# Efficient two-sample instrumental variable estimators with change points and
near-weak identification

Bertille Antoine
Simon Fraser University
<EMAIL_ADDRESS>Otilia Boldea
Tilburg University
<EMAIL_ADDRESS>Niccolò Zaccaria
Tilburg University
<EMAIL_ADDRESS>
###### Abstract

We consider estimation and inference in a linear model with endogenous
regressors where the parameters of interest change across two samples. If the
first-stage is common, we show how to use this information to obtain more
efficient two-sample GMM estimators than the standard split-sample GMM, even
in the presence of near-weak instruments. We also propose two tests to detect
change points in the parameters of interest, depending on whether the first-
stage is common or not. We derive the limiting distribution of these tests and
show that they have non-trivial power even under weaker and possibly time-
varying identification patterns. The finite sample properties of our proposed
estimators and testing procedures are illustrated in a series of Monte-Carlo
experiments, and in an application to the open-economy New Keynesian Phillips
curve. Our empirical analysis using US data provides strong support for a New
Keynesian Phillips curve with incomplete pass-through and reveals important
time variation in the relationship between inflation and exchange rate pass-
through.

Keywords: generalized method of moments; two-sample estimators, near-weak
identification; change points
JEL classification: C13, C22, C26, C36, C51.

##  1 Introduction

The evidence for change points is pervasive in macroeconomic applications
estimating key structural parameters for monetary policy rules: Clarida et al.
(1999), Stock and Watson (2002) and Ahmed et al. (2004), or when estimating
the relationship between inflation and unemployment via open or close-economy
New Keynesian Phillips curves - Hall et al. (2012), Magnusson and Mavroeidis
(2014), Antoine and Boldea (2018), Abbas (2023), Inoue et al. (2024a). There
is also strong evidence that many key macroeconomic variables responses to
fiscal and monetary policies change over the sample - Inoue et al. (2024b) or
depend on the state of the economy \- Auerbach and Gorodnichenko (2013),
Owyang et al. (2013), Ramey and Zubairy (2018), Barnichon and Matthes (2018),
Jordá et al. (2020), Alpanda et al. (2021), Alloza (2022), Jo and Zubairy
(2024), Klepacz (2021), and Cloyne et al. (2023), and the state dependence in
many of these models can be viewed as a change point upon reordering the data.

A model with endogenous regressors and a parameter change at a given point in
time is usually estimated by GMM separately over the two samples before and
after the change (we refer to this estimator as the split-sample estimator).
This practice can lead to large efficiency losses, because it ignores the fact
that a change in parameters over two samples need not imply that everything is
changing across these two samples. For example, external instruments - such as
the ones used in the local projection literature or for estimating structural
parameters - are constructed as shocks that are only correlated with the
endogenous regressors contemporaneously, and there is often no reason to
suspect that this correlation also changes over time. If, in fact, the first-
stage linking instruments and endogenous regressors has common parameters,
this information can be used to obtain more efficient estimators for the
parameters that do change across the two samples. Additionally, since in most
macroeconomic models, instruments are relatively weak - Mavroeidis (2005),
Dufour et al. (2006), Nason and Smith (2008), Kleibergen and Mavroeidis
(2009), Ramey and Zubairy (2018), Antoine and Boldea (2018), Barnichon (2020)
\- splitting the sample may make instruments appear even weaker than they are,
especially when ignoring information that is common across the two samples.

We start by showing how to use the information from a common first-stage to
increase asymptotic efficiency of the parameter estimates that do change
across samples. While a 2SLS estimator with common first-stage is available
for strong instruments in Hall et al. (2012), we show that it need not be more
efficient than our estimator or than the split-sample GMM estimator. We
instead develop a two-sample GMM estimator with a common first-stage, and
prove that it is asymptotically at least as efficient as the split-sample
estimator, even when the instruments are near-weak. The asymptotic variance we
derive allows us to construct a variance estimator that is guaranteed to be no
larger than that of the split-sample estimator in finite samples.111Note that
the estimator we derive is not a second-step GMM estimator with the 2SLS
estimator in Hall et al. (2012) as a first-step, because in the case of change
points, the relationship between the two breaks down - Hall et al. (2012).
Because we allow for autocorrelated errors, our framework can directly be
applied to local projection estimators - Stock and Watson (2018).

While our focus is on macroeconometric applications, when the change point is
known, our estimator can also be viewed as a new two-sample estimator where
both the first and the second-stages are observed, and therefore we also
contribute to the microeconometric literature on two-sample estimators. In the
original formulation of the two-sample estimators - Angrist and Krueger
(1992), Inoue and Solon (2010) \- the endogenous regressors and the
instruments are observed in the first sample, the outcome and the instruments
in the second sample, and the first-stage of the first sample is used as proxy
for the first-stage in the second sample to obtain IV and 2SLS estimators,
assuming that the two first-stages are common. In our case, both first-stages
and second-stages are observed, yielding new two-sample estimators.

Additionally, when the change point is unknown, we show that the implied
change fraction can be estimated consistently even when the instruments are
near-weak, and at the fastest available rate in the presence of exogenous
regressors such as an intercept. We also provide two Wald tests for a change
point in the presence of near-weak instruments. The first test assumes a
common first-stage: its asymptotic distribution is pivotal, and has non-
trivial power even under the alternative, although the implied change point
estimator need not be consistent under the alternative and this depends on how
much heterogeneity is allowed in the second moments of the data. The second
test assumes that the first-stage has a change point, and checks whether the
second-stage exhibits the same change point. Plugging in the implied OLS
change point estimator yields a chi-squared asymptotic distribution under the
null, with non-trivial power under the alternative.

We also discuss a general procedure for multiple change points, and show that
a two-sample estimator for the first-stage using information about the second-
stage is also more efficient than its OLS counterpart over the full-sample or
over sub-samples, generalizing Antoine and Boldea (2018).

A simulation study compares the performance of the split-sample GMM
estimators, the two-sample 2SLS estimators in Hall et al. (2012), and the
proposed two-sample GMM estimators in the case of a known change point, an
unknown change point, and pre-testing for a change point. The two-sample 2SLS
estimator has larger mean squared-error compared to the split-sample estimator
in several designs with heteroskedastic errors, while our two-sample GMM
estimator consistently displays both lower variance and lower mean squared
error than both.

We illustrate our method by estimating an open-economy New Keynesian Phillips
Curve (NKPC) with monthly US data. Our empirical results provide strong
support for imperfect exchange-rate pass-through - Monacelli (2005). We find
that this pass-through is significant and comparable in magnitude to that of
the unemployment coefficient. Our results also suggest that the relationship
between inflation and exchange rate pass-through exhibits important time
variation.

The rest of the paper is organized as follows. Section 2 motivates our
framework. Section 3 introduces our two-sample GMM estimator and derives its
asymptotic properties in presence of a known change point, a common first-
stage and near-weak instruments. Section 4 presents results for the change
point estimator and the two Wald tests for a change point. It also proposes a
general empirical approach for multiple change points. Section 6 contains the
simulations, and Section 7 the empirical application. Section 8 concludes. The
proofs of the theoretical results are in the Appendix.

##  2 Framework and examples

To motivate our framework, consider first the standard linear regression
model,

$\displaystyle y_{t}$ $\displaystyle=$ $\displaystyle
Z_{1t}^{\prime}\theta_{z}^{0}+X_{t}^{\prime}\theta_{x}^{0}+u_{t}$
$\displaystyle X_{t}^{\prime}$ $\displaystyle=$ $\displaystyle
Z_{t}^{\prime}\Pi^{0}+v_{t}^{\prime}\,,$

where $Z_{1t}$ are $p_{1}$ exogenous variables, $X_{t}$ are $p_{2}$ endogenous
variables, $Z_{t}$ are $q$ valid instruments (with $q\geq p_{2}$) that include
the exogenous regressors $Z_{1t}$ as the first elements, and $\Pi^{0}$ is a
full-rank matrix of size $(q,p_{2})$. The parameter of interest is the
$p\times 1$-vector $\theta^{0}=[\theta_{z}^{0^{\prime}}\
\theta_{x}^{0^{\prime}}]^{\prime}$ (with $p=p_{1}+p_{2}$).

We extend this model in three directions. First, we allow the parameters of
interest $\theta^{0}$ to change abruptly across sub-samples. Second, we allow
the second moments of the data - the variance of the moment conditions, the
second moment of the instruments, and the variance of the structural errors -
to change over the sample and therefore to exhibit substantial time
heterogeneity. Third, we allow for the instruments to be near-weak. For all
three extensions, we first consider the case when the relationship between the
instruments and endogenous regressors does not change over time.222In the
general procedure in Section 4, we allow for this relationship to also change.

Three examples below motivate the need for such an extension and, in
particular, why the first-stage may be common while the parameters of interest
may change over the sample.

New Keynesian Phillips curve. Consider a model where inflation depends on
current inflation predictions and unemployment gap, which are typically
endogenous and often instrumented by using lags of these variables - Hall et
al. (2012). As discussed in the introduction, there is strong evidence that
the parameter linking inflation to the unemployment gap changed over time,
while there is no reason to believe that the dynamics of measured inflation
expectations or unemployment gap changed at the same time as the unemployment
gap parameter. As shown in our empirical application, as well as in Abbas
(2023) \- there are sample periods over which the first-stage is common; that
is, the parameter $\Pi^{0}$ remains the same before and after the change
point. Even when external instruments are used instead of lags, as in Inoue et
al. (2024a), the same reasoning applies.

Local projections. Suppose we are interested in whether government spending
multipliers are different when interests rates are at the zero lower bound -
Section 5 in Ramey and Zubairy (2018). Since interest rates were close to the
zero lower bound from 2008 until recently, we can write this model as a
regression of output at different horizons on government spending before and
after 2008. Since government spending is endogenous, we can use external
instruments such as the government spending shocks in Ramey and Zubairy
(2018). Since the Blanchard and Perotti (2002) shock used in Ramey and Zubairy
(2018) as instrument is constructed to measure an unexpected discretionary
change in fiscal policy over the sample, it is unclear why the relationship of
the unexpected component of the fiscal policy with fiscal policy itself should
change exactly at the time the interest rates hit the lower bound.

Two-sample IV estimators in microeconometrics. Let $t$ above refer to
individuals instead of time, and suppose we are interested in the effect of
age at school entry on educational attainment - Angrist and Krueger (1992).
Because some parents may enroll children earlier, the age at school entry is
endogenous, and Angrist and Krueger (1992) use quarter of birth dummies as
instruments. They employed data on quarter of birth and age at school entry
for one sample of individuals, data on quarter of birth and educational
attainment on another sample, and introduced a two-sample IV estimator where
the first-stage is estimated in one sample, and plugged into the IV estimator
of the second sample. This plug-in approach is employed in all two-sample IV
estimators that we are aware of - Inoue and Solon (2010), Sanderson et al.
(2022) \- and assumes that the first-stage is common for the two populations.
However, if we observed the age at school entry and educational attainment in
both samples - which is likely the case for new administrative data - it is
conceivable that the relationship between educational attainment and age at
school entry would be heterogeneous across the two samples (and that the
parameters of interest would therefore change across the two samples), yet the
first-stage linking quarter of birth to age at school entry would have
constant parameters across the same two samples. In general, the causal effect
of a treatment on an outcome can be heterogeneous across groups of
populations, while the relationship between treatment uptake and intent to
treat could be homogeneous over certain groups.

##  3 Two-sample GMM estimators with a known change point

To simplify the exposition, consider first the case when the first-stage is
common; the case of change points in the first-stage is discussed in Section
4. For the remainder of this section, the reference model is:

$\displaystyle y_{t}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lclclccr}Z_{1t}^{\prime}\theta_{z,1}^{0}&+&X_{t}^{\prime}\theta_{x,1}^{0}&+&u_{t}&&,&t\leq\lfloor
T\lambda^{0}\rfloor\\\
Z_{1t}^{\prime}\theta_{z,2}^{0}&+&X_{t}^{\prime}\theta_{x,2}^{0}&+&u_{t}&&,&t>\lfloor
T\lambda^{0}\rfloor\end{array}\right.\,,$ (3.3) $\displaystyle X_{t}^{\prime}$
$\displaystyle=$ $\displaystyle
Z_{t}^{\prime}\frac{\Pi^{0}}{r_{T}}+v_{t}^{\prime}\,,t=1,\ldots,T$ (3.4)

where $\lfloor T\lambda_{0}\rfloor$ is a known change point with
$0<\lambda_{0}<1$, $Z_{1t}$ are $p_{1}$ exogenous variables, $X_{t}$ are
$p_{2}$ endogenous variables, $Z_{t}$ are $q$ valid instruments (with $q\geq
p_{2}$) that include the exogenous regressors $Z_{1t}$, and $\Pi^{0}$ is a
full-rank matrix of size $(q,p_{2})$. The instruments share the same
identification strength over the whole sample, which can range from strong
($r_{T}=1$) to near-weak ($\lim_{T\rightarrow\infty}r_{T}=\infty$ and
$r_{T}=o(\sqrt{T})$). The parameters of interest are the $p\times 1$ vectors
$\theta_{i}^{0}=[\theta_{z,i}^{0^{\prime}}\
\theta_{x,i}^{0^{\prime}}]^{\prime},i=1,2$, with $p=p_{1}+p_{2}$. For
simplicity, we assume that all elements of $\theta_{1}^{0}-\theta_{2}^{0}$ are
different than zero, so all parameters of interest change. To streamline
notations, let $\theta_{vec}^{0}=\mbox{vect
}(\theta_{z,1}^{0},\theta_{x,1}^{0},\theta_{z,2}^{0},\theta_{x,2}^{0})$,
$W_{t}=\mbox{vect }(Z_{1t,},X_{t})$, $T_{1}^{0}=T^{0}=\lfloor
T\lambda_{0}\rfloor$, and $T_{2}^{0}=T-T_{1}^{0}$.333$\mbox{vect }(\cdot)$ is
a generalization of the usual $vec(\cdot)$ operator, in the sense that for $S$
general matrices $A_{1},\ldots,A_{S}$, $\mbox{vect
}(A_{1},\ldots,A_{s})=((vec(A_{1}))^{\prime},\ldots,(vec(A_{S}))^{\prime})^{\prime}$.
This model can be estimated with the split-sample GMM estimator
$\hat{\theta}_{GMM,vec}(\lambda_{0})$ in Andrews (1993), which minimizes:

$\displaystyle O_{GMM}(\theta_{vec};\lambda_{0})$ $\displaystyle=$
$\displaystyle
g_{T}^{\prime}(\theta_{vec};\lambda_{0})\hat{S}_{u}^{-1}g_{T}(\theta_{vec};\lambda_{0}),$
where $\displaystyle\theta_{vec}=\mbox{vect
}(\theta_{1},\theta_{2})\,,\text{with }\theta_{i}=\mbox{vect
}(\theta_{z,i},\theta_{y,i})\,,i=1,2\,,$ $\displaystyle
g_{T}(\theta_{vec};\lambda_{0})=\left[\begin{array}[]{l}(T_{1}^{0})^{-1}\sum_{t=1}^{T_{1}^{0}}Z_{t}(y_{t}-W_{t}^{\prime}\theta_{1})\\\
(T_{2}^{0})^{-1}\sum_{t=T_{1}^{0}+1}^{T}Z_{t}\left(y_{t}-W_{t}^{\prime}\theta_{2}\right)\end{array}\right]\,,$
$\displaystyle\hat{S}_{u}=diag\
[\hat{S}_{u,1},\hat{S}_{u,2}]\stackrel{{\scriptstyle
p}}{{\rightarrow}}\mbox{AVar}\left[T^{1/2}g_{T}(\theta_{vec}^{0};\lambda_{0})\right]\,.$

Andrews (1993) introduced the split-sample GMM estimator to study the
properties of a change point test. Here, we derive its asymptotic distribution
under a constant first-stage and near-weak instruments: the structural
parameters associated with the endogenous variables are asymptotically
normally distributed at rate $\sqrt{T}/r_{T}$, whereas those associated with
the exogenous regressors are asymptotically normally distributed at the
standard rate $\sqrt{T}$, as shown in Theorem 1.444See AntoineRenault2009 who
derive asymptotic properties of 2SLS and GMM in stable models with near-weak
instruments. However, this GMM estimator is not the most efficient estimator
of (3.3) -(3.4) as it ignores the constancy of the first-stage across the two
samples.

An estimator that does take into account that the first-stage is common is the
two-sample 2SLS estimator (TS2SLS) estimator in Hall et al. (2012), which
estimates the first-stage on the full sample. This estimator
$\hat{\theta}_{TS2SLS,vec}(\lambda_{0})$ minimizes:

$\displaystyle O_{2SLS}(\theta_{vec};\lambda_{0})$ $\displaystyle=$
$\displaystyle\textstyle\sum_{t=1}^{T_{1}^{0}}(y_{t}-\hat{W}_{t}^{\prime}\theta_{1})^{2}+\textstyle\sum_{t=T_{1}^{0}+1}^{T}(y_{t}-\hat{W}_{t}^{\prime}\theta_{2})^{2}$
where
$\displaystyle\hat{W}_{t}^{\prime}=[Z_{1t}^{\prime}\,,\,Z_{t}^{\prime}\hat{\Pi}],\mbox{
and
}\hat{\Pi}=\textstyle(\sum_{t=1}^{T}Z_{t}Z_{t}^{\prime})^{-1}\textstyle(\sum_{t=1}^{T}Z_{t}X_{t}^{\prime}).$

With $\Pi^{z}=\begin{bmatrix}I_{p_{1}}\\\ O_{(q-p_{1})\times
p_{1}}\end{bmatrix}$, and $\Pi^{a}=[\Pi_{z},\Pi^{0}]$, such that
$W_{t}^{\prime}\frac{\Pi^{a}}{r_{T}}=(Z_{1t}^{\prime},Z_{t}^{\prime}\frac{\Pi^{0}}{r_{T}})$,
the implied moment conditions for this estimator are555A more efficient two-
step GMM estimator for the moment conditions (3.8) exists; however, because
the moment conditions are nonlinear, it will suffer from the curse of
dimensionality even with moderate number of instruments $Z_{t}$. We therefore
do not pursue it in this paper.

$\displaystyle\begin{bmatrix}\operatorname{E}[Z_{t}(y_{t}-W_{t}^{\prime}\frac{\Pi^{a}}{r_{T}}\theta_{1}^{0})]=0&t\leq
T^{0}\\\
\operatorname{E}[Z_{t}(y_{t}-W_{t}^{\prime}\frac{\Pi^{a}}{r_{T}}\theta_{2}^{0})]=0&t>T^{0}\\\
\operatorname{E}[\mbox{vect
}(Z_{t}(X_{t}^{\prime}-Z_{t}^{\prime}\frac{\Pi^{0}}{r_{T}}))]=0&t=1,\ldots
T\end{bmatrix}.$ (3.8)

To mimic the design of the TS2SLS estimator above and still obtain a quadratic
objective function, one may be tempted to add the $qp_{2}$ full-sample OLS
moment condition of the first-stage to the split-sample GMM estimator.
However, Ahn and Schmidt (1995) showed that adding a just-identified moment
condition with its own nuisance parameter does not change the asymptotic
distribution of the original estimator. Instead, we propose adding $2qp_{2}$
first-stage moments, for the samples before and after the change point, with
the restriction that $\Pi^{0}$ is the same across the two sub-samples. Let
$\Pi^{0}_{vec}=\mbox{vect }(\Pi^{0})$, and the new moment conditions be:

$\displaystyle\breve{g}_{T}(\theta_{vec},\Pi_{vec};\lambda_{0})=\left[\begin{array}[]{c}g_{T}(\theta_{vec};\lambda_{0})\\\
g_{T,2}(\Pi_{vec};\lambda_{0})\end{array}\right]\,,$

where $g_{T,2}(\Pi_{vec};\lambda_{0})$ stacks the OLS moment conditions from
the first-stage:

$\displaystyle
g_{T,2}(\Pi_{vec};\lambda_{0})=\begin{bmatrix}(T_{1}^{0})^{-1}\sum_{t=1}^{T_{1}^{0}}\mbox{vect
}(Z_{t}(X_{t}^{\prime}-Z_{t}^{\prime}\Pi))\\\
(T_{2}^{0})^{-1}\sum_{t=T_{1}^{0}+1}^{T}\mbox{vect
}(Z_{t}(X_{t}^{\prime}-Z_{t}^{\prime}\Pi))\end{bmatrix}.$ (3.10)

The two-sample GMM (TSGMM) estimators are defined as:

$\displaystyle\left[\begin{array}[]{c}\hat{\theta}_{TSGMM,vec}(\lambda_{0})\\\
\hat{\Pi}_{TSGMM,vec}(\lambda_{0})\end{array}\right]$ $\displaystyle=$
$\displaystyle\arg\min_{\theta_{vec},\Pi}\left[\breve{g}_{T}^{\prime}(\theta_{vec},\Pi_{vec};\lambda_{0})\
\hat{\mathcal{S}}^{-1}\
\breve{g}_{T}(\theta_{vec},\Pi_{vec};\lambda_{0})\right]\,,$
$\displaystyle\text{where }\ \hat{\mathcal{S}}$
$\displaystyle\stackrel{{\scriptstyle p}}{{\rightarrow}}$
$\displaystyle\mbox{AVar}[\sqrt{T}\breve{g}_{T}(\theta_{vec}^{0},\Pi_{vec}^{0};\lambda^{0})]\,.$

In Theorem 2 below, we show that the additional moment conditions (3.10) -
that now overidentify $\Pi^{0}$ \- are not redundant and deliver an
asymptotically more efficient estimator of $\theta^{0}_{vec}$. It is worth
pausing first to compare these estimators to the original two-sample IV
estimator in Angrist and Krueger (1992) and the two-sample 2SLS estimator in
Inoue and Solon (2010). These estimators were developed when $W_{t},Z_{t}$ are
observed in the first sample, and $y_{t},Z_{t}$ in the second sample. The two-
sample IV estimator uses the following moment condition:

$E(Z_{t}y_{t}1[t>T^{0}]-Z_{t}W_{t}^{\prime}\theta^{0}1[t\leq T^{0}])=0,$

which is very different from the moment conditions above. This estimator plugs
in the first-stage of the first sample $Z_{t}W_{t}^{\prime}1[t\leq T^{0}]$ for
the unavailable first-stage of the second sample
$Z_{t}W_{t}^{\prime}1[t>T^{0}]$. In contrast, the two-sample 2SLS estimator in
Inoue and Solon (2010) uses:

$\begin{bmatrix}\operatorname{E}[Z_{t}(y_{t}-W_{t}^{\prime}\frac{\Pi^{a}}{r_{T}}\theta_{2}^{0})]=0&t>T^{0}\\\
\operatorname{E}[\mbox{vect
}(Z_{t}(X_{t}^{\prime}-Z_{t}^{\prime}\frac{\Pi^{a}}{r_{T}}))]=0&t\leq
T^{0}\end{bmatrix}.$

Inoue and Solon (2010) also show that it is more efficient than the two-sample
IV in Angrist and Krueger (1992). Note that a first-step estimator of the
TSGMM estimator we proposed - based on two-stage least-squares - has four sets
of moment conditions, two of which are equivalent to the moment conditions
above, and two of which are the equivalent moment conditions in the other
samples which are observed in our paper.

Next, we derive the asymptotic distributions of the split-sample GMM, TS2SLS
and TSGMM. To do so, we impose the assumptions below.

###### Assumption 1

(Regularity of the error terms and first-stage equation (3.4))
(i) Let $h_{t}=Z_{t}\otimes\mbox{vect }(u_{t},v_{t})$ with $i^{th}$ element
$h_{t,i}$.
\- $E(h_{t})=0$.
\- The eigenvalues of $S=\mbox{AVar}\left(T^{-1/2}\sum_{t=1}^{T}h_{t}\right)$
are $\mathcal{O}(1)$.
\- $\operatorname{E}(h_{t,i})=0$ and for some $d>2$, $\|h_{t,i}\|_{d}<\infty$
for $t=1,\cdots,T$ and $i=1,\cdots,(p_{2}+1)q$.
\- $\\{h_{t,i}\\}$ is near epoch dependent with respect to some process
$\\{\xi_{t}\\}$, $\|h_{t}-E(h_{t}|\mathcal{G}_{t-m}^{t+m})\|_{2}\leq\nu_{m}$
with $\nu_{m}=\mathcal{O}(m^{-1/2})$ where $\mathcal{G}_{t-m}^{t+m}$ is a
$\sigma$-algebra based on $(\xi_{t-m},\cdots,\xi_{t+m})$.
\- $\\{\xi_{t}\\}$ is either $\phi$-mixing of size $m^{-d/[2(d-1)]}$ or
$\alpha$-mixing of size $m^{-d/(d-2)}$.
(ii) $\Pi^{0}$ is full column-rank equal to $p_{2}$.

Assumption 1 part (i) states that the instruments are valid, and allows for
general patterns of weak dependence in the data, including autocorrelated
errors. Therefore, our paper also applies to local projection IV estimators.
Assumption 1 part (ii) ensures that the instruments are not redundant.

###### Assumption 2

(Regularity of the change fraction and second moment matrices)
(i) $0<\lambda^{0}<1$.
(ii) Let $\sum_{1}=\sum_{t=1}^{T_{1}^{0}}$ and
$\sum_{2}=\sum_{t=T_{1}^{0}+1}^{T}$. Then $Q_{i}(\lambda_{0})$ and
$S_{i}(\lambda_{0})$ are positive definite for $i=1,2$, where

$\displaystyle
T^{-1}\textstyle\sum_{i}Z_{t}Z_{t}^{\prime}\stackrel{{\scriptstyle
p}}{{\rightarrow}}Q_{i}(\lambda_{0})=Q_{i}\,;\,S_{i}(\lambda_{0})=AVar\left[T^{-1/2}\textstyle\sum_{i}h_{t}\right]=\left[\begin{array}[]{cc}S_{u,i}(\lambda_{0})&S_{uv,i}^{\prime}(\lambda_{0})\\\
S_{uv,i}(\lambda_{0})&S_{u,i}(\lambda_{0})\end{array}\right]\,=S_{i}.$

Assumption 2(i) ensures that there are enough observations in each of the two
samples to estimate parameters consistently. Note that while we defined the
two samples based on a known change point, they could also be defined in terms
of a state variable splitting the two samples based on a known threshold
value, as in the local projection literature - Ramey and Zubairy (2018).
Additionally, they could just be two groups of observations with heterogeneous
parameters - Bonhomme and Manresa (2015). Assumption 2(ii) allows for general
time variation (heterogeneity) in the second moments of the data.

###### Assumption 3

(Regularity of the identification strength)
$r_{T}=o(\sqrt{T})$.

Assumption 3 ensures that the identification is sufficiently strong for the
structural parameters to be consistently estimated. Under these assumptions,
Theorem 1 derives the asymptotic distribution of the split-sample GMM, TS2SLS
and TSGMM estimators.

###### Theorem 1

(Asymptotic distributions of the split-sample GMM, TS2SLS and TSGMM
estimators)
Let
$\Lambda_{T}=\operatorname{diag}(T^{1/2}I_{p_{1}},T^{1/2}r_{T}^{-1}I_{p_{2}})$.
Under Assumptions 1 to 3,

(i)
$[I_{2}\otimes\Lambda_{T}]\,(\hat{\theta}_{TS2SLS,vec}(\lambda_{0})-\theta_{vec}^{0})\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,V_{TS2SLS,vec})$

(ii)
$[I_{2}\otimes\Lambda_{T}]\left(\hat{\theta}_{GMM,vec}(\lambda_{0})-\theta_{vec}^{0}\right)\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,V_{GMM,vec})$

(iii)
$\operatorname{diag}(I_{2}\otimes\Lambda_{T},T^{1/2}I_{p_{2}q})\begin{bmatrix}\hat{\theta}_{TSGMM,vec}(\lambda_{0})-\theta_{vec}^{0}\\\
\hat{\Pi}_{TSGMM,vec}(\lambda_{0})-\Pi_{vec}^{0}/r_{T}\end{bmatrix}\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,V_{TSGMM,vec}^{a}),$

and
$[I_{2}\otimes\Lambda_{T}]\left(\hat{\theta}_{TSGMM,vec}(\lambda_{0})-\theta_{vec}^{0}\right)\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,V_{TSGMM,vec})$, with
$V_{TSGMM,vec}=(V_{GMM,vec}^{-1}+\mathcal{G}^{\prime}\mathcal{G})^{-1}$.
The variances $V_{TS2SLS,vec}$, $V_{GMM,vec}$, $V_{TSGMM,vec}^{a}$ and
$V_{TSGMM,vec}$ and $\mathcal{G}$ are in Appendix Definition 1.

The estimators of the endogenous regressor parameters $\theta_{x,i}^{0}$
converge at rate $\sqrt{T}/r_{T}$, which is slower than usual: this rate is
due to the presence of near-weak instruments. The other parameter estimators
are not affected by the near-weak identification rate, and their estimators
are asymptotically normally distributed at the standard rate $\sqrt{T}$.666See
Antoine and Renault (2009) for related results without change points. More
importantly, the formula for the variance of the TSGMM estimator, containing
the positive semi-definite matrix $\mathcal{G}^{\prime}\mathcal{G}$, reveals
that this variance is at least as small as that of the split-sample GMM
estimator asymptotically. When plugging in a consistent estimator for
$\mathcal{G}$, this guarantees lower variance of the TSGMM estimator in finite
samples as well.

The next assumptions consider additional regularity conditions on homogeneity
of second moments and conditional homoskedasticity, which are key to derive
conditions under which the TSGMM estimators are strictly more efficient than
the TS2SLS estimators, and the TS2SLS estimators are strictly more efficient
than the split-sample GMM.

###### Assumption 4

(Homogeneity of the second moments)
(i) $Q_{1}(\lambda_{0})=\lambda_{0}Q$ and
$Q_{2}(\lambda_{0})=(1-\lambda_{0})Q$; (ii) $S_{1}(\lambda_{0})=\lambda_{0}S$
and $S_{2}(\lambda_{0})=(1-\lambda_{0})S$.

The homogeneity assumption prevents changes in the second moments of the
instruments and in their correlation with the error terms of the first-stage.
For example, Assumption 4(i) is violated when there is a change in
$E(Z_{t}Z_{t}^{\prime})$, while Assumption 4(ii) is violated when there is a
change in $\mbox{Var}(Z_{t}u_{t})$, for example.

###### Assumption 5

(Conditional homoskedasticity)

$S_{i}=\Phi\otimes Q_{i},\mbox{ where
}\Phi=\operatorname{E}\left[\left(\begin{array}[]{cc}u_{t}^{2}&u_{t}v_{t}^{\prime}\\\
u_{t}v_{t}&v_{t}v_{t}^{\prime}\\\
\end{array}\right)|\mathcal{F}_{t}\right]=\left(\begin{array}[]{cc}\Phi_{u}&\Phi_{uv}^{\prime}\\\
\Phi_{uv}&\Phi_{v}\\\ \end{array}\right),$

$Q_{1}=Q_{1}(\lambda_{0})$, $Q_{2}=Q_{2}(\lambda_{0})$, with $\mathcal{F}_{t}$
the $\sigma$-algebra generated by $\\{Z_{t},Z_{t-1},\ldots\\}$, and $i=1,2$.

The following theorem compares the asymptotic variance of the proposed
estimators with and without Assumptions 4 and 5.777For two square symmetric
matrices $A,B$ of size $p\times p$, we write $A\leq B$ iff $B-A$ is positive
semidefinite, and $A<B$ if $B-A$ is positive definite.

###### Theorem 2

(Efficiency comparison)
(i) Under Assumptions 1 to 3, $V_{TSGMM,vec}\leq V_{GMM,vec}$, with the
difference being of rank $min(p_{2}q,2p)$. If, additionally,
$S_{uv,1}(\lambda_{0})$, respectively $S_{uv,1}(1)-S_{uv,1}(\lambda_{0})$ ,
the covariances between the moment conditions in the first- and second-stages
of both samples, are of full column rank, and $p_{2}q\geq 2p$, then
$V_{TSGMM,vec}<V_{GMM,vec}$.
(ii) Under Assumptions 1 to 5, for $i=1,2$,

$\displaystyle V_{TS2SLS,i}<V_{GMM,i}$ $\displaystyle\Leftrightarrow$
$\displaystyle
2\Phi_{uv}^{\prime}\theta_{x,i}^{0}+\theta_{x,i}^{0^{\prime}}\Phi_{v}\theta_{x,i}^{0}<0$
(3.13) $\displaystyle V_{TSGMM,i}<V_{TS2SLS,i}$ $\displaystyle\Leftrightarrow$
$\displaystyle
2\Phi_{uv}^{\prime}\theta_{x,i}^{0}+\theta_{x,i}^{0^{\prime}}\Phi_{v}\theta_{x,i}^{0}>-\frac{\delta\Phi_{u}^{2}}{1+\delta\Phi_{u}}\,,$
$\displaystyle\text{with
$\delta=\Phi_{u}^{-2}\Phi_{uv}^{\prime}(\Phi_{v}-\Phi_{uv}\Phi_{u}^{-1}\Phi_{uv}^{\prime})^{-1}\Phi_{uv}$},$

with equalities instead of inequalities if (3.13), respectively (2), hold with
equality.

Theorem 2 part (i) shows that even when the first sample moments of the first-
stage are redundant for the second sample moments of the first-stage - as it
is the case under homogeneity Assumptions 4 and 5 \- they are not redundant
overall for the estimation of $\theta_{vec}^{0}$. The intuition is similar to
Theorem 4 in Breusch et al. (1999), where it is shown that with three moment
conditions, say $(g_{1},g_{2},g_{3})$, the redundancy of $g_{3}$ given $g_{2}$
does not imply that $g_{3}$ is redundant given $(g_{1},g_{2})$. This non-
redundancy is also related to recent results by Antoine and Renault (2017) who
extend Breusch et al. (1999) to frameworks that allow for heterogenous
identification strengths.

The proof relies on determining the rank of $\mathcal{G}$ defined in Theorem
1. We intentionally allowed in part (i) for moments of the data to be
heterogeneous over time for two reasons. First, because the second moments of
the data may also have change points at the same time as the parameters of
interest. Second, because the sample split need not be due to a change point,
but rather a threshold variable as in the literature on state-dependent local
projections - Ramey and Zubairy (2018), Alpanda et al. (2021). Reordering the
data according to whether the threshold variable is below or above a certain
known value can induce heterogeneity in the second moments of the data.
Nonetheless, with this reordering, we obtain the change point model in (3.3).

We also give conditions for strict efficiency of TSGMM over split-sample GMM
estimators in part (i). The first condition is natural if a first-stage
linking regressors and instruments exists: if the moment conditions of the
first and second-stage were not correlated, then $X_{t}$ would not be
endogenous according to that first-stage. The second condition is
automatically satisfied if we have two or more endogenous regressors, as is
the case in our empirical application, because then $p_{2}q\geq 2q\geq 2p$. If
we have only one endogenous regressor, then the condition becomes $q\geq 2p$,
meaning we need twice as many instruments as regressors in each of the two
samples.

Theorem 2 part (ii) compares the TS2SLS with the split-sample and the TSGMM
estimators, a comparison that is not possible in the general case due to the
moment conditions of the TS2SLS not being a special case of the moment
conditions of the other estimators.

We first show that TS2SLS estimators can be more efficient than the split-
sample GMM estimators. Condition (3.13),

$2\Phi_{uv}^{\prime}\theta_{x,i}^{0}+\theta_{x,i}^{0^{\prime}}\Phi_{v}\theta_{x,i}^{0}<0\
\Leftrightarrow\
\Phi_{u}+2\Phi_{uv}^{\prime}\theta_{x,i}^{0}+\theta_{x,i}^{0^{\prime}}\Phi_{v}\theta_{x,i}^{0}<\Phi_{u}$

is equivalent to
$\mbox{Var}(u_{t}+v_{t}^{\prime}\theta_{x,i}^{0}|\mathcal{F}_{t})<\mbox{Var}(u_{t}|\mathcal{F}_{t})$.
Heuristically, it states that TS2SLS estimators are more efficient when the
second-stage error - obtained after plugging in the full-sample first-stage -
has smaller variance than that of the structural error. This condition should
induce caution when using the TS2SLS estimators in practice, because it is not
satisfied, for example, if $Cov(u_{t},v_{t})$ is of the same sign as
$\theta_{x,i}^{0}$.

Additionally, the TS2SLS estimators are not a special case of the TSGMM
estimators either, for which a first-step estimator based on 2SLS can be
constructed.888The first-stage 2SLS estimator for TSGMM based on moment
conditions (3.10) is different.. Nevertheless, the TSGMM estimators are
strictly more efficient than the TS2SLS estimators despite second moment
homogeneity and conditional homoskedasticity, as long as (2) holds. This
condition is harder to interpret, but it is automatically satisfied for at
least one of the two TSGMM estimators in the case of a single endogenous
regressor and no exogenous regressors, that is $p_{2}=1$ and $p_{1}=0$. In
such a case,
$\Phi_{u}\delta=\Phi_{uv}^{2}(\Phi_{v}\Phi_{u}-\Phi_{uv}^{2})^{-1}$, and the
above condition becomes,

$\displaystyle
2\Phi_{uv}\theta_{x,i}^{0}+(\theta_{x,i}^{0})^{2}\Phi_{v}>-\frac{\Phi_{uv}^{2}}{\Phi_{v}}\
\Leftrightarrow\
\left(\theta_{x,i}^{0}+\frac{\Phi_{uv}}{\Phi_{v}}\right)^{2}>0,$

which has to hold for at least one of $i=1,2$ since we assumed there the
coefficients of the endogenous regressors change over the sample
($\theta_{x,1}^{0}\neq\theta_{x,2}^{0}$).

To interpret this condition, assume that the regressors are fixed. Then
$\displaystyle\beta^{0}=\frac{\Phi_{uv}}{\Phi_{v}}$ is the limiting
coefficient of a regression of $u_{t}$ on $v_{t}$, suggesting that the TSGMM
estimator “purges” $u_{t}$ of the true correlation with $v_{t}$. On the other
hand, the TS2SLS estimator transforms the error $u_{t}$ into
$(u_{t}+v_{t}^{\prime}\theta_{x,i}^{0})$ through an orthogonal projection, so
that $-\theta_{x,i}^{0}$ plays the role of $\beta^{0}$ for each sub-sample. As
a result, when these two are equal, say over the first sample, the two
associated estimators TS2SLS and TSGMM are asymptotically equivalent over this
sample, but the TSGMM estimator will be more efficient over the second sample.

##  4 Two-sample GMM estimators with unknown change points

In this section, we first explain how to deal with an unknown change point in
the equation of interest. First, we propose a consistent estimator of the
associated change fraction, and a new test to detect its presence. Second, we
propose a new test to detect whether the change point in the first also shows
up in the second-stage, and extend our inference procedure to the presence of
multiple change points.

###  4.1 Unknown change point estimation

####  4.1.1 Estimation of the change fraction $\lambda^{0}$

We extend the results in Hall et al. (2012) to show that minimizing a 2SLS
criterion delivers consistent estimators of both the change fraction
$\lambda^{0}$ and the parameters $\theta_{vec}^{0}$ in the presence of near-
weak instruments. In the first-stage, $X_{t}$ is estimated over the full-
sample by OLS to get $\hat{X}_{t}=\hat{\Pi}^{\prime}Z_{t}$, and the augmented
projected regressors $\hat{W}_{t}=\mbox{vect }(Z_{1t},\hat{X}_{t})$. In the
second-stage, consider the following 2SLS criterion given a candidate change
point $\lfloor T\lambda\rfloor$ and parameters $\theta_{vec}$,

$\displaystyle Q_{2SLS}(\lambda,\theta_{vec})=\sum_{t=1}^{\lfloor
T\lambda\rfloor}\left(y_{t}-\hat{W}_{t}^{\prime}\theta_{1}\right)^{2}+\sum_{t=\lfloor
T\lambda\rfloor+1}^{T}\left(y_{t}-\hat{W}_{t}^{\prime}\theta_{2}\right)^{2},$
with $\displaystyle\theta_{i}=\mbox{vect }(\theta_{z,i},\
\theta_{x,i})\,,i=1,2\,.$

To get the 2SLS estimators for each candidate change fraction $\lambda$, first
concentrate with respect to $\theta_{vec}$ to get
$\hat{\theta}_{vec}(\lambda)$ and, then, minimize
$Q_{2SLS}(\lambda,\hat{\theta}_{vec}(\lambda))$ over all partitions of the
sample defined by $\lfloor T\lambda\rfloor$ to get

$\displaystyle\hat{\lambda}$ $\displaystyle=$
$\displaystyle\arg\min_{\lambda\in\Lambda_{\epsilon}}Q_{2SLS}(\lambda,\hat{\theta}_{vec}(\lambda))\quad\text{with}\quad\hat{\theta}_{vec}(\lambda)\equiv\mbox{vect
}(\hat{\theta}_{1}(\lambda),\hat{\theta}_{2}(\lambda)),$ (4.1)

and $\Lambda_{\epsilon}=[\epsilon,1-\epsilon]$ for some $\epsilon>0$, a cut-
off to ensure enough observations in each of the two samples. The associated
2SLS estimators of the change point $\hat{T}$ and parameters
$\hat{\theta}_{vec}$ are then defined as:

$\displaystyle\hat{T}=\lfloor T\hat{\lambda}\rfloor\hskip
7.11317pt\text{and}\hskip
7.11317pt\hat{\theta}_{vec}\equiv\hat{\theta}_{vec}(\hat{\lambda})\equiv\mbox{vect
}(\hat{\theta}_{1}(\hat{\lambda}),\hat{\theta}_{2}(\hat{\lambda}))\equiv\mbox{vect
}(\hat{\theta}_{1},\hat{\theta}_{2}).$

To derive the asymptotic properties of $\hat{\lambda}$, impose the following
regularity assumptions, which extend the known change-point assumptions in
Section 3 to all change point candidates.

###### Assumption 6

(Regularity of the change fraction)
The candidate change points satisfy $\max(\lfloor T\lambda\rfloor,T-\lfloor
T\lambda\rfloor)>\max(p-1,\epsilon T)$ for some $\epsilon>0$ such that
$\epsilon<\min(\lambda^{0},1-\lambda^{0})$. Accordingly,
$\lambda\in\Lambda_{\epsilon}=[\epsilon,1-\epsilon]$.

###### Assumption 7

(Regularity of the instrumental variables)
Let $\hat{Q}_{1}(r)=T^{-1}\sum_{t=1}^{\lfloor Tr\rfloor}Z_{t}Z_{t}^{\prime}$.
Then $\hat{Q}_{1}(r)\stackrel{{\scriptstyle p}}{{\rightarrow}}Q_{1}(r)$,
uniformly in $r\in[0,1]$ where $Q_{1}(r)$ is positive definite and strictly
increasing in $r$.

###### Assumption 8

(Regularity of the variances)

$\mbox{AVar}\left[T^{-1/2}\sum_{t=1}^{\lfloor
Tr\rfloor}h_{t}\right]=S_{1}(r)=\left(\begin{array}[]{cc}S_{u,1}(r)&S_{uv,1}^{\prime}(r)\\\
S_{uv,1}(r)&S_{v,1}(r)\\\ \end{array}\right),$

uniformly in $r\in[0,1]$, where $S_{1}(r)$ is positive definite and strictly
increasing in $r$, with $S_{u,1}(r)$, $S_{v,1}(r)$ of size $q\times q$,
respectively $(p_{2}q)\times(p_{2}q)$.

Assumptions 6, 7 and 8 are typical for the change point literature. Assumption
6 ensures that there are enough observations in each sub-sample to identify
the true change point. Assumption 7 ensures that there is enough variation in
the instruments to identify the change point. It also allows for the second
moment of instruments to change over the sample. Assumption 8 allows for
(unconditional) heteroskedasticity in the sample moments of the first and the
second-stage. It also allows for a change in the variance of the moment
conditions.

###### Theorem 3

(Asymptotic properties of $\hat{\lambda}$ and associated estimators)
Under Assumptions 1 to 3 and 6 to 8,
(i) $\|\hat{\lambda}-\lambda^{0}\|=\mathcal{O}_{P}(T^{-1})$.
(ii) the asymptotic distribution of the split-sample GMM, TS2SLS and TSGMM
estimators is as if the change point was known, hence as described in Theorem
1.

Note that the estimator of the change fraction $\hat{\lambda}$ converges at
the fastest available rate $T$. This stems from the presence of fixed changes
in the parameters of the exogenous regressors; intuitively, the exogenous
regressors act as their own strong instruments, and the strongest instruments
determine the fast convergence rate of $\hat{\lambda}$.

####  4.1.2 Detection of the change point

In practice, the existence of the change point in the second-stage equation
often needs to be established. To that end, we consider the sup-Wald test of
Hall et al. (2012) for which the null and alternative hypotheses are:
$H_{0}:\,\mathcal{R}_{p}\theta^{0}=0$ versus
$H_{A}:\,\mathcal{R}_{p}\theta^{0}\neq 0$, with $\mathcal{R}_{p}=(1,-1)\otimes
I_{p}$. The associated test statistic is defined as:

$Sup-
Wald_{T}\;=\;\underset{\lambda\in\Lambda_{\epsilon}}{\sup}Wald_{T}(\lambda),$
(4.2) where $\displaystyle
Wald_{T}(\lambda)=T\hat{\theta}_{vec}^{\prime}(\lambda)\mathcal{R}_{p}^{\prime}\left[\mathcal{R}_{p}\,\hat{G}(\lambda)\,\mathcal{R}_{p}^{\prime}\right]^{-1}\mathcal{R}_{p}\hat{\theta}_{vec}(\lambda)\,,$
$\displaystyle\hat{G}=\operatorname{diag}[\
\hat{G}_{1}(\lambda),\hat{G}_{2}(\lambda)]\quad\text{with }\
\hat{G}_{i}(\lambda)=\hat{\mathcal{A}}_{i}^{-1}(\lambda)\hat{H}_{i}(\lambda)\hat{\mathcal{A}}_{i}^{-1}(\lambda)\,,$
$\displaystyle\hat{\mathcal{A}}_{i}(\lambda)=T^{-1}\sum_{t\in
I_{i}}\hat{X}_{t}\hat{X}_{t}^{\prime}\quad\,,I_{1}=\\{1,\ldots,[T\lambda]\\}\,,I_{2}=\\{[T\lambda]+1,\ldots,T\\}\,,$
and $\hat{H}_{i}(\lambda)$ is a HAC estimator such that
$\displaystyle\Lambda_{T}^{-1}\hat{H}_{i}(\lambda)\Lambda_{T}^{-1}\stackrel{{\scriptstyle
p}}{{\rightarrow}}H_{i}(\lambda)=\mbox{AVar}\left[\sum_{t\in
I_{i}}\Lambda_{T}^{-1}\hat{X}_{t}(u_{t}+v_{t}^{\prime}\theta_{x,i}^{0})\right]\,.$

We impose the following additional assumption under the null hypothesis, which
maintains the same homogeneity assumption as Assumption 4, but with respect to
all candidate change points. Without this assumption, the asymptotic
distribution of the test is not pivotal and needs to be simulated or
bootstrapped - Boldea et al. (2019).

###### Assumption 9

(Homogeneity of the second moments)
For any $r\in[0,1]$, we have: (i) $Q_{1}(r)=rQ$; (ii) $S_{1}(r)=rS$.

The following theorem provides the limiting distribution of the sup-Wald test
statistic.

###### Theorem 4

(Test for a change in the equation of interest)
(i) Under $H_{0}:\theta_{1}^{0}=\theta_{2}^{0}$, and Assumptions 1 and 6 to 9,

$Sup-
Wald_{T}\;\Rightarrow\;\sup_{\lambda\in\Lambda_{\epsilon}}\frac{||\mathcal{B}_{p}(\lambda)-\lambda\mathcal{B}_{p}(1)||^{2}}{\lambda(1-\lambda)},$

where $\mathcal{B}_{p}(\lambda)$ is a $p\times 1$ vector of independent
standard Brownian motions defined on $[0,1]$.
(ii) Under $H_{A}:\theta_{\Delta}^{0}=\theta_{1}^{0}-\theta_{2}^{0}\neq 0$,
and Assumptions 1 to 3 and 6 to 8, the test statistic diverges such that

$Sup-
Wald_{T}=\left\\{\begin{array}[]{ll}\mathcal{O}_{p}(T/r_{T}^{2})&\text{without
exogenous regressors ($p_{1}=0$)}\\\ \mathcal{O}_{p}(T)&\text{in presence of
exogenous regressors ($p_{1}\neq 0$)}\end{array}\right.$

In addition, if Assumption 9 holds and either $p_{1}=0,p_{2}=1$ (only one
endogenous regressor, no exogenous regressors) or
$\theta_{x,1}^{0}=\theta_{x,2}^{0}$, then the implicit change fraction
estimator is consistent,
$\hat{\lambda}^{W}=\arg\sup_{\lambda\in\Lambda_{\epsilon}}Wald_{T}(\lambda)\stackrel{{\scriptstyle
p}}{{\rightarrow}}\lambda^{0}$. Otherwise,
$\hat{\lambda}^{W}\stackrel{{\scriptstyle p}}{{\rightarrow}}\lambda^{0}$ is
not guaranteed.

We recommend using the sup-Wald test to test for a change point because of its
robustness to conditional heteroskedasticity and autocorrelation. We also note
that if Assumption 9 is violated, then the implicit change fraction estimator
may not be consistent, a result which to our knowledge was not previously
available and may be of independent interest.

Note that the global power of the test depends on the presence of exogenous
regressors: without exogenous regressors, the rate of divergence is affected
by the identification strength of the instruments and is equal to
$T/r_{T}^{2}$; in presence of exogenous regressors, the rate is standard and
equal to $T$, and thus is not affected by the identification strength.

###  4.2 General framework

We now discuss a generalization of the model considered in Section 3 to allow
for changes in the parameters of first and second-stages,

$\displaystyle y_{t}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lclclccr}Z_{1t}^{\prime}\theta_{z,1}^{0}&+&X_{t}^{\prime}\theta_{x,1}^{0}&+&u_{t}&&,&t\leq\lfloor
T\lambda^{0}\rfloor\\\
Z_{1t}^{\prime}\theta_{z,2}^{0}&+&X_{t}^{\prime}\theta_{x,2}^{0}&+&u_{t}&&,&t>\lfloor
T\lambda^{0}\rfloor\end{array}\right.$ (4.5) $\displaystyle X_{t}^{\prime}$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lclccl}\frac{Z_{t}^{\prime}\Pi_{1}^{0}}{r_{1T}}&+&v_{t}^{\prime}&&,&t\leq\lfloor
T\nu^{0}\rfloor\\\
\frac{Z_{t}^{\prime}\Pi_{2}^{0}}{r_{2T}}&+&v_{t}^{\prime}&&,&t>\lfloor
T\nu^{0}\rfloor\\\ \end{array}\right.\,,$ (4.8)

where $r_{iT}=1$, or $r_{iT}\rightarrow\infty,r_{iT}=o(\sqrt{(}T))$, with
$i=1,2$, and $Z_{t}$ is not correlated with $v_{t}$ or $u_{t}$. When there is
no change in the identification strength (that is, $r_{1T}\propto r_{2T}$),
(4.8) naturally extends the RF models considered in Hall et al. (2012) to
weaker identification patterns. Otherwise, (4.8) captures changes in
identification strength that may or may not be concomitant to those in the
parameter of interest $\theta^{0}_{vec}$. Our goal is to detect and locate
both parameter instability and changes in the identification strength, as well
as to provide correct and sharp inference on $\theta^{0}_{vec}$.

####  4.2.1 Test for common change

This section proposes a Wald test for a common change. Consider the case where
the RF change $T^{*}=\lfloor T\nu^{0}\rfloor$ has been detected and estimated
consistently by $\hat{T}^{*}$ (for example, using the methods developed in
Antoine and Boldea (2018) or Bai and Perron (1998)). To test whether $T^{*}$
is also a change point in the equation of interest, we test whether the 2SLS
parameter estimates defined over each corresponding sub-sample are equal to
each other using a standard Wald test. These parameter estimates are defined
as follows for $i=1,2$, and $\hat{I}_{1}^{*}=\\{1,\ldots,\hat{T}^{*}\\}$ and
$\hat{I}_{2}^{*}=\\{\hat{T}^{*}+1,\ldots,T\\}$,

$\displaystyle\hat{\theta}_{i}^{c}=\left(\sum_{\hat{I}_{i}^{*}}\hat{W}_{t,i}\hat{W}_{t,i}^{\prime}\right)^{-1}\left(\sum_{\hat{I}_{i}^{*}}\hat{W}_{t,i}y_{t}\right)\,,$
with $\displaystyle\hat{W}_{t,i}=\mbox{vect
}(Z_{1t},\hat{X}_{t,i})\,,\quad\text{and}\quad\hat{X}_{t,i}^{\prime}=Z_{t}^{\prime}\hat{\Pi}_{i}\,.$

The Wald test statistic for a common change is defined as:

$Wald_{T}^{c}\,=\,T\,\hat{\theta}_{vec}^{c\prime}\,\mathcal{R}_{p}^{\prime}\,(\mathcal{R}_{p}\,\hat{G}^{c}\,\mathcal{R}_{p}^{\prime})^{-1}\,\mathcal{R}_{p}^{\prime}\,\hat{\theta}^{c}_{vec},$
with $\displaystyle\hat{\theta}_{vec}^{c}=\mbox{vect
}(\hat{\theta}^{c}_{1},\hat{\theta}^{c}_{2})\,,\quad\mathcal{R}_{p}=(1,-1)\otimes
I_{p}\,,$
$\displaystyle\hat{G}^{c}=\operatorname{diag}\left[\hat{G}_{1}^{c},\hat{G}_{2}^{c}\right]\,,\quad\hat{G}_{i}^{c}=(\hat{A}_{i}^{c})^{-1}\hat{B}_{i}^{c}(\hat{A}_{i}^{c})^{-1}\,,$
$\displaystyle\hat{A}_{i}^{c}=T^{-1}\sum_{\hat{I}_{i}^{*}}\hat{X}_{t,i}\hat{X}_{t,i}^{\prime}\,,\quad\Lambda_{iT}=\operatorname{diag}(T^{1/2}I_{p_{1}},T^{1/2}r_{iT}^{-1}I_{p_{2}})\,,$
and
$\displaystyle\Lambda_{iT}^{-1}\hat{B}_{i}^{c}\Lambda_{iT}^{-1}\stackrel{{\scriptstyle
p}}{{\rightarrow}}B_{i}^{c}=\mbox{AVar}[T^{-1/2}\sum_{\hat{I}_{i}^{*}}\Lambda_{iT}^{-1}\hat{X}_{t,i}u_{t}]\,.$

The following theorem provides the limiting distribution of the above Wald
test statistic.

###### Theorem 5

(Wald test for common change)
If $\hat{T}-[T\nu^{0}]=\mathcal{O}_{P}(1)$, then:
(i) Under $H_{0}:\theta_{1}^{0}=\theta_{2}^{0}$, Assumptions 1 to 3 and 6 to
9, $Wald_{T}^{c}\stackrel{{\scriptstyle d}}{{\rightarrow}}\chi^{2}_{p}$.
(ii) Under $H_{A}:\theta_{1}^{0}\neq\theta_{2}^{0}$, Assumptions 1 to 3 and 6
to 9,

$Wald_{T}^{c}=\left\\{\begin{array}[]{ll}\mathcal{O}_{p}(T)&\text{when
$p_{1}\neq 0$ (with exogenous regressors)}\\\
\mathcal{O}_{p}(T/\overline{r}_{T}^{2})&\text{when $p_{1}=0$ (no exog.
regressors) and $\overline{r}_{T}=\max_{i}(r_{iT})$.}\end{array}\right.$

Unlike the test in the previous section, the common change point test is
simpler, because it is computed directly at the estimated change point coming
from the first-stage; however, note that its the rate of divergence is
different in absence of exogenous regressors, since it depends on the weakest
identification rate across the two samples $\overline{r}_{T}$.

####  4.2.2 Extension to multiple changes and inference procedure

Both the estimation and the change point detection methods can be extended to
multiple changes. For detecting the number of change points, we recommend the
sequential testing methods in Hall et al. (2012). Since we have shown that the
sup-Wald test of zero versus one change in Hall et al. (2012) remains valid in
the presence of near-weak instruments, we conjecture that the sequential tests
remain valid as well. Similarly, the TSGMM estimators can be extended to
multiple changes in a straightforward way.

We now present a four-stage inference procedure that builds on results
developed in Sections 3 and 4.

  1. 1.

First stage
Find the number changes in the first-stage via the methods in Hall et al.
(2012). Find their location via the methods in Hall et al. (2012) or Antoine
and Boldea (2018) and collect them in the set $\mathcal{B}_{S1}$. Use
$\mathcal{B}_{S1}$ to partition the sample, and construct $\hat{X}_{t}$ by OLS
(over sub-samples, if $\mathcal{B}_{S1}$ is not empty, and over the full-
sample, if $\mathcal{B}_{S1}$ is empty).

  2. 2.

Second stage
Partition the sample using $\mathcal{B}_{S1}$, and work over the associated
sub-samples. Find the number of changes in the equation of interest for each
of these sub-samples, via a sequential version of the sup-Wald test presented
in Section 4.1.2. Estimate them via the change point estimator of Section 3,
extended to multiple changes, and collect these changes in the set
$\mathcal{B}_{S2,NC}$.

  3. 3.

Third stage: common changes.
Partition the sample into sub-samples according to $\mathcal{B}_{S1,NC}$
again. Test in each of these for common changes to both the first and the
second-stages, via the Wald test in Section 4.2.1. Collect the common changes
in the set $\mathcal{B}_{S2,C}$, and obtain $B_{S2}=B_{S2,NC}\ \cup B_{S2,C}$.

  4. 4.

Forth stage: estimation.
Partition the sample according to $B_{S1}$: each resulting sub-sample first-
stage is stable. If there are second-stage changes in any of these sub-samples
according to $\mathcal{B}_{S2}$, use the TSGMM estimators as described in
Section 3; otherwise, use the standard full-sample GMM estimator.

Of course, the multiple pre-testing steps can affect the procedure, and to
safeguard against this, we suggest using a Bonferroni correction.

##  5 Efficient estimators of the first-stage parameters

This section shows that, in the presence of second-stage change, the TSGMM
estimators of the first-stage $\hat{\Pi}_{TSGMM,vec}$ introduced in Section 3
are also more efficient than the full-sample OLS estimates and the split-
sample OLS estimates that ignore the second-stage change point. To formalize
this result, consider the following stable first-stage, where we are
interested in estimating $\Pi^{0}$:

$X_{t}=Z_{t}^{\prime}\Pi^{0}+v_{t}\,.$

For simplicity, we consider one endogenous regressor $X_{t}$, no additional
exogenous regressor ($p_{1}=0,p_{2}=1$), and we impose strong identification,
i.e. $r_{T}=1$. Therefore, we drop the “vec” subscript on all estimators.

The parameter $\Pi$ is stable, but we allow for second-stage changes, as well
as potential changes in $\verb'Var'(v_{t}Z_{t})$,
$\verb'E'(Z_{t}Z_{t}^{\prime})$, or in all, at $T^{0}$, which is assumed known
for simplicity. Notice that a change in $\verb'Var'(v_{t}Z_{t})$ at $T^{0}$
means that $\verb'Var'(v_{t}Z_{t})$ changes once at $t=T^{0}$: as a result,
Assumption 4(ii) is violated. A change in $\verb'E'(Z_{t}Z_{t}^{\prime})$
implies that $\verb'E'(Z_{t}Z_{t}^{\prime})$ changes once as $t=T^{0}$, so
Assumption 4(i) is violated.

We introduce a third estimator that ignores the second-stage moment conditions
and uses the OLS sub-sample moment conditions before and after the change -
$g_{T,2}(\Pi;\lambda_{0})$, defined in Section 3. It corresponds to the
optimal (two-step) GMM estimator that uses the $2q$ moments
$g_{T,2}(\Pi;\lambda_{0})$ to estimate $q$ parameters and we denote it
$\hat{\Pi}_{GMM}$. The theorem below shows that $\hat{\Pi}_{TSGMM}$ is at
least as efficient as the other two estimators.

###### Theorem 6

(Efficiency of first-stage estimators)
(i) Under Assumptions 1 to 3, $\sqrt{T}(\hat{\Pi}-\Pi^{0})$,
$\sqrt{T}(\hat{\Pi}_{GMM}-\Pi^{0})$, and $\sqrt{T}(\hat{\Pi}_{TSGMM}-\Pi^{0})$
are asymptotically normally distributed with mean zero and respective
asymptotic variance-covariance matrices:

$\displaystyle V_{OLS,\Pi}$ $\displaystyle=$
$\displaystyle(Q_{1}+Q_{2})^{-1}(S_{v,1}+S_{v,2})(Q_{1}+Q_{2})^{-1}$
$\displaystyle V_{GMM,\Pi}$ $\displaystyle=$
$\displaystyle(Q_{1}S_{v,1}^{-1}Q_{1}+Q_{2}S_{v,2}^{-1}Q_{2})^{-1}$
$\displaystyle V_{TSGMM,\Pi}$ $\displaystyle=$
$\displaystyle(V_{GMM,\Pi}^{-1}+\mathcal{H}_{*}^{\prime}\mathcal{E}_{*}^{-1/2}\mathcal{M}_{\mathcal{J}_{*}}\mathcal{E}_{*}^{-1/2}\mathcal{H}_{*})^{-1},$

where
$\mathcal{E}_{*}=\operatorname{diag}(S_{u,1}-S_{uv,1}^{\prime}S_{v,1}^{-1}S_{uv,1}\
,\ S_{u,2}-S_{uv,2}^{\prime}S_{v,2}^{-1}S_{uv,2})$,
and
$\mathcal{M}_{\mathcal{J}_{*}}=I-\mathcal{J}_{*}(\mathcal{J}_{*}^{\prime}\mathcal{J}_{*})^{-1}\mathcal{J}_{*}^{\prime},\mathcal{J}_{*}=\mathcal{E}_{*}^{-1/2}\Gamma_{1},\mathcal{H}=\operatorname{diag}(S_{uv,1}^{\prime}S_{v,1}^{-1},S_{uv,2}^{\prime}S_{v,2}^{-1})\
\Gamma_{2}$.
(ii) Under Assumptions 1 to 4: $V_{TSGMM,\Pi}\leq V_{GMM,\Pi}\leq
V_{OLS,\Pi}$.
(iii) Under Assumptions 1 to 5: $V_{TSGMM,\Pi}\leq V_{GMM,\Pi}=V_{OLS,\Pi}$.

The restriction of a common first-stage delivers not only more efficient GMM
estimators of the parameters of interest, but also of the first-stage
parameters. Since the first-stage is part of a system of equations linked
through correlation between $u_{t}$ and $v_{t}$, the OLS estimators of the
first-stage are no longer the most efficient in presence of changes in the
second-stage, or in second moments of the instruments or in the variance of
the OLS moment conditions. It is important to note that OLS estimators remain
the most efficient among the class of estimators that ignore the second-stage,
and as long as no second moments change over the sample. Consequently, our
results do not conflict with the Gauss-Markov Theorem or classical results on
asymptotic efficiency of OLS.

##  6 Monte-Carlo study

In this section, we examine by simulation the small sample behavior of the
split-sample GMM estimators, TS2SLS and TSGMM estimators when the change point
is known, when it is unknown and needs to be estimated, and when pre-testing
for a change point.

###  6.1 Data generating process

We consider the following model:

$\displaystyle y_{t}$
$\displaystyle=\begin{cases}Z_{1t}^{\prime}\theta_{1,z}^{0}+X_{t}^{\prime}\theta_{x,1}^{0}+\sigma_{t}u_{t},\quad,\quad
t\leq\lfloor T\lambda^{0}\rfloor\\\
Z_{1t}^{\prime}\theta_{2,z}^{0}+X_{t}^{\prime}\theta_{x,2}^{0}+\sigma_{t}u_{t},\quad,\quad
t>\lfloor T\lambda^{0}\rfloor\end{cases}$ (6.1) $\displaystyle X_{t}$
${}^{\prime}=Z_{t}^{\prime}\Pi+v_{t},$ (6.2)

with $p$ explanatory variables $W_{t}$, $p_{1}$ exogeneous variables $Z_{1t}$,
$p_{2}$ endogeneous variables $X_{t}$, and $p$ parameters of interest
$\theta^{0^{\prime}}$. $Z_{t}$ is a vector of $q$ instruments, including the
exogeneous regressors $Z_{1t}$. $\sigma_{t}$ determines whether the structural
errors are homoskedastic or heteroskedastic.

We set $p=2$ with $p_{1}=p_{2}=1$. The exogenous regressor $Z_{1t}$ is a
constant, and the number of external instruments $n_{IV}=q-1$ are set to 1 or
4. The sample sizes are $T=400$ and $T=800$, and we implement $N=1,000$
repetitions. The true change fraction is $\lambda^{0}=0.4$, thus the (true)
change location is either $\lfloor T\lambda^{0}\rfloor=160$ or 320 depending
on the sample size. The change size is assumed to be equal to one, and the
true parameters are

$\displaystyle\theta^{0}_{vec}=\text{vect}(\theta_{1,z}^{0},\theta_{1,x}^{0},\theta_{2,z}^{0},\theta_{2,x}^{0})=(0,0,1,1)^{\prime},\quad\Pi=(1,1),$

where $\theta_{2}^{0}-\theta_{1}^{0}=1$ is referred to as the “‘change size”
in the Appendix Section A Results. The errors are jointly generated as
follows:

$\displaystyle\left(\begin{array}[]{c}u_{t}\\\
v_{t}\end{array}\right)\sim\mathcal{N}\left(\left(\begin{array}[]{c}0\\\
0\end{array}\right),\left[\begin{array}[]{cc}1&\rho\\\
\rho&1\end{array}\right]\right)$

with correlation coefficient $\rho=-0.5$. The RMSE is computed using the sum
of the average Monte carlo bias and of the average asymptotics-based standard
deviation. 999This standard deviation is calculated by plugging in the
Appendix, Definition 1, consistent and heteroskedasticity-robust estimators of
the quantities that enter the asymptotic variance. as

$RMSE=\sqrt{\left(\frac{1}{N}\sum_{n=1}^{N}\text{Bias}(\hat{\theta}^{n}_{j,i})\right)^{2}+\left(\frac{1}{N}\sum_{i=1}^{N}\text{Asympt.Std}(\hat{\theta}^{n}_{j,i})\right)^{2}}$

for $n=1,\dots,N$ repetitions, where $\hat{\theta}^{n}_{i,j}$ is short-hand
for the estimators in repetition $n$ of the coefficients on the endogenous
regressors in samples $i=1,2$ and of type $j\in\\{GMM,TS2SLS,TSGMM\\}$. All
GMM estimators are computed with a 2SLS first-step estimator based on their
respective moment conditions.

HOM corresponds to conditional homoskedasticity and thus $\sigma_{t}=1$. HET1
corresponds to conditional heteroskedasticity given by
$\sigma_{t}^{2}=(1+(\sum_{t}^{T}Z_{t})^{2})/2$, and HET2 to a GARCH(1,1) model
with $\sigma_{t}^{2}=0.1+0.6\epsilon_{t-1}^{2}+0.3\sigma_{t-1}^{2}$, where
$\epsilon_{t}=\sigma_{t}u_{t}$. Under HET2, we employ a burn-in sample of 100
observations for $T=400$, and 200 observations for $T=800$.

The data generating process satisfies Assumptions 1-4 and 6-9, and also
Assumption 5 in the HOM case. Condition 3.13 in Theorem 2 holds with equality,
so the asymptotic distributions of the TS2SLS and of the split-sample GMM are
the same under HOM, as shown in Theorem 2, and the ranking of the two
estimators based on asymptotic variances is unclear under HET1-HET2. However,
condition 2 holds as stated in Theorem 2, so TSGMM is asymptotically strictly
more efficient than both the TS2SLS and the split-sample GMM estimators under
HOM, and at least as efficient as the other two estimators under HET1-HET2.

###  6.2 Results with a known change point

We first consider the case where the DGP displays one change point which is
fully known: that is, both the existence of the change and its location are
known. Estimation results for GMM, TS2SLS, and TSGMM are reported in Tables 1,
2, and 3 with sample sizes $T=400$ and $800$ under HOM, HET1 and HET2,
respectively. We report average Monte-Carlo bias, Monte-Carlo standard
deviations, as well as asymptotics-based standard deviations and RMSE of
estimators $\theta^{n}_{i,j}$ (for simplicity we drop the superscript $n$ in
all tables). In addition, we report the average length of a 95% confidence
interval computed with the asymptotics-based standard deviations, and the
associated coverage rates.

Overall, the three estimation procedures are well-behaved, and fairly close to
each other: all display small biases, and coverage rates close to the nominal
95%. In addition, Monte-Carlo standard deviations are close to their
asymptotics-based counterparts, but they are slightly higher, so there are
benefits to using the asymptotic formulas in Appendix Definition 1. The TS2SLS
estimator does not uniformly dominate the split-sample GMM estimators in terms
of efficiency: the two asymptotics-based standard deviations are close as
predicted by Theorem 2 in the HOM case, but with more instruments, the
asymptotic standard deviations of the TS2SLS becomes larger than that of the
split-sample GMM in both HOM and HET cases. In contrast, the TSGMM
consistently display lower standard deviations compared to both TS2SLS and
split-sample GMM, as predicted by Theorem 2 and implied by the asymptotic
variance formulas. The bias of the TSGMM is also small, and as a result, TSGMM
has also consistently lower RMSE than the other two procedures, while the
empirical coverage of the confidence intervals is still close to the nominal
size in most cases.

###  6.3 Results with an unknown change point

Next, we assume the researcher knows that there is one change point, but
doesn’t know its location and estimates it via the method proposed in Section
4. To ensure there are enough observations in each sub-sample, the candidate
change points are taken to be between $[0.15T]$ and $[0.85T]$.

In Figure 1, we display the histograms of estimated change points with sample
sizes $T=400$ and $800$ under HOM (top row), HET1 (middle row), and HET2
(bottom row). Note that the change points are estimated accurately in the HOM
and HET1 cases, and are less accurate in the HET2 case due to the GARCH(1,1)
process chosen to be close to a unit root.

Estimation results are reported in Tables 4, 5, and 6 with sample sizes
$T=400$ and $800$ under HOM, and HET1-HET2. We also report the Monte-Carlo
mean of the associated estimated change points. Due to estimation error from
the change points, the differences between the three estimators are smaller,
but the TSGMM continues to consistently display lower standard deviations and
RMSE than the other two estimates.

As a robustness check, we also consider the case where there is no change
point in the DGP, but we estimate a change point, and we set $T=400$.
Estimation results are reported in Table 7, while histograms of associated
estimated change points are displayed in Figure 2. In all cases, the TSGMM
still has lower standard errors and RMSE than the other estimates, and its
coverage slightly deteriorates in the HET cases, which is to be expected due
to more moment conditions than the other estimators and a small sample size.

###  6.4 Detection of change point

Finally, we present results related to the detection of the change point.
Table 8 displays the probability of detecting a change, given that the DGP has
one change point, using the sup-Wald test in Section 4.1.2. The null
hypothesis of the test is that there is no change point in the data. The
critical values of limiting distribution of the test are from Bai and Perron
(2003). As can be seen from Table 8, the empirical size of the test is close
to the nominal size, and the power is also very good as long as the magnitude
of the change in both coefficients is large enough.

In Table 9, we pre-test for a change point, and subsequently estimate the
parameters. If the test rejects the null, we estimate the location of the
change point, and the parameters over the two sub-samples. If the test fails
to reject, we estimate the parameters over the whole sample. We report Monte-
Carlo bias, standard deviation, as well as asymptotic standard deviation
(computed using the asymptotic theory), and RMSE: each measure is obtained as
a weighted average of the corresponding quantity computed over each sub-
sample. We see that pre-testing does not affect the efficiency of the TS2GMM
estimators, which still displays lower standard deviations and RMSE. However,
it affects coverage for all estimators in small samples such as $T=400$,
especially in the case HET2, where the change point is estimated less
accurately.

##  7 Estimation of the NKPC in an open economy

The New Keynesian Phillips Curve (NKPC) played an important role in monetary
policy analysis: in an open economy, it relates inflation to the movements of
business cycles and expected future exchange rates. In this context, one
usually assumes instantaneous import price adjustment and full transformation
of exchange rate movements into consumer prices. However, it has been widely
documented that the exchange rate pass-through - that is, the real marginal
cost of importers - into consumer and import prices is incomplete, at least
for US and other developed countries -Campa and Goldberg (2005). Accordingly,
we focus here on estimating the NKPC under imperfect exchange rate pass-
through (as in Monacelli (2005)) which relates inflation to a domestic driving
force variable (e.g. unemployment rate) in addition to the pass-through rate
measured as the difference between the exchange rate and the terms of trade:

$\pi_{t}=\gamma_{f}\pi_{t+1}^{e}+\kappa_{u}u_{t}+\kappa_{\Psi}\Psi_{t}+\beta_{0}$
(7.1)

where $\pi_{t}$ is the consumer price inflation, $\pi_{t+1}^{e}$ is the
expected future inflation, $u_{t}$ is the unemployment rate, and $\Psi_{t}$ is
the pass-through rate.

Empirically, there does not seem to be a consensus on the exact definition of
the exchange rate pass-through, and whether - and how - it might affect the
Phillips curve. In the context of an incomplete pass-through, an increase in
the law of one price gap, i.e. $\kappa_{\Psi}$, causes an increase in
inflation, similar to the one caused by a cost-push shock. Hence, an
incomplete exchange rate pass-through could have relevant consequences for
monetary policy. At the same time, there is also some evidence suggesting that
the exchange rate pass-through has been declining, which brings into question
whether it should be taken into account in any policy consideration (see e.g.
Taylor (2000)).

Using monthly US data from 1990 to 2020, we find strong support for the NKPC
with imperfect pass-through. The results in the next section suggest that the
exchange rate pass-through plays a statistically significant role which is
comparable, in magnitude, to that of unemployment - at least over part of the
sample. We also find that relationship between inflation and exchange rate
pass-through exhibits important time variation.

###  7.1 Main empirical analysis

Our main specification considers the following NKPC model with imperfect pass-
through,

$\pi_{t}=\gamma_{f}\pi_{t+1}^{e}+\gamma_{1}\pi_{t-1}+\gamma_{2}\pi_{t-2}+\kappa_{u}u_{t}+\kappa_{\Psi}\Psi_{t}+\beta_{0}$
(7.2)

Notice that two lags of inflation are included as controls. To control for
potential endogeneity of (future) expected inflation and the unemployment
rate, we consider two first-stages with the following four additional
instruments: one lag of expected inflation, unemployment rate, interest rate
spread, and terms of trade. All variables are first de-trended using the
Hodrick-Prescott filter, and we only consider their cyclical component. Our
sample size consists of monthly observations over the period 1983m9-2020m4,
for a total of 440 observations. The data is:

  * •

inflation $\pi_{t}$: Consumer Price Index for All Urban Consumers: All Items
in U.S. City Average , Percent Change from Year Ago, Monthly, Seasonally
Adjusted.

  * •

expected inflation $\pi_{t+1}^{e}$: Michigan Consumer Survey, expected
inflation one-year ahead.

  * •

unemployment rate $u_{t}$: Unemployment Rate, Percent Change from Year Ago,
Monthly, Seasonally Adjusted.

  * •

pass-through rate $\Psi_{t}$: $q_{t}-(1-\gamma)s_{t}$, where $\gamma$ is
assumed to be 0.25, following Galí and Monacelli (2005) and Abbas (2023). The
pass-through rate is the difference between the real exchange rate $q_{t}$ and
the terms of trade $s_{t}$, with $\gamma$ indicating the degree of openness of
the economy.

  * •

real exchange rate $q_{t}$: US Exchange Rate: Relative Consumer Price Indices,
index with year 2010 as base 100, monthly.

  * •

terms of trade $s_{t}$ (partially interpolated): ratio between Export Price
Index (All Commodities) and Import Price Index (All Commodities). Both
variables are available monthly, starting from 1982-09-01 and 1983-09-01,
respectively. However, they both have missing values for the first few years,
hence we use cubic splines to interpolate the missing values.

  * •

interest rate spread: difference between 3-Months Treasury Bill Secondary
Market and Market Yield on U.S. Treasury Securities at 5-Year Constant
Maturity.

Figure 3 plots the time series of inflation and expected inflation,
unemployment rate, terms of trade, and real exchange rate.

We first estimate the change points in the two first-stage equations which we
call for simplicity reduced forms henceforth (RFs), i.e. expected inflation
and unemployment rate, using the Stata package Ditzen et al. (2021) that
implements Bai and Perron (2003)’s algorithm. Then, we consider the segments
over which both RFs are stable, and test in each for a change point in the
equation of interest using the sup-Wald test described in Section 4.1; if the
test rejects, wee estimate the change points in (7.2) over each such segment
using the estimator described in Section 3.101010Recall that both the sup-Wald
test employed and the change-point estimator is the same as in Hall et al.
(2012), and in this paper we showed that they are valid in the presence of
near-weak identification. Once the change location has been estimated, we
estimate the model parameters in (7.2) using either TS2SLS, split-sample GMM,
or TSGMM. The results are displayed in Tables 10 to 12.

In Table 10, we report the change point estimation results obtained over each
RF, respectively. The RF for expected inflation has one change at 2009m3
(significant at 1%), while the RF for unemployment rate has two changes,
respectively, at 2001m10 and 2008m12. Given the proximity of the first change
points obtained in each reduced form, it seems plausible and reasonable to
interpret them as the very same change. Accordingly, we consider the following
three segments over which both reduced forms are stable:
(i) 1983m9 – 2001m10; (ii) 2001m11 – 2009m3; and (iii) 2009m4 – 2020m4.

Table 11 reports the change point estimation results obtained in the
structural equation (equation of interest) using the proposed sup-Wald test in
Section 4.1; this analysis is conducted over each of the three aforementioned
(stable) RF segments. We find one change point in 1983m9 – 2001m10 significant
at $1\%$, one in 2001m11 – 2009m3 significant only at $5\%$, and no change
over the last period 2009m4 – 2020m4. Given the small sample size of the
second segment 2001m11 – 2009m3 (less than 90 observations) - which should
then be further split into two sub-periods - we limit our remaining analysis
to the first segment 1983m9 – 2001m9.111111The remaining results are presented
in the Supplementary appendix. The corresponding change point in the SF is
estimated at 1992m4 which is significant at 1%.

Finally, Table 12 collects the estimation results of the structural parameters
obtained by TS2SLS, GMM, and TSGMM over the two corresponding sub-samples, (i)
1983m9-1992m4 and (ii) 1992m5-2001m10, as well as GMM estimation results
obtained over the whole segment 1983m9-2001m10 (which effectively ignores the
change point discovered in the SF). These results are discussed in the next
subsection.

###  7.2 Discussion of the empirical results

The empirical analysis of the open-economy NKPC reveals three important
features with key policy implications: (i) the detection of structural change,
(ii) more precise estimation of the parameters of interest, and (iii)
significant time variation in these parameters of interest. If the structural
change discovered by our test was ignored, traditional GMM estimates -
computed over the whole sample - would imply a flat Phillips curve with
respect to unemployment rate, and limited importance of the exchange rate
pass-through: indeed, as can be seen from column (7) in Table 12, the estimate
of $\kappa_{u}$ is not found to be statistically significant, while the
estimate of $\kappa_{\Psi}$ is quite small. Instead, once the (structural)
change point is detected and estimated, and the sample is split accordingly,
both $\kappa_{u}$ and $\kappa_{\Psi}$ become statistically significant over
the second interval 1992m5–2001m9. Importantly, both estimates are
economically meaningful, e.g. $\kappa_{u}<0$ and $\kappa_{\Psi}>0$, and
somewhat comparable in terms of magnitude; see columns (5) and (6) in Table 12
for estimation results obtained with split-sample GMM and TSGMM respectively.
In addition, the estimate of $\kappa_{\Psi}$ is also significantly positive
over the first sample, though its magnitude is more modest. Noticeably, the
magnitude of each $\kappa_{\Psi}$ estimate obtained over a sub-sample remains
much larger than the (misleading) estimate obtained by GMM over the whole
sample: TSGMM estimates are respectively 0.0175 over 1983m9–1992m4 and 0.0266
over 1992m5-2001m9, while the full-sample GMM estimate is 0.0111.

Our analysis also suggests time-variation in the relationship between
inflation and unemployment, and this is not surprising, as it is a prevalent
finding in the NKPC literature. We also provide evidence of time-variation
between inflation and the exchange rate pass-through - confirming Abbas
(2023)’s findings - and the magnitude of both parameters increases from the
first sub-sample to the second. When comparing sub-sample estimates obtained
by TSGMM and GMM, both sets of estimates are relatively close to each other:
for example, over the second sub-sample, $\kappa_{\Psi}$ is estimated at
0.0266 with TSGMM and 0.0267 with GMM, while corresponding estimates for
$\kappa_{u}$ are -0.0427 and -0.0434, respectively; see columns (2), (3), (5)
and (6) in Table 12. One key difference between these two sets of estimates
relates to their precision: TSGMM delivers more precise estimates throughout.
For example, $\kappa_{\Psi}$ and $\kappa_{u}$ are significant at 1% with
TSGMM, and not with GMM. This is in line with our theoretical results which
show that TSGMM is more efficient than GMM as long as there is any change in
the structural equation over a stable reduced form. Intuitively, TSGMM
exploits (useful) information that is neglected by other estimators such as
split-sample GMM. Interestingly, with TSGMM, the 95%-confidence intervals
obtained for both $\kappa_{\Psi}$ and $\kappa_{u}$ over the two samples do not
intersect, suggesting significant changes in the relationship between
inflation and the exchange rate pass-through and between inflation and
unemployment. Specifically, the relation between inflation and unemployment
and inflation and exchange rate becomes more important in the second sample,
where the estimate of the exchange rate slope increases by over 50%.

Overall, the empirical analysis provides strong support for the NKPC with
imperfect pass-through.

###  7.3 Robustness checks

$\bullet$ Concerns about weak identification.
In Table 13, we report results obtained by the Kleibergen-Paap rank
test121212The traditional test of weak identification based on the F-test and
the rule-of-thumb only applies when there is one endogenous variable. for
different sub-samples. The matrix of reduced form coefficients is of size
$(2,4)$ \- since we have two reduced forms and four instruments - and we test
the null of a rank equal to 1 over each stable RF sub-sample: the associated
critical values are from the $\xi^{2}$ distribution with 3 degrees of freedom.
The null of under-identification is rejected at 5% over the first two sub-
samples, 1983m9 – 2001m10 and 2001m11 – 2009m3, while it is borderline at 10%
over the last sub-sample, 2009m4 – 2020m4. Overall, the identification seems
to be sufficiently strong to support our estimation results.

$\bullet$ Alternative specification.
We consider alternative choices of instruments: specifically, we report
results obtained with (i) only two instruments (one lag of unemployment rate
and expected inflation), as well as with (ii) five instruments (one lag of
unemployment rate, expected inflation, real exchange rate, terms of trade,
interest rate spread). These results are reported Tables 14 to 17 and Tables
18 to 21, respectively. Overall, these confirm our main results: the estimates
obtained with TSGMM are more precise and the parameters of interest are
subject to significant changes over the sample.

##  8 Conclusion

In this paper, we considered a linear model with endogenous regressors. We
first showed how to exploit the stability of the first-stage to deliver more
efficient estimators of the parameters of interest, even in the presence of
near-weak instruments. We also derived the limiting distributions of two tests
that detect parameter changes in the equation of interest: the first one
applies when the first-stage is stable, and the second one when the first-
stage exhibits a change point that is common to the equation of interest.
Third, we also showed how to exploit information from the equation of interest
to obtain more efficient estimators of the (stable) first-stage parameters
than the standard full-sample OLS and showed how using the TSGMM estimator can
provide tighter confidence intervals in an application to the open-economy
NKPC.

While the paper is focused on the efficiency of TSGMM, the moment conditions
of the TSGMM estimator could also be used for identification purposes, and
therefore our paper contributes to a large literature tackling identification
through stability restrictions - Rigobon (2003), Magnusson and Mavroeidis
(2014), Angelini et al. (2024), Angelini et al. (2024).

## Acknowledgement

We would like to thank conference and seminar participants for valuable
comments. B. Antoine acknowledges financial support from Social Sciences and
Humanities Research Council of Canada (SSHRC). A previous version of this
paper was circulated under the title “Inference in Linear Models with
Structural Changes and Mixed Identification Strength”.

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Appendix

##  Appendix A Results from the Monte-Carlo study

###  A.1 Known change point

Estimator | Bias | MC Std | As. Std. | RMSE | Length | Coverage
---|---|---|---|---|---|---
$T=400$, $\text{$n_{IV}$}=1$ |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0035 | 0.0818 | 0.0792 | 0.0793 | 0.3106 | 0.9400
$\theta_{TS2SLS,1}$ | 0.0015 | 0.0808 | 0.0786 | 0.0786 | 0.3080 | 0.9390
$\theta_{TSGMM,1}$ | -0.0008 | 0.0749 | 0.0724 | 0.0724 | 0.2838 | 0.9330
$\theta_{GMM,2}$ | 0.0009 | 0.0649 | 0.0646 | 0.0646 | 0.2532 | 0.9520
$\theta_{TS2SLS,2}$ | -0.0004 | 0.0634 | 0.0644 | 0.0644 | 0.2525 | 0.9550
$\theta_{TSGMM,2}$ | -0.0012 | 0.0607 | 0.0609 | 0.0609 | 0.2388 | 0.9570
$T=400$, $\text{$n_{IV}$}=4$ |  |  |  |  |  |
$\theta_{GMM,1}$ | -0.0008 | 0.0393 | 0.0384 | 0.0384 | 0.1506 | 0.9310
$\theta_{TS2SLS,1}$ | -0.0000 | 0.0385 | 0.0391 | 0.0391 | 0.1532 | 0.9420
$\theta_{TSGMM,1}$ | -0.0004 | 0.0371 | 0.0351 | 0.0351 | 0.1376 | 0.9250
$\theta_{GMM,2}$ | -0.0010 | 0.0329 | 0.0318 | 0.0318 | 0.1245 | 0.9320
$\theta_{TS2SLS,2}$ | -0.0009 | 0.0325 | 0.0321 | 0.0321 | 0.1260 | 0.9360
$\theta_{TSGMM,2}$ | -0.0010 | 0.0315 | 0.0299 | 0.0299 | 0.1173 | 0.9320
$T=800$, $\text{$n_{IV}$}=1$ |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0016 | 0.0573 | 0.0560 | 0.0561 | 0.2197 | 0.9440
$\theta_{TS2SLS,1}$ | 0.0004 | 0.0570 | 0.0559 | 0.0559 | 0.2190 | 0.9440
$\theta_{TSGMM,1}$ | 0.0003 | 0.0514 | 0.0514 | 0.0514 | 0.2013 | 0.9530
$\theta_{GMM,2}$ | 0.0034 | 0.0469 | 0.0459 | 0.0460 | 0.1800 | 0.9460
$\theta_{TS2SLS,2}$ | 0.0021 | 0.0471 | 0.0457 | 0.0458 | 0.1793 | 0.9480
$\theta_{TSGMM,2}$ | 0.0022 | 0.0441 | 0.0434 | 0.0435 | 0.1701 | 0.9440
$T=800$, $\text{$n_{IV}$}=4$ |  |  |  |  |  |
$\theta_{GMM,1}$ | -0.0016 | 0.0292 | 0.0275 | 0.0276 | 0.1078 | 0.9340
$\theta_{TS2SLS,1}$ | -0.0012 | 0.0289 | 0.0278 | 0.0278 | 0.1089 | 0.9400
$\theta_{TSGMM,1}$ | -0.0011 | 0.0272 | 0.0252 | 0.0252 | 0.0989 | 0.9340
$\theta_{GMM,2}$ | 0.0004 | 0.0229 | 0.0226 | 0.0226 | 0.0885 | 0.9510
$\theta_{TS2SLS,2}$ | 0.0004 | 0.0231 | 0.0227 | 0.0227 | 0.0891 | 0.9440
$\theta_{TSGMM,2}$ | 0.0003 | 0.0219 | 0.0213 | 0.0213 | 0.0836 | 0.9340

Table 1: Performance of the GMM, TS2SLS and TSGMM estimators under
homoskedasticity (HOM) when the change point is known. We consider either 1 or
4 instruments with a sample size of either 400 or 800.

Estimator | Bias | MC Std | As. Std. | RMSE | Length | Coverage
---|---|---|---|---|---|---
$T=400$, $\text{$n_{IV}$}=1$ |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0062 | 0.1145 | 0.1096 | 0.1098 | 0.4298 | 0.9340
$\theta_{TS2SLS,1}$ | 0.0036 | 0.1133 | 0.1087 | 0.1088 | 0.4262 | 0.9340
$\theta_{TSGMM,1}$ | 0.0007 | 0.1065 | 0.1005 | 0.1005 | 0.3940 | 0.9300
$\theta_{GMM,2}$ | 0.0011 | 0.0921 | 0.0896 | 0.0896 | 0.3510 | 0.9390
$\theta_{TS2SLS,2}$ | -0.0006 | 0.0867 | 0.0862 | 0.0862 | 0.3378 | 0.9500
$\theta_{TSGMM,2}$ | -0.0016 | 0.0865 | 0.0847 | 0.0847 | 0.3320 | 0.9440
$T=400$, $\text{$n_{IV}$}=4$ |  |  |  |  |  |
$\theta_{GMM,1}$ | -0.0023 | 0.0987 | 0.0907 | 0.0908 | 0.3557 | 0.9130
$\theta_{TS2SLS,1}$ | 0.0028 | 0.0978 | 0.0967 | 0.0967 | 0.3789 | 0.9340
$\theta_{TSGMM,1}$ | -0.0004 | 0.0939 | 0.0837 | 0.0837 | 0.3279 | 0.9180
$\theta_{GMM,2}$ | -0.0050 | 0.0811 | 0.0765 | 0.0766 | 0.2998 | 0.9260
$\theta_{TS2SLS,2}$ | -0.0013 | 0.0758 | 0.0767 | 0.0767 | 0.3006 | 0.9440
$\theta_{TSGMM,2}$ | -0.0043 | 0.0776 | 0.0725 | 0.0726 | 0.2842 | 0.9270
$T=800$, $\text{$n_{IV}$}=1$ |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0018 | 0.0814 | 0.0784 | 0.0784 | 0.3074 | 0.9340
$\theta_{TS2SLS,1}$ | 0.0002 | 0.0810 | 0.0782 | 0.0782 | 0.3065 | 0.9320
$\theta_{TSGMM,1}$ | 0.0001 | 0.0741 | 0.0722 | 0.0722 | 0.2830 | 0.9400
$\theta_{GMM,2}$ | 0.0044 | 0.0651 | 0.0644 | 0.0646 | 0.2525 | 0.9420
$\theta_{TS2SLS,2}$ | 0.0030 | 0.0628 | 0.0619 | 0.0620 | 0.2428 | 0.9430
$\theta_{TSGMM,2}$ | 0.0028 | 0.0616 | 0.0611 | 0.0612 | 0.2397 | 0.9430
$T=800$, $\text{$n_{IV}$}=4$ |  |  |  |  |  |
$\theta_{GMM,1}$ | -0.0046 | 0.0736 | 0.0671 | 0.0672 | 0.2630 | 0.9250
$\theta_{TS2SLS,1}$ | -0.0022 | 0.0731 | 0.0695 | 0.0696 | 0.2725 | 0.9450
$\theta_{TSGMM,1}$ | -0.0027 | 0.0698 | 0.0620 | 0.0621 | 0.2432 | 0.9190
$\theta_{GMM,2}$ | 0.0003 | 0.0580 | 0.0557 | 0.0557 | 0.2182 | 0.9380
$\theta_{TS2SLS,2}$ | 0.0014 | 0.0553 | 0.0546 | 0.0546 | 0.2140 | 0.9390
$\theta_{TSGMM,2}$ | 0.0005 | 0.0554 | 0.0529 | 0.0529 | 0.2075 | 0.9350

Table 2: Performance of the GMM, TS2SLS and TSGMM estimators under
heteroskedasticity of type 1 (HET1) when the change point is known. We
consider either 1 or 4 instruments with a sample size of either 400 or 800.

Estimator | Bias | MC Std | As. Std. | RMSE | Length | Coverage
---|---|---|---|---|---|---
$T=400$, $\text{$n_{IV}$}=1$ |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0062 | 0.0792 | 0.0768 | 0.0770 | 0.3009 | 0.9460
$\theta_{TS2SLS,1}$ | 0.0047 | 0.0784 | 0.0762 | 0.0763 | 0.2987 | 0.9440
$\theta_{TSGMM,1}$ | 0.0037 | 0.0750 | 0.0719 | 0.0720 | 0.2819 | 0.9420
$\theta_{GMM,2}$ | -0.0019 | 0.4586 | 0.4242 | 0.4242 | 1.6629 | 0.9550
$\theta_{TS2SLS,2}$ | -0.0059 | 0.4558 | 0.4193 | 0.4194 | 1.6438 | 0.9530
$\theta_{TSGMM,2}$ | -0.0091 | 0.4572 | 0.4187 | 0.4188 | 1.6411 | 0.9540
$T=400$, $\text{$n_{IV}$}=4$ |  |  |  |  |  |
$\theta_{GMM,1}$ | -0.0027 | 0.0366 | 0.0348 | 0.0349 | 0.1366 | 0.9440
$\theta_{TS2SLS,1}$ | -0.0017 | 0.0394 | 0.0375 | 0.0375 | 0.1470 | 0.9520
$\theta_{TSGMM,1}$ | -0.0027 | 0.0350 | 0.0325 | 0.0326 | 0.1275 | 0.9280
$\theta_{GMM,2}$ | -0.0001 | 0.1893 | 0.1675 | 0.1675 | 0.6565 | 0.9390
$\theta_{TS2SLS,2}$ | -0.0071 | 0.2341 | 0.2155 | 0.2157 | 0.8449 | 0.9600
$\theta_{TSGMM,2}$ | 0.0002 | 0.1886 | 0.1657 | 0.1657 | 0.6494 | 0.9400
$T=800$, $\text{$n_{IV}$}=1$ |  |  |  |  |  |
$\theta_{GMM,1}$ | -0.0091 | 0.3501 | 0.3164 | 0.3166 | 1.2403 | 0.9590
$\theta_{TS2SLS,1}$ | -0.0108 | 0.3488 | 0.3159 | 0.3161 | 1.2384 | 0.9580
$\theta_{TSGMM,1}$ | -0.0129 | 0.3471 | 0.3113 | 0.3116 | 1.2203 | 0.9580
$\theta_{GMM,2}$ | 0.0034 | 0.0601 | 0.0588 | 0.0589 | 0.2304 | 0.9510
$\theta_{TS2SLS,2}$ | 0.0021 | 0.0616 | 0.0595 | 0.0595 | 0.2332 | 0.9480
$\theta_{TSGMM,2}$ | 0.0022 | 0.0590 | 0.0572 | 0.0572 | 0.2241 | 0.9420
$T=800$, $\text{$n_{IV}$}=4$ |  |  |  |  |  |
$\theta_{GMM,1}$ | -0.0001 | 0.1275 | 0.1220 | 0.1220 | 0.4780 | 0.9590
$\theta_{TS2SLS,1}$ | -0.0052 | 0.1702 | 0.1581 | 0.1582 | 0.6198 | 0.9700
$\theta_{TSGMM,1}$ | 0.0010 | 0.1289 | 0.1200 | 0.1200 | 0.4702 | 0.9510
$\theta_{GMM,2}$ | 0.0015 | 0.0287 | 0.0275 | 0.0276 | 0.1079 | 0.9410
$\theta_{TS2SLS,2}$ | 0.0002 | 0.0313 | 0.0302 | 0.0302 | 0.1183 | 0.9490
$\theta_{TSGMM,2}$ | 0.0012 | 0.0278 | 0.0267 | 0.0267 | 0.1047 | 0.9460

Table 3: Performance of the GMM, TS2SLS and TSGMM estimators under
heteroskedasticity of type 2 (HET2) when the change point is known. We
consider either 1 or 4 instruments with a sample size of either 400 or 800.

###  A.2 Unknown change point

Figure 1: Histograms of the estimated change point for $\lambda^{0}=0.4$ and
$\text{$n_{IV}$}=4$.

Estimator | Bias | MC Std | As. Std. | RMSE | Length | Coverage
---|---|---|---|---|---|---
$T=400$, $\text{$n_{IV}$}=1$ |  |  |  |  |  |
Estimated change location: 161.021 |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0033 | 0.0819 | 0.0795 | 0.0796 | 0.3116 | 0.9460
$\theta_{TS2SLS,1}$ | 0.0013 | 0.0810 | 0.0788 | 0.0789 | 0.3091 | 0.9410
$\theta_{TSGMM,1}$ | -0.0009 | 0.0751 | 0.0726 | 0.0726 | 0.2845 | 0.9370
$\theta_{GMM,2}$ | -0.0031 | 0.0655 | 0.0655 | 0.0656 | 0.2567 | 0.9490
$\theta_{TS2SLS,2}$ | -0.0044 | 0.0639 | 0.0653 | 0.0654 | 0.2558 | 0.9480
$\theta_{TSGMM,2}$ | -0.0051 | 0.0612 | 0.0619 | 0.0621 | 0.2426 | 0.9460
$T=400$, $\text{$n_{IV}$}=4$ |  |  |  |  |  |
Estimated change location: 160.912 |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0042 | 0.0404 | 0.0397 | 0.0399 | 0.1557 | 0.9340
$\theta_{TS2SLS,1}$ | 0.0065 | 0.0400 | 0.0408 | 0.0413 | 0.1599 | 0.9410
$\theta_{TSGMM,1}$ | 0.0044 | 0.0384 | 0.0365 | 0.0367 | 0.1430 | 0.9260
$\theta_{GMM,2}$ | -0.0010 | 0.0330 | 0.0318 | 0.0318 | 0.1248 | 0.9350
$\theta_{TS2SLS,2}$ | -0.0008 | 0.0326 | 0.0322 | 0.0322 | 0.1263 | 0.9400
$\theta_{TSGMM,2}$ | -0.0009 | 0.0316 | 0.0300 | 0.0300 | 0.1175 | 0.9350
$T=800$, $\text{$n_{IV}$}=1$ |  |  |  |  |  |
Estimated change location: 321.038 |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0049 | 0.0575 | 0.0566 | 0.0568 | 0.2218 | 0.9490
$\theta_{TS2SLS,1}$ | 0.0037 | 0.0571 | 0.0564 | 0.0565 | 0.2209 | 0.9420
$\theta_{TSGMM,1}$ | 0.0035 | 0.0513 | 0.0520 | 0.0521 | 0.2037 | 0.9500
$\theta_{GMM,2}$ | 0.0035 | 0.0469 | 0.0459 | 0.0461 | 0.1801 | 0.9440
$\theta_{TS2SLS,2}$ | 0.0022 | 0.0472 | 0.0458 | 0.0458 | 0.1795 | 0.9470
$\theta_{TSGMM,2}$ | 0.0022 | 0.0442 | 0.0434 | 0.0435 | 0.1702 | 0.9400
$T=800$, $\text{$n_{IV}$}=4$ |  |  |  |  |  |
Estimated change location: 320.983 |  |  |  |  |  |
$\theta_{GMM,1}$ | -0.0016 | 0.0292 | 0.0275 | 0.0276 | 0.1078 | 0.9340
$\theta_{TS2SLS,1}$ | -0.0012 | 0.0289 | 0.0278 | 0.0278 | 0.1089 | 0.9400
$\theta_{TSGMM,1}$ | -0.0011 | 0.0272 | 0.0252 | 0.0252 | 0.0989 | 0.9340
$\theta_{GMM,2}$ | 0.0004 | 0.0229 | 0.0226 | 0.0226 | 0.0885 | 0.9510
$\theta_{TS2SLS,2}$ | 0.0004 | 0.0231 | 0.0227 | 0.0227 | 0.0891 | 0.9440
$\theta_{TSGMM,2}$ | 0.0003 | 0.0219 | 0.0213 | 0.0213 | 0.0836 | 0.9340

Table 4: Performance of the GMM, TS2SLS and TSGMM estimators under
homoskedasticity (HOM) when the change point is estimated. We consider either
1 or 4 instruments with a sample size of either 400 or 800.

Estimator | Bias | MC Std | As. Std. | RMSE | Length | Coverage
---|---|---|---|---|---|---
$T=400$, $\text{$n_{IV}$}=1$ |  |  |  |  |  |
Estimated change point: 161.071 |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0060 | 0.1147 | 0.1100 | 0.1101 | 0.4311 | 0.9380
$\theta_{TS2SLS,1}$ | 0.0034 | 0.1136 | 0.1091 | 0.1091 | 0.4276 | 0.9370
$\theta_{TSGMM,1}$ | 0.0006 | 0.1067 | 0.1008 | 0.1008 | 0.3950 | 0.9330
$\theta_{GMM,2}$ | -0.0029 | 0.0923 | 0.0901 | 0.0901 | 0.3531 | 0.9370
$\theta_{TS2SLS,2}$ | -0.0045 | 0.0869 | 0.0867 | 0.0868 | 0.3399 | 0.9480
$\theta_{TSGMM,2}$ | -0.0055 | 0.0866 | 0.0853 | 0.0855 | 0.3344 | 0.9470
$T=400$, $\text{$n_{IV}$}=4$ |  |  |  |  |  |
Estimated change point: 160.832 |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0036 | 0.0990 | 0.0912 | 0.0913 | 0.3576 | 0.9170
$\theta_{TS2SLS,1}$ | 0.0092 | 0.0978 | 0.0971 | 0.0976 | 0.3807 | 0.9350
$\theta_{TSGMM,1}$ | 0.0054 | 0.0944 | 0.0843 | 0.0844 | 0.3303 | 0.9130
$\theta_{GMM,2}$ | -0.0048 | 0.0813 | 0.0766 | 0.0768 | 0.3004 | 0.9290
$\theta_{TS2SLS,2}$ | -0.0012 | 0.0760 | 0.0768 | 0.0768 | 0.3012 | 0.9470
$\theta_{TSGMM,2}$ | -0.0041 | 0.0779 | 0.0726 | 0.0727 | 0.2847 | 0.9270
$T=800$, $\text{$n_{IV}$}=1$ |  |  |  |  |  |
Estimated change point: 321.04 |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0049 | 0.0813 | 0.0787 | 0.0789 | 0.3086 | 0.9360
$\theta_{TS2SLS,1}$ | 0.0033 | 0.0808 | 0.0785 | 0.0785 | 0.3075 | 0.9340
$\theta_{TSGMM,1}$ | 0.0032 | 0.0738 | 0.0726 | 0.0726 | 0.2845 | 0.9430
$\theta_{GMM,2}$ | 0.0045 | 0.0651 | 0.0645 | 0.0646 | 0.2527 | 0.9430
$\theta_{TS2SLS,2}$ | 0.0031 | 0.0629 | 0.0620 | 0.0621 | 0.2430 | 0.9430
$\theta_{TSGMM,2}$ | 0.0029 | 0.0617 | 0.0612 | 0.0612 | 0.2398 | 0.9400
$T=800$, $\text{$n_{IV}$}=4$ |  |  |  |  |  |
Estimated change point: 320.92 |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0010 | 0.0737 | 0.0673 | 0.0674 | 0.2640 | 0.9260
$\theta_{TS2SLS,1}$ | 0.0037 | 0.0733 | 0.0697 | 0.0698 | 0.2734 | 0.9400
$\theta_{TSGMM,1}$ | 0.0027 | 0.0699 | 0.0624 | 0.0624 | 0.2445 | 0.9170
$\theta_{GMM,2}$ | 0.0004 | 0.0580 | 0.0558 | 0.0558 | 0.2186 | 0.9400
$\theta_{TS2SLS,2}$ | 0.0016 | 0.0553 | 0.0547 | 0.0547 | 0.2144 | 0.9380
$\theta_{TSGMM,2}$ | 0.0007 | 0.0554 | 0.0530 | 0.0530 | 0.2078 | 0.9340

Table 5: Performance of the GMM, TS2SLS and TSGMM estimators under
heteroskedasticity of type 1 (HET1) when the change point is estimated. We
consider either 1 or 4 instruments with a sample size of either 400 or 800.

Estimator | Bias | MC Std | As. Std. | RMSE | Length | Coverage
---|---|---|---|---|---|---
$T=400$, $\text{$n_{IV}$}=1$ |  |  |  |  |  |
Estimated change location: 203.668 |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0127 | 0.0812 | 0.0783 | 0.0794 | 0.3071 | 0.9440
$\theta_{TS2SLS,1}$ | 0.0113 | 0.0804 | 0.0777 | 0.0785 | 0.3045 | 0.9440
$\theta_{TSGMM,1}$ | 0.0104 | 0.0775 | 0.0736 | 0.0743 | 0.2885 | 0.9410
$\theta_{GMM,2}$ | -0.0019 | 0.4604 | 0.4261 | 0.4261 | 1.6702 | 0.9550
$\theta_{TS2SLS,2}$ | -0.0059 | 0.4576 | 0.4211 | 0.4211 | 1.6507 | 0.9530
$\theta_{TSGMM,2}$ | -0.0092 | 0.4587 | 0.4204 | 0.4205 | 1.6481 | 0.9560
$T=400$, $\text{$n_{IV}$}=4$ |  |  |  |  |  |
Estimated change location: 180.481 |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0127 | 0.0812 | 0.0783 | 0.0794 | 0.3071 | 0.9440
$\theta_{TS2SLS,1}$ | 0.0113 | 0.0804 | 0.0777 | 0.0785 | 0.3045 | 0.9440
$\theta_{TSGMM,1}$ | 0.0104 | 0.0775 | 0.0736 | 0.0743 | 0.2885 | 0.9410
$\theta_{GMM,2}$ | -0.0019 | 0.4604 | 0.4261 | 0.4261 | 1.6702 | 0.9550
$\theta_{TS2SLS,2}$ | -0.0059 | 0.4576 | 0.4211 | 0.4211 | 1.6507 | 0.9530
$\theta_{TSGMM,2}$ | -0.0092 | 0.4587 | 0.4204 | 0.4205 | 1.6481 | 0.9560
$T=800$, $\text{$n_{IV}$}=1$ |  |  |  |  |  |
Estimated change location: 292.366 |  |  |  |  |  |
$\theta_{GMM,1}$ | -0.0152 | 0.2785 | 0.2503 | 0.2508 | 0.9811 | 0.9610
$\theta_{TS2SLS,1}$ | -0.0175 | 0.2784 | 0.2498 | 0.2505 | 0.9794 | 0.9590
$\theta_{TSGMM,1}$ | -0.0188 | 0.2738 | 0.2449 | 0.2457 | 0.9601 | 0.9580
$\theta_{GMM,2}$ | -0.1168 | 0.1766 | 0.1528 | 0.1923 | 0.5990 | 0.8420
$\theta_{TS2SLS,2}$ | -0.1179 | 0.1758 | 0.1519 | 0.1923 | 0.5956 | 0.8390
$\theta_{TSGMM,2}$ | -0.1179 | 0.1759 | 0.1517 | 0.1922 | 0.5948 | 0.8380
$T=800$, $\text{$n_{IV}$}=4$ |  |  |  |  |  |
Estimated change location: 302.204 |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0033 | 0.1274 | 0.1219 | 0.1219 | 0.4777 | 0.9560
$\theta_{TS2SLS,1}$ | -0.0018 | 0.1696 | 0.1578 | 0.1578 | 0.6185 | 0.9680
$\theta_{TSGMM,1}$ | 0.0043 | 0.1287 | 0.1199 | 0.1200 | 0.4699 | 0.9480
$\theta_{GMM,2}$ | 0.0015 | 0.0287 | 0.0276 | 0.0276 | 0.1080 | 0.9410
$\theta_{TS2SLS,2}$ | 0.0002 | 0.0313 | 0.0302 | 0.0302 | 0.1185 | 0.9480
$\theta_{TSGMM,2}$ | 0.0012 | 0.0278 | 0.0267 | 0.0268 | 0.1048 | 0.9470

Table 6: Performance of the GMM, TS2SLS and TSGMM estimators under
heteroskedasticity of type 2 (HET2) when the change point is estimated. We
consider either 1 or 4 instruments with a sample size of either 400 or 800.

Estimator | Bias | MC Std | As. Std. | RMSE | Length | Coverage
---|---|---|---|---|---|---
$T=400$, $\rho=-0.5$ and $\text{$n_{IV}$}=4$ |  |  |  |  |  |
HOM |  |  |  |  |  |
Estimated change location: 201.662 |  |  |  |  |  |
$\theta_{GMM,1}$ | -0.0006 | 0.0277 | 0.0274 | 0.0274 | 0.1072 | 0.9440
$\theta_{TS2SLS,1}$ | -0.0005 | 0.0276 | 0.0276 | 0.0276 | 0.1082 | 0.9420
$\theta_{TSGMM,1}$ | -0.0004 | 0.0275 | 0.0266 | 0.0266 | 0.1041 | 0.9330
$\theta_{GMM,2}$ | -0.0025 | 0.0607 | 0.0548 | 0.0548 | 0.2146 | 0.9230
$\theta_{TS2SLS,2}$ | 0.0002 | 0.0585 | 0.0567 | 0.0567 | 0.2222 | 0.9340
$\theta_{TSGMM,2}$ | -0.0022 | 0.0561 | 0.0475 | 0.0475 | 0.1861 | 0.9080
HET1 |  |  |  |  |  |
Estimated change location: 198.848 |  |  |  |  |  |
$\theta_{GMM,1}$ | -0.0030 | 0.0706 | 0.0666 | 0.0667 | 0.2612 | 0.9210
$\theta_{TS2SLS,1}$ | 0.0003 | 0.0698 | 0.0691 | 0.0691 | 0.2707 | 0.9450
$\theta_{TSGMM,1}$ | -0.0022 | 0.0703 | 0.0649 | 0.0649 | 0.2544 | 0.9190
$\theta_{GMM,2}$ | -0.0113 | 0.1470 | 0.1215 | 0.1220 | 0.4763 | 0.8850
$\theta_{TS2SLS,2}$ | 0.0007 | 0.1451 | 0.1361 | 0.1361 | 0.5333 | 0.9250
$\theta_{TSGMM,2}$ | -0.0074 | 0.1361 | 0.1063 | 0.1066 | 0.4168 | 0.8660
HET2 |  |  |  |  |  |
Estimated change location: 335.398 |  |  |  |  |  |
$\theta_{GMM,1}$ | -0.0008 | 0.0281 | 0.0267 | 0.0267 | 0.1046 | 0.9310
$\theta_{TS2SLS,1}$ | -0.0006 | 0.0303 | 0.0286 | 0.0286 | 0.1120 | 0.9390
$\theta_{TSGMM,1}$ | -0.0010 | 0.0275 | 0.0262 | 0.0262 | 0.1028 | 0.9350
$\theta_{GMM,2}$ | -0.0083 | 0.6605 | 0.5624 | 0.5624 | 2.2044 | 0.9340
$\theta_{TS2SLS,2}$ | -0.0248 | 0.8108 | 0.7269 | 0.7274 | 2.8495 | 0.9530
$\theta_{TSGMM,2}$ | -0.0014 | 0.6695 | 0.5263 | 0.5263 | 2.0632 | 0.8950

Table 7: Performance of the GMM, TS2SLS and TSGMM estimators under
homoskedasticity (HOM) when a change point is estimated but there is no change
point in the true DGP. We consider either 1 or 4 instruments with a sample
size of either 400 or 800.

Figure 2: Histograms of the estimated change point, when the true DGP has no
change points. Here $T=400$ and $n_{IV}=4$

###  A.3 Detecting a change point

Change size | Prob. of detecting change | Estimated change point
---|---|---
HOM |  |
1 | 1 | 160.912
0.5 | 1 | 160.705
0.3 | 1 | 160.728
0 | 0.07 | 196.174
HET1 |  |
1 | 1 | 160.832
0.5 | 0.99 | 160.012
0.3 | 0.924 | 162.458
0 | 0.065 | 198.848
HET2 |  |
1 | 0.883 | 180.481
0.5 | 0.293 | 262.182
0.3 | 0.028 | 309.525
0 | 0.001 | 335.398

Table 8: Probability of detecting a change point for different change sizes.
We consider one external instrument and a sample size of $400$. Note that the
change size equals zero implies the null hypothesis of no change.

Estimator | Bias | MC Std | As. Std. | RMSE | Length | Coverage
---|---|---|---|---|---|---
HOM |  |  |  |  |  |
Estimated change point: 160.912 |  |  |  |  |  |
Prob. of detecting the change: 1.000 |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0042 | 0.0404 | 0.0397 | 0.0399 | 0.1557 | 0.9340
$\theta_{TS2SLS,1}$ | 0.0065 | 0.0400 | 0.0408 | 0.0413 | 0.1599 | 0.9410
$\theta_{TSGMM,1}$ | 0.0044 | 0.0384 | 0.0365 | 0.0367 | 0.1430 | 0.9260
$\theta_{GMM,2}$ | -0.0010 | 0.0330 | 0.0318 | 0.0318 | 0.1248 | 0.9350
$\theta_{TS2SLS,2}$ | -0.0008 | 0.0326 | 0.0322 | 0.0322 | 0.1263 | 0.9400
$\theta_{TSGMM,2}$ | -0.0009 | 0.0316 | 0.0300 | 0.0300 | 0.1175 | 0.9350
HET1 |  |  |  |  |  |
Estimated change point: 160.832 |  |  |  |  |  |
Prob. of detecting the change: 1.000 |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0036 | 0.0990 | 0.0912 | 0.0913 | 0.3576 | 0.9170
$\theta_{TS2SLS,1}$ | 0.0092 | 0.0978 | 0.0971 | 0.0976 | 0.3807 | 0.9350
$\theta_{TSGMM,1}$ | 0.0054 | 0.0944 | 0.0843 | 0.0844 | 0.3303 | 0.9130
$\theta_{GMM,2}$ | -0.0048 | 0.0813 | 0.0766 | 0.0768 | 0.3004 | 0.9290
$\theta_{TS2SLS,2}$ | -0.0012 | 0.0760 | 0.0768 | 0.0768 | 0.3012 | 0.9470
$\theta_{TSGMM,2}$ | -0.0041 | 0.0779 | 0.0726 | 0.0727 | 0.2847 | 0.9270
HET2 |  |  |  |  |  |
Estimated change point: 180.481 |  |  |  |  |  |
Prob. of detecting the change: 0.883 |  |  |  |  |  |
$\theta_{GMM,1}$ | 0.0658 | 0.0542 | 0.0486 | 0.0986 | 0.1907 | 0.8480
$\theta_{TS2SLS,1}$ | 0.0707 | 0.0589 | 0.0585 | 0.1056 | 0.2292 | 0.8670
$\theta_{TSGMM,1}$ | 0.0659 | 0.0528 | 0.0468 | 0.0967 | 0.1834 | 0.8350
$\theta_{GMM,2}$ | -0.0439 | 0.3393 | 0.1593 | 0.1977 | 0.6242 | 0.8680
$\theta_{TS2SLS,2}$ | -0.0508 | 0.3476 | 0.2019 | 0.2336 | 0.7915 | 0.9110
$\theta_{TSGMM,2}$ | -0.0437 | 0.3389 | 0.1575 | 0.1959 | 0.6175 | 0.8680

Table 9: Pre-testing for a change point. If a change point is detected,
estimation is carried out across sub-samples as previously done; if no change
point is detected, the parameters are estimated over the entire sample. We
report a weighted average of the two specifications. We consider $T=400$ and
$\text{$n_{IV}$}=4$.

##  Appendix B Estimation of the NKPC

###  B.1 Data Plots

Figure 3: Time series of inflation and expected inflation (top left),
unemployment (top right), real exchange rate (bottom left), and terms of trade
(bottom right).

###  B.2 Estimation results in the benchmark model

Panel A: Expected inflation
---
|  | Bai and Perron (2003) critical values
| Test statistic |  $1\%$ critical value |  $5\%$ critical value |  $10\%$ critical value
$F(1|0)$ | 3.49 | 3.46 | 2.90 | 2.61
$F(2|1)$ | 2.77 | 3.63 | 3.15 | 2.89
Number of changes: |  | 2 | 2 | 2
Break points: |  | 2001m9 | 2009m3 |
Panel B: Unemployment rate
|  | Bai and Perron (2003) critical values
| Test statistic |  $1\%$ critical value |  $5\%$ critical value |  $10\%$ critical value
$F(1|0)$ | 3.62 | 3.46 | 2.90 | 2.61
$F(2|1)$ | 5.39 | 3.63 | 3.15 | 2.89
Number of changes: |  | 2 | 2 | 2
Break points: |  | 2001m9 | 2011m10 |

Table 10: Breaks in the Reduced forms using Bai and Perron (2003) critical
values. Trimming 0.2. $T=434$.

| Hall, Han and Boldea (2012) critical values
---|---
| Test statistic |  $1\%$ critical value |  $5\%$ critical value |  $10\%$ critical value
1983m9-2001m9: |  |  |  |
$F(1|0)$ | 36.86 | 24.45 | 20.08 | 17.95
Break point: | 1992m4 |  |  |
2001m10-2009m3: |  |  |  |
$F(1|0)$ | 21.22 | 24.45 | 20.08 | 17.95
Break point: | 2006m8 |  |  |
2009m4-2020m4: |  |  |  |
$F(1|0)$ | 5.20 | 24.45 | 20.08 | 17.95
Break point: | - |  |  |

Table 11: Breaks in the structural equation over each stable RF segment using
Hall et al. (2012) critical values. Trimming 0.15.

| 1983m9-1992m4 | 1992m5-2001m9 | 1983m9-2001m9
---|---|---|---
| TS2SLS | GMM | TSGMM | TS2SLS | GMM | TSGMM | GMM
| (1) | (2) | (3) | (4) | (5) | (6) | (7)
$\beta_{0}$ | -0.0024 | -0.0030 | -0.0022 | 0.0035 | -0.0001 | -0.0004 | 0.0001
| (0.0256) | (0.0264) | (0.0237) | (0.0199) | (0.0201) | (0.0198) | (0.0161)
$\gamma_{1}$ | 1.4323*** | 1.4751*** | 1.4748*** | 0.5369 | 0.6529*** | 0.6512*** | 1.2028***
| (0.2472) | (0.1190) | (0.1097) | (0.6315) | (0.1266) | (0.1218) | (0.0976)
$\gamma_{2}$ | -0.5522** | -0.5861*** | -0.5888*** | -0.1292 | -0.2287** | -0.2273** | -0.3989***
| (0.2302) | (0.1039) | (0.0931) | (0.3610) | (0.1003) | (0.0989) | (0.0854)
$\kappa_{\Psi}$ | 0.0150 | 0.0179* | 0.0175* | 0.0138 | 0.0267** | 0.0266*** | 0.0111*
| (0.0262) | (0.0131) | (0.0119) | (0.0370) | (0.0119) | (0.0113) | (0.0082)
$\gamma_{f}$ | -0.2618 | -0.2818* | -0.2814** | 0.2859 | -0.0817 | -0.0598 | -0.1915
| (0.3514) | (0.1793) | (0.1616) | (1.3275) | (0.4235) | (0.3828) | (0.1664)
$\kappa_{u}$ | 0.0150 | 0.0151 | 0.0143 | -0.0280* | -0.0434*** | -0.0427*** | -0.0036
| (0.0130) | (0.0111) | (0.0096) | (0.0209) | (0.0167) | (0.0156) | (0.0085)
$N$ |  | $104$ |  |  | $114$ |  | $218$

Table 12: *$p<0.10$, **$p<0.05$, ***$p<0.01$. Black stars indicate bi-
directional testing. Blue stars indicate right-tailed (positive) tests, and
Red stars left-tailed (negative) tests.

###  B.3 Robustness checks

####  B.3.1 Weak identification concerns

Sub-sample | KP statistic | N. Obs.
---|---|---
1983m9-2001m10 | 7.959 | 216
2001m11-2009m3 | 16.96 | 89
2009m4-2020m4 | 6.087 | 129

Table 13: Kleibergen and Paap (2006) KP statistics reported over the stable RF
subperiods. We test the null of rank deficiency (rank equal to 1 for the
(2,4)-matrix of RF coefficients) and the associated critical value are from
the Chi2(3): 7.81 at 5% and 6.25 at 10%.

####  B.3.2 Robustness checks: alternative instruments sets

In this section, we present the results obtained with alternate instrument
sets: 2 instruments (one lag of expected inflation and unemployment rate); 5
instruments (one lag of expected inflation, unemployment rate, real exchange
rate, terms of trade, short-long term interest rate spread). We follow exactly
the same steps as described in Section 7.1.

$\bullet$ Results obtained under just-identification with 2 instruments:

Panel A: Expected inflation
---
|  | Bai and Perron (2003) critical values
| Test statistic |  $1\%$ critical value |  $5\%$ critical value |  $10\%$ critical value
$F(1|0)$ | 2.88 | 3.97 | 3.26 | 2.90
$F(2|1)$ | 3.31 | 4.17 | 3.57 | 3.25
Number of changes: |  | 0 | 0 | 2
Break points: |  | 2001m10 | 2009m3 |
Panel B: Unemployment rate
|  | Bai and Perron (2003) critical values
| Test statistic |  $1\%$ critical value |  $5\%$ critical value |  $10\%$ critical value
$F(1|0)$ | 3.53 | 3.97 | 3.26 | 2.9
$F(2|1)$ | 4.02 | 4.17 | 3.57 | 3.25
Number of changes: |  | 0 | 2 | 2
Break points: |  | 2001m10 | 2008m12 |

Table 14: Breaks in the Reduced forms using Bai and Perron (2003) critical
values. Trimming 0.2. $T=434$.

|  Bai and Perron (2003) critical values
---|---
| Test statistic |  $1\%$ critical value |  $5\%$ critical value |  $10\%$ critical value
1983m9-2001m10: |  |  |  |
$F(1|0)$ | 35.43 | 24.50 | 20.08 | 17.95
Break point: | 1992m4 |  |  |
2001m11-2009m3: |  |  |  |
$F(1|0)$ | 17.91 | 24.50 | 20.08 | 17.95
Break point: | 2006m9 |  |  |
2009m3-2020m4: |  |  |  |
$F(1|0)$ | 4.35 | 24.50 | 20.08 | 17.95
Break point: | - |  |  |

Table 15: Breaks in the structural equation over each stable RF segment.
Trimming is 0.15. Note: the change point estimated at 2006m9 is at the margin
of significance; it is included in all our results.

Sub-sample | F statistic | N. Obs.
---|---|---
1983m9-2001m10 | 9.501 | 216
Table 16: Kleibergen and Paap (2006) KP statistics reported over the stable RF
subperiods. We test the null of rank deficiency (rank equal to 1 for the
(2,2)-matrix of RF coefficients) and the associated critical value are from
the Chi2(1): 3.84 at 5% and 2.71 at 10%.

| 1983m9-1992m4 | 1992m5-2001m10 | 1983m9-2001m10
---|---|---|---
| TS2SLS | GMM | TSGMM | TS2SLS | GMM | TSGMM | GMM
| (1) | (2) | (3) | (4) | (5) | (6) | (7)
$\beta_{0}$ | -0.0022 | -0.0050 | -0.0050 | 0.0041 | 0.0011 | 0.0011 | -0.0001
| (0.0256) | (0.0264) | (0.0242) | (0.0189) | (0.0232) | (0.0230) | (0.0163)
$\gamma_{1}$ | 1.4328*** | 1.4539*** | 1.4539*** | 0.5631 | 0.6641*** | 0.6641*** | 1.2018***
| (0.2510) | (0.1234) | (0.1158) | (0.4831) | (0.2320) | (0.2297) | (0.0983)
$\gamma_{2}$ | -0.5543** | -0.5786*** | -0.5786*** | -0.1353 | -0.2296 | -0.2296 | -0.4048***
| (0.2295) | (0.1049) | (0.0960) | (0.2706) | (0.1589) | (0.1568) | (0.0854)
$\kappa_{\Psi}$ | 0.0146 | 0.0161 | 0.0161* | 0.0141 | 0.0303 | 0.0303 | 0.0102
| (0.0262) | (0.0131) | (0.0124) | (0.0283) | (0.0457) | (0.0449) | (0.0083)
$\gamma_{f}$ | -0.2624 | -0.2432 | -0.2432 | 0.2187 | -0.2295 | -0.2295 | -0.1843
| (0.3696) | (0.1822) | (0.1666) | (1.0502) | (2.0702) | (2.0304) | (0.1716)
$\kappa_{u}$ | 0.0138 | 0.0140 | 0.0140 | -0.0278* | -0.0477 | -0.0477 | -0.0050
| (0.0127) | (0.0110) | (0.0097) | (0.0181) | (0.0725) | (0.0714) | (0.0088)
$N$ |  | $102$ |  |  | $114$ |  | $216$

Table 17: *$p<0.10$, **$p<0.05$, ***$p<0.01$. Black stars indicate bi-
directional testing. Blue stars indicate right-tailed (positive) tests, and
Red stars left-tailed (negative) tests.

$\bullet$ Results obtained with 5 instruments:

Panel A: Expected inflation
---
|  | Bai and Perron (2003) critical values
| Test statistic |  $1\%$ critical value |  $5\%$ critical value |  $10\%$ critical value
$F(1|0)$ | 3.20 | 3.30 | 2.77 | 2.49
$F(2|1)$ | 2.88 | 3.52 | 2.99 | 2.76
Number of changes: |  | 0 | 1 | 2
Break points: |  | 2009m3 |  |
Panel B: Unemployment rate
|  | Bai and Perron (2003) critical values
| Test statistic |  $1\%$ critical value |  $5\%$ critical value |  $10\%$ critical value
$F(1|0)$ | 3.47 | 3.30 | 2.77 | 2.49
$F(2|1)$ | 4.75 | 3.52 | 2.99 | 2.76
Number of changes: |  | 2 | 2 | 2
Break points: |  | 2001m10 | 2008m12 |

Table 18: Breaks in the Reduced forms. Trimming 0.2. $T=434$.

|  Bai and Perron (1998) critical values
---|---
| Test statistic |  $1\%$ critical value |  $5\%$ critical value |  $10\%$ critical value
1983m9-2001m10: |  |  |  |
$F(1|0)$ | 36.81 | 24.45 | 20.08 | 17.97
Break point: | 1992m6 |  |  |
2001m10-2009m3: |  |  |  |
$F(1|0)$ | 20.31 | 24.45 | 20.08 | 17.97
Break point: | 2006m9 |  |  |
2009m4-2020m4: |  |  |  |
$F(1|0)$ | 4.90 | 24.45 | 20.08 | 17.97
Break point: | - |  |  |

Table 19: Breaks in the equation of interest over each stable RF segment.
Trimming 0.15.

Sub-sample | F statistic | N. Obs.
---|---|---
1983m9 - 2001m10 | 4.26 | 216
Table 20: Kleibergen and Paap (2006) KP statistic reported over the first
stable RF period. We test the null of rank deficiency (rank equal to 1 for the
(2,3)-matrix of RF coefficients) and the associated critical value are from
the Chi2(4): 9.49 at 5% and 7.78 at 10%.

| 1983m9-1992m6 | 1992m7-2001m10 | 1983m9-2001m10
---|---|---|---
| TS2SLS | GMM | TSGMM | TS2SLS | GMM | TSGMM | GMM
| (1) | (2) | (3) | (4) | (5) | (6) | (7)
$\beta_{0}$ | -0.0051 | -0.0045 | 0.0000 | 0.0051 | 0.0022 | 0.0022 | 0.0002
| (0.0254) | (0.0261) | (0.0221) | (0.0238) | (0.0202) | (0.0164) | (0.0161)
$\gamma_{1}$ | 1.4289 *** | 1.4779 *** | 1.4779 *** | 0.5117 | 0.6402*** | 0.6207*** | 1.2038***
| (0.2515) | (0.1167) | (0.0965) | (0.8955) | (0.1244) | (0.1039) | (0.0973)
$\gamma_{2}$ | -0.5522 ** | -0.5951 *** | -0.5934 *** | -0.1149 | -0.2221** | -0.2356*** | -0.4006***
| (0.2398) | (0.1015) | (0.0824) | (0.5249) | (0.0995) | (0.0816) | (0.0850)
$\kappa_{\Psi}$ | 0.0136 | 0.0163 | 0.0178* | 0.0135 | 0.0262** | 0.0267*** | 0.0109*
| (0.0255) | (0.0131) | (0.0110) | (0.0459) | (0.0113) | (0.0093) | (0.0081)
$\gamma_{f}$ | -0.2576 | -0.2992 | -0.2812 | 0.3596 | -0.0481 | -0.0360 | -0.1786
| (0.3473) | (0.1775) | (0.1254) | (1.9307) | (0.3953) | (0.2906) | (0.1595)
$\kappa_{u}$ | 0.0136 | 0.0135 | 0.0125 | -0.0254 | -0.0418 ** | -0.0445 *** | -0.0036
| (0.0123) | (0.0112) | (0.0091) | (0.0264) | (0.0167) | (0.0129) | (0.0084)
$N$ |  | $106$ |  |  | $112$ |  | $218$

Table 21: *$p<0.10$, **$p<0.05$, ***$p<0.01$. Black stars indicate bi-
directional testing. Blue stars right-tailed (positive) tests, and red stars
(negative) left-tailed tests. One lag of expected inflation is included as
instrument.

##  Appendix C Proof of theoretical results

$\bullet$ Definition 1:

Recall that $\Pi_{z}=[I_{p_{1}},O_{(q-p_{1})\times p_{1}}]$ and
$\Pi^{a}\xlongequal{\text{def}}[\Pi_{z},\Pi^{0}]$ is the matrix that augments
$\Pi^{0}$ with zeros and ones such that
$Z_{t}^{\prime}\Pi^{a}=[Z_{t}^{\prime}\Pi_{z},Z_{t}^{\prime}\Pi^{0}]=[Z_{1t}^{\prime},Z_{t}^{\prime}\Pi^{0}]$.
Let $Q\xlongequal{\text{def}}Q_{1}+Q_{2}$. Then:

$\bullet$
$V_{TS2SLS,vec}=\begin{bmatrix}D_{1}^{\prime}\Omega_{1}D_{1}&D_{1}^{\prime}\Omega_{12}D_{2}\\\
D_{2}^{\prime}\Omega_{12}^{\prime}D_{1}&D_{2}^{\prime}\Omega_{2}D_{2}\end{bmatrix},$
$A_{i}=\Pi^{a^{\prime}}Q_{i}\Pi^{a}$,
$D_{i}\,=\,A_{i}^{-1}(\Pi^{a^{\prime}}M_{i}^{\prime})$,
$M_{1}^{\prime}=(I,Q_{2}Q^{-1},-Q_{1}Q^{-1})$,
$M_{2}^{\prime}=(I,Q_{1}Q^{-1},-Q_{2}Q^{-1})$, $a_{i}=\theta_{x,i}^{0}\otimes
I_{q}$,

$\Omega_{i}=\left(\begin{array}[]{ccc}S_{u,i}&S_{uv,i}^{\prime}a_{i}&O\\\
a_{i}^{\prime}S_{uv,i}&a_{i}^{\prime}S_{v,i}a_{i}&O\\\
O&O&a_{i}^{\prime}(S_{v}-S_{v,i})a_{i}\\\ \end{array}\right),\
\Omega_{12}=\left(\begin{array}[]{ccc}O&O&S_{uv,1}^{\prime}a_{2}\\\
O&O&a_{1}^{\prime}S_{v,1}a_{2}\\\
a_{1}S_{uv,2}&a_{1}^{\prime}S_{v,2}a_{2}&O\\\ \end{array}\right).$

$\bullet$ $V_{GMM,vec}=\operatorname{diag}(V_{GMM,1},V_{GMM,2})$ and
$V_{GMM,i}\,=\,\left[\,\Pi^{a^{\prime}}\,Q_{i}\,(S_{u,i})^{-1}\,Q_{i}\,\Pi^{a}\,\right]^{-1}$.

$\bullet$ $V_{TSGMM,vec}^{a}=[\Gamma^{\prime}\mathcal{S}^{-1}\Gamma]^{-1}$
with:

$\displaystyle\mathcal{S}=\begin{bmatrix}S_{u,1}&O&S_{uv,1}^{\prime}&O\\\
O&S_{u,2}&O&S_{uv,2}^{\prime}\\\ S_{uv,1}&O&S_{v,1}&O\\\
O&S_{uv,2}&O&S_{v,2}\end{bmatrix},\quad\Gamma$
$\displaystyle=-\begin{bmatrix}Q_{1}\Pi^{a}&O&O\\\ O&Q_{2}\Pi^{a}&O\\\
O&O&\begin{bmatrix}I_{p_{2}}\otimes Q_{1}\\\ I_{p_{2}}\otimes Q_{2}\\\
\end{bmatrix}\end{bmatrix}=-\operatorname{diag}(\Gamma_{1},\Gamma_{2}),$

with $\Gamma_{1}=\operatorname{diag}(Q_{1}\Pi^{a},Q_{2}\Pi^{a})$ and
$\Gamma_{2}=\begin{bmatrix}I_{p_{2}}\otimes Q_{1}\\\ I_{p_{2}}\otimes
Q_{2}\end{bmatrix}$.

$\displaystyle\bullet$ $\displaystyle
V_{TSGMM,vec}=(V_{GMM,vec}^{-1}+\mathcal{G}^{\prime}\mathcal{G})^{-1},\qquad\mathcal{G}=\mathcal{M}_{\mathcal{J}}\mathcal{E}^{-1/2}\mathcal{H},\qquad\mathcal{M}_{\mathcal{J}}=I-\mathcal{J}(\mathcal{J}^{\prime}\mathcal{J})^{-1}\mathcal{J}^{\prime},$
$\displaystyle\mathcal{J}=\mathcal{E}^{-1/2}\
\Gamma_{2},\mathcal{E}=\operatorname{diag}(S_{v,1}-S_{uv,1}S_{u,1}^{-1}S_{uv,1}^{\prime}\
,\ S_{v,2}-S_{uv,2}S_{u,2}^{-1}S_{uv,2}^{\prime}),$
$\displaystyle\mathcal{H}=[\operatorname{diag}\ (S_{uv,1}\
S_{u,1}^{-1},S_{uv,2}\ S_{u,2}^{-1})]\ \Gamma_{1}.$

\- Additional notation: Let
$\theta^{0}(t)\xlongequal{\text{def}}\theta_{1}^{0}\mathbf{1}[t\leq
T^{0}]+\theta_{2}^{0}\mathbf{1}[t>T^{0}]$,
$\widetilde{u}_{t}\xlongequal{\text{def}}y_{t}-\hat{W}_{t}^{\prime}\theta^{0}(t)=u_{t}+(W_{t}-\hat{W}_{t})^{\prime}\theta^{0}(t)=u_{t}+v_{t}^{\prime}\theta_{x}^{0}(t)+Z_{t}^{\prime}(\Pi_{T}-\hat{\Pi})$,
$\theta_{x}^{0}(t)\xlongequal{\text{def}}\theta_{x,1}^{0}\mathbf{1}[t\leq
T^{0}]+\theta_{x,2}^{0}\mathbf{1}[t>T^{0}]$,
$\Pi_{T}\xlongequal{\text{def}}\Pi^{0}/r_{T}$,
$\Pi_{T}^{a}\xlongequal{\text{def}}(\Pi_{z},\Pi_{T})$, and
$\hat{\Pi}^{a}\xlongequal{\text{def}}(\Pi_{z},\hat{\Pi})$ such that
$\hat{W}_{t}^{\prime}=[Z_{1t}^{\prime},\hat{X}_{t}^{\prime}]=Z_{t}^{\prime}[\Pi_{z},\hat{\Pi}]=Z_{t}^{\prime}\hat{\Pi}^{a}$.
Also, for simplicity, let
$\overline{R}_{T}=\operatorname{diag}(I_{p_{1}},r_{T}I_{p_{2}})$,
$\hat{A}_{1}(r)\xlongequal{\text{def}}\overline{R}_{T}\
T^{-1}\sum_{t=1}^{[Tr]}\hat{W}_{t}\hat{W}_{t}^{\prime}\ \overline{R}_{T}$,
$\hat{A}_{2}(r)\xlongequal{\text{def}}\hat{A}_{1}(1)-\hat{A}_{1}(r)$,
$\xi_{1}(r)\xlongequal{\text{def}}\overline{R}_{T}T^{-1/2}\sum_{t=1}^{[Tr]}\hat{W}_{t}\widetilde{u}_{t}$,
$\xi_{2}(r)\xlongequal{\text{def}}\xi_{1}(1)-\xi_{1}(r)$, and recall that
$\sum_{1}\xlongequal{\text{def}}\sum_{t=1}^{T_{1}^{0}}$,
$\sum_{2}\xlongequal{\text{def}}\sum_{t=T_{1}^{0}+1}^{T}$.

$\bullet$ Proof of Theorem 1:
For simplicity, in each of parts (i),(ii), (iii), we write
$\hat{\theta}_{vec}$ for each of $\hat{\theta}_{TS2SLS,vec}(\lambda_{0})$,
$\hat{\theta}_{GMM,vec}(\lambda_{0})$, and
$\hat{\theta}_{TSGMM,vec}(\lambda_{0})$, and
$\hat{\theta}_{vec}\xlongequal{\text{def}}\mbox{vect
}(\hat{\theta}_{1},\hat{\theta}_{2})$, where $\hat{\theta}_{i}$ are the
$p\times 1$ estimators in each of the two samples.
(i). Asymptotic properties of
$\hat{\theta}_{vec}=\hat{\theta}_{TS2SLS,vec}(\lambda_{0})$. This proof
generalizes Hall et al. (2012), Theorem 3, to near-weak instruments. Since
$\hat{\Pi}=\Pi_{T}+(T^{-1}\sum_{t=1}^{T}Z_{t}Z_{t}^{\prime})^{-1}T^{-1}\sum_{t=1}^{T}Z_{t}v_{t}^{\prime}$,
and $T^{-1}\sum_{t=1}^{T}Z_{t}Z_{t}^{\prime}\stackrel{{\scriptstyle
p}}{{\rightarrow}}Q$, $T^{-1}\sum_{t=1}^{T}Z_{t}v_{t}\stackrel{{\scriptstyle
p}}{{\rightarrow}}0$ by Assumptions 1-3 and a WLLN for NED processes,
$\hat{\Pi}\stackrel{{\scriptstyle p}}{{\rightarrow}}\Pi_{T}$, and
$\overline{R}_{T}\hat{\Pi}\stackrel{{\scriptstyle p}}{{\rightarrow}}\Pi^{a}$.
With the additional notation above,
$T^{1/2}\overline{R}_{T}^{-1}(\hat{\theta}_{i}-\theta_{i}^{0})=\hat{A}_{i}^{-1}(\lambda_{0})\
\xi_{i}(\lambda_{0})$, where:

$\displaystyle\hat{A}_{i}^{-1}(\lambda_{0})$ $\displaystyle=\overline{R}_{T}\
T^{-1}\textstyle\sum_{i}\hat{W}_{t}\hat{W}_{t}^{\prime}\
\overline{R}_{T}=\overline{R}_{T}\
\hat{\Pi}^{a^{\prime}}(T^{-1}\textstyle\sum_{i}Z_{t}Z_{t}^{\prime})\overline{R}_{T}\hat{\Pi}^{a}$
$\displaystyle=\overline{R}_{T}\
\Pi_{T}^{a^{\prime}}(T^{-1}\textstyle\sum_{i}Z_{t}Z_{t}^{\prime})\overline{R}_{T}\Pi_{T}^{a^{\prime}}+o_{P}(1)=\Pi^{a^{\prime}}Q_{i}\Pi^{a}+o_{P}(1)=A_{i}+o_{P}(1).$

On the other hand,
$\xi_{i}(\lambda_{0})=R_{T}T^{-1/2}\sum_{1}\hat{W}_{t}\widetilde{u}_{t}=(\Pi^{a}+o_{P}(1))^{\prime}T^{-1/2}\sum_{i}Z_{t}\widetilde{u}_{t}$.
Note that, for $i=1,2$, and $j=1,2,j\neq i$:

$\displaystyle\textstyle T^{-1/2}\sum_{i}Z_{t}\widetilde{u}_{t}$
$\displaystyle=\textstyle
T^{-1/2}\sum_{i}Z_{t}u_{t}+T^{-1/2}\sum_{i}Z_{t}v_{t}^{\prime}\theta_{x,i}^{0}-T^{-1}\sum_{i}Z_{t}Z_{t}^{\prime}(\sum_{t=1}^{T}Z_{t}Z_{t}^{\prime})^{-1}(T^{-1/2}\sum_{t=1}^{T}Z_{t}v_{t}^{\prime}\theta_{x,i}^{0})$
$\displaystyle=\textstyle T^{-1/2}\sum_{i}Z_{t}u_{t}+\textstyle
T^{-1/2}\sum_{i}Z_{t}v_{t}^{\prime}\theta_{x,1}^{0}-(Q_{i}^{-1}Q+o_{P}(1))\textstyle
T^{-1/2}\sum_{i}Z_{t}v_{t}^{\prime}\theta_{x,1}^{0}$
$\displaystyle-(Q_{i}^{-1}Q+o_{P}(1))\textstyle
T^{-1/2}\sum_{j}Z_{t}v_{t}^{\prime}\theta_{x,1}^{0}$
$\displaystyle=\Psi^{u}_{i}+(I-Q_{i}Q^{-1})\Psi_{1}^{v}\theta_{x,i}^{0}-(Q_{i}Q^{-1})\Psi_{2}^{v}\theta_{x,i}^{0}+o_{P}(1)$
$\displaystyle=M_{i}^{\prime}\mbox{vect
}(\Psi^{u}_{i},\Psi_{1}^{v}\theta_{i},\Psi_{2}^{v}\theta_{i}),$

where
$\Psi^{u}_{i}=T^{-1/2}\sum_{1}Z_{t}u_{t},\Psi^{v}_{i}=T^{-1/2}\sum_{i}Z_{t}v_{t}^{\prime},i=1,2$.
Note that the terms $\Psi_{1}^{v},\Psi_{2}^{v}$ are present because of the
full-sample first-stage. Under Assumptions 1-3, by the CLT for NED processes
(Theorem 2.11 in Wooldridge and White (1988)), $\mbox{vect
}(\Psi^{u}_{i},\Psi_{1}^{v},\Psi_{2}^{v})\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,\Omega_{i})$, where $\Omega_{i}$ is defined
above. Thus, $\textstyle
T^{-1/2}\sum_{i}Z_{t}\widetilde{u}_{t}\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,M_{i}^{\prime}\Omega_{i}M_{i})$, so
$\Lambda_{T}[\hat{\theta}_{i}-\theta_{i}^{0}]\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}\left(0,D_{i}^{\prime}\Omega_{1}D_{i}\right).$
Moreover, because of the full-sample first-stage, asymptotically,
$\Lambda_{T}[\hat{\theta}_{1}-\theta_{1}^{0}]\not\perp\Lambda_{T}[\hat{\theta}_{2}-\theta_{2}^{0}]$,
and
$\mbox{ACov}\left\\{\Lambda_{T}[\hat{\theta}_{1}-\theta_{1}^{0}],\Lambda_{T}[\hat{\theta}_{2}-\theta_{2}^{0}]\right\\}=D_{1}^{\prime}\Omega_{12}D_{2}.$
Putting these together, $[I_{2}\otimes\Lambda_{T}]\
[\hat{\theta}_{vec}-\theta_{vec}^{0}]\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}\left(0,V_{TS2SLS,vec}\right)$.

(ii). Asymptotic properties of
$\hat{\theta}_{vec}=\hat{\theta}_{GMM,vec}(\lambda_{0})$. Let
$\overline{W}^{0}\xlongequal{\text{def}}\operatorname{diag}(\overline{Z}_{1}^{0},\overline{Z}_{2}^{0})$,
where $\overline{Z}_{1}^{0},\overline{Z}_{2}^{0}$ are the first
$T_{1}^{0}\times q$, respectively the last $(T-T_{1}^{0})\times q$ blocks of
$Z$, the matrix with rows $Z_{t}^{\prime}$ stacked in order. Also let
$\overline{W}\xlongequal{\text{def}}\mbox{vect
}(\overline{W}_{1}^{0},\overline{W}_{2}^{0})$ be the partitions of $W$ (the
matrix with rows $W_{t}^{\prime}$) at $\hat{T}$, respectively $T^{0}$, and
similarly partition $y=\mbox{vect
}(\overline{y}_{1}^{0},\overline{y}_{2}^{0})$ and $u=\mbox{vect
}(\overline{U}_{1}^{0},\overline{U}_{2}^{0})$. Then for the weighting matrices
$\hat{S}_{u,i}^{-1}\stackrel{{\scriptstyle p}}{{\rightarrow}}S_{u,i}$,

$\displaystyle\hat{\theta}_{GMM,i}$
$\displaystyle=(\overline{W}_{i}^{0^{\prime}}\overline{Z}_{i}^{0}\,\hat{S}_{u,i}^{-1}\,\overline{Z}_{i}^{0^{\prime}}\overline{W}_{i}^{0})^{-1}\,\overline{W}_{i}^{0^{\prime}}\overline{Z}_{i}^{0}\,\hat{S}_{u,i}^{-1}\,\overline{Z}_{i}^{0^{\prime}}\overline{y}_{i}$
$\displaystyle=\theta_{i}^{0}+(\overline{W}_{i}^{0^{\prime}}\overline{Z}_{i}^{0}\,\hat{S}_{u,i}^{-1}\,\overline{Z}_{i}^{0^{\prime}}\overline{W}_{i}^{0})^{-1}\,\overline{W}_{i}^{0^{\prime}}\overline{Z}_{i}^{0}\,\hat{S}_{u,i}^{-1}\,\overline{Z}_{i}^{0^{\prime}}\overline{U}_{i}^{0}.$

The asymptotic distributions for the these split sample estimators follow from
the WLLN and CLT for NED processes, under Assumptions 103. In particular,
$\overline{R}_{T}T^{-1}\overline{W}_{i}^{0^{\prime}}\overline{Z}_{i}^{0}\stackrel{{\scriptstyle
p}}{{\rightarrow}}\Pi^{a^{\prime}}Q_{i}$,
$T^{-1/2}\overline{Z}_{i}^{0^{\prime}}\overline{U}_{i}^{0}\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,S_{u,i})$, for $i=1,2$, and since
$\hat{S}_{u,i}\stackrel{{\scriptstyle p}}{{\rightarrow}}S_{u,i}$,
$\Lambda_{T}[\hat{\theta}_{GMM,i}-\theta_{i}^{0}]\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,(\Pi^{a^{\prime}}Q_{i}S_{u,i}^{-1}Q_{i}\Pi^{a})^{-1})$,
and they are asymptotically independent by Assumption 1.

(ii). Asymptotic properties of
$\hat{\theta}_{vec}=\hat{\theta}_{TSGMM,vec}(\lambda_{0})$ and
$\hat{\Pi}_{vec}=\hat{\Pi}_{TSGMM,vec}(\lambda_{0})$. Let
$\breve{\Lambda}_{T}\xlongequal{\text{def}}\
\operatorname{diag}(I_{2}\otimes\Lambda_{T},T^{1/2}I_{qp2})$,
$\hat{\beta}\xlongequal{\text{def}}\begin{bmatrix}\hat{\theta}\\\
\hat{\Pi}_{TSGMM,vec}\end{bmatrix}$ and
$\beta_{T}^{0}\xlongequal{\text{def}}\begin{bmatrix}\theta_{vec}^{0}\\\
\Pi_{vec}^{0}/r_{T}\end{bmatrix}$. Then, the sample moment conditions for the
estimator $\hat{\beta}$ can be rewritten as: