Estimation of industry-level productivity with cross-sectional dependence by using spatial analysis

Abstract

In this paper, we incorporate spatial analysis to estimate industry-level productivity in the presence of inter-sectoral linkages. Since each industry plays a role in providing intermediate goods to other sectors, the interdependence of economic activities across industries is inevitable. We exploit the linkage patterns from the input-output relationship to define cross-industry dependencies in economic space. We propose a spatial stochastic frontier model, which extends the stochastic frontier model to a spatially dependent specification. The models are estimated using quasi-maximum likelihood methods. Applying the approach to U.S. industry-level data from 1947 to 2010, we find that sectoral dependencies are the consequences of indirect effects via the supply chain network of industries resulting in larger output elasticities as well as scale effects for the networked production processes. However, productivity growth is estimated comparably across different spatial and non-spatial model specifications.

Appendices

Appendix A: QML estimators for SARCSS

In Appendix A, we provide a detailed procedure of QML estimation for SARCSSW and SARCSSG.

Let $\psi ={(\beta ,\gamma ,\rho ,{\sigma }_{v}^{2})}^{{\prime} }$. The log-likelihood function of Eq. (4) is:

$$\begin{array}{ll}logL(\psi ,{\delta }_{i};y)=-\frac{NT}{2}log(2\pi {\sigma }_{v}^{2})+Tlog| {I}_{N}-\rho W| -\frac{1}{2{\sigma }_{v}^{2}}\mathop{\sum }\limits_{i=1}^{N}\mathop{\sum }\limits_{t=1}^{T}{\left({y}_{it}-\rho \mathop{\sum }\limits_{j = 1}^{N}{w}_{ij}{y}_{jt}-{X}_{it}^{{\prime} }\beta -{Z}_{i}^{{\prime} }\gamma -{R}_{t}^{{\prime} }{\delta }_{i}\right)}^{2}.\end{array}$$

(A1)

The first order condition of maximizing Eq. (A1) with respect to δ_i is

$$\frac{\partial logL}{\partial {\delta }_{i}}=\frac{1}{{\sigma }_{v}^{2}}\mathop{\sum }\limits_{i=1}^{N}\mathop{\sum }\limits_{t=1}^{T}{R}_{t}\left({y}_{it}-\rho \mathop{\sum }\limits_{j=1}^{N}{w}_{ij}{y}_{jt}-{X}_{it}^{{\prime} }\beta -{Z}_{i}^{{\prime} }\gamma -{R}_{t}^{{\prime} }{\delta }_{i}\right)=0.$$

(A2)

By solving for (A2), we can obtain

$${\hat{\delta }}_{i}={({R}_{t}{R}_{t}^{{\prime} })}^{-1}{R}_{t}\left({y}_{it}-\rho \mathop{\sum }\limits_{j=1}^{N}{w}_{ij}{y}_{jt}-{X}_{it}^{{\prime} }\beta -{Z}_{i}^{{\prime} }\gamma \right).$$

(A3)

Substituting (A3) into the log-likelihood function, (A1), we obtain the concentrated likelihood function

$$logL(y;\beta ,\rho ,{\sigma }_{v}^{2})=-\frac{NT}{2}log(2\pi {\sigma }_{v}^{2})+Tlog| {I}_{N}-\rho W| -\frac{1}{2{\sigma }_{v}^{2}}{\tilde{V}}^{{\prime} }\tilde{V},$$

(A4)

where $\tilde{V}={M}_{Q}y-\rho {M}_{Q}({W}_{N}\otimes {I}_{T})y-{M}_{Q}X\beta$, and ${M}_{Q}={I}_{NT}-Q{({Q}^{{\prime} }Q)}^{-1}{Q}^{{\prime} }$²².

1.1 Within estimator

Assuming T ≥ L, the projections onto the column space of Q and the null space of Q are denoted by ${P}_{Q}=Q{({Q}^{{\prime} }Q)}^{-1}{Q}^{{\prime} }$ and M_Q = I_NT − P_Q, respectively.²³ Let’s suppose the true value of ρ is known, say ρ^*. By pre-multiplying M_Q on (5), we have the within-transformed model

$${M}_{Q}y={\rho }^{* }{M}_{Q}({W}_{N}\otimes {I}_{T})y+{M}_{Q}X\beta +\tilde{V}.$$

(A5)

And the estimates of β(ρ^*) and of ${\sigma }_{v}^{2}({\rho }^{* })$ are derived by

$${\hat{\beta }}_{W}({\rho }^{* })={({X}^{{\prime} }{M}_{Q}X)}^{-1}{X}^{{\prime} }{M}_{Q}(y-{\rho }^{* }({W}_{N}\otimes {I}_{T})y),$$

(A6)

$${\hat{\sigma }}_{v}^{2}({\rho }^{* })=\frac{1}{N(T-L)-K}e{({\rho }^{* })}^{{\prime} }e({\rho }^{* }),$$

(A7)

respectively, where $e({\rho }^{* })=y-{\rho }^{* }({W}_{N}\otimes {I}_{T})y-X{\hat{\beta }}_{W}({\rho }^{* })$. By substituting the closed form solutions for the parameters β(ρ^*) and ${\sigma }_{v}^{2}({\rho }^{* })$ to Eq. (A4), we can concentrate out β and ${\sigma }_{v}^{2}$, and the concentrated log-likelihood function with single parameter ρ is of the form:

$$logL(y;\rho )=C-\frac{NT}{2}log\left[e{(\rho )}^{{\prime} }e(\rho )\right]+Tlog| {I}_{N}-\rho W| ,$$

(A8)

where C is a constant term that is not a function of ρ. By maximizing the concentrated log-likelihood function Eq. (A8) with respect to ρ, we can obtain the optimal solution for ρ. Even if there is no closed-form solution for ρ, we can find a numerical solution because the equation is concave in ρ. Finally, the estimators for β and σ² can be calculated by plugging ${\rho }^{* }=\hat{\rho }$ in Eq. (A6) and Eq. (A7).

The asymptotic variance-covariance matrix of parameters (β, ρ, σ²) is given by:

$$\begin{array}{rc}Asy.\,Var(\beta ,\rho ,{\sigma }_{v}^{2})={\left[\begin{array}{ccc}\frac{1}{{\sigma }_{v}^{2}}{\tilde{X}}^{{}^{{\prime} }}\tilde{X}&\frac{1}{{\sigma }_{v}^{2}}{\tilde{X}}^{{}^{{\prime} }}({W}^{* }\otimes {I}_{T})\tilde{X}\beta &{{{\boldsymbol{0}}}}\\ -&T\cdot tr({W}^{* }{W}^{* }+{W}^{{* }^{{\prime} }}{W}^{* })+\frac{1}{{\sigma }_{v}^{2}}{\beta }^{{}^{{\prime} }}{\tilde{X}}^{{}^{{\prime} }}({W}^{{* }^{{\prime} }}{W}^{* }\otimes {I}_{T})\tilde{X}\beta &\frac{T}{{\sigma }_{v}^{2}}tr({W}^{* })\\ -&-&\frac{NT}{2{\sigma }_{v}^{4}}\end{array}\right]}^{-1},\end{array}$$

(A9)

where $\tilde{X}={M}_{Q}X$ and ${W}^{* }=W{({I}_{N}-\rho W)}^{-1}$.

1.2 Generalized least squares estimator

Alternatively, we can estimate Eq. (5) by generalized least squares(GLS). Denote the variance-covariance matrix of the composite error ε = QU + V as cov(ε) = Ω.

The GLS estimator is the SAR estimator applied to the following transformed equation:

$$\begin{array}{ll}{\sigma }_{v}{{{\Omega }}}^{-1/2}y=\rho {\sigma }_{v}{{{\Omega }}}^{-1/2}({W}_{N}\otimes {I}_{T})y+{\sigma }_{v}{{{\Omega }}}^{-1/2}X\beta +{\sigma }_{v}{{{\Omega }}}^{-1/2}{{{\boldsymbol{Z}}}}\gamma +\, {\sigma }_{v}{{{\Omega }}}^{-1/2}{{{\boldsymbol{R}}}}{\delta }_{0}+{\sigma }_{v}{{{\Omega }}}^{-1/2}\varepsilon ,\end{array}$$

(A10)

where ε = QU + V, ${{\Omega }}=cov(\varepsilon )={\sigma }_{v}^{2}{I}_{NT}+Q({I}_{N}\otimes {{\Delta }}){Q}^{{\prime} }$. The estimation procedure of Eq. (A10) is same as the procedure for within-estimation. Let η = (β, γ, δ₀). Assuming we know the true value of ρ = ρ^*, the GLS estimators of η(ρ^*) are

$$\begin{array}{ll}{\hat{\eta }}_{G}({\rho }^{* })={[{(X,{{{\boldsymbol{Z}}}},{{{\boldsymbol{R}}}})}^{{\prime} }{\sigma }_{v}^{2}{{{\Omega }}}^{-1}(X,{{{\boldsymbol{Z}}}},{{{\boldsymbol{R}}}})]}^{-1}{(X,{{{\boldsymbol{Z}}}},{{{\boldsymbol{R}}}})}^{{\prime} }{\sigma }_{v}^{2}{{{\Omega }}}^{-1}(y-{\rho }^{* }({W}_{N}\otimes {I}_{T})y)\\ \qquad\quad\,\,={[{(X,{{{\boldsymbol{Z}}}},{{{\boldsymbol{R}}}})}^{{\prime} }{{{\Omega }}}^{-1}(X,{{{\boldsymbol{Z}}}},{{{\boldsymbol{R}}}})]}^{-1}{(X,{{{\boldsymbol{Z}}}},{{{\boldsymbol{R}}}})}^{{\prime} }{{{\Omega }}}^{-1}(y-{\rho }^{* }({W}_{N}\otimes {I}_{T})y)\\ \qquad\quad\,\,={[{(X,{{{\boldsymbol{Z}}}},{{{\boldsymbol{R}}}})}^{{\prime} }{{{\Omega }}}^{-1}(X,{{{\boldsymbol{Z}}}},{{{\boldsymbol{R}}}})]}^{-1}{(X,{{{\boldsymbol{Z}}}},{{{\boldsymbol{R}}}})}^{{\prime} }{{{\Omega }}}^{-1}y-{\rho }^{* }{[{(X,{{{\boldsymbol{Z}}}},{{{\boldsymbol{R}}}})}^{{\prime} }{{{\Omega }}}^{-1}(X,{{{\boldsymbol{Z}}}},{{{\boldsymbol{R}}}})]}^{-1}{(X,{{{\boldsymbol{Z}}}},{{{\boldsymbol{R}}}})}^{{\prime} }{{{\Omega }}}^{-1}({W}_{N}\otimes {I}_{T})y.\end{array}$$

(A11)

Hence, the GLS estimators of η can be represented as a difference of OLS estimators of regressing $\widetilde{y}$ on $(\widetilde{X},\widetilde{{{{\boldsymbol{Z}}}}},\widetilde{{{{\boldsymbol{R}}}}})$ and regressing $\widetilde{({W}_{N}\otimes {I}_{T})y}$ on $(\widetilde{X},\widetilde{{{{\boldsymbol{Z}}}}},\widetilde{{{{\boldsymbol{R}}}}})$ premultiplied by the spatial autoregressive coefficient ρ^*, where tilde represents GLS transformation. Ω can be estimated by:

$$\hat{{{\Omega }}}({\rho }^{* })={\hat{\sigma }}_{v}^{2}{I}_{NT}+Q({I}_{N}\otimes \hat{{{\Delta }}}({\rho }^{* })){Q}^{{\prime} }.$$

(A12)

Following Cornwell et al. (1990), Δ can be estimated as

$$\hat{{{\Delta }}}({\rho }^{* })=\frac{1}{N}\mathop{\sum }\limits_{i=1}^{N}\left[{({R}^{{\prime} }R)}^{-1}{R}^{{\prime} }{e}_{i}{e}_{i}^{{\prime} }R{({R}^{{\prime} }R)}^{-1}-{\hat{\sigma }}_{v}^{2}{({R}^{{\prime} }R)}^{-1}\right],$$

(A13)

where ${e}_{i}={M}_{R}y-{\rho }^{* }{M}_{R}({W}_{N}\otimes {I}_{T})y-{M}_{R}X{\hat{\beta }}_{W}({\rho }^{* }){| }_{i}$, which represents the IV residuals for individual i, and ${M}_{R}={{{\boldsymbol{R}}}}{({{{{\boldsymbol{R}}}}}^{{\prime} }{{{\boldsymbol{R}}}})}^{-1}{{{{\boldsymbol{R}}}}}^{{\prime} }$ is the projection onto the column space of R.

Consider the likelihood function of Eq. (A10). Since σ_vΩ^−1/2ε has mean zero and variance ${\sigma }_{v}^{2}$, the likelihood function is written in the form of

$$logL(\eta ,\rho ;y)=-\frac{NT}{2}log(2\pi {\sigma }_{v}^{2})+Tlog| {I}_{N}-\rho W| -\frac{1}{2}{\varepsilon }^{{\prime} }{{{\Omega }}}^{-1}\varepsilon ,$$

(A14)

where ε = y − ρ(W_N ⊗ I_T)y − Xβ − Zγ − Rδ₀. Substitution of Eq. (A11) and Eq. (A12) into Eq. (A14) gives a concentrated likelihood function as follows:

$$logL(\rho ;y)=C-\frac{NT}{2}log\left[e{(\rho )}^{{\prime} }e(\rho )\right]+Tlog| {I}_{N}-\rho W| ,$$

(A15)

where $e(\rho )={\hat{\sigma }}_{v}\hat{{{\Omega }}}{(\rho )}^{-1/2}y-\rho {\hat{\sigma }}_{v}\hat{{{\Omega }}}{(\rho )}^{-1/2}({W}_{N}\otimes {I}_{T})y-{\hat{\sigma }}_{v}\hat{{{\Omega }}}{(\rho )}^{-1/2}X\hat{\beta }-{\hat{\sigma }}_{v}\hat{{{\Omega }}}{(\rho )}^{-1/2}{{{\boldsymbol{Z}}}}\hat{\gamma }-{\hat{\sigma }}_{v}{\hat{{{\Omega }}(\rho )}}^{-1/2}{{{\boldsymbol{R}}}}\hat{{\delta }_{0}}$, and C is a constant term that is not a function of ρ. Finally, as is the case of within estimator, we can obtain the estimators for η and Ω using the estimate of ρ from Eq. (A15).

1.3 Implementation

For the implementation, we combine the procedure suggested by Elhorst (2014) and a typical two-stage approach of FGLS. In this section we discuss the implementation of within estimator and then turn to the implementation of the GLS estimator. The implementation consists of the following steps.

1.3.1 [Within Estimator]

From Eq. (A6), it is easy to show that ${\hat{\beta }}_{W}={b}_{0}-{\rho }^{* }{b}_{1}$, where b₀ and b₁ are the OLS estimators of regressing M_Qy and M_Q(W_N ⊗ I_T)y on M_QX, respectively. Similarly, the estimated residuals from Eq. (A5), e(ρ^*), can be expressed as e(ρ^*) = e₀ − ρ^*e₁, where e₀ and e₁ are the associated OLS residuals to b₀ and b₁, respectively. Hence the first step is obtaining b₀, b₁, e₀, and e₁. Second, we maximize Eq. (A8) with respect to ρ after replacing e(ρ) = e₀ − ρe₁, i.e.,

$$\mathop{\max }\limits_{\rho }logL(\rho | y)=C-\frac{NT}{2}log\left[{({e}_{0}-\rho {e}_{1})}^{{\prime} }({e}_{0}-\rho {e}_{1})\right]+Tlog| {I}_{N}-\rho W| .$$

(A16)

Third, by replacing ${\rho }^{* }=\hat{\rho }$ in Eq. (A6) and (A7) gives the within estimator, ${\hat{\beta }}_{W}$, and the estimated variance, ${\hat{\sigma }}^{2}$. Finally, the asymptotic variance-covariance matrix of parameters $({\hat{\beta }}_{W},\hat{\rho },{\hat{\sigma }}_{v})$ can be calculated by Eq. (A9).

1.3.2 [GLS Estimator]

Unlike the within estimator case, we are unable to find the separate OLS estimators of regressing σ_vΩ^−1/2y and σ_vΩ^−1/2(W_N ⊗ I_T)y on σ_vΩ^−1/2(X, Z, R) in advance of having $\hat{\rho }$, although Eq. (A11) is expressed as a subtraction of two terms. This is because the feasible Ω is obtainable only after we have a value for ρ. Instead of following the steps of within estimator, we can obtain $\hat{\rho }$ by simply maximizing the concentrated log-likelihood function (A15). Once we have $\hat{\rho }$, Eq. (A13), Eq. (A12), Eq. (A11) give ${{\Delta }}(\hat{\rho })$, ${{\Omega }}\hat{\rho }$, and ${\eta }_{G}(\hat{\rho })$ in order.

Appendix B: Estimation results with value-added dependent variable

Tables 8,9

Table 8 CSS vs SARCSS vs SDMCSS: Value-added

Table 9 Direct, Indirect, and Total Elasticity: Value-added

Appendix C: Sensitivity analyses

Table 10,11,12,13

Table 10 Direct, Indirect, and Total Elasticity: η = 0.1

Table 11 Direct, Indirect, and Total Elasticity: η = 0.5

Table 12 Direct, Indirect, and Total Elasticity: η = 1.0

Table 13 Direct, Indirect, and Total Elasticity: η = 1.5

Appendix D: Industry classifications and ISIC Rev.3 codes

Table 14

Table 14 Industry classifications and codes