# Detecting Learning by Exporting
and from Exporters††thanks: Acknowledgments: We would like to thank Subal
Kumbhakar and seminar participants at UNLV for many insightful comments on an
earlier draft of the paper. Correspondence: Emir Malikov, Lee Business School,
University of Nevada, Las Vegas, Las Vegas, NV 89154-6005. Email:
<EMAIL_ADDRESS>
Jingfang Zhang University of Kentucky Emir Malikov University of Nevada, Las
Vegas

(February 15, 2023)

###### Abstract

Existing literature at the nexus of firm productivity and export behavior
mostly focuses on “learning by exporting,” whereby firms can improve their
performance by engaging in exports. Whereas, the secondary channel of learning
via cross-firm spillovers from exporting peers, or “learning from exporters,”
has largely been neglected. Omitting this important mechanism, which can
benefit both exporters and non-exporters, may provide an incomplete assessment
of the total productivity benefits of exporting. In this paper, we develop a
unified empirical framework for productivity measurement that explicitly
accommodates both channels. To do this, we formalize the evolution of firm
productivity as an export-controlled process, allowing future productivity to
be affected by both the firm’s own export behavior as well as export behavior
of spatially proximate, same-industry peers. This facilitates a simultaneous,
“internally consistent” identification of firm productivity and the
corresponding effects of exporting. We apply our methodology to a panel of
manufacturing plants in Chile in 1995–2007 and find significant evidence in
support of both direct and spillover effects of exporting that substantially
boost the productivity of domestic firms.

Keywords: exporting, learning by exporting, learning from exporters,
production function estimation, productivity, spillovers

JEL Classification: D24, F10, L10

## 1\. Introduction

Governments in both developing and developed countries commonly pursue
policies aimed at promoting exports. In addition to boosting aggregate demand,
such policies are also routinely justified by arguing that domestic exporters
benefit from export-driven productivity improvements via absorption of new
technologies from abroad, learning of international best practices that lead
to improved business processes, productivity enhancements driven by exposure
to more competition, scale effects, quality and variety effects, etc. This
productivity-enhancing mechanism is usually referred to as “learning by
exporting” (LBE) (e.g., Clerides et al., 1998; Aw et al., 2000; Delgado et
al., 2002; Baldwin & Gu, 2004; Van Biesebroeck, 2005; De Loecker, 2007, 2013).
Such export-related productivity gains are facilitated by the firm’s own
direct access to foreign customers, partners and rivals.

Domestic firms, however, can learn not only from their own export experiences
but also from their local peers who engage in exports, and this indirect
learning opportunity is available to both exporters and non-exporters. These
external export-driven productivity spillovers are a type of cross-firm peer
effects, which we refer to as “learning from exporters” (LFE), and effectively
capture the secondary productivity effects of export engagement. Such cross-
firm spillovers may occur via labor turnover, learning by imitation, exposure
to affiliate and/or competitor products, etc. (see Greenaway et al., 2004;
Sala & Yalcin, 2015). For instance, the movement of workers from exporting
firms to other domestic firms may facilitate the dispersion of tacit knowledge
about more innovative/efficient foreign technologies or institutional
knowledge about foreign markets, which may help these firms improve their
productivity too.

Taking the indirect cross-firm productivity effect of export engagement for
granted, as customarily done in literature on the productivity–exports nexus,
will likely underestimate the total productivity benefits of exporting.
Besides, omitting this important mechanism may also contaminate the
measurement of more traditional LBE effects on firm productivity because it
may lead to an endogeneity-inducing omitted variable bias. In this paper, we
contribute to the literature by extending De Loecker (2013) to develop a
unified empirical framework for productivity measurement that explicitly
accommodates both the direct LBE channel taking place within the firm as well
as the indirect LFE channel working between spatially and industrially
proximate firms, which we then apply to a panel of manufacturing plants in
Chile in 1995–2007.

While the literature on (within-firm) LBE effects is rather well-established
(e.g., Kunst & Marin, 1989; Aw & Hwang, 1995; Bernard & Jensen, 1999; Baldwin
& Gu, 2003; Greenaway & Kneller, 2004; Keller, 2004; Blalock & Gertler, 2004;
De Loecker, 2007, 2013; Wagner, 2007; Salomon & Jin, 2008; Park et al., 2010;
Aw et al., 2011; Kasahara & Lapham, 2013; Manjón et al., 2013), the empirical
analysis of external effects of exporting on productivity in the industry
beyond exporter firms is practically non-existent. Existing empirical work on
“export spillovers” focuses mainly on how average export participation in the
industry affects the export status or marginal cost of non-exporters nearby
(e.g., Aitken et al., 1997; Clerides et al., 1998; Bernard & Jensen, 2004;
Greenaway et al., 2004; Greenaway & Kneller, 2008; Koenig, 2009; Koenig et
al., 2010; Alvarez et al., 2013; Poncet & Mayneris, 2013) whereas the
productivity implications of export spillovers in the domestic industry remain
understudied, essentially being limited to three studies (see Wei & Liu, 2006;
Alvarez & López, 2008; Jung, 2010).111Wei & Liu (2006) examine productivity
spillovers from exports in China’s manufacturing, Alvarez & López (2008) use
older Chilean manufacturing data, and Jung’s (2010) dissertation studies
Korean manufacturing plants. All three, however, employ empirical strategies
that are affected by the econometric issues we discuss here.

These empirical analyses of productivity spillovers from exporting are
typically done in one of two ways.222In fact, this is also the case for
studies of spillovers from imports, R&D, FDI and other production-related firm
activities in the market; see Malikov & Zhao (2021) for more discussion. The
first approach is two-step, whereby one first estimates unobserved firm
productivity via standard proxy methods while ignoring the dependence of the
former on exports under the assumption of exogenous first-order Markov
evolution of productivity and then examines spillover effects in a second step
by (linearly) regressing the already estimated firm productivity on the
average export spillover exposure (e.g., Alvarez & López, 2008; Jung, 2010).
Taken at its face value, such a second-step analysis is problematic because it
contradictorily postulates the existence of an endogenous exporting-
productivity relationship that is at odds with the assumption about firm
productivity being purely autoregressive in the first stage. De Loecker (2013)
makes the same argument in his critique of two-step analyses of LBE, but it is
equally important in the context of spillovers via LFE. Aside from the
econometric complications arising from its inherently contradictory setup, the
two-step approach also cannot provide structurally meaningful interpretation
of LFE effects on productivity because of its inability to distinguish between
different data-generating processes that all—and not only the one with non-
zero LFE—can produce a positive relation between the average export
participation of industry and firm productivity.

We avoid this internal inconsistency in our model by explicitly accounting for
potential export spillover-induced productivity improvements during the
estimation of firm productivity, which is partly what enables us to
consistently estimate LFE effects along with other production-function
components. Alternatively, estimates of both productivity and export effects
thereon can be biased due to omission of the relevant measurement of the
peers’ exporting activities from the productivity-proxy function that are
correlated with the firm’s own export behavior as well as quasi-fixed inputs
(via its latent productivity). Empirical findings of external export
spillovers based on a two-step estimation procedure may therefore be spurious.

The other popular approach to testing export spillovers involves a single step
and is based on estimation of the Griliches (1979)-style “augmented production
function” which, besides the conventional inputs, also admits—usually in a
linear fashion—measures of the firm’s own and others’ exporting activities
(e.g., Wei & Liu, 2006). Although such an approach explicitly recognizes the
existence of both LBE and LFE effects on productivity during the estimation,
it does so by restrictively assuming that productivity effects of exporting
are homogeneous across all firms, e.g., no matter their productivity levels.
Besides linearity of the log Cobb-Douglas form that rules out heterogeneous
effects, such an approach is also problematic because it assumes that the
relationship between exporting and productivity is deterministic and, more
importantly, it implies the possibility for a unit-elastic substitution
between inputs and export variables (see De Loecker, 2013, for more on these
points).

To improve upon the above methods, in this paper we seek to measure and test
the direct LBE and indirect LFE effects on productivity in a consistent
structural framework of firm production. Building upon Doraszelski &
Jaumandreu (2013) and De Loecker (2013), we formalize the evolution of firm
productivity as an endogenous export-controlled process, where we explicitly
accommodate two potential channels —internal and external (direct and
indirect)—by which exporting may impact productivity of domestic firms.
Specifically, we allow the firm’s future productivity to be affected not only
by its own past export behavior but also by that of its spatially proximate
peers in the industry. This facilitates a simultaneous, internally consistent
identification of firm productivity and corresponding LBE and LFE
effects.333While our approach is related to that by Malikov & Zhao (2021)
whose recent work also concerns an internally consistent measurement of cross-
firm productivity spillovers, it is distinct in that we focus on the
identification of “contextual” spillover effects—in the Manski (1993)
nomenclature—of peers’ activities (namely, exporting) on productivity whereas
they consider the measurement of “endogenous” effects of peers’ productivity.
The type of spillovers that we study here is more predominant in the
literature, particularly in the context of exporting.

Given the objective of our paper, in our analysis we abstract away from other
plausible and no less important channels of cross-firm interactions that may
lead to productivity spillovers, including local agglomeration, R&D
externalities via technology/knowledge sharing, spillovers along the supply
chains and material-product connections, and focus exclusively on spillovers
from exporting. Similar to De Loecker’s (2013) approach, ours is also silent
on the exact theoretical mechanism by which LBE and LFE occur. Instead of
pinpointing specific channels which is predictably more demanding on data and
experiment design, we pursue a simpler but more feasible goal of testing for
the presence of these export-driven effects on productivity in general.

To achieve identification in the wake of endogeneity-inducing correlation of
input allocations and, potentially, exporting with unobserved firm
productivity (Griliches & Mairesse, 1998), we rely on the popular proxy
variable technique. Our identification strategy utilizes a structural link
between the parametric production function and the firm’s first-order
condition for static inputs which helps us circumvent Ackerberg et al.’s
(2015) and Gandhi et al.’s (2018) non-identification critiques of conventional
proxy-based estimators à la Olley & Pakes (1996) and Levinsohn & Petrin
(2003). In addition, owing to a nonparametric treatment of the firm
productivity process, our model enables us to accommodate heterogeneity in
productivity effects of exporting across firms. This also lets us explore
potential nonlinearities in LBE and LFE effects whereby they can interact with
each other as well as, more importantly, with the firm’s own productivity thus
indirectly allowing for conditioning on the firm’s capacity to learn given its
proximity to the technology frontier. To this end, not only do we provide a
more comprehensive picture of the productivity effects of exporting, but we do
so in a robust way by dealing with internal inconsistency and identification
problems prevalent in prior literature.

We study the productivity-enhancing effects of exporting using plant-level
data on Chilean manufacturers during the 1995–2007 period, with exporters
accounting for about 21% of the sample. Using our semiparametric methodology,
we find that exporters enjoy a statistically significant productivity premium
over non-exporters along the entire distribution of productivity which,
however, can be reflective of both learning and self-selection effects.
Zooming in on the within-firm learning effects of exporting, we find
significant evidence of both LBE and LFE. Overall, the LBE productivity effect
is statistically significant for 93% of firms in our main specification. On
average, the size of LFE effect (cumulatively from all peers) is commensurate
to that of LBE but, at the observation level, LFE is significantly non-zero
for 69% of plants only, thereby suggesting that the indirect productivity-
boosting effect of exporting is less prevalent in Chilean manufacturing than
the direct learning effect taking place within the firm. We also document that
the LFE effect is stronger for the firms who also export themselves. This
empirical evidence therefore suggests that exporters benefit from the exposure
to peer exporters in their local industry more than non-exporters, plausibly
because there may be complementarities between internal/direct and
external/indirect learning from export experiences. Using simple back-of-
envelope calculations, we find that long-run total LBE and LFE effects on mean
firm productivity are, respectively, 0.70% and 0.63% per percentage point of
export intensity. Thus, in the long-run equilibrium a permanent 10 percentage
point increase in mean export intensity of all firms in the local industry is
roughly estimated to produce a sizable 13.3% boost to average firm
productivity through both direct LBE and indirect LFE channels.

The rest of the paper is organized as follows. Section 2 presents the
conceptual framework. Section 3 describes our identification and estimation
procedure. The data are discussed in Section 4. We report the empirical
results in Section 5. Section 6 concludes.

## 2\. Conceptual Framework

Consider the firm $i(=1,\dots,n)$ at time $t(=1,\dots,T)$. Following the
convention of productivity literature (e.g., Olley & Pakes, 1996; Blundell &
Bond, 2000; Levinsohn & Petrin, 2003; De Loecker & Warzynski, 2012;
Doraszelski & Jaumandreu, 2013; Ackerberg et al., 2015; Konings &
Vanormelingen, 2015; Jin et al., 2019), we assume that the firm employs
physical capital $K_{it}$, labor $L_{it}$ and an intermediate input such as
materials $M_{it}$ to produce the output $Y_{it}$ via the Cobb-Douglas
production technology subject to the Hicks-neutral productivity:

$Y_{it}=A_{0}K_{it}^{\alpha_{K}}L_{it}^{\alpha_{L}}M_{it}^{\alpha_{M}}\exp\left\\{\omega_{it}+\eta_{it}\right\\},$
(2.1)

where $A_{0}$ is a scalar constant;
$(\alpha_{K},\alpha_{L},\alpha_{M})^{\prime}$ are the input elasticities;
$\omega_{it}$ is the firm’s persistent productivity (TFP) which is known to
the firm at time $t$ but unknown to an econometrician; and $\eta_{it}$ is a
random i.i.d. productivity shock such that
$E[\eta_{it}|\mathcal{I}_{it}]=E[\eta_{it}]=0$, where $\mathcal{I}_{it}$ is
the $i$th firm’s information set in period $t$.

As in many studies in productivity literature (e.g., Gandhi et al., 2020;
Tsionas & Mallick, 2019; Hou et al., 2020; Malikov & Zhao, 2021; Malikov et
al., 2023), physical capital $K_{it}$ and labor $L_{it}$ are said to be
subject to adjustment frictions, and the firm optimizes them dynamically at
time $t-1$ rendering these inputs predetermined quasi-fixed state variables.
The materials input $M_{it}$ is freely varying and determined by the firm
statically at time $t$. Both $K_{it}$ and $L_{it}$ follow their respective
laws of motion:

$K_{it}=I_{it-1}+\left(1-\delta\right)K_{it-1}\quad\text{and}\quad
L_{it}=H_{it}+L_{it-1},$ (2.2)

where $I_{it}$, $H_{it}$ and $\delta$ respectively denote the gross
investment, net hiring and capital depreciation rate of the firm $i$ in period
$t$. We assume that the risk-neutral firm faces perfectly competitive output
and input markets and seeks to maximize a discounted stream of the expected
life-time profits subject to its state variables and expectations about the
market structure variables including prices that are common to all firms.

In this paper, our principal interest is in the measurement of internal and
external productivity effects of exporting in a domestic industry. Instead of
modeling the firm’s export behavior in a discrete fashion by focusing on its
“status” as popularly done in the literature (e.g., Blalock & Gertler, 2004;
Van Biesebroeck, 2005; Amiti & Konings, 2007; Kasahara & Lapham, 2013), we
formalize the firm’s exporting in a richer, continuous framework along the
lines of De Loecker (2007) and Malikov et al. (2020). We rely on the firm’s
export intensity as a measure of its own export behavior as well as to model
its exposure to peer exporters in the industry. Concretely, let
$X_{it}\in[0,1]$ denote the firm’s export intensity defined as the nominal
share of its total output produced for export abroad, with its boundary values
corresponding to wholly domestic and fully export-oriented firms.

We conceptualize the firm’s exporting decisions as a choice of the degree of
its export orientation $X_{it}$. Given the documented persistence of export
experience over time, we formalize these decisions as dynamic but, unlike
other dynamic production choices by the firm in our model such as capital
investment or hiring, they need not be subject to adjustment frictions that
would render them delayed and, hence, predetermined.444To make these
distinctions clearer: if the optimal decision concerning production in period
$t$ is affected by its history, then that decision is said to be “dynamic.”
If, due to adjustment frictions, a decision concerning production in period
$t$ is effectively made at $t-1$, then we say it is “predetermined.” In this
nomenclature, $K_{it}$ and $L_{it}$ are dynamic and predetermined, whereas
$X_{it}$ is dynamic but chosen at time $t$. This assumption is important
because it does not rule out a contemporaneous correlation between firm
productivity $\omega_{it}$ and its export orientation $X_{it}$. Thus, the
firm’s export intensity $X_{it}$ evolves according to the following dynamic
process:

$X_{it}=\mathcal{X}_{it}+X_{it-1},$ (2.3)

where $\mathcal{X}_{it}$ regulates the endogenous adjustment in the degree of
export orientation $X_{it}$ at $t$.

The firm’s dynamic optimization problem is then described by the following
Bellman equation:

$\displaystyle\mathbb{V}_{t}\big{(}\Xi_{it}\big{)}=\max_{I_{it},H_{it},\mathcal{X}_{it}}\Big{\\{}$
$\displaystyle\Pi_{t}(\Xi_{it})-\text{C}^{I}_{t}(I_{it},K_{it})-\text{C}^{H}_{t}(H_{it},L_{it})-\text{C}^{X}_{t}(\mathcal{X}_{it},X_{it-1})+\mathbb{E}\Big{[}\mathbb{V}_{t+1}\big{(}\Xi_{it+1}\big{)}\Big{|}\Xi_{it},I_{it},H_{it},\mathcal{X}_{it}\Big{]}\,\Big{\\}},$
(2.4)

where
$\Xi_{it}=(\omega_{it},K_{it},L_{it},X_{it-1})^{\prime}\in\mathcal{I}_{it}$
are the state variables; $\Pi_{t}(\Xi_{it})$ is the restricted profit function
derived as a value function corresponding to the static optimization problem
in (3.1); and $\text{C}^{\kappa}_{t}(\cdot)$ is the cost of adjustments in
capital ($\kappa=I$), labor ($\kappa=H$) and exporting ($\kappa=X$). From the
laws of motion in (2.2)–(2.3), in the above dynamic problem the firm’s
exporting behavior $X_{it+1}$ is chosen (via $\mathcal{X}_{it+1}$) in time
period $t+1$ unlike the amounts of dynamic inputs $K_{it+1}$ and $L_{it+1}$
that are chosen by the firm in time period $t$ (via $I_{it}$ and $H_{it}$,
respectively). Solving (2.4) for $I_{it}$, $H_{it}$ and $\mathcal{X}_{it}$
yields their respective optimal policy functions.

Next we formalize the productivity effects of exports. We do so by extending
De Loecker’s (2013) framework to accommodate not only the more traditional
direct LBE effects but also to allow for indirect effects via learning from
exporting peers. That is, we explicitly model two potential channels—internal
and external (direct and indirect)—by which exporting may impact the
productivity of domestic firms. The first channel, referred to as “learning by
exporting,” takes place within the firm internally and is commonly attributed
to the exporter firm’s absorption of new technologies from abroad, learning of
international best practices that lead to improved manufacturing processes,
productivity enhancements driven by the exposure to more competition, scale
effects, quality and variety effects, etc. These export-related productivity
gains are facilitated by the firm’s own direct access to foreign customers and
rivals.

The second channel is less obvious and oftentimes left unaccounted. Domestic
firms (both the exporters and non-exporters) can learn not only from their own
export behavior but also indirectly from their exporting peers’. These
external export-driven productivity spillovers are a type of cross-firm peer
effects, which we refer to as “learning from exporters,” and effectively
capture the secondary productivity effects of export engagement. Such cross-
firm spillovers may arise through pooling of workers, informal contacts (e.g.,
attendance of trade shows, exposure to affiliate and/or competitor products
and marketing, learning by imitation, customer-supplier discussions) or more
formal reverse engineering. For instance, by monitoring successful exporting
peers’ market behavior, both in domestic and foreign markets, domestic firms
can imitate and then adopt their business strategies to boost own
productivity. Alternatively, the movement of labor from exporting firms to
other domestic firms may facilitate the dispersion of tacit knowledge about
more innovative/efficient foreign technologies and better business practices
or institutional knowledge about foreign markets, which may help hiring firms
increase their productivity.

To capture export-driven productivity spillovers, we proxy each firm’s
exposure to exporters in the industry using the average export intensity of
its spatially proximate peers operating in the same industry defined as

$\overline{X}_{it}=\sum_{j\neq i}p_{ijt}X_{jt},$ (2.5)

where $\\{p_{ijt};\ j(\neq i)=1\dots,n\\}$ are the peer-firm weights
identifying exporters in the firm $i$’s industry and spatial locality in
period $t$. In our baseline specification, we construct the peer connection
weights as $p_{ijt}={1}(j\in\mathcal{L}_{it})\big{/}\sum_{k\left(\neq
i\right)=1}^{n}{1}(k\in\mathcal{L}_{it})$, where $\mathcal{L}_{it}$ denotes a
set of firms that are in the same industry and geographical region as is the
firm $i$ in time period $t$. Thus, for the $i$th firm at time $t$ this
weighting scheme assigns a uniform weight of $1/n_{it}$ to all other firms
from the same industry in the same region, where $n_{it}$ is the total number
of such peer firms. This scheme is based on the conventional argument that
geographical proximity and industry play a central role in productivity
spillovers. For example, it is technologically easier and less costly for
firms to monitor and mimic strategies of other exporters that operate within
the same industry and region. This is also in line with the literature on
export spillovers (e.g., see Bernard & Jensen, 2004; Greenaway & Kneller,
2008; Koenig, 2009; Koenig et al., 2010; Poncet & Mayneris, 2013). We weight
all peers equally, given that their export intensity which is being averaged
to compute $\overline{X}_{it}$ is already measured relative to the scale of
their production.

Note that our export exposure measure is firm-specific because it excludes the
$i$th firm. Thus, $\overline{X}_{it}$ captures the external export orientation
of the local industry which firm $i$ is exposed to. This measure varies across
both the firms and time. While closely related, $\overline{X}_{it}$ is
therefore not the “grand” industry average but the peer average in the
industry. This distinction is crucial for the separable identification of LBE
and LFE effects (more on this later).

We model firm productivity evolution as a controlled first-order Markov
process, whereby we allow firm $i$ to improve its future productivity not only
via learning by exporting but also via learning from exporting peers.
Generalizing Doraszelski & Jaumandreu’s (2013) and De Loecker’s (2013)
formulations to include the indirect cross-firm effects of exporting, the
productivity process is

$\omega_{it}=h\left(\omega_{it-1},X_{it-1},\overline{X}_{it-1}\right)+\zeta_{it},$
(2.6)

where $h(\cdot)$ is the conditional mean function of $\omega_{it}$; and
$\zeta_{it}$ is a zero-mean random innovation in persistent productivity that
is unanticipated by the firm at $t-1$:
$E\left[\zeta_{it}|\mathcal{I}_{it-1}\right]=0$. The LBE and LFE effects can
then be measured as $LBE_{it}={\partial
E\left[\omega_{it}|\cdot\right]}\big{/}{\partial X_{it-1}}$ and
$LFE_{it}={\partial
E\left[\omega_{it}|\cdot\right]}\big{/}{\partial\overline{X}_{it-1}}$,
respectively.

Identification of the productivity effects of exporting in our model is based
on several structural timing assumptions. The evolution process in (2.6)
assumes that both the internal and external learning associated with exports
occurs with a delay which is why the dependence of $\omega_{it}$ on controls
is lagged implying that export-driven improvements in firm productivity take a
period to materialize. Such a timing assumption is quite common in the LBE
literature (e.g., Van Biesebroeck, 2005; De Loecker, 2013; Malikov et al.,
2020, 2021, 2023). In $E[\zeta_{it}|\mathcal{I}_{it-1}]=0$, we also assume
that the firm does not experience changes in exporting in light of expected
future innovations in its productivity. This rules out the firm’s ability to
systematically predict future productivity shocks. Instead, the productivity
process in (2.6) says that firms anticipate the effect of their export
experience $X_{it-1}$ on future productivity $\omega_{it}$ when adjusting the
former in period $t-1$, and the conditional mean
$E[\omega_{it}|\omega_{it-1},X_{it-1},\overline{X}_{it-1}]$ captures that
expected productivity. But the actual firm productivity at time $t$ also
includes an unanticipated innovation $\zeta_{it}$. In essence, the error term
$\zeta_{it}$ represents unpredictable uncertainty that is naturally associated
with any productivity-modifying learning activities such as chance in
discovery, success in implementation, etc. The innovation $\zeta_{it}$ is
realized after $X_{it-1}$ will have already been chosen. This structural
timing assumption about the arrival of $\zeta_{it}$, which renders the firm’s
past export experience mean-orthogonal to a random innovation at time $t$,
helps us identify both the direct learning and external spillover effects. In
Appendix A, we also discuss how inclusion of the lagged firm productivity in
(2.6) indirectly allows us to, at least partly, control for self-selection of
firms into exporting based on their productivity levels.

Since, in the productivity process (2.6), exporting enters the conditional
mean of productivity via two variables, of natural interest is the ability of
our model to separate the direct LBE effect from indirect LFE spillovers.
Using simple calculus we can show that, owing to the definition of the average
peer export orientation which excludes the export information pertaining to
the $i$th firm:

$\displaystyle\frac{\partial E[\omega_{it}|\cdot]}{\partial X_{it-1}}$
$\displaystyle=\underbrace{\frac{\partial h(\cdot)}{\partial
X_{it-1}}}_{LBE_{it}}+\underbrace{\frac{\partial
h(\cdot)}{\partial\overline{X}_{it-1}}}_{LFE_{it}}\times\frac{\partial\overline{X}_{it-1}}{\partial
X_{it-1}}=LBE_{it},$ (2.7)

because ${\partial\overline{X}_{it-1}}\big{/}{\partial
X_{it-1}}=\partial\sum_{j\neq i}p_{ijt}X_{jt-1}\big{/}\partial X_{it-1}=0$.
Thus, $LBE_{it}$ is separably identifiable. Intuitively, all observable
variation in expected productivity due to a change in the firm’s own export
intensity is attributable to the direct learning effect because its exporting
does not immediately affect the export behavior of its peers. Obviously, the
separability of the two effects would be impossible if, in place of the
average peer export intensity, we would have used the total average of all
firms as oftentimes done in the literature (e.g., Alvarez & López, 2008; Jung,
2010).

## 3\. Empirical Strategy

Estimating the production function using least squares would result in a
simultaneity bias due to the dependence of freely varying inputs (regressors
in the production function) on unobserved firm productivity $\omega_{it}$
because the latter is a part of the firm’s information set $\mathcal{I}_{it}$
based upon which it makes optimal input allocation decisions at $t$. This
omitted variable bias is also known as a “transmission bias" (Griliches &
Mairesse, 1998). A proxy variable method proposed by Olley & Pakes (1996) and
extended by Levinsohn & Petrin (2003) tackles this endogenous problem by
proxying for unobservable $\omega_{it}$ via the observable static intermediate
input $M_{it}$ and then using weakly exogenous higher-order lags of inputs to
instrument for endogenous freely varying inputs. Recently, this methodology
has been critiqued for its lack of identification due to perfect functional
dependence between freely varying inputs and self-instrumenting quasi-fixed
factors (Ackerberg et al., 2015) and violation of the “rank condition” in
instrumentation of these endogenous freely varying inputs (Gandhi et al.,
2020). As a solution, Gandhi et al. (2020) have suggested employing the
information contained in the first-order condition for static inputs to
identify both the production function and latent firm productivity. However,
because their procedure is fully nonparametric, its implementation is three-
stage and quite computationally burdensome, especially in its requirement to
integrate the estimated static input elasticity function at each observation
in order to recover the unknown production function. In this paper, we
therefore rely on Malikov & Zhao’s (2021) more easy-to-implement
semiparametric adaptation of the Gandhi et al. (2020) methodology (which we
modify to suit our research question) that utilizes a prespecified parametric
form of the production function to derive the proxy function. This is similar
to the idea pursued by Doraszelski & Jaumandreu (2013). The semiparametric
adaptation can significantly ease the demand on data as well as the
computational burden of estimation.

Identification.—Consider the firm’s optimality condition for materials. Since
the intermediate input $M_{it}$ is freely varying and thus affects profits
only in the current period, the firm’s restricted expected profit maximization
problem with respect to $M_{it}$ is as follows:

$\underset{M_{it}}{\max}\
P_{t}^{Y}A_{0}K_{it}^{\alpha_{K}}L_{it}^{\alpha_{L}}M_{it}^{\alpha_{M}}\exp\left\\{\omega_{it}\right\\}\theta-
P_{t}^{M}M_{it},$ (3.1)

where $P_{t}^{Y}$ and $P_{t}^{M}$ respectively denote the output and material
input price, both of which are competitively determined. The constant $\theta$
is defined as $\theta\equiv
E\left[\exp\left\\{\eta_{it}\right\\}\mid\mathcal{I}_{it}\right]$.

Taking the log-ratio of the first-order condition with respect to $M_{it}$

$\alpha_{M}P_{t}^{Y}A_{0}K_{it}^{\alpha_{K}}L_{it}^{\alpha_{L}}M_{it}^{\alpha_{M}-1}\exp\left\\{\omega_{it}\right\\}\theta=P_{t}^{M}$
(3.2)

and the production function in (2.1) gives

$\ln\left(S_{it}^{M}\right)=\ln\left(\alpha_{M}\theta\right)-\eta_{it},$ (3.3)

where $S_{it}^{M}\equiv\frac{P_{t}^{M}M_{it}}{P_{t}^{Y}Y_{it}}$ is the
intermediate input share of output. Thus, we can identify a composite constant
$\alpha_{M}\theta$ from the unconditional moment $E\left[\eta_{it}\right]=0$,
from where we have that

$\ln\left(\alpha_{M}\theta\right)=E\left[\ln\left(S_{it}^{M}\right)\right].$
(3.4)

We can also separately identify the constant $\theta$ via $\theta\equiv
E\left[\exp\left\\{\eta_{it}\right\\}\right]=E\left[\exp\left\\{\ln\left(\alpha_{M}\theta\right)-\ln\left(S_{it}^{M}\right)\right\\}\right]=$
$E\left[\exp\left\\{E\left[\ln\left(S_{it}^{M}\right)\right]-\ln\left(S_{it}^{M}\right)\right\\}\right]$,
with equation (3.4) used to substitute for $\ln\left(\alpha_{M}\theta\right)$
in the third equality. Combining this result with (3.4), we identify the
firm’s material elasticity $\alpha_{M}$ as

$\alpha_{M}=\exp\left\\{E\left[\ln\left(S_{it}^{M}\right)\right]\right\\}/E\left[\exp\left\\{E\left[\ln\left(S_{it}^{M}\right)\right]-\ln\left(S_{it}^{M}\right)\right\\}\right],$
(3.5)

where it is a unique function of the first moments of data.

To identify the rest of production function as well as latent firm
productivity, we take the log of (2.1) on both sides to obtain

$y_{it}=\alpha_{0}+\alpha_{K}k_{it}+\alpha_{L}l_{it}+\alpha_{M}m_{it}+\omega_{it}+\eta_{it},$
(3.6)

where $\alpha_{0}\equiv\ln A_{0}$; and the lower-case variables correspond to
the log form of the respective upper-case variables. Exploiting the Markov
process of $\omega_{it}$ in (2.6) and bringing the already identified material
elasticity $\alpha_{M}$ to the left-hand side, we rewrite (3.6) as follows:

$y_{it}^{*}=\alpha_{K}k_{it}+\alpha_{L}l_{it}+g\left(\omega_{it-1},X_{it-1},\overline{X}_{it-1}\right)+\zeta_{it}+\eta_{it},$
(3.7)

where $y_{it}^{*}=y_{it}-\alpha_{M}m_{it}$ is fully identified and can be
treated as an observable, and $g\left(\cdot\right)\equiv h(\cdot)+\alpha_{0}$
is of unknown functional form.

Next, from equation (3.2) we derive the explicit form of the conditional
demand function for $M_{it}$, which we then invert to proxy for the
unobservable scalar $\omega_{it}$ in (B.5) in the spirit of material-based
proxy estimators:

$y_{it}^{*}=\alpha_{K}k_{it}+\alpha_{L}l_{it}+g\left(\left[m_{it-1}^{*}-\alpha_{K}k_{it-1}-\alpha_{L}l_{it-1}\right],X_{it-1},\overline{X}_{it-1}\right)+\zeta_{it}+\eta_{it},$
(3.8)

where
$m_{it-1}^{*}=\ln\left({P_{t-1}^{M}}\big{/}{P_{t-1}^{Y}}\right)-\ln\left(\alpha_{M}\theta\right)-\left(\alpha_{M}-1\right)m_{it-1}$
is also fully identified and treated as an observable. Since all regressors
appearing in (3.8) including $k_{it}$, $l_{it}$, $k_{it-1}$, $l_{it-1}$,
$m_{it-1}^{*}(m_{it-1})$, $X_{it-1}$ and $\overline{X}_{it-1}$ are weakly
exogenous based on our structural assumptions, there is no endogenous
covariate on the right-hand side of the equation. That is,

$E\left[\zeta_{it}+\eta_{it}\bigl{|}k_{it},l_{it},k_{it-1},l_{it-1},m_{it-1}^{*}(m_{it-1}),X_{it-1},\overline{X}_{it-1}\right]=0,$
(3.9)

and the equation (3.8) is identified.

From (3.8)–(3.9), it is obvious that, if there were indeed non-zero export
spillovers and we had failed to account for them in the firm’s productivity
evolution process, then $\overline{X}_{it-1}$ would have been omitted from the
proxy function $g(\cdot)$ in (3.8) and, consequently, been absorbed, along
with its interactions with other arguments of the proxy, into the error term.
In the latter case, the error term would then contain variation from the
firm’s quasi-fixed inputs, its own export intensity as well as the average
export orientation of its peers. Generally, these all would be correlated with
quasi-fixed inputs and the export variable included as regressors thus
violating the weak exogeneity condition, and the model would be unidentified
due to the omitted variable bias. This highlights the importance of embedding
the external spillover channel into the analytical framework explicitly.

Lastly, we can recover latent firm productivity up to a constant using the
identified production-function parameters and the productivity shock:
$\omega_{it}+\alpha_{0}=y_{it}-\alpha_{K}k_{it}-\alpha_{L}l_{it}-\alpha_{M}m_{it}-\eta_{it}$.

Estimation procedure.—The estimation is simple and involves a two-stage
procedure. In the first stage, we estimate $\alpha_{M}$ via a sample
counterpart of (3.5) constructed using sample averages computed from the raw
data on material share:

$\widehat{\alpha}_{M}=\exp\left\\{\frac{1}{nT}\sum_{i}\sum_{t}\ln\left(S_{it}^{M}\right)\right\\}\Bigg{/}\left[\frac{1}{nT}\sum_{i}\sum_{t}\exp\left\\{\left[\frac{1}{nT}\sum_{i}\sum_{t}\ln\left(S_{it}^{M}\right)\right]-\ln\left(S_{it}^{M}\right)\right\\}\right].$

As a by-product, we also have
$\ln\widehat{\left(\alpha_{M}\theta\right)}=\frac{1}{nT}\sum_{i}\sum_{t}\ln\left(S_{it}^{M}\right)$
and
$\widehat{\eta}_{it}=\ln\widehat{\left(\alpha_{M}\theta\right)}-\ln\left(S_{it}^{M}\right)$.
With these estimates in hand, we then construct estimates of $y_{it}^{*}$ and
$m_{it}^{*}$ as $\widehat{y}_{it}^{*}=y_{it}-\widehat{\alpha}_{M}m_{it}$ and
$\widehat{m}_{it-1}^{*}=\ln\left(\frac{P_{t-1}^{M}}{P_{t-1}^{Y}}\right)-\ln\widehat{\left(\alpha_{M}\theta\right)}-\left(\widehat{\alpha}_{M}-1\right)m_{it-1}$,
respectively.

The second-stage estimation requires the choice of an approximator for the
unknown $g(\cdot)$. We use the popular second-order polynomial sieves (e.g.,
Gandhi et al., 2020). Specifically, we approximate $g\left(\cdot\right)$ as
follows:

$\displaystyle g\left(\cdot\right)$
$\displaystyle\approx\left(1,W_{it-1}\left(\mathbf{\boldsymbol{\alpha}}\right),W_{it-1}^{2}\left(\boldsymbol{\alpha}\right),X_{it-1},X_{it-1}^{2},\overline{X}_{it-1},\overline{X}_{it-1}^{2},W_{it-1}\left(\boldsymbol{\alpha}\right)X_{it-1},W_{it-1}\left(\boldsymbol{\alpha}\right)\overline{X}_{it-1},X_{it-1}\overline{X}_{it-1}\right)^{\prime}\boldsymbol{\gamma}\equiv\boldsymbol{\lambda}_{it}\left(\boldsymbol{\alpha}\right)^{\prime}\boldsymbol{\gamma},$
(3.10)

where $\boldsymbol{\alpha}=\left(\alpha_{K},\alpha_{L}\right)^{\prime}$,
$W_{it-1}\left(\boldsymbol{\alpha}\right)=\widehat{m}_{it-1}^{*}-\alpha_{K}k_{it-1}-\alpha_{L}l_{it-1}$,
and $\boldsymbol{\gamma}$ is the unknown parameter vector.

We estimate (3.8) using a nonlinear least squares method to obtain the second-
stage estimates of $(\alpha_{K},\alpha_{L})^{\prime}$ and
$\boldsymbol{\gamma}$:

$\underset{\alpha_{K},\alpha_{L},\boldsymbol{\gamma}}{\min}\
\sum_{i}\sum_{t}\left(\widehat{y}_{it}^{*}-\alpha_{K}k_{it}-\alpha_{L}l_{it}-\boldsymbol{\lambda}_{it}\left(\alpha_{K},\alpha_{L}\right)^{\prime}\boldsymbol{\gamma}\right)^{2}.$

With the estimated production-function parameters in hand, we can compute the
productivity effects via
$\widehat{LBE}_{it}={\partial\widehat{g}\left(\cdot\right)}\big{/}{\partial
X_{it-1}}$ and
$\widehat{LFE}_{it}={\partial\widehat{g}\left(\cdot\right)}\big{/}{\partial\overline{X}_{it-1}}$,
where
$\widehat{g}\left(\cdot\right)=\boldsymbol{\lambda}_{it}\left(\widehat{\boldsymbol{\alpha}}\right)^{\prime}\widehat{\boldsymbol{\gamma}}$.555Note
that the definitions of the LBE and LFE effects are based on the gradients of
$h(\cdot)$ but, since $g(\cdot)$ and $h(\cdot)$ differ only by an additive
constant, their gradients are equal. We also recover $\omega_{it}$ up to a
constant via
$\widehat{\omega_{it}+\alpha_{0}}=y_{it}-\widehat{\alpha}_{K}{k}_{it}-\widehat{\alpha}_{L}{l}_{it}-\widehat{\alpha}_{M}{m}_{it}-\widehat{\eta}_{it}$.

For statistical inference, we employ accelerated biased-corrected percentile
bootstrap confidence intervals. Details are provided in Appendix C.

## 4\. Data

Our data come from the Encuesta Nacional Industrial Anual (ENIA), a national
industrial survey, conducted by the Chilean National Institute of Statistics
annually. The sample period runs from 1995 to 2007. Manufacturing plants are
classified into 22 industry groups according to the 2-digit International
Standard Industry Classification (ISIC) code. The dataset contains information
on plants from 13 regions including Tarapacá, Antofagasta, Atacama, Coquimbo,
Valparaíso, Libertador Gral. Bernardo O’Higgins, Maule, Biobío, La Araucanía,
Los Lagos, Aisén del Gral. Carlos Ibáñez del Campo, Magallanes and Chilean
Antarctica, and the Santiago Metropolitan region. Though each observation
represents a plant rather than a firm, single-plant establishments account for
over 90% of total units (also see Pavcnik, 2002).

The total output is defined as total revenues from the sale of products and
work done and, as such, our analysis is carried out using a revenue-based
productivity measure (TFPR). As in De Loecker (2013), the measured TFPR will
capture export-driven changes in both the cost and demand factors. Hence, a
positive LBE and/or LFE effect can be interpreted as working through either a
decline in production cost or an increase in demand, or both. For more on the
latter point, see Appendix B.

Capital is the fixed assets balance for buildings, machinery and vehicles at
the end of a period. Materials are defined as the total expenditure on
intermediate inputs consisting of raw materials and other intermediates. These
three variables are measured in hundred thousands of pesos, and we deflate
them using price deflators at the 4-digit ISIC level. We measure labor using
the total number of people working at the plant. We drop observations that
contain missing or negative values for these variables and exclude extreme
outliers lying outside the interval between the 1th and 99th percentiles of
these four variables. In the end, our sample consists of 8,353 manufacturing
plants with a total of 47,622 observations.

Export intensity is calculated as the nominal proportion of the firm’s exports
in its total sales, ranging from 0 to 1 by construction. Out of all plants,
exporters are 20.6% of the sample. As discussed earlier, we measure each
plant’s exposure to exporters using the average export intensity of its peers,
excluding the “recipient” plant itself. Peers are identified as operating in
the same one of 13 regions and the same 2-digit industry within the same
period (a total of 186 peer groups), but we remove the geographic limits in
one of the robustness checks. The exposure variable also ranges between 0 and
1 by construction. Table C.1 in Appendix D provides the summary statistics for
our data.

## 5\. Results

Our primary interest is in the estimates of productivity effects of exporting.
Owing to a nonparametric form of the conditional mean of $\omega_{it}$, we
obtain observation-specific estimates of the LBE and LFE effects. Since these
effects are defined as the gradients of expected future firm log-productivity
with respect to export intensity or the peer average thereof, both of which
are proportions, the reported $LBE_{it}$ ($LFE_{it}$) is a semi-elasticity
measuring percentage changes in productivity per unit percentage point change
in the firm’s export intensity (the average peer export orientation of the
local industry). Lastly, both effects measure “short-run” productivity
improvements per annum, which however can accumulate over time owing to the
persistent nature of productivity evolution.

### 5.1. Exporter productivity differential

As a preliminary analysis, we first examine the overall cross-firm
differential in productivity $\omega_{it}$ across exporters and non-exporters
in the Chilean manufacturing sector.666The associated production function
parameter estimates are $\widehat{\alpha}_{K}=0.21$,
$\widehat{\alpha}_{L}=0.46$ and $\widehat{\alpha}_{M}=0.29$. Scale elasticity,
defined as the sum of all three input elasticities, is 0.96 with the
corresponding 95% bootstrap confidence interval of (0.938, 0.975), indicating
the decreasing returns to scale, consistent with the profit-maximizing
behavior. Allowing for differences in estimators and sample periods, our
estimates are in the same ballpark as the input elasticity and returns to
scale values reported by other studies that used Chilean data. The mean
estimate of the (log)-productivity differential between exporters and non-
exporters is 0.28 with the corresponding 95% bootstrap percentile confidence
interval of (0.26, 0.31), which indicates that, on average, exporters are more
productive than non-exporters by roughly 28%.

We further investigate if a plant’s exporter status commands a productivity
premium of varying magnitude and significance at different points in the
productivity distribution. We do so by estimating a simple quantile
regression:
$Q_{\tau}\left[\omega_{it}|\cdot\right]=\beta_{0,\tau}+\beta_{1,\tau}1(X_{it}>0)\
\text{for}\ \tau\in(0,1)$. Employing a quantile regression enables us to
explore potential distribution heterogeneity in the exporter productivity
differential. We estimate this model for the quantile index $\tau$ taking
values from 0.2 to 0.8 (with 0.05 increments) thereby focusing on the central
portion of the productivity distribution. Figure 1 plots the quantile
regression estimates of the $\beta_{1,\tau}$ coefficient (the exporter
productivity differential) against $\tau$, along with the 95% confidence
intervals. The quantile productivity differentials are all significantly
positive and increasing with the quantile of $\omega$, indicating that the
productivity divergence between exporters and non-exporters is more prominent
magnitude-wise among more productive firms.777Including controls for the plant
size (proxied by the number of employees) as well as the region and year
effects produces similar findings.

Figure 1: Exporter Productivity Differential Estimates across Productivity
Quantiles
[Notes: Shaded bands correspond to 95% confidence intervals. Solid horizontal
lines correspond to the productivity differentials estimated at the
conditional mean, with their respective 95% confidence intervals shown using
dashed lines.] Figure 2: Distributions of log-Productivity by the Exporter
Status
[Notes: Vertical lines show respective sample means.]

For a more holistic look at the productivity differential between exporters
and non-exporters, we also plot the kernel density of firm log-productivity
for exporters and non-exporters, as shown in Figure 2. This allows us to
compare distributions of productivity estimates as opposed to merely focusing
on marginal moments or quantiles. The figure indicates that exporters appear
to enjoy a productivity premium over non-exporters distribution-wise. To
support this visual evidence, we do a formal test to check if exporters are
more productive than non-exporters along the entire distribution of
productivity. We utilize a generalization of the Kolmogorov-Smirnov test
proposed by Linton et al. (2005) to test the (first-order) stochastic
dominance of exporters’ productivity over non-exporters’. This test permits
the variables to be estimated latent quantities as opposed to observables from
the data and to also share dependence (in our case, the dependence is due to
their construction using the same set of parameter estimates). Employing the
sub-sampling procedure from Linton et al. (2005), we obtain a $p$-value for
the test statistic of 0.7789 and, thus, comfortably fail to reject the null
hypothesis that non-exporters’ productivity is stochastically dominated by
that of exporters.

Altogether, we can conclude that exporters enjoy a statistically significant
productivity premium over non-exporters along the entire distribution of firm
productivity.

### 5.2. Learning by exporting and from exporters

The exporter productivity premium can reflect both the learning and self-
selection effects. To zoom in on the within-firm evidence of productivity-
enhancing learning effects of exporting and to test if the firm’s past export
experience as well as the experience of its peers impact its future
productivity, we use $LBE_{it}={\partial
E\big{[}\omega_{it}|\omega_{it-1},X_{it-1},\overline{X}_{it-1}\big{]}}\big{/}{\partial
X_{it-1}}$ and $LFE_{it}={\partial
E\big{[}\omega_{it}|\omega_{it-1},X_{it-1},\overline{X}_{it-1}\big{]}}\big{/}{\partial\overline{X}_{it-1}}$
which are conditioned on the firm’s past productivity level $\omega_{it-1}$
thereby, as discussed in Appendix A, allowing us to account for self-selection
into exporting based on an a priori higher productivity. In effect, both LBE
and LFE measure a differential in mean future productivity identified from the
difference in current productivity between firms/peers with different export
experiences, holding their input use fixed. Table 1 reports a summary of point
estimates of these LBE and LFE effects for all firms as well as for exporters
and non-exporters only. We also test for the statistical significance of these
effects at each observation. The shares of observations for which each of the
two productivity effects of exporting is significant at the 5% level are
provided in the far right column of the table.

The LBE effect is estimated to average at 0.36 for the entire sample,
indicating that a percentage point increase in the firm’s own export intensity
raises its future productivity by 0.36%. The point estimates are between 0.32
and 0.45 within the inter-quartile range. Overall, the LBE productivity effect
is statistically significant for 93% of firms in our data set. We also
document notable differences in the magnitude and prevalence of LBE across
exporters and non-exporters. For exporters ($X_{it-1}>0$), the mean LBE effect
is significant at 0.158, albeit the observation-specific point estimates are
statistically non-zero only for 68% of the exporting firms. But the evidence
is more ubiquitous among non-exporters. While the latter category of firms
does not actually export, our methodology still produces an estimate of
$LBE_{it}={\partial E\big{[}\omega_{it}|\cdot\big{]}}\big{/}{\partial
X_{it-1}}$ for them which, in this case, is evaluated at $X_{it-1}=0$.
Essentially, these estimates of the LBE effect are “counterfactual” and
provide a measurement of how much non-exporters’ productivity would change if
they were to begin exporting (i.e., marginally increase their export intensity
from zero to a positive value).888We can estimate the LBE effect for non-
exporters directly by the virtue of algebra of the regression and, as pointed
out by a referee, this estimand is similar in character to the concept of a
treatment effect on the non-treated. When estimating the second-stage
regression (3.8) we approximate the nonlinear productivity process
$g(\omega_{it-1}X_{it-1},\overline{X}_{it-1})$ via quadratic polynomial given
in (3.10) and thus obtain heterogeneous estimates of the gradient function
measuring LBE, $\widehat{\partial g(\cdot)/\partial
X_{it-1}}=\widehat{\lambda}_{x}+0.5\widehat{\lambda}_{xx}X_{it-1}+\widehat{\lambda}_{x\omega}\widehat{\omega}_{it-1}+\widehat{\lambda}_{x\bar{x}}\overline{X}_{it-1}$
where $\widehat{\lambda}$’s are the estimated coefficients. It can be
evaluated at each observation, including for the firms that export
($X_{it-1}\neq 0$) and do not export ($X_{it-1}=0$) in period $t-1$. The
distinction between the LBE effects for exporters and non-exporters has been
typically under-emphasized in prior studies many of which relate firm
productivity and $X$ linearly, obtaining a singular estimate of the LBE
effect. Owing to the homogeneity of treatment effects implied by linearity,
such an LBE estimate applies to both exporters and non-exporters. We however
allow for heterogeneity in the effects.  The average LBE estimate for non-
exporters is 0.418 and significant. In fact, the point estimates of the LBE
effect are statistically significant for virtually all non-exporters (99%).
Magnitude-wise, the average effect size for non-exporters is about 2.6 times
larger than that for active exporters. These findings suggest that the bulk of
a productivity boost attributable to (internal) learning from exporting is
realized upon the domestic firm’s entry into in export markets as it gains
access to new technology and business practices, and the effectiveness of LBE
is diminishing as the firm further specializes in exporting.999The diminishing
productivity return to the own export experience is also corroborated by the
estimates of the LBE function which we discuss later. This initial effect
likely captures the productivity-boosting effects of various export-related
investments that new exporters normally undertake. Namely, the decisions to
start exporting tend to go together with other firm-level actions that can
enhance productivity such as the adoption of new/upgraded technologies,
quality upgrading or R&D spending (e.g., see Verhoogen, 2008; Aw et al., 2011;
Bustos, 2011). Due to the lack of richer data, we are unable to unbundle these
individual effects, and our LBE estimates ought to be interpreted as measuring
their composite export-focused effect attributable to multiple channels.

Table 1: The LBE and LFE Productivity Effect Estimates

Point Estimates Stat. Signif. Mean 1st Qu. Median 3rd Qu. (% Obs.) —Learning
by Exporting— All Firms 0.363 0.323 0.390 0.451 92.7 (0.189, 0.555) (0.139,
0.531) (0.207, 0.603) (0.250, 0.675) Exporters 0.158 –0.000 0.257 0.353 68.3
(0.068, 0.254) (–0.072, 0.067) (0.117, 0.419) (0.184, 0.546) Non-exporters
0.418 0.356 0.408 0.465 99.2 (0.234, 0.643) (0.170, 0.572) (0.220, 0.628)
(0.259, 0.695) —Learning from Exporters— All Firms 0.324 0.141 0.296 0.457
68.5 (0.116, 0.545) (–0.059, 0.363) (0.069, 0.520) (0.166, 0.729) Exporters
0.508 0.182 0.399 0.778 73.9 (0.302, 0.76) (–0.013, 0.432) (0.189, 0.618)
(0.508, 1.148) Non-exporters 0.275 0.132 0.28 0.42 67.1 (0.027, 0.489)
(–0.088, 0.34) (0.027, 0.497) (0.124, 0.701) Notes: Reported is a summary of
point estimates of the LBE and LFE effects tabulated by the firm’s exporter
status, with 95% bootstrap percentile confidence intervals in parentheses. The
far right column reports the share of (sub)sample for which the observation-
specific estimates are statistically significant at the 5% level.

On average, the size of the LFE effect is on par with that of LBE: a pooled
mean estimate of the LFE effect is statistically significant at 0.32 (vs. 0.36
for LBE) whereby a percentage point increase in the average peer export
intensity within a domestic firm’s local industry boosts its future
productivity by 0.32%. It may perhaps surprise the reader that the indirect
cross-firm learning channel is estimated to boost firm productivity by almost
as much as the LBE occurring internally. Of note, however, is that these two
effects are not commensurable with one another because, by construction, they
capture impacts of qualitatively different “treatments.” While $LBE_{it}$
measures the productivity effect of a unit change in one firm’s (own) export
intensity $X_{it-1}$, $LFE_{it}$ measures the effect of a unit change in the
average peer export intensity
$n_{it-1}^{-1}\sum_{j\in\mathcal{L}_{it-1}}X_{jt-1}$ of the entire local
industry, a change that is equivalent to an increase in the export orientation
of all peers by a unit. As such, the scale of two treatments is vastly
different. To wit, the LFE effect quantifies a total indirect/spillover effect
on firm $i$’s productivity at time $t$ from all its peers of whom, at time
$t-1$, it had $n_{it-1}$. The average spillover effect on future productivity
of firm $i$ from a percentage point increase in a single peer’s $X_{jt-1}$ is
a significantly more modest 0.013%. In what follows, we continue focusing on
the (total) LFE effect that aggregates spillovers from all peers—over the
average spillover across only a pair of firms—because its effect size is a
more policy-relevant quantity.

We find that the LFE effect is significantly non-zero only for 69% of
recipient firms, thus suggesting that the indirect cross-firm productivity-
boosting effect of exporting is less prevalent in Chilean manufacturing than
the direct effect taking place within a firm (recall, 93%). Of interest, we
also document that the LFE effect is stronger for firms who also export
themselves. Namely, the average estimate of the LFE effect for exporter firms
is estimated at 0.508, whereas the corresponding estimate for non-exporters is
half that at 0.275. In addition, non-zero learning from exporters is also more
prevalent for exporters (74%) than it is for non-exporters (67%). The
empirical evidence therefore suggests that exporters benefit from the exposure
to peer exporters in their local industry more than non-exporters. Plausibly,
there may be complementarities between internal/direct and external/indirect
learning from export experiences. This is reasonable because challenges
associated with engagements in the foreign market can make exporters more
motivated and pressured than their fully domestically-oriented non-exporting
peers to improve further and more intensely.

It is important to re-emphasize that, given an inherently “black box” nature
of production functions and the residual-based definition of TFP, magnitudes
of LFE point estimates ought to be taken with caution and, probably, not at
their face value. As noted earlier, interpretation of the LFE measure based on
spatio-industrial proximity of exporting peers as capturing, specifically, the
spillover effects of exporting requires that we assume away all other
plausible channels of cross-firm interactions that may also spur productivity
spillovers, including spatial agglomeration, R&D and FDI externalities,
spillovers along supply chains, etc. These other factors are expected to
correlate with exporting behavior, and the LFE effect that we measure may
partly reflect spillovers that they induce across firms as well. Unless all
plausible internal and external productivity modifiers are controlled for when
modeling firm productivity evolution, measured LFE effects will expectedly be
crude. This is an obvious limitation of our analysis but also happens to be
the case for any other study that has ever sought to measure productivity
spillovers. (The same arguments can also be made about LBE.) Consequently, we
mainly pursue a simpler, more feasible objective of testing for non-zero
export-driven spillover effects on productivity. That is, our focus is on the
“extensive margin,” i.e., whether productivity spillovers from peer exporters
occur in general, as opposed to how big these effects are. To this end, we
find that the productivity externalities from exporting peers are significant
in two-thirds of plants.

Cumulative Long-Run Effects.—As noted earlier, the reported point estimates of
the LBE and LFE effects are short-run and do not account for dynamic effects
over time. Obviously, owing to the persistence of productivity, the cumulative
implications of exporting for domestic firms’ productivity in the long run
will be more sizable. Because the estimated productivity process is nonlinear,
the computation of such cumulative effects is, however, not clear-cut. To
roughly size up the economic significance of LBE and LFE for productivity in
the domestic industry in the long-run equilibrium, we perform the following
back-of-envelope calculation. Approximating $h(\cdot)$ using a linear
expansion around the point such that its evaluated gradients are equal to the
mean estimates from our semiparametric model, i.e.,

$\omega_{it}\approx\text{const}+0.482\omega_{it-1}+0.363X_{it-1}+0.324\overline{X}_{it-1}+\text{error}_{it},$

under the temporal stationarity of log-productivity we have that the long-run
LBE and LFE effects are respectively equal to $0.363/(1-0.482)=0.701$ and
$0.324/(1-0.482)=0.627$. Thus, in the long-run equilibrium a permanent 10
percentage point increase in mean export intensity of all firms in a local
industry is roughly estimated to produce a sizable 13.3% boost to the average
productivity of domestic firms.101010The total long-run effect on $E[\omega]$
of a 10 percentage point increase in $E[X]$ is
$\text{d}E[\omega]/\text{d}E[X]\times 10=(\partial E[\omega]/\partial
E[X]+\partial E[\omega]/\partial E[\overline{X}]\times\partial
E[\overline{X}]/\partial E[X])\times 10=(0.701+0.627)\times 10=13.3$, where we
have made use of $\partial E[\overline{X}]/\partial E[X]=1$. This estimate
incorporates both direct LBE and indirect LFE channels as well as the temporal
multiplier effect due to persistence over time. Whether occurring through
technology/efficiency-driven cost savings or an increase in demand (or both),
this improvement in the long-run average TFPR, $E[\omega]$, associated with an
industry’s 10% pivot towards more exporting is roughly equivalent to
$13.3\%\times E[Y]=3,680$ thousands pesos more in revenues for an average
domestic firm, holding the input usage fixed.

Heterogeneity and Nonlinearity.—Although we find that average effect sizes of
LBE and LFE are commensurate, individual firms are highly heterogeneous across
many dimensions including their productivity, the degree of their export
orientation as well as the intensity of their exposure to other exporters in
the industry. It is therefore of interest to investigate if these
characteristics influence the effect size of their internal and external
learning from exporting.

Recall that we obtain the estimate of the productivity effects of exporting
via
$\widehat{LBE}_{it}={\partial\widehat{g}\left(\cdot\right)}\big{/}{\partial
X_{it-1}}$ and
$\widehat{LFE}_{it}={\partial\widehat{g}\left(\cdot\right)}\big{/}{\partial\overline{X}_{it-1}}$,
where we estimate $g\big{(}\omega_{it-1},X_{it-1},\overline{X}_{it-1}\big{)}$
using the second-order polynomial sieve approximation given in (3.10). Thus,
by derivation, both $\widehat{LBE}_{it}$ and $\widehat{LFE}_{it}$ are the
estimated linear functions of the “determinants” of firm productivity
$\big{(}\omega_{it-1},X_{it-1},\overline{X}_{it-1}\big{)}^{\prime}$.111111Concretely,
under the quadratic polynomial approximation of $g(\cdot)$ in (3.10), the
estimated gradient functions are
$\widehat{LBE}_{it}\equiv{\partial\widehat{g}(\cdot)/\partial
X_{it-1}}=\widehat{\lambda}_{x}+0.5\widehat{\lambda}_{xx}X_{it-1}+\widehat{\lambda}_{x\omega}\widehat{\omega}_{it-1}+\widehat{\lambda}_{x\bar{x}}\overline{X}_{it-1}$
and
$\widehat{LFE}_{it}\equiv{\partial\widehat{g}(\cdot)/\partial\overline{X}_{it-1}}=\widehat{\lambda}_{\bar{x}}+0.5\widehat{\lambda}_{\bar{x}\bar{x}}\overline{X}_{it-1}+\widehat{\lambda}_{\bar{x}\omega}\widehat{\omega}_{it-1}+\widehat{\lambda}_{x\bar{x}}X_{it-1}$
where $\widehat{\lambda}$’s are the estimated polynomial coefficients. Table 2
reports the coefficients on regressors in these functions. Table 2 reports the
estimates of parameters on these three variables for LBE and LFE.

Table 2: Estimates of the LBE and LFE Functions | LBE | LFE
---|---|---
$\omega_{it-1}\qquad$ | –0.113 | –0.346
| (–0.212, –0.039) | (–0.575, –0.134)
$X_{it-1}$ | –0.970 | 1.210
| (–1.454, –0.435) | (0.583, 2.008)
$\overline{X}_{it-1}$ | 1.210 | 0.220
| (0.583, 2.008) | (–0.959, 1.377)
Notes: Reported are the parameter estimates for the LBE and LFE functions
derived from the polynomial approximation of the conditional mean of
$\omega_{it}$, along with the 95% bootstrap percentile confidence intervals in
parentheses.

The coefficient estimates on $\omega_{it-1}$ for both LBE and LFE are
significantly negative, indicating that the magnitude of these effects
declines as firms get more productive. Thus, less productive plants benefit
more from export-driven learning, be it an internal or external channel. This
finding suggests that plants that are already highly productive have less to
gain from export-driven learning as well as less to learn from their exporting
peers, presumably because they’re much closer to the technology/knowledge
frontier compared to less productive domestic firms. We also find that a
firm’s own export intensity $X_{it-1}$ has a significantly negative effect on
learning by exporting but a significantly positive effect on learning from
exporters, indicating that less export-oriented plants improve more via
learning by exporting and less via learning from exporters (and vice versa).
Basically, this is indicative of the diminishing productivity return to the
own export experience: an increase in the degree of a firm’s export
orientation enhances its productivity at a decreasing rate. But we do not find
such a pattern for the LFE effect. On the contrary, the more export-oriented
the plant is, the higher the cross-firm export-driven productivity spillovers
are, which buttresses our earlier discussion of potential complementarities
between exporting and external learning from Table 1. Lastly, the results in
Table 2 suggest that the average of peer export orientation in local industry
$\overline{X}_{it-1}$ has a significantly positive influence on learning by
exporting, indicating that a greater exposure to exporters helps plants absorb
productivity improvements from their own export behavior. At the same time, we
find no significant effect of $\overline{X}_{it-1}$ on LFE effect size,
indicating that the average export participation in the industry improves
plant productivity at a constant rate.

Robustness Analysis.—We also assess the robustness of our main empirical
finding of significant LBE and LFE effects to the choice of how we measure (i)
export experience of firms and (ii) their exposure to export experiences of
their peers. Specifically, we re-estimate our model using a binary export-
status variable $1(X_{it}>0)$ as well as redefine $\overline{X}_{it}$ by no
longer restricting the pool of a firm’s peers to the same geographic region.
The latter is motivated by the plausible arguments that, unlike pooling of
workers, the knowledge diffusion mechanism need not be local and may spill
widely across space. These results are summarized in Table 3. Two observations
are in order here. First, using a discrete measure of export experience yields
smaller effect sizes for both LBE and LFE. Second, expanding the group of peer
firms to include all regions, thereby covering a wider spillover pool, results
in significantly larger estimates of the external LFE effect while expectedly
having no notable impact on the internal LBE effect.121212This is so despite
that the indirect LFE effect per a firm’s each peer actually gets diluted; see
tables in Appendix E. The latter indirectly lends support to the hypothesis
that export spillover effects may have a broad geographical scope. By
restricting the extent of spillovers to the same region in our main
specification, we also restrict the reach of cross-firm export externalities.

Table 3: Robustness to Alternative Measures of Exporting and of the Exposure
to Exporters

(I) (II) (III) (IV) LBE 0.363 0.379 0.147 0.151 (0.189, 0.555) (0.207, 0.575)
(0.117, 0.176) (0.121, 0.179) [92.7%] [89.4%] [99.6%] [99.6%] LFE 0.324 0.912
0.231 0.463 (0.116, 0.546) (0.635, 1.169) (0.117, 0.331) (0.321, 0.645)
[68.6%] [80.0%] [85.1%] [66.2%] Exporting Measure intensity ✓ ✓ $\times$
$\times$ status $\times$ $\times$ ✓ ✓ Pool of Peer Exporters same industry ✓ ✓
✓ ✓ same region ✓ $\times$ ✓ $\times$ Notes: Summarized are the results for
the LBE and LFE effects obtained using alternative variable specifications. We
report the mean point estimates along with 95% bootstrap percentile confidence
intervals (in parentheses) as well as the shares of sample for which the
observation-specific estimates are statistically significant at the 5% level
[in brackets]. Model (I) is our main specification, the results for which we
reproduce here for convenience.

As discussed earlier, to identify the spillover effects of exporting beyond
exporters, we rule out other potential channels of cross-firm interactions
within regions and industries that stem from spatio-industrial proximity of
firms to one another. However, we can somewhat relax this assumption by
allowing for peer network unobservables that, besides just exporting, may give
rise to spillovers in productivity as well (e.g., regional agglomerations).
The requirement is that these unobservables be time-invariant, in which case
we can control for potential region- and industry-level confounders using peer
group fixed effects. We consider group effects across both spatial and
industrial dimensions and re-estimate our baseline specification including
region and industry fixed effects. The corresponding results are summarized in
Table 4. While predictably the LBE estimates of within-firm effects of
exporting hardly change, the mean effect size of external spillovers (LFE)
increases notably when we rely solely on within-region or within-region-and-
industry variation in the estimation. The increase in the magnitude of
spillovers is consistent with significant between-region/industry
heterogeneity in (peer) firm exporting which, when omitting group effects,
“dilutes” the strength of cross-firm effects in the baseline model due to the
variation across peer groups.

Overall, the cross-specification variation in estimates is unsurprising and,
in fact, expected because models treat cross-firm interactions a bit
differently and/or utilize different variation in data to identify the
effects. The qualitative implication however remains unchanged across all
alternative specifications: both LBE and LFE are statistically significant and
widely prevalent. Also, consistently across these alternatives, we obtain LFE
effects that, on average, exceed LBE, implying that the total cross-firm
effect of exporting—after aggregating across all peers—may even out-size the
direct learning effect occurring within one firm.131313Analogues of Tables 3–4
that report average indirect LFE effects (per peer) are available in Appendix
E. As with our main specifications, these average indirect effects are much
smaller than the direct LBE effect. If anything, this further underscores the
importance of measuring both effects when seeking to quantify the total social
effects of exporting (and export-promoting policies, by extension) on domestic
industries.

Table 4: Robustness to Controlling for Peer Group Effects

(I) (II) (III) LBE 0.363 0.381 0.367 (0.189, 0.555) (0.202,0.577)
(0.189,0.559) [92.7%] [93.2%] [92.8%] LFE 0.324 0.711 0.496 (0.116, 0.546)
(0.512,0.958) (0.236,0.762) [68.6%] [94.6%] [83.6%] Region FE $\times$ ✓ ✓
Industry FE $\times$ $\times$ ✓ Notes: Summarized are the results for the LBE
and LFE effects obtained using alternative fixed-effect specifications. We
report the mean point estimates along with 95% bootstrap percentile confidence
intervals (in parentheses) as well as the shares of sample for which the
observation-specific estimates are statistically significant at the 5% level
[in brackets]. Model (I) is our main specification, the results for which we
reproduce here for convenience.

Alternative Two-Step Procedure.—Empirical analyses of export-driven spillovers
in productivity oftentimes employ a two-step methodology, whereby one first
estimates firm productivity via off-the-shelf proxy methods that assume—and
thus impose in the estimation—an exogenous Markov process for productivity and
then tests for spillovers in a second step by (linearly) regressing firm
productivity estimated in the first step on the export spillover exposure
variable(s). For instance, this is the approach taken by Alvarez & López
(2008). As discussed earlier, not only is it internally inconsistent
(contradictory) in that a non-zero internal LBE and/or external LFE effect
found in the second step must violate the key identifying assumption of the
first-step estimator, but it is also the case that it is typically implemented
using the export spillover exposure measures that average across all firms,
including the recipient of spillovers. The latter produces regression
coefficients that cannot separate the LBE from the LFE effect, muddying their
interpretation.141414For further discussion as well as an illustration for the
case of a Alvarez & López (2008)-type second-step regression in (5.1), see
Appendix F.

Notwithstanding these issues, we implement a two-step approach to estimate the
LFE effect and compare it with ours. More concretely, we re-estimate our model
to obtain firm productivity but, this time, assuming an exogenous Markov
process for productivity, whereby $\omega_{it}=h(\omega_{it-1})+\zeta_{it}$ is
imposed in place of (2.6), and then run the following Alvarez & López
(2008)-style second-step linear regression:

$E\left[\widetilde{\omega}_{it}|\cdot\right]=\beta_{0}+\beta_{x}X_{it-1}+\beta_{\bar{x}}\overline{X}_{it-1},$
(5.1)

where $\widetilde{\omega}_{it}$ is the first-step estimate of exogenously
evolving $\omega_{it}$, and the spillover pool $\overline{X}_{it-1}$ is
measured using the “grand” average $\sum_{j}p_{iit-1}X_{jt-1}$ that does not
exclude $X_{it-1}$. To maximize comparability, we specify the above second-
step regression in lags to match the dynamics of (2.6).

Figure 3: Alternative Two-Step Estimates of the LFE Effect
[Notes: “$\bullet$” plots two-step point estimates of the LFE effect from eq.
(5.1). “$\boxtimes$” plots the mean point estimates of the LFE effect obtained
via our methodology and reported in Table 4. Specifications are as follows:
(I) – no fixed effects, (II) – with region fixed effects, (III) – with both
region and industry fixed effects. Whiskers show 95% confidence intervals
constructed using robust standard errors clustered at the plant level.]

Figure 3 summarizes the estimates of $\beta_{\bar{x}}$ from (5.1) estimated
with and without peer group fixed effects. As we show in Appendix F, the
$\beta_{\bar{x}}$ parameter measures the LFE effect as defined by us in the
rest of the paper using the peer average measure of $\overline{X}_{it-1}$,
viz., $LFE\equiv{\partial
E\left[\omega_{it}|\cdot\right]}\big{/}{\partial\sum_{j{\neq
i}}p_{ijt-1}X_{jt-1}}$.151515The coefficient on the firm’s own $X_{it-1}$ is
significant and stable across all three specifications (ranging in
0.160–0.168) but we do not focus on it here due to the difficulty interpreting
the latter since it does not separately identify the LBE from the LFE. The
three fixed-effects specifications match those reported in Table 4.

Estimates from a two-step procedure and our model are non-negligibly
different. The specification with both region and industry fixed effects,
which drastically reduces the amount of identifying variation in
$\overline{X}_{it-1}$ defined at the region-industry level, leads to a noisy
estimate of $\beta_{\bar{x}}$ implying an insignificant LFE. Under the other
specifications, $\beta_{\bar{x}}$ estimates are all statistically significant
but much larger (even exceeding the unit elasticity) than the corresponding
mean LFE effects from our model. Given the lack of internal consistency and
the associated misspecification of a productivity process in the first and
second steps of the two-step procedure, it is difficult to precisely
rationalize why the LFE point estimates differ from ours. Having said that,
compared to the two-step procedure, our methodology exhibits a significantly
smaller variation in LFE estimates across specifications, lending assurance to
their robustness. On the other hand, the dramatic volatility of an Alvarez &
López (2008)-style approach is consistent with the simulation evidence
reported by Malikov & Zhao (2021) who find that two-step procedures suffer
from non-vanishing biases in spillover effect estimates and these biases can
be both positive and negative.

## 6\. Conclusion

This paper extends De Loecker (2013) to develop a unified empirical framework
for productivity measurement that explicitly accommodates both the direct LBE
channel taking place within a firm as well as the indirect LFE channel working
between firms, which enables us to robustly measure and test these two effects
in a consistent structural framework. We do so by formalizing the evolution of
firm productivity as an export-controlled process, with future productivity
potentially affected not only by the firm’s own export behavior but also by
that of its spatially proximate peers in the industry. This allows a
simultaneous, “internally consistent” identification of firm productivity and
the corresponding effects of exporting. Our identification strategy utilizes a
structural link between the parametric production function and the firm’s
first-order condition for static inputs which helps us circumvent recent non-
identification critiques of conventional proxy-based productivity estimators.
In addition, owing to a nonparametric treatment of the firm productivity
process, our model enables us to accommodate heterogeneity and nonlinearity in
productivity effects of exporting across firms.

We apply our semiparametric methodology to a panel of manufacturing plants in
Chile in 1995–2007 and find significant evidence of both LBE and LFE. Overall,
the LBE effect on productivity is statistically significant for 93% of firms
in our main specification. On average, the size of the LFE effect is
commensurate to that of LBE but, at the observation level, the LFE effect is
significantly non-zero for 69% of plants only, thereby suggesting that the
indirect productivity-boosting effect of exporting is less prevalent in
Chilean manufacturing than the direct learning effect taking place within the
firm. We also document that the LFE effect is stronger for firms who also
export. This empirical evidence therefore suggests that exporters benefit from
the exposure to peer exporters in their local industry more than non-
exporters, plausibly because there may be complementarities between
internal/direct and external/indirect learning from export experiences. Using
simple back-of-envelope calculation, we find that long-run LBE and LFE effects
on mean firm productivity are respectively 0.70% and 0.63% per percentage
point of export intensity. Thus, in the long-run equilibrium a permanent 10
percentage point increase in mean export intensity of all firms in the local
industry is roughly estimated to produce a sizable 13.3% boost to the average
firm productivity through both direct LBE and indirect LFE channels.

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## APPENDIX

##  Appendix A Controlling for Self-Selection into Exporting

The inclusion of the lagged firm productivity in the productivity process
(2.6) also indirectly allows us to, at least partly, control for self-
selection of firms into exporting based on their productivity levels.
Consistent with economic theory (Melitz, 2003) and vast empirical evidence
(e.g., Clerides et al., 1998; Delgado et al., 2002), engagement in export
markets is more likely among firms with higher productivity levels. However,
by including a firm’s own lagged productivity in the autoregressive
$\omega_{it}$ process, we are able to account (to an extent) for this
potential endogenous self-selection when measuring learning effects. That is,
both the LBE and LFE effects on future firm productivity in our model are
measured after partialling out the contribution of its own productivity,
thereby “fixing” the past productivity when comparing firms of different
export intensities.

Under our model structure, the $X_{it}$ decisions—while optimized
dynamically—are without delay which allows a contemporaneous correlation
between firm productivity $\omega_{it}$ and its export orientation $X_{it}$.
This implicitly accommodates the firm’s self-selection into exporting based on
its productivity. That is, from (2.3) and the policy function for
$\mathcal{X}_{it}$ implied by the Bellman equation in (2.4), we have that
$X_{it}(\omega_{it},K_{it},L_{it},X_{it-1})$ whereby the exporting behavior
$X_{it}$ contemporaneously and endogenously depends on $\omega_{it}$.
Substituting for firm productivity using its Markov process, the log
production function that we estimate becomes
$y_{it}=\alpha_{0}+\alpha_{K}k_{it}+\alpha_{L}l_{it}+\beta_{M}m_{it}+g(\omega_{it-1},X_{it-1},\overline{X}_{it-1})+\zeta_{it}+\eta_{it}$.
Recognizing that, per our assumptions,
$K_{it}(\omega_{it-1},K_{it-1},L_{it-1},X_{it-2})$ and
$L_{it}(\omega_{it-1},K_{it-1},L_{it-1},X_{it-2})$, it becomes apparent that
our measurement of the effect of $X_{it-1}$ on expected productivity
$g(\cdot)$ when estimating the production function is done while controlling
for its potential within-firm confounders
$(\omega_{it-1},K_{it}(\omega_{it-1},K_{it-1},L_{it-1},X_{it-2}),L_{it}(\omega_{it-1},K_{it-1},L_{it-1},X_{it-2}))^{\prime}$
that may impact the determination of $X_{it-1}$. De Loecker (2013, fn. 22)
makes an analogous argument but in the context of delayed exporting decisions.

##  Appendix B Using Deflated Revenues to Measure Output

Beyond just assuming a perfectly competitive output market as commonly done in
applied work on productivity, the structural identification outlined in
Sections 2–3 specifically assumes that (exogenous) prices are homogeneous
across all firms, namely, that $P^{Y}_{it}=P^{Y}_{t}$ for all $i$. The model
also implicitly assumes the availability of physical-quantity information
about firm outputs in the data. Such information, however, is oftentimes
lacking in production datasets, including ours: instead of the output quantity
and the corresponding price we only observe the firm’s revenue, which is their
product. The latter is a trivial point if output prices are indeed common to
all firms since, in this case, the output quantity $Y_{it}$ can be obtained by
simply deflating the revenue data using the (observable) output price index
$P^{Y}_{t}$. However, the assumption that competitive firms are price takers,
in theory, does not prevent them from being offered heterogeneous prices. If
output prices $P^{Y}_{it}$ do vary across individual firms and we continue
measuring the firm output using deflated revenues in the empirical analysis,
the obtained firm productivity will be revenue-based, capturing not only the
cost/production-side but also the price/demand-side heterogeneity across
firms. So long as we maintain the perfect competition assumption in that the
firms are price-takers who do not receive “quantity discounts,” which renders
output prices exogenous to the firm’s production decisions, the deviation from
the assumed price homogeneity does not invalidate our structural
identification arguments except altering the interpretation of a residual TFP
as reflecting both idiosyncratic supply and demand heterogeneity at the firm
level. In what follows, we show this more formally.

The quantity-based measure of total factor productivity (TFPQ)—that is,
$\omega_{it}$—appearing in the underlying “quantity production function” in
(3.6),
$y_{it}=\alpha_{0}+\alpha_{K}k_{it}+\alpha_{L}l_{it}+\alpha_{M}m_{it}+\omega_{it}+\eta_{it}$,
captures purely cost/production/supply-side heterogeneity across firms. When
using deflated revenues to measure the firm output, a feasible analogue of
this production function estimatable from the data is given by the following
“revenue production function:”

$r_{it}=\alpha_{0}+\alpha_{K}k_{it}+\alpha_{L}l_{it}+\alpha_{M}m_{it}+\underbrace{\big{(}\overbrace{\omega_{it}}^{\text{TFPQ}}+\pi_{it}\big{)}}_{\text{TFPR}}+\eta_{it},$
(B.1)

where $r_{it}=\ln\left(\frac{P_{it}^{Y}Y_{it}}{P_{t}^{Y}}\right)$ is the
deflated revenue, and
$\pi_{it}\equiv\ln\left(\frac{P_{it}^{Y}}{P_{t}^{Y}}\right)$ captures
heterogeneity in output prices at the firm level as measured by their log-
deviation from the common price index. It is easy to see that if
$P_{it}^{Y}=P_{t}^{Y}$ for all $i$, as in the main case discussed in our
paper, $\pi_{it}=0$ and $\text{TFPR}=\text{TFPQ}$ that simplifies (B.1) to
(3.6). Otherwise, the revenue-based firm productivity TFPR includes both the
cost heterogeneity $\omega_{it}$ and the demand heterogeneity $\pi_{it}$. As
such, in the context of productivity effects of exporting, the measured TFPR
will capture export-driven changes in both the cost and demand factors. Hence,
a positive LBE and/or LFE effect can be interpreted as working through either
a decline in production cost via the increasing technical productivity or the
price-raising increase in demand, or both.

Next, we show that our empirical strategy is generalizable to cases with non-
endogenous price heterogeneity across firms, i.e., when $\pi_{it}\neq 0$.
Without any actual changes in its implementation, our methodology structurally
identifies TFPR and LBE/LFE effects thereon if, instead of TFPQ
($\omega_{it}$), we assume that it is the composite TFPR
($\omega_{it}+\pi_{it}$) that is first-order Markov. This revision in
assumption is empirically immaterial since the latter continues to formalize
the evolution of an “observationally equivalent” object, namely, unobservable
scalar firm heterogeneity. As discussed earlier, the change is in the
interpretation of this heterogeneity only. Thus, we replace (2.6) with the
following controlled Markov process for TFPR:

$\psi_{it}=h\left(\psi_{it-1},X_{it-1},\overline{X}_{it-1}\right)+\zeta_{it},$
(B.2)

where, for convenience, $\psi_{it}\equiv\omega_{it}+\pi_{it}$. Clearly, (B.2)
nests (2.6) when prices are homogeneous.

Recall that our identification strategy starts with the firm’s static
optimality condition for materials. The analogue of (3.2) for price-taker
firms facing heterogeneous output prices $P_{it}^{Y}$ is

$\alpha_{M}P_{it}^{Y}A_{0}K_{it}^{\alpha_{K}}L_{it}^{\alpha_{L}}M_{it}^{\alpha_{M}-1}\exp\left\\{\omega_{it}\right\\}\theta=P_{t}^{M}$
(B.3)

Taking the log-ratio of this first-order condition and the underlying
“quantity production function” of the firm in (2.1), we obtain the same
material share equation as in (3.3):

$\ln\underbrace{\left(\frac{P_{t}^{M}M_{it}}{P_{it}^{Y}Y_{it}}\right)}_{S_{it}^{M}}=\ln\left(\alpha_{M}\theta\right)-\eta_{it},$
(B.4)

with the only difference being that the revenue entering the denominator in
$S_{it}^{M}$ now has a firm-specific output price. However, this has no
bearing on the first stage of our methodology, since $S_{it}^{M}$ is directly
measurable from the data regardless of whether the output price is homogeneous
or not. That is, the firm’s revenue is directly observable under both
scenarios, be it equal to $P_{it}^{Y}Y_{it}$ or $P_{t}^{Y}Y_{it}$. As such,
material elasticity continues to be identified via
$\alpha_{M}=\exp\left\\{E\left[\ln\left(S_{it}^{M}\right)\right]\right\\}/E\left[\exp\left\\{E\left[\ln\left(S_{it}^{M}\right)\right]-\ln\left(S_{it}^{M}\right)\right\\}\right]$
as it was before.

To identify the rest of the production function in the second stage, we now
need to work with the “revenue production function.” Using the Markov
assumption about $\psi_{it}=\omega_{it}+\pi_{it}$ and bringing the already
identified material elasticity $\alpha_{M}$ to the left-hand side of the
equation, from (B.1) we obtain

$r_{it}^{*}=\alpha_{K}k_{it}+\alpha_{L}l_{it}+\overbrace{g\left(\psi_{it-1},X_{it-1},\overline{X}_{it-1}\right)+\zeta_{it}}^{\psi_{it}}+\eta_{it},$
(B.5)

where $r_{it}^{*}=r_{it}-\alpha_{M}m_{it}$ is fully identified and can be
treated as an observable, and $g\left(\cdot\right)\equiv h(\cdot)+\alpha_{0}$
is some function.

To derive the proxy function for the TFPR, we invert the first-order condition
in (B.3) after rewriting it slightly (by dividing it through by the common
price index $P_{t}^{Y}$) as

$\alpha_{M}\times\underbrace{\frac{P_{it}^{Y}}{P^{Y}_{t}}}_{\exp\\{\pi_{it}\\}}\times
A_{0}K_{it}^{\alpha_{K}}L_{it}^{\alpha_{L}}M_{it}^{\alpha_{M}-1}\exp\left\\{\omega_{it}\right\\}\theta=\frac{P_{t}^{M}}{P_{t}^{Y}}$
(B.6)

and logging both sides to obtain

$\omega_{it}+\pi_{it}=\underbrace{\ln\left(\frac{P_{t}^{M}}{P_{t}^{Y}}\right)-\ln\left(\alpha_{M}\theta\right)-\left(\alpha_{M}-1\right)m_{it}}_{m_{it}^{*}}-\alpha_{K}k_{it}-\alpha_{L}l_{it}.$
(B.7)

The above material-based proxy for $\psi_{it}$, which includes unobservable
cross-firm price heterogeneity $\pi_{it}$, is now expressed in terms of the
observable common price index $P_{t}^{Y}$ that we also use when deflating
revenues to obtain $r_{it}$. Note that the proxy expression, i.e., the right-
hand side of (B.7) is the same as the one we use in the main body of the
paper, but now it controls for a composite heterogeneity
$\psi_{it}=\omega_{it}+\pi_{it}$.

The feasible second-stage revenue-based production function is then given by

$r_{it}^{*}=\alpha_{K}k_{it}+\alpha_{L}l_{it}+g\left(\left[m_{it-1}^{*}-\alpha_{K}k_{it-1}-\alpha_{L}l_{it-1}\right],X_{it-1},\overline{X}_{it-1}\right)+\zeta_{it}+\eta_{it},$
(B.8)

which identifies $\alpha_{K}$ and $\alpha_{L}$ as well as the LBE/LFE effects
on the mean TFPR function $g(\cdot)$ via least squares. The TFPR itself can
then be recovered (up to a constant) using the production-function parameters
and the shock $\eta_{it}$ from the first stage:
$\psi_{it}+\alpha_{0}=r_{it}-\alpha_{K}k_{it}-\alpha_{L}l_{it}-\alpha_{M}m_{it}-\eta_{it}$.

##  Appendix C Bootstrap

For statistical inference, we employ accelerated biased-corrected percentile
bootstrap confidence intervals proposed by Efron (1987), which can correct for
finite-sample biases and control for higher moments (skewness) of the sampling
distribution. Due to the panel structure of data, we employ wild residual
block bootstrap, which can preserve the within-firm correlation in the data,
to approximate the sampling distribution of the estimator. In addition, we
bootstrap both stages jointly because the estimation in the second stage is
based on the first-stage estimator.

Having obtained the bootstrap estimates of all parameters
$\big{\\{}\left(\widehat{\alpha}_{K}^{b},\widehat{\alpha}_{L}^{b},\widehat{\alpha}_{M}^{b},\widehat{\boldsymbol{\gamma}}^{b}\right)^{\prime};\
b=1,2,...B\big{\\}}$, we then use them to construct the bootstrap replications
of our main estimands of interest: $\big{\\{}\widehat{LBE}_{it}^{b}\big{\\}}$
and $\big{\\{}\widehat{LFE}_{it}^{b}\big{\\}}$, which we then use to construct
accelerated biased-corrected percentile confidence intervals for each
observation-specific $LBE_{it}$ and $LFE_{it}$. We use $B=500$ bootstrap
replications.

For simplicity, let $\widehat{Z}$ represent an estimate of some functional of
interest $Z$ and $\\{\widehat{Z}^{b}\\}$ be the set of its bootstrap
estimates. Then, a two-sided $(1-\alpha)100\%$ confidence intervals for $Z$ is
the $\left[\alpha_{1}\times 100\right]$th and $\left[\alpha_{2}\times
100\right]$th percentiles of the empirical distribution of
$\\{\widehat{Z}^{b}\\}$, where

$\alpha_{1}=\Phi\Bigg{(}\widehat{q}_{0}+\frac{\widehat{q}_{0}+q_{\alpha/2}}{1-\widehat{c}\left(\widehat{q}_{0}+q_{\alpha/2}\right)}\Bigg{)},\quad\alpha_{2}=\Phi\Bigg{(}\widehat{q}_{0}+\frac{\widehat{q}_{0}+q_{\left(1-\alpha/2\right)}}{1-\widehat{c}\left(\widehat{q}_{0}+q_{\left(1-\alpha/2\right)}\right)}\Bigg{)},$

and $\Phi$ denotes the standard normal cdf;
$q_{\alpha/2}=\Phi^{-1}(\alpha/2)$; $\widehat{q}_{0}$ denotes a bias-
correction factor defined as
$\widehat{q}_{0}=\Phi^{-1}\left({\\#\left\\{\widehat{Z}^{b}<\widehat{Z}\right\\}}\big{/}{B}\right)$;
$\widehat{c}$ denotes an acceleration parameter which, following the
literature (Shao & Tu, 1995), we estimate via jackknife:
$\widehat{c}={\sum_{j=1}^{J}\left(\sum_{s=1}^{J}\widehat{Z}^{s}-\widehat{Z}^{j}\right)^{3}}\big{/}\big{\\{}6\big{[}\sum_{j=1}^{J}\left(\sum_{s=1}^{J}\widehat{Z}^{s}-\widehat{Z}^{j}\right)^{2}\big{]}^{3/2}\big{\\}}$,
where $\big{\\{}\widehat{Z}^{j}\big{\\}}$ are the jackknife estimates of
$Z$.161616To account for the panel structure of data and to manage
computational time, we use a delete-$20T$ jackknife, i.e., we leave 20 cross-
sections out to obtain jackknife estimates.

##  Appendix D Data Summary Statistics

Table C.1: Data Summary Statistics, 1995–2007 Variable | Mean | 5th Perc. | Median | 95th Perc.
---|---|---|---|---
Output ($Y$) | 276.46 | 8.37 | 54.75 | 1,348.35
Capital ($K$) | 102.24 | 0.48 | 12.15 | 513.97
Labor ($L$) | 49.74 | 6.00 | 22.00 | 196.00
Materials ($M$) | 125.94 | 2.85 | 24.09 | 626.56
Export Intensity ($X$) | 0.056 | 0.000 | 0.000 | 0.499
Exposure to Exporter ($\overline{X}$) | 0.060 | 0.000 | 0.036 | 0.217
Notes: $Y$, $K$, $M$ are measured in hundred thousands of real pesos. $L$ is
measured in the number of people. $X$ and $\overline{X}$ are unit-free
proportions.

##  Appendix E Additional Results

Table D.1: Analogue of Table 3

(I) (II) (III) (IV) (direct) LBE 0.363 0.379 0.147 0.151 (0.189, 0.555)
(0.207, 0.575) (0.117, 0.176) (0.121, 0.179) (indirect) LFE 0.013 0.004 0.009
0.003 from one peer (0.005, 0.022) (0.003, 0.005) (0.004, 0.013) (0.002,
0.003) Notes: This table reproduces the results reported in Table 3 except
that indirect LFE effects are computed per peer. Presented are the mean point
estimates along with 95% bootstrap confidence intervals (in parentheses).
Model (I) is our main specification.

Table D.2: Analogue of Table 4

(I) (II) (III) (direct) LBE 0.363 0.381 0.367 (0.189, 0.555) (0.202,0.577)
(0.189,0.559) (indirect) LFE 0.013 0.030 0.020 from one peer (0.005, 0.022)
(0.021, 0.040) (0.010, 0.031) Notes: This table reproduces the results
reported in Table 4 except that indirect LFE effects are computed per peer.
Presented are the mean point estimates along with 95% bootstrap confidence
intervals (in parentheses). Model (I) is our main specification.

##  Appendix F Using “Grand” Averages to Measure the Spillover Exposure

As discussed in Section 2 and specifically shown in (2.7), by measuring the
$i$th firm’s exposure to exporters using the average peer export orientation
$\sum_{j{\neq i}}p_{ijt}X_{jt}$ which excludes exports of the firm $i$
“receiving” spillovers we are able to separately identify LBE from LFE. This
would not be the case if the export spillover exposure was measured using the
“grand” average export orientation $\sum_{j}p_{ijt}X_{jt}$ that aggregates
over all units in the peer group, including the firm $i$. We illustrate this
point for a second-step regression of the Alvarez & López (2008)-style
approach to testing for export spillovers given in (5.1):

$E\left[\widetilde{\omega}_{it}|\cdot\right]=\beta_{0}+\beta_{x}X_{it-1}+\beta_{\bar{x}}\overline{X}_{it-1}.$

In what follows, we abstract away from the identification problems associated
with measuring $\widetilde{\omega}_{it}$ in the first step without accounting
for its relation with $X_{it-1}$ and $\overline{X}_{it-1}$.

Let us first note that, had the peer average been used to measure
$\overline{X}_{it}$, the regression coefficients in (5.1) would have
identified both LBE and LFE effects. Namely, from

$\displaystyle E\left[\widetilde{\omega}_{it}|\cdot\right]$
$\displaystyle=\beta_{0}+\beta_{x}X_{it-1}+\beta_{\bar{x}}\Big{(}\sum_{j{\neq
i}}p_{ijt-1}X_{jt-1}\Big{)}$ (F.1)

we have that

$\displaystyle LBE$ $\displaystyle\equiv\frac{\partial
E\left[\widetilde{\omega}_{it}|\cdot\right]}{\partial
X_{it-1}}=\beta_{x}\qquad\text{and }\qquad LFE\equiv\frac{\partial
E\left[\widetilde{\omega}_{it}|\cdot\right]}{\partial\sum_{j{\neq
i}}p_{ijt-1}X_{jt-1}}=\beta_{\bar{x}}.$

However, because $\overline{X}_{it}$ is measured using the “grand” average
export intensity, the regression coefficients in (5.1) do not separately
identify LBE from LFE, viz., from

$\displaystyle E\left[\widetilde{\omega}_{it}|\cdot\right]$
$\displaystyle=\beta_{0}+\beta_{x}X_{it-1}+\beta_{\bar{x}}\Big{(}\sum_{j}p_{ijt-1}X_{jt-1}\Big{)}$
$\displaystyle=\beta_{0}+\beta_{x}X_{it-1}+\beta_{\bar{x}}\Big{(}p_{iit-1}X_{it-1}+\sum_{j{\neq
i}}p_{ijt-1}X_{jt-1}\Big{)}$ (F.2)

it immediately follows that

$\displaystyle LBE$ $\displaystyle\equiv\frac{\partial
E\left[\widetilde{\omega}_{it}|\cdot\right]}{\partial
X_{it-1}}=\beta_{x}+\beta_{\bar{x}}p_{iit-1}\neq\beta_{x}.$

The above shows that the coefficient on the firm’s own exporting $X_{it-1}$ in
(F.2) does not identify the LBE effect, albeit that is how it is typically
interpreted in such two-step analyses. This cross-contamination of the effects
vanishes only asymptotically as the peer group size diverges, forcing
$p_{iit-1}\to 0$. This is a nonissue in our case because, when using the peer
average $\sum_{j{\neq i}}p_{ijt-1}X_{jt-1}$, we a priori set $p_{iit-1}=0$.

Interestingly, the second-step regression in (F.2) does identify the LFE
effect, but as we define it in our paper:

$\displaystyle LFE$ $\displaystyle\equiv\frac{\partial
E\left[\widetilde{\omega}_{it}|\cdot\right]}{\partial\sum_{j{\neq
i}}p_{ijt-1}X_{jt-1}}=\beta_{\bar{x}},$

i.e., as the effect of a marginal change in the peers’ average exporting. This
is different from the typical conceptualization of an external cross-firm
“spillover effect” in two-step analyses that use “grand” average
$\overline{X}$, viz., $SPILL\equiv{\partial
E\left[\widetilde{\omega}_{it}|\cdot\right]}\big{/}{\partial\sum_{j}p_{ijt-1}X_{jt-1}}$.
It is easy to see that the regression coefficient on $\overline{X}_{it-1}$ in
(F.2) does not identify the $SPILL$ effect either. Specifically, from

$\displaystyle E\left[\widetilde{\omega}_{it}|\cdot\right]$
$\displaystyle=\beta_{0}+\beta_{x}X_{it-1}+\beta_{\bar{x}}\Big{(}\sum_{j}p_{ijt-1}X_{jt-1}\Big{)}$
$\displaystyle=\beta_{0}+\frac{\beta_{x}}{p_{iit-1}}\Big{(}\sum_{j}p_{ijt-1}X_{jt-1}-\sum_{j{\neq
i}}p_{ijt-1}X_{jt-1}\Big{)}+\beta_{\bar{x}}\Big{(}\sum_{j}p_{ijt-1}X_{jt-1}\Big{)}$
(F.3)

we get

$\displaystyle SPILL\equiv\frac{\partial
E\left[\widetilde{\omega}_{it}|\cdot\right]}{\partial\sum_{j}p_{ijt-1}X_{jt-1}}=\frac{\beta_{x}}{p_{iit-1}}+\beta_{\bar{x}}\neq\beta_{\bar{x}}.$

The above illustrates that not only is the $SPILL$ effect not identified by
the regression parameter $\beta_{\bar{x}}$ but it is also not well-defined to
begin with, because it explodes as the peer group size grows. As such, both
regression coefficients $\beta_{x}$ and $\beta_{\bar{x}}$ from the model
estimated using the “grand” average $\overline{X}$ lack a meaningful
interpretation within the model’s own conceptual paradigm.