Inference is performed using the bootstrap percentile confidence intervals.
Let the observation-specific estimand of interest be denoted by $\mathcal{E}$,
e.g., the firm $i$’s labor elasticity coefficient $\beta_{L}(S_{i})$ or the
returns to scale defined as the sums of $\beta_{K}(S_{i})$, $\beta_{L}(S_{i})$
and $\beta_{M}(S_{i})$. To test two-tailed hypotheses, we can use the
empirical distribution of
$\\{\widehat{\mathcal{E}}^{1},\dots,\widehat{\mathcal{E}}^{B}\\}$ to estimate
the two-sided $(1-\alpha)\times 100$% confidence bounds for $\mathcal{E}$ as
an interval between the $[\alpha/2\times 100]$th and $[(1-\alpha/2)\times
100]$th percentiles of the bootstrap distribution. Naturally, for one-tailed
hypotheses, to estimate the one-sided lower or upper $(1-\alpha)\times 100$%
confidence bound, we can use the $[\alpha\times 100]$th or $[(1-\alpha)\times
100]$th bootstrap percentiles, respectively.

Finite-Sample Performance.—Using the DGP described in Appendix C, we
investigate a finite-sample performance of the above bootstrap procedure in a
simulation. This is of interest because of the complexity of our multi-step
estimation procedure, which makes establishing the validity of bootstrap
nontrivial. We focus on the two-sided 95% confidence intervals ($\alpha=0.05$)
for (i) the average elasticity of capital across all locations
$\overline{\beta}_{K}=\frac{1}{D}\sum_{S_{i}\in\mathbb{S}}\beta_{K}(S_{i})$
and (ii) capital elasticity at select locations $\beta_{K}(S_{0})$. We choose
$S_{0}=\\{0.65,0.75,0.85\\}$ which roughly correspond to quartiles of
$\mathbb{S}$. To conserve computational time, the number of simulations is
$Q=300$ with $B=200$ bootstrap replications per each simulation.

Table D.1: Coverage Probability of the Two-Sided 95% Bootstrap Confidence Interval | $\quad\overline{\beta}_{K}\quad$ | $\beta_{K}(0.65)$ | $\beta_{K}(0.75)$ | $\beta_{K}(0.85)$
---|---|---|---|---
Panel A: True Parameters are Location-Specific
$n=200\qquad$ | 0.873 | 0.923 | 0.901 | 0.850
$n=400$ | 0.843 | 0.907 | 0.933 | 0.883
$n=800$ | 0.950 | 0.933 | 0.943 | 0.953
Panel B: True Parameters are Location-Invariant
$n=200$ | 0.937 | 0.943 | 0.943 | 0.950
$n=400$ | 0.963 | 0.957 | 0.970 | 0.960
$n=800$ | 0.963 | 0.967 | 0.963 | 0.977
Panel A (B) corresponds to the DGP with locationally-varying (location-
invariant) coefficients. In both cases, the estimation was performed using our
kernel-weighting estimator under the presumption that coefficients are
location-specific. $T=10$ throughout.

Figure D.1: Power of the Two-Sided 95% Bootstrap Confidence Interval
(Vertical line corresponds to the true $\beta_{K}(s)$ value)

Panel A in Table D.1 reports coverage probabilities of the bootstrap
confidence intervals for capital elasticity under our DGP for the sample size
$n=\\{200,400,800\\}$. The coverage probability is estimated as the relative
frequency (over $Q$ simulations) of the estimated 95% confidence interval
containing the true elasticity value. We also plot power curves for these
confidence intervals; see Figure D.1. Here, power is computed as the relative
rejection frequency against different nulls on the $x$-axis. The simulations
show a satisfactory performance of our bootstrap confidence intervals in
finite samples. The results indicate that there may be size distortions for
small $n$, which is common for nonparametric tests. However, for a sample size
modestly large enough, the estimated coverage is close to the correct
coverage. The intervals exhibit good power, which improves as $n$ grows as
anticipated of a consistent test.

When the DGP is such that the true parameters are actually location-invariant
(i.e., fixed), the performance of bootstrap improves and the coverage
probabilities match the nominal confidence level even for small $n$. See Panel
B of Table D.1. This is expected because the nonparametric estimation improves
significantly when the true process is parametric.

Bias-Corrected Inference.—Efron, ’s (1982) bias-corrected bootstrap percentile
confidence intervals provide means to robustify inference by correcting for
the estimator’s finite-sample bias. In this case, the bias-corrected two-sided
$(1-\alpha)\times 100$% confidence bounds for $\mathcal{E}$ are estimated as
an interval between the $[a_{1}\times 100]$th and $[a_{2}\times 100]$th
percentiles of the bootstrap distribution, where
$a_{1}=\Phi\left(2\widehat{z}_{0}+\Phi^{-1}(\alpha/2)\right)$ and
$a_{2}=\Phi\left(2\widehat{z}_{0}+\Phi^{-1}(1-\alpha/2)\right)$ with
$\Phi(\cdot)$ being the standard normal cdf along with its quantile function
$\Phi^{-1}(\cdot)$. Parameter
$\widehat{z}_{0}=\Phi^{-1}\left(\\#\big{\\{}\widehat{\mathcal{E}}^{b}<\widehat{\mathcal{E}}\big{\\}}/{B}\right)$
is a bias-correction factor measuring median bias, with $\\#\\{\mathcal{A}\\}$
being a count function that returns the number of times event $\mathcal{A}$ is
true. Analogously, to estimate the one-sided lower or upper $(1-\alpha)\times
100$% confidence bound with bias correction, we can respectively use the
$[o_{1}\times 100]$th or $[o_{2}\times 100]$th percentiles of the bootstrap
distribution, where
$o_{1}=\Phi\left(2\widehat{z}_{0}+\Phi^{-1}(\alpha)\right)$ and
$o_{2}=\Phi\left(2\widehat{z}_{0}+\Phi^{-1}(1-\alpha)\right)$. Note that the
bias-corrected confidence interval need not contain the point estimate if the
finite-sample bias is large.

##  Appendix E Specification Test of Location Invariance

Given that our semiparametric locationally varying production model nests a
more traditional fixed-parameter specification that implies locational
invariance/homogeneity of the production function and the productivity
evolution as a special case, we can formally discriminate between the two
models to see if the data support our more flexible modeling approach.

More concretely, we test the null hypothesis of a production model in which
the technology is common to firms across all locations:

$\ln F_{|S_{i}}(\cdot)=\ln
F(\cdot)=\beta_{K}k_{it}+\beta_{L}l_{it}+\beta_{M}m_{it},$ (E.1)

and firm productivity evolves according to a location-homogeneous first-order
Markov process with

$h^{\omega}_{|S_{i}}(\cdot)=h^{\omega}(\cdot)=\rho_{0}+\rho_{1}\omega_{it-1}+\rho_{2}G_{it-1}.$
(E.2)

The location-invariant fixed-coefficient analogue of our model under the null
of locational homogeneity is therefore given by

$\displaystyle
y_{it}=\beta_{K}k_{it}+\beta_{L}l_{it}+\beta_{M}m_{it}+\rho_{0}+\rho_{1}\omega_{it-1}+\rho_{2}G_{it-1}+\zeta_{it}+\eta_{it},$
(E.3)

which we test against our semiparametric varying-coefficient alternative in
(3.4). This is, essentially, the test of overall relevancy of $S_{i}$.

To test this hypothesis, we use Ullah, ’s (1985) nonparametric goodness-of-fit
test based on the comparison of the restricted (under H0) and unrestricted
(under H1) models. First, let the estimator under H0 be denoted by “tilde”
whereas the estimator under H1 be denoted by “hat.” Then, the residual-based
test statistic is $T_{n}=(RSS_{0}-RSS_{1})/RSS_{1}$, where
$RSS_{0}=\sum_{i}\sum_{t}(\widetilde{\zeta_{it}+\eta_{it}})^{2}$ and
$RSS_{1}=\sum_{i}\sum_{t}(\widehat{\zeta_{it}+\eta_{it}})^{2}$ are
respectively the second-step residual sum of squares under the (restricted
parametric) null and the (unrestricted semiparametric) alternative.191919We
use the second-step residuals because they already incorporate information
about the first-step estimation by virtue of sequential construction.
Intuitively, the test statistic is expected to converge to zero under the null
and is positive under the alternative; hence the test is one-sided. To
approximate the null distribution of $T_{n}$, we use wild panel-data block-
bootstrap by resampling residuals from the model under the null.

To approximate the null distribution of $T_{n}$, we use wild panel-data block-
bootstrap by resampling residuals from model under the null. The algorithm
builds on that described in Appendix D.

  1. 1.

Using the original data, compute the two steps of both the locationally-
invariant (under H0) and locationally varying (under H1) models. Denote the
estimates from the restricted model as
$\big{[}\widetilde{\beta}_{M},\widetilde{\theta},\widetilde{\Theta}^{\prime}\big{]}^{\prime}$
and the estimates from the unrestricted alternative as
$\big{[}\widehat{\beta}_{M}(S_{i}),\widehat{\theta},\widehat{\Theta}(S_{i})^{\prime}\big{]}^{\prime}$
for all $i=1,\dots,n$. Let the (negative of) first-step residuals under the
null be $\\{\widetilde{\eta}_{it}\\}$ and those under the alternative be
$\\{\widehat{\eta}_{it}\\}$. Also, obtain the second-step residuals under the
null $\\{\widetilde{\zeta_{it}+\eta_{it}}\\}$ and under the alternative
$\\{\widehat{\zeta_{it}+\eta_{it}}\\}$. Use the latter to compute the test
statistic $T_{n}$.

  2. 2.

Generate bootstrap weights $\xi_{i}^{b}$ for all cross-sectional units
$i=1,\dots,n$ from the Mammen,  (1993) two-point mass distribution in (D.1).
Next, for each observation $(i,t)$ with $i=1,\dots,n$ and $t=1,\dots,T$,
generate a new bootstrap first-step disturbance
$\eta_{it}^{b}=\xi_{i}^{b}\times\widetilde{\eta}_{it}$ and a new bootstrap
second-step disturbance
$(\zeta_{it}+\eta_{it})^{b}=\xi_{i}^{b}\times(\widetilde{\zeta_{it}+\eta_{it}})$.
When constructing these bootstrap disturbances, use re-centered residuals.

  3. 3.

Generate a new bootstrap first-step outcome variable based on the
specification under H0. From the first step, we have
$v_{it}^{b}=\ln[\widetilde{\beta}_{M}\widetilde{\theta}]-\eta_{it}^{b}$ for
all $i=1,\dots,n$ and $t=1,\dots,T$.

  4. 4.

Recompute the first step of both the locationally-invariant and locationally
varying models using $\\{v_{it}^{b}\\}$ in place of $\\{v_{it}\\}$ and denote
the obtained parameter estimates as
$\big{[}\widetilde{\beta}_{M}^{b},\widetilde{\theta}^{b}\big{]}$ under the
null and as
$\big{[}\widehat{\beta}_{M}^{b}(S_{i}),\widehat{\theta}^{b}\big{]}$ under the
alternative.

  5. 5.

Generate a new bootstrap second-step outcome variable based on the
specification under H0. From the second step, we have
$y_{it}^{*b}=\widetilde{\beta}_{K}k_{it}+\widetilde{\beta}_{L}l_{it}+\widetilde{\rho}^{\omega}_{0}+\widetilde{\rho}^{\omega}_{1}\Big{[}\widetilde{\nu}^{*}_{it-1}-\widetilde{\beta}_{K}k_{it-1}-\widetilde{\beta}_{L}l_{it-1}\Big{]}+\widetilde{\rho}^{\omega}_{2}G_{it-1}+(\zeta_{it}+\eta_{it})^{b}$
for all $i=1,\dots,n$ and $t=1,\dots,T$, where
$\widetilde{\nu}^{*}_{it-1}=\ln[P_{t-1}^{M}/P_{t-1}^{Y}]-\ln[\widetilde{\beta}_{M}\widetilde{\theta}]+[1-\widetilde{\beta}_{M}]m_{it-1}$
is constructed using the original parameter estimates.

  6. 6.

Recompute the second step of both models using $\\{y_{it}^{*b}\\}$ in place of
$\\{y_{it}^{*}\\}$. When recomputing the models, also use
$\widetilde{\nu}^{*b}_{it-1}$ in place of $\widetilde{\nu}^{*}_{it-1}$ for the
restricted model and $\widehat{\nu}^{*b}_{it-1}$ in place of
$\widehat{\nu}^{*}_{it-1}$ for the unrestricted model, where
$\widetilde{\nu}^{*b}_{it-1}=\ln[P_{t-1}^{M}/P_{t-1}^{Y}]-\ln[\widetilde{\beta}_{M}^{b}\widetilde{\theta}^{b}]+[1-\widetilde{\beta}_{M}^{b}]m_{it-1}$
and
$\widehat{\nu}^{*b}_{it-1}=\ln[P_{t-1}^{M}/P_{t-1}^{Y}]-\ln[\widehat{\beta}_{M}^{b}(S_{i})\widehat{\theta}^{b}]+[1-\widehat{\beta}_{M}^{b}(S_{i})]m_{it-1}$
are constructed using the bootstrap parameter estimates. Denote the obtained
parameter estimates as
$\widetilde{\Theta}^{b}=[\widetilde{\beta}_{K}^{b},\widetilde{\beta}_{L}^{b},\widetilde{\rho}^{\omega
b}_{0},\widetilde{\rho}^{\omega b}_{1},\widetilde{\rho}^{\omega
b}_{2}]^{\prime}$ and
$\widehat{\Theta}^{b}(S_{i})=[\widehat{\beta}_{K}^{b}(S_{i}),\widehat{\beta}_{L}^{b}(S_{i}),$
$\widehat{\rho}^{\omega b}_{0}(S_{i}),\widehat{\rho}^{\omega
b}_{1}(S_{i}),\widehat{\rho}^{\omega b}_{2}(S_{i})]^{\prime}$.

  7. 7.

Recompute the second-step bootstrap residuals under the null
$\\{\widetilde{\zeta_{it}+\eta_{it}}\\}^{b}=y_{it}-\widetilde{\beta}_{K}^{b}k_{it}-\widetilde{\beta}_{L}^{b}l_{it}-\widetilde{\beta}_{M}^{b}m_{it}-\widetilde{\rho}^{\omega
b}_{0}-\widetilde{\rho}^{\omega
b}_{1}\Big{[}\widehat{\nu}^{*b}_{it-1}-\widetilde{\beta}_{K}^{b}k_{it-1}-\widetilde{\beta}_{L}^{b}l_{it-1}\Big{]}-\widetilde{\rho}^{\omega
b}_{2}G_{it-1}$ and under the alternative
$\\{\widehat{\zeta_{it}+\eta_{it}}\\}^{b}=y_{it}-\widehat{\beta}_{K}^{b}(S_{i})k_{it}-\widehat{\beta}_{L}^{b}(S_{i})l_{it}-\widehat{\beta}_{M}^{b}(S_{i})m_{it}-\widehat{\rho}^{\omega
b}_{0}(S_{i})-\widehat{\rho}^{\omega
b}_{1}(S_{i})\Big{[}\widehat{\nu}^{*b}_{it-1}-\widehat{\beta}_{K}^{b}(S_{i})k_{it-1}-\widehat{\beta}_{L}^{b}(S_{i})l_{it-1}\Big{]}-\widehat{\rho}^{\omega
b}_{2}(S_{i})G_{it-1}$. Use these to compute the bootstrap test statistic
$T_{n}^{b}$.

  8. 8.

Repeat steps 2 through 7 of the algorithm $B$ times.

Use the empirical distribution of $B+1$ bootstrap statistics
$\left\\{T_{n}^{b}\right\\}$, where the first bootstrap replica is the test
statistic $T_{n}$ calculated from the original data in Step 1, to obtain the
$p$-value as $\sum_{b}\mathbbm{1}\left\\{T^{b}_{n}\geq T_{n}\right\\}/(B+1)$.
In our empirical application, the number of bootstrap iteration is set to
$B=999$.

Estimating the Location-Invariant Model.—The location-invariant model
specified in (E.1)–(E.2) is fully parametric and a special case of our
locationally-varying model when $S_{i}=S_{0}$ for all $i$. It is therefore can
be estimated following our methodology in (3.14)–(3.18) but by letting the
adaptive bandwidths in both steps [$R_{h_{1}}(s)$ and $R_{h_{2}}(s)$] diverge
to $\infty$ which would, in effect, obviate the need to locally weight the
data because all kernels will be the same. We can then set
$\mathcal{K}_{{h}_{1}}(S_{i},s)=\mathcal{K}_{{h}_{2}}(S_{i},s)=1$ for all $i$.

When the production technology and the productivity process are global and do
not vary across locations as described in (E.1)–(E.2), the location-invariant
analogue of the first-step material share equation in (3.6) is given by

$v_{it}=\ln[\beta_{M}\theta]-\eta_{it},$ (E.4)

and that of the second-step proxied production function in (3.11) is given by

$\displaystyle y_{it}^{*}$
$\displaystyle=\beta_{K}k_{it}+\beta_{L}l_{it}+\rho_{0}+\rho_{1}\Big{[}\nu^{*}_{it-1}-\beta_{K}k_{it-1}-\beta_{L}l_{it-1}\Big{]}+\rho_{2}G_{it-1}+\zeta_{it}+\eta_{it},$
(E.5)

where
$\nu^{*}_{it-1}=\ln[P_{t-1}^{M}/P_{t-1}^{Y}]-\ln[\beta_{M}\theta]+[1-\beta_{M}]m_{it-1}$.

To estimate the material elasticity, denoting the unknown
$\ln[\beta_{M}\theta]$ as some constant $b_{M}$, we have the following
counterpart of the estimator in (3.14) which is just a sample mean:

$\widehat{b}_{M}=\frac{1}{nT}\sum_{i}\sum_{t}v_{it}.$ (E.6)

With it, we obtain the counterpart of (3.17) that estimates $\beta_{M}$:

$\displaystyle\widehat{\beta}_{M}$
$\displaystyle=nT\exp\left\\{\widehat{b}_{M}\right\\}\Big{/}\sum_{i}\sum_{t}\exp\left\\{\widehat{b}_{M}-v_{it}\right\\}$
$\displaystyle=nT\exp\left\\{\frac{1}{nT}\sum_{i}\sum_{t}v_{it}\right\\}\Big{/}\sum_{i}\sum_{t}\exp\left\\{\frac{1}{nT}\sum_{i}\sum_{t}v_{it}-v_{it}\right\\}.$
(E.7)

Using $\widehat{y}_{it}^{*}\equiv y_{it}-\widehat{\beta}_{M}m_{it}$ and
$\widehat{\nu}^{*}_{it-1}=\ln[P_{t-1}^{M}/P_{t-1}^{Y}]-\ln[\widehat{\beta}_{M}\theta]+[1-\widehat{\beta}_{M}]m_{it-1}$,
we arrive at the location-invariant counterpart of the second-step estimator
in (3.2):

$\displaystyle\widehat{\Theta}=\arg\min_{\Theta}\ \sum_{i}\sum_{t}$
$\displaystyle\
\Big{(}\widehat{y}_{it}^{*}-\beta_{K}k_{it}-\beta_{L}l_{it}-\rho_{0}-\rho_{1}\Big{[}\widehat{\nu}^{*}_{it-1}-\beta_{K}k_{it-1}-\beta_{L}l_{it-1}\Big{]}+\rho_{2}G_{it-1}\Big{)}^{2},$
(E.8)

where $\Theta=[\beta_{K},\beta_{L},\rho_{0},\rho_{1},\rho_{2}]^{\prime}$ and
which can be estimated via the usual nonlinear least squares.

For inference, we follow the same bootstrap steps as those for our main model
in Appendix D except that the estimated location-invariant fixed coefficients
are used in place of the locationally-varying coefficients.

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