# Weak Instruments, First-Stage Heteroskedasticity, the Robust F-Test and a
GMM Estimator with the Weight Matrix Based on First-Stage Residuals

Frank Windmeijer
Department of Statistics and Nuffield College
University of Oxford, UK Email<EMAIL_ADDRESS>
(3 August, 2022)

###### Abstract

This paper is concerned with the findings related to the robust first-stage
F-statistic in the Monte Carlo analysis of Andrews (2018), who found in a
heteroskedastic grouped-data design that even for very large values of the
robust F-statistic, the standard 2SLS confidence intervals had large coverage
distortions. This finding appears to discredit the robust F-statistic as a
test for underidentification. However, it is shown here that large values of
the robust F-statistic do imply that there is first-stage information, but
this may not be utilized well by the 2SLS estimator, or the standard GMM
estimator. An estimator that corrects for this is a robust GMM estimator,
denoted GMMf, with the robust weight matrix not based on the structural
residuals, but on the first-stage residuals. For the grouped-data setting of
Andrews (2018), this GMMf estimator gives the weights to the group specific
estimators according to the group specific concentration parameters in the
same way as 2SLS does under homoskedasticity, which is formally shown using
weak instrument asymptotics. The GMMf estimator is much better behaved than
the 2SLS estimator in the Andrews (2018) design, behaving well in terms of
relative bias and Wald-test size distortion at more standard values of the
robust F-statistic. We show that the same patterns can occur in a dynamic
panel data model when the error variance is heteroskedastic over time. We
further derive the conditions under which the Stock and Yogo (2005) weak
instruments critical values apply to the robust F-statistic in relation to the
behaviour of the GMMf estimator.

JEL Classification: C12, C36

Keywords: Instrumental Variables, Weak Instruments, Heteroskedasticity,
F-Test, GMM, Grouped-Data IV, Dynamic Panel Data

##  1 Introduction

It is commonplace to report the first-stage F-statistic as a test for
underidentification in linear single endogenous variable models estimated by
two-stage least squares (2SLS). This could either be a non-robust or robust
version of the test, with robustness to for example heteroskedasticity, serial
correlation and/or clustering. Under maintained assumptions, these are valid
tests for the null $H_{0}:\pi=0$ in the first-stage linear specification
$x=Z\pi+v$, where $x$ is the endogenous explanatory variable in the model of
interest $y=x\beta+u$, and $Z$ are the instruments. If the null is not
rejected, then this is an indication that the relevance condition of the
instruments does not hold and that the 2SLS estimator does not provide a
meaningful estimate of the parameter of interest $\beta$. A rejection of the
null does, however, not necessarily imply that the 2SLS estimator is well
behaved. This follows the work of Staiger and Stock (1997) and Stock and Yogo
(2005), with the latter providing critical values for the first-stage non-
robust F-statistic for null hypotheses of weak instruments in terms of bias of
the 2SLS estimator relative to that of the OLS estimator and Wald test size
distortion. These non-robust weak instruments F-tests are valid only under
conditional homoskedasticity, no serial correlation and no clustering of both
the first-stage errors $v$ and the structural errors $u$, and do not apply to
the robust F-test in general designs, see Bun and de Haan (2010), Montiel Olea
and Pflueger (2013) and Andrews (2018). For general designs Montiel Olea and
Pflueger (2013) proposed the effective first-stage F-statistic and critical
values linked to the Nagar bias of the 2SLS estimator, whereas Andrews (2018)
obtained valid two-step identification robust confidence sets.

This paper is concerned with the findings related to the robust F-statistic in
the Monte Carlo analysis of Andrews (2018, Supplementary Appendix (SA)). In a
cross sectional heteroskedastic design he found that even for very large
values of the robust F-statistic, the standard 2SLS confidence intervals had
large coverage distortions. For example, for a high endogeneity design, “the
2SLS confidence set has a 15% coverage distortion even when the mean of the
first-stage robust F-statistic is 100,000”, Andrews (2018, SA, p 11). This is
a striking finding and appears to discredit the robust F-statistic as a test
for underidentification. However, it is shown here that large values of the
robust F-statistic do imply that there is first-stage information, but this
may not be utilized well by the 2SLS estimator, or GMM estimators that
incorporate heteroskedasticity in the structural error $u$ only.

The Andrews (2018) design is the same as a grouped data one, see Angrist
(1991) and the discussion in Angrist and Pischke (2009), where the instruments
are mutually exclusive group membership indicators. Denoting the groups by
$s=1,...,S$, the group specific concentration parameter values are determined
by the ratios $\pi_{s}^{2}/\sigma_{v,s}^{2}$, where $\sigma_{v,s}^{2}$ is the
group specific variance of the first-stage error $v$. The 2SLS estimator is a
weighted average of the group specific estimators of $\beta$, giving more
weight to large concentration parameter groups if $v$ is homoskedastic.
However, as shown in Section 2, this may not happen under heteroskedasticity,
where 2SLS gives more weight to high variance $\sigma_{v,s}^{2}$ groups,
everything else constant. In the design of Andrews (2018) we consider here,
there is one informative group, leading to the large value of the robust
F-statistic, but this group has a small variance $\sigma_{v,s}^{2}$, and
therefore gets only a relatively small weight in the 2SLS estimator.

An estimator that correctly gives larger weights to more informative groups is
a robust GMM estimator, not using the structural residuals $\widehat{u}$, but
the first-stage residuals $\widehat{v}$ in the robust weight matrix. This
estimator, called GMMf, is introduced in Section 3 and gives the weights to
the group-specific estimators according to the group-specific concentration
parameters in the same way as 2SLS does under homoskedasticity. This is
further formally shown using weak instrument asymptotics in Section 4. Section
5 discusses the potential problems of the standard GMM estimator that uses a
robust weight matrix based on the conditional variances of the structural
errors $u$. Monte Carlo results in Section 6 show that the GMMf estimator
exploits the available information well, with much better relative bias and
Wald test size properties than the 2SLS estimator for values of the robust
F-statistic in line with those of the non-robust F-statistic and behaviour of
the 2SLS estimator in the homoskedastic case.

Section 7 shows that similar patterns can occur when considering a simple
AR(1) dynamic panel data model with heteroskedasticity of the idiosyncratic
shocks in the time dimension, with estimation based on the forward orthogonal
deviations transformation.

For a general setting, we report in Section 8 the conditions under which the
Stock and Yogo (2005) critical values can be applied to the robust F-statistic
in relation to the behaviour of the GMMf estimator. These conditions are
derived in Appendix A.2. Whilst these have limited applicability, the fully
homoskedastic design is a special case.

##  2 Grouped-Data IV Model, First-Stage F-Statistic and 2SLS Weights

We consider the model as in Andrews (2018, SA C.3), which is the same as a
grouped-data IV setup,

$\displaystyle y_{i}$ $\displaystyle=$ $\displaystyle x_{i}\beta+u_{i}$
$\displaystyle x_{i}$ $\displaystyle=$ $\displaystyle
z_{i}^{\prime}\pi+v_{i},$

for $i=1,...,n$, where the $S$-vector
$z_{i}\in\left\\{e_{1},...,e_{S}\right\\}$, with $e_{s}$ an $S$-vector with
$s$th entry equal to $1$ and zeros everywhere else, for $s=1,\ldots,S$.
Assumptions for standard asymptotic normality results for the IV estimator
hold and the variance of the limiting distribution of the parameters can be
estimated consistently.

The variance-covariance structure for the errors is modeled fully flexibly by
group, and specified as

$\left(\left(\begin{array}[]{c}u_{i}\\\
v_{i}\end{array}\right)|z_{i}=e_{s}\right)\sim\left(0,\Sigma_{s}\right).$
$\Sigma_{s}=\left[\begin{array}[]{cc}\sigma_{u,s}^{2}&\sigma_{uv,s}\\\
\sigma_{uv,s}&\sigma_{v,s}^{2}\end{array}\right].$ (1)

At the group level, we therefore have for group member $j$ in group $s$

$\displaystyle y_{js}$ $\displaystyle=$ $\displaystyle x_{js}\beta+u_{js}$ (2)
$\displaystyle x_{js}$ $\displaystyle=$ $\displaystyle\pi_{s}+v_{js}$ (3)
$\left(\begin{array}[]{c}u_{js}\\\
v_{js}\end{array}\right)\sim\left(0,\Sigma_{s}\right)$

for $j=1,...,n_{s}$ and $s=1,...,S$, with $n_{s}$ the number of observations
in group $s$, $\sum_{s=1}^{S}n_{s}=n$, see also Bekker and Ploeg (2005). We
assume that $\lim_{n\rightarrow\infty}\frac{n_{s}}{n}=f_{s}$, with
$0<f_{s}<1$.

The OLS estimator of $\pi_{s}$ is given by
$\widehat{\pi}_{s}=\overline{x}_{s}=\frac{1}{n_{s}}\sum_{j=1}^{n_{s}}x_{js}$
and $Var\left(\widehat{\pi}_{s}\right)=\sigma_{v,s}^{2}/n_{s}$. The OLS
residual is $\widehat{v}_{js}=x_{js}-\overline{x}_{s}$ and the estimator for
the variance is given by
$V\widehat{a}r\left(\widehat{\pi}_{s}\right)=\widehat{\sigma}_{v,s}^{2}/n_{s}$,
where
$\widehat{\sigma}_{v,s}^{2}=\frac{1}{n_{s}}\sum_{j=1}^{n_{s}}\widehat{v}_{js}^{2}$.
Let $Z$ be the $n\times S$ matrix of instruments. For the vector $\pi$ the OLS
estimator is given by

$\widehat{\pi}=\left(Z^{\prime}Z\right)^{-1}Z^{\prime}x=\left(\overline{x}_{1},\overline{x}_{2},...,\overline{x}_{S}\right)^{\prime}.$

Let

$\displaystyle\widehat{\Omega}_{v}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\widehat{v}_{i}^{2}z_{i}z_{i}^{\prime}$
$\displaystyle=$
$\displaystyle\text{diag}\left(n_{s}\widehat{\sigma}_{v,s}^{2}\right),$

where $\text{diag}\left(q_{s}\right)$ is a diagonal matrix with $s$th diagonal
element $q_{s}$. Then the robust estimator of $Var\left(\widehat{\pi}\right)$
is given by

$\displaystyle V\widehat{a}r_{r}\left(\widehat{\pi}\right)$ $\displaystyle=$
$\displaystyle\left(Z^{\prime}Z\right)^{-1}\widehat{\Omega}_{v}\left(Z^{\prime}Z\right)^{-1}$
$\displaystyle=$
$\displaystyle\text{diag}\left(\widehat{\sigma}_{v,s}^{2}/n_{s}\right).$

The non-robust variance estimator is

$\displaystyle V\widehat{a}r\left(\widehat{\pi}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{n}\sum_{i=1}^{n}\widehat{v}_{i}^{2}\right)\left(Z^{\prime}Z\right)^{-1}$
$\displaystyle=$
$\displaystyle\left(\sum_{s=1}^{S}\frac{n_{s}}{n}\widehat{\sigma}_{v,s}^{2}\right)\text{diag}\left(\frac{1}{n_{s}}\right).$

The group (or instrument) specific IV estimators for $\beta$ are given by

$\widehat{\beta}_{s}=\frac{z_{s}^{\prime}y}{z_{s}^{\prime}x}=\frac{\overline{y}_{s}}{\overline{x}_{s}},$
(5)

with $\overline{y}_{s}=\frac{1}{n_{s}}\sum_{j=1}^{n_{s}}y_{js}$, and the 2SLS
estimator for $\beta$ is, with
$P_{Z}=Z\left(Z^{\prime}Z\right)^{-1}Z^{\prime}$,

$\displaystyle\widehat{\beta}_{2sls}$ $\displaystyle=$
$\displaystyle\left(x^{\prime}P_{Z}x\right)^{-1}x^{\prime}P_{Z}y$
$\displaystyle=$
$\displaystyle\frac{\sum_{s=1}^{S}n_{s}\overline{x}_{s}\overline{y}_{s}}{\sum_{s=1}^{S}n_{s}\overline{x}_{s}^{2}}$
$\displaystyle=$
$\displaystyle\frac{\sum_{s=1}^{S}n_{s}\overline{x}_{s}^{2}\left(\overline{y}_{s}/\overline{x}_{s}\right)}{\sum_{s=1}^{S}n_{s}\overline{x}_{s}^{2}}=\sum_{s=1}^{S}w_{2sls,s}\widehat{\beta}_{s}\text{,}$

the standard result that $\widehat{\beta}_{2sls}$ is a linear combination of
the instrument specific IV estimators, (see e.g. Windmeijer, 2019). The
weights are given by

$w_{2sls,s}=\frac{n_{s}\overline{x}_{s}^{2}}{\sum_{l=1}^{S}n_{l}\overline{x}_{l}^{2}}\geq
0$ (6)

and hence the 2SLS estimator is here a weighted average of the group specific
estimators.

For the group specific estimates, the first-stage F-statistics are equal to
the Wald statistics for testing the null hypotheses $H_{0}:\pi_{s}=0$, and are
given by

$F_{\pi_{s}}=\frac{\widehat{\pi}_{s}^{2}}{V\widehat{a}r\left(\widehat{\pi}_{s}\right)}=\frac{n_{s}\overline{x}_{s}^{2}}{\widehat{\sigma}_{v,s}^{2}}$
(7)

for $s=1,...,S$. For each group specific IV estimator $\widehat{\beta}_{s}$
the standard weak instruments results of Staiger and Stock (1997) and Stock
and Yogo (2005) apply. As these are just-identified models, we can relate the
values of the F-statistics to Wald-test size distortions.

The robust first-stage F-statistic for testing $H_{0}:\pi=0$ is given by

$\displaystyle F_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{S}\widehat{\pi}^{\prime}\left(V\widehat{a}r_{r}\left(\widehat{\pi}\right)\right)^{-1}\widehat{\pi}$
$\displaystyle=$
$\displaystyle\frac{1}{S}\sum_{s=1}^{S}\frac{n_{s}\overline{x}_{s}^{2}}{\widehat{\sigma}_{v,s}^{2}}=\frac{1}{S}\sum_{s=1}^{S}F_{\pi_{s}}.$

It is therefore clear, that if $F_{r}$ is large, then at least one of the
$F_{\pi_{s}}$ is large.

The non-robust F-statistic is given by

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{1}{S}\widehat{\pi}^{\prime}\left(V\widehat{a}r\left(\widehat{\pi}\right)\right)^{-1}\widehat{\pi}$
$\displaystyle=$
$\displaystyle\frac{1}{S}\frac{\sum_{s=1}^{S}n_{s}\overline{x}_{s}^{2}}{\left(\sum_{s=1}^{S}\frac{n_{s}}{n}\widehat{\sigma}_{v,s}^{2}\right)}=\frac{1}{S}\sum_{s=1}^{S}\frac{\widehat{\sigma}_{v,s}^{2}}{\left(\sum_{l=1}^{S}\frac{n_{l}}{n}\widehat{\sigma}_{v,l}^{2}\right)}F_{\pi_{s}}.$

From (6) and (7) it follows that the weights for the 2SLS estimator are
related to the individual F-statistics as follows

$w_{2sls,s}=\frac{n_{s}\overline{x}_{s}^{2}}{\sum_{l=1}^{S}n_{l}\overline{x}_{l}^{2}}=\frac{\widehat{\sigma}_{v,s}^{2}F_{\pi_{s}}}{\sum_{l=1}^{S}\widehat{\sigma}_{v,l}^{2}F_{\pi_{l}}}.$

Under first-stage homoskedasticity, $\sigma_{v,s}^{2}=\sigma_{v,l}^{2}$, for
$s,l=1,\ldots,S$. Then
$\widehat{\sigma}_{v,s}^{2}\approx\widehat{\sigma}_{v,l}^{2}$ for all $s,l$,
and hence $F\approx\frac{1}{S}\sum_{s=1}^{S}F_{\pi_{s}}$. Then the weights are
given by
$w_{2sls,s}\approx\frac{F_{\pi_{s}}}{\sum_{l=1}^{S}F_{\pi_{l}}}\approx\frac{F_{\pi_{s}}}{SF}$,
so we see that then the groups with the larger individual F-statistics get the
larger weights in the 2SLS estimator under homoskedasticity.

This is not necessarily the case under heteroskedasticity. For two groups with
equal value of the F-statistic, the group with the larger variance gets the
larger weight, and indeed, a large variance weakly identified group could
dominate the 2SLS estimator. As shown in the Monte Carlo exercises below, this
is exactly what happens in the design of Andrews (2018). The robust
F-statistic is large because one of the groups has a large value of the
individual F-statistic. However, this group has a very small variance
$\sigma_{v,s}^{2}$ and hence gets a small weight in the 2SLS estimator,
resulting in a poor performance of the estimator in terms of (relative) bias
and Wald-test size.

##  3 An Alternative GMM Estimator

Clearly, one would like to use an estimator that gives larger weights to more
strongly identified groups, independent of the value of $\sigma_{v,s}^{2}$,
mimicking the weights of the 2SLS estimator under homoskedasticity of the
first-stage errors. This is achieved by the following GMM estimator, denoted
GMMf, with the extension f for first stage,

$\displaystyle\widehat{\beta}_{gmmf}$ $\displaystyle=$
$\displaystyle\left(x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}x\right)^{-1}x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}y$
$\displaystyle=$
$\displaystyle\left(\widehat{\pi}^{\prime}Z^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}Z\widehat{\pi}\right)^{-1}\widehat{\pi}^{\prime}Z^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}y,$

with $\widehat{\Omega}_{v}=$
$\sum_{i=1}^{n}\widehat{v}_{i}^{2}z_{i}z_{i}^{\prime}$ as defined in (2). This
looks like the usual GMM estimator, but instead of the structural residuals
$\widehat{u}$, the first-stage residuals $\widehat{v}$ are used in the weight
matrix. It clearly links directly to the robust F-statistic, as the
denominator is equal to $SF_{r}$.

It follows that

$\displaystyle\widehat{\beta}_{gmmf}$ $\displaystyle=$
$\displaystyle\frac{\sum_{s=1}^{S}n_{s}\overline{x}_{s}\overline{y}_{s}/\widehat{\sigma}_{v,s}^{2}}{\sum_{s=1}^{S}n_{s}\overline{x}_{s}^{2}/\widehat{\sigma}_{v,s}^{2}}=\frac{\sum_{s=1}^{S}\left(n_{s}\overline{x}_{s}^{2}/\widehat{\sigma}_{v,s}^{2}\right)\widehat{\beta}_{s}}{\sum_{s=1}^{S}n_{s}\overline{x}_{s}^{2}/\widehat{\sigma}_{v,s}^{2}}$
$\displaystyle=$ $\displaystyle\sum_{s=1}^{S}w_{gmmf,s}\widehat{\beta}_{s},$

with

$w_{gmmf,s}=\frac{F_{\pi_{s}}}{\sum_{l=1}^{S}F_{\pi_{l}}}=\frac{F_{\pi_{s}}}{SF_{r}},$

and hence the groups with the larger F-statistics get the larger weights,
independent of the values of $\sigma_{v,s}^{2}$, mimicking the 2SLS weights
under homoskedasticity of the first-stage errors.

##  4 Weak Instrument Asymptotics

We can formalize the results obtained above further using weak instruments
asymptotics (WIA). For each group $s=1,...,S$ define

$\pi_{s}=\frac{c_{s}}{\sqrt{n_{s}}}.$

The limit for $n_{s}\rightarrow\infty$, $s=1,\ldots,S$, of the group specific
concentration parameters are then given by

$\mu_{s}^{2}=\frac{c_{s}^{2}}{\sigma_{v,s}^{2}}.$ (9)

Then

$\widehat{\pi}_{s}=\overline{x}_{s}=\frac{1}{n_{s}}\sum_{j=1}^{n_{s}}\left(\frac{c_{s}}{\sqrt{n_{s}}}+v_{js}\right)=\frac{c_{s}}{\sqrt{n_{s}}}+\overline{v}_{s},$

and

$\displaystyle n_{s}\overline{x}_{s}^{2}$ $\displaystyle=$ $\displaystyle
n_{s}\left(\frac{c_{s}}{\sqrt{n_{s}}}+\overline{v}_{s}\right)^{2}$
$\displaystyle=$
$\displaystyle\left(c_{s}^{2}+2c_{s}\sqrt{n_{s}}\overline{v}_{s}+\left(\sqrt{n_{s}}\overline{v}_{s}\right)^{2}\right)$
$\displaystyle\overset{d}{\rightarrow}\left(c_{s}+\sigma_{v,s}\mathcal{T}_{s}\right)^{2}=\sigma_{v,s}^{2}\left(\mu_{s}+\mathcal{T}_{s}\right)^{2}$

where $\mu_{s}=c_{s}/\sigma_{v,s}$ and $\mathcal{T}_{s}\sim
N\left(0,1\right)$. We get the standard WIA result that

$F_{\pi_{s}}=\frac{n_{s}\overline{x}_{s}^{2}}{\widehat{\sigma}_{v,s}^{2}}\overset{d}{\rightarrow}\left(\mu_{s}+\mathcal{T}_{s}\right)^{2}\sim\chi_{1,\mu_{s}^{2}}^{2},$

where $\chi_{1,\mu_{s}^{2}}^{2}$ is the non-central chi-squared distribution
with 1 degree of freedom and non-centrality parameter $\mu_{s}^{2}$.

From (6) it then follows that

$w_{2sls,s}\overset{d}{\rightarrow}\frac{\sigma_{v,s}^{2}\left(\mu_{s}+\mathcal{T}_{s}\right)^{2}}{\sum_{l=1}^{S}\sigma_{v,l}^{2}\left(\mu_{l}+\mathcal{T}_{l}\right)^{2}},$

with the $\mathcal{T}_{l}$ independent $N\left(0,1\right)$ variables, for
$l=1,...,S$.

For the weights $w_{gmmf,s}$,

$w_{gmmf,s}\overset{d}{\rightarrow}\frac{\left(\mu_{s}+\mathcal{T}_{s}\right)^{2}}{\sum_{l=1}^{S}\left(\mu_{l}+\mathcal{T}_{l}\right)^{2}}.$

Consider for illustration the case where there are two groups. Table 1
presents some results for the average values of $w_{2sls,1}$ and $w_{gmmf,1}$
after randomly drawing 100,000 values of $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$. In the first row, there is homoskedasticity,
$\sigma_{v,1}^{2}=\sigma_{v,2}^{2}=5$, and both groups have equal
concentration parameters, $\mu_{1}^{2}=\mu_{2}^{2}=5.76$, which is the value
of the concentration parameter for the group-specific Wald tests to have a
maximal rejection frequency of 10% at the 5% level. Then
$E\left(w_{2sls,1}\right)=E\left(w_{gmmf,1}\right)=0.5$ and both estimators
will give on average equal weight to the group specific estimators.

Table 1:  WIA weights for 2SLS and GMMf

$\sigma_{v,1}^{2}$ | $\sigma_{v,2}^{2}$ | $\mu_{1}^{2}$ | $\mu_{2}^{2}$ |  | $w_{2sls,1}$ | $w_{gmmf,1}$
---|---|---|---|---|---|---
5 | 5 | 5.76 | 5.76 |  | 0.50 | 0.50
5 | 0.1 | 5.76 | 5.76 |  | 0.95 | 0.50
5 | 0.1 | 1.96 | 5.76 |  | 0.84 | 0.32

Notes: Average weights from $100,000$ draws of $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$.

The second row considers the case where there is a large difference in the
variances, $\sigma_{v,1}^{2}=5$, and $\sigma_{v,2}^{2}=0.1$, but
$\mu_{1}^{2}=\mu_{2}^{2}=5.76$ as before. We find for this case that
$E\left(w_{2sls,1}\right)=0.95$, i.e. almost all weight will on average be
given to the high variance group 1. The expected weight for the GMMf estimator
is in this case not affected by the relative values of the $\sigma_{v,s}^{2}$
and remains at $E\left(w_{gmmf,1}\right)=0.5$. If we subsequently reduce the
value of $c_{1}$ such that $\mu_{1}^{2}=1.96$, then
$E\left(w_{2sls,1}\right)=0.84$, i.e. the 2SLS estimator will give more weight
to $\widehat{\beta}_{1}$, the estimator in the group with the smaller
concentration parameter, but larger variance. In contrast,
$E\left(w_{gmmf,1}\right)=0.32$ for this case, giving less weight to the less
informative group.

##  5 Variance of $u$

So far, focus has been on first-stage heteroskedasticity, with the robust GMMf
estimator exploiting the first-stage information by assigning larger weights
to the groups with larger group specific concentration parameters independent
of the values of $\sigma_{v,s}^{2}$. Consider next the infeasible robust GMM
group IV estimator, given by

$\displaystyle\widehat{\beta}_{gmm}$ $\displaystyle=$
$\displaystyle\frac{\sum_{s=1}^{S}n_{s}\overline{x}_{s}\overline{y}_{s}/\sigma_{u,s}^{2}}{\sum_{s=1}^{S}n_{s}\overline{x}_{s}^{2}/\sigma_{u,s}^{2}}=\frac{\sum_{s=1}^{S}\left(n_{s}\overline{x}_{s}^{2}/\sigma_{u,s}^{2}\right)\widehat{\beta}_{s}}{\sum_{s=1}^{S}n_{s}\overline{x}_{s}^{2}/\sigma_{u,s}^{2}}$
$\displaystyle=$ $\displaystyle\sum_{s=1}^{S}w_{gmm,s}\widehat{\beta}_{s}.$

Whereas $\widehat{\beta}_{gmm}$ is the best, normal, consistent and efficient
estimator under standard asymptotics, from the analysis above it is clear that
the weights may not be optimal under WIA. We have under WIA that

$\frac{n_{s}\overline{x}_{s}^{2}}{\sigma_{u,s}^{2}}\overset{d}{\rightarrow}\frac{\sigma_{v,s}^{2}}{\sigma_{u,s}^{2}}\left(\mu_{s}+\mathcal{T}_{s}\right)^{2}$

and so

$w_{gmm,s}\overset{d}{\rightarrow}\frac{\frac{\sigma_{v,s}^{2}}{\sigma_{u,s}^{2}}\left(\mu_{s}+\mathcal{T}_{s}\right)^{2}}{\sum_{l=1}^{S}\frac{\sigma_{v,l}^{2}}{\sigma_{u,l}^{2}}\left(\mu_{s}+\mathcal{T}_{s}\right)^{2}}.$

Clearly, if $u$ is homoskedastic, $\sigma_{u,s}^{2}=\sigma_{u,l}^{2}$ for
$\forall$$s,l$, then the infeasible GMM estimator has the same WIA limiting
distribution as the 2SLS estimator and suffers from the same problems as
described above for 2SLS. If $\sigma_{u,s}^{2}=\kappa\sigma_{v,s}^{2}$ for all
$s$ then $\widehat{\beta}_{gmm}$ behaves like the GMMf estimator, the latter
in that case also the efficient estimator under standard asymptotics. For
other cases the behaviour of $\widehat{\beta}_{gmm}$ depends on whether
$\sigma_{v,s}^{2}/\sigma_{u,s}^{2}$ assigns relatively larger or smaller
weights to the more informative groups.

An alternative is to weight by $\sigma_{u,s}^{2}\sigma_{v,s}^{2}$, such that

$\displaystyle\widehat{\beta}_{gmmuf}$ $\displaystyle=$
$\displaystyle\sum_{s=1}^{S}w_{gmmuf,s}\widehat{\beta}_{s},$ $\displaystyle
w_{gmmuf,s}$ $\displaystyle=$
$\displaystyle\frac{n_{s}\overline{x}_{s}^{2}/\left(\sigma_{u,s}^{2}\sigma_{v,s}^{2}\right)}{\sum_{l=1}^{S}n_{l}\overline{x}_{l}^{2}/\left(\sigma_{u,l}^{2}\sigma_{v,l}^{2}\right)}\overset{d}{\rightarrow}\frac{\frac{1}{\sigma_{u,s}^{2}}\left(\mu_{s}+\mathcal{T}_{s}\right)^{2}}{\sum_{l=1}^{S}\frac{1}{\sigma_{u,l}^{2}}\left(\mu_{l}+\mathcal{T}_{l}\right)^{2}}.$

The resulting weights are then as for the standard GMM estimator under first-
stage homoskedasticity. This would clearly improve efficiency if
$\sigma_{u,s}^{2}$ is relatively small for the more informative groups, but
can assign again less weight to more informative groups if their values of
$\sigma_{u,s}^{2}$ are relatively large.

##  6 Some Monte Carlo Results

We consider here the heteroskedastic design of Andrews (2018) with $S=10$
groups, $\beta=0$ and moderate endogeneity. Table 9 in the Supplementary
Appendix C.3 of Andrews (2018) presents the values of the conditional group
specific variance matrices $\Sigma_{s}$ as defined in (1) and the first-stage
parameters, denoted $\pi_{0s}$, for $s=1,\ldots,10$. Results for the high
endogeneity case are given in Appendix A.1. We multiply the first-stage
parameters $\pi_{0}$ by $0.04$, such that the value of the robust $F_{r}$ is
just over $80$ on average for $10,000$ replications and sample size
$n=10,000$. The group sizes are equal in expectation with
$P\left(z_{i}=e_{s}\right)=0.1$ for all $s$. The first two rows of Table 3
present the values of $\pi_{s}$ and $\sigma_{v,s}^{2}$ for $s=1,\ldots,10$.

Table 2 presents the estimation results. The non-robust F-statistic is small,
$F=1.41$ and the effective F-statistic of Montiel Olea and Pflueger (2013),
denoted $F_{eff}$, is equal to the non-robust F in this grouped-data IV
design. Although the robust F-statistic is large, $F_{r}=80.23$, the 2SLS
estimator $\widehat{\beta}_{2sls}$ is poorly behaved. Its relative bias equal
to $0.699$ and the Wald test rejection frequency for $H_{0}:\beta=0$ is equal
to $0.534$ at the 5% level. In contrast, the GMMf estimator is unbiased and
its Wald-test rejection frequency equal to $0.049$ at the 5% level.

Table 2:  Estimation results for $S=10$, moderate endogeneity

$F$ | $F_{eff}$ | $F_{r}$ | $\widehat{\beta}_{OLS}$ | $\widehat{\beta}_{2sls}$ | $\widehat{\beta}_{gmmf}$ | $W_{2sls}$ | $W_{gmmf}$
---|---|---|---|---|---|---|---
1.411 | 1.411 | 80.23 | -0.608 | -0.424 | -0.001 | 0.534 | 0.049
|  |  | (0.011) | (0.257) | (0.563) |  |

Notes: means and (st.dev.) of $10,000$ replications. Rej.freq. of robust Wald
tests at 5% level.

The details as given in Table 3 below make clear what is happening. It reports
the fixed values of $\pi_{s}$, $\sigma_{v,s}^{2}$,
$\mu_{n,s}^{2}=1000\pi_{s}^{2}/\sigma_{v,s}^{2}$ and the mean values of
$F_{\pi_{s}}$, $w_{2sls,s}$ and
$w_{gmmf,s}=F_{\pi_{s}}/\sum_{l=1}^{S}F_{\pi_{l}}$. Identification in the
first group is strong, with an average value of $F_{\pi_{1}}=789.5$.
Identification in all other 9 groups is very weak, with the largest average
value for $F_{\pi_{5}}=2.23$. But the variance in group 1 is very small, and
some of the variances in the other groups are quite large. This leads to the
low average value of $w_{2sls,1}=0.127$, showing that the 2SLS estimator does
not utilize the identification strength of the first group, with larger weight
given to higher variance, but lower concentration-parameter groups.

Table 3: Group information and estimator weights

$s$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
$\pi_{s}$ | 0.058 | -0.023 | 0.049 | 0.015 | 0.022 | 0.008 | -0.017 | 0.011 | -0.036 | -0.040
$\sigma_{v,s}^{2}$ | 0.004 | 2.789 | 4.264 | 0.779 | 0.395 | 7.026 | 1.226 | 0.308 | 1.709 | 6.099
$\mu_{n,s}^{2}$ | 785.7 | 0.184 | 0.556 | 0.284 | 1.190 | 0.009 | 0.236 | 0.387 | 0.770 | 0.266
$F_{\pi_{s}}$ | 789.5 | 1.170 | 1.564 | 1.279 | 2.225 | 0.997 | 1.203 | 1.372 | 1.798 | 1.246
$w_{2sls,s}$ | 0.126 | 0.098 | 0.178 | 0.035 | 0.031 | 0.180 | 0.049 | 0.015 | 0.096 | 0.192
$w_{gmmf,s}$ | 0.984 | 0.002 | 0.002 | 0.002 | 0.003 | 0.001 | 0.002 | 0.002 | 0.002 | 0.002

Notes: $\mu_{n,s}^{2}=1000\pi_{s}^{2}/\sigma_{v,s}^{2}$.

Table 3 further shows that for the GMMf estimator almost all weight is given
to the first group, with the average of $w_{gmmf,1}$ equal to $0.984$,
resulting in the good behaviour of the GMMf estimator in terms of bias and
Wald test size. In this case the standard deviation of the GMMf estimator is
quite large relative to that of the 2SLS estimator. This is driven by the
value of $\sigma_{u,1}^{2}$, which in this design is equal to $1.10$, much
larger than $\sigma_{v,1}^{2}$. Reducing the value of $\sigma_{u,1}^{2}$ (and
the value for $\sigma_{uv,1}$ accordingly to keep the same correlation
structure within group 1), will reduce the standard deviation of the GMMf
estimator.

Figure 1 displays the rejection frequencies of the robust Wald tests for
testing $H_{0}:\beta=0$ for varying values of the robust F-statistic $F_{r}$
for the 2SLS and GMMf estimators. Different values of $F_{r}$ are obtained by
different values of $d$ when setting the first-stage parameters
$\pi=d\pi_{0}$. It is clear that the Wald test based on the GMMf estimator is
much better behaved in terms of size than the test based on the 2SLS
estimator, with hardly any size distortion for mean values of $F_{r}$ larger
than 5. The right panel of Figure 1 shows that the bias of the GMMf estimator,
relative to that of the OLS estimator, is also substantially smaller than that
of the 2SLS estimator, with the relative bias smaller than $0.10$ for mean
values of $F_{r}$ larger than 9.

Figure 1:  Rejection frequencies of robust Wald tests and relative biases of
2SLS and GMMf estimators

##  7 Dynamic Panel Data Model

Next, consider the dynamic AR(1) panel data specification

$y_{it}=\gamma y_{i,t-1}+\eta_{i}+u_{it},$ (10)

for $i=1,...,n$, and $t=2,..,T$, and for $\left|\gamma\right|<1$. Let
$y_{it}^{\ast}$ and $y_{i,t-1}^{\ast}$ be the forward orthogonal deviations
transformed variables, see Arellano and Bover (1995), and $y_{i}^{\ast}$ and
$y_{i,-1}^{\ast}$ the associated $\left(T-2\right)$ vectors. Let the
$\left(T-2\right)\times\left(T-1\right)\left(T-2\right)/2$ matrix of
instruments $Z_{i}$ be defined as

$Z_{i}=\left[\begin{array}[]{cccccccc}y_{i1}&0&0&0&0&0&0&0\\\
0&y_{i1}&y_{i2}&0&0&0&0&0\\\ &&&\ddots\\\
0&0&0&0&y_{i1}&y_{i2}&\ldots&y_{i,T-2}\end{array}\right].$

Further, let the $n\left(T-2\right)$ vectors
$y^{\ast}=\left(y_{1}^{\ast\prime},y_{2}^{\ast\prime},...,y_{n}^{\ast\prime}\right)^{\prime}$
and
$y_{-1}^{\ast}=\left(y_{1,-1}^{\ast\prime},y_{2,-1}^{\ast\prime},...,y_{n,-1}^{\ast\prime}\right)^{\prime}$,
and the $n\left(T-2\right)\times\left(T-1\right)\left(T-2\right)/2$ instrument
matrix
$Z=\left[Z_{1}^{\prime},Z_{2}^{\prime},...,Z_{n}^{\prime}\right]^{\prime}$.
The 2SLS estimator is then given by

$\widehat{\gamma}_{2sls}=\left(y_{-1}^{\ast\prime}Z\left(Z^{\prime}Z\right)^{-1}Z^{\prime}y_{-1}^{\ast}\right)^{-1}y_{-1}^{\ast\prime}Z\left(Z^{\prime}Z\right)^{-1}Z^{\prime}y^{\ast}$

and is consistent and asymptotic normal under standard asymptotics, regularity
assumptions and the assumption of no serial correlation in $u_{it}$,
$E\left(u_{it}u_{is}\right)=0$ for $t\neq s$. It is efficient under
conditional homoskedasticity,
$E\left(u_{i}u_{i}^{\prime}|Z_{i}\right)=\sigma_{u}^{2}I_{T-1}$.

The first stage for the 2SLS estimator is here given by

$y_{i,-1}^{\ast}=Z_{i}^{\prime}\pi+v_{i}^{\ast}.$

The difference here compared to the grouped data IV example is that the
$v_{it}^{\ast}$ are not drawn separately, but the processes are driven by the
$u_{it}$ only. Let $\widehat{\pi}$ be the OLS estimator of $\pi$,
$\widehat{v}_{i}^{\ast}=y_{i,-1}^{\ast}-Z_{i}^{\prime}\widehat{\pi}$ and

$\widehat{\Omega}_{v}^{\ast}=\sum_{i=1}^{n}Z_{i}^{\prime}\widehat{v}_{i}^{\ast}\widehat{v}_{i}^{\ast\prime}Z_{i}.$

Then the robust F-statistic for $H_{0}:\pi=0$ is given by

$F_{r}=y_{-1}^{\ast\prime}Z\left(\widehat{\Omega}_{v}^{\ast}\right)^{-1}Z^{\prime}y_{-1}^{\ast}/k_{z},$

where $k_{z}=\left(T-1\right)\left(T-2\right)/2$. Accordingly, the GMMf
estimator is here given by

$\widehat{\gamma}_{gmmf}=\left(y_{-1}^{\ast\prime}Z\left(\widehat{\Omega}_{v}^{\ast}\right)^{-1}Z^{\prime}y_{-1}^{\ast}\right)^{-1}y_{-1}^{\ast\prime}Z\left(\widehat{\Omega}_{v}^{\ast}\right)^{-1}Z^{\prime}y^{\ast}.$

Under conditional heteroskedasticity the standard two-step GMM estimator is
efficient under standard asymptotics. Denote
$\widehat{u}_{i}^{\ast}=y_{i}^{\ast}-\widehat{\gamma}_{2sls}y_{i,-1}^{\ast}$,
and let

$\widehat{\Omega}_{u}^{\ast}=\sum_{i=1}^{n}Z_{i}^{\prime}\widehat{u}_{i}^{\ast}\widehat{u}_{i}^{\ast\prime}Z_{i}.$

The two-step GMM estimator is then given by

$\widehat{\gamma}_{gmm2}=\left(y_{-1}^{\ast\prime}Z\left(\widehat{\Omega}_{u}^{\ast}\right)^{-1}Z^{\prime}y_{-1}^{\ast}\right)^{-1}y_{-1}^{\ast\prime}Z\left(\widehat{\Omega}_{u}^{\ast}\right)^{-1}Z^{\prime}y^{\ast}.$

Following Arellano (2003, p 154), the 2SLS estimator can also be obtained as a
weighted average of cross-sectional 2SLS estimators.

$\displaystyle\widehat{\gamma}_{2sls}$ $\displaystyle=$
$\displaystyle\left(\sum_{t=2}^{T-1}y_{t,-1}^{\ast\prime}Z_{t}\left(Z_{t}^{\prime}Z_{t}\right)^{-1}Z_{t}^{\prime}y_{t,-1}^{\ast}\right)^{-1}\sum_{t=2}^{T-1}y_{t,-1}^{\ast\prime}Z_{t}\left(Z_{t}^{\prime}Z_{t}\right)^{-1}Z_{t}^{\prime}y_{t}^{\ast}$
$\displaystyle=$
$\displaystyle\sum_{t=2}^{T-1}w_{2sls,t}\widehat{\gamma}_{t},$

where

$\displaystyle\widehat{\gamma}_{t}$ $\displaystyle=$
$\displaystyle\left(y_{t,-1}^{\ast\prime}Z_{t}\left(Z_{t}^{\prime}Z_{t}\right)^{-1}Z_{t}^{\prime}y_{t,-1}^{\ast}\right)^{-1}y_{t,-1}^{\ast\prime}Z_{t}\left(Z_{t}^{\prime}Z_{t}\right)^{-1}Z_{t}^{\prime}y_{t}^{\ast}$
$\displaystyle w_{2sls,t}$ $\displaystyle=$
$\displaystyle\frac{\left(y_{t,-1}^{\ast\prime}Z_{t}\left(Z_{t}^{\prime}Z_{t}\right)^{-1}Z_{t}^{\prime}y_{t,-1}^{\ast}\right)}{\sum_{l=2}^{T-1}y_{l,-1}^{\ast\prime}Z_{l}\left(Z_{l}^{\prime}Z_{l}\right)^{-1}Z_{l}^{\prime}y_{l,-1}^{\ast}},$

with the $n$-vectors
$y_{t}^{\ast}=\left(y_{1t}^{\ast},...,y_{nt}^{\ast}\right)^{\prime}$,
$y_{t,-1}^{\ast}=$
$\left(y_{1,t-1}^{\ast},...,y_{n,t-1}^{\ast}\right)^{\prime}$, and the
$n\times\left(t-1\right)$ matrix
$Z_{t}=\left[\begin{array}[]{ccc}y_{1}&\ldots&y_{t-1}\end{array}\right]$, with
$y_{t}=\left(y_{1t},...,y_{nt}\right)^{\prime}$. We therefore see that we are
in a similar setup as the grouped-data IV example, with the group-specific IV
estimators here the cross-section specific ones. The 2SLS estimator may
therefore again give too much weight to less informative groups if the
associated cross-sectional variance of the first-stage error in the forward
orthogonal deviations transformed model is large.

We illustrate this with the following design. We set $n=200$, $T=5$,
$\gamma=0.9$, and draw $\eta_{i}\sim N\left(0,1\right)$ and $y_{i0}\sim
N\left(\frac{\eta_{i}}{1-\gamma},\frac{1}{1-\gamma^{2}}\right)$. The data are
then generated according to model (10) for $t=1,..,5$. The $u_{it}$ are
independently drawn, $u_{it}\sim N\left(0,\sigma_{u,t}^{2}\right)$, and so are
iid at the cross sectional level. Table 4 presents the estimation results for
the design with $\sigma_{u,t}=1$ for $t=1,2,4,5$, whereas $\sigma_{u,3}=4$.

The OLS estimator of $\gamma$ in the transformed model $y^{\ast}=\gamma
y_{-1}^{\ast}+u^{\ast}$ is denoted $\widehat{\gamma}_{ols}$. Whereas the GMMf
estimator takes fully account of the clustering of the first-stage errors, an
alternative estimator denoted $\widehat{\gamma}_{\widehat{\sigma}_{v}^{2}}$
takes account of the period specific variances only and is defined as

$\widehat{\gamma}_{\widehat{\sigma}_{v}^{2}}=\sum_{t=2}^{T-1}w_{\widehat{\sigma}_{v}^{2},t}\widehat{\gamma}_{t},$

with

$w_{\widehat{\sigma}_{v}^{2},t}=\frac{\left(y_{t,-1}^{\ast\prime}Z_{t}\left(Z_{t}^{\prime}Z_{t}\right)^{-1}Z_{t}^{\prime}y_{t,-1}^{\ast}\right)/\widehat{\sigma}_{v,t}^{2}}{\sum_{l=2}^{T-1}y_{l,-1}^{\ast\prime}Z_{l}\left(Z_{l}^{\prime}Z_{l}\right)^{-1}Z_{l}^{\prime}y_{l,-1}^{\ast}/\widehat{\sigma}_{v,l}^{2}},$

where $\widehat{\sigma}_{v,t}^{2}=\widehat{v}_{t}^{\prime}\widehat{v}_{t}/n$,
with $\widehat{v}_{t}=y_{t,-1}^{\ast}-Z_{t}\widehat{\pi}_{t}$.

Estimation results for this design are presented in Table 4. Due to the time
heteroskedasticity, the robust F-statistic has a larger mean than the non-
robust F-statistic, 6.74 vs 1.44. The OLS estimator is severely downward
biased. The 2SLS estimator also has a large downward bias and its relative
bias is 0.70. The two-step GMM estimator is less biased, but still has a large
relative bias of 0.45. The GMMf and
$\widehat{\gamma}_{\widehat{\sigma}_{v}^{2}}$ estimators perform better in
terms of bias, and have relative biases of 0.25 and 0.16 respectively.

Table 4: Estimation results for AR(1) panel data model

$F$ | $F_{r}$ | $\widehat{\gamma}_{ols}$ | $\widehat{\gamma}_{2sls}$ | $\widehat{\gamma}_{gmm2}$ | $\widehat{\gamma}_{gmmf}$ | $\widehat{\gamma}_{\widehat{\sigma}_{v}^{2}}$ |
---|---|---|---|---|---|---|---
1.440 | 6.741 | 0.371 | 0.527 | 0.662 | 0.768 | 0.815 |
|  | (0.017) | (0.231) | (0.254) | (0.287) | (0.219) |

Notes: $\gamma=0.9$, $n=200$, $T=5$. Means and (st.dev.) from $10,000$ MC
replications.

Table 5 displays the time specific information and paints a similar picture as
that given in Table 3 for the grouped-data IV case. The F-statistics for $t=2$
and $t=3$ are small and the first-stage variances are relatively large. The
average F-statistic for $t=4$ is relatively large, but the first-stage
variance is small. The 2SLS estimator gives a relatively large weight to the
uninformative periods, whereas the first-stage variance weighted estimator
gives a large weight to the informative $t=4$ period.

Table 5:  Period specific information and estimator weights

| $t$ |  | 2 | 3 | 4 |
---|---|---|---|---|---|---
| $\widehat{\sigma}_{v,t}^{2}$ |  | 5.620 | 9.780 | 0.516 |
| $F_{t}$ |  | 1.268 | 1.099 | 11.14 |
| $w_{2sls,t}$ |  | 0.126 | 0.393 | 0.480 |
| $w_{\widehat{\sigma}_{v}^{2},t}$ |  | 0.035 | 0.062 | 0.903 |

Notes: Averages over $10,000$ MC replications

Figure 2 displays the relative bias of the estimators for varying values of
the robust F-statistic $F_{r}$ by varying the value of
$\sigma_{u,3}=\left\\{1,1.3,1.6...,6.1\right\\}$. We see again that the GMMf
and $\widehat{\gamma}_{\widehat{\sigma}_{v}^{2}}$ (GMMfdiag) estimators
utilize the information as conveyed by the robust F-statistic well. The 2SLS
estimator has a large relative bias for all values of the mean of $F_{r}$.
Whereas the two-step GMM estimator’s performance improves with increasing
value of $F_{r}$ in this setting, its relative bias remains high at 0.385 at
the largest mean value of $F_{r}$ considered, $14.38$. The relative biases of
the GMMf and $\widehat{\gamma}_{\widehat{\sigma}_{v}^{2}}$ estimators are
respectively 0.119 and 0.069 at that value of the mean of $F_{r}$.

Figure 2: Relative bias

##  8 Testing for Weak Instruments

Using the GMMf estimator as a generalization of the 2SLS estimator to deal
with general forms of first-stage heteroskedasticity, we derive in the
Appendix under what conditions the weak-instruments Stock and Yogo (2005)
critical values derived for the non-robust F-test and the properties of the
2SLS estimator under full homoskedasticity apply to the robust F-test and the
properties of the GMMf estimator. We focus here on standard cross-sectional
heteroskedasticity, but results apply to cluster and/or serially correlated
designs.

Consider again the standard linear model

$\displaystyle y_{i}$ $\displaystyle=$ $\displaystyle x_{i}\beta+u_{i};$
$\displaystyle x_{i}$ $\displaystyle=$ $\displaystyle
z_{i}^{\prime}\pi+v_{i},$

where $z_{i}$ is a $k_{z}$-vector of instruments, and where other exogenous
variables, including the constant have been partialled out. General
conditional heteroskedasticity is specified as

$\displaystyle E\left[u_{i}^{2}|z_{i}\right]$ $\displaystyle=$
$\displaystyle\sigma_{u}^{2}\left(z_{i}\right);\,\,E\left[v_{i}^{2}|z_{i}\right]=\sigma_{v}^{2}\left(z_{i}\right);\,\,E\left[u_{i}v_{i}|z_{i}\right]=\sigma_{uv}\left(z_{i}\right).$

Further, let

$\Omega_{u}=E\left[\sigma_{u}^{2}\left(z_{i}\right)z_{i}z_{i}^{\prime}\right];\
\
\Omega_{v}=E\left[\sigma_{v}^{2}\left(z_{i}\right)z_{i}z_{i}^{\prime}\right];\
\ \Omega_{uv}=E\left[\sigma_{uv}\left(z_{i}\right)z_{i}z_{i}^{\prime}\right],$

and the unconditional variances and covariance

$\sigma_{u}^{2}=E_{z}\left[\sigma_{u}^{2}\left(z_{i}\right)\right];\ \
\sigma_{v}^{2}=E_{z}\left[\sigma_{v}^{2}\left(z_{i}\right)\right];\ \
\sigma_{uv}=E_{z}\left[\sigma_{uv}\left(z_{i}\right)\right].$

The robust F-statistic and GMMf estimator are given by

$\displaystyle F_{r}$ $\displaystyle=$ $\displaystyle
x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}x/k_{z}$
$\displaystyle\widehat{\beta}_{gmmf}$ $\displaystyle=$
$\displaystyle\left(x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}x\right)^{-1}x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}y$

Stock and Yogo (2005) derived critical values for the non-robust F-statistic
under homoskedasticity for the weak-instruments hypothesis on the relative
bias of the 2SLS estimator, relative to that of the OLS estimator. In Appendix
A.2 we show that these critical values apply to the robust F-statistic for
relative bias of the GMMf estimator, relative to that of the OLS estimator if
$\Omega_{uv}=\delta\Omega_{v}$ and $\sigma_{uv}=\delta\sigma_{v}^{2}$, where
$\delta$ is some arbitrary constant.

For the Wald test size distortion, we show in Appendix A.2 that the Stock and
Yogo (2005) critical values apply to the GMMf based Wald test if
$\Omega_{uv}=\delta\Omega_{v}$ and $\Omega_{u}=\kappa\Omega_{v}$, with
$\delta$ and $\kappa$ some arbitrary constants. The condition
$\Omega_{u}=\kappa\Omega_{v}$ implies that the GMMf estimator is also the
efficient estimator under standard asymptotics.

Whilst these conditions imply a limited applicability of the Stock and Yogo
(2005) critical values for the robust F-statistic in relation to the behaviour
of the GMMf estimator, it is a generalization of, and includes, the
homoskedastic case. It also encompasses the illustrative example of Montiel
Olea and Pflueger (2013, Section 3.1), where they considered a design with
$E\left[\left(u_{i}~{}v_{i}\right)^{\prime}\left(u_{i}~{}v_{i}\right)\right]=\Sigma$
and
$E\left[\left(\left(u_{i}~{}v_{i}\right)^{\prime}\left(u_{i}~{}v_{i}\right)\right)\otimes
z_{i}z_{i}^{\prime}\right]=a^{2}\Sigma\otimes I_{k_{z}}$, and where the non-
robust F-statistic gives an overestimate of the information content for the
2SLS estimator when $a>1$.

##  9 Conclusions

This paper has shown why large values of the first-stage robust F-statistic
may not translate in good behaviour of the 2SLS estimator. In the
heteroskedastic grouped-data design of Andrews (2018), this is the case
because a highly informative group had a relatively small first-stage
variance, and the 2SLS estimator gives more weight to groups with small
concentration parameters but large first-stage variances. A robust GMM
estimator, called GMMf, with the robust weight matrix estimated using the
first-stage residuals, remedies this problem and gives larger weights to more
informative groups. This is independent of the values of the first-stage
variances and is a generalization of the 2SLS estimator in that it mimics what
the 2SLS estimator does under first-stage homoskedasticity. A large value of
the robust F-statistic indicates that there is first-stage information
resulting in a well behaved GMMf estimator, also confirmed in an AR(1) dynamic
panel data model. We have provided the conditions under which the Stock and
Yogo (2005) weak instruments critical values developed for the non-robust
F-statistic and relative bias and Wald test size distortion of the 2SLS
estimator apply to the robust F-statistic and the behaviour of the GMMf
estimator.

## Appendix

###  A.1 Results for high endogeneity design

Tables A1 and A2 present estimation results for the $S=10$, high endogeneity
design of Andrews (2018, SA, Table 12). As in Section 6, the first-stage
parameters have been multiplied by a factor such that the robust F-statistic
has an average value of just over 80. As shown in Table A1, the pattern of
group information is similar to that in the moderate endogeneity case, with
one informative group, group $s=10$, with an average value of
$F_{\pi_{10}}=792.2$. However, the variance $\sigma_{v,10}^{2}$ is now so
small in relative terms, that the 2SLS weight for group 10 has an average
value of only $w_{2sls,10}=0.003$. The GMMf estimator corrects this, with the
average value of $w_{gmmf,10}=0.989$, and is again much better behaved than
the 2SLS estimator both in terms of (relative) bias and Wald test size, as
displayed in Table A2.

Table A1:  Group information and estimator weights, high endogeneity

$s$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
$100\cdot\pi_{s}$ | -0.021 | 0.095 | -0.484 | -0.069 | 0.159 | -0.028 | 0.101 | -0.418 | 0.450 | -0.546
$\sigma_{v,s}^{2}$ | 1.600 | 0.478 | 2.975 | 1.142 | 0.174 | 0.145 | 4.658 | 1.963 | 2.990 | 0.38$\cdot a$
$\mu_{n,s}^{2}$ | 0.28$\cdot a$ | 0.002 | 0.008 | 4.2$\cdot a$ | 0.015 | 5.6$\cdot a$ | 2.2$\cdot a$ | 0.009 | 0.007 | 789.9
$F_{\pi_{s}}$ | 0.998 | 1.017 | 0.979 | 1.010 | 1.034 | 0.984 | 0.977 | 1.031 | 0.997 | 792.2
$w_{2sls,s}$ | 0.111 | 0.040 | 0.177 | 0.085 | 0.016 | 0.013 | 0.242 | 0.134 | 0.181 | 0.003
$w_{gmmf,s}$ | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.989

Notes: $a=10^{-4}$; $\mu_{n,s}^{2}=1000\pi_{s}^{2}/\sigma_{v,s}^{2}$.

Table A2:  Estimation results for $S=10$, high endogeneity

$F$ | $F_{eff}$ | $F_{r}$ | $\widehat{\beta}_{OLS}$ | $\widehat{\beta}_{2sls}$ | $\widehat{\beta}_{gmmf}$ | $W_{2sls}$ | $W_{gmmf}$
---|---|---|---|---|---|---|---
0.994 | 0.994 | 80.12 | 0.754 | 0.749 | 0.007 | 1.000 | 0.067
|  |  | (0.000) | (0.023) | (0.029) |  |

Notes: means and (st.dev.) of $10,000$ replications. Rej.freq. of robust Wald
tests at 5% level.

Figure A1 shows the rejection frequencies of the robust Wald tests and the
relative bias of the 2SLS and GMMf estimators as a function of the value of
the robust F-statistic, showing a much better performance of the GMMf
estimator.

Figure A1: Rejection frequencies of robust Wald tests and relative bias, high
endogeneity

###  A.2 Testing for Weak Instruments

Using the GMMf estimator as a generalization of the 2SLS estimator to deal
with general forms of first-stage heteroskedasticity, we investigate here
under what conditions the Stock and Yogo (2005) weak instruments critical
values derived for the non-robust F-test and the properties of 2SLS estimator
under full homoskedasticity apply to the robust F-test and the properties of
the GMMf estimator.

Consider again the standard linear model

$\displaystyle y_{i}$ $\displaystyle=$ $\displaystyle x_{i}\beta+u_{i};$
$\displaystyle x_{i}$ $\displaystyle=$ $\displaystyle
z_{i}^{\prime}\pi+v_{i},$

with conditional heteroskedasticity specified as

$\displaystyle E\left[u_{i}^{2}|z_{i}\right]$ $\displaystyle=$
$\displaystyle\sigma_{u}^{2}\left(z_{i}\right);\,\,E\left[v_{i}^{2}|z_{i}\right]=\sigma_{v}^{2}\left(z_{i}\right);\,\,E\left[u_{i}v_{i}|z_{i}\right]=\sigma_{uv}\left(z_{i}\right),$

and, unconditionally,

$\sigma_{u}^{2}=E_{z}\left[\sigma_{u}^{2}\left(z_{i}\right)\right];\ \
\sigma_{v}^{2}=E_{z}\left[\sigma_{v}^{2}\left(z_{i}\right)\right];\ \
\sigma_{uv}=E_{z}\left[\sigma_{uv}\left(z_{i}\right)\right].$

Further, let

$\Omega_{u}=E\left[\sigma_{u}^{2}\left(z_{i}\right)z_{i}z_{i}^{\prime}\right];\
\
\Omega_{v}=E\left[\sigma_{v}^{2}\left(z_{i}\right)z_{i}z_{i}^{\prime}\right];\
\ \Omega_{uv}=E\left[\sigma_{uv}\left(z_{i}\right)z_{i}z_{i}^{\prime}\right],$

and assume that

$\left(\begin{array}[]{c}\frac{1}{\sqrt{n}}Z^{\prime}u\\\
\frac{1}{\sqrt{n}}Z^{\prime}v\end{array}\right)\overset{d}{\rightarrow}\left(\begin{array}[]{c}\psi_{zu}\\\
\psi_{zv}\end{array}\right)\sim N\left(\left(\begin{array}[]{c}0\\\
0\end{array}\right),\left(\begin{array}[]{cc}\Omega_{u}&\Omega_{uv}\\\
\Omega_{uv}^{\prime}&\Omega_{v}\end{array}\right)\right)$
$\left(\begin{array}[]{c}\Omega_{u}^{-1/2}\frac{1}{\sqrt{n}}Z^{\prime}u\\\
\Omega_{v}^{-1/2}\frac{1}{\sqrt{n}}Z^{\prime}v\end{array}\right)\overset{d}{\rightarrow}\left(\begin{array}[]{c}z_{u}\\\
z_{v}\end{array}\right)\sim N\left(\left(\begin{array}[]{c}0\\\
0\end{array}\right),\left(\begin{array}[]{cc}I_{k_{z}}&R\\\
R^{\prime}&I_{k_{z}}\end{array}\right)\right),$

where

$R=\Omega_{u}^{-1/2}\Omega_{uv}\Omega_{v}^{-1/2}.$

Let $\widehat{\pi}=\left(Z^{\prime}Z\right)^{-1}Z^{\prime}x$ be the OLS
estimator of $\pi$ and $\widehat{v}=x-Z\widehat{\pi}$ the OLS residual. For
the GMMf estimator, we have

$\widehat{\beta}_{gmmf}=\beta+\left(x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}x\right)^{-1}x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}u$

where, as before,
$\widehat{\Omega}_{v}=\sum_{i=1}^{n}\widehat{v}_{i}^{2}z_{i}z_{i}^{\prime}$.
Assume that conditions are such that

$\frac{1}{n}Z^{\prime}Z\overset{p}{\rightarrow}E\left[z_{i}z_{i}^{\prime}\right]=Q_{zz}$
$\frac{1}{n}\widehat{\Omega}_{v}\overset{p}{\rightarrow}\Omega_{v}=E\left[\sigma_{v}^{2}\left(z_{i}\right)z_{i}z_{i}^{\prime}\right].$

For weak instrument asymptotics, let

$\pi=\frac{c}{\sqrt{n}},$

then

$\displaystyle x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}x$
$\displaystyle=$
$\displaystyle\left(Z\frac{c}{\sqrt{n}}+v\right)^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}\left(Z\frac{c}{\sqrt{n}}+v\right)$
$\displaystyle=$
$\displaystyle\frac{1}{n}c^{\prime}Z^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}Zc+\frac{2}{\sqrt{n}}c^{\prime}Z^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}v+v^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}v$
$\displaystyle\overset{d}{\rightarrow}\left(\lambda+z_{v}\right)^{\prime}\left(\lambda+z_{v}\right)$

where

$\lambda=\Omega_{v}^{-1/2}Q_{zz}c.$

It follows that

$F_{r}=\frac{1}{k_{z}}x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}x\overset{d}{\rightarrow}\chi_{k_{z}}^{2}\left(\lambda^{\prime}\lambda\right)/k_{z}.$
(A.1)

For the numerator, we have

$\displaystyle x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}u$
$\displaystyle=$
$\displaystyle\left(Z\frac{c}{\sqrt{n}}+v\right)^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}u$
$\displaystyle=$
$\displaystyle\frac{2}{\sqrt{n}}c^{\prime}Z^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}u+v^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}u$
$\displaystyle\overset{d}{\rightarrow}\left(\lambda+z_{v}\right)^{\prime}\Omega_{v}^{-1/2}\Omega_{u}^{1/2}z_{u}.$

For the OLS estimator,

$\widehat{\beta}_{ols}-\beta=\frac{x^{\prime}u}{x^{\prime}x}=\frac{\frac{c^{\prime}}{\sqrt{n}}Z^{\prime}u+v^{\prime}u}{\frac{1}{n}c^{\prime}Z^{\prime}Zc+\frac{2}{\sqrt{n}}c^{\prime}Z^{\prime}v+v^{\prime}v}\overset{p}{\rightarrow}\frac{\sigma_{uv}}{\sigma_{v}^{2}}.$

As $E\left[z_{u}|z_{v}\right]=Rz_{v}$, it follows for the relative bias that

$\frac{E\left[\widehat{\beta}_{gmmf}-\beta\right]}{E\left[\widehat{\beta}_{ols}-\beta\right]}\rightarrow\frac{\sigma_{v}^{2}}{\sigma_{vu}}E\left[\frac{\left(\lambda+z_{v}\right)^{\prime}\Omega_{v}^{-1/2}\Omega_{u}^{1/2}Rz_{v}}{\left(\lambda+z_{v}\right)^{\prime}\left(\lambda+z_{v}\right)}\right]$
(A.2) $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
=\frac{\sigma_{v}^{2}}{\sigma_{uv}}E\left[\frac{\left(\lambda+z_{v}\right)^{\prime}\Omega_{v}^{-1/2}\Omega_{uv}\Omega_{v}^{-1/2}z_{v}}{\left(\lambda+z_{v}\right)^{\prime}\left(\lambda+z_{v}\right)}\right].$

Therefore it follows that if $\Omega_{uv}=\delta\Omega_{v}$ and
$\sigma_{uv}=\delta\sigma_{v}^{2}$, for an arbitrary constant $\delta$, then

$\frac{E\left[\widehat{\beta}_{gmmf}-\beta\right]}{E\left[\widehat{\beta}_{ols}-\beta\right]}\rightarrow
E\left[\frac{\left(\lambda+z_{v}\right)^{\prime}z_{v}}{\left(\lambda+z_{v}\right)^{\prime}\left(\lambda+z_{v}\right)}\right].$
(A.3)

The conditions $\Omega_{uv}=\delta\Omega_{v}$ and
$\sigma_{uv}=\delta\sigma_{v}^{2}$ are satisfied if
$\sigma_{uv}\left(z_{i}\right)=\delta\sigma_{v}^{2}\left(z_{i}\right)$ for all
$z_{i}\in\mathcal{Z}$.

The results (A.1) and (A.3) are the same as the Staiger and Stock (1997) and
Stock and Yogo (2005) results for the 2SLS estimator and full conditional
homoskedasticity,

$\left(\begin{array}[]{c}\psi_{zu}\\\ \psi_{zv}\end{array}\right)\sim
N\left(\left(\begin{array}[]{c}0\\\
0\end{array}\right),\left(\begin{array}[]{cc}\sigma_{u}^{2}&\sigma_{uv}\\\
\sigma_{uv}&\sigma_{v}^{2}\end{array}\right)\otimes Q_{zz}\right)$

and with $\lambda=\sigma_{v}^{-1}Q_{zz}^{1/2}c$. Therefore the Stock and Yogo
(2005) critical values apply to the robust F-statistic and relative bias (A.2)
of the GMMf estimator if $\Omega_{uv}=\delta\Omega_{v}$ and
$\sigma_{uv}=\delta\sigma_{v}^{2}$. For the grouped data IV example, this
condition is fulfilled if $\sigma_{uv,s}=\delta\sigma_{v,s}^{2}$ for all
$s=1,...,S$.

For the Wald test, we have

$V\widehat{a}r\left(\widehat{\beta}_{gmmf}\right)=\left(x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}x\right)^{-1}x^{\prime}Z\widehat{\Omega}_{v}^{-1}\widehat{\Omega}_{u}\widehat{\Omega}_{v}^{-1}Z^{\prime}x\left(x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}x\right)^{-1}$

and so

$W_{gmmf}\left(\beta\right)=\frac{\left(\widehat{\beta}_{gmmf}-\beta\right)^{2}\left(x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}x\right)^{2}}{x^{\prime}Z\widehat{\Omega}_{v}^{-1}\widehat{\Omega}_{u}\widehat{\Omega}_{v}^{-1}Z^{\prime}x}=\frac{\left(x^{\prime}Z\widehat{\Omega}_{v}^{-1}Z^{\prime}u\right)^{2}}{x^{\prime}Z\widehat{\Omega}_{v}^{-1}\widehat{\Omega}_{u}\widehat{\Omega}_{v}^{-1}Z^{\prime}x}.$

Then,

$\displaystyle\frac{1}{n}\widehat{\Omega}_{u}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\widehat{u}_{i}^{2}z_{i}z_{i}^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left(u_{i}-x_{i}\left(\widehat{\beta}_{gmmf}-\beta\right)\right)^{2}z_{i}z_{i}^{\prime}$
$\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left(u_{i}^{2}-2u_{i}x_{i}\left(\widehat{\beta}_{gmmf}-\beta\right)+x_{i}^{2}\left(\widehat{\beta}_{gmmf}-\beta\right)^{2}\right)z_{i}z_{i}^{\prime}$
$\displaystyle\overset{d}{\rightarrow}\Omega_{u}-2\Omega_{uv}\left(\widehat{\beta}_{gmmf}-\beta\right)+\Omega_{v}\left(\widehat{\beta}_{gmmf}-\beta\right)^{2},$

and so

$\displaystyle
x^{\prime}Z\widehat{\Omega}_{v}^{-1}\widehat{\Omega}_{u}\widehat{\Omega}_{v}^{-1}Z^{\prime}x\overset{d}{\rightarrow}\left(\lambda+z_{v}\right)^{\prime}\Omega_{v}^{-1/2}\Omega_{u}\Omega_{v}^{-1/2}\left(\lambda+z_{v}\right)$
$\displaystyle-2\left(\lambda+z_{v}\right)^{\prime}\Omega_{v}^{-1/2}\Omega_{uv}\Omega_{v}^{-1/2}\left(\lambda+z_{v}\right)\left(\widehat{\beta}_{gmmf}-\beta\right)$
$\displaystyle+\left(\lambda+z_{v}\right)^{\prime}\left(\lambda+z_{v}\right)\left(\widehat{\beta}_{gmmf}-\beta\right)^{2}.$

This results in

$W_{gmmf}\overset{d}{\rightarrow}\frac{q_{2}^{2}}{a-2bq_{2}/\eta_{1}+q_{2}^{2}/\eta_{1}},$

where

$\displaystyle q_{2}$ $\displaystyle=$
$\displaystyle\left(\lambda+z_{v}\right)^{\prime}\Omega_{v}^{-1/2}\Omega_{u}^{1/2}z_{u}$
$\displaystyle a$ $\displaystyle=$
$\displaystyle\left(\lambda+z_{v}\right)^{\prime}\Omega_{v}^{-1/2}\Omega_{u}\Omega_{v}^{-1/2}\left(\lambda+z_{v}\right)$
$\displaystyle b$ $\displaystyle=$
$\displaystyle\left(\lambda+z_{v}\right)^{\prime}\Omega_{v}^{-1/2}\Omega_{uv}\Omega_{v}^{-1/2}\left(\lambda+z_{v}\right)$
$\displaystyle\eta_{1}$ $\displaystyle=$
$\displaystyle\left(\lambda+z_{v}\right)^{\prime}\left(\lambda+z_{v}\right).$

If $\Omega_{u}=\kappa\Omega_{v}$ and $\Omega_{uv}=\delta\Omega_{v}$, so
$R=\frac{\delta}{\sqrt{\kappa}}I_{k_{z}}$, then

$W_{gmmf}\left(\beta\right)\overset{d}{\rightarrow}\frac{\kappa\eta_{2}^{2}}{\kappa\eta_{1}-2\delta\sqrt{\kappa}\eta_{2}+\kappa\eta_{2}^{2}/\eta_{1}}=\frac{\eta_{2}^{2}/\eta_{1}}{1-2\rho\eta_{2}/\eta_{1}+\left(\eta_{2}/\eta_{1}\right)^{2}}$
(A.4)

where

$\displaystyle\eta_{2}$ $\displaystyle=$
$\displaystyle\left(\lambda+z_{v}\right)^{\prime}z_{u}$ $\displaystyle\rho$
$\displaystyle=$ $\displaystyle\frac{\delta}{\sqrt{\kappa}}.$

Conditions $\Omega_{u}=\kappa\Omega_{v}$ and $\Omega_{uv}=\delta\Omega_{v}$
are satisfied if
$\sigma_{uv}\left(z_{i}\right)=\delta\sigma_{v}^{2}\left(z_{i}\right)$ and
$\sigma_{u}^{2}\left(z_{i}\right)=\kappa\sigma_{v}^{2}\left(z_{i}\right)$ for
all $z_{i}\in\mathcal{Z}$, and then
$\rho=\frac{\sigma_{uv}}{\sigma_{u}\sigma_{v}}$.

Result (A.4) is the same as that of the Staiger and Stock (1997) and Stock and
Yogo (2005) result for the 2SLS based Wald test under conditional
homoskedasticity with the maximum size distortion at $\rho^{2}=1$. Hence the
Stock and Yogo (2005) Wald size based critical values apply in the
heteroskedastic case to the GMMf based Wald test if
$\Omega_{u}=\kappa\Omega_{v}$ and $\Omega_{uv}=\delta\Omega_{v}$, with again
the maximum size distortion at $\delta^{2}/\kappa=1$.

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††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Resonant control of elastic collisions between 23Na40K molecules and 40K
atoms

Zhen Su  Hefei National Laboratory for Physical Sciences at the Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China  Shanghai Branch, CAS Center for Excellence and
Synergetic Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China
Shanghai Research Center for Quantum Sciences, Shanghai 201315, China  Huan
Yang  Hefei National Laboratory for Physical Sciences at the Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China  Shanghai Branch, CAS Center for Excellence and
Synergetic Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China
Shanghai Research Center for Quantum Sciences, Shanghai 201315, China  Jin Cao
Hefei National Laboratory for Physical Sciences at the Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China  Shanghai Branch, CAS Center for Excellence and
Synergetic Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China
Shanghai Research Center for Quantum Sciences, Shanghai 201315, China  Xin-Yao
Wang  Hefei National Laboratory for Physical Sciences at the Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China  Shanghai Branch, CAS Center for Excellence and
Synergetic Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China
Shanghai Research Center for Quantum Sciences, Shanghai 201315, China  Jun Rui
Hefei National Laboratory for Physical Sciences at the Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China  Shanghai Branch, CAS Center for Excellence and
Synergetic Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China
Shanghai Research Center for Quantum Sciences, Shanghai 201315, China  Bo Zhao
Hefei National Laboratory for Physical Sciences at the Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China  Shanghai Branch, CAS Center for Excellence and
Synergetic Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China
Shanghai Research Center for Quantum Sciences, Shanghai 201315, China  Jian-
Wei Pan  Hefei National Laboratory for Physical Sciences at the Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China  Shanghai Branch, CAS Center for Excellence and
Synergetic Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China
Shanghai Research Center for Quantum Sciences, Shanghai 201315, China

###### Abstract

We have demonstrated the resonant control of the elastic scattering cross
sections in the vicinity of Feshbach resonances between 23Na40K molecules and
40K atoms by studying the thermalization between them. The elastic scattering
cross sections vary by more than two orders of magnitude close to the
resonance, and can be well described by an asymmetric Fano profile. The
parameters that characterize the magnetically tunable s-wave scattering length
are determined from the elastic scattering cross sections. The observation of
resonantly controlled elastic scattering cross sections opens up the
possibility to study strongly interacting atom-molecule mixtures and improve
our understanding of the complex atom-molecule Feshbach resonances.

Feshbach resonances are of great importance to the study of ultracold gases
Chin et al. (2010). They occur when the energy of a bound state of a
collisional system coincides with that of a scattering state. Magnetic fields
can be used to tune the collisional system in and out of resonances if the
bound state and the scattering state have different magnetic moments. The
magnetically tunable Feshbach resonances have been used as a powerful tool to
study strongly interacting quantum gases Regal et al. (2004); Chin et al.
(2004); Zwierlein et al. (2006), Efimov resonances Kraemer et al. (2006), and
the association of diatomic molecules Regal et al. (2003); Herbig et al.
(2003). Resonant control of the s-wave scattering length is essential to these
applications Tiesinga et al. (1993); Inouye et al. (1998); Roberts et al.
(1998); Loftus et al. (2002). In the vicinity of atomic Feshbach resonances,
the s-wave scattering length can be described by a simple formula Chin et al.
(2010)

$a(B)=a_{\rm{bg}}(1-\frac{\Delta}{B-B_{0}-i\gamma/2}),$ (1)

where $a_{\rm{bg}}$ denotes the background scattering length, $B_{0}$
represents the resonance position, $\Delta$ is the resonance width, and
$\gamma$ is a decay term that is usually neglected in describing atomic
Feshbach resonances. These parameters are usually calculated by coupled-
channel calculations using precise Born-Oppenheimer potentials, which are
determined by comparing with the experiments.

Feshbach resonances involving ultracold molecules are extremely difficult to
understand Quéméner and Julienne (2012). For the heavy alkali-metal diatomic
molecules formed from ultracold atomic gases Ni et al. (2008); Molony et al.
(2014); Takekoshi et al. (2014); Park et al. (2015a); Guo et al. (2016);
Rvachov et al. (2017); Seeßelberg et al. (2018); Liu et al. (2019); Voges et
al. (2020), many rovibrational states of the molecules may contribute to
resonant states, and thus it is expected that scattering resonances play
crucial roles in ultracold collisions involving molecules. However, the large
number of collision channels, the uncertainties of the potential energy
surface, and the complexity of the three-body or four-body dynamics make the
coupled-channel calculations intractable Croft et al. (2017). Ultracold
collisions involving molecules are usually studied by analyzing the density of
short-range chaotic resonant states using the statistical method Mayle et al.
(2012, 2013). The universal loss rate coefficient is one of the main
predictions, which is usually compared with the experiments Ospelkaus et al.
(2010); Takekoshi et al. (2014); Park et al. (2015a); Guo et al. (2016);
Rvachov et al. (2017); Voges et al. (2020); Ye et al. (2018); Bause et al.
(2021); Gregory et al. (2021); Nichols et al. (2021). However, for atom-
molecule collisions, the estimated mean spacing between neighboring states is
actually much larger than the temperature of the ultracold gases Croft et al.
(2017); Christianen et al. (2019); Frye and Hutson (2021) and thus the
analysis based on the density of states and the thermal averaging over many
resonances is in principle not applicable.

Recently, magnetically tunable atom-molecule Feshbach resonances have been
observed in inelastic collisions between singlet ground-state 23Na40K
molecules and 40K atoms Yang et al. (2019); Wang et al. (2021), and reactive
collisions between triplet ground-state 23Na6Li molecules and 23Na atoms Son
et al. (2021). A detailed analysis of the resonance pattern in collisions of
23Na40K with 40K suggests that the resonant states associated with these
resonances may be the long-range bound states Wang et al. (2021), but not the
short-range chaotic bound states that are usually considered. The observation
of well-isolated individual atom-molecule Feshbach resonances opens up the
possibility of controlling the interactions in atom-molecule gases. However,
so far, only the resonantly enhanced losses of the molecules can be studied
near the Feshbach resonances. This largely limits the application of the atom-
molecule Feshbach resonance, since many important applications of Feshbach
resonances require the control of the s-wave scattering length or
equivalently, the elastic collision cross sections
$\sigma_{\rm{el}}=4\pi|a|^{2}$. Due to the large inelastic loss rate
coefficients, it is unclear whether tunable elastic collisions can be
observed. Although in the collisions between 23Na6Li and 23Na, a model based
on Fabry-Perot interferometer is used to extract the scattering length from
the enhanced loss rate coefficients Son et al. (2021), direct observation of
magnetically controlled elastic collisions or the scattering length remains
elusive.

In this Letter, we report on the study of resonant control of elastic
collisions in the vicinity of a Feshbach resonance between 23Na40K molecules
and 40K atoms. The 23Na40K molecules and 40K atoms are both prepared in the
maximally spin stretched state. We study the thermalization between 23Na40K
molecules and 40K atoms, from which the elastic scattering cross sections can
be obtained. We find that the elastic scattering cross sections vary by more
than two orders of magnitude close to the resonances. The elastic scattering
cross sections can be well described by an asymmetric Fano profile, which is a
hallmark of Feshbach resonance. The four parameters that characterize a
Feshbach resonance are directly obtained from the elastic scattering cross
sections. The demonstration of resonantly controlling the s-wave scattering
length paves the way towards studying strongly interacting atom-molecule
mixtures and improves the understanding of the complex atom-molecule
collisions.

Our experiment starts with an ultracold mixture of 23Na40K molecules and 40K
atoms. The experimental procedures for preparing the ultracold mixture have
been introduced in previous works Yang et al. (2022); Wang et al. (2022).
Briefly, we load laser-cooled 23Na and 40K atoms into a cloverleaf magnetic
trap to perform evaporative cooling. The atomic mixtures are then loaded into
a large-volume three-beam optical dipole trap for further evaporative cooling.
At the end of optical evaporative cooling, we create an ultracold mixture
containing about $3.0\times 10^{5}$ 23Na atoms and $2.3\times 10^{5}$ 40K
atoms. The trap frequencies for 40K atoms are
$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(238,71,24)$ Hz. The atomic
Feshbach resonance between the $|f,m_{f}\rangle_{\rm{Na}}=|1,1\rangle$ state
and the $|f,m_{f}\rangle_{\rm{K}}=|9/2,-9/2\rangle$ state at about 78.3 G is
used to form 23Na40K Feshbach molecules. We ramp the magnetic field from 80 G
to 77.6 G to create about $2.8\times 10^{4}$ 23Na40K Feshbach molecules. The
23Na40K Feshbach molecules are then transferred to the rovibrational ground
state by stimulated Raman adiabatic passage (STIRAP). We prepare the 23Na40K
molecules in the maximally polarized hyperfine level
$|v,N,m_{i_{\rm{Na}}},m_{i_{\rm{K}}}\rangle=|0,0,-3/2,-4\rangle$ of the
rovibrational ground state, where $v$ and $N$ represent the vibrational and
rotational quantum numbers, and $m_{i_{\rm{Na}}}$ and $m_{i_{\rm{K}}}$ denote
the nuclear spin projections of 23Na and 40K respectively. We use the
hyperfine level of the excited state $F_{1}\approx
3/2,m_{F_{1}}\approx-3/2,m_{\rm{K}}\approx-3$ as the intermediate state for
the STIRAP Park et al. (2015b); Liu and Zhao (2020). The $\sigma^{-}$
polarized pump laser and the $\sigma^{+}$ polarized Stokes laser propagate
along the direction of the magnetic field. The efficiency of a round-trip
STIRAP is about 50%. After removing the remaining 23Na atoms by resonant light
pulses, we prepare an ultracold mixture of 23Na40K molecules and 40K atoms.

We study the elastic collisions between 23Na40K molecules in the
$|0,0,-3/2,-4\rangle$ state and 40K atoms in the $|9/2,-9/2\rangle$ state in
the vicinity of an atom-molecule Feshbach resonance located at about 48 G.
After the ultracold atom-molecule mixture is prepared, we ramp the magnetic
field to a target value in 3 ms. We then apply a 50-$\mu$s weak resonant light
pulse to heat the 40K atoms. After the heating pulse, the temperatures of the
40K atoms and the 23Na40K molecules are about 500 nK and 300 nK respectively.
The heating light pulse also reduces the atom numbers by a factor of about 2.
The 23Na40K molecules and the 40K atoms are held at the target magnetic field
for a short time. The uncertainty of the magnetic field is about 200 mG. We
then remove the 40K atoms by a strong resonant light pulse and ramp the
magnetic field back to 77.6 G. The 23Na40K molecules are transferred back to
the Feshbach state and are detected by absorption imaging along the z
direction.

Figure 1:  The thermalization of the 23Na40K molecules following the heating
of the 40K atoms. The squares of the molecular cloud sizes $\sigma_{x}^{2}$
(red) and $\sigma_{y}^{2}$ (black) are plotted as a function of the hold time
at various target magnetic fields. The molecular cloud sizes are extracted
from the absorption images taken after a time of flight of 3-4 ms. The
increases of cloud sizes are due to the elastic collisions between the 23Na40K
molecules and the 40K atoms. The solid lines are the fits of the data points
to exponential functions $\sigma_{x,y}^{2}=A_{x,y}e^{-\Gamma_{x,y}t}+B_{x,y}$
where the parameters $A,B$ and $\Gamma$ are the fitting parameters. For the
magnetic fields close to 43 G at which the thermalization is very weak, the
parameters $B$ are fixed. The mean thermalization rate
$\Gamma_{\rm{th}}=(\Gamma_{x}+\Gamma_{y})/2$ is used to calculate the elastic
scattering cross section. Error bars represent the standard error of the mean.

During the hold time at the target magnetic field, the temperature of the
23Na40K molecules will increase due to elastic collisions with the 40K atoms.
Since the number of 40K atoms is about one order of magnitude larger than that
of the molecules, we may assume that the temperature of the 40K atoms is
constant and the temperature of the 23Na40K molecules changes according to
$T(t)=T_{f}-(T_{f}-T_{i})e^{-\Gamma_{\rm{th}}t}$, where $T_{i}$ and $T_{f}$
represent the initial and final temperatures respectively, and
$\Gamma_{\rm{th}}$ is the thermalization rate. The thermalization rate can be
extracted from the time evolution of the molecule cloud sizes. For a Maxwell-
Boltzmann distribution, the column density distributions obtained from the
absorption image can be fit to two-dimensional Gaussian functions $\propto
e^{-x^{2}/(2\sigma_{x}^{2})-y^{2}/(2\sigma_{y}^{2})}$, where the cloud sizes
along the $x$ and $y$ directions can be described by
$\sigma_{x,y}^{2}(t)=k_{\rm{B}}T(t)/m_{\rm{NaK}}(\tau_{\rm{tof}}^{2}+1/\omega_{x,y}^{2})$,
with $\tau_{\rm{tof}}$ being the time of flight.

Figure 2: The elastic scattering cross sections, calculated from the
thermalization rate, are plotted as a function of the magnetic fields. The
solid lines are the fit using the formula
$\sigma_{\rm{el}}=4\pi(\alpha^{2}+\beta^{2})$, where the scattering
$a=\alpha-i\beta$ is given by Eq. 1. The parameters obtained from the fit are
$a_{\rm{bg}}=-692(36)a_{0}$, $\Delta=-4.3(4)$ G, $B_{0}=47.5(3)$ G and
$\gamma=2.4(4)$ G. Error bars represent the standard error.

The squares of the molecule cloud sizes $\sigma_{x,y}^{2}$ extracted from the
absorption images are shown in Fig. 1 as a function of the hold time for
several target magnetic fields. The increases of the cloud sizes along both
the $x$ and $y$ directions versus the hold time reflect the thermalization due
to elastic collisions. It can be clearly seen that the thermalization rate
changes with the magnetic fields. The thermalization is very fast at 47 G,
which indicates that the elastic collisions are resonantly enhanced. However,
at 43 G the thermalization is largely suppressed. This indicates that the
elastic scattering cross section has a minimum at about 43 G. Note that at 48
G, it is difficult to determine the thermalization rate due to the strong
inelastic losses. We fit the molecule cloud sizes using an exponential
function $\sigma_{x,y}^{2}=A_{x,y}e^{-\Gamma_{x,y}t}+B_{x,y}$ where the
parameters $A,B$ and $\Gamma$ are the fitting parameters. For the magnetic
fields close to 43 G, the thermalization is very weak, and thus we set the
parameter $B$ to the mean value of the fit results obtained at other magnetic
fields to obtain a stable fit.

We determine the elastic scattering cross section from the measured
thermalization rate using the formula Mosk et al. (2001); Son et al. (2020);
Tobias et al. (2020)
$\Gamma_{\rm{th}}=n_{\rm{ov}}\sigma_{\rm{el}}v_{\rm{rel}}/(3/\xi)$, where
$n_{\rm{ov}}=(N_{\rm{NaK}}+N_{\rm{K}})[(\frac{2\pi
k_{\rm{B}}T_{\rm{K}}}{m_{\rm{K}}\bar{\omega}_{\rm{K}}^{2}})(1+\frac{m_{\rm{K}}T_{\rm{NaK}}}{m_{\rm{NaK}}T_{\rm{K}}\gamma_{t}^{2}})]^{-\frac{3}{2}}$
is the overlap density with $\gamma_{t}=0.79$ being the ratio between the trap
frequencies for 23Na40K molecules and 40K atoms,
$v_{\rm{rel}}=\sqrt{(8k_{\rm{B}}/\pi)(T_{\rm{K}}/m_{\rm{K}}+T_{\rm{NaK}}/m_{\rm{NaK}})}$
is the relative mean velocity, and the parameter
$\xi=4m_{\rm{NaK}}m_{\rm{K}}/(m_{\rm{NaK}}+m_{\rm{K}})^{2}$ represents the
mass effect. The elastic scattering cross sections determined in this way are
shown in Fig. 2 as a function of the magnetic fields, where the mean value
$\Gamma_{\rm{th}}=(\Gamma_{x}+\Gamma_{y})/2$ is used in the calculation.

The elastic scattering cross sections can be changed by more than two orders
of magnitude via varying the magnetic field. The elastic scattering cross
section shows a clear asymmetric Fano line shape Fano (1961); Chin et al.
(2010), which is a hallmark of Feshbach resonance. The minimum of the elastic
scattering cross section is caused by the Fano interference between the
background scattering amplitude in the open channel and the resonant
scattering amplitude originating from the bound state in the closed channel.
It represents the quantum control of the elastic atom-molecule collisions via
quantum interference.

Figure 3: The real (red) and imaginary (blue) parts of the scattering length
are plotted as a function of the magnetic field. The scattering length is
plotted according to Eq. 1, with the parameters determined from the elastic
scattering cross sections.

Although there is no theoretical model that can calculate the elastic
scattering cross sections, we may still use Eq. 1 to describe the s-wave
scattering length in the vicinity of the atom-molecule Feshbach resonance. In
this case, the s-wave elastic scattering cross section may be described by
$\sigma_{\rm{el}}=4\pi(\alpha^{2}+\beta^{2})$, where the complex scattering
length $a=\alpha-i\beta$ with $\alpha$ and $\beta$ real numbers. When using
this simple analytic model, we have neglected thermal average effects. The
thermal average will be important when the scattering length is comparable
with the de Broglie wavelength, which is about 3000 $a_{0}$. Therefore we
expect that the thermal average mainly affects the accuracy of the resonance
position. We fit the measured elastic scattering cross sections using this
analytic formula, where $a_{\rm{bg}},\Delta,B_{0}$ and $\gamma$ are the
fitting parameters.

The fit can only give the absolute value of the background scattering length,
but not its sign. The sign of the background scattering length can be
determined as follows. Since the minimum of the scattering cross section is
below the resonance position, there are two possibilities. If the bound state
is on the low field side of the resonance, the background scattering length is
negative. On the contrary, the background scattering length is positive.
Following the arguments in Ref. Wang et al. (2021), since both the molecules
and atoms are in the maximally polarized state, the triatomic bound states
accounting for this resonance must have a nonzero total spin-free rotational
angular momentum $J\geq 1$, and the bound state must approach the scattering
state from the low field side of the resonance. Therefore, the background
scattering length is negative. We obtain $a_{\rm{bg}}=-692(36)a_{0}$,
$\Delta=-4.3(4)$ G, $B_{0}=47.5(3)$ G and $\gamma=2.4(4)$ G. The magnetically
tunable scattering length is thus determined. The real and imaginary parts of
the scattering length are plotted in Fig. 3.

Figure 4: The loss rate coefficients obtained by measuring the losses of the
23Na40K molecules in the atom-molecule mixture. The solid line is a Lorentzian
fit to the data. Error bars represent the standard error.

The strong coupling between the bound state and the scattering state causes
avoided crossing, and thus the resonance position is different from the
intersection between the bare bound state and the scattering state. The
intersection between the bare states may be determined using the formula
Julienne and Gao (2006) $B_{c}=B_{0}-\Delta
r_{\rm{bg}}(1-r_{\rm{bg}})/[1+(1-r_{\rm{bg}})^{2}],$ where
$r_{\rm{bg}}=a_{\rm{bg}}/\bar{a}$ with $\bar{a}=74a_{0}$ the mean scattering
length. We obtain $B_{c}=43.7$ G, which is about 4 G lower than the resonance
position. This value is very close to the narrow resonances at about 43.4 G in
the collision channels $|0,0,-1/2,-4\rangle$\+ $|9/2,-9/2\rangle$ and
$|0,0,1/2,-4\rangle$+$|9/2,-9/2\rangle$ Wang et al. (2021). This indicates
that the bare bound states for these resonances may bunch together, even if
the projections of the total angular momentum is different. Therefore, the
determination of the bare bound state for broad resonances is important to
understand the resonance pattern. The resonance pattern may be used to
estimate the relative moment between the bound state and the scattering state.

It is interesting to compare the elastic collisions with the inelastic
collisions. To this end, we measure the loss rate coefficients of the 23Na40K
molecules in the atom-molecule mixtures. After ramping the magnetic field to a
target value, we monitor the time evolution of the 23Na40K molecules. Assuming
the loss of the 23Na40K molecules is caused by two-body inelastic collisions
with the 40K atoms, the loss rate coefficients are given by the ratio between
the measured decay rate and the density of 40K atoms. The measured loss rate
coefficients are shown in Fig. 4. The loss rate coefficients are fit to a
Lorentz function. The fit also gives the resonance position
$B^{\rm{in}}_{0}=48.8(3)$ G and the FWHW width $\gamma^{\rm{in}}=2.8(5)$G.
These values are very close to the values determined from the elastic
collision cross sections.

In summary, we have demonstrated the resonant control of elastic collisions
between 23Na40K molecules and 40K atoms. The observation of the magnetic
tunable elastic scattering cross sections will help to improve our
understanding of the complex atom-molecule Feshbach resonances. In previous
theoretical works, different theoretical models are developed to calculate the
loss rate coefficients and compare them with experiments. The tunable elastic
scattering cross sections near the Feshbach resonances thus provide another
important quantity that can be compared with the theory. The elastic
scattering cross sections may also be useful in determining the short-range
parameters required in the multichannel quantum-defect model for anisotropic
interactions Gao (2020).

The direct observation of the elastic scattering cross sections or the
scattering length opens up the possibility of studying strongly interacting
atom-molecule mixtures. Recently, ultracold mixtures of molecules and atoms
have been proposed to study a new type of impurity problem, i.e., angulon
Schmidt and Lemeshko (2015, 2016), which is a counterpart of the polaron. The
atom-molecule Feshbach resonances may be useful since it can provide the
strong interaction that is necessary in studying the impurity problem in
ultracold gases Schirotzek et al. (2009); Kohstall et al. (2012); Hu et al.
(2016); Jørgensen et al. (2016). Besides studying strongly interacting atom-
molecule mixtures, the universal binding energies of the triatomic bound
states close to the resonance position may be determined from the scattering
length, which can be compared with the binding energies obtained from the
radio-frequency association spectroscopy Yang et al. (2022).

###### Acknowledgements.

This work was supported by the National Key R&D Program of China (under Grant
No. 2018YFA0306502), the National Natural Science Foundation of China (under
Grant Nos. 11521063, 11904355), the Chinese Academy of Sciences, the Anhui
Initiative in Quantum Information Technologies, the Shanghai Municipal Science
and Technology Major Project (Grant No.2019SHZDZX01), the Shanghai Rising-Star
Program (Grant No. 20QA1410000).

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