Innovation and export activities in the German mechanical engineering sector: an application of testing restrictions in production analysis

Abstract

Since Solow (Q J Econ 70:65–94, 1956) the economic literature has widely accepted innovation and technological progress as the central drivers of long-term economic growth. From the microeconomic perspective, this has led to the idea that the growth effects on the macroeconomic level should be reflected in greater competitiveness of the firms. Although innovation effort does not always translate into greater competitiveness, it is recognized that innovation is, in an appropriate sense, unique and differs from other inputs like labor or capital. Nonetheless, often this uniqueness is left unspecified. We analyze two arguments rendering innovation special, the first related to partly non-discretionary innovation input levels and the second to the induced increase in the firm’s competitiveness on the global market. Methodologically the analysis is based on restriction tests in non-parametric frontier models, where we use and extend tests proposed by Simar and Wilson (Commun Stat Simul Comput 30(1):159–184, 2001; J Prod Anal, forthcoming, 2010). The empirical data is taken from the German Community Innovation Survey 2007 (CIS 2007), where we focus on mechanical engineering firms. Our results are consistent with the explanation of the firms’ inability to freely choose the level of innovation inputs. However, we do not find significant evidence that increased innovation activities correspond to an increase in the ability to serve the global market.

Appendices

Appendix 1: Aggregation of outputs

We only give a sketch of how to proceed for testing if some outputs can be aggregated. We again follow the ideas of Simar and Wilson (2001). Let $y=\left(y^1,y^2\right)$, where ${y^2\in {\mathbb R}_+^r}$ is the vector of the outputs we consider to aggregate. We denote as ${y^+=i_r^{\prime} y^2 \in {\mathbb R}_+}$ the resulting aggregated output. So we would like to solve the following test problem

$$ \begin{aligned} H_0\,& : \, y^2 \hbox{\,can\,be\,aggregated\,in\,}\,y^+, \\ H_1\,& : \, y^2 \hbox{\,cannot\,be\,aggregated\,in\,}\,y^+. \\ \end{aligned} $$

Now for any given point $(x,y)=(x,y^1,y^2)\in \Uppsi$ we define

$$ \theta(x,y)=\sup\{\theta|(\theta x,y^1,y^2)\in \Uppsi\} \\ $$

(A.1)

$$ \begin{aligned} \theta_0(x,y)&=\sup\left\{\theta|\left(\theta x,y^1,v\right) \in \Uppsi, \hbox{\,with\,} v\in {\mathbb R}_+^r,\right. \\ &\quad \quad\,\left. \left(x,y^1, v\right) \in \Uppsi \hbox{\,and\,} i^{\prime}_r v = y^+ \right\}\\ \end{aligned} $$

(A.2)

Clearly, we have in general $\theta_0(x,y)\le \theta(x,y)\le 1$. In addition, we have the following basic inequalities

$$ \begin{aligned} \hbox{if }\,H_0\,\hbox{ is\,true },\, \theta_0(x,y) & = \theta(x,y) \le 1,\hbox{\,for\,all\,} (x,y)\in\Uppsi \\ \hbox{if }\,H_1\,\hbox{ is\,true },\, \theta_0(x,y) & < \theta(x,y) \le 1,\hbox{\,for\,some\,} (x,y)\in\Uppsi. \\ \end{aligned} $$

(A.3)

Note that from a sample ${{\mathcal X}_n}$ , these two quantities are estimated by

$$ \begin{aligned} {\widehat \theta} (x,y) &= \max\left\{\theta | \theta x \ge \sum_{i=1}^n \gamma_i X_i,\, y^1\le \sum_{i=1}^n \gamma_i Y^1_i,\,y^2\le \sum_{i=1}^n \gamma_i Y^2_i\right.\\ & \quad \quad \quad \left. \sum_{i=1}^n \gamma_i=1, \,\gamma_i\ge 0\,\forall\,i=1,\,\ldots,\,n\right\}, \\ \end{aligned} $$

(A.4)

$$ \begin{aligned} {\widehat \theta}_0(x,y) &=\max\left\{\theta | \theta x \ge \sum_{i=1}^n \gamma_i X_i,\, y^1\le \sum_{i=1}^n \gamma_i Y^1_i,\,y^+\le \sum_{i=1}^n \gamma_i Y^+_i,\right.\\ & \quad \quad \quad \left. \sum_{i=1}^n \gamma_i=1, \,\gamma_i\ge 0\,\forall\,i=1,\,\ldots,\,n\right\},\\ \end{aligned} $$

(A.5)

whereas for $y^+, Y^+_i= i^{\ast}_rY^2_i$ for $i=1,\ldots,n$. For the same reasons as above, for all $(x,y)\in \widehat{\Uppsi}_{\hbox {DEA}}$ we have the basic inequality for the estimators

$$ {\widehat \theta}_0(x,y) \le {\widehat \theta} (x,y)\le 1. $$

(A.6)

Denote by ${{\mathbb P}_0}$ the restricted DGPs where the null hypothesis is true and ${{\mathbb P}_1}$ its complement, so we have ${{\mathbb P}_0 \cap {\mathbb P}_1 = \emptyset}$ and ${{\mathbb P}={\mathbb P}_0 \cup {\mathbb P}_1}$. In fact we want to test ${H_0 \,: \, P \in {\mathbb P}_0}$ versus ${H_1\, : \, P \in {\mathbb P}_1}$. Consider now a particular model ${P \in {\mathbb P}}$ and the model characteristic t(P) defined as

$$ t(P)=E \left( \frac{\theta(X,Y)}{\theta_0(X,Y)} -1 \right). $$

(A.7)

Due to the basic inequalities (A.3) discussed above, we have $t(P) \ge 0$ for all ${P\in {\mathbb P}}$, but t(P) = 0 if ${P\in {\mathbb P}_0}$ and t(P) > 0 if ${P\in {\mathbb P}_1}$.

A consistent estimator of t(P) is

$$ t_n({\mathcal X}_n)= \frac{1}{n} \sum_{i=1}^n \left( \frac{{\widehat \theta}(X_i,Y_i)}{\widehat\theta_0(X_i,Y_i)} -1 \right). $$

(A.8)

We know by construction, see (A.6), that ${t_n({\mathcal X}_n)\ge 0}$, and we will reject H ₀ if ${t_n({\mathcal X}_n)}$ is too large. The subsampling algorithms can be developed along the same lines as described above for the aggregation of inputs case.

Appendix 2: Some asymptotics

DEA estimators, as defined e.g. by (15), suffer from the curse of dimensionality shared by most of the nonparametric approaches, which means that to achieve the same accuracy, when the dimension (p + q) of the input–output space increases, we need much more data. Kneip et al. (2008) show that for every point $(x_0,y_0)\in\Uppsi$

$$n^{2/(p+q+1)}\left({\widehat \lambda}(x_0,y_0) -\lambda(x_0,y_0)\right) {\mathop {\longrightarrow}\limits^{\mathcal L}} Q(\cdot;\eta),$$

(B.1)

where $Q(\cdot;\eta)$ is a regular distribution function defined on ${{\mathbb R}_-}$ depending on a vector of unknown parameters η (these parameters depends on the DGP, like the density of (X, Y) near the frontier, the slope and curvature of the frontier, etc). Note that no closed form of this limiting distribution is available, but its existence is crucial for proving the consistency of bootstrap approximations. The curse of dimensionality is reflected by the fact that the rate of convergence n ^2/(p+q+1) is far below the usual parametric rate n ^1/2 when p or q increases with p + q > 3.

To derive the asymptotic distribution of our test statistics, we follow the same arguments as in Simar and Wilson (2010). Let Z = (X, Y) denote a generic observation and define

$$ T(Z)= \frac{\lambda_0(Z)}{\lambda(Z)} -1 \,\hbox{ and }\,{\widehat T} (Z;{\mathcal X}_n)=\frac{{\widehat \lambda}_0(Z)}{{\widehat \lambda}(Z)} -1, $$

(B.2)

where ${{\mathcal X}_n=\{Z_1,\ldots,Z_n\}}$. So that

$$ t(P) = E(T(Z))\,\hbox{ and }\, t_n({\mathcal X}_n)= \frac{1}{n} \sum_{i=1}^n \widehat T(Z_i;{\mathcal X}_n). $$

(B.3)

We know that for all ${P\in {\mathbb P}, T(Z) \stackrel{a.s.}{\ge} 0}$ and ${{\widehat T}(Z;{\mathcal X}_n)\stackrel{a.s.}{\ge} 0}$ but if ${P\in {\mathbb P}_0}$ , $T(Z) \stackrel{a.s.}{=} 0$ and still ${\widehat T(Z;{\mathcal X}_n)\stackrel{a.s.}{\ge} 0}$ . We know also from (B.1) that for all ${P\in{\mathbb P}}$ and for all $z=(x,y) \in \Uppsi$,

$$ n^{2/(p+q+1)}\left(\widehat T(z;{\mathcal X}_n) -T(z)\right)\stackrel{\mathcal L}{\mathop {\longrightarrow}} G(\cdot;z), $$

(B.4)

where G(.;z) is a nondegenerate distribution whose characteristics depends on z. By marginalizing on Z we have

$$ n^{2/(p+q+1)}\left({\widehat T}(Z;{\mathcal X}_n) -T(Z)\right)\stackrel{\mathcal L}{\mathop {\longrightarrow}} Q(\cdot), $$

(B.5)

where $Q(\cdot) = \int_z G(\cdot\mid z) f_Z(z) dz$ is, under regularity conditions, a nondegenerate distribution with finite mean μ₀ and finite variance σ ²₀  > 0.

Now, by using central limit theorem for triangular arrays (see e.g. Serfling 1980, Sect. 1.9.3), it is easy to show that if ${P\in {\mathbb P}_0}$,

$$ n^{2/(p+q+1)} \sqrt{n} \left(t_n({\mathcal L}_n) -\mu_0/n^{2/(p+q+1)}\right) \stackrel{\mathcal L}{\mathop {\longrightarrow}}{\mathcal N} (0,\sigma^2_0). $$

(B.6)

So the rate of convergence for obtaining a regular distribution of ${t_n({\mathcal X}_n)}$ under the null is $\tau_n=n^{2/(p+q+1)}\sqrt{n}$. Note that μ₀/n ^2/(p+q+1) acts as a bias term that can be neglected. It can also be proven that for all ${P\in {\mathbb P}, t_n({\mathcal X}_n)}$ converges in probability to t(P). For technical details and proofs, see Simar and Wilson (2010). So we can apply Theorem 3.1 of Politis et al. (2001) for using subsampling.