Multilevel heterogeneity of R&D cooperation and innovation determinants

Abstract

Assessing the impact of public support to innovation on R&D collaboration may require a more complex multilevel design, that describes the likely correlation present among firms characteristics within a particular sector. Using data from the 2006 edition of the Community Innovation Survey (CIS) for the Netherlands, we propose a methodology to study the effect of firm-level characteristics on the propensity to undertake a research collaborative agreement. In particular, we show that controlling for a richer variance structure yields a different picture with respect to simpler regression frameworks adopted in the literature of R&D cooperation determinants. Moreover, such a hierarchical framework can be generalized allowing for clustering at higher levels, such as sectors or geographical areas. Besides the link between public funding and R&D collaboration, our results confirm the findings of the literature: technological spillovers, risk and cost sharing rationales, firm’s size, and type of innovative activity are related to the decision of engaging in different sorts of research alliances.

Appendix 1: Multilevel heteroskedastic choice model

1.1 Appendix 1.1: Covariance structure

The firm- and sector-level random coefficients $\gamma _i^c$ and $\alpha ^c_j\equiv (\alpha _{1j},\dots ,\alpha ^c_{qj})^{\prime }$ are

$$\begin{aligned} \alpha ^c_j\sim N_q(0,W)\quad \text {and} \quad \gamma ^c_i\sim N(0,r), \end{aligned}$$

where q is the number of random effects included in the model. The variances of the firm- and sector-level random components, r and $W=diag(w_1,\dots ,w_q)$, respectively, are assumed to be invariant to cooperation choice c. We define the random effects for all sectors as $\alpha ^c\equiv ((\alpha ^{c}_1)^{\prime },\dots ,(\alpha ^{c}_J)^{\prime })^{\prime }$, for all firms as $\gamma ^c\equiv (\gamma ^c_1,\dots ,\gamma ^c_N)^{\prime }$, and for all c cooperation strategies as $\alpha \equiv \left( (\alpha ^{1})^{\prime },\dots ,(\alpha ^{C})^{\prime }\right) ^{\prime }$, and $\gamma \equiv \left( (\gamma ^{1})^{\prime },\dots ,(\gamma ^{C})^{\prime }\right) ^{\prime }$. We then assume that the vector of firm- and sector level random intercepts and slopes, $\gamma$ and $\alpha$ have the following covariance structure:

$$\begin{aligned} \gamma \sim N(\mathbf {0},\mathbf {G_1})\quad \text {and}\quad \alpha \sim N(\mathbf {0},\mathbf {G_2}). \end{aligned}$$

$\mathbf {G_1}$ and $\mathbf {G_2}$ are defined as the Kronecker product between matrices $\mathbf {A_1}$, and $\mathbf {A_2}$, and $\mathbf {V_1}$, and $\mathbf {V_2}$, i.e., $\mathbf {G_1=V_1}\otimes \mathbf {A_1}$, and $\mathbf {G_2=V_2}\otimes \mathbf {A_2}$ where

$$\begin{aligned} \qquad \mathbf {V_1}=\begin{pmatrix} \varsigma ^2_{11} &{} \varsigma _{12} &{}\dots &{} \varsigma _{1c} \\ \varsigma _{21} &{} \varsigma ^2_{22} &{}\dots &{} \varsigma _{2c} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \varsigma _{c1} &{} \varsigma _{c2} &{}\dots &{} \varsigma ^2_{cc} \\ \end{pmatrix} \qquad \text {and}\qquad \mathbf {V_2}=\begin{pmatrix} \sigma ^2_{11} &{} \sigma _{12} &{}\dots &{} \sigma _{1c} \\ \sigma _{21} &{} \sigma ^2_{22} &{}\dots &{} \sigma _{2c} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \sigma _{c1} &{} \sigma _{c2} &{}\dots &{} \sigma ^2_{cc} \\ \end{pmatrix}\end{aligned}$$

(3)

are the firm- and sector-level cooperation strategy-specific covariance matrices with elements $\varsigma _{c\tilde{c}}=\varsigma _{\tilde{c}c}$, and $\sigma _{c\tilde{c}}=\sigma _{\tilde{c}c}$, for $c\ne \tilde{c}$. In our application, these are $4\times 4$ matrices, as we consider 4 types of R&D cooperation ($C=4$), and where

$$\begin{aligned} \mathbf {A_1}=diag(r_1,\dots ,r_N)\quad \text {and}\quad \mathbf {A_2}=diag(W_1,\dots ,W_J). \end{aligned}$$

The matrices $W_1,\dots ,W_J$ have dimension $q\times q$, so that the dimension of $\mathbf {A_2}$ is $qJ\times qJ$.⁹ $\mathbf {G_1}$ and $\mathbf {G_2}$ are block matrices of dimension $4N\times 4N$ and $4qJ\times 4qJ$, respectively. When the hypothesis of constant coefficients across firms and sectors is supported, it could seem reasonable to allow variations in parameters across cross-sectional units in order to take into account the firm and sectoral heterogeneity. The model specification can be generalized so as to take into account for both random coefficients and heteroskedasticity, by using the following specification for the random intercept and coefficients:

$$\begin{aligned} \alpha ^c_j\sim N_q(0,W_j)\quad \text {and} \quad \gamma ^c_i\sim N(0,r_i). \end{aligned}$$

The variances of the firm- and sector-level random components, $r_i$ and $W_j=diag(w_{1j},\dots ,w_{qj})$, would then measure the degree of heterogeneity of each firm, nested in each sector. In this paper, the only source of heteroskedasticity which is explicitly taken into account is the one deriving from different cooperation alternatives, $\varsigma _{c\tilde{c}}$ and $\sigma _{c\tilde{c}}$.

1.2 Appendix 1.2: Bayesian estimation approach

Assuming conditional independence of firm’s choice probabilities given the covariates and the random effects, we can write the unconditional marginal probability¹⁰ of the response block matrix, $\mathbf {Y}\equiv \left[ y^c_{ij}\right] _{4\times N\times J}$, where $y^c_{ij}\in \{0,1\}$ is the observed research cooperation choice, as

$$\begin{aligned} L(\mathbf {Y}|\mathbf {G_1},\mathbf {G_2})=\int \int \prod _{j}\prod _{c}\prod _{i}f^c_{ij}(y^c_{ij}|\theta )\pi _1(\gamma |\mathbf {G_1})\pi _2(\alpha |\mathbf {G_2})d\gamma d\alpha . \end{aligned}$$

(5)

The maximum likelihood method is the standard approach for statistical inference in the mixed effects model. In order to maximize the sample likelihood, integration over the random-effects distribution must be performed. Yet, there exists no analytical solution for the intractable integral in Eq. (5). As a result, estimation is much more complicated than in models for continuous normally distributed outcomes where the solution can be expressed in closed form. Various approximations for evaluating the integral over the random-effects distribution have been proposed in the literature; many of these are reviewed in Rodríguez and Goldman (1995).

Simulation methods are also popular techniques to estimate mixed effects models (Train 2009). The unconditional probabilities in equation (5) are approximated through simulation for any given value $\varvec{\theta }$ of the parameters of the mixing distribution $f(\varvec{\alpha |\theta })$. Such methods fall under the rubric of Markov Chain Monte Carlo (MCMC) algorithms.

In this paper we adopt a Bayesian approach and explore the MCMC fitting of the multivariate mixed logit model. One advantage of the Bayesian approach over its frequentist counterpart includes the fact that the Bayesian procedures do not require maximization of any function. For complicated random effects structures, computation of a single maximum likelihood fit can be expensive, making the simulation of statistics of interest computationally prohibitive. Second, with Bayesian procedures, estimation properties, such as consistency and efficiency, can be attained under more relaxed conditions than with classical procedures. As shown in Train (2009) (Chapter 10), consistency of the Maximum Simulated Likelihood (MSL) estimator depends on the relationship between the number of draws that are used in the simulation and the sample size. If the number of draws is considered fixed, then the MSL estimator does not converge to the true parameters, because of the simulation bias. The simulation bias disappears as the sample size rises without bound together with the number of draws. In contrast, the Bayesian estimators are consistent for a fixed number of draws used in simulation and are efficient if the number of draws rises at any rate with sample size.

Following the Bayesian approach, the model parameters $\varvec{\beta }$, $\varvec{\alpha }$, $\mathbf G$, summarized in the vector $\varvec{\theta }$, are treated as random variables. The assumed distributions for the parameters, called prior distributions and denoted by $f(\varvec{\theta })$, borrow information from past studies, logic, or from the researcher’s ideas about the values of these parameters. Therefore, the prior distribution represents how likely the researcher thinks it is for the parameters to take a particular value, over all possible values that the parameters can take. Bayesian inference is based on the posterior distribution, $f(\varvec{\theta }|\mathbf y )$, which is the conditional distribution of the conjectured, but unknown, parameters $\varvec{\theta }$, given the observed data $\mathbf y ={y_1, \dots ,y_n}$.

The choice of a prior distribution $f(\varvec{\theta })$ affects Bayesian estimation. In other words, Bayesian inference may be influenced by a “strong” prior. In absence of any prior information, a non-informative prior is chosen ($f(\varvec{\theta }) \propto 1)$) and Bayesian inference is asymptotically equivalent to likelihood inference. In practice, we always specify a diffuse prior for $\varvec{\beta }$, and try different values of the set of parameters $\varvec{\alpha }$, $\mathbf G$, as a sensitivity analysis.

To estimate the parameters of the Generalized Linear Mixed Model (GLMM) defined in Sect. 4 following a Bayesian approach (Zeger and Karim 1991; Gelman et al. 2003), we use the R package MCMCglmm (Hadfield and Kruuk 2010). The default prior chosen by MCMCglmm for the regression model parameters $\beta ^c$ is a non-informative, normal distribution $N(0,1e+10)$,¹¹ while for both the residual and random-effect variance matrices a diffuse inverse-Wishart distribution is assumed, which is commonly used in practice. Then, assuming that the priors are independent,

$$\begin{aligned} f(\varvec{\beta },G)=f(\varvec{\beta })f(G), \end{aligned}$$

(6)

the posterior distribution can be written as

$$\begin{aligned} f(\varvec{\beta },G,\varvec{\alpha }|\mathbf y )\propto \prod ^n_{j=1} \prod ^4_{c=1}\prod ^{n^c_j}_{i=1}f^c_{ij}(y^c_{ij}|\varvec{\alpha }^c_j, \varvec{\beta }^c) f(\varvec{\beta }^c)\prod ^n_{j=1} \prod ^4_{c=1}f (\varvec{\alpha }^c_j|\mathbf {G})f(\mathbf {G}). \end{aligned}$$

(7)

The R package MCMCglmm generate samples from the posterior distribution using Metropolis–Hastings updates (for more details on the sampling schemes, see Hadfield and Kruuk 2010). Beginning with the starting values $(\varvec{\beta }^{(0)}, \varvec{\alpha }^{(0)},W^{(0)})$, after a warm-up (also called “burn-in”) period, we store a sample of $(\varvec{\beta }, \varvec{\alpha },W)$ from the posterior distribution. Once we generate a large number of samples, the posterior mean and posterior covariance can be approximated by the sample mean and the sample covariance based on the simulated samples. Convergence of the MCMC sampling scheme was assessed using empirical and test-based approaches (Heidelberger and Welch 1983; Geweke 1992). Results from convergence diagnostics indicated that it was sufficient to burn-in the first 15,000 samples and take the subsequent 1600 samples for inference.