might be
destructed by rescattering with other hadrons and also be regenerated by the
collisions of other hadrons, and decay daughter particles of resonance are
kicked by other hadrons causing the signal loss. The physical properties of
resonances, e.g. their masses and widths, might be modified by the surrounding
medium Lutz02npa ; Shuryak03Medium . In addition, the yields and momentum
spectra of resonances might be changed. The experimentally reconstructed
resonances are the synthetic results of the hadronization and hadronic
rescattering effects.

RHIC and SPS experiments have provided rich data of production of resonances
such as $K^{*}$, $\Sigma^{*}$ in relativistic heavy ion collisions
KstarRHIC62_200GeV ; StrangeBresonance200Gev ; kstarAuAuVpp05y ; Kstar158GeV .
The relative yield ratios of resonances to stable hadrons are studied
experimentally, which also invokes many theoretical explanations Bleicher02 ;
vogel06 ; VogelAndBleicher05 . This progress greatly promote our understanding
of QGP hadronization mechanism tested against stable hadrons and the hadronic
rescattering dynamics, e.g. the time span of hadronic stage and cross sections
of various hadronic interaction channels Rafelski01 ; Marker02 .

The production mechanism of resonances at QGP hadronization is hard to
identify due to the entanglement of hadronization and rescattering effects.
RHIC data, e.g. the phenomena of v2 quark number scaling and high $p/\pi$
ratio etc, show that the production of various stable hadrons at QGP
hadronization is realized by the combination of constituent (anti-)quarks
Fries:2003prc ; Greco2003prc ; Hwa:2004prc . Recently, STAR experiments
observed that the v2 of $K^{*0}$, the same as stable particles, follows the
constituent quark number scaling rule in the intermediate $p_{T}$ region
KstarRHIC62_200GeV . This provides an evidence for the quark
(re-)combination/coalescence mechanism of resonance production at
hadronization in relativistic heavy ion collisions at RHIC. The K* meson and
$\Sigma^{*}$ baryon are of particular interests due to their very short
lifetime ( 4fm/c) and strange valence quark content. The experimental data of
their midrapidity yields and $p_{T}$ spectra are all available recently
KstarRHIC62_200GeV ; StrangeBresonance200Gev . A systematical study of this
pair of meson and baryon resonance should be able to further test the quark
combination mechanism of the hadron production in relativistic heavy ion
collisions.

In this paper, we apply a quark (re-)combination model for the hadronization
of hot and dense quark matter and ART model LbaoART95 ; LbaoART01  for the
hadronic rescattering process to study the $K^{*0}$ and $\Sigma^{*}$
production in central Au+Au collisions at $\sqrt{s_{NN}}=$200 GeV and 62.4
GeV. The investigation strategy is divided into two steps. Firstly, we compare
directly the hadronization results with experimental data to investigate the
proportion/magnitude of hadronization exhibited in the final observation.
Here, as a tool, we use the quark combination model developed by Shandong
group (SDQCM) QBXie1988PRD ; Shao2005prc  to deal with QGP hadronization.
Secondly, we take into account effects of hadronic rescattering and study the
entanglement of hadronization and hadronic rescattering effects at RHIC 200
GeV and 62.4 GeV and time span of hadronic stage for the system produced at
high RHIC energies.

##  II Initial $K^{*0}$ production via hadronization

The performance of quark (re-)combination mechanism on explaining the
production of various stable hadrons in the intermediate $p_{T}$ region in
relativistic heavy ion collisions is pretty well Fries:2003prc ; Greco2003prc
; Hwa:2004prc ; ClwKoCM_phi_Omega . The mechanism can also well describe
$p_{T}$-integrated yields and rapidity spectra of hadrons at RHIC and high SPS
energies Shao2005prc ; shao2007prc ; CEShao2009PRC ; JSong2009MPA . There are
several popular recombination models at RHIC. Quark recombination model
Fries:2003prc ; Hwa:2004prc  and parton coalescence model Greco2003prc
inclusively describe the combination of quarks into hadrons. ALCOR alcor95
and SDQCM apply the exclusive description. The spirit of quark combination has
been extended to various transport, variation and statistic methods of hadron
production in relativistic heavy ion collisions RavaTS07 ; MHZpfDyCoal07 ;
Alaladyqcm08 ; Cassing09 ; Abir09 .

In this paper, we use SDQCM to treat the initial production of various
hadrons. Of all the “on market” combination models, SDQCM is unique for its
combination rule which guarantees that mesons and baryons exhaust the
probability of all the fates of the (anti)quarks in deconfined color-neutral
system at hadronization. The main idea of the combination rule is to line up
the (anti)quarks in a one-dimensional order in phase space, e.g., in rapidity,
and then let them combine into initial hadrons one by one according to this
order Shao2005prc . Three (anti)quarks or a quark-antiquark pair in the
neighborhood form a (anti)baryon or a meson, respectively. The exclusive
nature of the model make it convenient to predict the $K^{*0}$ and
$\Sigma^{*}$ production on the basis of the reproduction of the yields and
momentum spectra of various stable hadrons.

In the meson formation, the relative formation probability of the lowest lying
vector meson (V) to pseudo-scalar meson (P) with the same valance quark
composition is tuned by the parameter V/P ratio which can not be given from
first principles. Spin counting arguments would suggest a 3:1 mixture between
vector and pseudoscalar mesons. In Lund string fragmentation, the V/P ratio is
taken to be 1 for light mesons and 1.5 for strange mesons, which is based on
wave function overlap arguments And82a . The resulting $K^{*}/K$ yield ratio
is about 0.56 which is calculated using PYTHIA 6.4 with default settings
pythia6.4 . The data of $K^{*0}/K^{-}$ yield ratio in $pp$ collisions at RHIC
energies is about 0.35 kstarAuAuVpp05y . This will lead to a rough estimation
of the V/P ratio by the relationship V/(V+P) =0.35 after taking into account
$K^{*}$ decay, and the resulting V/P ratio is about 0.5. The SDQCM fit of the
value of $K^{*0}/K^{-}$ in $pp$ reactions gives V/P ratio of 0.45. The choice
of V/P ratio influence directly the predicted abundance of $K^{*0}$ at
hadronization, and thus influence the identification of the magnitude of
rescattering effects in hadronic stage in explaining the experimental data.
Here, we give the prediction of $K^{*0}$ at different V/P values to test its
influence quantitatively.

Figure 1: Panel (a): The $p_{T}$ spectra of stable hadrons at midrapidity in
central Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV. Panel (b): The $p_{T}$
spectrum of $K^{*0}$. Symbols are the experimental data abelev:152301 ;
Adams07hyperon ; Abelev07phiv2 ; KstarRHIC62_200GeV  and lines are the results
of SDQCM. Panel (c): Yield ratio of $K^{*0}/K^{-}$ at midrapidity. Open
squares are results by hadronization for different V/P ratios. Experimental
data of $pp$ collisions and central Au+Au collisions are shown as dashed line
and band area KstarRHIC62_200GeV , respectively. Panel (d): Elliptic flow v2
of $K^{*0}$ as the function of $p_{T}$ in minimum bias Au+Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV. Symbols are the experimental data KstarRHIC62_200GeV
and the solid line is the result of SDQCM. Dashed lines marked by $n=3$ and 4
are the guidance of the consequence of aggravating regeneration effects in
hadronic stage, which is from Ref. DongX04v2decay .

The left panel (a) in Fig.1 is the $p_{T}$ spectra of various stable hadrons
at midrapidity in central Au+Au at $\sqrt{s_{NN}}=$ 200 GeV. Symbols are the
experimental data and lines are results of SDQCM in Ref. CEShao2009PRC  in
which the $p_{T}$ spectra of their anti-particles are also studied in detail.
V/P ratio is taken to be 3 in the calculation. The input of model is the
$p_{T}$ spectra of light and strange quarks just before hadronization, i.e.
$f_{q}(p_{T})$ and $f_{s}(p_{T})$, which are fixed by the data of $\pi^{-}$
and $K^{-}$. Clearly, the transverse momentum spectra of these stable hadrons
can be self-consistently explained by two quark $p_{T}$ spectra via
combination. This means that constituent quark degrees of freedom play a
dominated role in the production of these thermal hadrons.

The middle panel (b) in Fig.1 shows midrapidity $p_{T}$ spectrum of $K^{*0}$
produced by hadronization based on the left panel results and parameters
(quark spectra and V/P ratio). The V/P ratio is sensitive to the $K^{*0}$
yield and is relatively less sensitive to the slope of $K^{*0}$ $p_{T}$
spectrum. The $K^{*0}$ spectrum at other V/P values, e.g. 1 and 0.45, are all
presented. Here, we stress that even at V/P=1 and 0.45 the model can still
reproduce the $p_{T}$ and rapidity spectra of stable hadrons shown left. It is
because the V/P ratio does not alter the nature of hadron formation, and it
just changes the decay contributions of resonance to stable hadrons. We find
that the slope of $K^{*0}$ $p_{T}$ spectrum in 1.5-3.0 GeV/c is roughly
consistent with the data but in the low $p_{T}$ region the directly produced
$K^{*0}$ by combination is above the data even for V/P=0.45.

The right panel (c) in Fig.1 is the yield ratio $N(K^{*0})/N(K^{-})$ at
midrapidity for different V/P values. Because kaon and $K^{*}$ both contain
same valence quarks, their ratio roughly cancel the effect of strangeness
enhancement in relativistic heavy ion collisions and thus is sensitive to the
mechanism of hadron production. The data of $N(K^{*0})/N(K^{-})$ is $0.2\pm
0.03\pm 0.03$ in central Au+Au collisions and is $0.34\pm 0.01\pm 0.05$ in
minimum bias $pp$ reactions at $\sqrt{s_{NN}}=$ 200 GeV KstarRHIC62_200GeV ,
respectively. We find that the calculated ratio $N(K^{*0})/N(K^{-})$ at V/P =3
and 1 are higher than the data of Au+Au collisions which is shown as the band
area. The result of V/P=0.45 is consistent with the data of $pp$ reactions
shown as dashed line but is also higher than the data of Au+Au collisions. The
over-prediction of hadronization in the low $p_{T}$ region indicates the
necessity of hadronic rescattering for the suppression of $K^{*0}$ yield.

To further manifest the hadronization effect in final state observation, we
present in panel (d) in Fig.1 the elliptic flow v2 of $K^{*0}$ at midrapidity
in minimum bias Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV. This result is
calculated using the extracted quark v2 in the previous study of v2 of various
stable hadrons in Ref. qcmv2 . V/P ratio does not influence the predicted v2
which is irrespective of particle abundance. One can see that the calculated
$K^{*0}$ v2, shown as solid line, is in well agreement with the experimental
data. The dashed lines are the parameterization of hadronic v2 as the function
of number of constituent quarks $n$ in Ref. DongX04v2decay . $n=3$ and 4 are
the guidance of the consequence of aggravating regeneration effects in
hadronic stage. The agreement of our result with the data means that the
$K^{*0}$ mesons observed experimentally mainly come from the hadronization.

##  III Hadronic rescattering effects on $K^{*0}$ production

The subsequent hadronic rescattering stage after hadronization is simulated by
A Relativistic Transport (ART) model LbaoART95 ; LbaoART01  which includes
baryon-baryon, baryon-meson, and meson-meson elastic and inelastic
scatterings. We apply the code in AMPT event generator V2.25t3 LinAMPT05
which has provided proper extension to higher RHIC energies from original SPS
energies. The time span of hadronic stage is important for the hadronic
rescattering effects such as the suppression magnitude of $K^{*0}$ yield. In
transport theory, the rescattering time should be infinite in principle. In
practice, one would like to choose a finite rescattering time, and this
choice, to some extent, is similar to the decouple criteria of kinetic freeze-
out in hydrodynamic theory of heavy ion collisions. In this work, we treat the
rescattering time as an adjustable parameter and study how long the time span
of hadronic phase is favored by the experimental data when the initial
produced $K^{*0}$ is fixed.

Figure 2: Top panel: The survived ratio of $K^{*}$yield as the function of
hadronic rescattering time $t_{res}$ in central Au+Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV. Bottom panel: The survived $K^{*0}$ as the function
of $p_{T}$ at different $t_{res}$.
Figure 3: Top panel: The final state $K^{*}$yield at midrapidity as the
function of hadronic rescattering time $t_{res}$ in central Au+Au collisions
at $\sqrt{s_{NN}}=$ 200 GeV. The symbols connected with lines are calculation
results corresponding three differen V/P ratios at hadronization. The
experimental data are shown as the band area KstarRHIC62_200GeV . Bottom
panel: The $p_{T}$ spectra of final state $K^{*}$ for three V/P values with
the corresponding rescattering times for yield reproduction in central Au+Au
collisions at 200 GeV. The experimental data are shown as symbols
KstarRHIC62_200GeV . The band area shows the result of $K^{*}$ just after
hadronization.

Fig.2 top panel shows the survived ratio of $K^{*}$ abundance
$N(K^{*})^{final}/N(K^{*})^{initial}$ after hadronic rescattering stage as the
function of rescattering time $t_{res}$. Here $N(K^{*})^{initial}$ is the
number of $K^{*}$ just after hadronization and $N(K^{*})^{final}$ is the
number of $K^{*}$ that can be experimentally reconstructed after hadronic
rescatterings. $K^{*}$ mesons are reconstructed from their hadronic decay
channels using pion-kaon invariant mass analysis, and the survived $K^{*}$
incorporates all effects of destruction, signal loss and regeneration. One can
see that the number of survived $K^{*}$ almost exponentially decreases with
the rescattering time. The time of $K^{*}$ number reducing by half is about 20
fm/c. Fig.2 bottom panel shows the number of survived $K^{*0}$ after hadronic
rescattering stage as the function of transverse momentum $p_{T}$ at different
rescattering time $t_{res}$. We find that the suppression of $K^{*0}$ caused
by hadronic rescattering effects is strong in low $p_{T}$ region. This is
qualitatively consistent with the result of UrQMD transport model
VogelAndBleicher05 ; Kstar158GeV .

Taking into account of effects of hadronic rescatterings on the yield
suppression and spectra distortion, we obtain the yield and $p_{T}$ spectrum
of final state $K^{*0}$ comparable to experimental data. The top panel in
Fig.3 shows the yield of final survived $K^{*0}$ at midrapidity as the
function of rescattering time $t_{res}$ in central Au+Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV. The directly produced $K^{*0}$ for different V/P
ratios at hadronization is taken to be the starting point of the hadronic
rescattering stage. For V/P=3 the rescattering time needed to reproduce the
experimental data is about 60 fm/c while for V/P=1 the needed time is about 30
fm/c. For V/P=0.45 favored by $pp$ $N(K^{*0})/N(K^{-})$ data the rescattering
time is about 13 fm/c. This value is consistent with the typical lifespan of
$13\pm 3$ fm/c in UrQMD simulation of hadronic rescattering stage Bleicher02 .
The estimation of time span between chemical and thermal freeze-out by a
thermal model using an additional pure rescattering phase is $2.5^{+1.5}_{-1}$
fm/c from the analysis of $K^{*0}/K^{-}$ yield ratio Rafelski01 ;
VogelAndBleicher06proc . The bottom panel in Fig.3 shows the $p_{T}$ spectrum
of final state $K^{*0}$ at midrapidity at three V/P values with the
corresponding rescattering times for yield reproduction in central Au+Au
collisions at 200 GeV. We find that the strong suppression of low $p_{T}$
$K^{*0}$ in hadronic rescattering process offset the over-predication of
initial $K^{*0}$ by hadronization shown as the band area, and this leads to an
obviously improved description of $p_{T}$ spectrum of $K^{*0}$.

The $K^{*0}$ data at different collision energies enable the further test of
hadronization mechanism and the investigation of the energy dependence of
hadronic rescattering effects. In Fig.4, we present the $p_{T}$ spectra of
various stable hadrons and $K^{*0}$ resonance at midrapidity in central Au+Au
collisions at $\sqrt{s_{NN}}=$ 62.4 GeV. Symbols are the experimental data and
lines are the results of SDQCM. The $p_{T}$ spectra of constituent quarks at
hadronization are taken to be the thermal exponential pattern
$\exp(-m_{T}/T_{s})$. The slope parameter $T_{s}$ is taken to be 0.31 GeV for
strange quarks and 0.29 GeV for light quarks, which are smaller than those at
200 GeV CEShao2009PRC . The numbers and rapidity spectra of constituent quarks
and antiquarks are fixed by the experimental data of pion and kaon rapidity
spectra Arsene62gev_y_spectra . One can see that, similar to Au+Au 200 GeV,
the model results of various stable hadrons are in well agreement with the
data. Then we can predict the $p_{T}$ spectrum of initial $K^{*0}$ just after
hadronization, and the results are presented in the right panel in Fig.4. The
degree of the agreement between the model result and experimental data is also
similar to that in Au+Au 200 GeV. The calculated $K^{*0}$ yield densities in
low $p_{T}$ region exceed the data and become to close to the data as $p_{T}$
rises to 2-3 GeV/c. This indicates the influence of the hadronic rescattering
on $K^{*0}$ production is still significant at intermediate RHIC energy.

Figure 4: The $p_{T}$ spectra of stable hadrons (left panel), their anti-
hadrons (middle panel) and $K^{*0}$ resonance (right panel) at midrapidity in
central Au+Au collisions at $\sqrt{s_{NN}}=$ 62.4 GeV. Symbols are the
experimental data piproton62gev ; multH62.4GeV  and lines are results of
SDQCM.
Figure 5: Top panel: The final state $K^{*}$ yield at midrapidity as the
function of hadronic rescattering time $t_{res}$ in central Au+Au collisions
at $\sqrt{s_{NN}}=$ 62.4 GeV. The symbols connected with lines are calculation
results corresponding three differen V/P ratios at hadronization. The
experimental data are shown as the band area KstarRHIC62_200GeV . Bottom
panel: The $p_{T}$ spectra of final state $K^{*}$ for three V/P values with
the corresponding rescattering times for yield reproduction in central Au+Au
collisions at 62.4 GeV. The experimental data are shown as symbols
KstarRHIC62_200GeV . The band area shows the result of $K^{*}$ just after
hadronization.

The top panel in Fig.5 shows the yield of final survived $K^{*0}$ at
midrapidity as the function of rescattering time $t_{res}$ in central Au+Au
collisions at $\sqrt{s_{NN}}=$ 62.4 GeV. For V/P=3 the rescattering time
needed to reproduce the experimental data is about 20 fm/c while for V/P=1 the
needed time is about 14 fm/c and for V/P=0.45 favored by $pp$
N($K^{*}$)/N($K^{-}$ ) data the time is about 5 fm/c. It is found that the
needed time at 62.4 GeV is about half of that at 200 GeV.

The bottom panel in Fig.5 presents the $p_{T}$ spectrum of final state
$K^{*0}$ at midrapidity at three V/P values with the corresponding
rescattering times for yield reproduction in central Au+Au collisions at 62.4
GeV. The strong suppression in low $p_{T}$ region in hadronic rescattering
process offset the over-predication of initial $K^{*}$by hadronization shown
as the band area, and this leads to an improved description of $p_{T}$
spectrum of $K^{*0}$.

##  IV $\Sigma^{*}$ production by hadronization at RHIC

The $\Sigma^{*}$ hyperon has a very short lifetime (4.5 fm/c). The measurement
of STAR Collaboration found that the $\Sigma^{*}/\Lambda$ yield ratio at
midrapidity in central Au+Au collisions is nearly the same as that in $pp$
collision at $\sqrt{s_{NN}}=$ 200 GeV StrangeBresonance200Gev . This indicates
that the net effects of rescattering loss and rescattering gain in hadronic
stage are very small. This poses serious constraint on cross sections of
various reaction channels of collision gain and collision loss. The small net
effect of hadronic rescattering has two possibilities. The first is that both
the effect of collision gain and that of collision loss on $\Sigma^{*}$ yield
are large but they offset with each other. The $p_{T}$ spectrum of
$\Sigma^{*}$ might change during hadronic rescattering in this case. The
second is that two components are all small and $p_{T}$ spectrum of
$\Sigma^{*}$ should not be changed dramatically compared with the result of
hadronization. Here, using the quark spectra fixed by stable hadrons in Fig.1,
we use SDQCM to study the yield and $p_{T}$ spectrum of $\Sigma^{*}$ at
midrapidity in central Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV. The yield
density $dN/dy$ of $\Sigma^{*}+\overline{\Sigma}^{*}$ at midrapidity just
after hadronization is 9.0, and the data of STAR Collaboration is $9.3\pm
1.4\pm 1.2$ StrangeBresonance200Gev . Here $\Sigma^{*}$ represents
$\Sigma^{*+}+\Sigma^{*-}$. The result of $p_{T}$ spectrum of $\Sigma^{*}$ just
after hadronization is presented in left panel in Fig.6 and is compared with
the experimental data. We find that the agreement between hadronization
results and the data is well. The result favors the second case. We further
predict the elliptic flow v2 of $\Sigma^{*}$ in minimum bias Au+Au collisions
at 200 GeV and the result is shown in middle panel in Fig. 6. It is well known
that the elliptic flow is sensitive to the mechanism of hadron production. The
elliptic flow of hadrons formed via quark combination mechanism follows the
constituent quark number scaling rule. The phenomenological regeneration of
$\Sigma^{*}$ by $\Lambda+\pi\rightarrow\Sigma^{*}$ will result in higher v2.
To test the effects of hadronic rescattering on $\Sigma^{*}$ production at
RHIC energies, we further predict the $\Sigma^{*}$ $p_{T}$ spectrum in central
Au+Au collisions at 62.4 GeV. The yield density of $\Sigma^{*}$ at midrapidity
is 4.3 just after hadronization.

Figure 6: Left panel: The $p_{T}$ spectrum of $\Sigma^{*}$ in central Au+Au
collisions at $\sqrt{s_{NN}}=$ 200 GeV. Symbols are the experimental data
StrangeBresonance200Gev  and the line is the result of SDQCM. Middle panel:
The prediction of elliptic flow v2 of $\Sigma^{*}$ in minimum bias Au+Au
collisions at 200 GeV. Right panel: The prediction of $p_{T}$ spectrum of
$\Sigma^{*}$ in central Au+Au collisions at $\sqrt{s_{NN}}=$ 62.4 GeV.

##  V summary

In this paper, we have used SDQCM model for the QGP hadronization and ART
model for the subsequent hadronic rescattering process to investigate the
production of $K^{*0}$ and $\Sigma^{*}$ resonances in central Au+Au collisions
at $\sqrt{s_{NN}}=$ 200 GeV and 62.4 GeV. The initial $K^{*0}$ produced by
quark combination mechanism is higher than the experimental data in the low
$p_{T}$ region and is close to data at 2-3 GeV/c. In the subsequent hadronic
rescattering stage, the number of $K^{*0}$ that can be reconstructed
experimentally exponentially decreases with the increasing rescattering time,
and the suppression of $K^{*0}$ yield focuses on low $p_{T}$ region, which
offsets the over-prediction of hadronization. Therefore the quark combination
hadronization plus hadronic rescattering can provide a well description of the
experimental data of $K^{*0}$ production. According to the suppression
magnitude of $K^{*0}$ yield, the time span of hadronic rescattering stage is
estimated. V/P ratio at hadronization is important for the extraction of
hadronic rescattering time from the data of $K^{*0}$ yield. For V/P =0.45
based on the data of $K^{*0}/K^{-}$ in $pp$ reaction, the time span of
hadronic rescattering stage is about 13 fm/c at 200 GeV and 5 fm/c at 62.4
GeV. Higher V/P ratio leads to the longer rescattering time. The yield density
and $p_{T}$ spectrum of $\Sigma^{*}$ directly from quark combination
hadronization in central Au+Au collisions at 200 GeV is found to be in well
agreement with the experimental data, which indicates a weak hadronic
rescattering effects. To make a further test, we predict the v2 of
$\Sigma^{*}$ in minimum bias Au+Au collisions at 200 GeV and $p_{T}$ spectrum
of $\Sigma^{*}$ at lower 62.4 GeV for the comparison with future STAR data .

## ACKNOWLEDGMENTS

The authors thank X. B. Xie, Z. T. Liang and G. Li for helpful discussions.
The work is supported in part by the National Natural Science Foundation of
China under grant 11175104 and 10947007, and by the Natural Science Foundation
of Shandong Province, China under grant ZR2011AM006.

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T_{9}&=&\frac{M}{N^{2}}\sum_{i=1}^{M}Z_{i}\sum_{j=1}^{n_{i}}\varepsilon_{ij}(1)\sum_{j=1}^{n_{i}}x_{ij}\\{\bar{\varepsilon}_{\mathcal{\scriptscriptstyle
T}}-\bar{x}_{\mathcal{\scriptscriptstyle
T}}\hat{Q}_{\textsc{i}}(1)\\},&&T_{10}&=&\frac{M}{N^{2}}\sum_{i=1}^{M}Z_{i}\sum_{j=1}^{n_{i}}x_{ij}\sum_{j=1}^{n_{i}}\varepsilon_{ij}(1)x_{ij}\hat{Q}_{\textsc{i}}(1),\\\
T_{11}&=&\frac{M}{N^{2}}\sum_{i=1}^{M}Z_{i}\sum_{j=1}^{n_{i}}x_{ij}\sum_{j=1}^{n_{i}}x_{ij}^{2}\hat{Q}_{\textsc{i}}(1)^{2},&&T_{12}&=&\frac{M}{N^{2}}\sum_{i=1}^{M}Z_{i}(\sum_{j=1}^{n_{i}}x_{ij})^{2}\hat{Q}_{\textsc{i}}(1)\\{\bar{\varepsilon}_{\mathcal{\scriptscriptstyle
T}}-\bar{x}_{\mathcal{\scriptscriptstyle T}}\hat{Q}_{\textsc{i}}(1)\\},\\\
T_{13}&=&\frac{M}{N^{2}}\sum_{i=1}^{M}Z_{i}n_{i}\sum_{j=1}^{n_{i}}\varepsilon_{ij}(1)x_{ij}\\{\bar{\varepsilon}_{\mathcal{\scriptscriptstyle
T}}-\bar{x}_{\mathcal{\scriptscriptstyle
T}}\hat{Q}_{\textsc{i}}(1)\\},&&T_{14}&=&\frac{M}{N^{2}}\sum_{i=1}^{M}Z_{i}n_{i}\sum_{j=1}^{n_{i}}x_{ij}^{2}\\{\bar{\varepsilon}_{\mathcal{\scriptscriptstyle
T}}-\bar{x}_{\mathcal{\scriptscriptstyle
T}}\hat{Q}_{\textsc{i}}(1)\\}\hat{Q}_{\textsc{i}}(1),\\\
T_{15}&=&\frac{M}{N^{2}}\sum_{i=1}^{M}Z_{i}n_{i}\sum_{j=1}^{n_{i}}x_{ij}\\{\bar{\varepsilon}_{\mathcal{\scriptscriptstyle
T}}-\bar{x}_{\mathcal{\scriptscriptstyle
T}}\hat{Q}_{\textsc{i}}(1)\\}^{2}.\end{array}$

We claim that all terms $T_{7}$–$T_{15}$ are of order
$O_{\mathbb{P}}\left(M\Omega\right)$. We show that
$T_{7}=O_{\mathbb{P}}\left(M\Omega\right)$ and omit the proofs for other
terms: by Assumption 2,

$\displaystyle|T_{7}|$ $\displaystyle\leq$
$\displaystyle\frac{M}{N^{2}}\sum_{i=1}^{M}\left|\sum_{j=1}^{n_{i}}\varepsilon_{ij}(1)\sum_{j=1}^{n_{i}}\varepsilon_{ij}(1)x_{ij}\right|$
$\displaystyle\leq$ $\displaystyle
O\left(\frac{M}{N^{2}}\right)\sum_{i=1}^{M}\left[\left\\{\sum_{j=1}^{n_{i}}\varepsilon_{ij}(1)\right\\}^{2}+\left\\{\sum_{j=1}^{n_{i}}\varepsilon_{ij}(1)x_{ij}\right\\}^{2}\right]$
$\displaystyle\leq_{\textsc{hi}}$ $\displaystyle
O\left(\frac{M}{N^{2}}\right)\sum_{i=1}^{M}\left\\{n_{i}\sum_{j=1}^{n_{i}}\varepsilon_{ij}(1)^{2}+n_{i}\sum_{j=1}^{n_{i}}\varepsilon_{ij}(1)^{2}x_{ij}^{2}\right\\}$
$\displaystyle\leq$ $\displaystyle
O\left(\frac{M}{N^{2}}\right)\sum_{i=1}^{M}n_{i}\sum_{j=1}^{n_{i}}\left\\{\varepsilon_{ij}(1)^{2}+\varepsilon_{ij}(1)^{4}+x_{ij}^{4}\right\\}\leq
O\left(M\Omega\right).$

Therefore, (A14) holds.

The proof of (A15) is similar to (A14), and we omit it. Therefore,
${H}=O_{\mathbb{P}}(M\Omega)$ follows from (A13)–(A15). ∎

Now we analyze the standard errors in Theorems 1 and 2.

###### Proof of the results about the standard errors in Theorem 1.

We first analyze
$\hat{\textup{se}}_{\textsc{lz}}^{2}(\hat{\tau}_{\textsc{i}})$. The matrix
$\tilde{X}^{\mathrm{\scriptscriptstyle T}}\tilde{X}$ simplifies to

$\tilde{X}^{\mathrm{\scriptscriptstyle
T}}\tilde{X}=\sum_{i=1}^{M}\tilde{X}_{i}^{\mathrm{\scriptscriptstyle
T}}\tilde{X}_{i}=\sum_{i=1}^{M}Z_{i}\begin{pmatrix}n_{i}&0\\\
0&0\end{pmatrix}+\sum_{i=1}^{M}(1-Z_{i})\begin{pmatrix}0&0\\\
0&n_{i}\end{pmatrix}=\begin{pmatrix}n_{\mathcal{\scriptscriptstyle T}}&0\\\
0&n_{\mathcal{\scriptscriptstyle C}}\end{pmatrix}.$

The residual from the OLS of $Y_{ij}$ on $(1,Z_{ij})$ is
$\hat{\varepsilon}_{ij}$ defined in Section 3.2 of the main text, so the
matrix $\sum_{i=1}^{M}\tilde{X}_{i}^{\mathrm{\scriptscriptstyle
T}}\hat{U}_{i}\tilde{X}_{i}$ simplifies to

$\displaystyle\sum_{i=1}^{M}\tilde{X}_{i}^{\mathrm{\scriptscriptstyle
T}}\hat{U}_{i}\tilde{X}_{i}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}Z_{i}\begin{pmatrix}\left(\sum_{j=1}^{n_{i}}\hat{\varepsilon}_{ij}\right)^{2}&0\\\
0&0\end{pmatrix}+\sum_{i=1}^{M}(1-Z_{i})\begin{pmatrix}0&0\\\
0&\left(\sum_{j=1}^{n_{i}}\hat{\varepsilon}_{ij}\right)^{2}\end{pmatrix}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}\sum_{i=1}^{M}Z_{i}\left(\sum_{j=1}^{n_{i}}\hat{\varepsilon}_{ij}\right)^{2}&0\\\
0&\sum_{i=1}^{M}(1-Z_{i})\left(\sum_{j=1}^{n_{i}}\hat{\varepsilon}_{ij}\right)^{2}\end{pmatrix}.$

These two matrices are both diagonal, and it is straightforward to use Lemma
A8 to obtain the formula of
$\hat{\textup{se}}_{\textsc{lz}}^{2}(\hat{\tau}_{\textsc{i}})$ in Theorem 1 of
the main text. The proof of its conservativeness is the same as that of
Theorem 2. So we omit it.

We then analyze
$\hat{\textup{se}}_{\textsc{hw}}^{2}(\hat{\tau}_{\textsc{i}})$. Angrist and
Pischke [2008, page 228] give

$N\times\hat{\textup{se}}^{2}_{\textsc{hw}}(\hat{\tau}_{\textsc{i}})=N\left({\sum_{ij\in\mathcal{T}}\hat{\varepsilon}_{ij}^{2}}/{n_{\mathcal{\scriptscriptstyle
T}}^{2}}+{\sum_{ij\in\mathcal{C}}\hat{\varepsilon}_{ij}^{2}}/{n_{\mathcal{\scriptscriptstyle
C}}^{2}}\right)=N\left\\{{\sum_{ij\in\mathcal{T}}(Y_{ij}-\bar{Y}_{\mathcal{\scriptscriptstyle
T}})^{2}}/{n_{\mathcal{\scriptscriptstyle
T}}^{2}}+{\sum_{ij\in\mathcal{C}}(Y_{ij}-\bar{Y}_{\mathcal{\scriptscriptstyle
C}})^{2}}/{n_{\mathcal{\scriptscriptstyle C}}^{2}}\right\\}.$

Therefore,

$\displaystyle
N{\sum_{ij\in\mathcal{T}}(Y_{ij}-\bar{Y}_{\mathcal{\scriptscriptstyle
T}})^{2}}/{n_{\mathcal{\scriptscriptstyle
T}}^{2}}=N{\sum_{ij\in\mathcal{T}}\\{\varepsilon_{ij}(1)-\bar{\varepsilon}_{\mathcal{\scriptscriptstyle
T}}\\}^{2}}/{n_{\mathcal{\scriptscriptstyle T}}^{2}}$ $\displaystyle=$
$\displaystyle
N\left\\{\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)^{2}-n_{\mathcal{\scriptscriptstyle
T}}\bar{\varepsilon}_{\mathcal{\scriptscriptstyle
T}}^{2}\right\\}/{n_{\mathcal{\scriptscriptstyle
T}}^{2}}=N\left\\{{\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)^{2}}/{n_{\mathcal{\scriptscriptstyle
T}}^{2}}-{\bar{\varepsilon}_{\mathcal{\scriptscriptstyle
T}}^{2}}/{n_{\mathcal{\scriptscriptstyle T}}}\right\\}.$

By Lemma A6 and Slutsky’s Theorem, we have
$\bar{\varepsilon}_{\mathcal{\scriptscriptstyle T}}=o_{\mathbb{P}}(1)$ and

$\displaystyle
N{\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)^{2}}/{n_{\mathcal{\scriptscriptstyle
T}}^{2}}$ $\displaystyle=$ $\displaystyle(N/n_{\mathcal{\scriptscriptstyle
T}})\left\\{n_{\mathcal{\scriptscriptstyle
T}}^{-1}\sum_{ij\in\mathcal{T}}{\varepsilon}_{ij}(1)^{2}\right\\}=\left\\{1/e+o_{\mathbb{P}}(1)\right\\}\left\\{N^{-1}\sum_{ij}{\varepsilon_{ij}(1)^{2}}+o_{\mathbb{P}}(1)\right\\}$
$\displaystyle=$
$\displaystyle(eN)^{-1}\sum_{ij}{\varepsilon_{ij}(1)^{2}}+o_{\mathbb{P}}(1).$

Therefore,
$N{\sum_{ij\in\mathcal{T}}(Y_{ij}-\bar{Y}_{\mathcal{\scriptscriptstyle
T}})^{2}}/{n_{\mathcal{\scriptscriptstyle
T}}^{2}}=(eN)^{-1}\sum_{ij}{\varepsilon_{ij}(1)^{2}}+o_{\mathbb{P}}(1).$ We
have the same result for control group. Therefore, we get the probability
limit of $\hat{\textup{se}}_{\textsc{hw}}^{2}(\hat{\tau}_{\textsc{i}})$. ∎

###### Proof of the standard error in Theorem 2.

Lemma A8 implies that

$M\times\hat{\textup{se}}_{\textsc{lz}}^{2}(\hat{\tau}_{\textsc{i}}^{\textup{adj}})=\left[{G}^{-1}{H}{G}^{-1}\right]_{(1,1)+(2,2)-2(1,2)}.$

So the key is to analyze ${G}^{-1}{H}{G}^{-1}$. Define

$\Lambda=\left(\begin{array}[]{ c c;{2pt/2pt} c c}e&0&0&0\\\ 0&1-e&0&0\\\
\hdashline[2pt/2pt]{0}&{0}&e\sum_{ ij}x_{ij} x_{ij}^{{
\mathrm{\scriptscriptstyle T} }} / { N}&0\\\ 0&0&0&(1-e)\sum_{ij} x_{ij}
x_{ij}^{{ \mathrm{\scriptscriptstyle T} }} / {
N}\end{array}\right)=\left(\begin{array}[]{c c}\Lambda_{11}&0\\\
0&\Lambda_{22}\end{array}\right)$

as the expectation of $G.$ We claim that

$\displaystyle\left[{G}^{-1}{H}{G}^{-1}\right]_{(1-2,1-2)}-\left[{\Lambda}^{-1}{H}{\Lambda}^{-1}\right]_{(1-2,1-2)}=o_{\mathbb{P}}(1),$
(A16)

which implies the conclusion because by Lemmas A9 and A10,

$\displaystyle\left[{{\Lambda}}^{-1}{{H}}{{\Lambda}}^{-1}\right]_{(1,1)+(2,2)-2(1,2)}$
$\displaystyle=$
$\displaystyle\frac{M\sum_{i=1}^{M}Z_{i}(\sum_{j=1}^{n_{i}}\hat{r}_{ij})^{2}}{N^{2}e^{2}}+\frac{M\sum_{i=1}^{M}(1-Z_{i})(\sum_{j=1}^{n_{i}}\hat{r}_{ij})^{2}}{N^{2}(1-e)^{2}}$
$\displaystyle=$
$\displaystyle\frac{M\sum_{i=1}^{M}\\{\sum_{j=1}^{n_{i}}{r}_{ij}(1)\\}^{2}}{N^{2}e}+\frac{M\sum_{i=1}^{M}\\{\sum_{j=1}^{n_{i}}{r}_{ij}(0)\\}^{2}}{N^{2}(1-e)}+o_{\mathbb{P}}(1).$

Now we complete the proof by showing (A16). Define ${\Delta}=G-\Lambda,$ and
$\Psi={{G}}^{-1}-{{\Lambda}}^{-1}.$ By Lemma A6,
${\Delta}=O_{\mathbb{P}}(\Omega^{1/2})$. By Assumptions 1 and 3, $\Lambda$
converges to a finite and invertible matrix. So Lemma A5 ensures
$\Psi=O_{\mathbb{P}}(\Omega^{1/2}).$ By definition,

$\displaystyle G^{-1}{H}{G}^{-1}-{\Lambda}^{-1}{H}{\Lambda}^{-1}$
$\displaystyle=$
$\displaystyle\left({\Lambda}^{-1}+{\Psi}\right){H}\left({\Lambda}^{-1}+{\Psi}\right)-{\Lambda}^{-1}{H}{\Lambda}^{-1}$
$\displaystyle=$
$\displaystyle{\Psi}{H}{\Lambda}^{-1}+{\Lambda}^{-1}{H}{\Psi}+{\Psi}{H}{\Psi},$

with

${\Psi}{H}{\Lambda}^{-1}=\left(\begin{array}[]{ l l
}{{\Psi}_{11}}&{{\Psi}_{12}}\\\
{{\Psi}_{21}}&{{\Psi}_{22}}\end{array}\right)\left(\begin{array}[]{ l l
}{{H}_{11}}&{{H}_{12}}\\\
{{H}_{21}}&{{H}_{22}}\end{array}\right)\left(\begin{array}[]{ c c
}{\Lambda_{11}^{-1}}&{0}\\\
{0}&{\Lambda_{22}^{-1}}\end{array}\right)=\left(\begin{array}[]{ c c c
}{{{\Psi}}_{11}{{H}}_{11}\Lambda_{11}^{-1}}&{+{{\Psi}}_{12}{{H}}_{21}\Lambda_{11}^{-1}}&{*}\\\
{*}&{}\hfil&{*}\end{array}\right),$

where $\Psi_{11},\Psi_{12},\Psi_{21}$, and $\Psi_{22}$ are $2\times 2,2\times
2p_{x},2p_{x}\times 2$, and $2p_{x}\times 2p_{x}$ submatrices of $\Psi$,
respectively, and the $*$ elements do not matter in the proof. Therefore, by
Lemma A10,

$\displaystyle\left[{\Psi}{H}{\Lambda}^{-1}\right]_{(1-2,1-2)}$
$\displaystyle=$
$\displaystyle{\Psi}_{11}{H}_{11}\Lambda_{11}^{-1}+{\Psi}_{12}{H}_{21}\Lambda_{11}^{-1}$
$\displaystyle=$ $\displaystyle
O_{\mathbb{P}}(\Omega^{1/2})O_{\mathbb{P}}\left(M\Omega\right)O_{\mathbb{P}}(1)+O_{\mathbb{P}}(\Omega^{1/2})O_{\mathbb{P}}\left(M\Omega\right)O_{\mathbb{P}}(1)=o_{\mathbb{P}}(1),$
$\displaystyle{\Psi}{H}{\Psi}$ $\displaystyle=$ $\displaystyle
O_{\mathbb{P}}(\Omega^{1/2})O_{\mathbb{P}}\left(M\Omega\right)O_{\mathbb{P}}(\Omega^{1/2})=O_{\mathbb{P}}(M\Omega^{2})=o_{\mathbb{P}}(1),$

which imply (A16). ∎

##  Regression estimators based on scaled cluster totals

For Theorems 3 and 4, because in $\hat{\tau}_{\textsc{t}}$ and
$\hat{\tau}_{\textsc{t}}^{\textup{adj}}$, we view
$\tilde{Y}_{i\cdot}=\sum_{j=1}^{n_{i}}Y_{ij}M/N$ as a new outcome, we only
need to check the regularity conditions on
$\tilde{Y}_{i\cdot}(z)=\sum_{j=1}^{n_{i}}Y_{ij}(z)M/N$ given the results in
completely randomized experiments.

###### Proof of Theorem 3.

The assumption $\Omega=o(1)$ implies

$\frac{1}{M}\sum_{i=1}^{M}\left\\{{\sum_{j=1}^{n_{i}}Y_{ij}(z)M/N}\right\\}^{2}\leq_{\textsc{hi}}\frac{M}{N^{2}}\sum_{i=1}^{M}n_{i}\sum_{j=1}^{n_{i}}Y_{ij}(z)^{2}=O\left(M\Omega\right)=o(M).$

The assumption $M^{2/3}\Omega=o(1)$ implies

$\frac{1}{M}\sum_{i=1}^{M}\left\\{{\sum_{j=1}^{n_{i}}Y_{ij}(z)}M/N\right\\}^{4}\leq_{\textsc{hi}}\frac{M^{3}}{N^{4}}\sum_{i=1}^{M}n_{i}^{3}\sum_{j=1}^{n_{i}}Y_{ij}(z)^{4}=O\left(M^{3}\Omega^{3}\right)=o(M).$

Lemma A2 guarantees the asymptotic property of $\hat{\tau}_{\textsc{t}}$. ∎

###### Proof of Theorem 4.

The assumption $\Omega=o(1)$ implies

$\frac{1}{M}\sum_{i=1}^{M}\left\\{{\sum_{j=1}^{n_{i}}Y_{ij}(z)}M/N\right\\}^{2}\leq_{\textsc{hi}}\frac{M}{N^{2}}\sum_{i=1}^{M}n_{i}\sum_{j=1}^{n_{i}}Y_{ij}(z)^{2}=O\left(M\Omega\right)=o(M).$

The assumption $M\Omega=O(1)$ implies

$\frac{1}{M}\sum_{i=1}^{M}\left\\{{\sum_{j=1}^{n_{i}}Y_{ij}(z)}M/N\right\\}^{4}\leq_{\textsc{hi}}\frac{M^{3}}{N^{4}}\sum_{i=1}^{M}n_{i}^{3}\sum_{j=1}^{n_{i}}Y_{ij}(z)^{4}=O\left(M^{3}\Omega^{3}\right)=O(1).$

Lemma A4 guarantees the asymptotic property of
$\hat{\tau}_{\textsc{t}}^{\textup{adj}}$. ∎

###### Proof of Proposition 1.

By definition,

$\displaystyle E(\hat{\tau}_{\textsc{i}}-\tau)^{2}$ $\displaystyle=$
$\displaystyle
E\left\\{{\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)}/{n_{\mathcal{\scriptscriptstyle
T}}}-{\sum_{ij\in\mathcal{C}}\varepsilon_{ij}(0)}/{n_{\mathcal{\scriptscriptstyle
C}}}\right\\}^{2}=E\left[\left\\{n_{\mathcal{\scriptscriptstyle
C}}\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)-n_{\mathcal{\scriptscriptstyle
T}}\sum_{ij\in\mathcal{C}}\varepsilon_{ij}(0)\right\\}\big{/}(n_{\mathcal{\scriptscriptstyle
T}}n_{\mathcal{\scriptscriptstyle C}})\right]^{2}$ (A17) $\displaystyle\geq$
$\displaystyle\left(\frac{2}{N}\right)^{4}E\left\\{n_{\mathcal{\scriptscriptstyle
C}}\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)-n_{\mathcal{\scriptscriptstyle
T}}\sum_{ij\in\mathcal{C}}\varepsilon_{ij}(0)\right\\}^{2},$

where the last inequality follows from $4n_{\mathcal{\scriptscriptstyle
T}}n_{\mathcal{\scriptscriptstyle C}}\leq(n_{\mathcal{\scriptscriptstyle
T}}+n_{\mathcal{\scriptscriptstyle C}})^{2}=N^{2}$.

If $e=1/2$ and $\bar{Y}(z)=0$ for $z=0,1$, then $\tau=0$ and

$\displaystyle
E(\hat{\tau}_{\textsc{t}}-\tau)^{2}=E\left({2\sum_{ij\in\mathcal{T}}Y_{ij}}/{N}-{2\sum_{ij\in\mathcal{C}}Y_{ij}}/{N}\right)^{2}=\left(\frac{2}{N}\right)^{2}E\left\\{\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)+\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(0)\right\\}^{2}.$
(A18)

From (A17) and (A18), we only need to prove

$\displaystyle
E\left\\{\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)+\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(0)\right\\}^{2}\left(\frac{N}{2}\right)^{2}\leq
E\left\\{n_{\mathcal{\scriptscriptstyle
C}}\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)-n_{\mathcal{\scriptscriptstyle
T}}\sum_{ij\in\mathcal{C}}\varepsilon_{ij}(0)\right\\}^{2}.$ (A19)

Because
$\sum_{ij}\varepsilon_{ij}(z)=\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(z)+\sum_{ij\in\mathcal{C}}\varepsilon_{ij}(z)=0$,
we have

$\displaystyle\left\\{n_{\mathcal{\scriptscriptstyle
C}}\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)-n_{\mathcal{\scriptscriptstyle
T}}\sum_{ij\in\mathcal{C}}\varepsilon_{ij}(0)\right\\}^{2}+\left\\{n_{\mathcal{\scriptscriptstyle
T}}\sum_{ij\in\mathcal{C}}\varepsilon_{ij}(1)-n_{\mathcal{\scriptscriptstyle
C}}\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(0)\right\\}^{2}$ $\displaystyle=$
$\displaystyle\left\\{n_{\mathcal{\scriptscriptstyle
C}}\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)+n_{\mathcal{\scriptscriptstyle
T}}\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(0)\right\\}^{2}+\left\\{n_{\mathcal{\scriptscriptstyle
T}}\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)+n_{\mathcal{\scriptscriptstyle
C}}\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(0)\right\\}^{2}$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\left\\{n_{\mathcal{\scriptscriptstyle
C}}\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)+n_{\mathcal{\scriptscriptstyle
T}}\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(0)+n_{\mathcal{\scriptscriptstyle
T}}\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)+n_{\mathcal{\scriptscriptstyle
C}}\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(0)\right\\}^{2}$ $\displaystyle=$
$\displaystyle\frac{N^{2}}{2}\left\\{\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(1)+\sum_{ij\in\mathcal{T}}\varepsilon_{ij}(0)\right\\}^{2}.$

Moreover, by $e=1/2$, the probabilities of receiving treatment assignments
$(Z_{i})_{1\leq i\leq M}$ and $(1-Z_{i})_{1\leq i\leq M}$ are the same.
Therefore, (A19) holds. ∎

##  General weighted estimand and regression-based estimators

###  A4.1 Proof of Proposition 2

Proposition 2 helps to simplify the proofs of Theorems 5 and 6. We prove it
now although it appears later than Theorems 5 and 6 in the main paper. We only
prove the result for $\hat{\tau}_{\textsc{a}\omega}^{\textup{adj}}$ because
the proof for $\hat{\tau}_{\textsc{a}\omega}$ is similar, and the proofs for
$\hat{\tau}_{\omega\textsc{a}}$ and
$\hat{\tau}_{\omega\textsc{a}}^{\textup{adj}}$ follow by definition. Also, we
assume $c_{i}=x_{ij}=\bar{x}_{i\cdot}$ in the following proof, so we can avoid
the notation $\bar{c}_{\omega}$ because
$\bar{c}_{\omega}=\sum_{i=1}^{M}\omega_{i}c_{i}=N^{-1}\sum_{i=1}^{M}n_{i}\bar{x}_{i\cdot}=N^{-1}\sum_{ij}x_{ij}=0.$

We can verify that the coefficients and heteroskedasticity-robust standard
errors are identical from WLS and OLS with both outcomes and regressors
multiply by $\omega_{i}^{1/2}$. So our proofs are based on the latter although
the theorems in the main paper are stated in terms of the former.

######  Lemma A11.

If $x_{ij}=\bar{x}_{i\cdot}$, then
$\hat{Q}_{\textsc{a}\omega}(1)=\hat{Q}_{\textsc{i}}(1)$ based on the treated
data, where $\hat{Q}_{\textsc{a}\omega}(1)$ equals the the coefficient of
$\omega_{i}^{1/2}\bar{x}_{i\cdot}$ in the OLS fit of
$\omega_{i}^{1/2}\bar{Y}_{i\cdot}$ on
$(\omega_{i}^{1/2},\omega_{i}^{1/2}\bar{x}_{i\cdot})$, and
$\hat{Q}_{\textsc{i}}(1)$ equals the coefficient of $x_{ij}$ in the OLS fit of
$Y_{ij}$ on $(1,x_{ij})$. The same conclusion holds for the control group.

###### Proof of Lemma A11.

We use the Frisch–Waugh–Lovell Theorem [Angrist and Pischke, 2008] for the
regression using the treated data. The coefficient of $\omega_{i}^{1/2}$ in
the OLS fit of $\omega_{i}^{1/2}\bar{Y}_{i\cdot}$ on $\omega_{i}^{1/2}$ is
${\sum_{i=1}^{M}Z_{i}\bar{Y}_{i\cdot}\omega_{i}^{1/2}\omega_{i}^{1/2}}\big{/}{\sum_{i=1}^{M}Z_{i}\omega_{i}^{1/2}\omega_{i}^{1/2}}=n_{\mathcal{\scriptscriptstyle
T}}^{-1}{\sum_{ij\in\mathcal{T}}Y_{ij}}=\bar{Y}_{\mathcal{\scriptscriptstyle
T}}$ with residual
$\check{Y}_{i}=\omega_{i}^{1/2}(\bar{Y}_{i\cdot}-\bar{Y}_{\mathcal{\scriptscriptstyle
T}})$. Similarly, the residual from the OLS fit of
$\omega_{i}^{1/2}\bar{x}_{i\cdot}$ on $\omega_{i}^{1/2}$ is
$\check{x}_{i}=\omega_{i}^{1/2}(\bar{x}_{i\cdot}-\bar{x}_{\mathcal{\scriptscriptstyle
T}})$. So
$\hat{Q}_{\textsc{a}\omega}(1)=(\sum_{i=1}^{M}Z_{i}\check{x}_{i}\check{x}_{i}^{{\mathrm{\scriptscriptstyle
T}}})^{-1}\sum_{i=1}^{M}Z_{i}\check{x}_{i}\check{Y}_{i}$. Its numerator
reduces to

$\displaystyle\sum_{i=1}^{M}Z_{i}\check{x}_{i}\check{Y}_{i}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\omega_{i}Z_{i}(\bar{x}_{i\cdot}-\bar{x}_{\mathcal{\scriptscriptstyle
T}})(\bar{Y}_{i\cdot}-\bar{Y}_{\mathcal{\scriptscriptstyle
T}})=N^{-1}\sum_{ij\in\mathcal{T}}(\bar{x}_{i\cdot}-\bar{x}_{\mathcal{\scriptscriptstyle
T}})(Y_{ij}-\bar{Y}_{\mathcal{\scriptscriptstyle T}}).$

Its denominator has a similar formula. Therefore,

$\displaystyle\hat{Q}_{\textsc{a}\omega}(1)$ $\displaystyle=$
$\displaystyle\left\\{\sum_{i=1}^{M}\omega_{i}Z_{i}(\bar{x}_{i\cdot}-\bar{x}_{\mathcal{\scriptscriptstyle
T}})(\bar{x}_{i\cdot}-\bar{x}_{\mathcal{\scriptscriptstyle
T}})^{{\mathrm{\scriptscriptstyle
T}}}\right\\}^{-1}\sum_{i=1}^{M}\omega_{i}Z_{i}(\bar{x}_{i\cdot}-\bar{x}_{\mathcal{\scriptscriptstyle
T}})(\bar{Y}_{i\cdot}-\bar{Y}_{\mathcal{\scriptscriptstyle T}})$
$\displaystyle=$
$\displaystyle\left\\{\sum_{ij\in\mathcal{T}}(\bar{x}_{i\cdot}-\bar{x}_{\mathcal{\scriptscriptstyle
T}})(\bar{x}_{i\cdot}-\bar{x}_{\mathcal{\scriptscriptstyle
T}})^{{\mathrm{\scriptscriptstyle
T}}}\right\\}^{-1}\sum_{ij\in\mathcal{T}}(\bar{x}_{i\cdot}-\bar{x}_{\mathcal{\scriptscriptstyle
T}})(Y_{ij}-\bar{Y}_{\mathcal{\scriptscriptstyle
T}})=\hat{Q}_{\textsc{i}}(1).$

∎

Because of the equivalence in Lemma A11, we use $\hat{Q}(1)$ in the following
proof for both $\hat{Q}_{\textsc{a}\omega}(1)$ and $\hat{Q}_{\textsc{i}}(1)$
when $x_{ij}=\bar{x}_{i\cdot}$.

###### Proof of Proposition 2.

By the property of the OLS using the treated data, the coefficient of
$\omega_{i}^{1/2}$ in the OLS fit of $\omega_{i}^{1/2}\bar{Y}_{i\cdot}$ on
$(\omega_{i}^{1/2},\omega_{i}^{1/2}\bar{x}_{i\cdot})$ is the same as the
coefficient of $\omega_{i}^{1/2}$ in the OLS fit of
$\omega_{i}^{1/2}\\{\bar{Y}_{i\cdot}-\bar{x}_{i\cdot}^{{\mathrm{\scriptscriptstyle
T}}}\hat{Q}(1)\\}$ on $\omega_{i}^{1/2}$:

${\sum_{i=1}^{M}Z_{i}\\{\bar{Y}_{i\cdot}-\bar{x}_{i\cdot}^{{\mathrm{\scriptscriptstyle
T}}}\hat{Q}(1)\\}\omega_{i}^{1/2}\omega_{i}^{1/2}}\big{/}{\sum_{i=1}^{M}Z_{i}\omega_{i}^{1/2}\omega_{i}^{1/2}}=n_{\mathcal{\scriptscriptstyle
T}}^{-1}{\sum_{ij\in\mathcal{T}}\\{Y_{ij}-\bar{x}_{i\cdot}^{{\mathrm{\scriptscriptstyle
T}}}\hat{Q}(1)\\}}.$

We have a similar result for the control group. Because
$\hat{\tau}_{\textsc{a}\omega}^{\textup{adj}}$ is the difference between the
coefficients of $\omega_{i}^{1/2}$ in the OLS fit of
$\omega_{i}^{1/2}\bar{Y}_{i\cdot}$ on
$(\omega_{i}^{1/2},\omega_{i}^{1/2}\bar{x}_{i\cdot})$ in treatment and control
groups, we have
$\hat{\tau}_{\textsc{a}\omega}^{\textup{adj}}=\hat{\tau}_{\textsc{i}}^{\textup{adj}}(\bar{x}_{i\cdot})$
based on the formula of $\hat{\tau}_{\textsc{i}}^{\textup{adj}}$ in the proof
of Theorem 2.

The residual from the OLS fit of $\omega_{i}^{1/2}\bar{Y}_{i\cdot}$ on
$(\omega_{i}^{1/2},\omega_{i}^{1/2}\bar{x}_{i\cdot})$ in treatment group is

$\omega_{i}^{1/2}\left\\{\bar{Y}_{i\cdot}-\bar{Y}_{\mathcal{\scriptscriptstyle
T}}-(\bar{x}_{i\cdot}-\bar{x}_{\mathcal{\scriptscriptstyle
T}})^{{\mathrm{\scriptscriptstyle
T}}}\hat{Q}(1)\right\\}=\omega_{i}^{1/2}{\sum_{j=1}^{n_{i}}\left\\{Y_{ij}-\bar{Y}_{\mathcal{\scriptscriptstyle
T}}-(\bar{x}_{i\cdot}-\bar{x}_{\mathcal{\scriptscriptstyle
T}})^{{\mathrm{\scriptscriptstyle
T}}}\hat{Q}(1)\right\\}}/n_{i}=\omega_{i}^{1/2}{\sum_{j=1}^{n_{i}}\hat{r}_{ij}}/n_{i},$

where $\hat{r}_{ij}$ is defined in Section A2.3. We have a similar result for
the control group. Therefore, the residual from the OLS fit of
$\omega_{i}^{1/2}\bar{Y}_{i\cdot}$ on
$(\omega_{i}^{1/2},\omega_{i}^{1/2}Z_{i},\omega_{i}^{1/2}\bar{x}_{i\cdot},\omega_{i}^{1/2}Z_{i}\bar{x}_{i\cdot})$
is $\omega_{i}^{1/2}{\sum_{j=1}^{n_{i}}\hat{r}_{ij}}/n_{i}$.

Let $\tilde{X}_{\textsc{a}\omega}$ be an $M\times(2+2p_{x})$ matrix with row
$i$ equaling $\tilde{x}_{i}^{\mathrm{\scriptscriptstyle
T}}=[\omega_{i}^{1/2}Z_{i},\omega_{i}^{1/2}(1-Z_{i}),\omega_{i}^{1/2}\bar{x}_{i\cdot}^{{\mathrm{\scriptscriptstyle
T}}}Z_{i},\omega_{i}^{1/2}\bar{x}_{i\cdot}^{{\mathrm{\scriptscriptstyle
T}}}(1-Z_{i})]$. Recall the definitions of $G,H_{i}$ and $H$ in Section A2.3.
We can verify that $\tilde{X}_{\textsc{a}\omega}^{{\mathrm{\scriptscriptstyle
T}}}\tilde{X}_{\textsc{a}\omega}=G,$ and

$\displaystyle\tilde{x}_{i}\left(\omega_{i}^{1/2}\sum_{j=1}^{n_{i}}\hat{r}_{ij}/n_{i}\right)^{2}\tilde{x}_{i}^{\mathrm{\scriptscriptstyle
T}}$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{ c c
}\omega_{i}^{1/2}Z_{i}\\\ \omega_{i}^{1/2}(1-Z_{i})\\\
\omega_{i}^{1/2}\bar{x}_{i\cdot}Z_{i}\\\
\omega_{i}^{1/2}\bar{x}_{i\cdot}(1-Z_{i})\end{array}\right)\left(\omega_{i}^{1/2}\sum_{j=1}^{n_{i}}\hat{r}_{ij}/n_{i}\right)^{2}\left(\omega_{i}^{1/2}Z_{i},\omega_{i}^{1/2}(1-Z_{i}),\omega_{i}^{1/2}\bar{x}_{i\cdot}^{{\mathrm{\scriptscriptstyle
T}}}Z_{i},\omega_{i}^{1/2}\bar{x}_{i\cdot}^{{\mathrm{\scriptscriptstyle
T}}}(1-Z_{i})\right)=H_{i}/N^{2}.$

So Lemma A8 implies

$\displaystyle
M\times\hat{\textup{se}}_{\textsc{hw}}^{2}(\hat{\tau}_{\textsc{a}\omega}^{\textup{adj}})$
$\displaystyle=$ $\displaystyle
M\left[\left(\tilde{X}_{\textsc{a}\omega}^{{\mathrm{\scriptscriptstyle
T}}}\tilde{X}_{\textsc{a}\omega}\right)^{-1}\left\\{\sum_{i=1}^{M}\tilde{x}_{i}\left(\omega_{i}^{1/2}\sum_{j=1}^{n_{i}}\hat{r}_{ij}/n_{i}\right)^{2}\tilde{x}_{i}^{{\mathrm{\scriptscriptstyle
T}}}\right\\}\left(\tilde{X}_{\textsc{a}\omega}^{{\mathrm{\scriptscriptstyle
T}}}\tilde{X}_{\textsc{a}\omega}\right)^{-1}\right]_{(1,1)+(2,2)-2(1,2)}$
$\displaystyle=$
$\displaystyle\left[MG^{-1}\sum_{i=1}^{M}H_{i}G^{-1}/N^{2}\right]_{(1,1)+(2,2)-2(1,2)}$
$\displaystyle=$
$\displaystyle[G^{-1}HG^{-1}]_{(1,1)+(2,2)-2(1,2)}=M\times\hat{\textup{se}}_{\textsc{lz}}^{2}\\{\hat{\tau}_{\textsc{i}}^{\textup{adj}}(\bar{x}_{i\cdot})\\}.$

∎

###  A4.2 Proofs of Theorems 5 and 6

Proposition 2 shows that when the cluster size is $n_{i}$, OLS estimators
based on individual-level data are equivalent to WLS estimators based on
cluster averages with $\omega_{i}$ as the weight. We can use this result
reversely to prove Theorems 5 and 6. If we want to analyze WLS estimators
based on cluster averages with $\pi_{i}$ as the weight, we can build a finite
population with cluster sizes $m_{i}$ so that $\pi_{i}=m_{i}/N_{\pi}$ with
$N_{\pi}=\sum_{i=1}^{M}m_{i}$, and establish the properties of
$\hat{\tau}_{\textsc{a}\pi}$ and $\hat{\tau}_{\textsc{a}\pi}^{\textup{adj}}$
by applying Theorems 1, 2 and Proposition 2 on this population. We can find
those $m_{i}$’s when the $\pi_{i}$’s are rational numbers, that is, they equal
ratios of positive integers. Although the $m_{i}$’s do not exist when the
$\pi_{i}$’s are general real numbers, we can use rational numbers to
approximate real numbers arbitrarily well. So the idea of the proof also
applies to general real $\pi_{i}$’s. This is a pure technical issue, but we
include the proof for completeness. We only prove Theorem 6 for
$\hat{\tau}_{\textsc{a}\pi}^{\textup{adj}}$ below because the proof of Theorem
5 for $\hat{\tau}_{\textsc{a}\pi}$ is similar.

###### Proof of Theorem 6.

We first prove the case with rational $\pi_{i}$’s. Let
$\pi_{i}=m_{i}/N_{\pi}$. Construct a finite population with $M$ clusters and
$N_{\pi}=\sum_{i=1}^{M}m_{i}$ units, where cluster $i$ has $m_{i}$ units. Unit
$(i,j)$ has potential outcomes $\bar{Y}_{i\cdot}(z)$ and unit-level covariates
$x_{ij}=\bar{x}_{i\cdot}=c_{i}-\bar{c}_{\pi}$ ($i=1,\ldots,M;\
j=1,\ldots,m_{i}$). Proposition 2 ensures that
$\hat{\tau}_{\textsc{a}\pi}^{\textup{adj}}=\hat{\tau}_{\textsc{i}}^{\textup{adj}}(c_{i})$.
Therefore, we only need to check the regularity conditions in Theorem 2 to
prove Theorem 6.

The regularity conditions in Theorem 6 and Assumptions 5 and 6 imply the
conditions for the consistency and asymptotic Normality of
$\hat{\tau}_{\textsc{i}}^{\textup{adj}}(c_{i})$ in Theorem 2: Assumption 1
holds automatically; Assumption 2 holds because

${N_{\pi}^{-1}\sum_{i=1}^{M}m_{i}\bar{Y}_{i\cdot}(z)^{4}}=\sum_{i=1}^{M}\pi_{i}\bar{Y}_{i\cdot}(z)^{4}=O(1),\quad{N_{\pi}^{-1}\sum_{i=1}^{M}m_{i}||\bar{x}_{i\cdot}||_{\infty}^{4}}=\sum_{i=1}^{M}\pi_{i}||c_{i}-\bar{c}_{\pi}||_{\infty}^{4}=O(1);$

Assumption 3 holds because
${N_{\pi}^{-1}\sum_{i=1}^{M}\sum_{j=1}^{m_{i}}(c_{i}-\bar{c}_{\pi})\bar{Y}_{i\cdot}(z)}=\sum_{i=1}^{M}\pi_{i}(c_{i}-\bar{c}_{\pi})\bar{Y}_{i\cdot}(z)$
converges to a finite vector, and
${N_{\pi}^{-1}\sum_{i=1}^{M}\sum_{j=1}^{m_{i}}(c_{i}-\bar{c}_{\pi})(c_{i}-\bar{c}_{\pi})^{\mathrm{\scriptscriptstyle
T}}}=\sum_{i=1}^{M}\pi_{i}(c_{i}-\bar{c}_{\pi})(c_{i}-\bar{c}_{\pi})^{\mathrm{\scriptscriptstyle
T}}$ converges to a finite and invertible matrix. We can also verify the
conditions on the maximum of the proportions of clusters by noticing that
$\max_{1\leq i\leq M}m_{i}/{N_{\pi}}=\max_{1\leq i\leq M}\pi_{i}$.

The asymptotic variance and estimated variance are $V\\{r_{i}(z)\\}$ and
${V}_{\textup{c}}\\{r_{i}(z)\\}$, respectively, where

$r_{i}(z)=\sum_{j=1}^{m_{i}}\\{\bar{Y}_{i\cdot}(z)-\bar{Y}_{\pi}(z)-(c_{i}-\bar{c}_{\pi})^{\mathrm{\scriptscriptstyle
T}}Q_{\textsc{a}\pi}(z)\\}M/N_{\pi}=M\pi_{i}\\{\bar{Y}_{i\cdot}(z)-\bar{Y}_{\pi}(z)-(c_{i}-\bar{c}_{\pi})^{\mathrm{\scriptscriptstyle
T}}Q_{\textsc{a}\pi}(z)\\}.$

We then prove the case with general real $\pi_{i}$’s. Assume there is a series
of $\epsilon_{M}$ which converges to 0 when $M\rightarrow+\infty$. Because the
population and the number of treatment assignments are finite, the formulas
and probability limits of OLS coefficients, variance and estimated variance
are continuous functions of $\pi_{i}$, we can find rational numbers
$\ddot{\pi}_{i}$ close to $\pi_{i}$ so that for all treatment assignments,
$\sum_{i=1}^{M}\ddot{\pi}_{i}\bar{Y}_{i\cdot}(z)^{4}=O(1),\
\sum_{i=1}^{M}\ddot{\pi}_{i}||c_{i}||_{\infty}^{4}=O(1),\ M^{2/3}\max_{1\leq
i\leq M}|\ddot{\pi}_{i}-\pi_{i}|=o(1)$,
$\sum_{i=1}^{M}\ddot{\pi}_{i}(c_{i}-\bar{c}_{\ddot{\pi}})\bar{Y}_{i\cdot}(z)$
converges to a finite vector, and
$\sum_{i=1}^{M}\ddot{\pi}_{i}(c_{i}-\bar{c}_{\ddot{\pi}})(c_{i}-\bar{c}_{\ddot{\pi}})^{{\mathrm{\scriptscriptstyle
T}}}$ converges to a finite and invertible matrix,
$M^{1/2}|\hat{\tau}_{\textsc{a}\ddot{\pi}}^{\textup{adj}}-\hat{\tau}_{\textsc{a}\pi}^{\textup{adj}}|<\epsilon_{M}$,
$M^{1/2}|\tau_{\ddot{\pi}}-\tau_{\pi}|<\epsilon_{M}$,
$M^{1/2}|\textup{se}(\hat{\tau}_{\textsc{a}\ddot{\pi}}^{\textup{adj}})-\textup{se}(\hat{\tau}_{\textsc{a}\pi}^{\textup{adj}})|<\epsilon_{M}$,
$M|\hat{\textup{se}}_{\textsc{hw}}^{2}(\hat{\tau}_{\textsc{a}\ddot{\pi}}^{\textup{adj}})-\hat{\textup{se}}_{\textsc{hw}}^{2}(\hat{\tau}_{\textsc{a}\pi}^{\textup{adj}})|<\epsilon_{M}$,
$|V_{\textup{c}}\\{\ddot{r}_{i}(z)\\}-V_{\textup{c}}\\{r_{i}(z)\\}|<\epsilon_{M}$,
where those notations with double dots are corresponding values derived from
the OLS fit of $\ddot{\pi}_{i}^{1/2}\bar{Y}_{i\cdot}$ on
$\\{\ddot{\pi}_{i}^{1/2},\ddot{\pi}_{i}^{1/2}Z_{i},\ddot{\pi}_{i}^{1/2}(c_{i}-\bar{c}_{\ddot{\pi}}),\ddot{\pi}_{i}^{1/2}Z_{i}(c_{i}-\bar{c}_{\ddot{\pi}})\\}$.
By the results for rational $\ddot{\pi}_{i}$’s and Slutsky’s Theorem, Theorem
6 holds. ∎

###  A4.3 Proofs of Theorems 7 and 8

Finally, we prove Theorems 7 and 8. Because $\hat{\tau}_{\pi\textsc{a}}$ and
$\hat{\tau}_{\pi\textsc{a}}^{\textup{adj}}$ treat $M\pi_{i}\bar{Y}_{i\cdot}$
as a new outcome, we only need to check the regularity conditions on
$M\pi_{i}\bar{Y}_{i\cdot}(z)$ given the results in completely randomized
experiments.

###### Proof of Theorem 7.

The assumption $\sum_{i=1}^{M}\pi_{i}^{2}\bar{Y}_{i\cdot}(z)^{2}=o(1)$ implies
$M^{-1}\sum_{i=1}^{M}\left\\{M\pi_{i}\bar{Y}_{i\cdot}(z)\right\\}^{2}=M\sum_{i=1}^{M}\pi_{i}^{2}\bar{Y}_{i\cdot}(z)^{2}=o(M).$
The assumption $\sum_{i=1}^{M}M^{2}\pi_{i}^{4}\bar{Y}_{i\cdot}(z)^{4}=o(1)$
implies
$M^{-1}\sum_{i=1}^{M}\left\\{M\pi_{i}\bar{Y}_{i\cdot}(z)\right\\}^{4}=M^{3}\sum_{i=1}^{M}\pi_{i}^{4}\bar{Y}_{i\cdot}(z)^{4}=o(M).$
Lemma A2 guarantees the asymptotic properties of $\hat{\tau}_{\pi\textsc{a}}$.
∎

###### Proof of Theorem 8.

The assumption $\sum_{i=1}^{M}\pi_{i}^{2}\bar{Y}_{i\cdot}(z)^{2}=o(1)$ implies
$M^{-1}\sum_{i=1}^{M}\left\\{M\pi_{i}\bar{Y}_{i\cdot}(z)\right\\}^{2}=M\sum_{i=1}^{M}\pi_{i}^{2}\bar{Y}_{i\cdot}(z)^{2}=o(M).$
The assumption $\sum_{i=1}^{M}M^{3}\pi_{i}^{4}\bar{Y}_{i\cdot}(z)^{4}=O(1)$
implies
$M^{-1}\sum_{i=1}^{M}\left\\{M\pi_{i}\bar{Y}_{i\cdot}(z)\right\\}^{4}=M^{3}\sum_{i=1}^{M}\pi_{i}^{4}\bar{Y}_{i\cdot}(z)^{4}=O(1).$
Lemma A4 guarantees the asymptotic properties of
$\hat{\tau}_{\pi\textsc{a}}^{\textup{adj}}$. ∎

##  More detailed results

Tables A1 and A2 give more details for Section 6. Table A3 gives more details
for Section 7.

Table A1: Simulation: Part I

(a) Simulation in Section 6.1 | $\hat{\tau}_{\textsc{i}}$ | $\hat{\tau}_{\textsc{i}}^{\textup{adj}}$ | $\hat{\tau}_{\textsc{i}}^{\textup{adj}}(\bar{x}_{i\cdot})$ | $\hat{\tau}_{\textsc{i}}^{\textup{ancova}}$ |  |  |
---|---|---|---|---|---|---|---
bias ($\times$100) | $0.21$ | $0.02$ | $-0.11$ | $0.18$ |  |  |
se ($\times$100) | $7.89$ | $6.39$ | $6.32$ | $6.79$ |  |  |
$\hat{\textup{se}}_{\textsc{ols}}$ ($\times$100) | $4.66$ | $4.48$ | $4.56$ | $4.55$ |  |  |
$\hat{\textup{se}}_{\textsc{hw}}$ ($\times$100) | $4.86$ | $4.48$ | $4.75$ | $4.60$ |  |  |
$\hat{\textup{se}}_{\textsc{lz}}$ ($\times$100) | $8.75$ | $7.59$ | $7.39$ | $7.80$ |  |  |
rmse ($\times$100) | $7.88$ | $6.38$ | $6.32$ | $6.79$ |  |  |
coverage rate${}_{\textsc{ols}}$ | $0.749$ | $0.837$ | $0.841$ | $0.815$ |  |  |
coverage rate${}_{\textsc{hw}}$ | $0.774$ | $0.835$ | $0.855$ | $0.817$ |  |  |
coverage rate${}_{\textsc{lz}}$ | $0.968$ | $0.978$ | $0.966$ | $0.968$ |  |  |
| $\hat{\tau}_{\textsc{t}}$ | $\hat{\tau}_{\textsc{t}}^{\textup{adj}}(n_{i})$ | $\hat{\tau}_{\textsc{t}}^{\textup{adj}}(n_{i},\tilde{x}_{i\cdot})$ | $\hat{\tau}_{\textsc{a}}$ | $\hat{\tau}_{\textsc{a}}^{\textup{adj}}(\bar{x}_{i\cdot})$ | $\hat{\tau}_{\textsc{a}\omega}$ | $\hat{\tau}_{\textsc{a}\omega}^{\textup{adj}}(\bar{x}_{i\cdot})$
bias ($\times$100) | $0.46$ | $0.18$ | $-0.11$ | $-8.74$ | $-8.96$ | $0.21$ | $-0.11$
se ($\times$100) | $12.80$ | $6.42$ | $3.48$ | $8.14$ | $6.58$ | $7.89$ | $6.32$
$\hat{\textup{se}}_{\textsc{ols}}$ ($\times$100) | $10.70$ | $6.87$ | $4.85$ | $7.95$ | $6.39$ | $7.72$ | $6.23$
$\hat{\textup{se}}_{\textsc{hw}}$ ($\times$100) | $13.87$ | $6.91$ | $4.31$ | $9.07$ | $7.68$ | $8.75$ | $7.39$
rmse ($\times$100) | $12.80$ | $6.42$ | $3.48$ | $11.94$ | $11.12$ | $7.88$ | $6.32$
coverage rate${}_{\textsc{ols}}$ | $0.892$ | $0.963$ | $0.994$ | $0.811$ | $0.711$ | $0.935$ | $0.936$
coverage rate${}_{\textsc{hw}}$ | $0.957$ | $0.961$ | $0.987$ | $0.869$ | $0.824$ | $0.968$ | $0.966$

(b) Simulation in Section 6.2 | $\hat{\tau}_{\textsc{i}}$ | $\hat{\tau}_{{\textsc{i}}}^{{\textup{adj}}}$ | $\hat{\tau}_{{\textsc{i}}}^{{\textup{adj}}}(\bar{x}_{i\cdot})$ | $\hat{\tau}_{{\textsc{i}}}^{\textup{ancova}}$ | $\hat{\tau}_{{\textsc{t}}}$ | $\hat{\tau}_{{\textsc{t}}}^{\textup{adj}}(n_{i})$ | $\hat{\tau}_{{\textsc{t}}}^{\textup{adj}}(n_{i},\tilde{x}_{i\cdot})$ | $\hat{\tau}_{\textsc{a}}$ | $\hat{\tau}_{\textsc{a}}^{{\textup{adj}}}(\bar{x}_{i\cdot})$
---|---|---|---|---|---|---|---|---|---
bias ($\times$100) | $-0.11$ | $-0.11$ | $-0.15$ | $-0.11$ | $0.15$ | $-0.17$ | $-0.21$ | $-10.17$ | $-10.04$
se ($\times$100) | $6.09$ | $6.09$ | $6.05$ | $6.08$ | $11.58$ | $3.24$ | $3.28$ | $6.32$ | $6.28$
$\hat{\textup{se}}$ ($\times$100) | $7.68$ | $7.67$ | $7.61$ | $7.67$ | $13.48$ | $4.63$ | $4.62$ | $7.97$ | $7.89$
rmse ($\times$100) | $6.09$ | $6.08$ | $6.04$ | $6.08$ | $11.58$ | $3.24$ | $3.28$ | $11.97$ | $11.84$
coverage rate | $0.982$ | $0.982$ | $0.979$ | $0.982$ | $0.973$ | $0.993$ | $0.99$ | $0.808$ | $0.813$

Table A2: Simulation: Part II

(c) Simulation in Section 6.3 | $\hat{\tau}_{\textsc{i}}$ | $\hat{\tau}_{\textsc{i}}^{\textup{adj}}$ | $\hat{\tau}_{\textsc{i}}^{\textup{ancova}}$ | $\hat{\tau}_{\textsc{t}}$ | $\hat{\tau}_{\textsc{t}}^{\textup{adj}}(n_{i})$ | $\hat{\tau}_{\textsc{t}}^{\textup{adj}}(n_{i},\tilde{x}_{i\cdot})$ | $\hat{\tau}_{\textsc{a}}$ | $\hat{\tau}_{\textsc{a}}^{\textup{adj}}(\bar{x}_{i\cdot})$
---|---|---|---|---|---|---|---|---
bias ($\times$10) | $0.06$ | $0.06$ | $0.05$ | $0.01$ | $-0.05$ | $0.01$ | $7.51$ | $7.44$
se ($\times$10) | $1.56$ | $1.69$ | $1.68$ | $1.55$ | $0.65$ | $0.36$ | $2.50$ | $1.82$
$\hat{\textup{se}}$ ($\times$10) | $2.18$ | $2.31$ | $2.25$ | $2.17$ | $0.87$ | $0.50$ | $3.49$ | $2.52$
rmse ($\times$10) | $1.56$ | $1.69$ | $1.68$ | $1.55$ | $0.65$ | $0.36$ | $7.92$ | $7.66$
coverage rate | $0.989$ | $0.991$ | $0.988$ | $0.990$ | $0.992$ | $0.991$ | $0.400$ | $0.099$

(d) Simulation in Section 6.4 | $\hat{\tau}_{\textsc{i}}$ | $\hat{\tau}_{\textsc{i}}^{\textup{adj}}$ | $\hat{\tau}_{\textsc{i}}^{\textup{ancova}}$ | $\hat{\tau}_{\textsc{t}}$ | $\hat{\tau}_{\textsc{t}}^{\textup{adj}}(n_{i})$ | $\hat{\tau}_{\textsc{t}}^{\textup{adj}}(n_{i},\tilde{x}_{i\cdot})$ | $\hat{\tau}_{\textsc{t}}^{\textup{adj}}(\bar{x}_{i\cdot})$ | $\hat{\tau}_{\textsc{a}}$ | $\hat{\tau}_{\textsc{a}}^{\textup{adj}}(\bar{x}_{i\cdot})$
---|---|---|---|---|---|---|---|---|---
bias ($\times$10) | $0.04$ | $0.00$ | $-0.03$ | $0.01$ | $-0.31$ | $-0.31$ | $-0.05$ | $6.53$ | $6.46$
se ($\times$10) | $1.93$ | $1.88$ | $1.88$ | $1.92$ | $1.90$ | $1.61$ | $1.69$ | $2.50$ | $1.82$
$\hat{\textup{se}}$ ($\times$10) | $2.66$ | $2.55$ | $2.59$ | $2.65$ | $1.91$ | $1.39$ | $2.30$ | $3.49$ | $2.52$
rmse ($\times$10) | $1.93$ | $1.87$ | $1.88$ | $1.92$ | $1.92$ | $1.64$ | $1.69$ | $6.99$ | $6.71$
coverage rate | $0.971$ | $0.985$ | $0.983$ | $0.977$ | $0.668$ | $0.505$ | $0.986$ | $0.549$ | $0.223$

(e) Simulation in Section 6.5 | $\hat{\tau}_{\textsc{i}}$ | $\hat{\tau}_{\textsc{i}}^{\textup{adj}}$ | $\hat{\tau}_{\textsc{i}}^{\textup{ancova}}$ | $\hat{\tau}_{\textsc{t}}$ | $\hat{\tau}_{\textsc{t}}^{\textup{adj}}(n_{i})$ | $\hat{\tau}_{\textsc{t}}^{\textup{adj}}(n_{i},\tilde{x}_{i\cdot})$ | $\hat{\tau}_{\textsc{a}}$ | $\hat{\tau}_{\textsc{a}}^{\textup{adj}}(\bar{x}_{i\cdot})$
---|---|---|---|---|---|---|---|---
bias ($\times$10) | $-0.30$ | $-0.26$ | $-0.06$ | $-0.01$ | $-0.59$ | $-0.40$ | $0.01$ | $-0.01$
se ($\times$10) | $3.41$ | $1.64$ | $1.28$ | $7.13$ | $2.97$ | $2.03$ | $0.70$ | $0.60$
$\hat{\textup{se}}$ ($\times$10) | $1.96$ | $0.70$ | $0.69$ | $6.93$ | $0.92$ | $0.90$ | $0.70$ | $0.60$
rmse ($\times$10) | $3.42$ | $1.66$ | $1.28$ | $7.13$ | $3.03$ | $2.07$ | $0.70$ | $0.60$
coverage rate | $0.720$ | $0.383$ | $0.628$ | $1.000$ | $0.186$ | $0.459$ | $0.948$ | $0.951$

Table A3: Data analysis and additional simulation in Section 7

(a) Point estimate, standard error and confidence interval from Guiteras et al. [2015] | $\hat{\tau}_{\textsc{i}}$ | $\hat{\tau}_{{\textsc{i}}}^{{\textup{adj}}}$ | $\hat{\tau}_{{\textsc{i}}}^{{\textup{adj}}}(\bar{x}_{i\cdot})$ | $\hat{\tau}_{{\textsc{i}}}^{\textup{ancova}}$ | $\hat{\tau}_{{\textsc{t}}}$ | $\hat{\tau}_{{\textsc{t}}}^{\textup{adj}}(n_{i})$ | $\hat{\tau}_{{\textsc{t}}}^{\textup{adj}}(n_{i},\tilde{x}_{i\cdot})$ | $\hat{\tau}_{\textsc{a}}$ | $\hat{\tau}_{\textsc{a}}^{{\textup{adj}}}(\bar{x}_{i\cdot})$
---|---|---|---|---|---|---|---|---|---
est | $0.178$ | $0.159$ | $0.157$ | $0.160$ | $0.231$ | $0.180$ | $0.152$ | $0.182$ | $0.177$
se | $0.044$ | $0.025$ | $0.023$ | $0.025$ | $0.085$ | $0.044$ | $0.021$ | $0.038$ | $0.031$
lower CI | $0.092$ | $0.110$ | $0.111$ | $0.110$ | $0.064$ | $0.094$ | $0.111$ | $0.108$ | $0.117$
upper CI | $0.263$ | $0.208$ | $0.203$ | $0.210$ | $0.399$ | $0.265$ | $0.192$ | $0.256$ | $0.237$

(b) Simulation in Section 7.2 | $\hat{\tau}_{\textsc{i}}$ | $\hat{\tau}_{{\textsc{i}}}^{{\textup{adj}}}$ | $\hat{\tau}_{{\textsc{i}}}^{{\textup{adj}}}(\bar{x}_{i\cdot})$ | $\hat{\tau}_{{\textsc{i}}}^{\textup{ancova}}$ | $\hat{\tau}_{{\textsc{t}}}$ | $\hat{\tau}_{{\textsc{t}}}^{\textup{adj}}(n_{i})$ | $\hat{\tau}_{{\textsc{t}}}^{\textup{adj}}(n_{i},\tilde{x}_{i\cdot})$ | $\hat{\tau}_{\textsc{a}}$ | $\hat{\tau}_{\textsc{a}}^{{\textup{adj}}}(\bar{x}_{i\cdot})$
---|---|---|---|---|---|---|---|---|---
bias ($\times$100) | $0.05$ | $0.00$ | $-0.01$ | $0.00$ | $-0.03$ | $0.21$ | $-0.07$ | $-0.90$ | $-1.01$
se ($\times$100) | $3.04$ | $1.26$ | $1.26$ | $1.26$ | $8.61$ | $3.20$ | $1.41$ | $2.96$ | $1.83$
$\hat{\textup{se}}$ ($\times$100) | $3.24$ | $1.98$ | $1.95$ | $2.00$ | $8.57$ | $3.19$ | $1.89$ | $3.25$ | $2.22$
rmse ($\times$100) | $3.04$ | $1.26$ | $1.26$ | $1.26$ | $8.61$ | $3.21$ | $1.41$ | $3.09$ | $2.09$
coverage rate | $0.951$ | $0.996$ | $0.994$ | $0.995$ | $0.950$ | $0.931$ | $0.986$ | $0.961$ | $0.952$