Estimating production functions for the US states: the role of public and human capital

Abstract

This paper estimates production functions for the 48 contiguous U.S. states over the 1970–2000 period employing recently developed panel estimators that tackle simultaneously parameter heterogeneity, cross-sectional dependence and non-stationarity. The findings suggest that labor, private capital and, when controlling for cross-sectional dependence, average schooling years exert a positive and significant effect on state income. In contrast, the income effect of public capital stock is negative. The state-specific coefficients indicate that this effect likely stems from the negative elasticity of income with respect to public capital in the states located primarily in the Snow Belt region.

Appendix

Column (1) of Table 7 reports the estimation results using specification (3a) modified in the form of a spatial autoregressive (SAR) model (LeSage and Pace 2009, chapter 1). In effect, we estimate:

$$\begin{aligned} y_{ it } =\rho {\mathop {\sum }\limits _{\begin{array}{c} i,j=1\\ i\ne j \end{array}}^{48}} W_{ ij } y_{ jt } +\alpha _{i} k_{ it } +\beta _{i} l_{ it } +\gamma _{i} g_{ it } +\delta _{i} h_{ it } +\mu _{i} +v_t +u_{ it } \end{aligned}$$

(13)

where both i and $j=1,{\ldots },48$ denote a state, and $t=1,\ldots ,31$ a time period. The SAR model posits that the dependent variable is a function of the dependent variable observed in neighboring states and a set of observed local characteristics. Specifically, rho is the spatial autoregressive coefficient and $W_{ ij }$ is the lag weight matrix of dimension $48\times 48$ (there are 48 states in the sample), which measures the geographical distance between any pair of states i, j. We assume that the elements of the weight matrix are inversely related to the distance between states, i.e., $\omega _{ ij }=(1/d_{ ij })$, where $d_{ ij }$ is the Euclidean distance between the centroids of states i and j. Also, we posit that the maximum distance where spillovers take place is equal to the $1{\mathrm{st}}$ quartile of the distribution of all distance pairs between any pair of U.S. states, so that spillovers do not occur among very distant states. This structure of weights is such that states contribute to the geographical spillovers proportionally to their distance, so that weights penalize more the distant states. Spatial weights are row-standardized, so that for each state the sum of weights equals one.¹⁵ Thus, column (1) posits that real personal income in each state depends on that of its neighboring states up to a threshold distance, at a rate which decreases proportionally to the distance of each neighboring state from the state in question. We believe that this is a more flexible assumption than that of Dall’erba and Llamosas-Rosas (2015), who only allow for spillovers from the four closest neighbors. We include time and state fixed effects to allow for time trend and state-specific unobserved characteristics¹⁶ and compute bootstrapped cluster-robust standard errors according to Pace and LeSage (2009).

Column (2) reports the estimates of the spatial autocorrelation (SAC) model (LeSage and Pace 2009, chapter 2), which incorporates both spatial effects of the dependent variable and spatial effects among the error terms. In effect, we estimate:

$$\begin{aligned} y_{ it } =\rho {\mathop {\sum }\limits _{\begin{array}{c} i,j=1\\ i\ne j \end{array}}^{48}} W_{ ij } y_{ jt } +a_{i} k_{ it } +\beta _{i} l_{ it } +\gamma _{i} g_{ it } +\delta _{i} h_{ it } +\mu _{i} +v_t +\lambda {\mathop {\sum }\limits _{\begin{array}{c} i,j=1\\ i\ne j \end{array}}^{48}} W_{ ij } \varepsilon _{ jt } +u_{ it }\nonumber \\ \end{aligned}$$

(14)

Thus, SAC nests the SAR model and assumes that real personal income in each state depends on (1) real personal income of its neighbors as defined above and (2) a set of unobserved, common characteristics omitted from the model. The estimated values of rho and lambda are the spatial parameters of the spatially lagged dependent variable and spatially autocorrelated error terms, respectively. We estimate both SAR and SAC models with maximum likelihood. According to the results, the rho parameter is statistically insignificant in both specifications. In contrast, the lambda parameter is highly significant in the SAC model (column 2). In that sense, the SAC model seems to be a better choice than SAR in our setup and provides evidence for the existence of significant spatial effects in the U.S. states working primarily through the error terms.

Table 6 Variable definitions and sources

Table 7 Spatial estimation results in levels

Table 8 Educational quality as a determinant of states’ income

The upper panel of Table 7 reports the coefficient estimates of the total effects of the explanatory variables (labor, private capital, total public capital stock and average schooling years), while the lower panels report the corresponding direct and indirect estimates. The total effects measure the impact of a unit change in the above variables in all states on personal income in state i, averaged over all states. The direct effects measure the impact of a unit change in state i on personal income in state i, again averaged over all states. The indirect impact estimates are the difference of total and direct effects, which economists usually refer to as spillovers and depend on geographical distance in our specifications. When considering the direct effects, we observe that in our setup all four state inputs exert a sizeable, positive and statistically significant impact on the respective state incomes. These findings are in line with Dall’erba and Llamosas-Rosas (2015) who also find positive direct effects for private capital, public capital and human capital (although the latter is statistically insignificant). The presence or absence of significant spillovers depends on whether the indirect effects that arise result in statistically significant estimates. We observe that all indirect effects are statistically insignificant in both specifications, thereby casting doubt on the geographical proximity as the main driver for the presence of spillovers across U.S. states. Thus, observable production factors causing spillovers across U.S. states do not seem to be related to distance between states.