Estimation of firm productivity in the presence of spillovers and common shocks

Abstract

Productivity is largely estimated ignoring the potential impact of spillovers and common shocks in the literature, and therefore, the estimates may be subject to the omitted variable bias and internal inconsistency. In this paper, we estimate a nonparametric production function, in which technology spillovers and common shocks have persistent effects on productivity and are controlled for through spatial networks and a factor structure in the productivity evolution process. We synthesize the proxy variable method to structurally identifying the production functions using the semiparametric common correlated effect estimator. The proposed model is then applied to the Chinese computer and peripheral equipment firms. We find that the annual productivity growth rate in this high-technology sector is about 15%. While firms are cross-sectionally dependent via both spatial and non-spatial connections, the productivity growth is largely explained by firms’ own effort, and mildly explained by the neighbors’ activities. Productivity is found to be higher in the areas of agglomeration, and the common shock effects on productivity are not necessarily correlated with the spatial variables.

Appendices

Appendix: Bootstrap inference

In this appendix, we discuss the bootstrap method for statistical inference. To allow for cross-sectional dependence among idiosyncratic errors, we follow a multiple-step procedure similar to Gonçalves and Perron (2020). For a random variable $x_{it}$, denote $x_{t}=(x_{1t}, \ldots , x_{Nt})'$. Specifically, the bootstrap algorithm to get the standard errors is as follows.

• 1. For $t=1, \ldots , T$, let

  $$\begin{aligned} lns_{M,t}^* = ln({\hat{\beta }}_{Mt}(K_{t},L_{t},M_{t}){\hat{\varOmega }}) - \eta _{t}^*, \end{aligned}$$

  (A.1)

  where $\eta _t^*$ is a wild bootstrap resampled version of ${\hat{\eta }}_t=({\hat{\eta }}_{1t}, \ldots , {\hat{\eta }}_{Nt})'$ and ${\hat{\eta }}_{it} = ln({\hat{\beta }}_{Mt}(K_{it},L_{it}, M_{it}){\hat{\varLambda }})-lns_{M,it}$. That is

  $$\begin{aligned} \eta _t^* = {\hat{\eta }}_t v_t, \end{aligned}$$

  where the external random variable $v_t$ is i.i.d. (0, 1) over t.

• 2. Estimate Eq. (2.9) with $lns_{M,it}$ replaced with $lns_{M,it}^*$ to obtain the bootstrapped ${\hat{\beta }}_{Mt}^*$.

• 3. Let

  $$\begin{aligned} {\mathscr {Y}}_{t}^*= & {} -{\hat{\phi }}_t(K_{t}, L_{t})+{\hat{\psi }}_t(K_{t-1}, L_{t-1}, M_{t-1}, z_{t-1})\nonumber \\&+\,{\hat{\lambda }}'{\bar{z}}_{-,t-1}+ {\hat{\gamma }} {\hat{f}}_{t-1} + \xi _{t}^*, \end{aligned}$$

  (A.2)

  where $\xi _t^*$ is a wild bootstrap resampled version of ${\hat{\xi }}_t$, i.e., $\xi _t^* = {\hat{\xi }}_t v_t$, and ${\hat{\xi }}_t = \hat{{\mathscr {Y}}}_{t} - \Big (-{\hat{\phi }}_t(K_{t}, L_{t}) + {\hat{\psi }}_t(K_{t-1}, L_{t-1}, M_{t-1}, z_{t-1})+{\hat{\lambda }}'{\bar{z}}_{-,t-1}+ {\hat{\gamma }} {\hat{f}}_{t-1}\Big )$.

• 4. Estimate Eq. (2.14) with ${\mathscr {Y}}_{it}$ replaced with ${\mathscr {Y}}_{it}^*$ to get the bootstrapped ${\hat{\lambda }}^*$, ${\hat{\phi }}_t^*(K_{it}, L_{it})$, and ${\hat{\psi }}_t^*(K_{it-1}, L_{it-1}, M_{it-1}, z_{it-1})$. Combined with Eq. (2.11), we have the bootstrapped ${\hat{F}}_t^*(K_{it},L_{it},M_{it})$.

• 5. Let $\zeta _{it}^*={\mathscr {Y}}^*_{it}+{\hat{\phi }}_t^*(K_{it}, L_{it})-{\hat{\psi }}_t^*(K_{it-1}, L_{it-1}, M_{it-1}, z_{it-1})-{\hat{\lambda }}^{*\prime }{\bar{z}}_{-i,t-1}$. Estimate Eq. (2.17) using $\zeta _{it}^*$ to have the bootstrapped ${\hat{\gamma }}_i^*$ and ${\hat{f}}_t^*$.

• 6. Last, we can use the conventional wild boostrap for ${\hat{u}}_{it}$ to get the standard error of $\varphi$ based on the reduced form Eq. (3.11).

Appendix: A translog production function

In the main text, we do not assume a specific functional form for the production function and estimate the production technology semi/nonparametrically. One can however employ a parametric production function if certain parametric forms are desirable. Here, we show how to estimate our model when the widely used translog production function is assumed. Specifically, consider the following production function in logarithm, i.e.,

$$\begin{aligned} y_{it}&=lnT(k_{it}, l_{it}, m_{it})+\omega _{it}+\eta _{it}, \nonumber \\&=\beta _k k_{it} + \frac{1}{2}\beta _{kk}k_{it}^2 + \beta _l l_{it} + \frac{1}{2}\beta _{ll}l_{it}^2 + \beta _m m_{it} + \frac{1}{2}\beta _{mm}m_{it}^2\nonumber \\&\quad + \beta _{kl}k_{it}l_{it} + \beta _{km}k_{it}m_{it} + \beta _{lm}l_{it}m_{it} +\omega _{it}+\eta _{it}. \end{aligned}$$

(B.1)

Step one For the above translog production, the material elasticity is derived as $\frac{\partial y_{it}}{\partial m_{it}} = \beta _m + \beta _{mm}m_{it} + \beta _{km}k_{it} + \beta _{lm}l_{it}$. Therefore, Eq. (2.9), the firm’s first-order condition with respect to $M_{it}$, now takes the following form:

$$\begin{aligned} lns_{M,it} = ln\left( [\beta _m + \beta _{mm}m_{it} + \beta _{km}k_{it} + \beta _{lm}l_{it}]\varLambda \right) - \eta _{it}. \end{aligned}$$

(B.2)

Estimating Eq. (B.2) using NLS gives estimates of all the coefficients related to $m_{it}$ in the translog function.

Step two Define ${\mathscr {Y}}_{it} = y_{it} - \beta _m m_{it} - \frac{1}{2}\beta _{mm}m_{it}^2 - \beta _{km}k_{it}m_{it} - \beta _{lm}l_{it}m_{it} - \eta _{it}$. We can then rewrite the production function as

$$\begin{aligned} {\mathscr {Y}}&= \beta _k k_{it} + \frac{1}{2}\beta _{kk}k_{it}^2 + \beta _l l_{it} + \frac{1}{2}\beta _{ll}l_{it}^2 + \beta _{kl}k_{it}l_{it} +\omega _{it}, \nonumber \\&=\beta _k k_{it} + \frac{1}{2}\beta _{kk}k_{it}^2 + \beta _l l_{it} + \frac{1}{2}\beta _{ll}l_{it}^2 \nonumber \\&\quad + \beta _{kl}k_{it}l_{it}+g(\omega _{it-1}, z_{it-1}) +\lambda '{\bar{z}}_{-i,t-1}+\gamma _i' f_{t-1} + \xi _{it}, \nonumber \\&=\beta _k k_{it} + \frac{1}{2}\beta _{kk}k_{it}^2 + \beta _l l_{it} + \frac{1}{2}\beta _{ll}l_{it}^2 \nonumber \\&\quad + \beta _{kl}k_{it}l_{it}+\psi _t(k_{it-1}, l_{it-1}, m_{it-1}, z_{it-1})+\lambda '{\bar{z}}_{-i,t-1}+ \gamma _i' f_{t-1} + \xi _{it}, \end{aligned}$$

(B.3)

where the second equality is derived using the Markov process assumption of $\omega _{it}$, and the third equality is derived using the inverse material demand function. Similar derivations are employed in the main text to get Eq. (2.14). The above equation is a semiparametric partially linear regression model with an unobserved factor structure $\gamma _i'f_t$, in which $\psi _t(\cdot )$ is to be estimated nonparametrically. Following Su and ** (2012), we can estimate all other parameters in the production function, $\lambda$, and $\psi _t(\cdot )$ by using the cross-sectional averages of inputs as proxies of $f_t$ and approximating $\psi _t(\cdot )$ with polynomial bases.

After estimating the translog production function, we can follow the same steps as in the main text to estimate the factor structure ($\gamma _i'f_t$) and the spatial dependence parameter ($\varphi$) in Eq. (3.12).