A space-time analysis of knowledge production

Abstract

Regional growth models often emphasize the importance of research and development activities leading to technological progress. The role of knowledge production and spatiotemporal spillover effects is investigated using a space-time panel data set covering 49 US states over the period 1994–2005. The aim is to test for the existence of regional knowledge spillovers in the context of a space-time dynamic suggested by the knowledge production function. A space-time specification is set forth that can be applied to panel data models that include random effects. We compare alternative models that have been proposed in the panel data literature to provide a better understanding of how new ideas diffuse across space and time. The results indicate that the space-time panel data set is consistent with the presence of strong spatiotemporal regional spillovers of knowledge. The empirical findings are interpreted in light of the existing theoretical and empirical literature on endogenous growth.

Appendices

Appendix 1: Endogenous specification

The extension exploits spatial dependence between each observation/region. In the empirical application, the first cross-section is treated as unconditional (endogenous) and it is assumed that the sample $t=1, \ldots, T$ observations and presample values (those prior to time 1) were generated by the same process.

To motivate the spatial extension of the method, Bhargava and Sargan (1983) note that recursive substitution of the spatial dynamic panel data model described by Eq. 8 leads to an expression for the observation y _t equal to:

$$ \begin{aligned} y_t &= A^{s+1}y_{t-(s+1)} +\sum_{s=0}^{+\infty} A^{s}x_{t-s-1} \\ &\quad+(I_N-A)^{-1}(\iota_{N} \alpha + \mu)+\sum_{s=0}^{+\infty} A^{s}\varepsilon_{t-s} \end{aligned} $$

(18)

where A = ϕI _N  + θW.

The stationarity assumption |A| < 1 is equivalent to |ϕ| + |θ| < 1 if the spatial weight matrix is row normalized so the maximum eigenvalue equals one. This implies the first right-hand side variable approaches zero as s approaches infinity. Bhargava and Sargan (1983) note that since $\sum_{s=0}^{+\infty} x_{-s-1}$ is not observed, the covariance structure for this observation and other time periods, var − cov(y ₀) is undetermined. We follow their suggestion and rely on an optimal predictor for $\tilde{y}_0$ under the stationary assumption:

$$ \tilde{y}_{0} = E(\tilde{y}_{0} | x) + \xi $$

(19)

$$ = \iota_N \pi_0 + x^{\ast}\pi + \xi, $$

(20)

where π₀ is an intercept coefficient, $x^{\ast}=(x_0,\ldots,x_{T-1})$ is the N × kT matrix of explanatory variables, $\pi = (\pi_1',\ldots,\pi_{T}')'$ is a kT vector of unknown parameters and $\xi \sim N(0,\sigma_{\xi}^{2}I_N).$ ¹⁰ The initial observation y ₀ can be approximated using:

$$ \begin{aligned} y_{0} &= \tilde{y}_0+\xi_0 \\ y_{0}&= \iota_N \pi_0 +x^{\ast}\pi+ \xi_0, \end{aligned} $$

The error term ξ₀ is the sum of three components: the prediction error ξ for $\tilde{y}_{0},$ the individual effects μ, and the cumulative shocks before time zero. Thus,

$$ \xi_0 = \xi +(I_N-A)^{-1}\mu + \sum_{s=0}^{+\infty} A^{s}\varepsilon_{-s}. $$

(21)

and the predictor of $\tilde{y}_0$ is defined as in Eq. 20.

The log likelihood for this model involves the N (T + 1) × N (T + 1) variance-covariance of $u = (\xi_0,\eta')',$ which we denote $\Upomega^{\ast}:$

$$ \Upomega^{\ast} = \left( \begin{array}{cc} w_{11} & w_{12} \\ w_{21} & \Upomega \end{array} \right), $$

(22)

$$ w_{11}=E(\xi_0\xi_0') = \sigma_{\xi}^{2}I_N+ \sigma_{\varepsilon}^2 \left[I_N-AA'\right]^{-1}+ \sigma_{\mu}^2\left[(I_N-A)^{ \prime}(I_N-A)\right]^{-1} $$

(23)

$$ w_{12}=E( \xi_0\eta') = \sigma_{\mu}^2\iota_T'\otimes (I_N-A)^{-1} $$

(24)

where $w_{21}=w_{12}'$ and $\Upomega$ is defined in Eq. 9.

The log likelihood for this endogenous treatment of the initial period observations has the following form:

$$ \begin{aligned} \log L(\zeta) &= -\frac{N(T+1)}{2}\log(2\pi)-\frac{1}{2}\log|\Upomega^{\ast}|- \frac{1}{2}u^{\ast \prime}\Upomega^{\ast -1}u^{\ast} \\ \zeta &=(\beta',\alpha,\sigma_{\varepsilon}^{2},\sigma_{\mu}^2,\sigma_{\xi}^2,\pi_0,\pi',\phi,\theta)'\\ u^{\ast}&=(y_0'-(\iota_N \pi_0 + x^{\ast}\pi)',u')' \\ u&=Y-[\phi I_{NT} + \theta (I_T\otimes W)] Y_{-1} - X\beta - \iota_{NT}\alpha \end{aligned} $$

(25)

which differs from Eq. 10 due to treatment of the initial conditions. Therefore, an incorrect specification choice regarding the first cross-section will yield estimates that are not asymptotically correct (see Hsiao 2003).

Appendix 2: Data

The dependent variable corresponds to the number of U.S. patents granted by the U.S. Patent and Trademark Office during the period 1994–2005. ¹¹ We assume that ideas production in a given year is reflected in research activities undertaken previously. We calculate the stock of knowledge for each year using a three-year average (A _i,t  = (Pat _i,t  + Pat _i,t−1 + Pat _i,t−2)/3), where Pat _i,t represents the number of patent applications filed with the US patent office by inventors living in region i during year t. This captures the time it takes for innovation to materialize through the registration of a patent. Even though less accurate, this method is often used when only data on patents granted are available (see Autant-Bernard 2001; Parent and LeSage 2011b). Note that in case of multi-inventor teams, the first-named inventor is used to determine the location of innovative activities. This neglects the region cooperative activities. A complete analysis of the accumulation process of new ideas using citations and cross-region inventor teams can be found in Fischer et al. (2009).

National Patterns of R&D Resources can be downloaded from the National Science Foundation Website. They describe and analyze patterns of research and development (R&D) in the United States every year since 1994. The vast majority of scientists working in the laboratory are postdocs and doctoral students (Stephan 1996). Data related to the number of doctorates awarded are derived from the fall 2005 National Science Foundation-National Institutes and the Survey of Earned Doctorates. This survey is designed to obtain characteristics of doctorate recipients from U.S. institutions. The measure of output is Gross State Product (GSP) in manufacturing from the BEA Website.