Estimation of endogenous firm productivity without instruments: an application to foreign investment

Abstract

We consider identification and estimation of the firm-level gross production function with a controlled productivity evolution process and endogenous contemporaneous productivity determinants in the absence of instrumental variables (IVs). We allow the joint determination of firm productivity and its determinants, such as research and development, exports, and foreign direct investment, which is motivated by the recognition that certain unobserved confounders, such as CEO ability and managerial strategies, may simultaneously affect both productivity and these determinants. This assumption is in sharp contrast to those used in previous studies on proxy-variable estimation of production functions, in which productivity processes without determinants or with exogenous/predetermined determinants are often assumed. Since IVs for endogenous productivity determinants are in general difficult to obtain in the production context, we propose an IV-free generalized method of moments (GMM) estimator based on Lewbel et al. (2023) and use higher-order moments for unobserved errors in addition to conventional orthogonality conditions. We demonstrate the finite sample performance of the proposed estimator through Monte Carlo simulations. We also apply the methodology to investigate the impact of foreign equity on the productivity of Chinese manufacturing firms.

Appendix

1.1 Additional empirical results

This section includes additional empirical results of firm/industry productivity in Chinese chemical manufacturing sector over the period 2004–2006.

Figure 3 provides the boxplots of firm productivity estimated from our model during the sample period and by ownership. It is apparent that both types of firms become more productive over our sample period – their median values and interquartile boxes shift upward. The estimated annual productivity of the foreign firms is on average larger than that of the domestic firms.

Fig. 3

Boxplots of Productivity

Figure 4 shows the density plots of estimated productivity by ownership. Foreign firms have a slightly more left-skewed distribution compared with that of domestic firms, although the distribution range of these two types of firms are similar.

Fig. 4

Density of Productivity

To further explore the productivity differentials between domestic and foreign firms at different quantiles of the productivity distribution, we estimate the deciles of estimated productivity for domestic and foreign firms respectively, and then calculate their differences. The numeric results of both models are reported in Table 7. Both methods show that foreign equity can bring productivity gains to foreign-invested firms that are located at different places of the productivity distribution; the productivity differential between foreign and domestic firms becomes larger as the decile increases.

Table 7 Productivity Quantile Comparison

We also estimate the following quantile regression which corresponds to the analysis (5.1) in the main text but controls for the plant size (proxied by the number of employees) as well as the region and year effects:

$${Q}_{t}[{\omega }_{it}]={\beta }_{0,\tau }+{\beta }_{1,\tau }{D}_{F}+{\beta }_{2,\tau }{X}_{it}\ \,{{\mbox{for}}}\,\ \tau \in (0,1),$$

(C.1)

where D_F is a dummy variable which equals 1 if firm i is a foreign firm at year t and 0 for a domestic firm, X contains the control variables, e.g., the plant size. We let the quantile index τ take values from 0.1 to 0.9 with 0.05 increments. In Figure 5, the solid horizontal line shows the conditional mean productivity difference with its 95% confidence interval denoted by the dash lines. The quantile productivity differentials are all significantly positive and increasing with the productivity quantile, indicating that the productivity divergence between the foreign and the domestic is more prominent among the more productive firms.

Fig. 5

Foreign Productivity Differential Estimates across Productivity Quantiles. Notes: The shaded band corresponds to a 95% confidence interval. The solid horizontal line shows the conditional mean, with the 95% confidence interval denoted by dashed lines

Table 8 reports the Olley and Pakes (1996) decomposition of the output-weighted aggregate industry productivity, Φ, following (5.2) which use Gandhi et al. (2020) estimates.¹⁸ Similar to the findings in the main text, the production distribution shift dominates the effect of market share reallocation efficiency in both firm groups, though foreign firms perform slightly better in terms of the productivity level and reallocation efficiency.

Table 8 Weighted Productivity Decomposition – Gandhi et al. (2020) Estimator

1.2 Heteroskedasticity adapted estimator

We first extend Lemma 1 of Lewbel et al. (2023) to a conditional version. The proof follows the same arguments used by Lewbel et al. (2023) except that everything now is conditional on X, a vector of exogenous variables. Using the same notation as Lewbel et al. (2023), let Q = W − γY = βU + R, P = W − (γ + β)Y = − βV + R, and Y = U + V. The conditional version of the first equation in Lemma 1 of Lewbel et al. (2023) holds trivially, i.e., ${\mathbb{E}}(QPY| X)=0$. For the conditional version of the second equation, the left hand side is

$$\begin{array}{ll}Cov(QP,{Y}^{2}| X)&=Cov\left[(\beta U+R)(-\beta V+R),{(U+V)}^{2}| X\right]\\ &=Cov\left[{R}^{2}+RU\beta -RV\beta -UV{\beta }^{2},{U}^{2}+2UV+{V}^{2}| X\right]\\ &=Cov\left(-UV{\beta }^{2},{U}^{2}+2UV+{V}^{2}| X\right)\\ &=-{\beta }^{2}E\left[UV({U}^{2}+2UV+{V}^{2})| X\right]\\ &=-2{\beta }^{2}E({U}^{2}| X)E({V}^{2}| X),\end{array}$$

where the third equality holds because U, V and R are mutually independent conditional on X, the fourth equality holds because E(UV∣X) = E(U∣X)E(V∣X) = 0, and the last equality holds because E(U³V∣X) = E(U³∣X)E(V∣X) = 0 and E(UV³∣X) = E(U∣X)E(V³∣X) = 0. Meanwhile, notice that

$$\begin{array}{ll}E(QY| X)E(PY| X)\,=E\left[(\beta U+R)(U+V)| X\right]E\left[(-\beta V+R)(U+V)| X\right]\\ \qquad\qquad\qquad\qquad\quad=E\left[\beta {U}^{2}| X\right]E\left[-\beta {V}^{2}| X\right]\\ \qquad\qquad\qquad\qquad\quad=-{\beta }^{2}E({U}^{2}| X)E({V}^{2}| X)\\ \qquad\qquad\qquad\qquad\quad=-{\beta }^{2}{\sigma }_{U}^{2}(X){\sigma }_{V}^{2}(X).\end{array}$$

Hence,

$$Cov(QP,{Y}^{2}| X)=2E(QY| X)E(PY| X).$$

(D.1)

Note that Eq. (D.1) does not imply the following equality regarding unconditional moments

$$Cov(QP,{Y}^{2})=2E(QY)E(PY)$$

because

$$Cov(QP,{Y}^{2})\,\ne \,E\left[Cov(QP,{Y}^{2}| X)\right]$$

$$E(QY)E(PY)\,\ne \,E\left[E(QY| X)E(PY| X)\right]$$

To show that the estimator in our main text has the potential to be generalized to allow for heteroskedasticity, we demonstrate a simple estimator based on Eq. (D.1) in the case where the exogenous variable X is a scalar discrete variable through simulations.

To sketch ideas, rewrite Eq. (D.1) as

$$E\left[QP{Y}^{2}-2E(QY| X)E(PY| X)| X\right]=0.$$

(D.2)

Two issues emerge regarding the moment condition in Eq. (D.2). First, this is a conditional moment condition, and it implies that any functions of X can be used as IVs to convert Eq. (D.2) into a set of unconditional moments. The literature on how to construct the optimal IV then is relevant in selecting the IVs. Secondly, the conditional expectations E(QY∣X) and E(PY∣X) in Eq. (D.2) need to be estimated. Some candidate methods include using polynomials of X, using local kernel polynomials, or in the case that X is discrete, simply evaluating the two conditional means by X. Note that the complexity of the second issue compound with the first because finding the optimal IV requires the knowledge of the functional forms of E(QY∣X) and E(PY∣X).

Next, We develop a simple extension of the estimator in the main text to the case with a scalar discrete variable X. This simple setup allows us to estimate E(QY∣X) and E(PY∣X) non-parametrically using their sample counterparts. We do not dive into the optimal IV issue but simply select the function A(X) = aX/(1 + δX²) as the IV to use for the Eq. (D.2). This functional form is motivated by the heteroskedasticity functional form often used in the weighted least squares. Simulation results show that the proposed extension works.

The simulation design is similar to that in Section 4 except that the models for Z and ω are heteroskedastic. Specifically, the ω process now is

$${\omega }_{it}={\gamma }_{0}+{\gamma }_{1}{\omega }_{it-1}+{\gamma }_{2}{Z}_{it}+{\gamma }_{X}X+{\xi }_{it},$$

and the three error terms are generated according to

$${\zeta }_{it}={\zeta }_{0}+{\zeta }_{1}(1+\delta X),$$

$${u}_{it}={u}_{0}+{u}_{1}(1+\delta X),$$

$${\nu }_{it}={\nu }_{0}+{\nu }_{1}(1+\delta X),$$

where ζ₀ ~ LogNormal (−0.5, 1) − e and e is the exponential constant (approximately, 2.718), u₀ and ν₀ ~ Gumbel − Γ and Γ is the gamma constant (approximately, 0.577), ζ₁, u₁ and ν₁ ~ N (0, 1), and ζ₀, u₀, ν₀, ζ₁, u₁, ν₁ and X are mutually independent. That is, each error term is a sum of a homoskedastic part and a heteroskedastic part. The homoskedastic parts are essentially the same as those used in Section 4 except that they are now demeaned. The heteroskedastic parts are normal distributions of which the variances depends on the exogenous variable X. The coefficient δ controls the degree of heteroskedasticity. In the design reported here, δ = 2 and X ~ Bernoulli(0.5). Other designs where X is still discrete but takes on more than two outcomes (say uniform discrete random variable over {0, 1,.., 10}) have also been investigated, and the results are similar and thus are committed. The true parameter values are β_k = 0.25, β_m = 0.65, γ₀ = 0.2, γ₁ = 0.5, γ₂ = 0.5, γ_X = 0.1, ρ = 0.5, θ₀ = 0.01, and θ₁ = 0.3.

We use discrete X in this simulation design to simplify the estimation of E(e_1,ite_3,it∣X) and E(e_2,ite_3,it∣X), as our purpose here is to show that the proposed method has the potential to be generalized to cases with heteroskedasticity. With discrete X, E(e_1,ite_3,it∣X) and E(e_2,ite_3,it∣X) can simply be estimated by their sample analogs. Estimating E(e_1,ite_3,it∣X) and E(e_2,ite_3,it∣X) when X is continuous is more difficult and deserves dedicated work in future studies.

We select the unity and a(X) = X/(1 + 2X)² as the instruments for the conditional moment condition in (D.2). We chose them to (naively) capture the general pattern of the heteroskedasticity in the design. In particular, the denominator (1 + 2X)² in a(X) is motivated by the fact that the variance of the heteroskedastic part of each error term is proportional to (1 + 2X)². To some extent it mimics the idea of weighted least squares which downweights those less informative observations with larger noises. Deriving the optimal instruments is not easy here as we do not know the closed functional form of E(e_1,ite_3,it∣X) and E(e_2,ite_3,it∣X). Hence, the resulting GMM estimator is consistent but not efficient. These instruments suggest that we replace the fourth equation in (3.9) with the the following two conditional moments

$$E\left[{e}_{1,it}{e}_{2,it}{e}_{3,it}^{2}-2E({e}_{1,it}{e}_{3,it}| X)E({e}_{2,it}{e}_{3,it}| X)\right]=0,$$

$$E\left\{\left[{e}_{1,it}{e}_{2,it}{e}_{3,it}^{2}-2E({e}_{1,it}{e}_{3,it}| X)E({e}_{2,it}{e}_{3,it}| X)\right]a(X)\right\}=0.$$

It is worth noting that the first moment above is different from the fourth equation in (3.9) because $E\left[E({e}_{1,it}{e}_{3,it}| X)E({e}_{2,it}{e}_{3,it}| X)\right]\,\ne \,E({e}_{1,it}{e}_{3,it})E({e}_{2,it}{e}_{3,it})$ in general.

The simulation results are reported in Table 9. The Gandhi et al. estimators using Z_it still use the tilde notations, and our GMM estimators use the hat notations. The cross-sectional number of observations is 10,000, and the number of time periods is 5. The number of replications in the Monte Carlo simulation is 1000. The finite sample performance of the heteroskedasticity adapted estimator shows that it is robust to heteroskedasticity at least in the current simple setup.

Table 9 Heteroskedasticity Adapted Estimator

As a comparison, the simulation result for the homoskedastic version proposed in the main text is reported in Table 10. The results show large finite sample biases in the estimate for γ₂ (see, e.g., ${\hat{\gamma }}_{2}$ for n = 20,000 in Table 10).

Table 10 Homoskedasticity Estimator

1.3 A translog production function

The proposed model in the main text can be modified to identify and estimate production functions that take more flexible functional forms. In this appendix, we consider a translog (parametric log-quadratic) production function, viz.,

$$\begin{array}{ll}{y}_{it}=&{\beta }_{K}{k}_{it}+{\beta }_{L}{l}_{it}+{\beta }_{M}{m}_{it}+\frac{1}{2}{\beta }_{KK}{k}_{it}^{2}+\frac{1}{2}{\beta }_{LL}{l}_{it}^{2}+\frac{1}{2}{\beta }_{MM}{m}_{it}^{2}\\ &+{\beta }_{KL}{k}_{it}{l}_{it}+{\beta }_{KM}{k}_{it}{m}_{it}+{\beta }_{LM}{l}_{it}{m}_{it}+{\omega }_{it}+{\eta }_{it}.\end{array}$$

(A.1)

The first-order condition of the firm’s short-run profit maximization problem suggests that

$$\ln {s}_{M,it}=\ln \left[\left({\beta }_{M}+{\beta }_{MM}{m}_{it}+{\beta }_{KM}{k}_{it}+{\beta }_{LM}{l}_{it}\right){{\Lambda }}\right]-{\eta }_{it}.$$

(A.2)

It is the counterpart of (3.2) in the main text. Since ${\mathbb{E}}({\eta }_{it}| {k}_{it},{l}_{it},{m}_{it})=0$, the above equation can be estimated from the conventional nonlinear least squares (NLS) method.¹⁹ This is the first-step of our estimation, from which we get ${\widehat{\beta }}_{M}$, ${\widehat{\beta }}_{MM}$, ${\widehat{\beta }}_{KM}$, and ${\widehat{\beta }}_{LM}$.

In the second step, we redefine ${y}_{it}^{* }$ as ${y}_{it}^{* }={y}_{it}-{\beta }_{M}{m}_{it}-\frac{1}{2}{\beta }_{MM}{m}_{it}^{2}-{\beta }_{KM}{k}_{it}{m}_{it}-{\beta }_{LM}{l}_{it}{m}_{it}$. By definition, we have

$${y}_{it}^{* }={\beta }_{K}{k}_{it}+{\beta }_{L}{l}_{it}+\frac{1}{2}{\beta }_{KK}{k}_{it}^{2}+\frac{1}{2}{\beta }_{LL}{l}_{it}^{2}+{\beta }_{KL}{k}_{it}{l}_{it}+{\gamma }_{0}+{\gamma }_{1}{\omega }_{it-1}+{\gamma }_{2}{Z}_{it}+\rho {\zeta }_{it}+{\nu }_{it}+{\eta }_{it}.$$

(A.3)

The above equation can be used to replace the first equation in system (3.7). We still have the second-step estimator defined in (3.10) with the updated ${x}_{it}={(1,{k}_{it},{l}_{it},{k}_{it}^{2},{l}_{it}^{2},{k}_{it}{l}_{it},{\omega }_{it-1})}^{{\prime} }$. Additional lags of capital and labor can be added into x_it. For the unobserved ω_it−1, we can get its control function from the firm’s first-order condition with the translog production function form, viz.,

$$\begin{array}{r}{\omega }_{it-1}={m}_{it-1}^{* }-{\beta }_{K}{k}_{it-1}-{\beta }_{L}{l}_{it-1}-\frac{1}{2}{\beta }_{KK}{k}_{it-1}^{2}-\frac{1}{2}{\beta }_{LL}{l}_{it-1}^{2}-{\beta }_{KL}{k}_{it-1}{l}_{it-1},\end{array}$$

(A.4)

where

$$\begin{array}{ll}{m}_{it-1}^{* }\,=\,\ln ({P}_{t}^{M}/{P}_{t}^{Y})-\ln \left[\left({\beta }_{M}+{\beta }_{MM}{m}_{it-1}+{\beta }_{KM}{k}_{it-1}+{\beta }_{LM}{l}_{it-1}\right){{\Lambda }}\right]\\ \qquad\qquad+\,(1-{\beta }_{M}){m}_{it-1}-\frac{1}{2}{\beta }_{MM}{m}_{it-1}^{2}-{\beta }_{KM}{k}_{it-1}{m}_{it-1}-{\beta }_{LM}{l}_{it-1}{m}_{it-1}\end{array}$$

(A.5)

and ${m}_{it-1}^{* }$ is a function with parameters that have already been identified in the first step. Using (A.4) and (A.5), we replace the unobserved ω_it−1 with observed data, estimated parameters in the first step, and unknown parameters, when the moment conditions involved in (3.10) are constructed.

1.4 A general framework with nonlinear DGPs

For the ease of notation, we assume linearity for the production function in (2.1), the productivity law of motion in (2.2), and the productivity determinant process in (2.3) in the main text. Although these linear equations are widely-used in the literature of production function and productivity estimation, they are not identification assumptions in our paper. To see that, in this appendix, we consider a general framework, in which the above linearity assumptions are relaxed.

Consider a general production process of firm i in time period t as follows:

$${Y}_{it}=F({K}_{it},{L}_{it},{M}_{it};\beta ){e}^{{\omega }_{it}+{\eta }_{it}},$$

(B.1)

where production function F( ⋅ ) is defined with a finite number of unknown parameters β. Apparently, both the Cobb-Douglas and translog production functions discussed previously are included in (B.1). In general, we can write the ω process as

$${\omega }_{it}=h({\omega }_{it-1},{Z}_{it};\gamma )+{\xi }_{it},$$

(B.2)

and the Z process as

$${Z}_{it}=g({\omega }_{it-1};\alpha )+{\epsilon }_{it},$$

(B.3)

where h( ⋅ ; γ) and g( ⋅ ; α) can be nonlinear functions of their respective arguments, and γ and α are two vectors of unknown parameters of finite dimensions. Note that although we allow the endogenous variable Z_it to enter function h( ⋅ ) in a nonlinear way, we may not have enough moment conditions for identification if multiple unknown parameters are attached to Z_it. For example, empirical studies often use the second-order polynomial functions to approximate smooth but flexible functional forms. In that case, we have $h(\cdot ;\gamma )={\gamma }_{0}+{\gamma }_{1}{\omega }_{it-1}+{\gamma }_{2}{Z}_{it}+{\gamma }_{3}{\omega }_{it-1}^{2}$ and $g(\cdot ;\alpha )={\alpha }_{0}+{\alpha }_{1}{\omega }_{it-1}+{\alpha }_{2}{\omega }_{it-1}^{2}$. We drop ${Z}_{it}^{2}$ and Z_itω_it−1 from a complete polynomial to approximate function h( ⋅ ) since that leads to multiple endogenous variables and we may not have enough moment conditions.

Make use of the first-order condition, and production function (B.1) implies that the counterpart of (3.6) takes the following form:

$${y}_{it}^{* }=\ln {F}^{* }({K}_{it},{L}_{it};{\beta }^{* })+h({\omega }_{it-1},{Z}_{it};\gamma )+{\xi }_{it}+{\eta }_{it},$$

(B.4)

where the superscript * means that the impact of the flexible input M_it has been removed from the production function after the first step estimation. Using the general processes of ω_it and Z_it, we can also get the counterpart of (3.8) as

$${y}_{it}^{* }=\ln {F}^{* }({K}_{it},{L}_{it};{\beta }^{* })+h({\omega }_{it-1},{Z}_{it};\gamma )+\rho {Z}_{it}-\rho g({\omega }_{it-1};\alpha )-\rho {u}_{it}+{\nu }_{it}+{\eta }_{it}$$

(B.5)

Update the definitions of the error terms e_1,it, e_2,it, and e_3,it, we have

$$\begin{array}{ll}{e}_{1,it}\,\equiv \rho {\zeta }_{it}+{\nu }_{it}+{\eta }_{it}={y}_{it}^{* }-\ln {F}^{* }({K}_{it},{L}_{it};{\beta }^{* })-h({\omega }_{it-1},{Z}_{it};\gamma ),\\ {e}_{2,it}\,\equiv -\rho {u}_{it}+{\nu }_{it}+{\eta }_{it}={y}_{it}^{* }-\ln {F}^{* }({K}_{it},{L}_{it};{\beta }^{* })-h({\omega }_{it-1},{Z}_{it};\gamma )-\rho {Z}_{it}+\rho g({\omega }_{it-1};\alpha ),\\ {e}_{3,it}\,\equiv {\zeta }_{it}+{u}_{it}={Z}_{it}-g({\omega }_{it-1};\alpha ).\end{array}$$

We can show that the moment conditions in (3.9) do not rely on the functional forms of F^*( ⋅ ; β^*), h( ⋅ ; γ), and g( ⋅ ; α). Specifically, ${\mathbb{E}}({e}_{1,it}\cdot {x}_{it})=0$ and ${\mathbb{E}}({e}_{3,it}\cdot {x}_{it})=0$ directly follow from our structural model assumption that ${({\zeta }_{it},{\nu }_{it},{u}_{it},{\eta }_{it})}^{{\prime} }$ are contemporaneous errors and uncorrelated with predetermined variables in x_it. To verify the condition ${\mathbb{E}}({e}_{1,it}\cdot {e}_{2,it}\cdot {e}_{3,it})=0$, we use the definitions of the error terms:

$${\mathbb{E}}({e}_{1,it}\cdot {e}_{2,it}\cdot {e}_{3,it})={\mathbb{E}}\left((\rho {\zeta }_{it}+{\nu }_{it}+{\eta }_{it})(-\rho {u}_{it}+{\nu }_{it}+{\eta }_{it})({\zeta }_{it}+{u}_{it})\right).$$

Since ${({\zeta }_{it},{\nu }_{it},{u}_{it},{\eta }_{it})}^{{\prime} }$ are mutually independent, after removing the parentheses inside the expectation operator, all terms have the expected value of zero. Then, the condition ${\mathbb{E}}({e}_{1,it}\cdot {e}_{2,it}\cdot {e}_{3,it})=0$ holds. For the last condition ${\mathbb{C}}{{{\rm{ov}}}}({e}_{1,it}\cdot {e}_{2,it},\,{e}_{3,it}^{2})-2\,{\mathbb{E}}({e}_{1,it}\cdot {e}_{3,it})\,{\mathbb{E}}({e}_{2,it}\cdot {e}_{3,it})=0$, we can show that

$${\mathbb{C}}{{{\rm{ov}}}}({e}_{1,it}\cdot {e}_{2,it},\,{e}_{3,it}^{2})={\mathbb{C}}{{{\rm{ov}}}}\left((\rho {\zeta }_{it}+{\nu }_{it}+{\eta }_{it})(-\rho {u}_{it}+{\nu }_{it}+{\eta }_{it}),{({\zeta }_{it}+{u}_{it})}^{2}\right)=-2{\rho }^{2}{\mathbb{E}}({\zeta }_{it}^{2}){\mathbb{E}}({u}_{it}^{2}),$$

and

$${\mathbb{E}}({e}_{1,it}\cdot {e}_{3,it})\,{\mathbb{E}}({e}_{2,it}\cdot {e}_{3,it})={\mathbb{E}}\left((\rho {\zeta }_{it}+{\nu }_{it}+{\eta }_{it})({\zeta }_{it}+{u}_{it})\right){\mathbb{E}}\left((-\rho {u}_{it}+{\nu }_{it}+{\eta }_{it})({\zeta }_{it}+{u}_{it})\right)=-{\rho }^{2}{\mathbb{E}}({\zeta }_{it}^{2}){\mathbb{E}}({u}_{it}^{2}).$$

Then, the last condition also holds. The specific functional forms of F^*( ⋅ ; β^*), h( ⋅ ; γ), and g( ⋅ ; α) play no role in the above proof.