# On the Estimation of Cross-Firm Productivity
Spillovers with an Application to FDI††thanks: Email<EMAIL_ADDRESS>(Malikov) and<EMAIL_ADDRESS>(Zhao).

We benefited greatly from feedback and discussions with Paul Grieco, Jordi
Jaumandreu, Devesh Raval, Chris Vickers, Nic Ziebarth, seminar participants at
Auburn, Binghamton, Nebraska, TAMU and UNLV as well as participants at the
2019 Midwest Econometrics Group meeting at the Ohio State and the North
American Productivity Workshop XI at the University of Miami.

Emir Malikov University of Nevada, Las Vegas Shunan Zhao Oakland University

(June 20, 2021)

###### Abstract

We develop a novel methodology for the proxy variable identification of firm
productivity in the presence of productivity-modifying learning and spillovers
which facilitates a unified "internally consistent" analysis of the spillover
effects between firms. Contrary to the popular two-step empirical approach,
ours does not postulate contradictory assumptions about firm productivity
across the estimation steps. Instead, we explicitly accommodate cross-
sectional dependence in productivity induced by spillovers which facilitates
identification of both the productivity and spillover effects therein
simultaneously. We apply our model to study cross-firm spillovers in China’s
electric machinery manufacturing, with a particular focus on productivity
effects of inbound FDI.

Keywords: productivity spillovers, production function, proxy variable, FDI
spillovers

JEL Classification: C14, C23, D24, F21, L20, O30

##  1 Introduction

Since its popularization by Marshall (1890), the concept of cross-firm
knowledge, or technology, spillovers has increasingly become a central fixture
in many economic theories, including of long-run growth, spatial
agglomeration, research and innovation, international trade and more. The idea
is simple: firms improve their productivity by learning from one another, with
the most commonly conjectured drivers of these knowledge exchanges (technology
transfers) being human interaction along with spatial and
industrial/technological proximity. These productivity spillovers can also
propel significant positive externalities in many productivity-enhancing
activities such as research and development (R&D), foreign direct investment
(FDI) or exporting. In this paper, we develop a new methodology for the proxy
variable structural identification of firm productivity in the presence of
productivity-modifying learning and spillovers which facilitates a unified
“internally consistent” analysis of the spillover effects between peer firms.

Although productivity is straightforward in concept, its measurement is not
trivial for a multitude of reasons among which is the inherent latency of firm
productivity/efficiency. Naturally, the identification of productivity
spillovers across firms is even more challenging a task because, as Krugman
(1991, p.53) points out, “knowledge flows … are invisible; they leave no paper
trail by which they may be measured and tracked.” On this account, most
empirical work on cross-firm technology spillovers abstracts away from
pinpointing specific mechanisms by which such spillovers occur111Nuanced
empirical studies of spillovers are rare and require detailed matched (and
usually proprietary or confidential) datasets and, by design, have a limited
identifying ability restricted to particular channels/mechanisms of knowledge
diffusion such as labor turnover (e.g., Balsvik, 2011; Poole, 2013; Stoyanov &
Zubanov, 2012) or coauthorship networks (e.g., Zacchia, 2020). Others resort
to limiting the scope of analyzed spillovers: e.g., Jiang et al. (2019)
restrict FDI spillovers to direct links between joint ventures and their
domestic partners, whereas Newman et al. (2015) focus on productivity
spillovers along vertical supply chains. and instead focuses on a simpler but
more feasible objective of testing for the presence of cross-firm spillovers
in general. The most common frameworks either (i) focus squarely on
“productivity spillovers” by seeking to identify how a firm’s productivity is
affected by that of its peers—the “endogenous effect” in the Manski (1993)
nomenclature—or (ii) take a more reduced-form approach centered only on
measuring “contextual” spillover effects of various productivity-modifying
activities (FDI, R&D, exporting, etc.) facilitated by cross-firm spillovers in
productivity. Recent examples of the first include Bazzi et al. (2017) and
Serpa & Krishnan (2018) who study vertical productivity spillovers along
supply chains and material-product connections. As it happens, the literature
embracing the second framework is more predominant and has a longer history:
e.g., see Alvarez & López (2008) on spillovers from exporting; Javorcik
(2004), Javorcik & Spatareanu (2008), Haskel et al. (2007), Blalock & Gertler
(2008), Keller & Yeaple (2009), Barrios et al. (2011), Lu et al. (2017) on FDI
spillovers; Branstetter (2001), Griffith et al. (2006), Bloom et al. (2013),
Zacchia (2020) on technology spillovers from R&D; and Acharya & Keller (2008)
on productivity spillover effects of imports.

Both empirical frameworks are usually operationalized in two steps, whereby
one first recovers firm productivity from the production function estimates
and then examines spillovers in the second step by (linearly) regressing these
productivity estimates on various peer-group averages capturing firms’
exposure to potential spillovers. Owing to its popularity and ease of
implementation, most studies estimate firm productivity in the first step via
the proxy variable approach à la Olley & Pakes (1996) and Levinsohn & Petrin
(2003) that typically assumes that each firm’s productivity process is an
independent (over firms) exogenous Markov chain. However, if present,
spillovers would generate cross-sectional dependence among firms, which is
nonetheless being overlooked in the first-step estimation of productivity. Not
only does this raise reservations about the identification of production
function (and hence, productivity) econometrically, but more importantly, such
a two-step procedure suffers from the conceptual “internal inconsistency”
because the second-step regressions, in effect, contradictorily postulate the
existence of spillover-induced cross-firm dependence in productivity which is
at odds with the identifying assumptions used in the first step. As such,
conclusions about spillovers based on a two-step procedure may be spurious.

With the above in mind, we provide a novel (semiparametric) methodology for
the estimation of productivity spillovers. In line with the existence of
cross-firm spillovers, in building our model, we explicitly accommodate cross-
sectional peer dependence in firm-specific (latent) productivity that such
spillovers induce. This is fundamentally different from the aforementioned
traditional two-step approach.222We should note that a two-step framework is
not universal across empirical studies of productivity spillovers. The
exceptions are predominantly from the literature on R&D-borne spillovers that
favors the estimation of “augmented production functions.” We discuss the
benefits of our methodology over the latter in Appendix A.

To keep our methodology amenable to a wide range of contexts, we conceptualize
peer dependence in firm performance via spatiotemporal spillovers in latent
productivity itself. We generalize the conventional setup of firm production
assumed in the literature to introduce the dependence of each firm’s
productivity on its (geographically and industrially proximate) peer-group
average productivity. To that end, we dispense with the standard assumption of
independent (over $i$) exogenous Markov process for latent productivity (e.g.,
Olley & Pakes, 1996; Levinsohn & Petrin, 2003; Ackerberg et al., 2015; Gandhi
et al., 2020, and others) in favor of a controlled productivity process with
explicitly incorporated cross-sectional dependence. This permits the firm to
improve its productivity by learning not only directly from its own
productivity-modifying activities but also indirectly from the activities of
its peers.

Explicit modeling of cross-sectional dependence in firm productivity directly
affecting its evolution (along with a structural timing assumption about
learning process) enables us to build upon the popular proxy variable
technique to develop a unified identification scheme for both the latent
productivity and spillover effects therein simultaneously that is also robust
to Ackerberg et al.’s (2015) and Gandhi et al.’s (2020) critiques. In fact, as
we show in the paper, estimating the firm production function or productivity
using traditional proxy methods while ignoring the spillover-induced cross-
sectional dependence, as customarily done in the literature, likely leads to
misspecification and omitted variable bias. This underscores the key practical
advantage of our proposed methodology. Also, by virtue of a nonparametric
formulation of the firm productivity process, we transcend restrictive
additively linear specifications favored in the spillovers literature which
lets us accommodate heterogeneous spillover effects.

Because our methodology can be easily adapted to admit various spillover
origins, it is fit to investigate productivity spillovers in many contexts,
including spatial agglomeration, R&D externalities, learning from exporters,
and others. In our paper, for example, we consider an application to the FDI
inflows.

We also contribute to the literature on proxy-based identification of
production functions more broadly, by providing a practical, easy-to-implement
semiparametric adaptation of Gandhi et al.’s (2020) estimator. Our point of
departure is a parametric assumption about the functional form of production
function, which is the predominant modeling strategy in productivity
literature with the Cobb-Douglas specification being the most popular among
researchers. Along the lines of Doraszelski & Jaumandreu (2013), our modeling
approach fully embraces the assumed parametric specification of the production
function by explicitly utilizing a known functional form of the static first-
order condition for materials and the inverse conditional input demand
function that it implies. By doing so, we circumvent the need to integrate the
estimated material elasticity function at each observation in order to recover
the unknown production function required by Gandhi et al.’s (2020) more
computationally demanding, albeit admittedly less restrictive, nonparametric
methodology. In contrast, our parametric inversion of the material demand
yields a much simpler semiparametric estimator. We also show how to extend our
methodology to more flexible specifications of the firm’s production function
such as translog.

Besides the empirical application of our methodology, we also demonstrate its
ability to successfully identify firm productivity and cross-firm spillovers
therein in a set of Monte Carlo experiments. The results are encouraging and
show that our approach recovers the true parameters well, thereby lending
strong support to the validity of our identification strategy. We also use the
simulations to show how estimating spillovers via the popular but internally
inconsistent two-step procedure can lead to spurious and misleading results.

The rest of the paper unfolds as follows. Section 2 provides context for our
application centered on FDI spillovers. Section 3 describes a generic model of
firm production with productivity-modifying learning and spillovers. We
discuss identification and estimation in Sections 4 and 5, respectively.
Section 6 reports simulation results. We present our empirical application in
Section 7 and conclude in Section 8.

##  2 Application to FDI

Cross-firm productivity spillovers can propel significant positive
externalities in many productivity-enhancing activities, which are especially
important from a policy perspective. Take, for instance, inbound foreign
direct investment.

Public policies aimed at attracting FDI are commonplace both in developing and
developed economies. Besides immediate returns in the form of capital inflows
and employment gains, the primary justification for the FDI-promoting
government incentives is mostly centered around gaining access to intangible
productive “knowledge” assets from abroad such as new technologies,
proprietary know-hows, more efficient and innovative marketing and management
practices, established relational networks, reputation, etc., which can boost
productivity of domestic firms. More crucially, productivity-enhancing effects
of inbound FDI are widely believed to realize broadly beyond immediate
recipients, who benefit from direct learning of foreign knowledge, by also
benefiting many other domestic firms via productivity spillovers. These
spillovers may occur via informal contacts (e.g., attendance of trade shows,
exposure to affiliate and/or competitor products and marketing, learning by
imitation, customer-supplier discussions), more formal reverse engineering, or
labor turnover, eventually yielding large within- and/or cross-industry
productivity gains. Measuring the extent and significance of these “social
returns” of FDI is therefore imperative for the design of effective industrial
policy.

To empirically showcase our estimator, we apply it to study horizontal
productivity spillovers in China’s electric machinery manufacturing industry
in 1998–2007, with a particular focus on the technology-transfer effects of
inbound FDI on productivity via domestic firms’ learning of more
advanced/efficient foreign knowledge to which they may gain access directly
through their own foreign investors and indirectly through spillovers from
their foreign-invested peers. Among the world’s top destinations for foreign
investment, China presents a natural environment for the analysis of broad
productivity effects of FDI on domestic firms especially because of its “open
door” policies aimed at promoting foreign investment (e.g., special economic
zones with regulatory environments favorable to foreign capital) and its
fairly recent accession to the World Trade Organization in 2001. Focus on the
electric machinery industry in particular is motivated by its being
historically one of China’s most fundamental manufacturing sectors and among
the largest FDI recipients (see Appendix B for more on the choice of the
industry).

The empirical literature on FDI spillovers has generally produced mixed
findings, especially for the long-sought-after horizontal productivity
spillovers (see Keller, 2008, 2010, for excellent surveys). Few earlier
studies that have analyzed external productivity spillovers from inbound FDI
in China (see Jiang et al., 2019, and the references therein) have done so
using the two-step approach or by “augmenting” the firm’s production function
with the associated methodological issues as discussed earlier. The results
have been mixed, further heightening the appeal of our study based on a new
methodology. The reanalysis of FDI-borne technology spillovers in China is
also timely and relevant in light of the ongoing trade disputes between the
U.S. and China fostered, among other things, by grievances of the former
against China’s “unfair technology transfer regime” for foreign companies.
Investigating the extent of external spillovers can therefore provide an
informative context for a more holistic understanding of the FDI environment
in the country and the implications of its technology-transfer rules and
regulations.

To briefly preview our key results, we find that at least 87% of manufacturers
of electric machinery in China enjoy significant productivity-boosting effects
of inbound FDI, both directly and indirectly. At the median, an increase of
the foreign share in all firms’ equity by 10 percentage points, in the short
run, improves each firm’s productivity by 1.4% via direct learning and by 0.4%
via external effects. The latter indirect effect of FDI is facilitated by
substantial cross-firm productivity spillovers in the industry, with the
median spillover elasticity estimated at 0.33. These productivity spillovers
are significantly positive for about 84% of firms in the industry.

##  3 Production with Learning and Spillovers

We now describe a generic paradigm of production in the presence of
productivity-modifying learning and spillovers. Consider the production
process of a firm $i=1,\dots,n$ in time period $t=1,\dots,T$ in which physical
capital $K_{it}$, labor $L_{it}$ and an intermediate input such as materials
$M_{it}$ are transformed into the output $Y_{it}$ via production function,
given the log-additive Hicks-neutral firm productivity. Following the popular
convention in the literature (e.g., Olley & Pakes, 1996; Levinsohn & Petrin,
2003; Topalova & Khandelwal, 2011; Doraszelski & Jaumandreu, 2013), we assume
that the firm’s stochastic production process is Cobb-Douglas:

$Y_{it}=K_{it}^{\beta_{K}}L_{it}^{\beta_{L}}M_{it}^{\beta_{M}}\exp\left\\{\omega_{it}+\eta_{it}\right\\},$
(3.1)

where the exponent $\omega_{it}+\eta_{it}$ is the latent “composite”
productivity residual consisting of (i) the firm $i$’s persistent productivity
$\omega_{it}$ and (ii) a random transitory productivity shock $\eta_{it}$. Our
methodology can also adopt more flexible specifications of the firm’s
production function such as the log-quadratic translog specification. See
Appendix C for this extension of our model.

We assume that $K_{it}$ and $L_{it}$ are subject to adjustment frictions
(e.g., time-to-install, hiring and training costs) and thus are quasi-fixed,
whereas $M_{it}$ is freely varying. That is, $M_{it}$ is chosen in period $t$,
whereas $K_{it}$ and $L_{it}$ are determined in period $t-1$. Both $K_{it}$
and $L_{it}$ are the state variables with dynamic implications and follow
their respective deterministic laws of motion:

$K_{it}=I_{it-1}+(1-\delta)K_{it-1}\quad\text{and}\quad
L_{it}=H_{it-1}+L_{it-1},$ (3.2)

where $I_{it}$, $H_{it}$ and $\delta$ are the gross investment, net hiring and
the depreciation rate, respectively. The firm maximizes a discounted stream of
expected life-time profits in perfectly competitive output and factor markets.
Also, for convenience, let $\mathcal{I}_{it}$ denote the information set
available to the firm $i$ for making period $t$ production decisions.

Our main objective is to study the role of learning and spillovers in the
evolution of firm productivity. To that end, we need to dispense with the
standard assumption of exogenous Markov process for $\omega_{it}$ in favor of
a controlled productivity process and, more importantly, to explicitly
recognize the potential for cross-sectional dependence therein. For generality
sake, we denote the productivity-modifying “controls” via a (vector of)
generic variable(s) $G_{it}$. This variable may measure the firm’s deliberate
activities aimed at improving its productivity such as R&D expenditures
(Doraszelski & Jaumandreu, 2013) or some other aspects of its behavior in the
marketplace that have productivity implications such as exporting (De Loecker,
2013). Depending on the application of interest, $G_{it}$ may also admit
measures of the firm’s exposure to technological innovations from investors or
partners—the focus of our empirical illustration—or its access to public
subsidies and other forms of favorable treatment from the government owing to
political connections, etc. In the end, no matter the choice of $G_{it}$, the
rationale for its inclusion in the firm productivity evolution is to capture
within-firm “learning” facilitated by the firm’s own productivity-modifying
activities or characteristics.

Next, we permit the firm $i$ to improve its productivity by learning not only
from its own activities but also from its peer firms. We do so by relaxing the
usual assumption of firm productivity being an independent (over $i$) Markov
chain to allow for cross-sectional dependence. More concretely, we assume that
the $i$th firm’s productivity $\omega_{it}$ evolves according to the following
controlled first-order process:

$\omega_{it}=\mathbb{E}\Bigg{[}\omega_{it}\Bigg{|}\
\omega_{i,t-1},G_{i,t-1},\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}\Bigg{]}+\zeta_{it},$ (3.3)

where $\\{s_{ij,t-1};\ j(\neq i)=1,\dots,n\\}$ are the peer-identifying
weights (from the perspective of firm $i$), and $\zeta_{it}$ is a mean-
independent unanticipated random innovation in persistent productivity
normalized to have a zero mean:
$\mathbb{E}[\zeta_{it}|\mathcal{I}_{i,t-1}]=\mathbb{E}[\zeta_{it}|\omega_{i,t-1},G_{i,t-1},\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}]=0$.

While the exact choice of how to construct peer weights $\\{s_{ijt}\\}$
depends on the empirical context, for a general baseline case here, we let the
peers be identified based on their spatial vicinity and industrial similarity
to firm $i$. Thus, letting $\mathcal{L}(i,t)$ represent a set of spatially
proximate “neighbors” of the firm $i$ in time period $t$ that also operate in
the same industry, peer weights $\\{s_{ijt}\\}$ are constructed for each
$(i,t)$ as follows:

$s_{ijt}=\frac{\mathbbm{1}\big{\\{}(j,t)\in\mathcal{L}(i,t)\big{\\}}}{\sum_{k(\neq
i)=1}^{n}\mathbbm{1}\big{\\{}(k,t)\in\mathcal{L}(i,t)\big{\\}}},$ (3.4)

where the normalization in the denominator yields a convenient interpretation
of $\sum_{j(\neq i)}s_{ijt}\omega_{jt}$ as the average peer productivity.
Focusing on geographically proximate peers within the industry fits a broader
narrative in regional and urban economics about the scopes of agglomeration
economies and the localized cross-firm productivity spillovers (due to
technology and knowledge diffusion, labor market interactions, etc.) being one
of the main sources of such externalities (e.g., see Duranton & Puga, 2004;
Rosenthal & Strange, 2004). By restricting the scope to the same industry we
effectively focus on intra-industry horizontal productivity spillovers.

######  Remark 1.

The weighing scheme in (3.4) treats cross-firm spillovers symmetrically in
that all members of a peer group affect each other’s productivity. Not only is
this a standard approach to measuring “peer effects,” but in doing so we also
wish to remain as agnostic about spillovers as possible and avoid imposing
priors about the directionality of peer dependence. But should one choose to
regulate the direction of spillovers by restricting them to occur, say, from
more productive to less productive firms only, our framework can be modified
to accommodate that too. For more discussion, see Appendix D.

######  Remark 2.

In (3.4), a uniform weighting is applied across all peers of the firm $i$ that
are located in its spatial proximity. This implicitly assumes that within
boundaries of the firm’s spatial “neighborhood” the distance gradient is of
second-order importance for knowledge spillovers. The main benefit of
postulating such a feature of peer networks is that it does not require
granular geographic data about individual firms and can be operationalized
using coarse location information such as ZIP code, census track, city,
region. The degree to which this is a reasonable weighting scheme obviously
depends on the selected “level” of neighborhoods as well as the application-
specific institutional context. If desired and feasible, peer weights
$\\{s_{ijt}\\}$ can be appended to incorporate a (decay) function of the
distance between $i$ and its peers $\\{j\\}$.

The innovativeness of our model in the context of a broader literature on the
structural proxy variable estimation of production functions is as follows.
The “controlled” formulation in (3.3) is more general than the most commonly
assumed exogenous Markov process à la Olley & Pakes (1996) whereby
$\omega_{it}=\mathbb{E}\left[\omega_{it}|\omega_{i,t-1}\right]+\zeta_{it}$
because it enables the firm to influence the evolution of its productivity via
its own productivity-enhancing activities/characteristics as well as by
interacting with other local firms in the industry as captured by $G_{i,t-1}$
and $\sum_{j(\neq i)}s_{ij,t-1}\omega_{j,t-1}$, respectively. While controlled
Markov processes for $\omega_{it}$ are not novel to the literature (e.g.,
Doraszelski & Jaumandreu, 2013; De Loecker, 2013), all such studies have
focused exclusively on an independently (over $i$) distributed $\omega_{it}$
having the latter depend on the firm’s own productivity and productivity-
modifying variables. Our important generalization is that we permit cross-
sectional dependence in firm productivity within peer networks due to
agglomeration.

Consider the spatiotemporal autoregressive conditional mean of $\omega_{it}$
in (3.3) that represents the $i$th firm’s expected one-period-ahead
productivity at time $t-1$. First, by letting it depend on the firm’s own
productivity modifier $G_{i,t-1}$, we are able to account for (internal)
direct learning taking place within the firm, with the corresponding estimand
of interest being

$DL_{it}=\frac{\partial\mathbb{E}[\omega_{it}|\cdot]}{\partial G_{i,t-1}}.$
(3.5)

Second, in including the spatial average of other firms’ productivities
$\sum_{j(\neq i)}s_{ij,t-1}\omega_{j,t-1}$, not only can we accommodate
potential agglomeration externalities facilitated by productivity spillovers
across firms, but we are also able to capture the (external) indirect learning
whereby the productivity-modifying activities may have secondary effects on
firms beyond their immediate (and intended) beneficiary. Concretely, defining
the cross-firm productivity spillovers as

$SP_{it}=\frac{\partial\mathbb{E}[\omega_{it}|\cdot]}{\partial\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}},$ (3.6)

the measure of firm $i$’s indirect learning from firm $j$’s productivity-
modifying activities is

$\displaystyle IL_{ijt}$
$\displaystyle=\frac{\partial\mathbb{E}[\omega_{it}|\cdot]}{\partial
G_{j,t-2}}=\frac{\partial\mathbb{E}[\omega_{it}|\cdot]}{\partial\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}}\left(s_{ij,t-1}\frac{\partial\omega_{j,t-1}}{\partial
G_{j,t-2}}\right)=SP_{it}\times s_{ij,t-1}\times DL_{j,t-1}.$ (3.7)

The $IL_{ijt}$ effect in (3.7) is defined for an $(i,j)$ pair of firms, and we
can aggregate it to the total indirect learning of firm $i$ from all of its
peers as

$\displaystyle TIL_{it}$ $\displaystyle=\sum_{j(\neq
i)}IL_{ijt}=SP_{it}\times\sum_{j(\neq i)}s_{ij,t-1}DL_{j,t-1}.$ (3.8)

As defined in (3.5)–(3.8), the learning and spillover effects on firm
productivity are “short-run,” but they accumulate and diffuse over time owing
to a persistent nature of the firm’s productivity evolution. Indirectly, this
dynamic feature permits two peer firms that are separated temporally to
continue to affect one another with the effect size attenuating over time.
Such time-separated interactions characterize the temporal scope of
productivity spillovers which helps propel dynamic agglomeration economies.
The underlying idea here is that the knowledge acquired either through
internal learning or from peers takes time to accumulate. Together with the
geographic and industrial scopes of spillovers embedded in the definition of
peer weights $\\{s_{ijt}\\}$, the autoregressiveness of productivity
specification covers the three main dimensions of external economies (see
Rosenthal & Strange, 2004).

######  Remark 3.

The total indirect learning effect in (3.8) is, effectively, a measure of
spillovers specifically in $G$. In this, our conceptualization of external
effects of $G$ as operating through the firm’s exposure to the aggregate of
its peers’ unobservable productivities—that is, via “productivity spillovers”
more broadly—fundamentally differs from the conventional approach to measuring
spillovers in productivity-modifying activities (think, FDI, R&D or export
spillovers) that relies on observable industry aggregates of $G$. That is, we
measure the $i$th firm’s exposure to the external knowledge using an aggregate
of $\\{\omega_{jt};\ j(\neq i)=1,\dots,n\\}$ as opposed to an aggregate of
$\\{G_{jt};\ j(\neq i)=1,\dots,n\\}$. Our formulation is more flexible because
it does not restrict the origins of cross-firm productivity spillovers to $G$
alone. It is also more realistic and conceptually congruous because it
incorporates secondary information about the peer firms’ own direct/internal
learning facilitated by the productivity-modifying activities they undertake:
namely, to learn from one’s peers’ $G$, peers themselves should learn from
their own $G$ first.

The productivity evolution process in (3.3) characterizes the peer interaction
between firms through their productivity. Each firm $i$ has a “reference
group” of spatially proximate peers from the same industry $\mathcal{L}(i,t)$
with which it interacts. The identification of such cross-peer relations in
networks is a notoriously challenging problem (e.g., see Manski, 1993, 2000;
Moffitt, 2001; Blume et al., 2011). The potential obstacles include (i) the
perfect functional dependence between the average outcome of the group and its
mean characteristics due to the so-called “reflection problem” which may leave
no exogenous variation excluded to instrument the endogenous peer behavior
when there is more than one channel for the peer effects, (ii) the confounding
presence of unobserved “correlated” group effects, and (iii) the endogenous
group membership (or network structure). In our case, the additional layer of
complexity is the latency of firm productivity. This aspect is addressed in
the proxy variable framework by making a full use of the behavioral model of
firm production, and we discuss this in detail in Section 4. We now consider
the issues pertaining to the measurement of peer effects between firms.

The identification of learning and spillover effects on firm productivity in
our model is based on several structural assumptions about the timing as well
as the underlying form of peer interactions and network organization. To begin
with, the productivity process in (3.3) is a dynamic analogue of a “pure
endogenous-effects model” in Manski’s nomenclature. It postulates that the
cross-firm peer interactions occur only through the outcomes (i.e., $\omega$)
whereby each firm’s productivity is affected by the mean productivity of the
peers in its reference group. As noted in Remark 3, we effectively assume away
the “contextual effects” of the peers’ productivity modifiers and, in doing
so, address the first of two Manski’s (1993) unidentification results about
the indistinguishability of endogenous and exogenous peer effects.333His
second result is about the difficulty to distinguish “real” peer interactions
through observables from the unobservable “correlated effects;” more on this
later. The latter issue becomes moot because in the absence of contextual
effects of $\\{G_{j,t-1};\ j(\neq i)=1,\dots,n\\}$ on $\omega_{it}$ our model
postulates a single channel of cross-peer dependence. Appendix E discusses how
our setup may be augmented to allow for such contextual effects.

The evolution process in (3.3) also implicitly assumes that learning occurs
with a delay which is why the dependence of $\omega_{it}$ on both its own
productivity-modifying controls and peers’ productivity is lagged, implying
that the improvements in firm productivity take a period to materialize.
Furthermore, in $\mathbb{E}[\zeta_{it}|\mathcal{I}_{i,t-1}]=0$ we assume that
firms do not experience changes in their location and/or productivity
modifiers in light of expected future innovations in productivity. This timing
assumption about the arrival of $\zeta_{it}$ renders both the lagged
$G_{i,t-1}$ and a set of spatially proximate peers $\mathcal{L}(i,t-1)$ at
time $t-1$ that defines the peer weights $\\{s_{ij,t-1};\ j(\neq
i)=1,\dots,n\\}$ predetermined (weakly exogenous) with respect to the firm
$i$’s productivity innovation at time $t$, which helps identify both the
learning and spillover effects on firm productivity.

When it comes to internal learning effects (via own $G_{i,t-1}$), such a
timing assumption is quite common in the productivity literature (e.g., see
Van Biesebroeck, 2005; De Loecker, 2013; Doraszelski & Jaumandreu, 2013;
Malikov et al., 2020). More specifically,
$\mathbb{E}[\zeta_{it}|\mathcal{I}_{i,t-1}]=0$ rules out the firm’s ability to
systematically predict future productivity shocks. Instead, the Markovian
process in (3.3) states that the firm anticipates the effect of its $G$
productivity modifier on $\omega_{it}$ in period $t$ when adjusting the former
in period $t-1$, and the conditional mean
$\mathbb{E}\left[\omega_{it}|\omega_{i,t-1},G_{i,t-1},\right.$
$\left.\sum_{j(\neq i)}s_{ij,t-1}\omega_{j,t-1}\right]$ is what captures that
expected productivity. But the actual firm productivity at time $t$ also
includes a random innovation $\zeta_{it}$. Essentially, the conditional-
expectation-function error $\zeta_{it}$ represents unpredictable uncertainty
that is naturally associated with productivity-modifying activities (new R&D
investments, entering export markets or attracting new foreign investors) such
as chance in discovery, success in implementation, etc. This productivity
innovation $\zeta_{it}$ is realized after $G_{i,t-1}$ is fully
determined.444Depending on the source of learning, it may be possible to
reasonably relax the assumption of a delayed learning effect of $G_{it}$ on
firm productivity. See discussion in Appendix E.

In our paper, we also extend this timing assumption to external cross-firm
learning via spillovers, which yields mean-orthogonality of the spatiotemporal
“lag” of peers’ productivities $\sum_{j(\neq i)}s_{ij,t-1}\omega_{j,t-1}$ and
the innovation $\zeta_{it}$. That is, the assumed is weak exogeneity of the
location-dependent peer weights $\\{s_{ij,t-1}\\}$, according to which firms
do not relocate in anticipation of future productivity shocks because such
shocks are purely random. This rules out endogeneity of the firm’s peer
network in period $t-1$ with respect to the productivity shock $\zeta_{it}$ it
experiences at time $t$. The plausibility of this is further buttressed by the
fact that firm relocation in most industries (e.g., agriculture,
manufacturing, utilities) is highly, if not prohibitively, costly. In fact,
our assumption about the weak exogeneity of group membership is not as strong
as the standard assumption of fixed (non-random) networks commonly made in the
(empirical) social-effects or spatial literature.

Note that our timing assumption about learning and spillover effects does not
rule out a contemporaneous correlation between firm productivity and its
productivity modifiers or even the location. That is, we do not assume that
$\mathbb{E}[\zeta_{it}|\mathcal{I}_{it}]=0$. Consequently, firms are permitted
to endogenously update their $G_{it}$ as well as to change their locations
based on the (observable by firms) period $t$ level of their productivity
$\omega_{it}$. For instance, in the presence of inbound FDI opportunities that
can help improve a domestic firm’s productivity (when $G_{it}$ measures the
firm’s exposure to investors from abroad), the more productive firms are more
likely to be attractive for investors, and the corresponding non-zero
$\text{Cov}[G_{it},\omega_{it}]$ is well within our framework.

An important implication of our structural assumption about
$\mathbb{E}[\zeta_{it}|\sum_{j(\neq i)}s_{ij,t-1}\omega_{j,t-1}]=0$ is that
the innovation in the productivity evolution process (3.3) does not contain
any unobservable “correlated effects” at the reference group level—to borrow
Manski’s terminology—the presence of which can complicate, if not hinder, the
identification of peer effects occurring through the group mean productivity
$\sum_{j(\neq i)}s_{ij,t-1}\omega_{j,t-1}$. Effectively, we attribute all
cross-firm dependence in productivity to the within-group dependence of each
firm’s underlying productivity on that of its peers as opposed to the tendency
of all group firm-members to see their productivities evolve in a similar
fashion due to the influence of common group unobservables such as shared
locational/institutional environments. This is an admittedly strong but fairly
common working assumption in the literature, given the well-known challenges
in tackling group-level unobservables in network models (for an excellent
review, see Blume et al., 2011). Our no-group-effects assumption echoes the
existing studies of R&D/FDI/export spillovers and the productivity literature
more broadly, and we maintain it to maximize comparability with the commonly
used methodologies. Having said that, this assumption can be relaxed—we do so
in our robustness checks—if we restrict the group-level unobservables to be
time-invariant à la Graham & Hahn (2005) and Bramoullé et al. (2009).

Fundamentally, the potential threats to identification of the spillover
effects posed by the correlated group effects can otherwise be cast as a
spatial selection/sorting problem, whereby more productive firms may be ex
ante sorting into the what-then-become high productivity locations. Under this
scenario, when we compare the firm to its spatial peers, we may mistakenly
attribute any future productivity improvements to spillovers from the peers
(i.e., agglomeration), while in actuality it merely reflects the underlying
propensity of all firms in this location to be more productive and,
consequently, more apt at improving their productivity. While there has
recently been notable progress in formalizing and understanding these
coincident phenomena theoretically (e.g., Behrens et al., 2014; Gaubert,
2018), disentangling firm sorting and agglomeration remains a non-trivial task
empirically.555Urban economics literature also distinguishes the third
endogenous process usually referred to as the “selection” which differs from
sorting in that it occurs ex post after the firms had self-sorted into
locations and which determines their continuing survival. We abstract away
from this low-productivity-driven attrition issue in the light of the growing
empirical evidence suggesting that it explains none of spatial productivity
differences which, in contrast, are mainly driven by agglomeration economies
(see Combes et al., 2012). Relatedly, the firm attrition out of the sample has
also become commonly accepted as a practical non-issue in the productivity
literature so long as the data are kept unbalanced. For instance, Levinsohn &
Petrin (2003, p.324) write: “The original work by Olley and Pakes devoted
significant effort to highlighting the importance of not using an artificially
balanced sample (and the selection issues that arise with the balanced
sample). They also show once they move to the unbalanced panel, their
selection correction does not change their results.” However, by including the
firm’s own lagged productivity in the autoregressive $\omega_{it}$ process in
(3.3), we are able (at least to some extent) to account for this potential
self-sorting because sorting into locations is heavily influenced by the
firm’s own productivity (oftentimes stylized as the “talent” or “efficiency”
in theoretical models). That is, the spillover effect $SP_{it}$ on future firm
productivity in our model is measured after partialling out the contribution
of its own productivity. Incidentally, De Loecker (2013) argues the same in
the context of export-based learning and self-selection of exporters.

We maintain the i.i.d. assumption about the random transitory shock
$\eta_{it}$, from where it follows that
$\mathbb{E}[\eta_{it}|\mathcal{I}_{it}]=\mathbb{E}[\eta_{it}]=0$ with the mean
normalized to zero. The latter implies that the shock $\eta_{it}$ is
observable to firms in period $t$ only ex post after all production decisions.

##  4 A System Approach to Identification via Proxy Variables

Logging the production function in (3.1) and making use of the Markovian
nature of $\omega_{it}$ from (3.3), we obtain

$y_{it}=\beta_{K}k_{it}+\beta_{L}l_{it}+\beta_{M}m_{it}+h\left(\omega_{i,t-1},G_{i,t-1},\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}\right)+\zeta_{it}+\eta_{it},$ (4.1)

where the lower-case variables denote the logs of the respective upper-case
variables, and $h[\cdot]\equiv\mathbb{E}[\omega_{it}|\cdot]$ is some unknown
function. Under our structural assumptions about firm behavior, all right-
hand-side covariates in (4.1) are predetermined and thus mean-independent of
$\zeta_{it}+\eta_{it}$, except for the freely varying input $m_{it}$ that the
firm chooses in time period $t$ conditional on $\omega_{it}$ (among other
state variables including quasi-fixed inputs) thereby making it a function of
$\zeta_{it}$. That is, the materials variable is endogenous.

To consistently estimate (4.1), we first need to address the latency of firm
productivity $\omega_{it}$. A widely popular solution to this problem in the
literature is to adopt a proxy variable approach à la Levinsohn & Petrin
(2003) whereby unobservable $\omega_{it}$ is proxied by inverting the firm’s
conditional demand for an observable static input $m_{it}$. However, Gandhi et
al. (2020) show that identification generally fails under such a standard
estimation procedure due to the lack of a valid instrument for the endogenous
$m_{it}$ despite the abundance of predetermined higher-order lags of inputs.
Therefore, the production function remains unidentified in the flexible input.
To solve this problem, they suggest exploiting a structural link between the
production function and the firm’s (static) optimality condition. In what
follows, we build on this idea which we adapt in the spirit of Doraszelski &
Jaumandreu (2013), whereby we explicitly make use of the assumed functional
form of the production function to streamline identification of the material
elasticity and to ease computational burden of estimation (also see Remark 4).

We first focus on the identification of the production function in its
flexible input $M_{it}$. Specifically, given the Cobb-Douglas form, we seek to
identify the material elasticity parameter $\beta_{M}$. To do so, we consider
an equation for the firm’s first-order condition with respect to $M_{it}$.
Since it is a static input, the firm’s optimal choice of $M_{it}$ can be
modeled as the restricted expected profit-maximization problem666Under the
risk neutrality of firms. subject to the (already) optimal allocation of
quasi-fixed inputs:

$\displaystyle\max_{M_{it}}\
P_{t}^{Y}K_{it}^{\beta_{K}}L_{it}^{\beta_{L}}M_{it}^{\beta_{M}}\exp\\{\omega_{it}\\}\theta-
P_{t}^{M}M_{it},$ (4.2)

where $P_{t}^{Y}$ and $P_{t}^{M}$ are respectively the output and material
prices that, under the commonly invoked assumption of perfect competition,
need not vary across firms; and $\theta\equiv\mathbb{E}[\exp\\{\eta_{it}\\}|\
\mathcal{I}_{it}]$. The first-order condition is given by

$\beta_{M}P_{t}^{Y}K_{it}^{\beta_{K}}L_{it}^{\beta_{L}}M_{it}^{\beta_{M}-1}\exp\\{\omega_{it}\\}\theta=P_{t}^{M},$
(4.3)

which can be transformed via dividing it by the production function in (3.1)
to obtain the following stochastic material share equation (in logs):

$\ln V_{it}=\ln(\beta_{M}\theta)-\eta_{it},$ (4.4)

where $V_{it}\equiv P_{t}^{M}M_{it}/(P_{t}^{Y}Y_{it})$ is the nominal share of
material costs in total revenue. This share is readily observable in the data,
and the construction thereof does not require firm-level prices.

Intuitively, equation (4.4) says that unobservable material elasticity of the
production function $\beta_{M}$ can be identified from observable material
share $V_{it}$ because the two must be equal on average (in logs) to maximize
profits. Specifically, it identifies $\beta_{M}\times\theta$ (and the random
productivity residual $\eta_{it}$) based on $\mathbb{E}[\eta_{it}]=0$:

$\ln(\beta_{M}\theta)=\mathbb{E}[\ln V_{it}].$ (4.5)

To identify the material elasticity $\beta_{M}$ net of constant $\theta$,
recognize that $\theta$ can be identified via
$\theta=\mathbb{E}[\exp\\{\eta_{it}\\}]=\mathbb{E}\left[\exp\\{\ln(\beta_{M}\theta)-\ln
V_{it}\\}\right]=\mathbb{E}\left[\exp\\{\mathbb{E}[\ln V_{it}]-\ln
V_{it}\\}\right]$. Then, we have that

$\beta_{M}=\exp\left\\{\mathbb{E}[\ln
V_{it}]\right\\}\big{/}\mathbb{E}\left[\exp\\{\mathbb{E}[\ln V_{it}]-\ln
V_{it}\\}\right].$ (4.6)

With $\beta_{M}$ identified from (4.6), we have thus identified the production
function in the dimension of its endogenous freely varying input $M_{it}$
thereby effectively circumventing Gandhi et al.’s (2017) critique. To see the
latter, we rewrite (4.1) as follows:

$y_{it}^{*}=\beta_{K}k_{it}+\beta_{L}l_{it}+h\left(\omega_{i,t-1},G_{i,t-1},\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}\right)+\zeta_{it}+\eta_{it},$ (4.7)

where $y_{it}^{*}\equiv y_{it}-\beta_{M}m_{it}$ is already identified and thus
observable, and our model in (4.7) no longer contains endogenous variables
needing instrumentation.

To identify the remaining parameters of the production function
$(\beta_{K},\beta_{L})^{\prime}$ as well as latent firm productivity
$\omega_{it}$ in (4.7), we make use of the known parametric form of the
conditional material demand function
$M_{it}=\mathbb{M}(\omega_{it},K_{it},L_{it},P_{t}^{Y},P_{t}^{M})$ implied by
the first-order condition in (4.3) which we invert for $\omega_{it}$. Under
our standard assumptions about firm behavior and regularity conditions on the
production function, $\mathbb{M}(\cdot)|M_{it}>0$ must be strictly monotonic
in $\omega_{it}$ for any given $(K_{it},L_{it},P_{t}^{Y},P_{t}^{M})$, and
hence we can invert $\mathbb{M}(\cdot)$ to control for unobserved persistent
productivity via
$\omega_{it}=\mathbb{M}^{-1}(M_{it},K_{it},L_{it},P_{t}^{Y},P_{t}^{M})$.
Specifically, substituting for $\omega_{i,t-1}$ and $\omega_{j,t-1}$ using the
inverted material function derived analytically from (4.3), from (4.7) we get

$\displaystyle y_{it}^{*}=$ $\displaystyle\
\beta_{K}k_{it}+\beta_{L}l_{it}+h\left(\omega_{i,t-1}^{*}\left(\beta_{K},\beta_{L}\right),G_{i,t-1},\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}^{*}\left(\beta_{K},\beta_{L}\right)\right)+\zeta_{it}+\eta_{it},$
(4.8)

where

$\displaystyle\omega_{it}^{*}\left(\beta_{K},\beta_{L}\right)=\Big{[}(1-\beta_{M})m_{it}-\ln(\beta_{M}\theta)-\ln(P_{t}^{Y}/P_{t}^{M})\Big{]}-\beta_{K}k_{it}-\beta_{L}l_{it}\quad\forall
i,t$ (4.9)

is the inverted material demand function in which the bracketed component is
already observable and therefore the only remaining unknown parameters in it
are $(\beta_{K},\beta_{L})^{\prime}$. All right-hand-side covariates in the
semiparametric model (4.8) are weakly exogenous and thus self-instrument. The
model is thus identified on the basis of

$\mathbb{E}\left[\zeta_{it}+\eta_{it}\Bigg{|}k_{it},l_{it},k_{i,t-1},l_{i,t-1},m_{i,t-1},G_{i,t-1},\sum_{j\neq
i}s_{ij,t-1}k_{j,t-1},\sum_{j\neq i}s_{ij,t-1}l_{j,t-1},\sum_{j\neq
i}s_{ij,t-1}m_{j,t-1}\right]=0,$ (4.10)

where we have made explicit use of the variables entering the proxy function
$\omega_{it}^{*}\left(\beta_{K},\beta_{L}\right)$.

The appearance of group averages of the peers’ predetermined inputs in (4.10)
is akin to the idea of instrumenting the endogenous group mean of an outcome
with the exogenous group mean characteristics, which is a common
identification strategy in both the social-effects and spatial econometrics
literature (e.g., see LeSage & Pace, 2009; Bramoullé et al., 2009). The
critical distinction here is that, in our case, the “group mean of an outcome”
$\sum_{j\neq i}s_{ij,t-1}\omega_{j,t-1}$ is not endogenous with respect to
$\zeta_{it}+\eta_{it}$ and therefore needs no instrumentation. In contrast,
our use of the “group mean characteristics” $\big{(}\sum_{j\neq
i}s_{ij,t-1}k_{j,t-1},\sum_{j\neq i}s_{ij,t-1}l_{j,t-1},\sum_{j\neq
i}s_{ij,t-1}m_{j,t-1}\big{)}^{\prime}$ is effectively in their proxy-variable
capacity given latency of $\omega_{j,t-1}$.

######  Remark 4.

Following the steps of Doraszelski & Jaumandreu (2013), our approach fully
embraces the assumed parametric specification of the firm’s production
function by explicitly utilizing the known functional form of the first-order
condition for materials and the inverse conditional input demand function that
it implies. By doing so, we circumvent the need to integrate the estimated
material elasticity function at each observation in order to recover the
unknown production function required by Gandhi et al.’s (2017) nonparametric
methodology. Importantly, by relying on parameter restrictions between the
production function and inverted material demand function in (4.8), we do not
have to rely on nonparametric methods to estimate the unknown proxy function
for $\omega$ that appears inside the also unknown $h(\cdot)$ function.
Otherwise, identification of (4.8) would have been complicated by the presence
of a nonparametric $\mathbb{M}^{-1}(\cdot)$ function (evaluated at multiple
data points777That is, evaluated at $(m_{i,t-1},k_{i,t-1},l_{i,t-1})$ to proxy
for $\omega_{i,t-1}$ as well as at $(m_{j,t-1},k_{j,t-1},l_{j,t-1})\ \forall\
j$ to proxy for $\omega_{j,t-1}$ entering the spillover-capturing peer group
average.) inside another nonparametric function $h(\cdot)$. Our parametric
inversion of $\mathbb{M}^{-1}(\cdot)$ yields a much simpler semiparametric
estimator.

######  Remark 5.

Our model is also robust to Ackerberg et al.’s (2015) critique that focuses on
the potential inability of structural proxy variable estimators to separably
identify the production function and productivity proxy. This issue arises in
the wake of perfect functional dependence between variable inputs appearing
both inside the unknown production function and productivity proxy function.
Our second-stage equation (4.8) does not suffer from such a problem because it
contains no (endogenous) variable input on the right-hand side, the
corresponding parameter of which has already been identified from the share
equation in the first stage.

Lastly, with all parameters of the production function
$(\beta_{K},\beta_{L},\beta_{m})^{\prime}$ and the transitory productivity
shock ${\eta}_{it}$ successfully identified in the two stages, we readily
identify latent firm productivity $\omega_{it}$ from the production function
in logs:
$\omega_{it}=y_{it}-\beta_{K}k_{it}-\beta_{L}l_{it}-\beta_{M}m_{it}-\eta_{it}$.

##  5 Estimation Procedure

We implement our identification strategy in two stages. In the first stage, we
estimate the material elasticity of the production function. Based on (4.6),
the consistent estimator of $\beta_{M}$ is

$\widehat{\beta}_{M}=\exp\Big{\\{}\frac{1}{N}\sum_{i}\sum_{t}\ln
V_{it}\Big{\\}}\Big{/}\Big{[}\frac{1}{N}\sum_{i}\sum_{t}\exp\Big{\\{}\Big{(}\frac{1}{N}\sum_{i}\sum_{t}\ln
V_{it}\Big{)}-\ln V_{it}\Big{\\}}\Big{]},$ (5.1)

where $N$ is the total number of observations, which equals $nT$ in the case
of a balanced panel.

We then estimate $y_{it}^{*}$ via
$\widehat{y}_{it}^{*}=y_{it}-\widehat{\beta}_{M}m_{it}$ and also construct
“partial” estimates of the productivity proxy function
$\omega_{it}^{*}\left(\beta_{K},\beta_{L}\right)$ in (4.9) as

$\displaystyle\widehat{\omega}_{it}^{*}\left(\beta_{K},\beta_{L}\right)=\underbrace{\Big{[}(1-\widehat{\beta}_{M})m_{it}-\ln(\widehat{\beta_{M}\theta})-\ln(P_{t}^{Y}/P_{t}^{M})\Big{]}}_{\textstyle\widehat{\varkappa}_{it}}-\beta_{K}k_{it}-\beta_{L}l_{it},$

where $\ln(\widehat{\beta_{M}\theta})=\frac{1}{N}\sum_{i}\sum_{t}\ln V_{it}$
on the basis of (4.5). Note that
$\widehat{\omega}_{it}^{*}\left(\beta_{K},\beta_{L}\right)$ still contains two
unknowns which enter the function linearly: $(\beta_{K},\beta_{L})^{\prime}$.
For convenience, let the already identified portion of productivity be denoted
by
$\widehat{\varkappa}_{it}=(1-\widehat{\beta}_{M})m_{it}-\ln(\widehat{\beta_{M}\theta})-\ln(P_{t}^{Y}/P_{t}^{M})$.

With $\widehat{y}_{it}^{*}$ and
$\widehat{\omega}_{it}^{*}\left(\beta_{K},\beta_{L}\right)$ from the first
stage in hand, we proceed to the second-stage estimation of (4.8), where we
approximate unknown function $h(\cdot)$ using polynomial sieves. Specifically,
recognize that
$\widehat{\omega}_{it}^{*}\left(\beta_{K},\beta_{L}\right)=\widehat{\varkappa}_{it}-\beta_{K}k_{it}-\beta_{L}l_{it}$
and let
$\widehat{\mathbf{z}}_{i,t-1}(\boldsymbol{\beta})=([\widehat{\varkappa}_{i,t-1}-\beta_{K}k_{i,t-1}-\beta_{L}l_{i,t-1}],$
$G_{i,t-1},\sum_{j(\neq
i)}s_{ij,t-1}[\widehat{\varkappa}_{j,t-1}-\beta_{K}k_{j,t-1}-\beta_{L}l_{j,t-1}])^{\prime}$
be a $3\times 1$ vector with
$\boldsymbol{\beta}=(\beta_{K},\beta_{L})^{\prime}$. Then, for each
$\widehat{\boldsymbol{z}}(\boldsymbol{\beta})$, we approximate the unknown
function $h\left(\widehat{\boldsymbol{z}}(\boldsymbol{\beta})\right)$ in (4.8)
as follows:

$h\left(\widehat{\boldsymbol{z}}(\boldsymbol{\beta})\right)\approx\mathcal{A}_{L_{n}}\left(\widehat{\boldsymbol{z}}(\boldsymbol{\beta})\right)^{\prime}\boldsymbol{\gamma},$
(5.2)

where
$\mathcal{A}_{L_{n}}\left(\cdot\right)=\left(A_{1}\left(\cdot\right),\dots,A_{L_{n}}\left(\cdot\right)\right)^{\prime}$
is an $L_{n}\times 1$ vector of known basis functions of
$\widehat{\boldsymbol{z}}(\boldsymbol{\beta})$ including a vector of ones,
$\boldsymbol{\gamma}$ is a conformable vector of parameters, and
$L_{n}\to\infty$ slowly with $n$.

Given the orthogonality conditions in (4.10), we estimate $\boldsymbol{\beta}$
and $\boldsymbol{\gamma}$ via nonparametric nonlinear least squares. Letting
$\mathbf{x}_{it}=(k_{it},l_{it})^{\prime}$, the parameter estimators are given
by

$\displaystyle\left(\widehat{\beta}_{K},\widehat{\beta}_{L},\widehat{\boldsymbol{\gamma}}^{\prime}\right)^{\prime}=\operatorname*{argmin}_{\boldsymbol{\beta},\boldsymbol{\gamma}}\
\sum_{i}\sum_{t}\left[\widehat{y}_{it}^{*}-\mathbf{x}_{it}^{\prime}\boldsymbol{\beta}-\mathcal{A}_{L_{n}}\left(\widehat{\mathbf{z}}_{i,t-1}(\boldsymbol{\beta})\right)^{\prime}\boldsymbol{\gamma}\right]^{2},$
(5.3)

where the minimand is the sum of squared errors corresponding to our sieve
estimator of (4.8).

Using the already estimated
$(\widehat{\beta}_{K},\widehat{\beta}_{L},\widehat{\beta}_{M})^{\prime}$ and
$\widehat{\boldsymbol{\gamma}}$, we then readily have the estimators for our
primary estimands of interest:
$\widehat{DL}_{it}\equiv\partial\widehat{h}_{it}(\cdot)/\partial G_{i,t-1}$,
$\widehat{SP}_{it}\equiv\partial\widehat{h}_{it}(\cdot)/\partial\sum_{j(\neq
i)}s_{ij,t-1}\widehat{\omega}_{j,t-1}$ and
$TIL_{it}=\widehat{SP}_{it}\times\sum_{j(\neq
i)}s_{ij,t-1}\widehat{DL}_{j,t-1}$ respectively measuring the direct learning,
cross-firm spillover and total indirect learning effects, where
$\widehat{h}_{it}(\cdot)=\mathcal{A}_{L_{n}}\big{(}\widehat{\mathbf{z}}_{it}(\widehat{\boldsymbol{\beta}})\big{)}^{\prime}\widehat{\boldsymbol{\gamma}}$
and
$\widehat{\omega}_{j,t-1}=\widehat{\varkappa}_{j,t-1}-\widehat{\beta}_{K}k_{j,t-1}-\widehat{\beta}_{L}l_{j,t-1}$.
Lastly, the estimator of latent firm productivity is
$\widehat{\omega}_{it}=y_{it}-\widehat{\beta}_{K}k_{it}-\widehat{\beta}_{L}l_{it}-\widehat{\beta}_{M}m_{it}-\widehat{\eta}_{it}$,
where $\widehat{\eta}_{it}=\ln(\widehat{\beta_{M}\theta})-\ln V_{it}$ from the
first stage.

For the limit results, our sequential estimation methodology can be recast as
a moment-based semiparametric sieve M-estimation problem. Specifically,
letting

$\mathbf{z}_{i,t-1}\left(\beta_{M},\beta_{K},\beta_{L},\theta\right)=\begin{bmatrix}(1-\beta_{M})m_{it-1}-\ln(\beta_{M}\theta)-\ln(P_{t-1}^{Y}/P_{t-1}^{M})-\beta_{K}k_{it-1}-\beta_{L}l_{it-1}\\\
G_{it-1}\\\ \sum_{j(\neq
i)}s_{ijt-1}\left[(1-\beta_{M})m_{jt-1}-\ln(\beta_{M}\theta)-\ln(P_{t-1}^{Y}/P_{t-1}^{M})-\beta_{K}k_{jt-1}-\beta_{L}l_{jt-1}\right]\end{bmatrix}$

and
$r_{it}\left(\beta_{M},\beta_{K},\beta_{L},\theta\right)=y_{it}-\beta_{M}m_{it}-\beta_{K}k_{it}-\beta_{L}l_{it}-\mathcal{A}_{L_{n}}\big{(}\mathbf{z}_{i,t-1}\left(\beta_{M},\beta_{K},\beta_{L},\theta\right)\big{)}^{\prime}\boldsymbol{\gamma}$,
we can rewrite our two estimation stages in the form of the following
multiple-equation moment restrictions:

$\mathbb{E}\begin{bmatrix}\ln V_{it}-\ln(\beta_{M}\theta)\\\
\exp\left\\{\ln(\beta_{M}\theta)-\ln V_{it}\right\\}-\theta\\\
r_{it}\left(\beta_{M},\beta_{K},\beta_{L},\theta\right)\begin{bmatrix}\partial
r_{it}\left(\beta_{M},\beta_{K},\beta_{L},\theta\right)\big{/}\partial\left(\beta_{K},\beta_{L}\right)^{\prime}\\\
-\mathcal{A}_{L_{n}}\big{(}\mathbf{z}_{i,t-1}\left(\beta_{M},\beta_{K},\beta_{L},\theta\right)\big{)}\end{bmatrix}\end{bmatrix}=\mathbf{0}_{4+L_{n}},$
(5.4)

consisting of two blocks, where the first two moments correspond to the
estimator of the material elasticity (first block) and the remaining
orthogonality conditions correspond to the nonlinear sieve least-squares
estimation of the proxied production function and productivity in (5.3).

In the above, $\mathcal{A}_{L_{n}}(\cdot)$ is a sieve approximation of the
unknown infinite-dimensional nonparametric function $h(\cdot)$, and
$\left(\beta_{M},\beta_{K},\beta_{L},\theta\right)^{\prime}$ are the unknown
parameters of fixed dimension. Thus, our estimator falls within Ai & Chen’s
(2003) general framework for the minimum distance estimation based on the
conditional moment restrictions of a generic form
$\mathbb{E}[\rho(X,\delta_{0},g_{0}(\cdot))|Z]=0$, where $X$ and $Z$ are data
and $\rho(\cdot)$ is a vector of “residual functions” with finite-dimensional
unknown parameters $\delta$ and infinite-dimensional unknown functions $g$.
The large-sample limit results from Ai & Chen (2003) and Chen et al. (2003)
therefore extend to our two-step estimator. Inference can be asymptotic or via
bootstrap; we discuss both in detail in Appendix F.

##  6 Simulations

We conduct a set of Monte Carlo experiments. Our data generating process draws
from those used by Grieco et al. (2016) and Gandhi et al. (2020).
Specifically, we consider a balanced panel of $n=\\{100,200,400\\}$ firms
operating during $T=10$ periods.888We have also experimented with 5 and 50
periods. The results are qualitatively unchanged. Each panel is simulated
1,000 times. To simplify matters, we dispense with labor and consider the
production process only with the quasi-fixed dynamic $K_{it}$ and freely-
varying static $M_{it}$. The production technology is

$Y_{it}=K_{it}^{\beta_{K}}M_{it}^{\beta_{M}}\exp\\{\omega_{it}+\eta_{it}\\},$
(6.1)

where we set $\beta_{K}=0.25$ and $\beta_{M}=0.65$, and the noise
$\eta_{it}\sim\text{i.i.d.}\ \mathbb{N}(0,\sigma_{\eta}^{2})$ with
$\sigma_{\eta}=\sqrt{0.07}$.

The firm’s capital is set to evolve according to
$K_{it}=I_{i,t-1}+(1-\delta_{i})K_{i,t-1}$, with the firm-specific
depreciation rates $\delta_{i}\in\\{0.05,0.075,0.10,0.125,0.15\\}$ distributed
uniformly across $i$. The initial levels of capital $K_{i0}$ is drawn from
i.i.d. $\mathbb{U}(10,200)$. The investment function takes the following form:
$I_{i,t-1}=K_{i,t-1}^{\alpha_{1}}\exp\\{\alpha_{2}\omega_{it-1}\\}$, where
$\alpha_{1}=0.8$ and $\alpha_{2}=0.1$.

The materials $M_{it}$ series is generated solving the firm’s restricted
expected profit maximization problem along the lines of (4.2). The conditional
demand for $M_{it}$ is given by

$\displaystyle M_{it}$
$\displaystyle=\operatorname*{argmax}_{\mathcal{M}_{it}}\
\Big{\\{}P_{t}^{Y}K_{it}^{\beta_{K}}\mathcal{M}_{it}^{\beta_{M}}\exp\\{\omega_{it}\\}\theta-
P_{t}^{M}\mathcal{M}_{it}\Big{\\}}=\left(\beta_{M}K_{it}^{\beta_{K}}\exp\\{\omega_{it}\\}\right)^{1/(1-\beta_{M})},$
(6.2)

where, in the second equality, we have normalized $P_{t}^{M}=\theta\ \forall\
t$ and have assumed no temporal variation in output prices: $P^{Y}_{t}=1$ for
all $t$.

We assume that the firm’s productivity modifier $G_{it}$ is autoregressively
persistent. More specifically, we consider two laws of motion for $G_{it}$:
(a) an exogenous autoregressive process whereby
$G_{it}=\gamma_{0}+\gamma_{1}G_{i,t-1}+\epsilon_{it}$ and (b) a controlled
autoregressive process, contemporaneously conditional on the firm’s latent
productivity:
$G_{it}=\gamma_{0}+\gamma_{1}G_{i,t-1}+\gamma_{2}\omega_{it}+\epsilon_{it}$,
where $\gamma_{0}=0.01$, $\gamma_{1}=0.6$, $\gamma_{2}=0.3$ and
$\epsilon_{it}\sim\text{i.i.d.}\ \mathbb{N}(0,\sigma_{\epsilon}^{2})$ with
$\sigma_{\epsilon}=0.1$. Of the two processes, the second one assumes that
more productive firms engage in higher levels of the productivity-modifying
activities. The process (b) permits firms to endogenously update their
$G_{it}$ based on the (observable by them) period $t$ level of their
productivity $\omega_{it}$. For example, if $G_{it}$ measures the firm’s
exposure to investors from abroad, this accommodates the scenario when foreign
investors choose to invest in more productive domestic firms in the first
place.

We let firm productivity $\omega_{it}$ be a linear spatiotemporal first-order
autoregressive process:

$\displaystyle\omega_{it}=\rho_{0}+\rho_{1}\omega_{i,t-1}+\rho_{2}\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}+\rho_{3}G_{i,t-1}+\zeta_{it},$ (6.3)

where, unless stated otherwise, we set $\rho_{0}=0.2$, $\rho_{1}=0.55$,
$\rho_{2}=0.4$ and $\rho_{3}=0.5$. The innovation is generated as
$\zeta_{it}\sim\text{i.i.d.}\ \mathbb{N}(0,\sigma_{\zeta}^{2})$ with
$\sigma_{\zeta}=0.2$. The initial level of productivity
$\omega_{i0}\sim\text{i.i.d.}~{}\mathbb{U}(1,3)$ over $i$. In Appendix G, we
also present the results for a nonlinear specification for $\omega_{it}$.

To keep matters simple, we consider one common spatial region for all firms
and assume that all firms belong to the same industry. Hence, cardinality of
the set $\mathcal{L}(i,t)$ is the same across all $i$ and equals $n-1$. The
peer weights $\\{s_{ijt};\ j(\neq i)\\}$ are constructed according to (3.4)
and, given the setup, are equal to $1/(n-1)$ for all firms and time periods.

#### Proposed Methodology.

First, we evaluate the performance of our proposed estimator with the focus on
its ability to successfully identify productivity spillovers across firms. For
each combination of the $G$ and $\omega$ processes, we consider the following
three DGP scenarios: (i) a general case scenario in which firm productivity is
modified via both the direct learning ($DL_{it}\neq 0$) and cross-firm
spillovers ($SP_{it}\neq 0$); (ii) a special case scenario in which we assume
no direct learning ($DL_{it}=0$ globally) in order to focus our attention
exclusively on the agglomeration-driven learning via spillovers ($SP_{it}\neq
0$); (iii) an even more special case scenario in which firm productivity
evolves exogenously (both $DL_{it}=0$ and $SP_{it}=0$ globally) as
traditionally assumed in the proxy variable production function estimation
literature. The special case scenarios are implemented by setting the
appropriate coefficients in the productivity process (6.3) to zero.

We estimate the model via the two-stage estimation algorithm outlined in
Section 5, where we approximate unknown $h(\cdot)$ using second-degree
polynomial sieves. Table 1 reports the simulation results for our proposed
estimator, when $G_{it}$ evolves exogenously [top panel] and following an
$\omega_{it}$-dependent controlled process [bottom panel]. Each of these two
panels includes the results from the three different scenarios. Reported are
the mean, root mean squared error (RMSE) and mean absolute error (MAE) of the
fixed-parameter $\beta_{K}$ estimates999We omit the results corresponding to
the material elasticity $\beta_{M}$ from the first stage because the estimator
yields very precise estimates of $\beta_{M}$ via (5.1) with the MSE and MAE
being at least as small as $10^{-10}$ owing to the small sampling error
induced by $\eta_{it}$ in our DGPs. We also experimented with much larger
values of $\sigma_{\eta}$ with no significant changes to the qualitative
results. and the averages (across simulation iterations) of these metrics
corresponding for each iteration computed using observation-specific
nonparametric estimates of the autoregressive gradient $AR_{it}=\partial
h(\cdot)/\partial\omega_{i,t-1}$, $DL_{it}=\partial h(\cdot)/\partial
G_{i,t-1}$, $SP_{it}=\partial h(\cdot)/\partial\sum_{j\neq i}\omega_{j,t-1}$
and $TIL_{it}=SP_{it}\times\sum_{j(\neq i)}s_{ij,t-1}DL_{j,t-1}$.

The results in Table 1 are encouraging and show that our methodology recovers
the true parameters remarkably well, thereby lending strong support to the
validity of our identification strategy. As expected of a consistent
estimator, the estimation becomes more stable as $n$ grows. Same is the case
when the productivity DGP in nonlinear (see Table G.1 in Appendix G).

#### Alternative Procedures.

Next, to demonstrate the advantage of our internally consistent methodology,
we inspect the performance of a widely used alternative procedure for
estimating spillovers via a two-step approach. In this case, the unobserved
firm productivity $\omega_{it}$ is first estimated via the standard proxy
variable estimator (which assumes that the productivity process is an
exogenous Markov chain and thus ignores spillovers) and then linearly
regressed on some measure of the firm’s exposure to its peers in the second
step. As already discussed at length, such second-step regressions are
inconsistent with the assumptions made in the first step because they
contradictorily postulate the existence of cross-peer dependence which was
assumed away when recovering firm productivity in the first place.
Consequently, the productivity estimates (by means of the production function)
obtained via such an approach are prone to biases due to the endogeneity-
inducing misspecification of the productivity proxy. The empirical evidence of
spillovers can thus be spurious. This is unsurprising because the unaccounted
cross-sectional dependence is a hindrance to identification in general (see
Pesaran, 2006; Bai, 2009).

The second-step regressions used in spillovers literature have numerous
variations but can be by and large categorized into two distinct types: those
that measure the firm’s exposure to spillovers from peers using the group
means of characteristics which are said to facilitate such spillovers (FDI,
R&D, exports, etc.), and those that measure the firm’s exposure to spillovers
using the peer group mean of an outcome (that is, firm productivity).
Essentially, the first type of regressions focuses on the “contextual effects”
while the second type models cross-peer dependence via “endogenous effects.”
Rarely do researchers allow for both effects at the same time. The first type
is arguably the predominant choice in spillovers literature. Such studies
overwhelmingly estimate linear specifications, and virtually all omit the
temporal lag of the firm’s own productivity in the second-step analysis.

We consider alternative methodologies with the second-step regressions of both
these types. To facilitate a level-playing-field comparison between these and
our models in the ability to identify spillovers, we specify the second-step
regressions in lags. This is to ensure the maximal compatibility of the
second-step regressions with the fashion in which learning and spillovers
occur in the DGP. For concreteness, we run the following two second-step
regressions:

$\displaystyle[\text{ALT1}]\qquad\widehat{\omega}_{it}$
$\displaystyle=\alpha_{11}+\alpha_{12}\sum_{j(\neq
i)}s_{ij,t-1}G_{j,t-2}+\alpha_{13}G_{i,t-1}+e_{1,it},$ (6.4)
$\displaystyle[\text{ALT2}]\qquad\widehat{\omega}_{it}$
$\displaystyle=\alpha_{21}+\alpha_{22}\sum_{j(\neq
i)}s_{ij,t-1}\widehat{\omega}_{j,t-1}+\alpha_{23}G_{i,t-1}+e_{2,it},$ (6.5)

using $\widehat{\omega}_{it}$ recovered in the first step via our
semiparametric production function estimator but assuming an exogenous Markov
process for productivity
$\omega_{it}=\mathbb{E}[\omega_{it}|\omega_{i,t-1}]+\zeta_{it}$.101010Essentially,
firm productivity here is estimated via the semiparametric adaptation of the
original Gandhi et al. (2020) procedure modified to take advantage of the
“known” parametric form of the production technology. Here we also permit the
$DL$ effects as oftentimes done in this literature. Because regressors in both
alternative procedures in (6.4)–(6.5) are all weakly exogenous per our DGP and
the assumptions, these second-step regressions are estimates via least
squares.

To be able to meaningfully analyze the performance of alternative models as
well as to fairly compare them to our methodology (especially, in case of the
popular ALT1 specification), we focus on the estimands that match in terms of
their qualitative interpretations. Instead of looking at specific parameters
that may not always be directly comparable across the models and with the DGP,
we consider the derived measures of $DL$, $SP$ and $TIL$ as
appropriate/available. For instance, of the two alternative methodologies,
only the ALT2 specification postulates cross-firm spillovers via the mean peer
productivity as in our proposed conceptualization in Section 3 and the DGP.
Therefore, $\alpha_{22}$ is essentially comparable to $\rho_{2}$ in the DGP:
both measure the $SP$ effect. This is however not the case with the ALT1
specification which only models the contextual effect. Hence, we cannot
contrast $\alpha_{12}$ to the true $\rho_{2}$ value in the DGP. Having said
that, $\alpha_{12}$ measuring the (twice lagged) total indirect effect of the
peers’ $G$ can indeed be meaningfully compared to the similarly interpretable
$TIL=SP\times DL=\rho_{2}\times\rho_{3}$ effect derived from the DGP that also
occurs over two periods. When it comes to direct learning, both $\alpha_{13}$
and $\alpha_{23}$ are comparable to the true $DL=\rho_{3}$ from the DGP in
(6.3). Tables 2–3 summarize these results.

To examine the ability of alternative models to identify firm productivity, we
first study if they can consistently estimate the production function
coefficients (here $\beta_{K}$) because $\widehat{\omega}_{it}$ is a direct
construct of these parameters:
$\widehat{\omega}_{it}=y_{it}-\widehat{\beta}_{K}k_{it}-\widehat{\beta}_{M}m_{it}-\widehat{\eta}_{it}$.111111The
estimates of $\widehat{\beta}_{M}$ and $\widehat{\eta}_{it}$ are obtained from
the material revenue share regression which does not depend on the Markovian
assumption about $\omega_{it}$. Hence, they are exactly the same as those in
our methodology. The corresponding estimates of $\beta_{K}$ are reported in
Table G.2 in Appendix G. These first-step results apply to both the ALT1 and
ALT2 models and are obtained assuming that $\omega_{it}$ is an exogenous
first-order Markov process. As expected, the estimation of production-function
parameters (and hence, firm productivity) becomes biased with no tangible
improvement following the growth in $n$ as soon as we deviate from the
exogenous productivity process [scenarios (i) and (ii)]. In the latter case,
biases originate from misspecification of the productivity proxy function that
is missing relevant controls pertaining to productivity-modifying learning
and/or spillovers.

As seen in Tables 2–3, the misestimation of productivity feeds into the
second-step regressions. Across all experiments, both the ALT1 and ALT2 models
exhibit non-vanishing biases in the estimation of spillovers. The same is also
generally the case for estimation of within-firm direct learning, with the
exception of the ALT2 estimator in the least probable scenarios when $G$ is an
irrelevant uncorrelated covariate (i.e., when $DL=0$ and $G$ evolves
exogenously in population). Notably and perhaps more importantly, the
alternative estimators fail at identifying (zero) cross-firm spillovers even
when exogeneity of the Markov productivity process assumed in the first step
is true [scenario (iii)]. This is because the second-step regressions remain
misspecified due to their omission of the lagged productivity as customarily
done in spillovers studies. Thus, if the first-step assumption of exogenous
productivity in such analyses is indeed correct, the “evidence” of cross-firm
spillovers uncovered in the second step is likely spurious and effectively
driven by the missing autoregressive dynamics in productivity within the firm.
This is not just a feature of specifications in (6.4)–(6.5). In Appendix G, we
consider their multiple variants drawn from the literature. Those results
provide further evidence of the potential for spurious findings of spillovers
using the popular two-step analysis procedure.

##  7 Empirical Application

We apply our methodology to study cross-firm spillovers with a particular
focus on the productivity effects of inbound FDI via the domestic firms’
learning of more advanced/efficient foreign knowledge. We proxy the firm’s
exposure to foreign knowledge using information on the share of foreign
capital in its equity. This is a standard measure of foreign knowledge
exposure in the literature. Thus, the foreign equity share $G_{it}\in[0,1]$ is
our productivity modifier of interest.

Our objective is to study two potential channels—direct and indirect—of the
productivity-boosting effects of inbound FDI. First, domestic firms may boost
their productivity levels via “importing” better/new technology and learning
more efficient management and marketing practices from abroad that they gain
direct access to through foreign investors; these are direct technology
transfers. A second mechanism by which domestic firms may indirectly improve
their productivity is by learning from other spatially proximate foreign-
invested/owned firms in the industry and then adopting their superior
practices already imported into the country. The latter channel is indirect
and works through cross-firm peer effects. To model these indirect
productivity effects of FDI, we need to explicitly recognize the potential for
cross-sectional dependence in productivity which would permit FDI spillovers
capable of influencing the domestic firms’ productivity levels (and hence
their output) beyond the immediate recipients. Our proposed model in Section 3
readily provides an empirical framework for this analysis. It allows
identification of both the direct/internal ($DL$) and indirect/external
($TIL$) effects of inbound FDI in the presence of non-zero productivity
spillovers ($SP$) among peer firms. In line with Remark 3, we model “FDI
spillovers” as operating through the firm’s exposure to the average peer
productivity, i.e., via “productivity spillovers” due to agglomeration
externalities more broadly.

#### Data.

Our data come from the Chinese Industrial Enterprises Database survey
conducted by China’s National Bureau of Statistics. We focus on the electric
machinery and equipment manufacturing industry, SIC 2-digit code 39. The
rationale behind the choice of this industry is discussed in Appendix B. Our
sample period runs from 1998 to 2007, and the operational sample is an
unbalanced panel of 23,720 firms with a total of 73,095 observations. In
Appendix H, we provide the details of variable construction and describe the
data.

We use postal codes to identify spatial neighbors included in each firm’s peer
group $\mathcal{L}(i,t)$. Peers are defined at the city level and at the level
of the upper administrative division (provinces, autonomous regions,
municipalities under the direct rule of government and special administrative
regions) to allow for a broader geographical extent of spillovers while also
respecting regulatory, administrative and cultural heterogeneity across
regions. For the baseline results, the industrial scope of peer effects is
defined at the level of the whole 2-digit industry. We consider a more
granular definition of industrial similarity at the 4-digit level in
robustness checks.

###  7.1 Results

Owing to the nonparametric specification of the firm productivity process, we
obtain observation-specific heterogeneous estimates of $SP_{it}$, $DL_{it}$
and $TIL_{it}$. We estimate the unknown $h(\cdot)$ via sieve methods using the
popular second-degree polynomial series.121212For instance, see De Loecker et
al. (2016) or Gandhi et al. (2020). We have also experimented with higher-
order polynomials, and the results are very similar except somewhat noisier,
as expected. All estimations include time effects (the quadratic time trend
yields qualitatively similar results). Also note that, because $\omega_{it}$
is the log-productivity, $SP_{it}$ is an elasticity measured in percents per
unit percent of firm productivity, whereas both the $DL_{it}$ and $TIL_{it}$
are semi-elasticities measured in percents per unit percentage point change in
the firm’s foreign equity share. The measured learning effects on productivity
are short-run and partial (i.e., for one given firm only). They do not capture
mutual peer effects of an FDI injection across the network of firms, and
neither do they account for dynamic effects over time. Obviously, owing to the
persistence and cross-peer dependence of productivity, the cumulative
implications of FDI for domestic firms’ productivity in the long run
equilibrium will be more sizable due to accumulation and diffusion over time
and space.

Table 4 summarizes semiparametric point estimates of cross-firm productivity
spillovers along with the direct and indirect effects of FDI on the
productivity of domestic firms from our baseline specification,131313The
associated production function parameter estimates are
$\widehat{\beta}_{M}=0.74$, $\widehat{\beta}_{K}=0.05$ and
$\widehat{\beta}_{L}=0.12$ with the implied scale elasticity of $0.91$. These
are in line with the Cobb-Douglas estimates for Chinese manufacturing reported
in the literature (e.g., see Brandt et al., 2017) and suggest that the
industry exhibits the decreasing returns to scale. in which each firm’s peer
group is restricted to the same province and the industrial scope of
spillovers is defined at the level of the entire 2-digit industry. All
reported estimates are accompanied by the two-tailed 95% bootstrap intervals.
In addition, we formally test for significantly positive productivity effects
at each observation using the one-sided 95% bootstrap lower bounds.
Throughout, we use accelerated bias-corrected bootstrap percentile confidence
intervals (see Appendix F). The share of firms for which the estimates
statistically exceed zero are reported in the last column of Table 4. In
Appendix I, we also summarize these productivity effect estimates graphically
via empirical distributions across firms.

The estimated median $DL$ effect of own FDI is 0.14, whereby an increase of
the foreign share in the median firm’s equity by 10 percentage points boosts
its productivity next year by 1.4%. Expectedly, the $TIL$ effect of peers’ FDI
is smaller in magnitude—0.04 at the median—so a 10 percentage point increase
in the peer group average of the foreign equity share boosts the firm’s
productivity by only 0.4%. Overall, at least 87% of firms enjoy significant
productivity-boosting effects of inbound FDI, both directly and indirectly.

The non-zero external/indirect learning effect of FDI is facilitated by the
presence of substantial and positive cross-firm productivity spillovers in the
industry, with the median spillover elasticity $SP$ estimated at 0.33 along
with the corresponding interquartile range of 0.18–0.45. Thus, a 10%
improvement in the average productivity of the firm’s peers is estimated to
increase the median firm’s own productivity by about 3.3%. These productivity
spillovers are significantly positive for roughly 84% of firms in the
industry. We examine their geographic distribution in Appendix I.

#### Heterogeneity and Nonlinearity.

Even within a given industry, firms are highly heterogeneous across many
dimensions including their productivity and the extent of their exposure to
foreign investors, both direct and through their peers. These characteristics
can influence the effect size of spillovers and learning. Conveniently, our
model readily facilitates testing of that.

Recall that we estimate the productivity effects of interest via
$\widehat{SP}_{it}=\partial\widehat{h}_{it}(\cdot)/\partial\sum_{j(\neq
i)}s_{ij,t-1}{\omega}_{j,t-1}$ and
$\widehat{DL}_{it}=\partial\widehat{h}_{it}(\cdot)/\partial G_{i,t-1}$, where
we estimate $h\left(\cdot\right)$ using the second-order polynomial sieve
approximation. Thus, by analytical derivation, the estimated
$\widehat{SP}_{it}$ and $\widehat{DL}_{it}$ are the linear functions of the
“determinants” of firm productivity $(\omega_{i,t-1},G_{i,t-1},\sum_{j(\neq
i)}s_{ij,t-1}{\omega}_{j,t-1})^{\prime}$. Table 5 reports the estimates of
sieve coefficients on these three variables in the $SP$ and $DL$ functions.

Consider the spillovers first. The coefficient on the firm’s own productivity
is negative, indicating that the spillover effects decline in magnitude as
firms themselves become more productive. Thus, less productive manufacturers
have a greater potential to benefit from positive peer effects. Also
consistent with economic intuition, the effect size of spillovers increases
with the average productivity of peers: there is more to learn from highly
productive neighbors in the industry. We also find a negative relationship
between the firm’s foreign equity share and the effect size of spillovers.
This suggests that the domestic firms experiencing larger productivity
improvements via indirect learning from their foreign-invested peers—thanks to
positive productivity spillovers—are those with limited direct access to
foreign knowledge through their own investors (i.e., low-foreign-equity-share
firms).

In the case of FDI effects on productivity, results in the far right column of
Table 5 suggest that the direct learning effects diminish as the firm’s
productivity rises, implying that the more productive firms have less
absorptive capacity to learn. The foreign share in a firm’s equity negatively
affects the learning effect size, which basically indicates the diminishing
productivity returns to receiving FDI. Lastly, there is evidence that the
higher the average of peer productivity, the lesser the productivity boosts
from FDI. Thus, positive spillovers from highly productive neighbors
essentially diminish the importance of direct FDI effects.

#### Robustness Analysis.

We first assess robustness of our empirical findings of significantly positive
productivity spillovers and learning effects of FDI to the following modeling
choices: (i) the inclusion of peer group effects to control for unobservable
“correlated effects;” (ii) the composition of a reference peer group
$\mathcal{L}(i,t)$; and (iii) the peer-weighting scheme $\\{s_{ijt}\\}$.

As discussed in Section 3, to structurally identify productivity spillovers
$SP$, we rule out unobservable “correlated effects” at the peer group level.
However, we can replace this no-group-effects assumption with a much milder
assumption allowing for network unobservables but having them be time-
invariant. In this case, we can control for the potential network confounders
using group-level fixed effects (see Graham & Hahn, 2005; Bramoullé et al.,
2009). We consider group effects across both the spatial and industrial
dimensions. Specifically, we re-estimate our baseline specification by adding
fixed effects at the level of the entire peer group as well as more granular
subgroups.141414These peer group effects are included in the second-stage
estimation that models the productivity process. The corresponding results are
summarized in columns F1–F4 of Table 6 (see table notes for the details on
group fixed effects). While predictably there is no dramatic change in the
$DL$ estimates of within-firm learning, the median effect size of cross-firm
spillovers $SP$ increases notably when we rely solely on the within-group
variation over time to estimate the productivity peer effects.151515Larger
magnitudes of $TIL$ are a direct result of the increased $SP$ estimates. The
latter is especially true when the correlated group effects are defined
narrowly at the 4-digit sub-industry level. In this case, the median spillover
effect is 0.61 (against the baseline estimate of 0.33) and the effect is
statistically positive for almost all firms (99%). The increase in the effect
size of spillovers is indicative of the substantial between-group
heterogeneity in (peer) firm productivity, which is consistent with the well-
documented differential in productivity levels across regions in China. Thus,
when omitting group-specific effects, the measure of the strength of peer
dependence across firms gets “diluted” in the baseline model due to the
variation across groups.161616This is the case even if the strength of within-
group spillovers is the same for all groups.

In columns P1–P3 of Table 6, we estimate productivity spillovers under three
alternative definitions of who the firm’s relevant peers are. Each one
presumes a much smaller reference group than the baseline. Namely, we consider
narrowing the scope of local spillovers to the level of city and/or the
4-digit sub-industry. The direct effect of the firm’s own FDI expectedly
continues to stay largely unchanged, but the estimates of productivity
spillovers diminish in size significantly. The latter indirectly corroborates
the rationale of our baseline specification in that the agglomeration effects
have broad geographical and industrial scopes. By restricting the extent of
spillovers to the local city and/or the firm’s sub-industry only, we also
restrict the reach of cross-firm externalities in productivity. Intuitively,
when restricting the firm’s learning opportunities to a narrower group of
neighbors, a 10% improvement in the average peer productivity is estimated to
help boost the firm’s own productivity by only about 0.6–0.9% at the median.
In contrast, if the relevant peer reference group is actually larger in scope,
the same 10% improvement across all peers (as in the baseline specification)
implies a bigger industry-wide aggregate effect and, consequently, a larger
estimated spillover effect on the firm of 3.3%.

In Table 6, we also consider an alternative way of weighting peers, whereby
bigger neighbors get assigned larger relative weights (column W1). The
spillover effects only modestly decline in size. Overall, the cross-model
variation in the spillover estimates we observe in Table 6 is unsurprising
and, in fact, expected because each model treats peer interactions a bit
differently and/or utilizes different variation in data to identify
productivity spillovers. Having said that, the $SP$ point estimates across all
models are highly positively correlated, with the rank correlation coefficient
being 0.81 on average. Consistently across all specifications, we continue to
find that the overwhelming majority of the electric machinery manufacturers in
China enjoy positive and significant productivity spillovers, in general, and
FDI spillovers, specifically.

Appendix I contains additional robustness checks, including to the potential
violation of the weak exogeneity of the lagged foreign equity share.
Controlling for potential endogeneity of the FDI exposure, we continue to find
strong evidence in support of significantly positive productivity spillovers
for 80% of firms or more, with our findings remaining qualitatively unchanged.
In the same appendix, we also explore potential heterogeneity in the external
productivity spillovers from the peers conditional on their FDI status, which
gives rise to bidimensional spillovers. We find evidence of heterogeneity in
the strength of spillovers from wholly-domestic versus foreign-invested peers
but, overall, our main findings stay the same: productivity spillovers are
positive and significant for most firms. For the details, see Appendix I.

##  8 Conclusion

This paper develops a novel methodology for the proxy variable structural
identification of (latent) firm productivity in the presence of learning and
cross-firm spillovers which allows a unified one-step analysis of the
knowledge-transfer effects between peer firms. Our framework is fundamentally
different from the popular empirical approach traditionally implemented in two
steps, whereby one first recovers firm productivity using the available
standard proxy variable estimators and then tests for spillovers in the second
step by regressing these productivity estimates on various peer-group averages
capturing firms’ exposure to potential spillovers. Contrary to such an
approach, our methodology is “internally consistent” in that it does not
postulate contradictory assumptions. In building our model, we explicitly
accommodate cross-sectional dependence in firm productivity induced by
spillovers. We also show that estimating the firm production function or
productivity using traditional proxy methods while ignoring the spillover-
induced cross-sectional dependence, as customarily done in the literature,
likely leads to misspecification and endogeneity-generating omitted variable
bias. Because our methodology can be easily adapted to admit various spillover
origins such as spatial agglomeration, R&D, FDI, exporting, etc., it is fit to
investigate cross-firm productivity spillovers in many contexts.

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Table 1: Simulation Results for Our Estimation Methodology | True | $n=100$ | $n=200$ | $n=400$
---|---|---|---|---
| Value | Mean | RMSE | MAE | Mean | RMSE | MAE | Mean | RMSE | MAE
| DGP with $G$ Evolving Exogenously
Scenario (i): $DL\neq 0$ and $SP\neq 0$
$\beta_{K}$ | 0.25 | 0.249 | 0.055 | 0.043 | 0.250 | 0.038 | 0.030 | 0.252 | 0.027 | 0.021
$AR$ | 0.55 | 0.546 | 0.080 | 0.065 | 0.547 | 0.054 | 0.044 | 0.549 | 0.038 | 0.031
$DL$ | 0.50 | 0.500 | 0.164 | 0.134 | 0.502 | 0.113 | 0.091 | 0.501 | 0.079 | 0.064
$SP$ | 0.40 | 0.399 | 0.144 | 0.116 | 0.400 | 0.102 | 0.082 | 0.399 | 0.070 | 0.057
$TIL$ | 0.20 | 0.199 | 0.121 | 0.097 | 0.201 | 0.085 | 0.069 | 0.199 | 0.059 | 0.047
Scenario (ii): $DL=0$ and $SP\neq 0$
$\beta_{K}$ | 0.25 | 0.249 | 0.055 | 0.042 | 0.250 | 0.037 | 0.029 | 0.252 | 0.027 | 0.021
$AR$ | 0.55 | 0.546 | 0.085 | 0.070 | 0.546 | 0.059 | 0.048 | 0.549 | 0.042 | 0.034
$DL$ | 0 | –0.002 | 0.154 | 0.124 | 0.001 | 0.106 | 0.087 | 0.000 | 0.075 | 0.060
$SP$ | 0.40 | 0.395 | 0.172 | 0.140 | 0.402 | 0.124 | 0.102 | 0.400 | 0.085 | 0.069
$TIL$ | 0 | 0.004 | 0.067 | 0.049 | 0.002 | 0.045 | 0.034 | 0.000 | 0.030 | 0.024
Scenario (iii): $DL=0$ and $SP=0$
$\beta_{K}$ | 0.25 | 0.248 | 0.051 | 0.040 | 0.250 | 0.035 | 0.028 | 0.251 | 0.025 | 0.020
$AR$ | 0.55 | 0.546 | 0.083 | 0.067 | 0.546 | 0.058 | 0.047 | 0.549 | 0.040 | 0.033
$DL$ | 0 | 0.003 | 0.155 | 0.125 | 0.003 | 0.106 | 0.086 | 0.001 | 0.075 | 0.061
$SP$ | 0 | –0.031 | 0.157 | 0.131 | –0.009 | 0.108 | 0.091 | –0.004 | 0.078 | 0.065
$TIL$ | 0 | –0.004 | 0.028 | 0.018 | –0.002 | 0.013 | 0.008 | –0.001 | 0.006 | 0.004
| DGP with $G$ Following an $\omega$-Controlled Process
Scenario (i): $DL\neq 0$ and $SP\neq 0$
$\beta_{K}$ | 0.25 | 0.249 | 0.060 | 0.047 | 0.251 | 0.040 | 0.032 | 0.252 | 0.029 | 0.023
$AR$ | 0.55 | 0.544 | 0.110 | 0.089 | 0.546 | 0.073 | 0.060 | 0.548 | 0.053 | 0.043
$DL$ | 0.50 | 0.503 | 0.148 | 0.120 | 0.502 | 0.099 | 0.081 | 0.502 | 0.070 | 0.057
$SP$ | 0.40 | 0.404 | 0.058 | 0.047 | 0.403 | 0.041 | 0.033 | 0.401 | 0.028 | 0.023
$TIL$ | 0.20 | 0.201 | 0.069 | 0.056 | 0.201 | 0.047 | 0.038 | 0.201 | 0.032 | 0.026
Scenario (ii): $DL=0$ and $SP\neq 0$
$\beta_{K}$ | 0.25 | 0.251 | 0.058 | 0.044 | 0.250 | 0.037 | 0.029 | 0.252 | 0.027 | 0.021
$AR$ | 0.55 | 0.544 | 0.105 | 0.087 | 0.545 | 0.070 | 0.058 | 0.548 | 0.051 | 0.042
$DL$ | 0 | 0.004 | 0.134 | 0.108 | 0.003 | 0.092 | 0.076 | 0.002 | 0.067 | 0.055
$SP$ | 0.40 | 0.387 | 0.224 | 0.183 | 0.400 | 0.158 | 0.130 | 0.399 | 0.116 | 0.094
$TIL$ | 0 | –0.007 | 0.060 | 0.043 | –0.003 | 0.038 | 0.028 | –0.001 | 0.027 | 0.021
Scenario (iii): $DL=0$ and $SP=0$
$\beta_{K}$ | 0.25 | 0.248 | 0.052 | 0.041 | 0.250 | 0.035 | 0.028 | 0.251 | 0.025 | 0.019
$AR$ | 0.55 | 0.546 | 0.101 | 0.083 | 0.546 | 0.069 | 0.056 | 0.548 | 0.050 | 0.040
$DL$ | 0 | –0.001 | 0.134 | 0.109 | 0.000 | 0.092 | 0.075 | 0.001 | 0.068 | 0.054
$SP$ | 0 | –0.031 | 0.155 | 0.128 | –0.008 | 0.105 | 0.088 | –0.004 | 0.075 | 0.063
$TIL\qquad$ | 0 | 0.000 | 0.022 | 0.015 | 0.000 | 0.011 | 0.007 | 0.000 | 0.005 | 0.003
Notes: Owing to linearity of the productivity process in (6.3), the true
values of $AR$, $DL$, $SP$ and $TIL$ are the fixed coefficients for all $i$
and $t$, and $TIL=SP\times DL$ is derived indirectly. Throughout, $T=10$.

Table 2: Simulation Results for the Two-Step Alternative Estimator of Spillovers ALT1 | True | $n=100$ | $n=200$ | $n=400$
---|---|---|---|---
| Value | Mean | RMSE | Mean | RMSE | Mean | RMSE
| DGP with $G$ Evolving Exogenously
Scenario (i): $DL\neq 0$ and $SP\neq 0$
$DL$ | 0.50 | 0.167 | 0.355 | 0.171 | 0.340 | 0.173 | 0.333
$TIL$ | 0.20 | –0.577 | 0.793 | –0.578 | 0.787 | –0.577 | 0.781
Scenario (ii): $DL=0$ and $SP\neq 0$
$DL$ | 0 | –0.202 | 0.225 | –0.200 | 0.211 | –0.200 | 0.206
$TIL$ | 0 | –0.348 | 0.372 | –0.345 | 0.361 | –0.352 | 0.357
Scenario (iii): $DL=0$ and $SP=0$
$DL$ | 0 | 0.461 | 0.470 | 0.464 | 0.468 | 0.465 | 0.467
$TIL$ | 0 | 0.165 | 0.204 | 0.170 | 0.191 | 0.169 | 0.180
| DGP with $G$ Following an $\omega$-Controlled Process
Scenario (i): $DL\neq 0$ and $SP\neq 0$
$DL$ | 0.50 | 1.651 | 1.152 | 1.650 | 1.151 | 1.652 | 1.152
$TIL$ | 0.20 | 0.582 | 0.387 | 0.585 | 0.387 | 0.586 | 0.387
Scenario (ii): $DL=0$ and $SP\neq 0$
$DL$ | 0 | 0.416 | 0.420 | 0.420 | 0.422 | 0.424 | 0.425
$TIL$ | 0 | –0.112 | 0.119 | –0.113 | 0.117 | –0.115 | 0.117
Scenario (iii): $DL=0$ and $SP=0$
$DL$ | 0 | 0.616 | 0.619 | 0.617 | 0.619 | 0.621 | 0.621
$TIL\qquad$ | 0 | –0.362 | 0.365 | –0.361 | 0.362 | –0.362 | 0.363
Notes: Reported are the results from the second-step regression in (6.4)
estimated with $\widehat{\omega}_{it}$ obtained in the first step using the
standard proxy estimator under the assumption of exogenous Markov productivity
process. $DL$ and $TIL$ are respectively measured by $\alpha_{13}$ and
$\alpha_{12}$, with the latter capturing spillovers. Throughout, $T=10$.

Table 3: Simulation Results for the Two-Step Alternative Estimator of Spillovers ALT2 | True | $n=100$ | $n=200$ | $n=400$
---|---|---|---|---
| Value | Mean | RMSE | Mean | RMSE | Mean | RMSE
| DGP with $G$ Evolving Exogenously
Scenario (i): $DL\neq 0$ and $SP\neq 0$
$DL$ | 0.50 | 0.800 | 0.313 | 0.814 | 0.320 | 0.817 | 0.320
$SP$ | 0.40 | 1.074 | 0.676 | 1.083 | 0.684 | 1.090 | 0.690
Scenario (ii): $DL=0$ and $SP\neq 0$
$DL$ | 0 | –0.036 | 0.093 | –0.015 | 0.063 | –0.008 | 0.044
$SP$ | 0.40 | 0.751 | 0.518 | 0.874 | 0.502 | 0.924 | 0.527
Scenario (iii): $DL=0$ and $SP=0$
$DL$ | 0 | 0.005 | 0.084 | 0.004 | 0.059 | 0.003 | 0.042
$SP$ | 0 | 0.539 | 0.541 | 0.545 | 0.546 | 0.548 | 0.548
| DGP with $G$ Following an $\omega$-Controlled Process
Scenario (i): $DL\neq 0$ and $SP\neq 0$
$DL$ | 0.50 | 1.149 | 0.650 | 1.152 | 0.652 | 1.156 | 0.656
$SP$ | 0.40 | 0.580 | 0.182 | 0.579 | 0.180 | 0.577 | 0.178
Scenario (ii): $DL=0$ and $SP\neq 0$
$DL$ | 0 | 0.369 | 0.376 | 0.376 | 0.379 | 0.375 | 0.377
$SP$ | 0.40 | –0.708 | 1.304 | –0.635 | 1.121 | –0.583 | 1.020
Scenario (iii): $DL=0$ and $SP=0$
$DL$ | 0 | 0.406 | 0.406 | 0.405 | 0.406 | 0.407 | 0.407
$SP\qquad$ | 0 | 0.390 | 0.393 | 0.395 | 0.396 | 0.397 | 0.397
Notes: Reported are the results from the second-step regression in (6.5)
estimated with $\widehat{\omega}_{it}$ obtained in the first step using the
standard proxy estimator under the assumption of exogenous Markov productivity
process. $DL$ and $SP$ are respectively measured by $\alpha_{23}$ and
$SP=\alpha_{22}$. Throughout, $T=10$.

Table 4: Estimates of the Productivity Effects

Point Estimates Statistically $>0$ Estimand 1st Qu. Median 3rd Qu. (% Obs.)
$SP$ 0.182 0.327 0.452 83.84 (0.108, 0.240) (0.211, 0.399) (0.302, 0.543) $DL$
0.096 0.138 0.168 89.13 (0.076, 0.118) (0.118, 0.164) (0.146, 0.197) $TIL$
0.022 0.037 0.053 86.71 (0.011, 0.032) (0.022, 0.050) (0.034, 0.068) Notes:
Reported are the results based on our baseline specification of the
productivity process in (3.3). The left panel summarizes point estimates of
$SP_{it}$, $DL_{it}$ and $TIL_{it}$ with the corresponding two-sided 95%
bootstrap percentile confidence intervals in parentheses. The last column
reports the share of observations for which the point estimates are
statistically positive at the 5% significance level using one-sided bootstrap
confidence intervals.

Table 5: Heterogeneity and Nonlinearity in the Productivity Effects | $SP$ | $DL$
---|---|---
$\omega_{i,t-1}$ | –0.735 | –0.164
| (–0.890, –0.550) | (–0.203, –0.126)
$G_{i,t-1}$ | –0.198 | –0.155
| (–0.403, –0.103) | (–0.184, –0.127)
$\sum_{j}s_{ij,t-1}\omega_{j,t-1}$ | 1.249 | –0.198
| (0.423, 1.527) | (–0.403, –0.103)
Notes: Reported are the parameter estimates for the $SP$ and $DL$ functions
derived from the polynomial approximation of the conditional mean of
$\omega_{it}$ in the productivity process formulation in (3.3). Two-sided 95%
bootstrap percentile confidence intervals in parentheses. These correspond to
our baseline specification.

Table 6: Estimates of the Productivity Effects: Robustness Analysis

Estimand F1 F2 F3 F4 W1 P1 P2 P3 —Median Estimates— $SP$ 0.499 0.410 0.607
0.605 0.222 0.076 0.085 0.064 (0.386, 0.535) (0.313, 0.428) (0.540, 0.614)
(0.524, 0.647) (0.186, 0.254) (0.048, 0.100) (0.049, 0.122) (0.051, 0.077)
$DL$ 0.167 0.176 0.171 0.177 0.144 0.146 0.145 0.149 (0.145, 0.194) (0.155,
0.205) (0.149, 0.198) (0.156, 0.205) (0.123, 0.171) (0.123, 0.172) (0.121,
0.171) (0.128, 0.174) $TIL$ 0.071 0.060 0.089 0.092 0.009 0.009 0.010 0.008
(0.050, 0.079) (0.046, 0.067) (0.074, 0.100) (0.076, 0.106) (0.006, 0.014)
(0.006, 0.013) (0.006, 0.016) (0.006, 0.011) —Statistically $>0$ (% Obs.)—
$SP$ 95.98 94.19 98.78 99.15 77.96 92.25 89.69 88.89 $DL$ 90.24 90.39 90.34
90.43 89.11 89.43 89.69 89.57 $TIL$ 96.31 94.72 99.07 99.31 72.46 93.11 89.34
94.97 Notes: Reported are the results for the productivity process formulation
in (3.3). The two-sided 95% bootstrap percentile confidence intervals for the
median point estimates are in parentheses. Statistical positiveness is at the
5% significance level, using one-sided bootstrap percentile confidence
intervals. In the baseline specification: (i) each firm’s peers are restricted
to the firms located in the same province and the industrial scope of
spillovers defined at the level of the entire 2-digit industry, (ii) the
technical change is flexibly controlled for using a series of year effects.
The other model specifications are based on the baseline except for the
following alternate features. F1: includes location fixed effects at the level
of peer groups (here, province); F2: includes location fixed effects at the
level of peer subgroups (here, city); F3: includes location-industry fixed
effects at the level of peer subgroups (here, province and 4-digit sub-
industry); F4: includes spatial and industry fixed effects at the level of
peer subgroups (here, city and 4-digit sub-industry); W1: peers are size-
weighted, i.e.,
$s_{ijt}={L_{jt}\mathbbm{1}\big{\\{}(j,t)\in\mathcal{L}(i,t)\big{\\}}}\big{/}{\sum_{k}L_{kt}\mathbbm{1}\big{\\{}(k,t)\in\mathcal{L}(i,t)\big{\\}}}$;
P1: peers in $\mathcal{L}(i,t)$ are restricted to the 4-digit sub-industry;
P2: peers in $\mathcal{L}(i,t)$ are restricted to the same city; P3: peers in
$\mathcal{L}(i,t)$ are restricted to the 4-digit sub-industry and the same
city.

Online Appendix to

On the Estimation of Cross-Firm Productivity
Spillovers with an Application to FDI

Emir Malikov1 Shunan Zhao2

1University of Nevada, Las Vegas

2Oakland University

##  Appendix A Relation to the Augmented Production Function Approach

A two-step framework, which we seek to improve upon in this paper, is not
universal across empirical studies of productivity spillovers. The exceptions
are predominantly from the literature on R&D-borne productivity spillovers,
where some studies instead adopt a singe-step methodology centered on the
estimation of the Griliches (1979)-style “augmented production function”
which, besides the conventional inputs, also explicitly admits the firm’s own
and external knowledge capital stock. Seemingly, such a model readily provides
estimates of the “contextual” spillover effects of R&D on firm production in
one step. However, this framework is rather unique to the studies of
spillovers in R&D, because this productivity-enhancing activity is the most
input-accumulation-like in that it is an investment into the knowledge
capital. Augmenting the firm’s production function to include the
FDI/exports/imports variables and their respective spillover pool measures is
however not as conceptually unambiguous. Specifically, this is problematic on
at least two fronts. First, the spillover effects on firm productivity in such
a setup is essentially assumed to be deterministic, whereby the impact on
productivity is improbably the same for all firms without a possibility of the
varying degree of success (say, due to random luck or misfortune). Second,
including internal and external measures of productivity modifiers directly
into the production function effectively implies substitutability of the
firm’s inputs with not only its own productivity-enhancing activities such FDI
or exporting but also—and perhaps more eyebrow-raising—with those of its
peers. This remark equally applies to the case of R&D spillovers111More
recently, the literature has been gravitating towards embedding the firm’s R&D
behavior into the productivity process and taking it out of the production
function itself; e.g., see Doraszelski & Jaumandreu (2013, 2018). and is along
the lines of De Loecker’s (2013) critique in the context of estimating the
learning-by-exporting effects on firm productivity. Not least importantly,
identification of productivity spillovers in prior studies (including those
via R&D) may also be seriously hindered by the well-known econometric problems
with standard proxy-based or (dynamic panel) fixed-effects production function
estimators.222See Griliches & Mairesse (1998), Ackerberg et al. (2015) and
Gandhi et al. (2020) for the discussion of various production-function
estimators and identification challenges associated with them. This further
highlights the practical usefulness of our proposed methodology.

##  Appendix B China’s Electric Machinery Manufacturing

Our empirical analysis focuses on China’s electric machinery and equipment
manufacturing industry which includes manufacturing of generators and motors,
power transmission and distribution equipment, wires and cables, batteries,
household electric and non-electric appliances, lighting appliances, etc. We
select this industry because it is histrionically one of the country’s most
fundamental manufacturing sectors. The development of this industry has been
closely related to the growth of GDP and the ever-expanding demand of
electricity. By its very nature, the industry has thus been crucial for
promoting the overall industrialization in China. Besides that, electric
machinery and equipment is also China’s most exported product (Euro Exim Bank,
2020), and the industry amounts to over a quarter of global sales (Deloitte
China Manufacturing Industry Group, 2013). It is also one of the manufacturing
industries that receive most of FDI. For instance, in 2005 alone (near the end
of our sample period) the machinery and equipment industry in China attracted
$4 billion in foreign investment, which was about 10% of FDI inflows to
Chinese manufacturing that year (Ihrcke & Becker, 2006).

Foreign-invested firms are the dominant players in this industry. Arguably,
this is mainly due to China’s lack of domestic innovation capabilities,
excessive failure rates of R&D and, consequently, high dependence on new
technologies imported from abroad, particularly during the first decade
following renewed privatization efforts in the late 1990s (the period of our
analysis). For example, according to the Xiamen Bureau of Statistics, in
Xiamen (a large and important port-city on the East coast) just 48 large
foreign-invested firms owned 82% of fixed assets and produced 79% of the
output value in the local electric machinery industry in 2005.

More generally, the Chinese electric machinery and equipment manufacturing
industry is characterized by a high degree of spatial clustering and
industrial agglomeration [mainly on the coast; see Figure H.2(a) in Appendix
H] typical for such technology- and skill-intensive industries. Along with the
government’s emphasis on innovations and new technologies as a means for
sustainable development of this industry, this makes it an interesting
application for studying productivity effects of inbound FDI and the
associated spillovers across firms.

##  Appendix C Translog Production Function

Our methodology can adapt more flexible specifications of the firm’s
production function. The log-quadratic translog specification provides a
natural extension of the log-linear Cobb-Douglas form that we have assumed in
(3.1). The former is more flexible and implies input and scale elasticities
that vary both over time and across firms thereby being more robust to firm
heterogeneity. For instance, see De Loecker & Warzynski (2012) and De Loecker
et al. (2016) for recent applications of the translog production functions in
the structural proxy estimation.

Let the firm’s stochastic production function takes the following form in
logs:

$\displaystyle y_{it}=$ $\displaystyle\
\beta_{0}+\beta_{K}k_{it}+\tfrac{1}{2}\beta_{KK}k_{it}^{2}+\beta_{L}l_{it}+\tfrac{1}{2}\beta_{LL}l_{it}^{2}+\beta_{M}m_{it}+\tfrac{1}{2}\beta_{MM}m_{it}^{2}\
+$ $\displaystyle\
\beta_{KL}k_{it}l_{it}+\beta_{KM}k_{it}m_{it}+\beta_{LM}l_{it}m_{it}+\omega_{it}+\eta_{it}$
$\displaystyle\equiv$ $\displaystyle\
T(k_{it},l_{it},m_{it})+\omega_{it}+\eta_{it},$ (C.1)

where $T(k_{it},l_{it},m_{it})$ is a shorthand for the translog expansion of
inputs. All the remaining assumptions about the market environment,
productivity processes, timing of production decisions and learning, etc. stay
unchanged.

The firm’s static optimization problem with respect to materials now is

$\displaystyle\max_{M_{it}}\
P_{t}^{Y}\exp\\{T(k_{it},l_{it},m_{it})\\}\exp\\{\omega_{it}\\}\theta-
P_{t}^{M}M_{it},$ (C.2)

with the corresponding first-order condition given by

$P_{t}^{Y}\exp\\{T(k_{it},l_{it},m_{it})\\}\frac{\beta_{M}+\beta_{MM}m_{it}+\beta_{KM}k_{it}+\beta_{LM}l_{it}}{M_{it}}\exp\\{\omega_{it}\\}\theta=P_{t}^{M}.$
(C.3)

Dividing (C.3) by the translog production function expressed in levels and
then taking logs of both sides, we obtain the following material share
equation:

$\ln
V_{it}=\ln(\left[\beta_{M}+\beta_{MM}m_{it}+\beta_{KM}k_{it}+\beta_{LM}l_{it}\right]\theta)-\eta_{it},$
(C.4)

where $\beta_{M}+\beta_{MM}m_{it}+\beta_{KM}k_{it}+\beta_{LM}l_{it}$ is the
material elasticity function. Analogous to the discussion in Section 4, the
above share equation identifies the material-related production-function
parameters $(\beta_{M},\beta_{MM},\beta_{KM},\beta_{LM})^{\prime}$ as well as
the mean of exponentiated shocks $\theta=\mathbb{E}[\exp\\{\eta_{it}\\}]$
based on the mean-orthogonality condition $\mathbb{E}[\eta_{it}|\
\mathcal{I}_{it}]=\mathbb{E}[\eta_{it}]=0$. These parameters are to be
estimated in the first stage via nonlinear least squares on (C.4).

Having identified the production function in the dimension of its endogenous
static input $m_{it}$, we focus on the remaining production-function
parameters as well as the nonparametric evolution process for $\omega_{it}$.
With the already identified $y_{it}^{*}\equiv
y_{it}-\beta_{M}m_{it}-\tfrac{1}{2}\beta_{MM}m_{it}^{2}-\beta_{KM}k_{it}m_{it}-\beta_{LM}l_{it}m_{it}$
and using the Markovian process for productivity, we now have the analogue of
(4.7):

$y_{it}^{*}=\beta_{K}k_{it}+\tfrac{1}{2}\beta_{KK}k_{it}^{2}+\beta_{L}l_{it}+\tfrac{1}{2}\beta_{LL}l_{it}^{2}+\beta_{KL}k_{it}l_{it}+h\left(\omega_{i,t-1},G_{i,t-1},\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}\right)+\zeta_{it}+\eta_{it}$ (C.5)

that contains no endogenous variables on the right-hand side. Next, proxying
for $\omega_{i,t-1}$ and $\omega_{j,t-1}$ via the inverted material function
derived from (C.3), we obtain

$\displaystyle y_{it}^{*}=$ $\displaystyle\
\beta_{K}k_{it}+\tfrac{1}{2}\beta_{KK}k_{it}^{2}+\beta_{L}l_{it}+\tfrac{1}{2}\beta_{LL}l_{it}^{2}+\beta_{KL}k_{it}l_{it}\
+$ (C.6) $\displaystyle\
h\Bigg{(}\omega_{i,t-1}^{*}\left(\beta_{K},\beta_{L},\beta_{KK},\beta_{LL},\beta_{KL}\right),G_{i,t-1},\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}^{*}\left(\beta_{K},\beta_{L},\beta_{KK},\beta_{LL},\beta_{KL}\right)\Bigg{)}+\zeta_{it}+\eta_{it},$

where the productivity proxy function is given by

$\displaystyle\omega_{it}^{*}\left(\beta_{K},\beta_{L},\beta_{KK},\beta_{LL},\beta_{KL}\right)=\varkappa_{it}^{*}-\beta_{K}k_{it}-\tfrac{1}{2}\beta_{KK}k_{it}^{2}-\beta_{L}l_{it}-\tfrac{1}{2}\beta_{LL}l_{it}^{2}-\beta_{KL}k_{it}l_{it}\quad\forall
i,t,$ (C.7)

with

$\displaystyle\varkappa_{it}^{*}=$ $\displaystyle\
\ln(P_{t}^{M}/P_{t}^{Y})-\ln(\left[\beta_{M}+\beta_{MM}m_{it}+\beta_{KM}k_{it}+\beta_{LM}l_{it}\right]\theta)\
-$ $\displaystyle\
(1-\beta_{M})m_{it}-\tfrac{1}{2}\beta_{MM}m_{it}^{2}-\beta_{KM}k_{it}m_{it}-\beta_{LM}l_{it}m_{it}$

being a function of the parameters that have already been identified in the
first stage.

A semiparametric model in (C.6) is then identified based on the same moment
restriction as in (4.10), with all right-hand-side covariates being weakly
exogenous and thus self-instrumenting. Approximating the unknown $h(\cdot)$
via linear sieves, (C.6) is to be estimated in the second stage via
semiparametric nonlinear least-squares. The remaining aspects closely follow
the estimation procedure outlined in Section 5.

##  Appendix D Asymmetric Productivity Spillovers

Our baseline peer weighing scheme in (3.4) treats cross-firm spillovers
symmetrically in that all members of a peer group affect each other’s
productivity. That is, each $i$th firm’s productivity is influenced by the
average productivity of all its peers: those that are more and those that are
less productive than the firm $i$ itself. Given that we have no prior beliefs
about the directionality of productivity spillovers in China’s electric
machinery manufacturing that we study in our empirical application, we opt for
a symmetric specification. But should one choose to regulate the direction of
productivity spillovers by restricting them to occur from more productive to
less productive firms, our framework can be modified to accommodate that too.

The latter case however implies a somewhat different conceptualization of
cross-firm dependence in which firms are said to learn exclusively from
(relative) productivity “leaders.” The identification of such asymmetric
spillovers, which are conditional on the firm’s own productivity relative to
that of its peers, generally requires additional structural/timing
assumptions.

To model productivity spillovers between firms asymmetrically, we can redefine
peer weights $\\{s_{ijt}\\}$ as follows:

$s_{ijt}^{*}=\frac{\mathbbm{1}\big{\\{}(j,t)\in\mathcal{L}(i,t)\ \text{and}\
\omega_{j,t-1}>\omega_{i,t-1}\big{\\}}}{\sum_{k(\neq
i)=1}^{n}\mathbbm{1}\big{\\{}(k,t)\in\mathcal{L}(i,t)\ \text{and}\
\omega_{j,t-1}>\omega_{i,t-1}\big{\\}}},$ (D.1)

so that only the neighbors who are more productive than the firm $i$ are
identified as its peers for external cross-firm learning. Note that, in the
above, the relevant peers at time $t$ are selected based on their relative
productivity superiority in the previous period $t-1$. Without this, we would
not be able to separate the cross-firm spillover effect from the firm’s own
autoregressive effect, conflating the two. The latter becomes obvious when we
substitute (D.1) into the Markov productivity process (3.3) that describes the
evolution of firm $i$’s productivity over time:

$\omega_{it}=\mathbb{E}\Bigg{[}\omega_{it}\Bigg{|}\
\omega_{i,t-1},G_{i,t-1},\sum_{j(\neq
i)}\underbrace{\frac{\mathbbm{1}\big{\\{}(j,t-1)\in\mathcal{L}(i,t-1)\
\text{and}\ \omega_{j,t-2}>\omega_{i,t-2}\big{\\}}}{\sum_{k(\neq
i)=1}^{n}\mathbbm{1}\big{\\{}(k,t-1)\in\mathcal{L}(i,t-1)\ \text{and}\
\omega_{j,t-2}>\omega_{i,t-2}\big{\\}}}}_{s_{ij,t-1}^{*}}\omega_{j,t-1}\Bigg{]}+\zeta_{it}.$
(D.2)

By making the asymmetry in external learning be a function of the twice-lagged
pair-wise productivity differentials between the firm and its peers, we avoid
the appearance of $\omega_{i,t-1}$ in two places thereby allowing us to
partial out the cross-firm spillovers from the autoregressive persistence in
productivity. Thus, to separably identify asymmetric spillovers in
productivity, in addition to assuming that (both the internal and external)
learning occurs with a delay, one also requires an assumption that the firm
takes an additional period to identify more productive peers. However, we do
not need this additional timing assumption in our baseline analysis (with
symmetric interactions).

##  Appendix E Additional Modeling Considerations

#### Contextual Effects.

Because we assume delayed cross-firm peer interactions, as noted by Manski
(1993), the dynamic nature of productivity model (3.3) can potentially provide
an additional avenue to circumvent the unidentification problems and separate
different types of peer effects, should one be interested in also modeling the
“contextual effects” on productivity via $\sum_{j(\neq
i)}s_{ij,t-1}G_{j,t-1}$. In such a setup, per the results in Bramoullé et al.
(2009), the separable identification of “endogenous” and “contextual” effects
can also be achieved by relying on variation in the size of peer reference
groups, so long as the firm $i$ is excluded when computing group means, as is
in our case. Alternatively, (co)variance-based quadratic moment conditions may
be used to aid identification (Kelejian & Prucha, 1999; Lee, 2007; Kuersteiner
& Prucha, 2020).

#### Contemporaneous Effects.

Depending on the particular source of learning, it may sometimes be possible
to reasonably relax the timing assumption that the learning effect of $G_{it}$
on firm productivity be with a delay. Take, for example, the firm’s export
status in the context of “learning by exporting.” Consistent with much
theoretical and empirical work in international trade, the decision to start
exporting is usually associated with large sunk entry costs, which would
impede firms from adjusting their export status immediately after experiencing