an improvement in their productivity. Analogous arguments can be made about
costliness of swift geographic relocations. If so, it may be feasible to
replace weak exogeneity of lagged $G_{i,t-1}$ and $\\{s_{ij,t-1}\\}$ with a
stronger assumption of weak exogeneity of $G_{it}$ and $\\{s_{ijt}\\}$. The
productivity process in (3.3) can then be modified as follows:
$\omega_{it}=\mathbb{E}\left[\omega_{it}|\ \omega_{i,t-1},G_{it},\sum_{j(\neq
i)}s_{ijt}\omega_{j,t-1}\right]+\zeta_{it}$, where the implied mean-
orthogonality of $\zeta_{it}$ and $(G_{it},\sum_{j(\neq
i)}s_{ijt}\omega_{j,t-1})^{\prime}$ is effectively paramount to assuming that,
due to adjustment costs, both the $G_{it}$ and firm location in period $t$ are
determined in period $t-1$ based on $\omega_{i,t-1}$ just like the dynamic
inputs are.

##  Appendix F Inference

#### Asymptotic Inference.

Let the moment vector in (5.4) be concisely written as
$\mathbb{E}[\boldsymbol{\rho}({\Theta})]=\mathbf{0}$, where $\Theta$ is a
collection of both the finite-dimensional coefficients
$\left(\beta_{M},\beta_{K},\beta_{L},\theta\right)^{\prime}$ and nonparametric
sieve “parameters” $\boldsymbol{\gamma}$. Given the just-identification of the
model and so long as we use linear sieves for
$\mathcal{A}_{L_{n}}\left(\cdot\right)$ such as polynomial or B-spline series,
we can make use of the numerical equivalence (see Hahn et al., 2018) between
the consistent estimator of the asymptotic variance of semiparametric
“parameters” $\widehat{{\Theta}}$ and a consistent estimator of the asymptotic
variance derived for these parameter estimators as if the estimated model were
of a parametric form specified in (5.1) and (5.3). Thus, in practice, one can
use the variance formula for a parametric two-step estimator to consistently
estimate the variance of a semiparametric sieve estimator.333Note that this
equivalence applies to finite samples only because, asymptotically, the number
of sieve “parameters” will diverge to infinity with the sample size whereas
the number of parameters in a parametric specification will stay a finite
constant. Furthermore, the numerical equivalence holds more generally for
fully nonparametric two-step sieve estimators. In our case, the estimator is
semiparametric, with the first step implemented using the known parametric
form. Since ours is a special case of the nonparametric setup studied by Hahn
et al. (2018), their results continue to apply. The asymptotic variance for
such a parametric two-step estimator can in turn be derived following Newey’s
(1984) suggestion by making use of the optimal GMM covariance formula:
$\mathbb{V}ar\big{[}\widehat{{\Theta}}\big{]}=\big{[}\mathbb{E}\frac{\partial\boldsymbol{\rho}({\Theta})}{\partial{\Theta}^{\prime}}\big{]}^{-1}\mathbb{E}[\boldsymbol{\rho}({\Theta})\boldsymbol{\rho}({\Theta})^{\prime}]\big{[}\mathbb{E}\frac{\partial\boldsymbol{\rho}({\Theta})}{\partial{\Theta}}\big{]}^{-1}$.
This streamlines asymptotic inference.

#### Bias-Corrected Bootstrap Inference.

However, because asymptotic inference for semi- and nonparametric estimators
is well-known to perform unreliably due to finite-sample biases as well as the
first-order asymptotic theory’s poor ability to approximate the distribution
of estimators in finite samples (Horowitz, 2001), for hypothesis testing, we
therefore rely on Efron’s (1987) accelerated bias-corrected bootstrap
percentile confidence intervals, which are second-order accurate and provide
means not only to correct for the estimator’s finite-sample bias but also to
account for higher-order moments (particularly, skewness) in the sampling
distribution.

We approximate sampling distributions of the estimator via wild residual block
bootstrap that takes into account a panel structure of the data, with both
stages resampled jointly owing to a sequential nature of our estimation
procedure. More specifically, when constructing wild bootstrap residuals, we
work with the joint distribution of firm-specific time series of
$\\{\widehat{\eta}_{it}\\}$ and $\\{\widehat{\zeta}_{it}\\}$, with the
auxiliary random variable drawn from the Mammen (1993) two-point distribution
independently over $i$. Note that this independence over $i$ is consistent
with our model’s assumption about random productivity shocks. We set the
number of bootstrap replications to $B=400$. Having first obtained bootstrap
parameter estimates
$\\{(\widehat{\beta}_{K}^{b},\widehat{\beta}_{L}^{b},\widehat{\beta}_{M}^{b})^{\prime};\
b=1,\dots,B\\}$ and $\\{\widehat{\boldsymbol{\gamma}}^{b};\ b=1,\dots,B\\}$,
we then obtain bootstrap values for our main estimands of interest:
$\widehat{DL}_{it}^{b}$, $\widehat{SP}_{it}^{b}$ and $\widehat{TIL}_{it}^{b}$
for $b=1,\dots,B$ (at each observation). Next, we use the accelerated bias-
correction method to make inference about $DL$, $SP$ and $TIL$.

To make matters concrete, let the (observation-specific) estimand of focus be
denoted by $\widehat{E}$. We use the empirical distribution of $B$ bootstrap
estimates $\big{\\{}\widehat{E}^{1},\dots,\widehat{E}^{B}\big{\\}}$ to
estimate $(1-a)\times 100$% confidence bounds for $\widehat{E}$ as intervals
between the $[a_{1}\times 100]$th and $[a_{2}\times 100]$th percentiles of its
bootstrap distribution with

$\displaystyle
a_{1}=\Phi\left(\widehat{\phi}_{0}+\frac{\widehat{\phi}_{0}+\phi_{a/2}}{1-\widehat{c}\left(\widehat{\phi}_{0}+\phi_{a/2}\right)}\right)\quad\text{and}\quad
a_{2}=\Phi\left(\widehat{\phi}_{0}+\frac{\widehat{\phi}_{0}+\phi_{(1-a/2)}}{1-\widehat{c}\left(\widehat{\phi}_{0}+\phi_{(1-a/2)}\right)}\right),$
(F.1)

where $\Phi(\cdot)$ is the standard normal cdf, $\phi_{\alpha}$ is the
$(\alpha\times 100)$th percentile of the standard normal distribution,

$\displaystyle\widehat{\phi}_{0}=\Phi^{-1}\left(\\#\big{\\{}\widehat{E}^{b}<\widehat{E}\big{\\}}/B\right)$
(F.2)

is a bias-correction factor, and $\widehat{c}$ is an acceleration parameter
which, following the literature, is estimated via jackknife as follows (e.g.,
see Shao & Tu, 1995):

$\displaystyle\widehat{c}=\frac{\sum_{j=1}^{J}\left(\sum_{s=1}^{J}\widehat{E}^{s}-\widehat{E}^{j}\right)^{3}}{6\Big{[}\sum_{j=1}^{J}\left(\sum_{s=1}^{J}\widehat{E}^{s}-\widehat{E}^{j}\right)^{2}\Big{]}^{3/2}},$
(F.3)

where $\widehat{E}^{j}$ is the $j(=1,\dots,J)$th jackknife estimate of
$E$.444We have tried different versions of jackknife with similar results. We
settle on a delete-$50T$ jackknife (i.e., leave-$50$-cross-sections-out) which
respects the panel structure of our data while yielding a reasonable number of
subsamples the estimation on which is not computationally prohibitive.

Note that both the acceleration and bias-correction factors are different for
each estimator, denoted here generically by $\widehat{E}$. That is, the bias-
correction procedure is not only estimand-specific but may also be
observation-specific as is in our case. Also, the estimated confidence
intervals may not contain the original estimates if the finite-sample bias is
large.

##  Appendix G Additional Simulation Results

Table G.1: Simulation Results for Our Estimator under the Nonlinear Productivity Process | Mean 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 $SP\neq 0$
$\beta_{K}$ | 0.250 | 0.237 | 0.077 | 0.059 | 0.247 | 0.046 | 0.035 | 0.251 | 0.031 | 0.024
$AR$ | 0.596 | 0.593 | 0.081 | 0.066 | 0.592 | 0.054 | 0.044 | 0.594 | 0.038 | 0.031
$DL$ | 0.412 | 0.416 | 0.157 | 0.128 | 0.418 | 0.106 | 0.087 | 0.414 | 0.075 | 0.061
$SP$ | 0.301 | 0.274 | 0.313 | 0.257 | 0.303 | 0.200 | 0.166 | 0.302 | 0.138 | 0.115
$TIL$ | 0.124 | 0.102 | 0.144 | 0.112 | 0.119 | 0.085 | 0.069 | 0.120 | 0.059 | 0.048
Scenario (ii): $DL=0$ and $SP\neq 0$
$\beta_{K}$ | 0.250 | 0.233 | 0.082 | 0.063 | 0.247 | 0.049 | 0.037 | 0.251 | 0.033 | 0.025
$AR$ | 0.609 | 0.606 | 0.081 | 0.067 | 0.606 | 0.055 | 0.045 | 0.608 | 0.039 | 0.032
$DL$ | 0 | 0.003 | 0.158 | 0.127 | 0.003 | 0.107 | 0.087 | 0.001 | 0.076 | 0.061
$SP$ | 0.274 | 0.088 | 0.301 | 0.246 | 0.222 | 0.198 | 0.162 | 0.257 | 0.130 | 0.107
$TIL$ | 0 | –0.011 | 0.075 | 0.049 | –0.005 | 0.038 | 0.026 | –0.002 | 0.023 | 0.017
Scenario (iii): $DL=0$ and $SP=0$
$\beta_{K}$ | 0.250 | 0.246 | 0.064 | 0.051 | 0.249 | 0.044 | 0.035 | 0.251 | 0.032 | 0.025
$AR$ | 0.672 | 0.623 | 0.078 | 0.064 | 0.623 | 0.054 | 0.044 | 0.626 | 0.038 | 0.031
$DL$ | 0 | 0.003 | 0.155 | 0.126 | 0.003 | 0.106 | 0.087 | 0.001 | 0.075 | 0.061
$SP$ | 0 | –0.034 | 0.152 | 0.126 | –0.010 | 0.106 | 0.086 | –0.004 | 0.073 | 0.060
$TIL$ | 0 | –0.004 | 0.027 | 0.017 | –0.002 | 0.012 | 0.008 | –0.001 | 0.006 | 0.004
| DGP with $G$ Following an $\omega$-Controlled Process
Scenario (i): $DL\neq 0$ and $SP\neq 0$
$\beta_{K}$ | 0.250 | 0.249 | 0.028 | 0.022 | 0.250 | 0.020 | 0.015 | 0.251 | 0.014 | 0.011
$AR$ | 0.338 | 0.328 | 0.078 | 0.060 | 0.334 | 0.054 | 0.043 | 0.338 | 0.038 | 0.030
$DL$ | 1.148 | 1.157 | 0.119 | 0.093 | 0.152 | 0.083 | 0.064 | 0.148 | 0.058 | 0.046
$SP$ | 0.695 | 0.697 | 0.036 | 0.029 | 0.697 | 0.025 | 0.021 | 0.695 | 0.017 | 0.014
$TIL$ | 0.798 | 0.808 | 0.294 | 0.166 | 0.803 | 0.207 | 0.119 | 0.798 | 0.146 | 0.082
Scenario (ii): $DL=0$ and $SP\neq 0$
$\beta_{K}$ | 0.250 | 0.233 | 0.086 | 0.065 | 0.245 | 0.053 | 0.039 | 0.251 | 0.033 | 0.025
$AR$ | 0.609 | 0.609 | 0.102 | 0.084 | 0.607 | 0.068 | 0.055 | 0.608 | 0.050 | 0.040
$DL$ | 0 | –0.006 | 0.135 | 0.109 | –0.002 | 0.092 | 0.075 | 0.000 | 0.066 | 0.054
$SP$ | 0.274 | 0.085 | 0.341 | 0.277 | 0.207 | 0.215 | 0.175 | 0.259 | 0.152 | 0.124
$TIL$ | 0 | 0.023 | 0.069 | 0.045 | 0.005 | 0.034 | 0.023 | 0.003 | 0.021 | 0.015
Scenario (iii): $DL=0$ and $SP=0$
$\beta_{K}$ | 0.250 | 0.246 | 0.068 | 0.053 | 0.249 | 0.044 | 0.035 | 0.251 | 0.031 | 0.025
$AR$ | 0.627 | 0.624 | 0.100 | 0.082 | 0.623 | 0.067 | 0.054 | 0.626 | 0.049 | 0.040
$DL$ | 0 | –0.001 | 0.134 | 0.108 | 0.001 | 0.092 | 0.075 | 0.001 | 0.066 | 0.054
$SP$ | 0 | –0.035 | 0.147 | 0.122 | –0.009 | 0.101 | 0.084 | –0.004 | 0.071 | 0.060
$TIL\qquad$ | 0 | 0.002 | 0.021 | 0.014 | 0.000 | 0.010 | 0.007 | 0.000 | 0.005 | 0.003
Notes: Owing to nonlinearity of the productivity process in (G), $AR$, $DL$,
$SP$ and $TIL$ are all observation-specific. With the sole exception for the
fixed parameter $\beta_{K}=0.25$, the mean true values are the averages
(across simulation repetitions) of the mean simulated values over $i$ and $t$.
Throughout, $T=10$.

#### Nonlinear Productivity Process.

Table G.1 presents the results for our proposed estimator when the
productivity DGP is nonlinear. Specifically, we consider the following
nonlinear productivity process:

$\displaystyle\omega_{it}=$ $\displaystyle\
\rho_{0}+\rho_{11}\omega_{i,t-1}+\rho_{12}\omega_{i,t-1}^{2}+\rho_{21}\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}+\rho_{22}\Bigg{(}\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}\Bigg{)}^{2}\ +$ $\displaystyle\
\rho_{31}G_{i,t-1}+\rho_{32}G_{i,t-1}^{2}+\varrho_{12}\omega_{i,t-1}\Bigg{(}\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}\Bigg{)}+\varrho_{13}\omega_{i,t-1}G_{i,t-1}\ +$
$\displaystyle\ \lambda_{23}G_{i,t-1}\Bigg{(}\sum_{j(\neq
i)}s_{ij,t-1}\omega_{j,t-1}\Bigg{)}+\zeta_{it},$ (G.1)

where $\rho_{0}=0.2$, $\rho_{11}=0.65$, $\rho_{12}=-0.015$, $\rho_{21}=0.18$,
$\rho_{22}=0.025$, $\rho_{31}=0.37$, $\rho_{32}=0.12$, $\varrho_{12}=0.006$,
$\varrho_{13}=-0.06$ and $\lambda_{23}=0.07$. The rest of the DGP is kept
unchanged (see Section 6).

Table G.1 essentially replicates Table 1 using this new productivity process.
The simulation results remain encouraging and show that our estimation
methodology is consistent and recovers the true parameters well.

Table G.2: Simulation Results for the Alternative Estimator of $\beta_{K}$ | 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 $SP\neq 0$ | 0.25 | 0.376 | 0.136 | 0.373 | 0.128 | 0.374 | 0.126
Scenario (ii): $DL=0$ and $SP\neq 0$ | 0.25 | 0.438 | 0.193 | 0.435 | 0.188 | 0.436 | 0.187
Scenario (iii): $DL=0$ and $SP=0$ | 0.25 | 0.251 | 0.040 | 0.250 | 0.029 | 0.251 | 0.020
| DGP with $G$ Following an $\omega$-Controlled Process
Scenario (i): $DL\neq 0$ and $SP\neq 0$ | 0.25 | 0.118 | 0.152 | 0.113 | 0.148 | 0.109 | 0.146
Notes: Reported are the first-step results for $\widehat{\beta}_{K}$ from the
alternative estimators which proxy for latent productivity under the
assumption of exogenous Markov process for $\omega_{it}$. The results
corresponding to scenarios (ii) and (iii) of the second DGP [bottom panel] are
omitted because they are identical to those for the first DGP [top panel].
This is because not only does $G$ not enter the alternative estimator but it
also does not affect the evolution of firm productivity by design ($DL=0$) in
these two scenarios. Throughout, $T=10$.

#### First-Step Estimates of $\beta_{K}$ from the Two-Step Estimator.

To examine the ability of alternative models to identify firm productivity, we
first study if these two-step estimators can consistently estimate the
production function coefficients (here $\beta_{K}$) because
$\widehat{\omega}_{it}$ is a direct construct of these parameters. The
corresponding estimates of $\beta_{K}$ are reported in Table G.2. 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.

Table G.3: Simulation Results for Different Variants of the Two-Step
Alternative Estimator

Results for $\alpha_{12}$ ($TIL$) in eq. (6.4) Results for $\alpha_{22}$
($SP$) in eq. (6.5) I II III IV V VI VII VIII IX X XI Mean Estimate $n=100$
1.154 0.558 1.057 0.620 –0.359 0.378 –0.362 0.990 0.591 0.540 0.390 $n=200$
1.157 0.556 1.064 0.624 –0.355 0.386 –0.361 0.995 0.597 0.546 0.395 $n=400$
1.167 0.560 1.073 0.628 –0.356 0.387 –0.362 0.998 0.599 0.548 0.397 Root Mean
Squared Error $n=100$ 1.171 0.596 1.072 0.647 0.359 0.416 0.365 0.987 0.595
0.541 0.393 $n=200$ 1.166 0.575 1.072 0.638 0.356 0.405 0.362 0.995 0.599
0.547 0.396 $n=400$ 1.170 0.570 1.077 0.635 0.357 0.398 0.363 0.998 0.600
0.549 0.397 Rejection Frequency for $H_{0}:\text{Spillover Parameter}=0$
$n=100$ 1.00 0.97 1.00 0.99 1.00 0.90 1.00 1.00 0.99 1.00 1.00 $n=200$ 1.00
1.00 1.00 1.00 1.00 0.98 1.00 1.00 1.00 1.00 1.00 $n=400$ 1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 Variables in the Second-Step Regression
Spillover Variable $\overline{G}_{it}$ $\overline{G}_{it}$
$\overline{G}_{it-1}$ $\overline{G}_{it-1}$ $\overline{G}_{it-2}$
$\overline{G}_{it-2}$ $\overline{G}_{it-2}$ $\overline{\omega}_{it}$
$\overline{\omega}_{it}$ $\overline{\omega}_{it-1}$ $\overline{\omega}_{it-1}$
DL Variable – $G_{it}$ – $G_{it-1}$ – $G_{it-2}$ $G_{it-1}$ – $G_{it}$ –
$G_{it-1}$ Notes: Reported are the results for “spillovers” (defined as either
$SP$ or $TIL$) from different variants of the second-step regressions in
(6.4)–(6.5) estimated with $\widehat{\omega}_{it}$ obtained in the first step
using the standard proxy estimator under the assumption of exogenous Markov
productivity. The data are simulated assuming linear productivity process
under scenario (iii) with true $DL=0$ and $SP=0$. Thus, the true spillovers
are zero. For the estimation of second-step regressions, the $G$ series is
generated as an $\omega$-controlled process. The specifications containing
contemporaneously endogenous regressors are estimated using first lags as
instruments. $\overline{G}_{it}\equiv\sum_{j}s_{ijt}G_{jt}$ and
$\overline{\omega}_{it}\equiv\sum_{j}s_{ijt}\omega_{jt}$. Throughout, $T=10$.

#### Alternative Two-Step Estimators of Spillovers.

Table G.3 reports the results for the “spillovers” estimator (defined as
either $SP$ or $TIL$) from different variants of the second-step regressions
in (6.4)–(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. The data are simulated assuming a linear
productivity process under scenario (iii) with the true $DL=0$ and $SP=0$.
Thus, the first-step estimation of productivity is correctly specified and
consistent. For the estimation of second-step regressions, the $G$ series is
generated as an $\omega$-controlled process (b). The specifications containing
contemporaneously endogenous regressors are estimated using their respective
first lags as predetermined instruments.

Examining the results in Table G.3, we find that, across all specifications,
the second-step estimator exhibits non-vanishing biases in the estimation of
spillovers. All models spuriously fail at identifying zero cross-firm
spillovers. Here we also report the rejection frequencies (over simulation
repetitions) for the asymptotic $z$-test of the null that the coefficient of a
spillover variable in the model is zero at the 95% confidence level. Had the
second-step been correctly specified and consistent, we were to expect these
rejection frequencies to be all around 0.05. Consistent with our expectations,
the results in the table indicate drastic size distortions due to
misspecification of the two-step approach. The results are qualitatively the
same when we also control for firm and/or time fixed effects.

##  Appendix H Data

Our data are drawn from the Chinese Industrial Enterprises Database survey
conducted by China’s National Bureau of Statistics (NBS). This database covers
all firms with sales above 5 million yuan (about 0.6 million in U.S. dollar)
and includes most industries including mining, manufacturing and public
utilities. We focus on the electric machinery and equipment manufacturing
industry, SIC 2-digit code 39.

Table H.4: Data Summary Statistics

Variable Mean 1st Qu. Median 3rd Qu. $Y$ 80,031 9,165 19,577 51,320 $K$ 14,330
1,039 2,957 8,929 $L$ 3,627 533 1,128 2,707 $M$ 55,761 6,372 13,640 35,878 $G$
0.09 0 0 0 $\mathbbm{1}\\{G>0\\}\quad$ 0.19 Notes: $Y$ – gross output; $K$ –
capital stock; $L$ – labor; $M$ – materials; $G$ – foreign equity share;
$\mathbbm{1}\\{G>0\\}$ – binary indicator signifying foreign-invested firms.
$Y$, $K$, $L$ and $M$ are measured in thousands of RMB; $G$ is a unit-free
proportion.

The production variables are as follows. The firm’s capital stock ($K_{it}$)
is the net fixed assets deflated by the price index of investment into fixed
assets. Labor ($L_{it}$) is measured as the total wage bill plus benefits
deflated by the GDP deflator. Materials ($M_{it}$) are the total intermediate
inputs, including raw materials and other production-related inputs, deflated
by the purchasing price index for industrial inputs. The output ($Y_{it}$) is
defined as the gross industrial output value deflated by the producer price
index. The price indices are obtained from NBS and the World Bank. The four
variables are measured in thousands of real RMB. In addition, the foreign
equity share ($G_{it}$) is a bounded proportion that lies between zero and
one, by construction.

Figure H.1: Empirical Distribution of the Foreign Equity Share:
(a) All Firms; (b) Foreign-Invested Firms Only

Figure H.2: Geographic Distribution of Firms (a) and the Foreign Equity Share
(b)
Note: Shown are the number of firms in the sample from each province (a) and
the sample mean at the province level (b), with the darker areas corresponding
to higher values.

We exclude observations with missing values for these variables as well as a
small number of likely erroneous observations with the foreign equity share
values outside the unit interval. With the sample period running from 1998 to
2007, the operational sample is an unbalanced panel of 23,720 firms with a
total of 73,095 observations. Table H.4 reports summary statistics for these
data.

Figure H.1(a) plots a histogram of $G_{it}$ across firms which expectedly has
a zero mode because the manufacturing sector in China is dominated by wholly
domestic firms. Figure H.1(b) on the right plots the distribution of
$G_{it}|G_{it}>0$, i.e., for foreign-invested firms only. Overall, 81% firms
in our sample are wholly domestically-owned, 1% are pure foreign
multinationals, with the remaining 18% being (partially) foreign-invested
domestic firms. The map in Figure H.2(b) shows the spatial distribution of the
(average) foreign equity share across regions where, consistent with one’s
priors, we see the heightened concentration of FDI along the coast.

##  Appendix I Additional Empirical Results

#### Baseline Results.

Figure I.3 plots empirical histograms of the point estimates of productivity
effects under the baseline specification. These estimates are the same as
those summarized in Table 4. Subfigure I.3(a) shows the distribution of
productivity spillover elasticities $SP$; the direct/internal and
indirect/external learning effects of FDI ($DL$ and $TIL$, respectively) are
presented in subfigure I.3(b).

Figure I.3: Distributions of the Productivity Effects: (a) $SP$, (b) $DL$ and
$TIL$
Note: Plotted are the point estimates based on our baseline specification of
the productivity process in (3.3). Figure I.4:  Spatial Distribution of the
Productivity Spillover Effects
Note: Plotted are the sample medians at the province level, with the darker
areas corresponding to higher values.

We also examine the geographic distribution of productivity spillovers in
Figure I.4. The map shows by-province median estimates of productivity
spillovers. We observe that productivity spillovers are stronger in the highly
industrialized, fast-growing provinces in the Southeast and near the coast.
Interestingly, comparing Figure I.4 with the spatial distribution of the
industry in Figure H.2(a) in Appendix H, we find that productivity spillovers
are comparable in strength (little, if any, shade gradient) across most
Southeastern and coastal provinces and thus extend beyond the Shanghai and
Guangzhou areas where the majority of the electric machinery manufacturing
industry is concentrated. Therefore, the evidence of productivity spillovers
that we find is not just a “mechanical” function of the spatial density of
data.

#### Endogenous Exposure to FDI.

The structural identification of our model only requires that the lagged
foreign equity share $G_{i,t-1}$ be weakly exogenous with respect to the
future productivity innovation $\zeta_{it}$. Therefore, firms in our framework
may experience endogenous updates to their exposure to foreign knowledge
$G_{it}$ and even relocate based on the contemporaneous productivity
$\omega_{it}$. Nonetheless, however mild an assumption, predeterminedness of
$G_{i,t-1}$ may still be violated in case the firms—or their foreign
investors—can forecast their future productivity (shocks). As a robustness
check to this potential violation of the weak exogeneity of lagged foreign
equity share, we re-estimate our baseline model using the inverse probability
weighting (IPW) procedure and instrumentation. In our case, we need to weight
only the second-stage regression since the first-stage share equation contains
neither the FDI variable nor a productivity innovation. Analogously,
instrumenting the lagged foreign equity share affects only the second stage.
The results are summarized in Table I.5.

Table I.5: Estimates of the Productivity Effects: Endogeneity of Lagged FDI

Exogenous IPW IV-External IV-Lewbel —Median Estimates— $SP$ 0.327 0.477 0.466
0.414 (0.211, 0.399) (0.413, 0.512) (0.411, 0.585) (0.276, 0.515) $DL$ 0.138
0.128 0.171 0.191 (0.118, 0.164) (0.105, 0.158) (0.138, 0.241) (0.053, 0.341)
$TIL$ 0.037 0.055 0.067 0.066 (0.022, 0.050) (0.044, 0.069) (0.056, 0.100)
(0.019, 0.126) —Statistically $>0$ (% Obs.)— $SP$ 83.84 98.55 97.17 79.56 $DL$
89.13 91.40 78.48 82.30 $TIL$ 86.71 98.99 97.64 83.48 Notes: Reported are the
results for the productivity process in (3.3) under baseline specification.
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 confidence intervals. The
“Exogenous” column corresponds to our proposed estimation procedure under the
structural assumption of weak exogeneity of $G_{i,t-1}$, with the second-stage
eq. (4.8) estimated via least squares. The three models on the right allow for
the violation of $\mathbb{E}[\zeta_{it}+\eta_{it}|G_{i,t-1}]=0$ and address
the potential endogeneity of lagged FDI in (4.8) via (i) inverse probability
weighting [IPW], (ii) instrumentation using external IVs including coastal
province dummy, province-level openness measure and their interactions with
predetermined $L_{i,t-1}$ [IV-External], (iii) instrumentation with the
heteroskedasticity-based internal IV à la Lewbel (2012) [IV-Lewbel].

By means of IPW, we seek to account for the potential selection (on
observables) of firms by their foreign investors based on their future
productivity. We use the stabilized IPWs which are typically more numerically
stable and produce narrow confidence bounds (Hernán & Robins, 2020). Also,
note that we deal with a continuous “treatment” $G_{it}$ which is why our
approach is different from the more conventional propensity score estimation
suitable for binary treatments (e.g., Imbens & Wooldridge, 2009). The IPWs for
a continuous treatment $G_{it}$ are given by
$f_{G}(G_{it})/f_{G|\textbf{d}}(G_{it}|\textbf{d}_{i,t+1})$, where $f(\cdot)$
is a pdf. [For more, also see Hirano & Imbens (2004) and Hernán & Robins
(2020).] Note that the vector of observables $\textbf{d}_{i,t+1}$ includes
firm characteristics reflective of its performance next period, because of
concern is the selection into treatment based on the future productivity. We
include the following correlates of the firm productivity that may influence
its foreign exposure: size proxied by the logged labor, age, state equity
share, government subsidy receipts, export intensity, normalized profits, the
return on assets, leverage, logged total assets as well as the East coast
dummy and the time trend.

To avoid the curse of dimensionality as well as the problem of near-zero
extreme values associated with nonparametric estimation of densities, we
employ a parametric maximum-likelihood approach to estimate $f_{G}$ and
$f_{G|\textbf{d}}$. Given the bounded nature of a fractional variable
$G_{it}\in[0,1]$, we assume it is Beta-distributed. Because beta distribution
is not trivial to estimate, we impose few data-motivated restrictions on it.
More specifically, we let $G_{it}\sim\text{Beta}(\alpha,\beta)$ and
$G_{it}|\mathbf{d}_{i,t+1}\sim\text{Beta}(\alpha,\beta(\mathbf{d}_{i,t+1}))$
where, to match the data, we restrict the first shape parameter to a unit
value ($\alpha=1$) and the second parameter/function $\beta(\cdot)$ to be
greater than 1 so that the distribution of $G_{it}$ is unimodal with a zero
mode in both instances.555Recall that the mode of the
$\text{Beta}(\alpha,\beta)$ distribution is 0 for $0<\alpha\leq 1$ and
$\beta>1$. The densities are estimated via maximum likelihood (ML), although
the method of moments provides an alternative route.

Since we seek to address the potential endogeneity of lagged $G_{i,t-1}$ in
the second-stage least squares regression, we also lag the estimated IPWs to
have them match the time period of “treatment.” In other words, we weight each
observation in the second stage by
$\widehat{f}_{G}(G_{i,t-1})/\widehat{f}_{G|\textbf{d}}(G_{i,t-1}|\textbf{d}_{it})$.666More
concretely, the IPW function is
$\displaystyle\frac{f_{G}(G_{i,t-1})}{f_{G|\textbf{d}}(G_{i,t-1}|\textbf{d}_{it})}=\frac{(1-G_{i,t-1})^{\beta_{0}-1}\Gamma(1+\beta_{0})}{\Gamma(\beta_{0})}\times\left[\frac{(1-G_{i,t-1})^{\beta(\textbf{d}_{it})-1}\Gamma(1+\beta(\textbf{d}_{it}))}{\Gamma(\beta(\textbf{d}_{it}))}\right]^{-1},$
where $\beta_{0}$ is a scalar shape parameter estimated via ML using
$G_{it}\sim\text{Beta}(1,\beta_{0})$; $\beta(\textbf{d}_{it})$ is a scalar
function estimated via ML using
$G_{it}|\textbf{d}_{i,t+1}\sim\text{Beta}(1,\beta(\textbf{d}_{i,t+1}))$ with
$\beta(\textbf{d}_{it})$ parameterized using the exponential function of a
single index; and $\Gamma(\cdot)$ is the Gamma function. The results are
summarized in the IPW column of Table I.5.

Table I.5 also reports the estimates of productivity effects from the second
stage estimated via generalized method of moments using external instruments
(the IV-External column) and Lewbel’s (2012) heteroskedasticity-based internal
instruments (the IV-Lewbel column) for $G_{i,t-1}$. The external instruments
include the East coast dummy, a province-level measure of openness (defined as
the ratio of the sum of imports and exports to the gross domestic product) and
their interactions with the firm-level lagged labor input. The two external
instruments are motivated by the previous studies, such as Eichengreen & Tong
(2007) and Keller & Yeaple (2009), and are selected to proxy friendly regional
policies towards foreign capital, shipping costs and the overall ease of
engaging in international trade and finance. We interact these instruments
with the predetermined labor at the firm level to gain variation. For
identification of the firm’s productivity process based on heteroskedasticity,
adapting Lewbel (2012) we first estimate an auxiliary equation for the
endogenous regressor by regressing $G_{i,t-1}$ on all the other exogenous
variables in the $\omega_{it}$ process, namely, $\omega_{i,t-1}$ and
$\sum_{j\neq i}s_{ij,t-1}\omega_{j,t-1}$. The residuals from this auxiliary
regression are then interacted with the demeaned $\omega_{i,t-1}$ and used to
instrument for $G_{i,t-1}$. Here, we use the firm’s predetermined lagged
productivity $\omega_{i,t-1}$ as a “$Z$” variable that is uncorrelated with
the product of productivity innovation $\zeta_{it}$ (with which $G_{i,t-1}$ is
suspected to be correlated) and the error in the auxiliary equation for
$G_{i,t-1}$. For more details on instrumentation via heteroskedasticity, see
Lewbel (2012).

Controlling for potential endogeneity of the FDI exposure, we continue to find
strong empirical evidence in support of significantly positive productivity
spillovers for 80% of firms or more. At least 78% of manufacturers benefit
from significant productivity boosts associated with receiving FDI. The
changes in the effect sizes are not out of the ordinary either, and the rank
correlation coefficient of the point estimate of productivity effects across
these estimators is at least as high as 0.71. All in all, our findings remain
qualitatively unchanged.

#### Bidimensional Spillovers.

In our main analysis, per the productivity process (3.3), spillovers from all
spatially proximate peers in the industry have the potential to affect the
recipient firm’s productivity in the same manner, no matter the exposure of
these peers to foreign knowledge. Given the documented impact of FDI on firm
productivity, it may also be of interest to allow for heterogeneity in the
external productivity spillovers from the peers conditional on their FDI
status. To this end, we adapt our methodology to allow a more general
evolution process for productivity that permits bidimensional spillovers by
means of two spatiotemporal lags. Thus, we now consider the following
evolution process of firm productivity:

$\omega_{it}=\mathbb{E}\left[\omega_{it}\Bigg{|}\
\omega_{i,t-1},G_{i,t-1},\sum_{j(\neq
i)}s_{ij,t-1}^{0}\omega_{j,t-1},\sum_{j(\neq
i)}s_{ij,t-1}^{1}\omega_{j,t-1}\right]+\zeta_{it},$ (I.1)

with the distinction between peer weights $\\{s_{ijt}^{0}\\}$ and
$\\{s_{ijt}^{1}\\}$ based on the peers’ FDI status:

$\displaystyle s_{ijt}^{0}=\frac{\mathbbm{1}\big{\\{}G_{jt}=0\ \text{and}\
(j,t)\in\mathcal{L}(i,t)\big{\\}}}{\sum_{k(\neq
i)=1}^{n}\mathbbm{1}\big{\\{}G_{kt}=0\ \text{and}\
(k,t)\in\mathcal{L}(i,t)\big{\\}}},$ (I.2) $\displaystyle
s_{ijt}^{1}=\frac{\mathbbm{1}\big{\\{}G_{jt}>0\ \text{and}\
(j,t)\in\mathcal{L}(i,t)\big{\\}}}{\sum_{k(\neq
i)=1}^{n}\mathbbm{1}\big{\\{}G_{kt}>0\ \text{and}\
(k,t)\in\mathcal{L}(i,t)\big{\\}}}.$ (I.3)

The cross-firm productivity spillovers are now bidimensional. The null
intersection of the two sets of peers ensures separable identifiability of
heterogeneous spillovers from (i) the wholly domestically owned peers
$SP_{it}^{0}=\partial\mathbb{E}[\omega_{it}|\cdot]\big{/}\partial\sum_{j(\neq
i)}s_{ij,t-1}^{0}\omega_{j,t-1}$ and (ii) foreign-invested peers
$SP_{it}^{1}=\partial\mathbb{E}[\omega_{it}|\cdot]\big{/}\partial\sum_{j(\neq
i)}s_{ij,t-1}^{1}\omega_{j,t-1}$.

Table I.6 summarizes point estimates of bidimensional productivity spillovers
under our baseline specification, whereby the firm’s peer group
$\mathcal{L}(i,t)$ is defined at the level of the same province and the entire
2-digit industry. Just like in the case of our main model with the
unidimensional cross-firm dependence, we continue to find substantial and
positive productivity spillovers in the industry. However, by disentangling
the peer effects of wholly domestically owned and foreign-invested neighbors,
we find that the former group has a significantly larger effect on its
neighbors. The median spillover elasticity from fully domestic firms $SP^{0}$
is 0.30, whereas the counterpart estimate $SP^{1}$ from the foreign-invested
peers is 0.17 only. Regardless of the peer group, the productivity spillovers
are statistically positive for the overwhelming majority of firms in the
industry (83% or more). This is on par with the extent of spillovers that we
have found in our main model with unidimensional spillovers.

Table I.6: Estimates of Bidimensional Productivity Spillovers

Point Estimates Statistically $>0$ Estimand 1st Qu. Median 3rd Qu. (% Obs.)
$SP^{0}$ 0.153 0.300 0.428 82.58 (0.090, 0.180) (0.198, 0.349) (0.302, 0.504)
$SP^{1}$ 0.158 0.172 0.185 98.89 (0.119, 0.188) (0.129, 0.204) (0.137, 0.216)
Notes: Reported are the semiparametric estimates of bidimensional productivity
spillovers from (I.1) under our baseline specification. The left panel
summarizes point estimates of $SP_{it}^{0}$ and $SP_{it}^{1}$ 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 percentile confidence intervals.

The documented heterogeneity in the magnitudes of spillovers across the two
types of peers suggests that it is relatively “easier” for Chinese
manufacturers to learn from other domestic firms that are not recipients of
FDI. This may be because foreign-invested firms are more protective of their
newly adopted foreign technologies/knowledge, which makes it more difficult to
learn from them. At the same time, it also may be that learning from these
foreign-invested firms is simply more difficult because their practices are
too advanced and biased towards more productive/efficient firms in the first
place. If so, firms that are already foreign-invested—and, hence, are more
productivity due to direct learning effects of FDI—are to enjoy larger
spillovers from their peers who have also received foreign investments. This
is corroborated by the evidence in Table I.7: the coefficient on the firms’
own FDI exposure is negative for $SP^{0}$ and positive for $SP^{1}$.

Table I.7: Heterogeneity in Bidimensional Productivity Spillovers | $SP^{0}$ | $SP^{1}$
---|---|---
$\omega_{i,t-1}$ | –0.720 | –0.070
| (–0.882, –0.571) | (–0.125, –0.019)
$G_{i,t-1}$ | –0.185 | 0.036
| (–0.355, –0.073) | (–0.151, 0.130)
$\sum_{j}s_{ij,t-1}^{0}\omega_{j,t-1}$ | 1.206 | –0.147
| (0.591, 1.459) | (–0.276, –0.023)
$\sum_{j}s_{ij,t-1}^{1}\omega_{j,t-1}$ | –0.147 | 0.144
| (–0.276, –0.023) | (0.101, 0.178)
Notes: Reported are the parameter estimates for the $SP^{0}$ and $SP^{1}$
functions derived from the polynomial approximation of the conditional mean of
$\omega_{it}$ in the productivity process formulation with bidimensional
spillovers in (I.1). Two-sided 95% bootstrap percentile confidence intervals
in parentheses. These correspond to our baseline specification, with (i) each
firm’s peers 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 flexibly controlled for using a series of
year effects.

The results in Table I.7 continue to indicate that the more productivity firms
have less absorptive capacity to learn from their peers. For both the $SP^{0}$
and $SP^{1}$ spillovers, the effect size increases with the average
productivity of peers from whom spillovers originate. But on the other hand,
the strength of spillovers from one peer group declines with the average
productivity of the other group (note the negative coefficient on the cross-
peer productivity). Taken together, these two findings suggest some
substitutability between learning from the two groups along with the recipient
firm’s finite capacity to absorb such spillovers from the peers in a given
period. Thus, if the foreign-invested peers improve their productivity, the
firm starts learning more from them and less from the non-foreign-invested
peers, and vice versa.

To conclude, although we find evidence of heterogeneity in the strength of
spillovers from wholly-domestic versus foreign-invested peers (with those from
the former being relatively stronger), in the grand scheme of things, our main
findings stay the same: productivity spillovers are positive and significant
for most firms in the industry.

#### Asymmetric Spillovers.

As we explain in Appendix D, the peer weighing scheme that we use in our
analysis 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 that the firm $i$ itself. 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, albeit with additional timing
assumptions.

Table I.8: Estimates of the Productivity Effects with Asymmetric Spillovers

Point Estimates Statistically $>0$ Estimand 1st Qu. Median 3rd Qu. (% Obs.)
$SP$ 0.301 0.331 0.362 99.54 (0.277, 0.328) (0.306, 0.361) (0.335, 0.400) $DL$
0.083 0.133 0.169 87.01 (0.058, 0.106) (0.104, 0.156) (0.135, 0.192) Notes:
Reported are the results based on the productivity process formulation with
asymmetric productivity spillovers in (D.2) using our baseline specification
of $\mathcal{L}(i,t)$. The left panel summarizes point estimates of $SP_{it}$
and $DL_{it}$ 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 percentile confidence
intervals.

We estimate such an asymmetric specification given in (D.2). Table I.8
summarizes the corresponding estimates of productivity effects. Comparing
these results with our main estimates in Table 4, we see that the estimates of
direct learning stay by and large unchanged, as expected. While comparable at
the median, the asymmetric spillover effect estimates exhibit a much smaller
variation in effect size (perhaps, because the peer pool is more homogeneous
now) but are as prevalent as they are when we model them symmetrically.

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