Recasting the trade impact on labor share: a fixed-effect semiparametric estimation study

Abstract

The cross-country declined labor share has been partially attributed to rising trade openness. However, the role of exports and imports in the literature was not studied separately and assumed to be homogeneous across countries. We propose two hypotheses for how exports and imports can affect labor share differently and nonlinearly. We empirically test the hypotheses by re-examining the trade–labor share nexus across 96 countries during 1970–2009, and we employ a partially linear model with fixed effect that allows a general functional form of trade variables to be estimated. Results are fairly consistent with our hypotheses, showing that while export (import) share significantly declines (raises) labor share, both effects diminish as the level of export or import share increases. The indicated nonlinear effects are significant and robust by controlling for related economic, social, and political factors. Also, we find a significant heterogeneous impact of export and import across OECD and non-OECD countries, with its implication also discussed.

Appendices

Appendix 1

Following Racine (1997), we evaluate the relevance of Z (i.e., Openness, Exs, or Ims) in the conditional mean function $G(\cdot )$ by testing the following null and alternative hypothesis:

$$\begin{aligned} {\left\{ \begin{array}{ll} H_0: \lambda =E\left[ \frac{\partial E(\tilde{Y}|Z)^2}{\partial Z}\right] =0\\ H_1: \lambda =E\left[ \frac{\partial E(\tilde{Y}|Z)^2}{\partial Z}\right] > 0 \end{array}\right. } \end{aligned}$$

where the vector $\tilde{Y}=Y-X\hat{\varvec{\beta }}_\mathrm{PLM}$ or $\tilde{Y}=Y-X\hat{\varvec{\beta }}_\mathrm{FEPLM}$ is discussed in Sect. 4.2. To estimate $\lambda$, we replace the unknown gradients $G^{'}(Z)\equiv \frac{\partial G(Z)}{\partial Z}$ with its nonparametric estimator $\hat{G}^{'}(Z)$ as discussed in Sect. 4.2.2. to construct our feasible pivoted test statistic:

$$\begin{aligned} \hat{t}=\frac{\hat{\lambda }}{\hbox {sd}(\hat{\lambda })} \end{aligned}$$

where

$$\begin{aligned} \hat{\lambda }=\frac{1}{NT}\sum _{i=1}^{N}\sum _{t=1}^{T}\left[ \frac{\hat{G}^{'}(Z_{it})}{\hbox {sd}(\hat{G}^{'}(Z_{it}))}\right] ^2 \end{aligned}$$

and $\hbox {sd}(\cdot )$ is the sample standard deviation. The following outlines its bootstrap procedure:

• 1. Given the original sample $\{\tilde{Y}_{it},Z_{it}\}_{i=1,t=1}^{N,T}$, calculate $\hat{\lambda }$. Re-sample with replacement the original sample to obtain a new dataset $\{\tilde{Y}_{it}^*,Z_{it}^*\}_{i=1,t=1}^{N,T}$ to compute $\hat{\lambda }$. Repeat this step $B_2$ times to obtain re-sampled test statistics $\hat{\lambda ^*_1},\hat{\lambda ^*_2},\ldots ,\hat{\lambda ^*_{B2}}$, which is used to compute the standard error $\hbox {sd}(\hat{\lambda })$ and test statistic $\hat{t}$.

• 2. Re-sampling with replacement the sample residual $\{\hat{u}_{it}\}_{i=1,t=1}^{N,T}$ to get re-sampled residual $\{\hat{u}_{it}^{*}\}_{i=1,t=1}^{N,T}$ and center it, where $\hat{u}_{it}=\tilde{Y}_{it}-\bar{Y}$ with $\bar{Y}$ the sample mean of $\tilde{Y}$ to impose the null where Z does not show up in $G(\cdot )$ so that $\beta (Z_{it})=0$$\forall i=1,\ldots ,N, t=1,\ldots ,T$.

• 3. Obtain bootstrap sample $\{\tilde{Y}_{{it}}^*, Z_{it}\}_{i=1,t=1}^{N,T}$, where $\tilde{Y}_{{it}}^{*}=\bar{Y}+u_{it}^{*}$. Compute $\hat{\lambda }^{*}$, $\hbox {sd}(\lambda ^{*})$, and $\hat{t}^{*}$ as in step 1, except with the bootstrap sample used.

• 4. Repeat the steps 2–3 $B_1$ times to obtain $\hat{t}^*_1$,$\hat{t}^*_2$,...,$\hat{t}^*_{B_1}$. We reject $H_0$ if $\hat{t} > \hat{t}^*_{1-\alpha }$, where $\hat{t}^*_{1-\alpha }$ is the upper $(1-\alpha )$ percentile value of its empirical distribution, and $\alpha$ is the significance level.

We set $\alpha =0.05$ and set $B_1=1000$, and $B_2=100$ to be in line with Racine (1997). We choose the 2nd-order Gaussian kernel function (i.e., the standard normal p.d.f.) and follow Silverman (1986) to select adaptive rule-of-thumb bandwidth $h_\mathrm{AROT}=1.059\;A(NT)^{-1/5}$, where $A = \min \{\hat{\sigma }, \hbox {IQR}_z/1.34\}$ and $\hbox {IQR}_z$ is the inter-quantile range of the variable Z.

Appendix 2

We employ a model specification test by Zheng (1996) to test the significant nonlinearity of Z in $G(\cdot )$, whose null hypothesis states whether the functional form of $G(\cdot )$ can be parametrically specified, such as linear or quadratic function. Equivalently, we test whether $E(U|Z)=0$ under the null. By the law of iteration, we have the following infeasible test statistic under the null and the alternative:

$$\begin{aligned} {\left\{ \begin{array}{ll} H_0: I=E(UE(U|Z)f(Z))=0\\ H_1: I=E(UE(U|Z)f(Z))>0 \end{array}\right. } \end{aligned}$$

where $f(\cdot )$ is the density of its argument, which can be consistently estimated by its kernel estimator $\hat{f}(Z)=(1/NT)\sum _{i=1}^{N}\sum _{t=1}^{T} K_h(Z_{it}, Z)$. We estimate I by replacing U with $\hat{U}$ the residual from the null regression (i.e., the OLS residuals), and replacing the conditional mean with the local constant estimator of U on Z:

$$\begin{aligned} \hat{E}(U|Z)=\frac{\sum \nolimits _{i=1}^{N}\sum \nolimits _{t=1}^{T} K_h(Z_{it}, Z) U_{it}}{\sum \nolimits _{i=1}^{N}\sum \nolimits _{t=1}^{T} K_h(Z_{it},Z)} \end{aligned}$$

By leaving out the diagonal terms to lower its finite-sample bias, we construct the feasible test statistic

$$\begin{aligned} \hat{I}=\frac{1}{N^2T^2h}\sum _{i=1}^{N}\sum _{t=1}^{T}\mathop {\sum _{j=1}^{N}\sum _{k=1}^{T}}_{\{j,k\}\ne \{i,t\}}\hat{u}_{it}\hat{u}_{jk}K_h(Z_{it}, Z_{jk}) \end{aligned}$$

with its variance:

$$\begin{aligned} \hat{\sigma }^{2}=\frac{2}{N^2T^2h}\sum _{i=1}^{N}\sum _{t=1}^{T}\mathop {\sum _{j=1}^{N}\sum _{k=1}^{T}}_{\{j,k\}\ne \{i,t\}}\hat{u}_{it}^{2}\hat{u}_{jk}^{2}K_h^{2}(Z_{it}, Z_{jk}) \end{aligned}$$

so we obtain the standardized version of test statistic:

$$\begin{aligned} \hat{J}=(NT\sqrt{h})\frac{\hat{I}}{\hat{\sigma }} \end{aligned}$$

(6)

which converges to the standard normal distribution under the null. The following illustrates its bootstrap procedure.

• 1. Given the original sample $\{\tilde{Y}_{it},Z_{it}\}_{i=1,t=1}^{N,T}$, calculate $\hat{J}$.

• 2. Generate centered bootstrapped residual $\{\hat{u}_{it}^*\}_{i,t=1}^{N,T}$, where for each observation $i =1,\ldots ,N$ and $t=1,\ldots ,T$, $\hat{u}_{it}^*=\frac{1-\sqrt{5}}{2}(\hat{u}_{it}-\bar{\hat{u}})$ with probability $\frac{1+\sqrt{5}}{2\sqrt{5}}$ and $\hat{u}_{it}^*=\frac{1+\sqrt{5}}{2}(\hat{u}_{it}-\bar{\hat{u}})$ with probability $\frac{1-\sqrt{5}}{2\sqrt{5}}$, where $\bar{\hat{u}}$ refers to the mean of $\hat{u}$.

• 3. Compute $\hat{J}^{*}$ using the bootstrap sample $\{\tilde{Y}_{{it}}^*, Z_{it}\}_{i=1,t=1}^{N,T}$, where $\tilde{Y}_{{it}}^{*}=m(Z_{it};\varvec{\beta })+u_{it}^{*}$, with $m(Z_{it};\varvec{\beta })$ the parametric functional form of our choice, including linear, quadratic, or cubic function, that imposes the null.

• 4. Repeat the steps 2–3 B times to obtain $\hat{J}^*_1$,$\hat{J}^*_2$,...,$\hat{J}^*_{B}$. Reject $H_0$ if $\hat{J} > \hat{J}^*_{1-\alpha }$, where $\hat{J}^*_{1-\alpha }$ is the upper $(1-\alpha )$ percentile value of its empirical distribution, and $\alpha$ is the significance level.

We choose the second-order Gaussian kernel function, set $(\alpha , B)=(0.05, 399)$, and implement the adaptive rule-of-thumb bandwidth as in “Appendix 1.”

Appendix 3

We employ a nonparametric test by Li (1996) for the distribution equality of estimated gradients of exports and imports between OECD and non-OECD countries. Let $\mathbf g ^{1}$ and $\mathbf g ^{2}$ be vectors that contain the first-order gradient of export (or import) $\{g^{1}_{it}\}_{i,t=1}^{N_1,T}$ in one group of countries with observations $N_1$ and $\{g^{2}_{it}\}_{i,t=1}^{N_2,T}$ in another one with observations $N_2$. We also assume that $g^{m}$ is i.i.d. distributed for fixed t and strictly stationary for fixed i for $m=1,2$. Define their associated density function $q_1(\cdot )$ and $q_2(\cdot )$, we test the null hypothesis $H_{04}: q_1(\mathbf g ^{1})=q_2(\mathbf g ^{1})$ for almost all continuous variables $\mathbf g ^{1}$. A test statistic with nonzero center term is constructed on the basis of integrated squared error between two different densities evaluating at the same random variable:

$$\begin{aligned} T=\int _{g^{1}}\left[ q_1(\mathbf g ^{1})-q_2(\mathbf g ^{1})\right] ^2 \hbox {d}{} \mathbf g ^{1} \end{aligned}$$

which can be extended to:

$$\begin{aligned} T=\int _\mathbf{g ^{1}}\left[ q_1(\mathbf g ^{1})\hbox {d}Q_1(\mathbf g ^{1})+q_2(\mathbf g ^{1})\hbox {d}Q_2(\mathbf g ^{1})-q_1(\mathbf g ^{1})\hbox {d}Q_2(\mathbf g ^{1})-q_2(\mathbf g ^{1})\hbox {d}Q_1(\mathbf g ^{1})\right] \end{aligned}$$

(7)

where $Q_1(\cdot )$ and $Q_2(\cdot )$ are the c.d.f. of $\mathbf g ^{1}$ and $\mathbf g ^{2}$, respectively, and T converges to zero if and only if the null is true. We replace the density $q_1(\cdot )$ and $q_2(\cdot )$ with their univariate kernel density estimator:

$$\begin{aligned} \hat{q}_1(g^{1})= & {} \frac{1}{N_1Th_1}\sum _{i=1}^{N_1}\sum _{t=1}^{T}k_h\left( \hat{g}^{1}_{it},\hat{g}^{1}\right) \end{aligned}$$

(8)

$$\begin{aligned} \hat{q}_2(g^{2})= & {} \frac{1}{N_2Th_2}\sum _{i=1}^{N_2}\sum _{t=1}^{T}k_h\left( \hat{g}^{2}_{it},\hat{g}^{2}\right) \end{aligned}$$

(9)

and select the bandwidth $h_1$ and $h_2$ based on adaptive rule-of-thumb$h_\mathrm{AROT}$. Replacing (8) and (9) into (7), we have the following feasible center-free test statistic:

$$\begin{aligned} \widehat{T}&=\frac{1}{N_1^2T^2|H|}\sum _{i=1}^{N_1}\sum _{t=1}^{T}\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{N_1}\sum _{\begin{array}{c} k=1\\ k\ne t \end{array}}^{T}k_h\left( \hat{g}^{1}_{it},\hat{g}^{1}_{jk}\right) +\frac{1}{N_2^2T^2|H|}\sum _{i=1}^{N_2}\sum _{t=1}^{T}\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{N_2}\sum _{\begin{array}{c} k=1\\ k\ne t \end{array}}^{T}k_h\left( \hat{g}^{2}_{it},\hat{g}^{2}_{jk}\right) \\&\quad -\frac{1}{N_1N_2T^2|H|}\sum _{i=1}^{N_1}\sum _{t=1}^{T}\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{N_2}\sum _{\begin{array}{c} k=1\\ k\ne t \end{array}}^{T}k_h\left( \hat{g}^{1}_{it},\hat{g}^{2}_{jk}\right) \\ {}&\quad -\frac{1}{N_2N_1T^2|H|}\sum _{i=1}^{N_2}\sum _{t=1}^{T}\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{N_1}\sum _{\begin{array}{c} k=1\\ k\ne t \end{array}}^{T}k_h\left( \hat{g}^{2}_{it},\hat{g}^{1}_{jk}\right) \end{aligned}$$

where $|H|=h_1h_2$. The standardized version of T is given by:

$$\begin{aligned} \hat{J}=\sqrt{N_1N_2T|H|}\left( \frac{\widehat{T}}{\hat{\sigma }}\right) \xrightarrow {d} N(0,1) \end{aligned}$$

(10)

where

$$\begin{aligned} \hat{\sigma }^2&=\frac{2}{N_1^2T^2|H|}\sum _{i=1}^{N_1}\sum _{t=1}^{T}\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{N_1}\sum _{\begin{array}{c} k=1\\ k\ne t \end{array}}^{T}k_h^{2}\left( \hat{g}^{1}_{it},\hat{g}^{1}_{jk}\right) +\frac{2}{N_2^2T^2|H|}\sum _{i=1}^{N_2}\sum _{t=1}^{T}\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{N_2}\sum _{\begin{array}{c} k=1\\ k\ne t \end{array}}^{T}k_h^{2}\left( \hat{g}^{2}_{it},\hat{g}^{2}_{jk}\right) \\&\quad -\frac{2}{N_1N_2T^2|H|}\sum _{i=1}^{N_1}\sum _{t=1}^{T}\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{N2}\sum _{\begin{array}{c} k=1\\ k\ne t \end{array}}^{T}k_h^{2}\left( \hat{g}^{1}_{it},\hat{g}^{2}_{jk}\right) \\ {}&\quad -\frac{2}{N_2N_1T^2|H|}\sum _{i=1}^{N_2}\sum _{t=1}^{T}\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^{N1}\sum _{\begin{array}{c} k=1\\ k\ne t \end{array}}^{T}k_h^{2}\left( \hat{g}^{2}_{it},\hat{g}^{1}_{jk}\right) \end{aligned}$$

The following provides procedure for a density equality test by Li (1996):

• 1. Given the original sample of gradients estimated in two different samples $\{\hat{g}^{1}_{it},\hat{g}^{2}_{jt}\}_{i,j,t=1}^{N_1,N_2,T}$ obtained by the local linear estimator of $G(\cdot )$, compute $\hat{J}$.

• 2. To impose the null where two densities are identical in an $L_2$ norm sense, pool the original sample into $\varvec{\Theta }=\{\hat{g}^{1}_{11},\ldots ,\hat{g}^{1}_{N_1T}, \hat{g}^{2}_{11},\ldots ,\hat{g}^{2}_{N_2T}\}$, call it pool sample. Randomly select with replacement $N_1$ observations from the pool sample $\varvec{\Theta }$ for the estimation of $q_1(\cdot )$, and also randomly select with replacement $N_2$ observations from the pool sample $\varvec{\Theta }$ for the estimation of $q_2(\cdot )$. Call these samples $\{\hat{g}^{1*}_{it}\}_{i,t=1}^{N_1,T}$ and $\{\hat{g}^{2*}_{it}\}_{i,t=1}^{N_1,T}$, respectively.

• 3. Compute $\hat{J}^{*}$ in the same way as in step 1, except that the original sample $\{\hat{g}^{1}_{it},\hat{g}^{2}_{jt}\}_{i,j,t=1}^{N_1,N_2,T}$ is replaced with the sample $\{\hat{g}^{1*}_{it}\}_{i,t=1}^{N_1,T}$ and $\{\hat{g}^{2*}_{it}\}_{i,t=1}^{N_2,T}$ in step 2.

• 4. Repeat the steps 2–3 B times to obtain $\hat{J}^*_1$,$\hat{J}^*_2$,...,$\hat{J}^*_{B}$. Reject $H_0$ if $\hat{J} > \hat{J}^*_{1-\alpha }$, where $\hat{J}^*_{1-\alpha }$ is the upper $(1-\alpha )$ percentile value of its empirical distribution, and $\alpha$ is the significance level.

We choose the second-order Gaussian kernel function, set $(\alpha , B)=(0.05, 399)$, and implement the adaptive rule-of-thumb bandwidth as in “Appendix 1.”

Appendix 4

See Table 9.

Table 9 Selected sample countries: whole sample

Appendix 5

See Table 10.

Table 10 Robustness check: whole sample