# Inferential Theory for Granular Instrumental Variables in High
Dimensions111We are grateful to Xu Cheng, Gloria Gonzalez-Rivera, Bruce
Hansen, Jean Helwege, Bo Honoré, Andrew Patton, Hashem Pesaran, Ruoyao Shi,
Aman Ullah, Qiankun Zhou, seminar participants at UC Riverside and JSM 2021
for helpful comments and suggestions. We thank Michael Bates for providing
access to UCR’s High Performance Computing Center to carry out simulations
efficiently.

Saman Banafti† Tae-Hwy Lee‡

The Granular Instrumental Variables (GIV) methodology exploits panels with
factor error structures to construct instruments to estimate structural time
series models with endogeneity even after controlling for latent factors. We
extend the GIV methodology in several dimensions. First, we extend the
identification procedure to a large $N$ and large $T$ framework, which depends
on the asymptotic Herfindahl index of the size distribution of $N$ cross-
sectional units. Second, we treat both the factors and loadings as unknown and
show that the sampling error in the estimated instrument and factors is
negligible when considering the limiting distribution of the structural
parameters. Third, we show that the sampling error in the high-dimensional
precision matrix is negligible in our estimation algorithm. Fourth, we
overidentify the structural parameters with additional constructed
instruments, which leads to efficiency gains. Monte Carlo evidence is
presented to support our asymptotic theory and application to the global crude
oil market leads to new results.

Keywords: Interactive effects, Factor error structure, Simultaneity, Power-law
tails, Asymptotic Herfindahl index, Global crude oil market, Supply and demand
elasticities, Precision matrix.
JEL Classifications: C26, C36, C38, C46, C55

##  1 Introduction

In the absence of randomized control trials, finding valid and strong
instruments to circumvent unobserved confounders is a very challenging task.
The Granular Instrumental Variables, hereafter GIV, methodology that [39]
propose, establishes a systematic way to construct instruments from suitably
weighted idiosyncratic shocks, from observational datasets and use them as
instruments for aggregate endogenous variables. Constructing instruments.
There are some existing methodologies which seek to eliminate the need to find
an instrument. A leading example is the [5] framework in the context of
estimating the speed of adjustment or state dependence parameters using
dynamic panel data models with fixed effects, in which higher order lags of
the dependent variable serve as instruments for the included lags of the
dependent variable. The [17] methodology (aka shift-share estimators) where
instruments are constructed from identities involving the (endogenous)
explanatory variable whose shift component is interacted with shares. The [68]
setting exploits the existence of structural breaks in the conditional
heteroskedasticity regime, which is common place in many applications of
interest. This allows one to bring a system with less equations than unknowns
to a just identified system with as many equations as unknowns. The [12]
methodology lays out a panel simultaneous equations model (similar to the
model analyzed in this paper) where the estimated (strong) factors can be used
as instrumental variables under certain conditions. We will return to the [12]
methodology when we overidentify the structural parameters of interest as it
is inspired by their framework. The vast methodological refinements cited
within the papers referenced above are not listed here for brevity.
Microeconomic (granular) origins of aggregate fluctuations. How can
idiosyncratic shocks be relevant for endogenous aggregate variables? The
literature on "granularity" traces back to historic debates in macroeconomics;
no attempt to fully catalog this debate is made here, rather a concise summary
is offered. [59] demonstrate that in a multisector stochastic neoclassical
growth model, sectoral shocks (as opposed to aggregate shocks) can potentially
lead to GDP fluctuations. Intuitively, complex production processes form
sectoral linkages which in turn provide a transmission mechanism of shocks
across sectors. Subsequently, [45] and [30] debate whether sectoral shocks
decay according to $\frac{1}{\sqrt{N}}$ as the central limit theorem would
suggest. [38] provides an initial theoretical solution to the debate by
showing that when the firm size distribution is heavy tailed, the central
limit theorem does not apply and sectoral volatility decays much slower than
$\frac{1}{\sqrt{N}}$. [38] coins this mechanism as the so-called "granular"
hypothesis, in which the economy is composed of incompressible grains as
opposed to infinitesimally small micro units. [1] formulates a network
approach to demonstrate that sectoral idiosyncratic shocks generate non-
negligible aggregate volatility when there exists sufficient asymmetry in the
input-output relationships. [67] build off of the theoretical approach of [1]
and develop econometric theory to measure the degree of network dominance and
in their application they find some evidence of sector-specific shock
propagation albeit not overwhelmingly strong for the US input-output accounts
data over the period 1972-2002. More empirical evidence for such propagation
mechanism is presented in [40], [24], [55], [19], [60], [76], [38], [25],
[72], [2], [49] and [57]. GIV, Gabaix and Koijen (2021). In an econometric
framework, GK illustrate that when the market under consideration is
sufficiently concentrated, then one can use the the collection of
idiosyncratic shocks to individual micro units, at each time period $t$, as an
instrument for endogenous aggregate variables. The instrumental relevance
follows heuristically from the paragraphs above. The exogeneity condition, as
in any instrumental variables procedure, requires assumptions on unobserved
random variables. However it should be noted that the exogeneity condition
exploited in this framework is a relatively mild assumption that is often made
in factor models (e.g. [10]) for identification purposes. The insight and
contribution of GK opens the doors to a wide possibility of ways in which one
can continue building on the promising new GIV methodology. Contributions of
this paper. Our contributions to the GIV methodology are primarily focused on
the underlying econometric issues. First, we naturally extend GK’s
identification procedure to a large $N$ and large $T$ framework (GK formally
introduced GIV for a fixed $N$ and large $T$) by establishing and restricting
the asymptotic behavior of the Herfindahl index for large $N$ markets as a
function of the tail index of the size distribution. Given the large $N$ and
large $T$ framework, we treat both the factors and loadings as unknown and
allow the idiosyncratic error term to be weakly cross-sectionally
correlated.222GK treat the factor loadings as known and extract the factors
via period-by-period cross-sectional regressions. While they advocate
extraction of latent factors via principal components analysis when loadings
are unknown, they abstract away from the corresponding sampling error. We will
show that the sampling error is indeed negligible. As such, from our
preliminary stage, we extract not only the estimated factors but also the
estimated loadings via principal components analysis, PCA hereafter, or
depending on the generality of the model ($k_{x}\neq 0$ in our notation from
Section 2), we use the iterative OLS-PCA method of [7]. Second, we show that
the sampling error in the estimated instrument and estimated factors is
negligible when considering the limiting distribution of the structural
parameters of interest; that is, the estimator is robust to the latent factor
structure. Moreover, the exogeneity requirement for one of the structural
parameters generally depends on a potentially high dimensional precision
matrix (the inverse of the covariance matrix). Third, we show that the
sampling error in the high dimensional precision matrix is negligible in our
iterative estimation algorithm for said structural parameter. Fourth, we
overidentify the structural parameters which leads to efficiency gains. This
leads to new and improved results in our empirical application of GIV to the
global crude oil markets. Monte Carlo evidence is presented to confirm the
finite sample behavior of our estimators are well approximated by the
asymptotic distributions. We label our refinement to the GIV methodology as
Feasible Granular Instrumental Variables or FGIV for short. Finally, an
empirical application of the estimation methods to estimate demand and supply
elasticities of the global crude oil markets are presented to demonstrate the
estimation procedures. Notation. We distinguish vectors and matrices from
scalars by making an object bold. Let $\\{X_{it},i=1,\dots,N;t=1,\dots,T\\}$
be a double index process of random variables where $N$ denotes the number of
cross-sectional units and $T$ denotes the number of time periods. We
frequently stack across $i$, in which we obtain $\underset{N\times
1}{\boldsymbol{X}_{\cdot
t}}:=\begin{pmatrix}X_{1t}&\dots&X_{Nt}\end{pmatrix}^{\prime}$. Similarly, if
we stack across $t$ we obtain $\underset{T\times
1}{\boldsymbol{X}_{i\cdot}}:=\begin{pmatrix}X_{i1}&\dots&X_{iT}\end{pmatrix}^{\prime}$.
When $\boldsymbol{X}_{it}$ is itself a vector, say of dimension $k$, then we
obtain a matrix when we stack across $i$ or $t$, e.g. $\underset{N\times
k}{\boldsymbol{X}_{\cdot t}}$ or $\underset{T\times
k}{\boldsymbol{X}_{i\cdot}}$. Define $X_{\boldsymbol{w}t}$ as the cross-
sectionally weighted average of $X_{it}$, that is
$X_{\boldsymbol{w}t}:=\boldsymbol{w}^{\prime}X_{\cdot
t}=\sum^{N}_{i=1}w_{i}X_{it}$. Common weights, $\underset{N\times
1}{\boldsymbol{w}}=(w_{i})$, used frequently throughout the paper are (1) the
precision weights,
$\boldsymbol{E}:=\frac{\boldsymbol{\Sigma}^{-1}_{u}\boldsymbol{\iota}}{\boldsymbol{\iota}^{\prime}\boldsymbol{\Sigma}^{-1}_{u}\boldsymbol{\iota}}$
where $\underset{N\times
N}{\boldsymbol{\Sigma}_{u}}:=\mathbbm{E}(\boldsymbol{u}_{\cdot
t}\boldsymbol{u}_{\cdot t}^{\prime})$ is the covariance matrix of the
idiosyncratic error term, $u_{it}$, $\underset{N\times 1}{\boldsymbol{\iota}}$
is a vector of ones and (2) the share weights, which we simply refer to as
size weights,
$\boldsymbol{S}:=\begin{pmatrix}S_{1}&\dots&S_{N}\end{pmatrix}^{\prime}$. Let
$\widetilde{X}_{it}=X_{it}-\bar{X}_{t}$, where
$\bar{X}_{t}=\frac{1}{N}\sum_{i=1}^{N}X_{it}$, denote a cross-sectionally
demeaned variable. Unless otherwise specified, we denote the $L^{2}$-norm as
$||\cdot||$ or sometimes explicitly as $||\cdot||_{2}$, the $L^{1}$-norm as
$||\cdot||_{1}$ and the Frobenius norm as $||\cdot||_{F}$; if another norm is
used, it will be explicitly noted. Given a square matrix $\boldsymbol{A}$, let
$\gamma_{max}(\boldsymbol{A})$ denote the maximum eigenvalue of
$\boldsymbol{A}$. Joint convergence of $N$ and $T$ will be denoted as
$(N,T)\overset{j}{\rightarrow}\infty$ without any restriction on the relative
rates; whenever restrictions on relative rates of convergence are imposed, it
will be explicitly noted. The expression $\overset{p}{\rightarrow}$ denotes
convergence in probability while $\overset{d}{\rightarrow}$ denotes
convergence in distribution. The equation
$\boldsymbol{y}=\mathcal{O}_{p}(\boldsymbol{x})$ states that the vector of
random variables $\boldsymbol{y}$ is at most of order $\boldsymbol{x}$ in
probability. The equation $a=\Theta_{p}(b)$ states that $a$ is stochastically
bounded by $b$ and $b$ is stochastically bounded by $a$, hence $a$ and $b$
rise jointly proportionally.

##  2 Model

A general formulation of the model examined in this paper is given in the
following panel simultaneous equations model with factor error structure
$\displaystyle\boldsymbol{y}_{it}$
$\displaystyle=\boldsymbol{B}\boldsymbol{x}_{it}+\boldsymbol{C}\boldsymbol{a}_{t}+\boldsymbol{v}_{it},$
$\displaystyle\boldsymbol{v}_{it}$
$\displaystyle=\boldsymbol{\Lambda}_{i}^{\prime}\boldsymbol{F}_{t}+\boldsymbol{u}_{it},$
where
$\boldsymbol{y}_{it}=\begin{pmatrix}y_{1,it}&\dots&y_{G,it}\end{pmatrix}^{\prime}$
is a $G\times 1$ vector of dependent variables,
$\boldsymbol{x}_{it}=\begin{pmatrix}x_{1,it}&\dots&x_{k_{x},it}\end{pmatrix}^{\prime}$
is a $k_{x}\times 1$ vector of strictly exogenous variables (which can be
arbitrarily correlated with the common factors, $\boldsymbol{F}_{t}$, and/or
the loadings, $\boldsymbol{\Lambda}_{i}$),
$\boldsymbol{a}_{t}=\begin{pmatrix}a_{1,t}&\dots&a_{k_{a},t}\end{pmatrix}^{\prime}$
is a $k_{a}\times 1$ vector of potentially endogenous aggregate variables,
$\boldsymbol{v}_{it}$ is a $G\times 1$ vector of composite error terms which
admit a low-rank plus sparse (factor structure) error decomposition, where
$\boldsymbol{\Lambda}_{i}$ is an $r\times G$ matrix of latent factor loadings
and $\boldsymbol{F}_{t}$ is an $r\times 1$ vector of latent factors. In our
exposition, we focus on the canonical setting of estimating the supply and
demand elasticities in the global crude oil market, so we set the dimension of
$G=2$ for supply and demand variables respectively. We take $k_{x}=0$ for ease
of exposition but we present a general estimation algorithm for when
$k_{x}\neq 0$. Moreover, we assume that only one of the $G=2$ variables has a
panel structure, whereas the other variable is an aggregate time series. That
is, $\boldsymbol{y}_{it}=\begin{pmatrix}d_{t}&y_{it}\end{pmatrix}^{\prime}$
where $d_{t}$ is the log change of aggregate crude oil consumption and
$y_{it}$ is the log change of country $i$’s crude oil production,
$a_{t}=p_{t}$, with $k_{a}=1$, is the log change of real crude oil price
(where we deflate the nominal oil price with the U.S. general price
index).333One may wonder why $p_{t}$ is not disaggregated; in fact the $p_{t}$
we use can be considered as the weighted average of country specific real oil
prices (in changes). As shown in [62], for a proper global analysis, deflating
the nominal oil price in U.S. dollars by the U.S. price index is generally
theoretically invalid unless the law of one price holds universally. Namely,
let $P_{it}$ denote the general price index faced by country $i$, $E_{it}$
denotes country $i$’s exchange rate measured as units of country $i$’s
currency per U.S. dollar, $p_{it}$ denote country specific log of real oil
prices and $\tilde{p}_{t}$ denotes nominal oil prices in U.S. dollars, if
$E_{it}P_{US,t}=P_{it}\,\,\forall i$; then it follows that
$\sum_{i=1}^{N}w_{i}p_{it}=\tilde{p}_{t}+\sum_{i=1}^{N}w_{i}\text{log}(E_{it}/P_{it})=\tilde{p}_{t}+\sum_{i=1}^{N}w_{i}\text{log}(1/P_{US,t})=\tilde{p}_{t}-p_{US,t}:=p_{t}$.
As it turns out, $p_{t}=\tilde{p}_{t}-p_{US,t}$ is an appropriate
approximation as documented in [62] for their long run analysis, in the sense
that it respects the long-run equilibrium relationships. We assume it is an
appropriate approximation for our short-run analysis.,444The main results
extend relatively naturally to the case where both variables have a panel
model. Given our stylizations the coefficient matrix $\boldsymbol{C}$ and
composite error, $\boldsymbol{v}_{it}$ becomes $\displaystyle\boldsymbol{C}$
$\displaystyle=\begin{pmatrix}\phi^{d}&0\\\ 0&\phi^{s}\end{pmatrix},$
$\displaystyle\boldsymbol{v}_{it}$
$\displaystyle=\boldsymbol{\Lambda}_{i}^{\prime}\boldsymbol{F}_{t}+\boldsymbol{u}_{it}=\begin{pmatrix}1&0\\\
0&\boldsymbol{\lambda}_{i}\end{pmatrix}\begin{pmatrix}\varepsilon_{t}\\\
\boldsymbol{\eta}_{t}\end{pmatrix}+\begin{pmatrix}0\\\ u_{it}\end{pmatrix},$
where the coefficients $\phi^{d}$ and $\phi^{s}$ denote the crude oil demand
and supply elasticities, respectively, and
$\boldsymbol{\eta}_{t},\boldsymbol{\lambda}_{i}$ are $r\times 1$ vectors of
latent factors and latent loadings, respectively. Our stylized simultaneous
equations model takes the simple form $\displaystyle d_{t}$
$\displaystyle=\phi^{d}p_{t}+\varepsilon_{t}$ (1) $\displaystyle y_{it}$
$\displaystyle=\phi^{s}p_{t}+\boldsymbol{\lambda}_{i}^{\prime}\boldsymbol{\eta}_{t}+u_{it}.$
(2) The global market clearing condition is given by $y_{St}=d_{t}$, where
$y_{St}:=\boldsymbol{S}^{\prime}\boldsymbol{y}_{\cdot
t}=\sum_{i=1}^{N}S_{i}y_{it}$, $\boldsymbol{S}$ is the $N\times 1$ vector of
shares that are normalized such that $\sum_{i=1}^{N}S_{i}=1$ and $i$ and $t$
take the values $i=1,\dots,N$ and $t=1,\dots,T$, respectively.555As oil is a
storable good, one could easily allow oil prices to adjust to the gap between
supply and demand, e.g. as in [62]. This introduces more complex notations
without adding any substance to the main points of the paper. Making use of
the global market clearing condition we see that $\displaystyle
p_{t}=\frac{1}{\phi^{d}-\phi^{s}}\left(u_{St}+\boldsymbol{\lambda}_{S}^{\prime}\boldsymbol{\eta}_{t}-\varepsilon_{t}\right),$
(3) which makes the simultaneity clear, e.g., that prices are composed of
size-weighted idiosyncratic shocks, aggregate supply shocks and the demand
shock. The objective of the GIV methodology is to extract the idiosyncratic
shocks and use them as instruments for price. Demand estimation in the case of
uniform loadings
($\boldsymbol{\lambda}_{i}=\boldsymbol{\lambda}\,\forall\,i$). To momentarily
fix ideas, it is helpful to consider a major simplification when constructing
the instrument. Suppose that the loadings are uniform,
$\boldsymbol{\lambda}_{i}=\boldsymbol{\lambda}\,\forall i$. Then, the
instrument$,z_{t}$, can be formed as $\displaystyle z_{t}$
$\displaystyle=y_{St}-\frac{1}{N}\sum_{i=1}^{N}y_{it}=(\phi^{d}p_{t}+\boldsymbol{\lambda}^{\prime}\boldsymbol{\eta}_{t}+u_{\boldsymbol{S}t})-(\phi^{d}p_{t}+\boldsymbol{\lambda}^{\prime}\boldsymbol{\eta}_{t}+\frac{1}{N}\sum_{i=1}^{N}u_{it}),$
$\displaystyle=u_{St}-\frac{1}{N}\sum_{i=1}^{N}u_{it}:=u_{\Gamma t},$ (4)
where $\boldsymbol{\Gamma}:=\boldsymbol{S}-\boldsymbol{\iota}/N$ is an
$N\times 1$ random vector such that
$\boldsymbol{\iota}^{\prime}\boldsymbol{\Gamma}=\sum_{i=1}^{N}\Gamma_{i}=0$,
by construction. $\Gamma_{i}$ is random because we assume the shares follow a
fat-tailed distribution, see Assumption 4. Identification and estimation of
demand by GIV requires that
$\displaystyle\mathbbm{E}(z_{t}\varepsilon_{t})=\mathbbm{E}\left(\sum_{i=1}^{N}\Gamma_{i}\mathbbm{E}(u_{it}\varepsilon_{t}|\boldsymbol{\Gamma})\right)=0.$
(5) (5) is our exogeneity condition and (3) gives $\mathbbm{E}(z_{t}p_{t})\neq
0$, relevance. A sufficient condition for the moment condition in (5) to be
zero is $\mathbbm{E}(u_{it}\varepsilon_{t}|\boldsymbol{\Gamma})=0$, which
effectively requires that conditional on size, $u_{it}$ and $\varepsilon_{t}$
are uncorrelated. Given relevance, exogeneity implies the following demand
elasticity estimator
$\widehat{\phi}^{d}=\frac{\sum_{t}d_{t}z_{t}}{\sum_{t}p_{t}z_{t}}$.
Intuitively, $z_{t}$ places larger weights on the idiosyncratic shocks to
larger oil producers, these granular shocks will shift the supply curve while
keeping the aggregate demand curve fixed since demand responds to these shocks
only through their affects on prices. This allows for consistent estimation of
the demand elasticity. The uniform loadings assumption in this case
tremendously facilitate the analysis. Uniform loadings allow one to construct
the instrument, as in (4), from observables. Supply estimation in the case of
uniform loadings and $u_{it}\,\,\boldsymbol{i.i.d.}$ Continuing on with the
uniform loadings case, remarkably, GK show that one can use the same
instrument, $z_{t}$, to also estimate the supply elasticity using a cross-
sectionally aggregated supply equation. Now, GK further assume that $u_{it}$
are $i.i.d.$, $\mathbbm{E}(\boldsymbol{u}_{\cdot t}\boldsymbol{u}_{\cdot
t}^{\prime}):=\boldsymbol{\Sigma}_{u}=\sigma^{2}_{u}\boldsymbol{I}_{N}$, where
$\boldsymbol{u}_{\cdot
t}:=\begin{pmatrix}u_{1t}&\dots&u_{Nt}\end{pmatrix}^{\prime}$ and
$\boldsymbol{I}_{N}$ is the identity matrix and define the $N\times 1$
precision weight vector
$\boldsymbol{E}:=\frac{\boldsymbol{\Sigma}^{-1}_{u}\boldsymbol{\iota}}{\boldsymbol{\iota}^{\prime}\boldsymbol{\Sigma}_{u}^{-1}\boldsymbol{\iota}}$
which reduces to $\boldsymbol{\iota}/N$ when $u_{it}$ are $i.i.d.$ across $i$.
Aggregation of the supply equation is performed using the vector
$\boldsymbol{E}$, we have that
$y_{Et}=\phi^{s}p_{t}+\boldsymbol{\lambda}^{\prime}\boldsymbol{\eta}_{t}+u_{Et}.$
Identification and estimation of supply by GIV requires that the instrument
satisfies exogeneity with respect to the composite error term
$\displaystyle\mathbbm{E}((\boldsymbol{\lambda}^{\prime}\boldsymbol{\eta}_{t}+u_{Et})z_{t})=0.$
(6) The first term in (6) has similar interpretation as in (5), i.e., size-
weighted idiosyncratic supply shocks are uncorrelated with the aggregate
supply component, $\boldsymbol{\lambda}^{\prime}\boldsymbol{\eta}_{t}$.
Miraculously, the second term is exactly zero
$\displaystyle\mathbbm{E}(u_{Et}z_{t})$
$\displaystyle=\mathbbm{E}(\boldsymbol{E}^{\prime}\boldsymbol{u}_{\cdot
t}\boldsymbol{u}_{\cdot
t}^{\prime}\boldsymbol{\Gamma})=\mathbbm{E}(\boldsymbol{E}^{\prime}\mathbbm{E}(\boldsymbol{u}_{\cdot
t}\boldsymbol{u}_{\cdot
t}^{\prime}|\boldsymbol{\Gamma})\boldsymbol{\Gamma})=\frac{\sigma^{2}_{u}}{N}\mathbbm{E}(\boldsymbol{\iota}^{\prime}\boldsymbol{\Gamma})=0.\text{}$
(7) The moment condition (7) is zero due to independence of $\Gamma_{i}$ and
$u_{it}$ by assumption and the sum-to-zero property of $\boldsymbol{\Gamma}$.
For identification with large $N$, we assume size to follow a power law in
tail (see Assumption 4), thus $\Gamma_{i}$ is stochastic and assumed to be
independent of $u_{it}$.777For a fixed $N$, it is not required to assume
independence of $\Gamma_{i}$ and $u_{it}$ because $\Gamma_{i}$ can be treated
as constant and (7) is zero solely by virtue of the fact that
$\boldsymbol{\iota}^{\prime}\boldsymbol{\Gamma}=0$. So again, we have
$\mathbbm{E}(z_{t}p_{t})\neq 0$ and for this simplified example, we avoid the
need to estimate the factor structure since (i) due to uniform loadings,
$z_{t}$ is constructed from observables and (ii) $z_{t}$ is uncorrelated with
the composite error term. If either of (i) or (ii) fails to hold, estimation
of the factor structure becomes a preliminary step, as in our general
procedure. Nevertheless, (6) leads to the following simple supply elasticity
estimator $\widehat{\phi}^{s}=\frac{\sum_{t}y_{Et}z_{t}}{\sum_{t}p_{t}z_{t}}$.
The intuition here is that, again, $z_{t}$ places larger weights on the
idiosyncratic shocks to larger oil producers, these granular shocks keep the
simple average (or more generally precision-weighted, i.e., weighted heavily
towards more stable oil producers) supply curve fixed. That is, on average,
precision-weighted supply responds to these granular shocks only through their
effects on prices (due to
$\mathbbm{E}((\boldsymbol{\lambda}^{\prime}\boldsymbol{\eta}_{t}+u_{Et})z_{t})=0$)
and at the same time since smaller oil producers take as given price changes
caused by these granular shocks, it will shift their demand curves which
enables consistent estimation of the supply elasticity. In the case of uniform
loadings and $u_{it}$ $i.i.d.$, the vector $\boldsymbol{E}$ and the instrument
are constructed from observables, the large sample properties of
$\widehat{\phi}^{s}$ and $\widehat{\phi}^{d}$ only entail fixed $N$, large $T$
asymptotics for which GK have laid out. In general, however, the cross-section
will need to be exploited to estimate $\boldsymbol{E}$ since one can not know
if $u_{it}$ are $i.i.d.$ across $i$. Indeed, the factors typically take care
of a substantial portion of the cross-sectional correlations but it is prudent
to allow for cross correlations in $u_{it}$ since the exogeneity condition for
estimation of the supply elasticity heavily exploits the structure of
$\boldsymbol{\Sigma}_{u}$. Therefore, it will be important to generally allow
for some weak cross correlations in $\boldsymbol{\Sigma}_{u}$, which our
algorithm accommodates, as discussed in Section 3 and Section 4. Although
homogeneous loadings was only an abstraction to illustrate the instrument, GK
advocate the use of $y_{\Gamma t}=y_{St}-\frac{1}{N}\sum_{i}y_{it}$ in
practice even when the loadings are not uniform. In the general heterogeneous
loadings case, their instrument becomes $\displaystyle Z_{t}$
$\displaystyle:=y_{\Gamma t}=u_{\Gamma
t}+\boldsymbol{\lambda}_{\Gamma}^{\prime}\boldsymbol{\eta}_{t}.$ (8) They
label this instrument with a capital case convention, to distinguish it
because it is no longer solely composed of weighted idiosyncratic shocks,
$u_{\Gamma t}$, as the
$\boldsymbol{\lambda}_{\Gamma}^{\prime}\boldsymbol{\eta}_{t}$ term is
contaminating the instrument. However, this clever formulation is possible
because they advocate estimation of the factors in practice, which they
augment to their structural equations, thereby controlling for the second term
which can potentially make their moment conditions different from zero.

##  3 Feasible Granular Instrumental Variables

Homogeneous loadings are overly restrictive but relaxing this can be easily
accommodated in practice via PCA or iterative OLS-PCA methods, e.g., [6] or
[7] in a preliminary stage to construct an estimate of the instrument.888For
our theory, we assume a balanced panel. However, in the case of unbalanced
panels with data missing at random (which is beyond the scope of this paper)
one can instead use the [9] method or [14] method to estimate the factor
structure and the instrument. Although in GK’s asymptotic theory they assume
homogeneous loadings and that the instrument is exogenous with respect to the
composite error, which circumvents the need to estimate the factor structure,
they indeed advocate augmenting their structural equations with estimated
factors either via period-by-period cross sectional regressions when the
loadings are known or via PCA in the case of non-parametric (unknown)
loadings. GK abstract away from the sampling error in suggesting the use of
augmented factors, which only vanishes for both large $N$ and $T$. [11] and
[42] have developed the asymptotic distribution for structural parameters in
factor augmented regressions in time series and panel models respectively. In
this paper, a variant of their corresponding result is established in showing
the sampling error from estimating the high dimensional precision matrix, the
factors, as well as the instrument is negligible in the asymptotic
distribution of the structural parameters. The general heterogeneous loadings
case and $u_{it}$ non-$\boldsymbol{i.i.d.}$ Now we formulate the estimation
approach in the general case, which makes much heavier use of the cross-
section. When we cross-sectionally demean the supply equation and stack across
$i$ we obtain (recall $\widetilde{\boldsymbol{X}}$ denotes a generic demeaned
variate) $\displaystyle\boldsymbol{\widetilde{y}}_{\cdot
t}=\boldsymbol{\widetilde{\Lambda}}\boldsymbol{\eta}_{t}+\boldsymbol{\widetilde{u}}_{\cdot
t},$ (9) which is estimable with vanilla PCA when the factor structure is
strong.999Strong factors in the sense that
$\boldsymbol{\Lambda}^{\prime}\boldsymbol{\Lambda}/N\overset{p}{\rightarrow}\boldsymbol{\Sigma}_{\Lambda}>0$;
thus we assume the factors are strong/pervasive in the sense that a
significant fraction of cross-sectional units are affected by their presence.
Consistent estimation of weak factors is beyond the scope of this paper, see
for example [65], [15] or [35] for suitable conditions for which it is
possible. Even when estimable, their convergence rates are slower relative to
estimates of strong factors, e.g., see [13]. This will generally require
modifications to the limiting distributions we derive in this paper. Letting
$\boldsymbol{Q}=(\boldsymbol{I}_{N}-\boldsymbol{\widetilde{\Lambda}(\widetilde{\Lambda}^{\prime}\widetilde{\Lambda})^{-1}\widetilde{\Lambda}}^{\prime})$,
then $\boldsymbol{Q}\boldsymbol{\widetilde{y}}_{\cdot
t}=\boldsymbol{Q}\boldsymbol{\widetilde{u}}_{\cdot t}$, completely purges the
process of the common factors through the loading space. Premultiplying the
share weights gives the instrument $\displaystyle z_{t}$
$\displaystyle:=\boldsymbol{S}^{\prime}\boldsymbol{Q}\boldsymbol{\widetilde{y}}_{\cdot
t},$ (10)
$\displaystyle=\boldsymbol{S}^{\prime}\boldsymbol{Q}\boldsymbol{\widetilde{u}}_{\cdot
t}:=\boldsymbol{\Gamma}^{\prime}\boldsymbol{\widetilde{u}}_{\cdot t},$ (11)
where $\boldsymbol{\Gamma}:=\boldsymbol{QS}$ is unknown because
$\boldsymbol{Q}$ is unknown, but $\boldsymbol{Q}$ is easily estimated from
data. Once we have $\boldsymbol{\widehat{Q}}$, we form
$\widehat{z}_{t}=\boldsymbol{S}^{\prime}\boldsymbol{\widehat{Q}}\boldsymbol{\widetilde{y}}_{\cdot
t}$ from observables. Importantly, when
$\boldsymbol{\lambda}_{i}=\boldsymbol{\lambda}\,\forall i$, then
$\boldsymbol{\Gamma}=(\boldsymbol{I}_{N}-\boldsymbol{\widetilde{\Lambda}(\widetilde{\Lambda}^{\prime}\widetilde{\Lambda})^{-1}\widetilde{\Lambda}}^{\prime})\boldsymbol{S}=\boldsymbol{S}-\boldsymbol{\iota}/N$
as in the previous case with homogeneous loadings. This gives rise to a more
general demand elasticity estimator
$\displaystyle\widehat{\phi}^{d}=\widehat{\phi}^{d}(\boldsymbol{\widehat{z}})$
$\displaystyle=\frac{\sum_{t}d_{t}\widehat{z}_{t}}{\sum_{t}p_{t}\widehat{z}_{t}}.$
(12) In Section 6, we show that the demand elasticity can be estimated as if
the infeasible instrument, $z_{t}$, is used. In the case of the supply
elasticity, the estimator will additionally depend on the estimated
(potentially high dimensional) precision matrix. That is,
$\widehat{\phi}^{s}=\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{\widehat{\Sigma}}^{-1}_{u})$.
This creates the need to jointly estimate
$\boldsymbol{\widehat{\Sigma}}^{-1}_{u}$ to form $\boldsymbol{\widehat{E}}$ in
order to aggregate the panel to estimate $\widehat{\phi}^{s}$. We propose a
simple iterative procedure and show that the supply elasticity can be
estimated as if the infeasible precision matrix,
$\boldsymbol{\Sigma}_{u}^{-1}$, and instrument, $z_{t}$, were used. More
specifically, let
$y_{Et}=\phi^{s}p_{t}+\boldsymbol{\lambda}_{E}^{\prime}\boldsymbol{\eta}_{t}+u_{Et}:=\boldsymbol{f}_{t}^{\prime}\boldsymbol{\theta}^{s}+u_{Et},$
where
$\boldsymbol{\theta}^{s}=\begin{pmatrix}\phi^{s}&\boldsymbol{\lambda}_{E}^{\prime}\end{pmatrix}^{\prime}$
and
$\boldsymbol{f}_{t}=\begin{pmatrix}p_{t}&\boldsymbol{\eta}^{\prime}_{t}\end{pmatrix}^{\prime}$
are $(1+r)\times 1$ vectors. The remarkable result
$\mathbbm{E}(z_{t}u_{Et})=0$, shown in (7) for the previous simple example
with homogeneous loadings, continues to hold in this setting as well, with
$z_{t}=\boldsymbol{S}^{\prime}\boldsymbol{Q}\boldsymbol{\tilde{y}_{\cdot
t}}=\boldsymbol{S}^{\prime}\boldsymbol{Q}\boldsymbol{\tilde{u}_{\cdot t}}$ and
$\boldsymbol{\Gamma}=\boldsymbol{Q}\boldsymbol{S}$ (recall that
$\boldsymbol{\iota}^{\prime}\boldsymbol{\Gamma}=0$)
$\displaystyle\mathbbm{E}(u_{Et}z_{t})$
$\displaystyle=\mathbbm{E}\left(\boldsymbol{E}^{\prime}\boldsymbol{u}_{\cdot
t}\boldsymbol{\tilde{u}}_{\cdot
t}^{\prime}\boldsymbol{\Gamma}\right)=\mathbbm{E}(\boldsymbol{E}^{\prime}\boldsymbol{u}_{\cdot
t}(\boldsymbol{u}_{\cdot
t}-\bar{u}_{t}\boldsymbol{\iota})^{\prime}\boldsymbol{\Gamma})=\mathbbm{E}(\boldsymbol{E}^{\prime}\boldsymbol{u}_{\cdot
t}\boldsymbol{u}_{\cdot
t}^{\prime}\boldsymbol{\Gamma})-\mathbbm{E}(\boldsymbol{E}^{\prime}\boldsymbol{u}_{\cdot
t}\bar{u}_{t}\boldsymbol{\iota}^{\prime}\boldsymbol{\Gamma})$
$\displaystyle=\mathbbm{E}(\boldsymbol{E}^{\prime}\,\mathbbm{E}(\boldsymbol{u}_{\cdot
t}\boldsymbol{u}_{\cdot
t}^{\prime}|\boldsymbol{\Gamma})\boldsymbol{\Gamma})-0=\frac{1}{\boldsymbol{\iota}^{\prime}\boldsymbol{\Sigma}_{u}^{-1}\boldsymbol{\iota}}\mathbbm{E}\left(\boldsymbol{\iota}^{\prime}\boldsymbol{\Gamma}\right)=0.$
(13) So we have that (where the estimated factors self-instrument)
$\displaystyle\mathbbm{E}\left[\begin{pmatrix}z_{t}\\\
\boldsymbol{\eta}_{t}\end{pmatrix}\cdot u_{Et}\right]$
$\displaystyle=\mathbbm{E}\left[\begin{pmatrix}z_{t}\\\
\boldsymbol{\eta}_{t}\end{pmatrix}\cdot\left(y_{Et}-\phi^{s}p_{t}-\boldsymbol{\lambda}_{E}^{\prime}\boldsymbol{\eta}_{t}\right)\right]=\boldsymbol{0}.$
(14) However, given our interest lies in inference for $\phi^{s}$, it is
useful to stack over $t$,
$\boldsymbol{y}_{E}=\boldsymbol{p}\,\phi^{s}+\boldsymbol{\eta}\,\boldsymbol{\lambda}_{E}+\boldsymbol{u}_{E},$
where $\boldsymbol{y}_{E},\boldsymbol{p},$ and $\boldsymbol{u}_{E}$ are
$T\times 1$ vectors and $\boldsymbol{y}_{\widehat{E}}$ is the feasible
counterpart of $\boldsymbol{y}_{E}$. Let
$\boldsymbol{M}_{\widehat{\eta}}=(\boldsymbol{I}_{T}-\boldsymbol{\widehat{\eta}(\widehat{\eta}^{\prime}\widehat{\eta})^{-1}\widehat{\eta}}^{\prime})$,
then it follows from standard partitioned regression results that
$\displaystyle\widehat{\phi}^{s}=\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$
$\displaystyle=\dfrac{\boldsymbol{\widehat{z}}^{\,{}^{\prime}}\,\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{y}_{\widehat{E}}}{\boldsymbol{\widehat{z}}^{\,{}^{\prime}}\,\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{p}}.$
(15) As $\boldsymbol{\widehat{\Sigma}}_{u}^{-1}$ depends on
$\widehat{\phi}^{s}$, (15) generally requires an iterative estimation
procedure. To that end, note that if $\phi^{s}$ were known,
$y_{it}-p_{t}\phi^{s}=\boldsymbol{\lambda}_{i}^{\prime}\boldsymbol{\eta}_{t}+u_{it}$
follows an approximate factor structure. Thus, a covariance estimator,
$\boldsymbol{\widehat{\Sigma}}_{u}$, for the idiosyncratic part can be
obtained following [32] by applying thresholding to the eigenvalue
decomposition, $\frac{1}{T}\sum_{t=1}^{T}(\boldsymbol{y}_{\cdot
t}-\boldsymbol{\iota}p_{t}\phi^{s})(\boldsymbol{y}_{\cdot
t}-\boldsymbol{\iota}p_{t}\phi^{s})^{\prime}=\sum_{i=1}^{N}\gamma_{i}\boldsymbol{\xi}_{i}\boldsymbol{\xi}_{i}^{\prime},$
where $\gamma_{i}$ and $\boldsymbol{\xi}_{i}$ are the eigenvalues (sorted in
decreasing order) and corresponding eigenvectors of
$\frac{1}{T}\sum_{t=1}^{T}(\boldsymbol{y}_{\cdot
t}-\boldsymbol{\iota}p_{t}\phi^{s})(\boldsymbol{y}_{\cdot
t}-\boldsymbol{\iota}p_{t}\phi^{s})^{\prime}$ respectively. More specifically,
if $\phi^{s}$ were known, we have
$\displaystyle\boldsymbol{\widehat{\Sigma}}_{(\boldsymbol{y}_{\cdot
t}-\boldsymbol{\iota}p_{t}\phi^{s})}:=\sum_{i=1}^{r}\widehat{\gamma}_{i}\boldsymbol{\widehat{\xi}}_{i}\boldsymbol{\widehat{\xi}}_{i}^{\prime}+\boldsymbol{\widehat{\Sigma}_{u}}^{\mathbfcal{T}},$
(16) where
$\boldsymbol{\widehat{\Sigma}}_{u}^{\mathbfcal{{T}}}=\sum_{i=r+1}^{N}\widehat{\gamma}_{i}\boldsymbol{\widehat{\xi}}_{i}\boldsymbol{\widehat{\xi}}_{i}^{\prime}=(\widehat{\sigma}^{\mathcal{T}}_{u,ij})_{N\times
N}$,
$\displaystyle\widehat{\sigma}^{\mathcal{T}}_{u,ij}=\begin{cases}\widehat{\sigma}_{u,ii},\hskip
99.58464pti=j,\\\ h_{ij}(\widehat{\sigma}_{u,ij}),\hskip 79.66771pti\neq
j,\end{cases}$ (17) and $h_{ij}(\cdot)$ is a generalized shrinkage function of
[4].101010Examples of $h_{ij}(\cdot)$ include hard thresholding
$h_{ij}(x)=x\mathbbm{1}(|x|\geq\tau_{ij})$ and soft thresholding
$h_{ij}(x)=\text{sgn}(x)(|x|-\tau_{ij})_{+}$. The entry dependent threshold,
$\tau_{ij}>0$, can be defined as $C\omega_{T}\sqrt{\widehat{\alpha}_{ij}}$,
where
$\widehat{\alpha}_{ij}=\frac{1}{T}\sum_{t=1}^{T}(\widehat{u}_{it}\widehat{u}_{jt}-\widehat{\sigma}_{u,ij})^{2}$,
$\widehat{\sigma}_{u,ij}=\frac{1}{T}\sum_{t=1}^{T}\widehat{u}_{it}\widehat{u}_{jt}$
and
$\widehat{u}_{it}=y_{it}-\phi^{s}p_{t}-\boldsymbol{\widehat{\lambda}}_{i}^{\prime}\boldsymbol{\widehat{\eta}}_{t}$
for some predetermined decreasing sequence $\omega_{T}>0$ and $C>0$. The
choice of $C$ can be data driven; [32] choose $C$ through multifold cross-
validation to maintain positive definiteness of
$\boldsymbol{\widehat{\Sigma}}_{u}^{\mathbfcal{{T}}}(C)$. In our algorithm
below, we make use of the R package for POET, written by the authors [32]. Of
course, $\phi^{s}$ can not be known as it requires an estimate of
$\boldsymbol{\Sigma}_{u}^{-1}$. Thus, we now address joint estimation of
$\phi^{s}$ and $\boldsymbol{\Sigma}_{u}^{-1}$ in what follows and subsequently
establish that the sampling error in $\boldsymbol{\widehat{E}}$ is negligible
given some regularity conditions. The iterative procedure is summarized in
Algorithm 1 presented below. Algorithm 1 FGIV for $\phi^{s}$ (when $k_{x}=0$):
• Step 1: Run PCA on (9) and obtain
$\widehat{z}_{t}=\boldsymbol{S}^{\prime}\boldsymbol{\widehat{Q}}\boldsymbol{\widetilde{y}}_{\cdot
t}$ as the sample counterpart of (10). • Step 2: Initialize
$\boldsymbol{\widehat{\Sigma}}_{u}^{-1}=\boldsymbol{I}_{N}$. • Step 3: Obtain
$\boldsymbol{y}_{\widehat{E}}(\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$ and
$\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$
as in (15). • Step 4: Update $\boldsymbol{\widehat{\Sigma}}_{u}^{-1}$ by
inverting $\boldsymbol{\widehat{\Sigma}}_{u}^{\mathbfcal{T}}$ defined in (17),
$\boldsymbol{y}_{\widehat{E}}(\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$ and
$\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$.
• Step 5: Iterate Step 3 and Step 4 until convergence. When $r$ is unknown,
one can augment Step 1 and estimate $r$ using a procedure as in [10], [64] or
[3]; we use the $ER$ and $GR$ methods of [3] (hereafter AH). For more details
of the $ER$ and $GR$ methods, see Section C of the Supplementary Appendix.
FGIV algorithm accommodating cross-section specific covariates. When
$k_{x}\neq 0$ then the demeaning transformation from (9) results in
$\boldsymbol{\widetilde{y}}_{\cdot
t}=\boldsymbol{\widetilde{\Lambda}}\boldsymbol{\eta}_{t}+\boldsymbol{\widetilde{x}}_{\cdot
t}\boldsymbol{\beta}+\boldsymbol{\widetilde{u}}_{\cdot t},$ where
$\boldsymbol{\widetilde{x}}_{\cdot t}$ is an $N\times k_{x}$ matrix, which
leaves $\underset{k_{x}\times 1}{\boldsymbol{\beta}}$ as an additional
parameter to estimate. $\boldsymbol{\beta}$ can be easily estimated by
adapting the procedure of [8], which is generalizing [7], to handle
endogeneity of prices even after controlling for latent common factors. More
specifically,
$\displaystyle\boldsymbol{\beta}(\boldsymbol{\widetilde{\Lambda}},\boldsymbol{\eta}_{t},\boldsymbol{\Sigma}_{u}^{-1})$
$\displaystyle=\left(\sum_{t=1}^{T}\boldsymbol{\widetilde{x}}_{\cdot
t}^{\prime}\boldsymbol{\Sigma}_{u}^{-1}\boldsymbol{\widetilde{x}}_{\cdot
t}\right)^{-1}\sum_{t=1}^{T}\boldsymbol{\widetilde{x}}_{\cdot
t}^{\prime}\boldsymbol{\Sigma}_{u}^{-1}(\boldsymbol{\widetilde{y}}_{\cdot
t}-\boldsymbol{\widetilde{\Lambda}}\boldsymbol{\eta}_{t}),$ (18)
$\displaystyle(\boldsymbol{\widetilde{y}}_{\cdot
t}-\boldsymbol{\widetilde{x}}_{\cdot t}\boldsymbol{\beta})$
$\displaystyle=\boldsymbol{\widetilde{\Lambda}}\boldsymbol{\eta}_{t}+\boldsymbol{\widetilde{u}}_{\cdot
t},$ (19) since (19) follows a factor structure, the $T\times r$ factor
matrix, $\boldsymbol{\eta}(\boldsymbol{\beta},\boldsymbol{\Sigma}_{u}^{-1})$,
can be estimated using the principal components estimator whose columns are
the eigenvectors corresponding to the largest $r$ eigenvalues of the $T\times
T$ matrix
$(\boldsymbol{\widetilde{y}}_{\cdot\cdot}-\boldsymbol{\widetilde{x}}_{\cdot\cdot}(\boldsymbol{\beta}))\boldsymbol{\Sigma}_{u}^{-1}(\boldsymbol{\widetilde{y}}_{\cdot\cdot}-\boldsymbol{\widetilde{x}}_{\cdot\cdot}(\boldsymbol{\beta}))^{\prime}$,
where the $T\times N$ matrix
$\boldsymbol{\widetilde{x}}_{\cdot\cdot}(\boldsymbol{\beta}):=\begin{pmatrix}\boldsymbol{\widetilde{x}}_{1\cdot}\boldsymbol{\beta}&\dots&\boldsymbol{\widetilde{x}}_{N\cdot}\boldsymbol{\beta}\end{pmatrix}$
and
$\boldsymbol{\widetilde{\Lambda}}(\boldsymbol{\beta},\boldsymbol{\Sigma}_{u}^{-1})=\frac{1}{T}\sum_{t=1}^{T}(\boldsymbol{\widetilde{y}}_{\cdot
t}-\boldsymbol{\widetilde{x}}_{\cdot
t}\boldsymbol{\beta})\boldsymbol{\eta}_{t}^{\prime}(\boldsymbol{\beta},\boldsymbol{\Sigma}_{u}^{-1}).$
Thus, to deal with general (strictly exogenous) covariates,
$\boldsymbol{x}_{it}$, Algorithm 2 can be applied. Algorithm 2 FGIV for
$\phi^{s}$ (when $k_{x}\neq 0$): • Step 1: Initialize
$\boldsymbol{\widehat{\beta}}=\boldsymbol{0}$,
$\boldsymbol{\widehat{\Sigma}}_{u}^{-1}=\boldsymbol{I}_{N}$. • Step 2: Run PCA
on (19) to obtain
$\boldsymbol{\widehat{\eta}}_{t}(\boldsymbol{\widehat{\beta}},\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$
and
$\boldsymbol{\widehat{\widetilde{\Lambda}}}(\boldsymbol{\widehat{\beta}},\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$
as explained above. • Step 3: Update $\boldsymbol{\widehat{\beta}}$ as the
sample counterpart of (18). • Step 4: Obtain
$\widehat{z}_{t}=\boldsymbol{S}^{\prime}\boldsymbol{\widehat{Q}}(\boldsymbol{\widetilde{y}}_{\cdot
t}-\boldsymbol{\widetilde{x}}_{\cdot t}\boldsymbol{\widehat{\beta}})$. • Step
5: Initialize
$\boldsymbol{y}_{\widehat{E}}(\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$ and
$\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{\widehat{\Sigma}}_{u}^{-1})=\left(\boldsymbol{\widehat{z}}^{\,{}^{\prime}}\,\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{p}\right)^{-1}\boldsymbol{\widehat{z}}^{\,{}^{\prime}}\,\boldsymbol{M}_{\widehat{\eta}}\,(\boldsymbol{y}_{\widehat{E}}-\boldsymbol{x}_{i\cdot}\boldsymbol{\beta})$.
• Step 6: Update $\boldsymbol{\widehat{\Sigma}}_{u}^{-1}$ by inverting
$\boldsymbol{\widehat{\Sigma}}_{u}^{\mathbfcal{T}}$ defined in (17), where
$\widehat{\gamma}_{i}$ and $\boldsymbol{\widehat{\xi}}_{i}$ are the
eigenvalues and eigenvectors (sorted in decreasing order) corresponding to the
sample analog of
$\frac{1}{T}\sum_{t=1}^{T}(\boldsymbol{y}_{\cdot
t}-\boldsymbol{\iota}p_{t}\phi^{s}-\boldsymbol{x}_{\cdot
t}\boldsymbol{\beta})(\boldsymbol{y}_{\cdot
t}-\boldsymbol{\iota}p_{t}\phi^{s}-\boldsymbol{x}_{\cdot
t}\boldsymbol{\beta})^{\prime}$ respectively. • Step 7: Iterate Step 2 through
Step 6 until convergence. When $r$ is unknown, one can augment Step 2 and
iteratively estimate $r$ using the $ER$ and $GR$ methods of [3]. The main
takeaway is that when both $(N,T)$ are large, one can generalize the GIV
estimators proposed by GK along different dimensions; here we accommodate
latent heterogeneous loadings, latent factors and latent precision matrix
(e.g., $u_{it}$ can be weakly cross-correlated and heteroskedastic). As
mentioned earlier, we call the proposed estimators of the elasticities in
(12), Algorithm 1 and Algorithm 2 as FGIV estimators.

###### Remark 1

In principle, the theory for the estimators proposed in this paper allows for
$N\gg T$. This case is relevant in many empirical settings (e.g., empirical
industrial organization and finance). However, it may be beneficial to avoid
estimating the precision matrix for cases where $N\ll T$ (e.g., empirical
macro). But, as (7) and (13) show, to have a valid instrument for which the
moment equation is exactly zero, we must specify $\boldsymbol{\Sigma}_{u}$
correctly. This is the primary motivation for estimating the general precision
matrix in Algorithms 1 and 2. In order to avoid estimating the precision
matrix, we must assume (potentially erroneously) $u_{it}$ are cross-
sectionally independent. We now analyze the consequences of making this
assumption when in fact $u_{it}$ are cross-sectionally correlated. Suppose we
erroneously assume cross-sectional independence, then the vector
$\boldsymbol{E}$ reduces to $\boldsymbol{\iota}/N$ and we end up with the
following moment equation $\displaystyle\mathbbm{E}(u_{Et}z_{t})$
$\displaystyle=\mathbbm{E}\left(\boldsymbol{E}^{\prime}\boldsymbol{u}_{\cdot
t}\boldsymbol{\tilde{u}}_{\cdot
t}^{\prime}\boldsymbol{\Gamma}\right)=\mathbbm{E}(\boldsymbol{E}^{\prime}\boldsymbol{u}_{\cdot
t}(\boldsymbol{u}_{\cdot
t}-\bar{u}_{t}\boldsymbol{\iota})^{\prime}\boldsymbol{\Gamma})$
$\displaystyle=\mathbbm{E}(\boldsymbol{E}^{\prime}\mathbbm{E}(\boldsymbol{u}_{\cdot
t}\boldsymbol{u}_{\cdot
t}^{\prime}|\boldsymbol{\Gamma})\boldsymbol{\Gamma})-\mathbbm{E}(\boldsymbol{E}^{\prime}\boldsymbol{u}_{\cdot
t}\boldsymbol{\iota}^{\prime}\boldsymbol{\Gamma})\bar{u}_{t}=\frac{1}{N}\mathbbm{E}(\boldsymbol{\iota}^{\prime}\boldsymbol{\Sigma}_{u}\boldsymbol{\Gamma})-0=o(1).$
(20) Hence, $z_{t}$ is not a valid instrument in the traditional sense because
we allow $\mathbbm{E}(u_{Et}z_{t})\neq 0$ for any given sample. Nevertheless,
this moment converges to zero for large $N$. Indeed, the moment satisfies
$\mathbbm{E}(u_{Et}z_{t})=o\left(1\right)$ under our regularity assumptions,
and thus, $z_{t}$ is asymptotically a valid instrument.111111It can be shown
that
$\boldsymbol{\iota}^{\prime}\boldsymbol{\Sigma}_{u}\boldsymbol{\Gamma}=\boldsymbol{\iota}^{\prime}\boldsymbol{\Sigma}_{u}\boldsymbol{QS}\leq\boldsymbol{\iota}^{\prime}\boldsymbol{\Sigma}_{u}\boldsymbol{S}\gamma_{max}(\boldsymbol{Q})=\sum_{i,j}\sigma_{u,ij}S_{j}\leq\left(\sum_{i,j}\sigma^{2}_{u,ij}\right)^{1/2}||\boldsymbol{S}||^{2}_{2}=||\boldsymbol{\Sigma}_{u}||_{F}||\boldsymbol{S}||^{2}_{2}\leq||\boldsymbol{\Sigma}_{u}||_{1}||\boldsymbol{S}||^{2}_{2}\leq\mathcal{O}(m_{N})\Theta_{p}(1)=o(N)\Theta_{p}(1)=o(N)$,
where $m_{N}$ is defined in Assumption 3 and $m_{N}=o(N)$. This insight
reveals that this moment is approaching zero, hence it may prove to be
beneficial to aggregate the panel, $y_{it}$, using weights
$\boldsymbol{\iota}/N$ regardless of the covariance structure. The immediate
implication is that
$\widehat{\phi}^{s}=\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{I}_{N})$,
so there is no need for an algorithmic estimation procedure, the simple
analytical formula for the supply elasticity estimator with potentially
misspecified covariance structure for $u_{it}$ is given by
$\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{I}_{N})=\frac{\boldsymbol{\widehat{z}}^{\,{}^{\prime}}\,\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{\bar{y}}}{\boldsymbol{\widehat{z}}^{\,{}^{\prime}}\,\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{p}},$
where $\boldsymbol{\bar{y}}$ stacks
$\bar{y}_{t}=\frac{1}{N}\sum_{i=1}^{N}y_{it}$ for each $t=1,\dots,T$; this
estimator is essentially Step 2 and Step 3 of Algorithm 1. Asymptotically, it
holds that
$\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{\widehat{\Sigma}}_{u}^{-1})=\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{I}_{N})+o_{p}(1).$
However, regarding performance in finite samples, when $u_{it}$ are not
$i.i.d.$ and when $N\ll T$, it is not clear ex-ante if
$\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{I}_{N})$ will
outperform
$\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$.
When $N\gg T$ one would expect ex-ante that
$\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{I}_{N})$ will be less
efficient than
$\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$
since the former is not optimally weighting the observations, whereas the
latter is. When $u_{it}$ are indeed $i.i.d.$ we would expect
$\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{I}_{N})$ to perform
better.121212In unreported simulations where $N\ll T$ and $u_{it}$ are
non-i.i.d., we find that
$\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$
typically has a smaller bias than
$\widehat{\phi}^{s}(\boldsymbol{\widehat{z}},\boldsymbol{I}_{N})$ (in absolute
terms, the bias of both estimators are very small) but with a slightly larger
variance.

##  4 Efficient GMM Estimation: Factor-Augmented FGIV

We now proceed to overidentify the elasticities, which yields overidentified
FGIV estimators. We will refer to the overidentified FGIV estimators simply as
efficient GMM estimators and the just identified FGIV estimators simply as
FGIV estimators. It will be of interest to practitioners to see if
overidentification is possible for the supply and demand equations. In this
section, we show that the system is indeed overidentified to varying degrees
for the supply and demand equations. Demand. It is common practice to assume
uncorrelated aggregate supply and aggregate demand shocks, that is
$\mathbbm{E}(\boldsymbol{\eta}_{t}\varepsilon_{t})=0$. When we are willing to
entertain this, then our supply factors, estimated via principal components,
serve as valid instruments in estimation of the demand elasticity, rendering
an overidentified parameter. In fact, the theory for using principal
components as instruments was laid out in [12] under strong instrument
asymptotics, as well as [51] under many/weak instrument asymptotics. In the
remainder of this section, we let the GIV be denoted as $z_{t,GIV}:=z_{t}$ to
distinguish it from the full instrument vector we introduce with upper case
conventions. Our full instrument matrix for the demand equation is
$\underset{T\times(1+r)}{\boldsymbol{Z}_{d}}:=\begin{pmatrix}\boldsymbol{z}_{GIV}&\boldsymbol{\eta}\end{pmatrix}$
with $\mathbbm{E}(\boldsymbol{Z}_{dt}\varepsilon_{t})=0$;
$\boldsymbol{Z}_{dt}$ simply augments factors to be used as instruments.
Making use of the $(1+r)\times 1$ dimensional moment condition, the efficient
GMM demand elasticity estimator is defined as
$\displaystyle\widehat{\phi}^{d}_{GMM}$
$\displaystyle=\underset{\phi^{d}}{\text{argmin}}\,\frac{\boldsymbol{\varepsilon}^{\prime}\boldsymbol{Z}_{d}}{T}\,\boldsymbol{W}_{d}\,\frac{\boldsymbol{Z}_{d}^{\prime}\boldsymbol{\varepsilon}}{T},$
$\displaystyle=\left(\boldsymbol{p}^{\prime}\,\boldsymbol{\widehat{Z}}_{d}\,\boldsymbol{\widehat{\Omega}}_{d}^{-1}\,\boldsymbol{\widehat{Z}}^{\,{}^{\prime}}_{d}\boldsymbol{p}\right)^{-1}\boldsymbol{p}^{\prime}\,\boldsymbol{\widehat{Z}}_{d}\,\boldsymbol{\widehat{\Omega}}_{d}^{-1}\,\boldsymbol{\widehat{Z}}^{\,{}^{\prime}}_{d}\boldsymbol{d},$
(21) where $\underset{(1+r)\times(1+r)}{\boldsymbol{W}_{d}}$ is an arbitrary
positive definite weight matrix, but is optimally set as
$\boldsymbol{\widehat{W}}_{d}=\boldsymbol{\widehat{\Omega}}_{d}^{-1}$, where
$\boldsymbol{\widehat{\Omega}}_{d}=\frac{1}{T}\sum_{t=1}^{T}\boldsymbol{\widehat{Z}}_{dt}\boldsymbol{\widehat{Z}}_{dt}^{\prime}(d_{t}-p_{t}\widehat{\phi}^{d}_{2SLS})^{2}$.
It is clear that (21) nests the FGIV estimator for the demand elasticity as a
special case. In this sense, $\widehat{\phi}^{d}_{GMM}$ will be robust to
scenarios where $z_{t}$ is weaker. Supply. In the same vein, the supply
elasticity can always be overidentified given our identifying assumptions
because $\mathbbm{E}(\varepsilon_{t}u_{Et})=0$ and thus $\varepsilon_{t}$ can
serve as an additional instrument. To estimate the entire parameter vector for
the supply equation, let
$\underset{T\times(2+r)}{\boldsymbol{Z}_{s}}:=\begin{pmatrix}\boldsymbol{z}_{GIV}&\boldsymbol{\varepsilon}&\boldsymbol{\eta}\end{pmatrix}$,
where the augmented factors self-instrument as they are part of the supply
equation. Then
$\boldsymbol{y}_{E}=\boldsymbol{f}\boldsymbol{\theta}^{s}+\boldsymbol{u}_{E}$
and recall
$\boldsymbol{\theta}^{s}=\begin{pmatrix}\phi^{s}&\boldsymbol{\lambda}_{E}^{\prime}\end{pmatrix}^{\prime}$
and
$\boldsymbol{f}_{t}=\begin{pmatrix}p_{t}&\boldsymbol{\eta}^{\prime}_{t}\end{pmatrix}^{\prime}$
are $(1+r)\times 1$ vectors and the matrix $\boldsymbol{f}$ is $T\times(1+r)$,
which stacks $\boldsymbol{f}_{t}$. We have
$\mathbbm{E}(\boldsymbol{Z}_{st}\boldsymbol{u}_{Et})=0$; hence, making use of
the $(2+r)\times 1$ dimensional moment conditions, the efficient GMM supply
elasticity estimator is defined as
$\displaystyle\boldsymbol{\widehat{\theta}}^{s}_{GMM}$
$\displaystyle=\underset{\boldsymbol{\theta}^{s}}{\text{argmin}}\,\frac{\boldsymbol{u}_{E}^{\prime}\boldsymbol{Z}_{s}}{T}\,\boldsymbol{W}_{s}\,\frac{\boldsymbol{Z}_{s}^{\prime}\boldsymbol{u}_{E}}{T},$
$\displaystyle=\left(\boldsymbol{\widehat{f}}^{\prime}\boldsymbol{\widehat{Z}}_{s}\,\boldsymbol{\widehat{\Omega}}_{s}^{-1}\,\boldsymbol{\widehat{Z}}^{\,{}^{\prime}}_{s}\boldsymbol{\widehat{f}}\right)^{-1}\boldsymbol{\widehat{f}}^{\prime}\boldsymbol{\widehat{Z}}_{s}\,\boldsymbol{\widehat{\Omega}}_{s}^{-1}\,\boldsymbol{\widehat{Z}}^{\,{}^{\prime}}_{s}\boldsymbol{y}_{\widehat{E}}.$
(22) where $\underset{(2+r)\times(2+r)}{\boldsymbol{W}_{s}}$ is an arbitrary
positive definite weight matrix, but is also optimally set as
$\boldsymbol{\widehat{W}}_{s}=\boldsymbol{\widehat{\Omega}}_{s}^{-1}$, where
$\boldsymbol{\widehat{\Omega}}_{s}=\frac{1}{T}\sum_{t=1}^{T}\boldsymbol{\widehat{Z}}_{st}\boldsymbol{\widehat{Z}}_{st}^{\prime}(y_{\widehat{E}t}-\widehat{\boldsymbol{f}}^{\prime}_{t}\boldsymbol{\widehat{\theta}}^{s}_{GMM})^{2}$.131313In
the case of the demand elasticity estimator in (21) we use 2SLS residuals to
construct $\boldsymbol{\widehat{\Omega}}_{d}$. However, we implement (22) via
Algorithm 3 which, by iteration, renders the residuals used to construct
$\boldsymbol{\widehat{\Omega}}_{s}$ to be GMM residuals. It is clear that (22)
nests the FGIV estimator for the supply equation as a special case. As in the
just identified case in (15), $\widehat{\boldsymbol{\theta}}^{s}_{GMM}$ in
(22) depends on $\widehat{\boldsymbol{\Sigma}}_{u}^{-1}$, hence, will
generally require an iterative estimation procedure. Algorithm 3 below
generalizes Algorithm 1 by extending the joint estimation of the supply
elasticity estimator and the precision matrix to the overidentified case for
when $k_{x}=0$. In view of Algorithm 2, Algorithm 3 can be further extended to
the case when $k_{x}>0$, but we omit the details for brevity. Algorithm 3
Efficient GMM for $\phi^{s}$ (when $k_{x}=0$): • Step 1: Run PCA on (9) and
obtain
$\widehat{z}_{t}=\boldsymbol{S}^{\prime}\boldsymbol{\widehat{Q}}\boldsymbol{\widetilde{y}}_{\cdot
t}$ as the sample counterpart of (10). • Step 2: Initialize
$\boldsymbol{\widehat{\Sigma}}_{u}^{-1}=\boldsymbol{I}_{N}$. • Step 3:
Estimate (21) to obtain $\boldsymbol{\widehat{\varepsilon}}$, initialize
$\boldsymbol{\widehat{W}}_{s}=(\boldsymbol{\widehat{Z}}_{s}^{\prime}\boldsymbol{\widehat{Z}}_{s})^{-1}$
and obtain
$\boldsymbol{\widehat{\theta}}_{2SLS}^{s}(\boldsymbol{\widehat{Z}}_{s},\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$.
• Step 4: Obtain
$\boldsymbol{y}_{\widehat{E}}(\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$. • Step
5: Update
$\boldsymbol{\widehat{W}}_{s}=\left(\frac{1}{T}\sum_{t=1}^{T}\boldsymbol{\widehat{Z}}_{st}\boldsymbol{\widehat{Z}}^{\prime}_{st}\widehat{u}^{2}_{\widehat{E}t}\right)^{-1}$,
where
$\widehat{u}_{\widehat{E}t}=y_{\widehat{E}t}-\boldsymbol{\widehat{\theta}}_{GMM}^{s}(\boldsymbol{\widehat{Z}}_{s},\boldsymbol{\widehat{\Sigma}}_{u}^{-1})^{\prime}\boldsymbol{\widehat{f}}_{t}$
and construct
$\boldsymbol{\widehat{\theta}}_{GMM}^{s}(\boldsymbol{\widehat{Z}}_{s},\boldsymbol{\widehat{\Sigma}}_{u}^{-1})$
as the sample counterpart of (22). • Step 6: Update
$\boldsymbol{\widehat{\Sigma}}_{u}^{-1}$ by inverting
$\boldsymbol{\widehat{\Sigma}}_{u}^{\mathbfcal{T}}$ defined in (17). • Step 7:
Iterate Step 4 through Step 6 until convergence. In addition to efficiency
gains, the efficient GMM estimators exhibit superior finite sample properties
and are also robust to the GIV itself being a weak instrument. We illustrate
these points in greater detail in Remark 5 and Section 7. The intuition for
the overidentified estimators can be seen from observing the reduced form
equation for (equilibrium) prices,
$p_{t}=\frac{1}{\phi^{d}-\phi^{s}}\left(u_{St}+\boldsymbol{\lambda}_{S}^{\prime}\boldsymbol{\eta}_{t}-\varepsilon_{t}\right)$.
Clearly $\mathbbm{E}(p_{t}\boldsymbol{\eta}_{t})\neq 0$ and
$\mathbbm{E}(p_{t}\varepsilon_{t})\neq 0$ and so instrumental relevancy is
established. Thus, we are effectively back to the classical approach of
finding exogenous supply shifters, in this case $\varepsilon_{t}$, to estimate
the supply elasticity and finding exogenous demand shifters, in this case
$\boldsymbol{\eta}_{t}$, to estimate the demand elasticity. With the exception
that these shifters, $\boldsymbol{\eta}_{t}$ and $\varepsilon_{t}$ are
unobserved. In what follows, we show that estimating $\boldsymbol{\eta}_{t}$
and $\varepsilon_{t}$ has a negligible effect on the limiting distributions of
the estimators of demand and supply elasticities, respectively.

##  5 Assumptions

Below we lay out the assumptions needed to derive our main results. Assumption
1, Assumption 2 and Assumption 3 are standard in the literature; see, for
example, [6], [32] and [8], and so we just state them for completeness.
Whereas, Assumption 4 parts ii.) and iii.) are new so we provide more details.

######  Assumption 1Factor Error Structure

The composite error term in (2) is assumed to admit an (approximate) factor
structure representation
$v_{it}:=\boldsymbol{\lambda}^{\prime}_{i}\boldsymbol{\eta}_{t}+u_{it},$ where
$\boldsymbol{\eta}_{t}=\begin{pmatrix}\eta_{1t}&\dots&\eta_{rt}\end{pmatrix}^{\prime}$
is an $r\times 1$ vector of latent common factors and
$\boldsymbol{\lambda}_{i}=\begin{pmatrix}\lambda_{1i}&\dots&\lambda_{ri}\end{pmatrix}^{\prime}$
is an $r\times 1$ vector of latent factor loadings. We assume the factors are
pervasive in the sense that
$\boldsymbol{\Lambda}^{\prime}\boldsymbol{\Lambda}/N$ converges to some
$r\times r$ positive definite matrix.

###### Assumption 2

_(Strict Stationarity, Exponential Tails & Strong Mixing)_
$\textbf{(A2i.)}\,\,\\{\boldsymbol{\eta}_{t},u_{it},\varepsilon_{t}\\}_{t\geq
1}$ is strictly stationary and each with a zero mean.
$\textbf{(A2ii.)}\,\,\exists\,\,c_{1},c_{2}>0$ with
$\gamma_{\emph{min}}(\boldsymbol{\Sigma}_{u})>c_{2}$, $\underset{j\leq
N}{\emph{max}}||\gamma_{j}||<c_{1}$,
$c_{2}<\gamma_{\emph{min}}(\mathbbm{cov}(\boldsymbol{\eta}_{t}))\leq\gamma_{\emph{max}}(\mathbbm{cov}(\boldsymbol{\eta}_{t}))<c_{1}.$
(A2iii.) Exponential tail: $\exists\,\,r_{1},r_{2}>0$ and $b_{1},b_{2}>0$,
such that for any $s>0$, $i\leq N$ and $j\leq r$,
$\mathbbm{P}(|u_{it}|>s)\leq\emph{exp}(-(s/b_{1})^{r_{1}})),$ and
$\mathbbm{P}(|\eta_{t,j}|>s)\leq\emph{exp}(-(s/b_{2})^{r_{2}})).$
(A2iv.) Strong Mixing: $\exists\,\,r_{3},\,C>0$
$\forall\,\,T>0,\,\,r_{1}^{-1}+r_{2}^{-1}+r_{3}^{-1}>1$,
$\underset{A\in\mathcal{F}_{-\infty}^{0},\,\,B\in\mathcal{F}_{T}^{\infty}}{\emph{sup}}|\mathbbm{P}(A)\mathbbm{P}(B)-\mathbbm{P}(AB)|<\emph{exp}(-CT^{r_{3}}),$
where $\mathcal{F}_{-\infty}^{0}$ and $\mathcal{F}_{T}^{\infty}$ denote the
$\sigma$-algebras generated by
$\\{(\boldsymbol{\eta}_{t},u_{it},\varepsilon_{t}):t<0\\}$ and
$\\{(\boldsymbol{\eta}_{t},u_{it},\varepsilon_{t}):t>T\\}$ respectively.

######  Assumption 3Sparsity on $\boldsymbol{\Sigma}_{u}$

Let $\boldsymbol{\Sigma}_{u}=(\sigma_{u,ij})$, for some $q\in[0,1/2),$ define
$\displaystyle m_{N}=\underset{i\leq
N}{\emph{max}}\sum_{j=1}^{N}|\sigma_{u,ij}|^{q}.$ (23) We require that there
is $q\in[0,1/2)$ such that $m_{N}\omega_{N,T}^{1-q}=o(1)$, where
$\omega_{N,T}=\sqrt{\frac{\emph{log}(N)}{T}}+\frac{1}{\sqrt{N}}$.

###### Assumption 4

_(Identification by GIV)_
$\textbf{(A4i.)}\,\,\,\mathbbm{E}(z_{t}u_{Et})=\mathbbm{E}(z_{t}\varepsilon_{t})=\mathbbm{E}(\boldsymbol{Z}_{st}u_{Et})=\mathbbm{E}(\boldsymbol{Z}_{dt}\varepsilon_{t})=0$.
(A4ii.) The sizes $\mathscr{S}_{1},\dots,\mathscr{S}_{N}$ are drawn $i.i.d.$
from an arbitrary distribution for which the tail of the size distribution
(i.e. above some threshold) follows a power law, with tail index, $\mu>0$
$\displaystyle\mathbbm{P}(\mathscr{S}>s)=cs^{-{\mu}}.$ The tail index $\mu$
determines the probability of observing extreme values. We assume that
$\mathscr{S}_{i}$ is independent of $u_{it}$.
(A4iii.) Suppose the sizes are ordered in decreasing fashion as such:
$\mathscr{S}_{1}\geq\mathscr{S}_{2}\geq\dots\geq\mathscr{S}_{N-1}\geq\mathscr{S}_{N},$
and we partition the cross-section as,
$\mathcal{N}_{dominant}:=\\{1,\dots,N_{1}\\}$ and
$\mathcal{N}_{fringe}:=\\{N_{1}+1,\dots,N\\}$ such that
$\mathcal{N}_{dominant}\cup\mathcal{N}_{fringe}:=\mathcal{N}_{full}$. Let
$S_{i}=\frac{\mathscr{S}_{i}}{\sum_{j=1}^{N}\mathscr{S}_{j}}$ denote the
normalized shares such that $\sum_{i}S_{i}=1$. We assume
$\forall\,i\,\in\mathcal{N}_{dominant}$, $\,S_{i}=\Theta_{p}(1)$, and
$\forall\,i\in\mathcal{N}_{fringe},$
$S_{i}=\mathcal{O}_{p}\left(\frac{1}{N}\right)$. We further assume that the
cardinality of the dominant units is fixed as $N\rightarrow\infty$, that is,
$|\mathcal{N}_{dominant}|=N_{1}$ and $N_{1}$ does not rise with $N$ while the
cardinality of the fringe grows with $N$,
$|\mathcal{N}_{fringe}|=N-N_{1}\rightarrow\infty$ as $N\rightarrow\infty$.

###### Remark 2

The first condition gives us instrumental exogeneity for the FGIV and
efficient GMM estimators. The second condition allows for instrumental
relevance in the extension of a large N framework. An important implication of
the second condition is that the Herfindahl index, $h_{N,\mu}$, has the
following asymptotic property
$\displaystyle\sqrt{h_{N,\mu}}=\left|\big{|}\boldsymbol{S}\big{|}\right|_{2}=\begin{cases}\Theta_{p}\left(1\right)\hskip
38.12666pt\text{for}\hskip 14.22636pt\mu\in[0,1),\\\
\mathcal{O}_{p}\left(g_{N,\mu}\right)\hskip 22.76219pt\text{for}\hskip
14.22636pt\mu\in[1,2),\end{cases}$ with
$\mathcal{O}_{p}\left(g_{N,\mu}\right)\gg 1/\sqrt{N}$. The variance of the
just identified estimators is inversely proportional to the Herfindahl index,
that is $\mathbbm{V}(\widehat{\phi}^{j}_{FGIV})=\mathcal{O}(h_{N,\mu}^{-1})$
for $j=s,d$, reflecting the fact that the more concentrated the market, the
more precise the GIV methodology will be and also reflecting the fact that if
the Herfindahl converges to zero in the limit, the variance will
diverge.141414The derivation of the asymptotic behavior of $h_{N,\mu}$ can be
found in Supplementary Appendix B. However, if $\mu$ is slightly greater than
1, theoretically identification breaks down for large $N$ but in any finite
sample the GIV could be relevant (precisely due to
$\mathcal{O}_{p}\left(g_{N,\mu}\right)\gg 1\sqrt{N}$). Nevertheless, we rule
this case out for the purpose of asymptotic inference.151515For more details
on instrumental relevance for large $N$, see Section 7. Note, that the third
condition is consistent with $\mu\in(0,1)$, but is slightly stronger. The
third condition is also a generalization of the so-called "granular" weights
in the panel data literature, say $\underset{{N\times 1}}{\boldsymbol{w}}$,
which are typically assumed to satisfy
$||\boldsymbol{w}||_{2}=\mathcal{O}\left(\frac{1}{\sqrt{N}}\right)$ and
$\frac{w_{i}}{||\boldsymbol{w}||_{2}}=\mathcal{O}\left(\frac{1}{\sqrt{N}}\right)\,\,\forall\,i$.
The third condition allows the share vector to be partitioned into a dominant
part and a fringe part. That is,
$\boldsymbol{S}=\begin{pmatrix}\boldsymbol{S}^{\prime}_{d}&\boldsymbol{S}^{\prime}_{f}\end{pmatrix}^{\prime}$
where $\boldsymbol{S}_{d}$ is $N_{1}\times 1$, is the dominant part and
$\boldsymbol{S}_{f}$ is $N_{2}\times 1$, is the fringe part; with
$N_{1}+N_{2}=N$, the key being that $N_{1}(N)=N_{1}$ is fixed while
$N_{2}(N)\rightarrow\infty$ as $N\rightarrow\infty$. This assumption can be
empirically justified in concentrated markets, see Section 9 as an example; as
well as mathematically justified, see [58].

###### Remark 3

Taking the variance of the equilibrium price process (assuming the covariances
to be zero for simplicity) we obtain
$\mathbbm{V}(p_{t})=\frac{1}{(\phi^{d}-\phi^{s})^{2}}(\mathbbm{V}(u_{St})+\mathbbm{V}(\boldsymbol{\lambda}_{S}^{\prime}\boldsymbol{\eta}_{t})+\mathbbm{V}(\varepsilon_{t}))=\Theta(1),$
where the last equality follows by the second and the third conditions in
Assumption 4, details can be found in Lemma 1 in the Appendix. Without these
conditions, one would obtain the unsatisfactory result that
$\mathbbm{V}(p_{t})=\mathcal{O}(N)$, that is, the variance of the price
process is unbounded for each $t$ as $N\rightarrow\infty$. Effectively,
Assumption 4 allows the coexistence of a finite number of dominant units, in
terms of size, whose cardinality can not grow with $N$, while at the same time
allowing for a bounded variance for the aggregate endogenous variable $p_{t}$.

##  6 Limiting Distributions

In this section, we first present the limiting distributions of the FGIV
elasticity estimators, corresponding to (12) and (15) with Algorithm 1. We
then move on to the limiting distributions of the efficient GMM elasticity
estimators, corresponding to (21) and (22) with Algorithm 3. Just identified
demand elasticity. The just identified demand elasticity estimator in (12) is
given by
$\displaystyle\widehat{\phi}^{d}(\boldsymbol{\widehat{z}})=\dfrac{\sum_{t=1}^{T}\widehat{z}_{t}d_{t}}{\sum_{t=1}^{T}\widehat{z}_{t}p_{t}}=\dfrac{\sum_{t=1}^{T}\sum_{i,j}S_{i}\widehat{Q}_{ij}\tilde{y}_{jt}d_{t}}{\sum_{t=1}^{T}\sum_{i,j}S_{i}\widehat{Q}_{ij}\tilde{y}_{jt}p_{t}}.$
Hence, $\displaystyle\widehat{\phi}^{d}-\phi^{d}$
$\displaystyle=\left(\sum_{t}\widehat{z}_{t}p_{t}\right)^{-1}\left(\sum_{t}\widehat{z}_{t}\varepsilon_{t}\right),$
$\displaystyle=\left(T^{-1}\sum_{t}z_{t}p_{t}+T^{-1}\sum_{t}(\widehat{z}_{t}-z_{t})p_{t}\right)^{-1}\left(T^{-1}\sum_{t}z_{t}\varepsilon_{t}+T^{-1}\sum_{t}(\widehat{z}_{t}-z_{t})\varepsilon_{t}\right).$
From above, it is apparent we need to show
$\frac{1}{T}\sum_{t=1}^{T}(\widehat{z}_{t}-z_{t})\varepsilon_{t}=\frac{1}{T}\sum_{t=1}^{T}\boldsymbol{S}^{\prime}(\boldsymbol{\widehat{Q}}-\boldsymbol{Q})\boldsymbol{\tilde{y}_{\cdot
t}}\varepsilon_{t}=o_{p}(1)$ and
$\frac{1}{T}\sum_{t=1}^{T}(\widehat{z}_{t}-z_{t})p_{t}=\frac{1}{T}\sum_{t=1}^{T}\boldsymbol{S}^{\prime}(\boldsymbol{\widehat{Q}}-\boldsymbol{Q})\boldsymbol{\tilde{y}}_{\cdot
t}p_{t}=o_{p}(1)$. Indeed, we show in Lemma 2, in the Appendix, that
$\displaystyle
T^{-1}\sum_{t=1}^{T}\boldsymbol{S}^{\prime}(\boldsymbol{\widehat{Q}}-\boldsymbol{Q})\boldsymbol{\tilde{y}}_{\cdot
t}\varepsilon_{t}=\mathcal{O}_{p}\left(\frac{1}{\text{min}\\{N,T\\}}\right)+\mathcal{O}_{p}\left(\frac{1}{\text{min}\\{N,\sqrt{NT}\\}}\right)$
$\displaystyle=o_{p}(1),$ (24) $\displaystyle
T^{-1}\sum_{t=1}^{T}\boldsymbol{S}^{\prime}(\boldsymbol{\widehat{Q}}-\boldsymbol{Q})\boldsymbol{\tilde{y}}_{\cdot
t}p_{t}=\mathcal{O}_{p}\left(\frac{1}{\text{min}\\{N,T\\}}\right)+\mathcal{O}_{p}\left(\frac{1}{\text{min}\\{N,\sqrt{NT}\\}}\right)$
$\displaystyle=o_{p}(1).$ (25) Thus, making use of (24) and (25) we obtain
$\displaystyle\widehat{\phi}^{d}-\phi^{d}$
$\displaystyle=\left(\sum_{t}\widehat{z}_{t}p_{t}\right)^{-1}\left(\sum_{t}\widehat{z}_{t}\varepsilon_{t}\right)=\left(T^{-1}\sum_{t}z_{t}p_{t}\right)^{-1}T^{-1}\sum_{t}z_{t}\varepsilon_{t}+o_{p}(1).$
(26) The order of the sampling error generally relies, in part, on the order
of the Herfindahl. The order of the Herfindahl, in turn, critically depends on
$\mu$, the tail index of the size distribution. Results on the order of the
Herfindahl as a function of the tail index parameter $\mu$ entails a total of
six possible cases. The results can be found in Table 6 of Supplementary
Appendix B. However, for inference, we require $\mu\in(0,1)$ (regularly
varying tails) or $\mu\rightarrow 0$ (slowly varying tails) as discussed in
detail in the previous section’s remarks. Given this, even after pinning down
the order of the Herfindahl, the panel dimensions can distinguish more cases
as seen above. Nevertheless, as (24), (25) and (26) indicate, for consistency
we have the following result:

######  Theorem 1 (Consistency of $\widehat{\phi}^{d}$)

Under Assumptions 1-4, as $(N,T)\overset{j}{\rightarrow}\infty$, we have
$\displaystyle\widehat{\phi}^{d}-\phi^{d}\overset{p}{\rightarrow}0.$ (27) Now,
multiplying (26) by $\sqrt{T}$
$\displaystyle\sqrt{T}(\widehat{\phi}^{d}-\phi^{d})$
$\displaystyle=\left(T^{-1}\sum_{t}z_{t}p_{t}\right)^{-1}\left(\frac{1}{\sqrt{T}}\sum_{t}z_{t}\varepsilon_{t}+\mathcal{O}_{p}\left(\frac{\sqrt{T}}{\text{min}\\{N,T\\}}\right)+\mathcal{O}_{p}\left(\frac{\sqrt{T}}{\text{min}\\{N,\sqrt{NT}\\}}\right)\right),$
(28) we can state the following result for the limiting distribution:

######  Theorem 2 (Limiting distribution for $\widehat{\phi}^{d}$)

Under Assumptions 1-4 as $(N,T)\overset{j}{\rightarrow}\infty$, we have that
when $N\geq T$ or $N<T$ and $\sqrt{T}/N\rightarrow 0$
$\displaystyle\sqrt{T}(\widehat{\phi}^{d}-\phi^{d})\overset{d}{\rightarrow}\mathcal{N}\left(0,\mathbbm{v}_{d}\right),$
(29) where
$\mathbbm{v}_{d}:=\mathbbm{m}_{zp}^{-2}\,\mathbbm{v}_{z\varepsilon}$,
$\mathbbm{v}_{z\varepsilon}:=\mathbbm{E}(z^{2}_{t}\varepsilon^{2}_{t})$ and
$\mathbbm{m}_{zp}:=\mathbbm{E}(z_{t}p_{t})$. $\mathbbm{v}_{z\varepsilon}$ can
be consistently estimated with
$\displaystyle\widehat{\mathbbm{v}}_{z\varepsilon}=\begin{cases}T^{-1}\sum_{t=1}^{T}\widehat{z}_{t}^{\,2}\,\widehat{\varepsilon}_{t}^{\,2}\hskip
252.37593pt\text{HC},\vspace{3mm}\\\
T^{-1}\sum_{t=1}^{T}\widehat{z}_{t}^{\,2}\,\widehat{\varepsilon}_{t}^{\,2}+2\cdot
T^{-1}\sum_{j=1}^{m}\left(1-\dfrac{j}{m+1}\right)\sum_{t=j+1}^{T}\widehat{z}_{t}\widehat{\varepsilon}_{t}\widehat{z}_{t-j}\widehat{\varepsilon}_{t-j}\hskip
14.22636pt\text{HAC},\end{cases}$ (30) where HC and HAC denote
heteroskedasticity-consistent and heteroskedasticity and autocorrelation
consistent estimators, respectively. Hence
$\frac{\sqrt{T}(\widehat{\phi}^{d}-\phi^{d})}{\widehat{\mathbbm{v}}_{d}^{1/2}}\sim\text{t}_{df}\overset{d}{\rightarrow}\mathcal{N}(0,1)$,
where
$\widehat{\mathbbm{v}}_{d}^{1/2}=\widehat{\mathbbm{m}}_{zp}^{-2}\widehat{\mathbbm{v}}_{z\varepsilon}^{1/2}$,
with $\widehat{\mathbbm{m}}_{zp}=T^{-1}\sum_{t=1}^{T}\widehat{z}_{t}p_{t}$
also consistent for $\mathbbm{m}_{zp}$. We will see in Section 8 that the
asymptotic theory provides good approximations to the finite sample
distribution.

###### Remark 4

As in GK, we express $\mathbbm{v}_{d}$ as inversely related to the Herfindahl,
$h_{N,\mu}$, as claimed in Remark 2, for insights on the role of market
concentration on precision of the GIV. Assuming conditional homoskedasticity
of $\varepsilon_{t}$ and homoskedasticity of $u_{it}$, we have that
$\displaystyle\mathbbm{v}_{z\varepsilon}$
$\displaystyle=\mathbbm{E}(z_{t}^{2}\varepsilon_{t}^{2})=\sigma^{2}_{\varepsilon}\cdot\sigma^{2}_{\tilde{u}}\cdot\mathbbm{E}(\boldsymbol{S}^{\prime}\boldsymbol{Q}\boldsymbol{S}).$
(31) If $\boldsymbol{\lambda}_{i}=\boldsymbol{\lambda}\,\,\forall i$, then
there is no need to purge the factor structure through the loading space. That
is, a simple cross-sectional demeaning transformation will suffice,
$\boldsymbol{Q}=(\boldsymbol{I}_{N}-\boldsymbol{\tilde{\Lambda}(\tilde{\Lambda}^{\prime}\tilde{\Lambda})^{-1}\tilde{\Lambda^{\prime}}})=(\boldsymbol{I}_{N}-\boldsymbol{\frac{\iota\iota^{\prime}}{N}})$.
We can simplify equation (31) to (where we make use of the normalization that
$\boldsymbol{S}^{\prime}\boldsymbol{\iota}=1$)
$\displaystyle\mathbbm{v}_{z\varepsilon}$
$\displaystyle=\sigma^{2}_{\varepsilon}\cdot\sigma^{2}_{\tilde{u}}\cdot\left(\mathbbm{E}(\boldsymbol{S^{\prime}S})-\frac{1}{N}\right)=\sigma^{2}_{\varepsilon}\cdot\underbrace{\sigma^{2}_{\tilde{u}}\cdot\left(\mathbbm{E}(h_{N,\mu})-\frac{1}{N}\right)}_{\text{$\mathbbm{E}(z_{t}^{2})$}},$
whereas,
$\mathbbm{m}_{zp}=\mathbbm{E}(p_{t}z_{t})\propto\mathbbm{E}(z_{t}^{2})$.
Hence, $\displaystyle\mathbbm{v}_{d}$
$\displaystyle\propto\dfrac{\sigma^{2}_{\varepsilon}\cdot\sigma^{2}_{\tilde{u}}\cdot\left(\mathbbm{E}(h_{N,\mu})-\frac{1}{N}\right)}{\left[\sigma^{2}_{\tilde{u}}\cdot\left(\mathbbm{E}(h_{N,\mu})-\frac{1}{N}\right)\right]^{2}}=\dfrac{\sigma^{2}_{\varepsilon}}{\sigma^{2}_{\tilde{u}}\cdot\left(\mathbbm{E}(h_{N,\mu})-\frac{1}{N}\right)}.$
(32) Thus, the more concentrated the market, the more precise the estimator.
See Section 7 for a more general treatment. Just identified supply elasticity.
For the just identified supply elasticity estimator in (15), upon convergence
of Algorithm 1, we have that $\displaystyle\widehat{\phi}^{s}-\phi^{s}$
$\displaystyle=\underbrace{\left(T^{-1}\boldsymbol{\widehat{z}}^{\,{}^{\prime}}\,\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{p}\right)^{-1}}_{\text{$\widehat{A}^{-1}$}}\,\underbrace{T^{-1}\boldsymbol{\widehat{z}}^{\,{}^{\prime}}\,\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{\eta}\cdot\boldsymbol{\lambda}_{\widehat{E}}}_{\text{$\widehat{B}$}}+\left(T^{-1}\boldsymbol{\widehat{z}}^{\,{}^{\prime}}\,\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{p}\right)^{-1}\,\underbrace{T^{-1}\boldsymbol{\widehat{z}}^{\,{}^{\prime}}\,\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{u}_{\widehat{E}}}_{\text{$\widehat{C}$}}.$
(33) We can write the scalars $\widehat{A}$, $\widehat{B}$ and $\widehat{C}$
as follows $\displaystyle\widehat{A}$
$\displaystyle=T^{-1}\boldsymbol{z}^{\prime}\,\boldsymbol{M}_{\eta}\,\boldsymbol{p}+\underbrace{T^{-1}(\boldsymbol{\widehat{z}}-\boldsymbol{z})^{\prime}\,\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{p}}_{\text{$a_{1}$}}+\underbrace{T^{-1}\boldsymbol{z}^{\prime}\,(\boldsymbol{M}_{\widehat{\eta}}-\boldsymbol{M}_{\eta})\,\boldsymbol{p}}_{\text{$a_{2}$}},$
(34) $\displaystyle\widehat{B}$
$\displaystyle=\underbrace{T^{-1}(\boldsymbol{\widehat{z}}-\boldsymbol{z})^{\prime}\,\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{\eta}\lambda_{\widehat{E}}}_{\text{$b_{1}$}}+\underbrace{T^{-1}\boldsymbol{z}^{\prime}\,(\boldsymbol{M}_{\widehat{\eta}}-\boldsymbol{M}_{\eta})\,\boldsymbol{\eta}\boldsymbol{\lambda}_{\widehat{E}}}_{\text{$b_{2}$}},$
(35) $\displaystyle\widehat{C}$
$\displaystyle=T^{-1}\boldsymbol{z}^{\prime}\,\boldsymbol{M}_{\eta}\,\boldsymbol{u}_{E}+\underbrace{T^{-1}(\boldsymbol{\widehat{z}}-\boldsymbol{z})^{\prime}\,\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{u}_{\widehat{E}}}_{\text{$c_{1}$}}+\underbrace{T^{-1}\boldsymbol{z}^{\prime}\,(\boldsymbol{M}_{\widehat{\eta}}-\boldsymbol{M}_{\eta})\,\boldsymbol{u}_{\widehat{E}}}_{\text{$c_{2}$}}+\underbrace{T^{-1}\boldsymbol{z}^{\prime}\,\boldsymbol{M}_{{\eta}}\,(\boldsymbol{u}_{\widehat{E}}-\boldsymbol{u}_{E})}_{\text{$c_{3}$}}.$
(36) It is shown in Lemma 3 of the Appendix that the terms $a_{i},b_{i},c_{j}$
are $o_{p}(1)$ for $i=1,2$; $j=1,2,3$, such that
$\displaystyle\widehat{\phi}^{s}-\phi^{s}$
$\displaystyle=\left(T^{-1}\boldsymbol{z}^{\prime}\,\boldsymbol{M}_{\eta}\,\boldsymbol{p}\right)^{-1}\left(T^{-1}\boldsymbol{z}^{\prime}\,\boldsymbol{M}_{\eta}\,\boldsymbol{u}_{E}\right)+o_{p}(1).$
(37) We can now state the following result:

######  Theorem 3 (Consistency of $\widehat{\phi}^{s}$)

Under Assumptions 1-4, as $(N,T)\overset{j}{\rightarrow}\infty$, we have
$\displaystyle\widehat{\phi}^{s}-\phi^{s}\overset{p}{\rightarrow}0.$ (38) Now,
multiplying (37) by $\sqrt{T}$
$\displaystyle\sqrt{T}(\widehat{\phi}^{s}-\phi^{s})$
$\displaystyle=\left(T^{-1}\boldsymbol{z}^{\prime}\,\boldsymbol{M}_{\eta}\,\boldsymbol{p}\right)^{-1}\left(\frac{\boldsymbol{z}^{\prime}\,\boldsymbol{M}_{\eta}\,\boldsymbol{u}_{E}}{\sqrt{T}}+\mathcal{O}_{p}\left(\frac{\sqrt{T}}{\text{min}\\{N,T\\}}\right)+\mathcal{O}_{p}\left(\frac{\sqrt{T}}{\text{min}\\{N,\sqrt{NT}\\}}\right)+\mathcal{O}_{p}\left(\frac{m_{N}\omega^{1-q}_{N,T}}{\sqrt{T}}\right)\right),$
(39) we can state the following result for the limiting distribution:

######  Theorem 4 (Limiting distribution for $\widehat{\phi}^{s}$)

Under Assumptions 1-4, as $(N,T)\overset{j}{\rightarrow}\infty$, we have that
when $N\geq T$ or $N<T$ and $\sqrt{T}/N\rightarrow 0$
$\displaystyle\sqrt{T}(\widehat{\phi}^{s}-\phi^{s})\overset{d}{\rightarrow}\mathcal{N}\left(0,\mathbbm{v}_{s}\right),$
(40) where $\mathbbm{v}_{s}:=\mathbbm{m}_{z\tilde{p}}^{-2}\,\mathbbm{v}_{zu}$,
$\mathbbm{v}_{zu}:=\mathbbm{E}(z_{t}^{\,2}\,(\boldsymbol{M}_{\eta}\,\boldsymbol{u}_{E})_{t}^{\,2})$
and
$\mathbbm{m}_{z\tilde{p}}=\mathbbm{E}(z_{t}(\boldsymbol{M}_{{\eta}}\,\boldsymbol{p})_{t})$.
$\mathbbm{v}_{zu}$ can be consistently estimated with
$\displaystyle\widehat{\mathbbm{v}}_{zu}=\begin{cases}T^{-1}\sum_{t=1}^{T}\widehat{z}_{t}^{\,2}\,(\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{\widehat{u}}_{E})_{t}^{\,2}\hskip
301.31467pt\text{HC,}\vspace{3mm}\\\
T^{-1}\sum_{t=1}^{T}\widehat{z}_{t}^{\,2}(\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{\widehat{u}}_{\widehat{E}})_{t}^{2}+2\cdot
T^{-1}\sum_{j=1}^{m}\left(1-\dfrac{j}{m+1}\right)\sum_{t=j+1}^{T}\widehat{z}_{t}(\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{\widehat{u}}_{\widehat{E}})_{t}\widehat{z}_{t-j}(\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{\widehat{u}}_{\widehat{E}})_{t-j}\hskip
10.5275pt\text{HAC.}\end{cases}$ (41) Hence
$\frac{\sqrt{T}(\widehat{\phi}^{\,s}-\phi^{s})}{\widehat{\mathbbm{v}}_{s}^{\,\,1/2}}\sim\text{t}_{df}\overset{d}{\rightarrow}\mathcal{N}(0,1)$,
where
$\widehat{\mathbbm{v}}_{s}^{1/2}=\widehat{\mathbbm{m}}_{z\tilde{p}}^{-2}\widehat{\mathbbm{v}}_{zu}$,
with
$\widehat{\mathbbm{m}}_{z\tilde{p}}=T^{-1}\sum_{t=1}^{T}\widehat{z}_{t}(\boldsymbol{M}_{\widehat{\eta}}\,\boldsymbol{p})_{t}$
consistent for $\mathbbm{m}_{z\tilde{p}}$. We will see in Section 8 that
asymptotic theory provides good approximations to the finite sample
distribution. Overidentified demand elasticity. For the overidentified demand
elasticity estimator in (21), recall the estimated instrument matrix consists
of
$\boldsymbol{\widehat{Z}}_{d}=\begin{pmatrix}\boldsymbol{\widehat{z}}_{GIV}&\boldsymbol{\widehat{\eta}}\end{pmatrix}$.
For the strong factors, $\boldsymbol{\widehat{\eta}}$, estimated via PCA, [12]
showed that the generated regressors problem of [66] does not arise when both
$N$ and $T$ are large. Thus, the sampling error in
$\boldsymbol{\widehat{\eta}}$ is negligible in consideration of the limiting
distribution of the overidentified demand elasticity estimate. In the previous
section, we established that estimation of $\boldsymbol{\widehat{z}}_{GIV}$ is
also negligible under regularity, we can then use standard asymptotic theory
to also obtain asymptotic normality of the efficient GMM estimator in the case
of demand since $\displaystyle\widehat{\phi}^{d}_{GMM}-\phi^{d}$
$\displaystyle=\left(\boldsymbol{p}^{\prime}\,\boldsymbol{\widehat{Z}}_{d}\,\boldsymbol{\widehat{\Omega}}_{d}^{-1}\,\boldsymbol{\widehat{Z}}^{\prime}_{d}\,\boldsymbol{p}\right)^{-1}\boldsymbol{p}^{\prime}\,\boldsymbol{\widehat{Z}}_{d}\,\boldsymbol{\widehat{\Omega}}_{d}^{-1}\,\boldsymbol{\widehat{Z}}^{\,{}^{\prime}}_{d}\,\boldsymbol{\varepsilon},$
$\displaystyle=\left(\boldsymbol{p}^{\prime}\,\boldsymbol{Z}_{d}\,\boldsymbol{\Omega}_{d}^{-1}\,\boldsymbol{Z}^{\,{}^{\prime}}_{d}\,\boldsymbol{p}\right)^{-1}\boldsymbol{p}^{\prime}\,\boldsymbol{Z}_{d}\,\boldsymbol{\Omega}_{d}^{-1}\,\boldsymbol{Z}^{\,{}^{\prime}}_{d}\,\boldsymbol{\varepsilon}+o_{p}(1).$
(42) We can now state the following theorem:

######  Theorem 5 (Limiting distribution for $\widehat{\phi}^{d}_{GMM}$)

Under Assumptions 1-4, with
$\mathbbm{E}(\varepsilon_{t}\boldsymbol{\eta}_{t})=0\,\forall t$, as
$(N,T)\overset{j}{\rightarrow}\infty$, we have that when $N\geq T$ or $N<T$
and $\sqrt{T}/N\rightarrow 0$
$\displaystyle\sqrt{T}(\widehat{\phi}^{d}_{GMM}-\phi^{d})\overset{d}{\rightarrow}\mathcal{N}\left(0,\mathbbm{V}(\widehat{\phi}^{d}_{GMM})\right),$
(43) where $\displaystyle\mathbbm{V}(\widehat{\phi}^{d}_{GMM})$
$\displaystyle=\left(\boldsymbol{m}^{\prime}_{{Z}_{d}p}\,\boldsymbol{\Omega}_{d}^{-1}\boldsymbol{m}_{{Z}_{d}p}\right)^{-1},$
(44) with $\boldsymbol{m}_{{Z}_{d}p}=\mathbbm{E}(\boldsymbol{Z}_{dt}\,p_{t})$
and $\boldsymbol{\Omega}_{d}=\emph{plim
}T^{-1}\sum_{t=1}^{T}\boldsymbol{\widehat{Z}}_{dt}\boldsymbol{\widehat{Z}}_{dt}^{{}^{\prime}}(d_{t}-p_{t}\widehat{\phi}^{d}_{2SLS})^{2}$.
$\mathbbm{V}(\widehat{\phi}^{d}_{GMM})$ can be consistently estimated using
$2SLS$ residuals with
$\displaystyle\widehat{\mathbbm{V}}(\widehat{\phi}^{d}_{GMM})=\left(\frac{\boldsymbol{p}^{\,{}^{\prime}}\boldsymbol{\widehat{Z}}_{d}}{T}\,\boldsymbol{\widehat{\Omega}}_{d}^{-1}\,\frac{\boldsymbol{\widehat{Z}}_{d}^{\,{}^{\prime}}\,\boldsymbol{p}}{T}\right)^{-1},$
(45) where
$\boldsymbol{\widehat{\Omega}}_{d}=T^{-1}\sum_{t=1}^{T}\boldsymbol{\widehat{Z}}_{dt}\boldsymbol{\widehat{Z}}_{dt}^{\,{}^{\prime}}\,(d_{t}-p_{t}\widehat{\phi}^{d}_{2SLS})^{2}$.
It is well known that $\mathbbm{V}(\widehat{\phi}^{d}_{GMM})$ attains the
semiparametric efficiency bound, as shown by [26], which reduces to (44) in
the linear model. Standard overidentification tests can be carried out since
$\displaystyle
J_{d}=T\cdot\left(T^{-1}\sum_{t=1}^{T}\boldsymbol{Z}_{dt}\,\varepsilon_{t}(\widehat{\phi}_{GMM}^{d})\right)^{\prime}\,\boldsymbol{\widehat{\Omega}}_{d}^{-1}\,\left(T^{-1}\sum_{t=1}^{T}\boldsymbol{Z}_{dt}\,{\varepsilon}_{t}(\widehat{\phi}_{GMM}^{d})\right)\overset{d}{\rightarrow}\chi^{2}_{df_{d}},$
(46) where the degrees of freedom is given by $df_{d}=(1+r)-k_{d}=r$ and
$k_{d}=1$ is the number of endogenous regressors. We will see that simulation
evidence shows that the size of the $J$-test is near the nominal size when the
true $r$ is used and when $rmax>r$ factors are used; which is important in
empirical work when $r$ is typically estimated and it is generally known that
an overestimate of $r$ is preferred in order to prevent an effect akin to
omitted variable bias, see [63] who formalize this notion. Overidentified
supply elasticity. The full instrument matrix for the overidentified supply
elasticity estimator in (22) consists of
$\boldsymbol{\widehat{Z}}_{s}=\begin{pmatrix}\boldsymbol{\widehat{z}}_{GIV}&\boldsymbol{\widehat{\varepsilon}}&\boldsymbol{\widehat{\eta}}\end{pmatrix}$,
(recall the factors self instrument here, as they are part of the supply
equation). We show that the sampling error in $\widehat{\boldsymbol{Z}}_{s}$
is indeed negligible. This is again due to both large $N$ and $T$. As a
result, $\underset{(2+r)\times(2+r)}{\boldsymbol{\Omega}_{s}}=\text{plim
}\,T^{-1}\sum_{t=1}^{T}\boldsymbol{\widehat{Z}}_{st}\boldsymbol{\widehat{Z}}^{{}^{\prime}}_{st}(y_{\widehat{E}t}-\boldsymbol{\widehat{f}}_{t}^{\prime}\boldsymbol{\widehat{\theta}}^{s}_{GMM})^{2}$
is sufficient when constructing the efficient weighting matrix, even though it
does not take the sampling error in our estimate of ${\phi}^{d}$ into account
(since
$\boldsymbol{\widehat{\varepsilon}}=\boldsymbol{\varepsilon}(\widehat{\phi}_{GMM}^{d})$).
That is
$\displaystyle\boldsymbol{\widehat{\theta}}^{s}_{GMM}-\boldsymbol{\theta}^{s}$
$\displaystyle=\left(\boldsymbol{f}^{\prime}\,\boldsymbol{\widehat{Z}}_{s}\,\boldsymbol{\widehat{\Omega}}_{s}^{-1}\,\boldsymbol{\widehat{Z}}^{\,{}^{\prime}}_{s}\boldsymbol{f}\right)^{-1}\boldsymbol{f}^{\prime}\,\boldsymbol{\widehat{Z}}_{s}\,\boldsymbol{\widehat{\Omega}}_{s}^{-1}\,\boldsymbol{\widehat{Z}}^{\,{}^{\prime}}_{s}\boldsymbol{\varepsilon},$
$\displaystyle=\left(\boldsymbol{f}^{\prime}\,\boldsymbol{Z}_{s}\,\boldsymbol{\Omega}_{s}^{-1}\,\boldsymbol{Z}^{\,{}^{\prime}}_{s}\boldsymbol{f}\right)^{-1}\boldsymbol{f}^{\prime}\,\boldsymbol{Z}_{s}\,\boldsymbol{\Omega}_{s}^{-1}\,\boldsymbol{Z}^{\,{}^{\prime}}_{s}\boldsymbol{\varepsilon}+o_{p}(1).$
(47) We can now state the following theorem:

######  Theorem 6 (Limiting distribution for
$\boldsymbol{\widehat{\theta}}^{s}_{GMM}$)

Under Assumptions 1-4, as $(N,T)\overset{j}{\rightarrow}\infty$, we have that
when $N\geq T$ or $N<T$ and $\sqrt{T}/N\rightarrow 0$
$\displaystyle\sqrt{T}(\boldsymbol{\widehat{\theta}}^{s}_{GMM}-\boldsymbol{\theta}^{s})$
$\displaystyle\overset{d}{\rightarrow}\mathcal{N}\left(\boldsymbol{0},\mathbbm{V}(\boldsymbol{\widehat{\theta}}^{s}_{GMM})\right),$
(48) where $\displaystyle\mathbbm{V}(\boldsymbol{\widehat{\theta}}^{s}_{GMM})$
$\displaystyle=\left(\boldsymbol{m}^{\prime}_{{Z}_{s}f}\,\boldsymbol{\Omega}_{s}^{-1}\boldsymbol{m}_{{Z}_{s}f}\right)^{-1},$
(49) with
$\boldsymbol{m}_{{Z}_{s}f}=\mathbbm{E}(\boldsymbol{Z}_{st}\boldsymbol{f}^{\prime}_{t})$
and $\boldsymbol{\Omega}_{s}=\emph{plim
}T^{-1}\sum_{t=1}^{T}\boldsymbol{\widehat{Z}}_{st}\boldsymbol{\widehat{Z}}_{st}^{{}^{\prime}}(y_{\widehat{E}t}-\boldsymbol{\widehat{f}}_{t}^{\prime}\boldsymbol{\widehat{\theta}}^{s}_{GMM})^{2}$.
$\mathbbm{V}(\boldsymbol{\widehat{\theta}}^{s}_{GMM})$ can be consistently
estimated using $GMM$ residuals with
$\displaystyle\widehat{\mathbbm{V}}(\boldsymbol{\widehat{\theta}}^{s}_{GMM})=\left(\frac{\boldsymbol{\widehat{f}}^{\,{}^{\prime}}\boldsymbol{\widehat{Z}}_{s}}{T}\,\boldsymbol{\widehat{\Omega}}_{s}^{-1}\,\frac{\boldsymbol{\widehat{Z}}_{s}^{\,{}^{\prime}}\boldsymbol{\widehat{f}}}{T}\right)^{-1},$
(50) where
$\boldsymbol{\widehat{\Omega}}_{s}=T^{-1}\sum_{t=1}^{T}\boldsymbol{\widehat{Z}}_{st}\boldsymbol{\widehat{Z}}_{st}^{\,{}^{\prime}}\,(y_{\widehat{E}t}-\widehat{\boldsymbol{f}}_{t}^{\prime}\boldsymbol{\widehat{\theta}}^{s}_{GMM})^{2}$.
Just as in the case of the overidentified demand elasticity estimator, (49)
achieves the semiparametric efficiency bound. Overidentification tests can be
carried out since $\displaystyle
J_{s}=T\cdot\left(T^{-1}\sum_{t=1}^{T}\boldsymbol{Z}_{st}\,u_{\widehat{E}t}(\widehat{\boldsymbol{\theta}}_{GMM}^{s})\right)^{\prime}\,\boldsymbol{\widehat{\Omega}}_{s}^{-1}\,\left(T^{-1}\sum_{t=1}^{T}\boldsymbol{Z}_{st}\,u_{\widehat{E}t}(\widehat{\boldsymbol{\theta}}_{GMM}^{s})\right)\overset{d}{\rightarrow}\chi^{2}_{df_{s}},$
(51) where the degrees of freedom are given by $df_{s}=2-k_{s}=2-1=1$ and
$k_{s}=1$ is the number of endogenous regressors for the supply equation.

###### Remark 5

The asymptotic distribution of the FGIV and efficient GMM estimators of demand
and supply elasticities are established. However, the finite sample moments of
these estimators, are unbounded to different degrees. The extensive literature
on the classic simultaneous equations model has documented this result in many
forms, see [61], [43], [71], [74], [75], [70], [37] and [44]. A complete
representation of the above results was given by [53]. Kinal’s result for 2SLS
states that, if the dependent variable, explanatory variables and instruments
are jointly normal, then $\mathbbm{E}||\widehat{\phi}^{j}_{2SLS}||^{m}<\infty$
for $m<\ell_{j}-k_{j}+1$, $j=d,s$, where $\ell_{j}$ is the number of
instruments and $k_{j}$ is the number of endogenous regressors. Thus, the FGIV
estimators for supply and demand exhibit no bounded absolute moments since
$\ell_{j}=k_{j}=1$, $j=d,s$. Whereas, the efficient GMM estimators exhibits
$\ell_{d}-k_{d}=(1+r)-1=r$ bounded absolute moments in finite samples for the
case of demand and $\ell_{s}-k_{s}=2-1=1$ bounded absolute moment in finite
samples for the case of supply. Hence, the efficient GMM elasticity estimators
(overidentified FGIV estimators) exhibit superior finite sample properties
relative to their (just identified) FGIV counterparts. Of course, in general,
just identified instrumental variables estimators (with strong instruments)
exhibit nice properties asymptotically.

##  7 Weak Instruments

The classical weak instruments framework introduced by [73] has its analog in
this framework. Interestingly, here the "weak" aspect is partially linked to
the Herfindahl index without making the usual local-to-zero assumption as in
[73]. Moreover, the traditional notion of local-to-zero with
$\frac{1}{\sqrt{T}}$ scaling which matches the rate of convergence of the
estimator need not necessarily apply here for weak instruments to arise. More
specifically, the locality to zero can be expressed as decaying functions of
$N$, except in the case of $\mu\in(0,1)$, which we require for inference under
our maintained strong instruments assumption; whereas the rate of convergence
is at the $\sqrt{T}$ rate. To make things more clear, it is useful to see the
reduced form, equilibrium price equation again. Recall from (3), we have that
$p_{t}=\dfrac{1}{\phi^{d}-\phi^{s}}\left(u_{St}+\boldsymbol{\lambda}^{\prime}_{S}\boldsymbol{\eta}_{t}-\varepsilon_{t}\right)$.
Thus, it is clear that for finite $N$, $\mathbbm{Cov}(p_{t},z_{t})>0$, which
automatically renders the GIV as relevant. However, for large $N$, writing
$z_{t}=\boldsymbol{S}^{\prime}\boldsymbol{Q}\boldsymbol{\tilde{u}}_{\cdot t}$
we observe that
$\displaystyle\mathbbm{V}(\boldsymbol{S}^{\prime}\boldsymbol{Q}\boldsymbol{\tilde{u}}_{\cdot
t})$
$\displaystyle=\mathbbm{E}(\boldsymbol{S}^{\prime}\boldsymbol{Q}\mathbbm{E}(\boldsymbol{\tilde{u}}_{\cdot
t}\boldsymbol{\tilde{u}}_{\cdot
t}^{\prime}|\boldsymbol{\Gamma})\boldsymbol{Q}\boldsymbol{S})=\mathbbm{E}(\boldsymbol{S}^{\prime}\boldsymbol{Q}\,\boldsymbol{\Sigma}_{\tilde{u}}\,\boldsymbol{Q}\boldsymbol{S}),$
(52) where
$\boldsymbol{\Sigma}_{\tilde{u}}:=\mathbbm{E}(\boldsymbol{\tilde{u}}_{\cdot
t}\boldsymbol{\tilde{u}}_{\cdot t}^{\prime})$. The term inside the expectation
can be simplified to
$\displaystyle\boldsymbol{S}^{\prime}\boldsymbol{Q}\,\boldsymbol{\Sigma}_{\tilde{u}}\,\boldsymbol{Q}\boldsymbol{S}$
$\displaystyle=\boldsymbol{S}^{\prime}\boldsymbol{Q}\,\boldsymbol{\Sigma}_{u}\,\boldsymbol{Q}\boldsymbol{S}+\mathcal{O}_{p}\left(N^{-1}\right)$
$\displaystyle\leq\boldsymbol{S}^{\prime}\boldsymbol{Q}\boldsymbol{S}\,\gamma_{max}(\boldsymbol{\Sigma}_{u})+\mathcal{O}_{p}\left(N^{-1}\right)\leq\boldsymbol{S}^{\prime}\boldsymbol{S}\cdot\gamma_{max}(\boldsymbol{\Sigma}_{u})\cdot\gamma_{max}(\boldsymbol{Q})+\mathcal{O}_{p}\left(N^{-1}\right)$
$\displaystyle=\mathcal{O}\left(1\right)h_{N,\,\mu}+\mathcal{O}_{p}\left(N^{-1}\right),$
(53) where we make use of
$\gamma_{max}(\boldsymbol{\Sigma}_{u})=\mathcal{O}\left(1\right)$ and the fact
that a symmetric idempotent matrix, such as $\boldsymbol{Q}$, has eigenvalues
of $0$ or $1$ and so $\gamma_{max}(\boldsymbol{Q})=1$. Taken together, (52)
and (53) imply
$\displaystyle\mathbbm{V}(z_{t})\leq\mathcal{O}\left(1\right)\mathbbm{E}(h_{N,\mu})+\mathcal{O}\left(N^{-1}\right).$
(54) As such, only when we are in a tail regime indexed by $\mu\in(0,1)$ do we
avoid the locality to zero. For example, when $\mu>2$, we have that
$\mathbbm{V}(\boldsymbol{S}^{\prime}\boldsymbol{Q}\boldsymbol{\tilde{u}}_{\cdot
t})\leq\mathcal{O}\left(1\right)\mathbbm{E}(h_{N,\,\mu>2})=\mathcal{O}\left(\frac{1}{N}\right)$.
As a result, when $\mu>2$, we have that
$z_{t}=\boldsymbol{S}^{\prime}\boldsymbol{Q}\boldsymbol{\tilde{u}}_{\cdot
t}=\mathcal{O}_{p}\left(\frac{1}{\sqrt{N}}\right)$, so our equilibrium price
equation simplifies to the following large $N$ representation $\displaystyle
p_{t}=\dfrac{1}{\phi^{d}-\phi^{s}}\left(\boldsymbol{\lambda}_{S}^{\prime}\boldsymbol{\eta}_{t}-\varepsilon_{t}\right)+\mathcal{O}_{p}\left(\dfrac{1}{\sqrt{N}}\right).$
(55) This would render the GIV as very weak since
$\mathbbm{Cov}(p_{t},u_{St})=\mathbbm{Cov}(p_{t},z_{t})=\mathcal{O}\left(\frac{1}{N}\right)$.
Note that the $\mathbbm{Cov}(p_{t},u_{St})$ and $\mathbbm{Cov}(p_{t},z_{t})$
are of the same order precisely because $\gamma_{max}(\boldsymbol{Q})=1$. (55)
is effectively the relationship that was exploited by [62], who assumed the
so-called "granular" weights of order $\mathcal{O}\left(\frac{1}{N}\right)$
and used this weak correlation for large $N$ to ultimately deduce that prices
can be treated as weakly exogenous.161616”Granular” has a different definition
in the panel data literature, which is referring to properties of weights and
heuristically, rules out the existence of dominant units, see e.g., [62] and
Remark 2. On the contrary, our usage of the term ”granular” follows [38] and
is essentially referring to the existence of dominant cross-sectional units,
see Section 1. Consider the well documented and empirically relevant case
where $\mu$ is just above $1$ (Zipf’s law corresponds to $\mu=1$); when
$\mu\in(1,2)$ we have
$h_{N,\,\mu\in(1,2)}=\mathcal{O}_{p}\left(1/(N^{2-\frac{2}{\mu}})\right)$. So,
$\displaystyle
p_{t}=\dfrac{1}{\phi^{d}-\phi^{s}}\left(\boldsymbol{\lambda}^{\prime}_{S}\boldsymbol{\eta_{t}}-\varepsilon_{t}\right)+\mathcal{O}_{p}\left(\dfrac{1}{N^{1-\frac{1}{\mu}}}\right).$
(56) Therefore, even though
$\mathbbm{Cov}(p_{t},z_{t})=\mathcal{O}\left(1/(N^{2-\frac{2}{\mu}})\right)$,
it is in fact decaying to zero so slowly for $\mu$ near 1, that this
potentially corresponds to a highly relevant instrument in any finite sample.
That is,
$\mathbbm{Cov}(p_{t},z_{t})=\mathcal{O}\left(1/(N^{2-\frac{2}{\mu}})\right)$,
is potentially consistent with $z_{t}$ accounting for large fractions of
aggregate variation, see [38]. However, the case we theoretically entertain,
for consistency and valid asymptotic inference, requires $\mu\in(0,1)$, which
in conjunction with the additional regularity assumptions, renders
$\mathbbm{Cov}(p_{t},z_{t})=\Theta(1)$ even as $N\rightarrow\infty$.
Rothemberg Representations. Moreover, to further assess the likelihood of weak
instruments, we can analyze the efficient GMM estimator of the demand
elasticity which uses both the GIV and the factors as instruments and for
comparison we can analyze the just identified demand elasticity estimator
which uses only the GIV as an instrument. We analyze the overidentified case
with conditional homoskedasticity (assuming only for remainder of this
section). Define the projection matrix
$\boldsymbol{P}_{Z_{d}}=\boldsymbol{Z}_{d}(\boldsymbol{Z}_{d}^{\prime}\boldsymbol{Z}_{d})^{-1}\boldsymbol{Z}_{d}^{\prime}$,
the 2SLS estimator takes the form
$\displaystyle\widehat{\phi}^{d}_{GMM}-\phi^{d}$
$\displaystyle=\frac{\boldsymbol{p}^{\prime}\,\boldsymbol{P}_{Z_{d}}\,\boldsymbol{\varepsilon}}{\boldsymbol{p}^{\prime}\,\boldsymbol{P}_{Z_{d}}\,\boldsymbol{p}}.$
(57) Write the structural and reduced form equations as
$\displaystyle\boldsymbol{d}=\boldsymbol{p}\,\phi^{d}+\boldsymbol{\varepsilon}$
$\displaystyle\boldsymbol{p}=\boldsymbol{z}\boldsymbol{\pi}^{\prime}+\boldsymbol{v},$
(58) where
$\underset{T\times(1+r)}{\boldsymbol{z}}=\begin{pmatrix}\boldsymbol{u}_{S}&\boldsymbol{\eta}\end{pmatrix}$,
$\underset{(1+r)\times
1}{\boldsymbol{\pi}}=\begin{pmatrix}\frac{1}{\phi^{d}-\phi^{s}}&\frac{1}{\phi^{d}-\phi^{s}}\cdot\boldsymbol{\lambda}^{\prime}_{S}\end{pmatrix}^{\prime}$
and $\underset{T\times
1}{\boldsymbol{v}}=\frac{1}{\phi^{d}-\phi^{s}}\cdot\boldsymbol{\varepsilon}$.

###### Remark 6

The difference between $\boldsymbol{z}$ in (58) and our actual instrument,
$\boldsymbol{Z}_{d}$, boils down to the difference between their first
columns,
$\boldsymbol{Z}_{d}[\cdot,1]=\boldsymbol{z}_{GIV}=\boldsymbol{\tilde{u}}_{\cdot\cdot}\boldsymbol{Q}\boldsymbol{S}$
and
$\boldsymbol{z}[\cdot,1]=\boldsymbol{u}_{S}=\boldsymbol{u}_{\cdot\cdot}\boldsymbol{S}$.
In the case of the demand equation, $\boldsymbol{u}_{S}$ is ideal, whereas
$\boldsymbol{z}_{GIV}$ is a proxy. The reason the proxy is used is simply due
to a simpler theoretical exposition than a direct estimate for the ideal.
Indeed $\boldsymbol{z}_{GIV}$ is in fact a good proxy. For example, the
correlation between $\boldsymbol{u}_{S}$ and $\boldsymbol{z}_{GIV}$ is over
90% regardless of the complexity of our DGP in Monte Carlo simulations even
for small configurations of $(N,T)$. Moreover, in the case of the supply
equation, $\boldsymbol{u}_{S}$ is no longer valid, whereas
$\boldsymbol{z}_{GIV}$ is; see (13). Mathematically,
$\boldsymbol{S}^{\prime}\boldsymbol{u}_{\cdot
t}-\boldsymbol{S}^{\prime}\boldsymbol{Q}\boldsymbol{\tilde{u}}_{\cdot
t}=\boldsymbol{S}^{\prime}\boldsymbol{P}_{\tilde{\Lambda}}\boldsymbol{u}_{\cdot
t}=\boldsymbol{S}^{\prime}\boldsymbol{P}_{\tilde{\Lambda}}\boldsymbol{P}_{\tilde{\Lambda}}\boldsymbol{u}_{\cdot
t}$ for each $t$, where $\boldsymbol{P}_{\tilde{\Lambda}}$ is the symmetric
and idempotent projection matrix in the demeaned loading space. Hence,
$\boldsymbol{S}^{\prime}\boldsymbol{P}_{\tilde{\Lambda}}\boldsymbol{P}_{\tilde{\Lambda}}\boldsymbol{u}_{\cdot
t}$ is zero when the loadings and the share vector are asymptotically
uncorrelated and/or the loadings and idiosyncratic errors are asymptotically
uncorrelated; which explains why our simulations exhibit near perfect
correlation. Then, it follows from [69] that (57) has the following
illustrative representation
$\displaystyle\mu_{d,GMM}(\widehat{\phi}^{d}_{GMM}-\phi^{d})$
$\displaystyle=\left(\frac{\sigma^{2}_{\varepsilon}}{\sigma^{2}_{v}}\right)^{\frac{1}{2}}\frac{X+(\omega_{1}/\mu_{d,GMM})}{1+2Y/\mu_{d,GMM}+(\omega_{2}/\mu_{d,GMM}^{2})},$
(59) where
$X=\boldsymbol{\pi}^{\prime}\boldsymbol{z}^{\prime}\boldsymbol{P}_{Z_{d}}\boldsymbol{\varepsilon}/(\sigma^{2}_{\varepsilon}\boldsymbol{\pi}^{\prime}\boldsymbol{z}^{\prime}\boldsymbol{z}\boldsymbol{\pi})^{\frac{1}{2}}$
and
$Y=\boldsymbol{\pi}^{\prime}\boldsymbol{z}^{\prime}\boldsymbol{P}_{Z_{d}}\boldsymbol{v}/(\sigma^{2}_{v}\boldsymbol{\pi}^{\prime}\boldsymbol{z}^{\prime}\boldsymbol{z}\boldsymbol{\pi})^{\frac{1}{2}}$
are bivariate standard normal variates with correlation coefficient $\rho$.
The random variable
$\omega_{1}=\boldsymbol{v}^{\prime}\boldsymbol{P}_{Z_{d}}\boldsymbol{\varepsilon}/(\sigma^{2}_{\varepsilon}\sigma^{2}_{v})^{\frac{1}{2}}$
has mean equal to $\text{rank}(P_{Z_{d}})\rho=(r+1)\rho$ and variance equal to
$(r+1)(1+\rho^{2})$. The random variable
$\omega_{2}=\boldsymbol{v}^{\prime}\boldsymbol{P}_{Z_{d}}\boldsymbol{v}/\sigma^{2}_{v}$
has mean equal to $\text{rank}(\boldsymbol{P}_{Z_{d}})=(r+1)$ and variance
equal to $2(r+1)$. Finally, $\mu_{d,GMM}$ is the square root of the so-called
concentration parameter
$\mu_{d,GMM}^{2}=\boldsymbol{\pi}^{\prime}\boldsymbol{z}^{\prime}\boldsymbol{z}\boldsymbol{\pi}/\sigma^{2}_{v}$
for the demand equation. $\mu_{d,GMM}$ plays the role of $\sqrt{T}$, that is,
when $\mu_{d,GMM}$ is large, $\mu_{d,GMM}(\widehat{\phi}^{d}_{GMM}-\phi^{d})$
is well approximated by a $\mathcal{N}(0,1)$ variate. Large values of
$\mu_{d,GMM}$ are consistent with large values of $T$, i.e., our typical large
sample approximations. However, large values of $\mu_{d,GMM}$ are also
consistent with small values of $\sigma^{2}_{v}$, regardless of the value of
$T$, i.e., small-$\sigma$ asymptotics, as introduced originally by [50]. More
insights can be gained by simplifying the concentration parameter for the
demand elasticity $\displaystyle\mu_{d,GMM}^{2}$
$\displaystyle=\boldsymbol{\pi}^{\prime}\boldsymbol{z}^{\prime}\boldsymbol{z}\boldsymbol{\pi}/\sigma^{2}_{v}=\frac{1}{\sigma^{2}_{v}}\cdot\begin{pmatrix}\pi_{1}&\boldsymbol{\pi_{2}}^{\prime}\end{pmatrix}\begin{pmatrix}\boldsymbol{u}^{\prime}_{S}\boldsymbol{u}_{S}&\boldsymbol{u}^{\prime}_{S}\boldsymbol{\eta}\\\
\boldsymbol{\eta}^{\prime}\boldsymbol{u}_{S}&\boldsymbol{\eta}^{\prime}\boldsymbol{\eta}\end{pmatrix}\begin{pmatrix}\pi_{1}\\\
\boldsymbol{\pi_{2}}\end{pmatrix}\approx\frac{\boldsymbol{u}^{\prime}_{S}\,\boldsymbol{u}_{S}+\boldsymbol{\lambda}^{\prime}_{S}\boldsymbol{\lambda}_{S}}{\sigma^{2}_{\varepsilon}},$
(60) where the approximation is due to ignoring the terms involving
$\boldsymbol{\eta}^{\prime}\boldsymbol{u}_{S}$, which are zero only in
expectation. (60) is very intuitive, if the proportion of the volatility in
the GIV and size-weighted common components dominate the volatility of the
demand shocks, so that the ratio in (60) is large, then the concentration
parameter $\mu_{d,GMM}$ will be large and one should expect good
approximations to the finite sampling distributions. On the other hand, when
only the GIV is used as an instrument, if we redefine $\underset{T\times
1}{\boldsymbol{z}}=\boldsymbol{u}_{S}$, $\underset{1\times
1}{\boldsymbol{\pi}}=\frac{1}{\phi^{d}-\phi^{s}}$ and $\underset{T\times
1}{\boldsymbol{v}}=\frac{1}{\phi^{d}-\phi^{s}}\cdot(\boldsymbol{\varepsilon}+\boldsymbol{\eta}\boldsymbol{\lambda}_{S})$
from (58) and simply follow the logic above through (60), we arrive at the
following concentration parameter for the FGIV estimator
$\displaystyle\mu^{2}_{d,FGIV}\approx\frac{\boldsymbol{u}^{\prime}_{S}\boldsymbol{u}_{S}}{\boldsymbol{\lambda}^{\prime}_{S}\boldsymbol{\lambda}_{S}+\sigma^{2}_{\varepsilon}}.$
(61) Thus, by inspection of (60) against (61) we can see that in the case of
the just identified FGIV estimator, we would need the volatility of just the
GIV to drive up the ratio of the concentration parameter, $\mu^{2}_{d,GIV}$,
and the size-weighted common component would be working against us (in the
denominator), in this case, instead of working for us as in $\mu^{2}_{d,GMM}$.
Although the literature on granularity has demonstrated that idiosyncratic
shocks alone can be quite volatile, in this context, we advocate starting with
the efficient GMM estimators, since the $J$-test is well sized as illustrated
with simulation evidence, because the efficient GMM estimators can exhibit
substantially improved finite sample properties relative to the just
identified estimators and are less likely to suffer from weak instrument
issues as well.

##  8 Monte Carlo

We simulate the following panel simultaneous equations system with latent
factor structure that was analyzed in the theoretical sections: $\displaystyle
d_{t}$
$\displaystyle=\phi^{d}p_{t}+\varepsilon_{t},\,\,y_{it}=\phi^{s}p_{t}+\lambda_{1i}\eta_{1t}+\lambda_{2i}\eta_{2t}+u_{it},\,\,y_{\boldsymbol{S}t}=d_{t},$
$\displaystyle p_{t}$
$\displaystyle=\frac{1}{\phi^{d}-\phi^{s}}\left(u_{St}+\boldsymbol{\lambda}^{\prime}_{S}\boldsymbol{\eta_{t}}-\varepsilon_{t}\right),\,\,\mathscr{S}_{i}=\left(\frac{i}{N}\right)^{-\frac{1}{\mu}},\,\,S_{i}=\frac{\mathscr{S}_{i}}{\sum_{j=1}^{N}\mathscr{S}_{j}}.$
We consider two sets of simulated experiments. In Design 1, we let $u_{it}$ be
$i.i.d.$ to establish a set of baseline results. In Design 2, we allow for
sparse cross-sectional dependence in $u_{it}$. In addition, in unreported
simulations, we simulate $z_{t,GIV}$ to be a weak instrument to illustrate
that the efficient GMM estimators are robust to this as they optimally shift
their weights away from this point of weakness, whereas the just identified
estimators of GK and our FGIV will substantially deteriorate in their
performances. Design 1 - $u_{it}\,\,\boldsymbol{i.i.d.}$ case. We set
$\phi^{s}=0.1$ and $\phi^{d}=-0.3$. We draw the supply factors and loadings
as, $\underset{T\times
r}{\boldsymbol{\eta}}\overset{i.i.d.}{\sim}\mathcal{N}(0,\boldsymbol{I}_{r})$
and $\underset{N\times
r}{\boldsymbol{\Lambda}}\overset{i.i.d.}{\sim}\mathcal{N}(0,\sigma^{2}_{\Lambda}\boldsymbol{I}_{N})$,
respectively, with $r=2$.171717The results do not change significantly if we
draw the loadings from a Uniform distribution with non-zero mean. We draw the
idiosyncratic supply shocks as $\underset{T\times
N}{\boldsymbol{u}}\overset{i.i.d.}{\sim}\mathcal{N}(0,\sigma^{2}_{u}\boldsymbol{I}_{N}\otimes\boldsymbol{I}_{T})$
and aggregate demand shocks as $\underset{T\times
1}{\boldsymbol{\varepsilon}}\overset{i.i.d.}{\sim}\mathcal{N}(0,\sigma^{2}_{\varepsilon}\boldsymbol{I}_{T})$.
Design 2 - $u_{it}$ non-$\boldsymbol{i.i.d.}$ case. Everything is identical to
Design 1, except that we no longer set
$\boldsymbol{\Sigma}_{u}=\sigma^{2}_{u}\boldsymbol{I}_{N}$ for each
$t=1,\dots,T$. We generate a non-diagonal banded covariance matrix; as such,
it satisfies the sparsity requirement from Assumption 3.181818In unreported
simulations, we find that the results do not change significantly if we
generate a dense, non-diagonal $\boldsymbol{\Sigma}_{u}$, such as one arising
from a cross-sectional AR(p) process. This is because although the cross-
sectional AR(p) generates a dense matrix, it is not too dense since the off-
diagonals decay exponentially fast to 0 as $|i-j|\rightarrow\infty$. We
consider the following banded idiosyncratic covariance matrix with cross-
sectional dependence and heteroskedasticity
$\displaystyle\sigma_{u,ij}=\begin{cases}\tau^{|i-j|}\sqrt{\sigma_{u,i}\sigma_{u,j}}\hskip
46.94704pt|i-j|\leq k;k\geq 0,\\\ 0\hskip 110.25443pt|i-j|>k,\\\ \end{cases}$
(62) with bandwidth $k=3$ and $\sigma^{2}_{u,i}$ are drawn from
$\mathcal{U}[0.5,1]$. Target parameterizations. The variance of the price
process takes the following form
$\mathbbm{V}(p_{t})=c\cdot\left(\mathbbm{V}(u_{St})+\mathbbm{V}(\boldsymbol{\lambda}^{\prime}_{S}\boldsymbol{\eta}_{t})+\mathbbm{V}(\varepsilon_{t})\right)$,
where $c=\dfrac{1}{(\phi^{d}-\phi^{s})^{2}}$. This conveniently allows us to
parameterize the relative volatilities of the various components of
equilibrium prices. We parameterize the individual variances, $\sigma^{2}_{u}$
(for Design 1), $\sigma^{2}_{\Lambda}$ and $\sigma^{2}_{\varepsilon}$ such
that $\psi_{u}:=\dfrac{\mathbbm{V}\left(\sqrt{c}\cdot
u_{St}\right)}{\mathbbm{V}(p_{t})}\in(0.15,0.35),$
$\psi_{u+\eta}:=\dfrac{\mathbbm{V}\left(\sqrt{c}\cdot(u_{St}+\boldsymbol{\lambda}^{\prime}_{S}\boldsymbol{\eta}_{t})\right)}{\mathbbm{V}(p_{t})}\in(0.45,0.65),$
and
$\psi_{u+\varepsilon}:=\dfrac{\mathbbm{V}\left(\sqrt{c}\cdot(u_{St}+\varepsilon_{t})\right)}{\mathbbm{V}(p_{t})}\in(0.45,0.65).$191919The
interval for $\psi_{u}$ is consistent with the literature on granularity,
which has documented the proportion of aggregate fluctuations traced back to
idiosyncratic shocks falling in this specified range. In Design 1, we achieve
an average across simulations of $\bar{\psi}_{u}\approx 0.23$,
$\bar{\psi}_{u+\eta}\approx 0.58$ and $\bar{\psi}_{u+\varepsilon}\approx 0.65$
which implies that $\bar{\psi}_{\eta}=0.35$ and
$\bar{\psi}_{\varepsilon}=0.42$. That is, the idiosyncratic shocks are not the
dominating force in terms of observed price volatility; however, their
granular role is still substantial enough to draw inferences from when used as
instruments. In Design 2, we achieve an average across simulations of
$\bar{\psi}_{u}\approx 0.27$, $\bar{\psi}_{u+\eta}\approx 0.64$ and
$\bar{\psi}_{u+\varepsilon}\approx 0.63$. Let $\widehat{\phi}^{j}(m)$,
$j=d,s$, denote the estimate in the $m^{th}$ monte carlo repetition,
$m=1,...,M$. We report the monte carlo bias:
$\text{Bias}(\widehat{\phi}^{j})=\left(\dfrac{1}{M}\sum_{m=1}^{M}\widehat{\phi}^{j}(m)-\phi^{j}\right)$
for $j=d,s;$ and square root of the monte carlo MSE:
$\text{RMSE}(\widehat{\phi}^{j})=\sqrt{\dfrac{1}{M}\sum_{m=1}^{M}(\widehat{\phi}^{j}-\phi^{j})^{2}}$
for $j=s,d.$ Additionally we report the size of the $t$-test for all
estimators and size of the $J$-test for the efficient GMM estimators. The
results are reported in Table 1 (Design 1) and Table 2 (Design 2). In Table 1,
we multiply the bias by 100 because all the estimators perform quite well in
this ideal setting. For nearly each configuration of $(N,T)$ the GMM
estimators perform the best, in terms of bias and RMSE, as the theory
suggests. Importantly, the $t$-test and $J$-test is well sized even when
$rmax=3>r=2$ factors are used. In Table 2, we report the bias as is, and we
find that for a given configuration of $(N,T)$ the bias is two orders of
magnitude larger in Design 2 relative to Design 1. Nevertheless, as the theory
would suggest, the efficient GMM estimators perform the best in terms of bias
and RMSE. There are some size distortions for the supply side estimators but
the distortions are decreasing in $N$ for a given $T$.

##  9 Application to Global Crude Oil Markets

The data construction follows the recent literature: [52], [22], [18]
(hereafter BH) and GK. The following is a breakdown of the raw variables
collected for Jan. 1985 - Dec. 2015 $(T=372$ months$)$: monthly oil production
for $N=22$ countries from the U.S. Energy Information Administration
(hereafter EIA); world oil production from the U.S. EIA; monthly oil prices
based on the refiner acquisition cost of imported crude oil from the U.S. EIA;
U.S. CPI from the St. Louis FRED database; monthly change in inventories from
BH; monthly industrial production index from BH. The CPI is used to deflate
nominal oil prices to arrive at the real price of oil, which is highly non-
stationary. Following the aforementioned literature, we take the logarithm of
the real price of oil series and then take first differences. We apply the
same transformation to the monthly oil production for each country. These
transformations render the production and price series stationary as confirmed
by a host of Dickey-Fuller tests. For ensuring the tail index of the size-
distribution, $\mu$, is in the region the theory requires, we provide visual
evidence along with 6 estimates of $\mu$ that all fall beneath 1, see Table 3.
Also, see Figure 1, Figure 2 & Figure 3. Let $y_{it}$ denote the log
difference of the oil supply for country $i$ at time $t$ and $p_{t}$ denote
the log difference of the real price of oil. Following GK, we estimate an OPEC
factor using information on the cross-section of countries (i.e., known
loadings). To that end, let $o_{it}$ denote a dummy variable equal to 1 if
country $i$ is an OPEC member at time $t$ and note that $o_{it}=o_{i}$ for
most $i$, with the exception of Gabon and Ecuador in our sample. Finally,
$\boldsymbol{c}_{t-1}$ denotes a $4\times 1$ vector containing: lagged
$p_{t}$, lagged world supply growth, lagged change in inventories, and lagged
growth in industrial production. The system is given as follows $\displaystyle
y_{it}$
$\displaystyle=\phi^{s}p_{t}+\boldsymbol{\gamma_{s}}^{\prime}\boldsymbol{c}_{t-1}+o_{it}\eta_{OPEC,t}+\boldsymbol{\lambda}_{i}^{\prime}\boldsymbol{\eta}_{t}+u_{it},$
(63) $\displaystyle d_{t}$
$\displaystyle=\phi^{d}p_{t}+\boldsymbol{\gamma}_{d}^{\prime}\boldsymbol{c}_{t-1}+\lambda_{d}\eta_{OPEC,t}+\varepsilon_{t},$
(64) $\displaystyle\sum_{i=1}^{N}S_{it}\,y_{it}$ $\displaystyle=d_{t};$ (65)
where we lose the observation $t=1$ due to differencing. The cross-sectionally
demeaned supply equation is given by the approximate factor model,
$\displaystyle\tilde{y}_{it}=y_{it}-\frac{1}{N}\sum_{i}y_{it}$
$\displaystyle=\tilde{o}_{it}\eta_{OPEC,t}+\boldsymbol{\tilde{\lambda}}_{i}^{\prime}\boldsymbol{\eta}_{t}+\tilde{u}_{it}=\tilde{o}_{it}\eta_{OPEC,t}+\tilde{e}_{it},$
(66) where
$\tilde{e}_{it}:=\boldsymbol{\tilde{\lambda}}_{i}^{\prime}\boldsymbol{\eta}_{t}+\tilde{u}_{it}$.
Note that (66) implies we can obtain the OPEC factor, $\eta_{OPEC,t}$, via
cross-sectional regression, for each $t>1$, that is
$\widehat{\eta}_{OPEC,t}=(\boldsymbol{\tilde{o}}_{\cdot
t}^{\prime}\boldsymbol{\tilde{o}}_{\cdot t})^{-1}\boldsymbol{\tilde{o}}_{\cdot
t}^{\prime}\boldsymbol{\tilde{y}}_{\cdot t}$. Hence, in our preliminary stage,
we extract $\widehat{\eta}_{OPEC,t}$ and then run PCA on
$\widehat{\tilde{e}}_{it}=\tilde{y}_{it}-\widehat{\eta}_{OPEC,t}$ to extract
the latent demeaned loadings and latent factors. Define
$\boldsymbol{y}^{*}_{\cdot t}:=\boldsymbol{\tilde{y}}_{\cdot
t}-\boldsymbol{\tilde{x}}_{\cdot t}\widehat{\eta}_{OPEC,t}$, then we purge the
latent factors via $\boldsymbol{Q}$ as in the main text:
$\boldsymbol{{Q}}\,\boldsymbol{y}^{*}_{\cdot
t}=\boldsymbol{{Q}}\,\boldsymbol{\tilde{u}}_{\cdot t}$. However, when forming
the GIV, there is a minor difference that we have time-varying size-weights,
so we no longer construct $\boldsymbol{z}_{GIV}$ with a time-invariant share
vector $S_{i}$, but rather we weight each idiosyncratic component at time $t$
with its corresponding share from time $t-1$ to avoid endogeneity issues
arising from contemporaneous weighting $\displaystyle\underset{(T-1)\times
1}{\boldsymbol{z}_{GIV}}=\begin{pmatrix}\boldsymbol{S}^{\prime}_{\cdot
1}\boldsymbol{Q}\,\boldsymbol{\tilde{y}}^{*}_{\cdot 2}\\\
\boldsymbol{S}^{\prime}_{\cdot
2}\boldsymbol{Q}\,\boldsymbol{\tilde{y}}^{*}_{\cdot 3}\\\ \vdots\\\
\boldsymbol{S}^{\prime}_{\cdot
T-1}\boldsymbol{Q}\,\boldsymbol{\tilde{y}}^{*}_{\cdot T}\end{pmatrix}.$ (67)
Besides these modifications from the stylized model in the theory, we estimate
the elasticities using the estimators outlined in the main text. The number of
factors, $r$, is estimated via the AH procedures (as outlined in the
Supplementary Appendix C). The $ER$ method of AH estimated
$\widehat{r}_{ER}=1$, while the $GR$ method estimated $\widehat{r}_{GR}=3$;
with $k_{max}=10$. To be safe, we take $\widehat{r}=\widehat{r}_{GR}+1$.
Supply results. The results for the supply elasticity are presented in Table
4. In Table 4, the 2nd column displays GK’s results. The instrument GK use is
given by (8) and their dependent variable is simply the cross-sectional
average of the log difference of oil supply (i.e.,
$\boldsymbol{E}=\boldsymbol{\iota}/N$). Our results are in columns 3 and 4. In
contrast to (8), the instrument we use in column 3 purges the common factors
through the loading space. The instrument we use in column 4 also adds an
estimate of the unobserved aggregate demand shocks,
$\widehat{\varepsilon}_{t}$, to our FGIV. Moreover, the dependent variable we
use is weighted using the estimated precision vector
$\boldsymbol{\widehat{E}}$, which allows for cross-sectional correlations and
heteroskedasticity in $u_{it}$. These differences lead to significantly
different results. Columns 2 and 3, which attempt to use only GIVs as
instruments, both lead to weak instruments as indicated by the first-stage
$F$-statistics less than the rule of thumb, 10. Nevertheless, the FGIV supply
elasticity estimate (0.016) from column 3 (estimated via Algorithm 1) is
roughly one third that of GK’s (0.044). Whereas, our efficient GMM supply
elasticity estimate (0.005) from column 4 (estimated via Algorithm 3) is
highly significant at the 1% level. Additionally, our results reveal that
using estimates of unobserved aggregate demand shocks as supply instruments
indeed renders a strong instrument as indicated by the first-stage $F$-stat of
14.33 in column 4. Moreover, the $p$-value for the $J$-statistic (0.11) fails
to reject the null hypothesis of a valid model. An $F$-stat greater than 10,
coupled with a small $J$-statistic provides statistical evidence in favor of
our efficient GMM point estimate for the supply elasticity. Demand results.
Turning now to the demand elasticity in Table 5, the dependent variable GK use
in this case is the same as the one we use. However, the instruments are
different. Column 2 displays GK’s demand elasticity (-0.463), again using the
instrument as in (8). Column 3 presents our result when using only the FGIV as
an instrument (-.0009), which is roughly 400 times smaller than GK’s estimate.
Columns 4 through 7 sequentially add principal components to the instrument
vector for our efficient GMM estimator from column 3 until 4 principal
components are used. Here we find that none of the models yield first-stage
$F$-statistics greater than 10. It is reassuring, however, that the
$J$-statistic for columns 4 through 7 all fail to reject the null of a valid
model. Lastly, column 8 presents the [12] estimator which only includes the
four principal components but not the FGIV as instruments, nearly all
statistics remain unchanged except that inclusion of the FGIV increases the
$t$-stat by about 25%. Taken together, our empirical results suggest that
supply shocks, whether they be aggregate or idiosyncratic supply shocks,
albeit valid, do not serve as strong instruments for estimation of the demand
elasticity. Whereas, aggregate demand shocks indeed seem to be a strong source
of exogenous variation to tease out the supply elasticity.

##  10 Concluding Remarks

In this paper, we have further developed the GIV methodology introduced by
[39], which takes advantage of panel data to construct instruments for
estimation of structural time series regression models that involve endogenous
regressors. This paper focuses on the underlying econometric issues involved
in developing FGIV in a large $N$ and large $T$ framework where the loadings
are treated as unknown parameters to be estimated before constructing the FGIV
instrument. We further demonstrate that the sampling error arising from
estimating the instrument, factors and a high dimensional precision matrix
does not affect the limiting distribution for the structural parameters of
interest. We also overidentify the structural parameters, which leads to new
and improved results in the crude oil markets application and demonstrate that
the $J$-test is well sized with simulation evidence. Our Monte Carlo study
illustrates that our estimators and algorithms exhibit desirable performance
with the finite sample distributions being well approximated by the asymptotic
distributions. More fruitful areas of research would be empirical applications
of the theoretical results derived in this paper. Interesting theoretical
extensions would be to allow for random slope coefficients with correlated
heterogeneity and / or the presence of weak factors. The authors are currently
pursuing the inclusion of dynamics (lagged dependent variable) which induces a
many instrument problem. Table 1: Bias$\times 100$, RMSE, Size of $t$-test and
Size of $J$-test for Design 1. Finite sample properties for Design 1. $N$ $T$
$\mu$ $\widehat{\phi}^{s}_{FGIV}$ $\widehat{\phi}^{s}_{GK}$
$\widehat{\phi}^{s}_{GMM}$ $\widehat{\phi}^{\,s,rmax}_{GMM}$
$\widehat{\phi}^{d}_{FGIV}$ $\widehat{\phi}^{d}_{GK}$
$\widehat{\phi}^{d}_{GMM}$ $\widehat{\phi}^{d,rmax}_{GMM}$ 1 30 400 0.92
0.0612 0.0910 0.0164 0.0273 -0.2760 -0.2723 0.1100 0.2100 (0.0344) (0.0330)
(0.0204) (0.0204) (0.0152) (0.0173) (0.0079) (0.0080) $[0.0635]$ $[0.1205]$
$[0.0830]$ $[0.0830]$ $[0.0570]$ $[0.0510]$ $[0.0685]$ $[0.0735]$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0540\\}$ $\\{0.0625\\}$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0490\\}$ $\\{0.0465\\}$ 2 50 400
0.85 0.0071 0.0774 0.0174 0.0224 -0.1803 -0.1649 0.1462 0.2306 (0.0313)
(0.0292) (0.0200) (0.0200) (0.0106) (0.0186) (0.0058) (0.0059) $[0.0650]$
$[0.2685]$ $[0.0710]$ $[0.0715]$ $[0.0555]$ $[0.0555]$ $[0.0700]$ $[0.0740]$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0520\\}$ $\\{0.0510\\}$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0480\\}$ $\\{0.0550\\}$ 3 100
400 0.80 0.0059 0.0159 0.0070 0.0102 -0.1886 -0.1558 0.1449 0.2330 (0.0287)
(0.0271) (0.0196) (0.0197) (0.0068) (0.0112) (0.0039) (0.0040) $[0.0610]$
$[0.2505]$ $[0.0585]$ $[0.0615]$ $[0.0515]$ $[0.0540]$ $[0.0705]$ $[0.0790]$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0495\\}$ $\\{0.0485\\}$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0440\\}$ $\\{0.0425\\}$ 4 200
400 0.77 0.0192 0.0246 -0.006 0.0010 -0.1896 -0.1713 0.0524 0.1409 (0.0276)
(0.0263) (0.0188) (0.0188) (0.0046) (0.0080) (0.0027) (0.0027) $[0.0600]$
$[0.2660]$ $[0.0545]$ $[0.0535]$ $[0.0410]$ $[0.0450]$ $[0.0625]$ $[0.0635]$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0590\\}$ $\\{0.0620\\}$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0495\\}$ $\\{0.0425\\}$ 5 500
400 0.75 -0.0013 -0.0078 -0.0097 -0.0102 0.2501 0.2255 -0.0893 -0.1747
(0.0287) (0.0280) (0.0188) (0.0187) (0.0028) (0.0030) (0.0016) (0.0016)
$[0.0545]$ $[0.0840]$ $[0.062]$ $[0.0625]$ $[0.0540]$ $[0.0590]$ $[0.0680]$
$[0.0730]$ $\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0560\\}$
$\\{0.0535\\}$ $\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.054\\}$
$\\{0.051\\}$ • Notes: We report Bias$\times 100$, (RMSE), [$t$-test],and
$\\{$$J$-test$\\}$ (if applicable) with a nominal size of 5%.
$\widehat{\phi}^{s}_{FGIV}$, $\widehat{\phi}^{s}_{GMM}$,
$\widehat{\phi}^{s,rmax}_{GMM}$ estimated with Algorithm 1, 3, 3 with $r=2$
and $rmax=3$, respectively and $\widehat{\phi}^{s}_{GK}$,
$\widehat{\phi}^{d}_{GK}$ both use (8) as an instrument.
$\widehat{\phi}^{d}_{FGIV}$, $\widehat{\phi}^{d}_{GMM}$,
$\widehat{\phi}^{d,rmax}_{GMM}$ estimated with (12), (21), (21) with $r=2$ and
$rmax=3$, respectively. $\mu$ set to maintain $h_{N,\mu}=0.12$ across all
configurations of $(N,T)$. $\bar{\psi}_{u}=0.23$, $\bar{\psi}_{u+\eta}=0.58$,
$\bar{\psi}_{u+\varepsilon}=0.65$. Table 2: Bias, RMSE, Size of $t$-test and
Size of $J$-test for Design 2. Finite sample properties for Design 2. $N$ $T$
$\mu$ $\widehat{\phi}^{s}_{FGIV}$ $\widehat{\phi}^{s}_{GK}$
$\widehat{\phi}^{s}_{GMM}$ $\widehat{\phi}^{\,s,rmax}_{GMM}$
$\widehat{\phi}^{d}_{FGIV}$ $\widehat{\phi}^{d}_{GK}$
$\widehat{\phi}^{d}_{GMM}$ $\widehat{\phi}^{d,rmax}_{GMM}$ 1 30 400 0.92
0.0263 0.0225 0.0080 0.0089 -0.0021 -0.0016 0.0003 0.0009 (0.0365) (0.0259)
(0.0150) (0.0151) (0.0371) (0.0368) (0.0148) (0.0161) $[0.2730]$ $[0.2895]$
$[0.12450]$ $[0.1410]$ $[0.0500]$ $[0.0510]$ $[0.0560]$ $[0.0600]$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.1895\\}$ $\\{0.3035\\}$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.05200\\}$ $\\{0.0465\\}$ 2 50
400 0.85 0.0144 0.0136 0.0058 0.0063 -0.0009 -0.0007 0.0007 0.0013 (0.0245)
(0.0219) (0.0154) (0.0155) (0.0381) (0.0327) (0.0112) (0.0117) $[0.4030]$
$[0.4090]$ $[0.1245]$ $[0.1410]$ $[0.0465]$ $[0.0490]$ $[0.0705]$ $[0.0735]$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.1185\\}$ $\\{0.1615\\}$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0425\\}$ $\\{0.0405\\}$ 3 100
400 0.80 0.0070 0.0065 0.0029 0.0031 -0.0006 -0.0004 0.0009 0.0014 (0.0221)
(0.0196) (0.0137) (0.0139) (0.0117) (0.0196) (0.0068) (0.0069) $[0.1365]$
$[0.3865]$ $[0.0925]$ $[0.0915]$ $[0.0465]$ $[0.0455]$ $[0.0530]$ $[0.0590]$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0895\\}$ $\\{0.0955\\}$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0495\\}$ $\\{0.0510\\}$ 4 200
400 0.77 0.0031 0.0029 0.0015 0.0016 -0.0011 -0.0009 0.0006 0.0010 (0.0216)
(0.0197) (0.0139) (0.0139) (0.0071) (0.0130) (0.0044) (0.0045) $[0.0950]$
$[0.3645]$ $[0.0745]$ $[0.0760]$ $[0.0400]$ $[0.0455]$ $[0.0640]$ $[0.0665]$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0590\\}$ $\\{0.0615\\}$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0510\\}$ $\\{0.0515\\}$ 5 500
400 0.75 0.0014 0.0014 0.0006 0.0006 -0.0011 -0.0009 0.0004 0.0008 (0.0218)
(0.0209) (0.0141) (0.0141) (0.0037) (0.0042) (0.0024) (0.0024) $[0.0690]$
$[0.1125]$ $[0.0670]$ $[0.0690]$ $[0.0515]$ $[0.0570]$ $[0.0605]$ $[0.0675]$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0535\\}$ $\\{0.0520\\}$
$\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $\\{0.0575\\}$ $\\{0.0555\\}$ • Notes:
We report Bias, (RMSE), [$t$-test],and $\\{$$J$-test$\\}$ (if applicable) with
a nominal size of 5%. $\widehat{\phi}^{s}_{FGIV}$, $\widehat{\phi}^{s}_{GMM}$,
$\widehat{\phi}^{s,rmax}_{GMM}$ estimated with Algorithm 1, 3, 3 with $r=2$
and $rmax=3$, respectively and $\widehat{\phi}^{s}_{GK}$,
$\widehat{\phi}^{d}_{GK}$ both use (8) as an instrument.
$\widehat{\phi}^{d}_{FGIV}$, $\widehat{\phi}^{d}_{GMM}$,
$\widehat{\phi}^{d,rmax}_{GMM}$ estimated with (12), (21), (21) with $r=2$ and
$rmax=3$, respectively. $\mu$ set to maintain $h_{N,\mu}=0.12$ across all
configurations of $(N,T)$. $\bar{\psi}_{u}=0.23$, $\bar{\psi}_{u+\eta}=0.58$,
$\bar{\psi}_{u+\varepsilon}=0.65$. Tail index estimator $\widehat{\mu}$ MLE
0.4216 OLS 0.5095 Percentiles Method 0.8987 Modified Percentiles Method 0.9000
Geometric Percentiles Method 0.5208 Weighted Least Squares 0.3725 • Notes: The
estimates are for a month selected at random. However, the estimates do not
change significantly if we estimate $\widehat{\mu}$ for each month and average
across months. Table 3: Tail Index Estimates by Various Methods
$\widehat{\phi}^{s}_{GK}$ $\widehat{\phi}^{s}_{FGIV}$
$\widehat{\phi}^{s}_{GMM}$ Supply instruments $\boldsymbol{Z}_{GIV}$
$\boldsymbol{z}_{GIV}$
$\boldsymbol{Z}_{s}=(\boldsymbol{z}_{GIV},\boldsymbol{\varepsilon})$ Dep.
variable $\boldsymbol{\bar{y}}$ $\boldsymbol{y}_{\widehat{E}}$
$\boldsymbol{y}_{\widehat{E}}$ $\boldsymbol{p}$ $0.044$ $0.016$ $0.005$
$t$-stat $(1.43)$ $(1.35)$ $(4.32)$ $(N,T)$ $(21,370)$ $(21,370)$ $(21,370)$
$J$-stat $p$-value $\\{\text{N.A.}\\}$ $\\{\text{N.A.}\\}$ $0.11$ First stage
$F$-stat $<10$ $<10$ $14.33$ First stage $R^{2}$ $0.26$ $0.14$ $0.21$ • Notes:
$\widehat{\phi}^{s}_{GK}$ is estimated using (8) as the instrument; whereas
$\widehat{\phi}^{s}_{FGIV}$, and $\widehat{\phi}^{s}_{GMM}$ are estimated
using Algorithm 1 and 3 respectively. See Section 9 for more details. The
$t$-stat is reported in parenthesis below coefficient estimates. The
coefficient estimates on
$\boldsymbol{\widehat{\eta}_{t}}$,$\widehat{\eta}_{OPEC,t}$ and
$\boldsymbol{c}_{t-1}$ are omitted for brevity. Table 4: Global crude oil
market: supply elasticity $\widehat{\phi}^{d}_{GK}$
$\widehat{\phi}^{d}_{FGIV}$ $\widehat{\phi}^{d}_{GMM}(r)$ FGMM, BN Demand
instruments $\boldsymbol{Z}_{GIV}$ $\boldsymbol{z}_{GIV}$
$(\boldsymbol{z}_{GIV},\boldsymbol{\eta}[,1])$
$(\boldsymbol{z}_{GIV},\boldsymbol{\eta}[,1:2])$
$(\boldsymbol{z}_{GIV},\boldsymbol{\eta}[,1:3])$
$(\boldsymbol{z}_{GIV},\boldsymbol{\eta}[,1:4])$ $\boldsymbol{\eta}[,1:4]$
Dep. variable $\boldsymbol{d}$ $\boldsymbol{d}$ $\boldsymbol{d}$
$\boldsymbol{d}$ $\boldsymbol{d}$ $\boldsymbol{d}$ $\boldsymbol{d}$
$\boldsymbol{p}$ $-0.463$ $-0.0009$ $-0.0009$ $-0.0003$ $-0.0003$ $-0.0003$
$-0.0003$ $t$-stat $(-3.54)$ $(-0.88)$ $(-0.89)$ $(-0.87)$ $(-0.93)$ $(-1.01)$
$(-0.80)$ $(N,T)$ $(21,370)$ $(21,370)$ $(21,370)$ $(21,370)$ $(21,370)$
$(21,370)$ $(21,370)$ $J$-stat $p$-value $\\{\text{N.A.}\\}$
$\\{\text{N.A.}\\}$ $0.83$ $0.67$ $0.85$ $0.93$ $0.98$ First stage $F$-stat
$<10$ $<10$ $<10$ $<10$ $<10$ $<10$ $<10$ First stage $R^{2}$ $0.58$ $0.12$
$0.12$ $0.15$ $0.15$ $0.16$ $0.16$ • Note: $\widehat{\phi}^{d}_{GK}$ is
estimated using (8) as the instrument; whereas $\widehat{\phi}^{d}_{FGIV}$,
and $\widehat{\phi}^{d}_{GMM}$ are estimated using (12) and (21) respectively.
See Section 9 for more details. The $t$-stat is reported in parenthesis below
coefficient estimates. The coefficient estimates on $\widehat{\eta}_{OPEC,t}$
and $\boldsymbol{c}_{t-1}$ are omitted for brevity. The final column
represents the [12] factor GMM estimator (FGMM). Table 5: Global crude oil
market: demand elasticity

In this appendix, we prove Theorems 1-6, which require 4 lemmas. Lemmas 1, 2
and 3 are included in this appendix while Lemma 4 is deferred to Appendix B of
the Supplementary Appendix.

###### Lemma 1

Under Assumptions 1-4, we have that
$\displaystyle(i.)\,\,\,\underset{i}{\emph{max}}\left(\frac{1}{T}\sum_{t=1}^{T}\tilde{y}_{it}\varepsilon_{t}\right)^{2}=\mathcal{O}_{p}\left(1\right)$
(68) $\displaystyle(ii.)\,\,\,\mathbbm{V}(p_{t})=\Theta(1)$ (69)
$\displaystyle(iii.)\,\,\,\underset{i}{\emph{max}}\left(\frac{1}{T}\sum_{t=1}^{T}\tilde{y}_{it}p_{t}\right)^{2}=\mathcal{O}_{p}\left(1\right)$
(70)
$\displaystyle(iv.)\,\,\,\sum_{i=1}^{N}\sum_{j=1}^{N}\left(\frac{1}{T}\sum_{t=1}^{T}S_{i}\tilde{y}_{jt}\varepsilon_{t}\right)^{2}=\mathcal{O}_{p}(1).$
(71) Proof of Lemma 1: For $(i.)$, by application of Cauchy-Schwarz, we have
that
$\displaystyle\left(\frac{1}{T}\sum_{t=1}^{T}\tilde{y}_{it}\varepsilon_{t}\right)$
$\displaystyle\leq\left(\frac{1}{T}\sum_{t=1}^{T}\tilde{y}^{2}_{it}\right)^{\frac{1}{2}}\cdot\left(\frac{1}{T}\sum_{t=1}^{T}\varepsilon^{2}_{t}\right)^{\frac{1}{2}}=\left(\frac{1}{\sqrt{T}}||\boldsymbol{\tilde{y}}_{i\cdot}||\right)\cdot\left(\frac{1}{\sqrt{T}}||\boldsymbol{\varepsilon}||\right)=\mathcal{O}_{p}(1).$
For $(ii.)$, note that by Assumption 3 we can decompose our share vector into
a dominant and a fringe part:
$\boldsymbol{S}=\begin{pmatrix}\boldsymbol{S}^{\prime}_{d}&\boldsymbol{f}^{\prime}_{G}\end{pmatrix}^{\prime}$
where $\boldsymbol{S}_{d}$ is $N_{1}\times 1$, is the dominant part and
$\boldsymbol{S}_{f}$ is $N_{2}\times 1$, is the fringe part; with
$N_{1}+N_{2}=N$. The key being that $N_{1}(N)=N_{1}$ is fixed while
$N_{2}(N)\rightarrow\infty$ as $N\rightarrow\infty$. Recall that prices are
given by
$p_{t}=\dfrac{1}{\phi^{d}-\phi^{s}}\left(u_{St}+\boldsymbol{\lambda}^{\prime}_{S}\boldsymbol{\eta}_{t}-\varepsilon_{t}\right)$.
For simplicity, suppose that supply and demand shocks are uncorrelated, so
that (ignoring the squared constant term) $\displaystyle\mathbbm{V}(p_{t})$
$\displaystyle=\mathbbm{E}(\boldsymbol{S}^{\prime}\boldsymbol{\Sigma}_{u}\boldsymbol{S})+\mathbbm{E}(\boldsymbol{S}^{\prime}\boldsymbol{\Lambda}\boldsymbol{\Lambda}^{\prime}\boldsymbol{S})+\mathbbm{V}(\varepsilon_{t})=\mathbbm{E}(\boldsymbol{S}^{\prime}\boldsymbol{\Sigma}_{u}\boldsymbol{S})+\mathbbm{E}(\boldsymbol{S}_{d}^{\prime}\boldsymbol{\Lambda}_{d}\boldsymbol{\Lambda}_{d}^{\prime}\boldsymbol{S}_{d})+\mathbbm{E}(\boldsymbol{S}_{f}^{\prime}\boldsymbol{\Lambda}_{f}\boldsymbol{\Lambda}_{f}^{\prime}\boldsymbol{S}_{f})+\mathbbm{V}(\varepsilon_{t})$
$\displaystyle\leq\mathbbm{E}(||\boldsymbol{S}||_{2}^{2}\gamma_{max}(\boldsymbol{\Sigma}_{u}))+\mathbbm{E}(||\boldsymbol{S}_{d}||_{2}^{2}\gamma_{max}(\boldsymbol{\Lambda}_{d}\boldsymbol{\Lambda}_{d}^{\prime}))+\mathbbm{E}(||\boldsymbol{S}_{f}||_{2}^{2}\gamma_{max}(\boldsymbol{\Lambda}_{f}\boldsymbol{\Lambda}_{f}^{\prime}))+\mathcal{O}(1),$
by Assumption 3 and Assumption 4; the first term consists of
$||\boldsymbol{S}||^{2}$, which is $\Theta_{p}(1)$ for $\mu\in(0,1)$, see
Lemma 4 in Appendix B of the Supplementary Appendix, and
$\gamma_{max}(\boldsymbol{\Sigma}_{u})=\mathcal{O}(1)$ by assumption, the
second term is $\mathcal{O}(1)$ by Assumption 4 and the third term is
$\mathcal{O}(\frac{1}{N})\cdot\mathcal{O}(N)=\mathcal{O}(1)$ by Assumption 4.
For part (iii.),
$\displaystyle\left(\frac{1}{T}\sum_{t=1}^{T}\tilde{y}_{it}p_{t}\right)$