spefa: spatial stochastic frontier with fixed effects and endogeneity

1. What this package estimates

spefa implements the maximum-likelihood estimator developed in

Giannini, M. (2025). A spatial stochastic frontier model with fixed effects and endogenous environmental variables. Spatial Economic Analysis, 20(3), 420–441. doi:10.1080/17421772.2024.2414962

The model is a stochastic production (or cost) frontier on panel data that simultaneously accounts for three features that the earlier literature treated separately:

individual fixed effects $\alpha_i$, removed by first differencing (following Wang and Ho, 2010), which avoids the incidental-parameters problem and sidesteps the singularity of the within transformation in spatial panels;
endogenous regressors and environmental variables, corrected through a Gaussian control function obtained from a Cholesky factorisation of the error covariance (following Kutlu, Tran and Tsionas, 2020);
spatial spillovers, through a spatial-autoregressive (SAR) term $\lambda W y$.

For unit $i$ and period $t$ the frontier is

\[ y_{it} = \alpha_i + \lambda \sum_j w_{ij} y_{jt} + x_{it}'\beta - u_{it} + v_{it}, \qquad u_{it} = h_{it}\,u_i^{*}, \]

with $h_{it} = \exp(z_{it}'\phi) > 0$ a multiplicative scaling and a time-invariant $u_i^{*} \sim N^{+}(\mu,\sigma_u^2)$ (half-normal when $\mu = 0$). Endogenous variables $q_j$ obey a reduced form $q_{j,it} = z_{j,it}'\delta_j + \varepsilon_{j,it}$, and endogeneity is the correlation $\rho_j = \mathrm{corr}(\varepsilon_j, v)$. The full derivation of the likelihood is in Giannini (2025); the package implements it in closed form and maximises it numerically.

The implementation depends only on base R (the stats package).

2. Installation

install.packages("spefa_0.1.0.tar.gz", repos = NULL, type = "source")
library(spefa)

3. The data you supply

The estimator needs a balanced panel and a spatial weight matrix:

a data.frame in long format with a unit identifier and a time identifier;
the outcome y and the frontier regressors X (these may include endogenous variables);
for each endogenous variable, one or more instruments;
the variables that enter the inefficiency scaling $h_{it}$;
an $N \times N$ weight matrix W, with rows and columns ordered by the sorted unit id, and (typically) row-normalised.

4. The `spefa()` function and its options

spefa(formula, data, index, W,
      endogenous = NULL, scaling = NULL, mu = FALSE,
      control = list(maxit = 500, reltol = 1e-9))

Argument	Meaning
`formula`	Frontier equation, e.g. `y ~ x1 + q1`. The intercept is removed by first differencing, so it is irrelevant. Endogenous regressors are listed here like any other regressor.
`data`	A balanced panel `data.frame`.
`index`	Length-2 character vector `c(unit, time)`, e.g. `c("id","time")`.
`W`	The $N\times N$ spatial weight matrix (sorted-id order).
`endogenous`	Named list mapping each endogenous variable to its instruments, e.g. `list(q1 = ~ z1, q2 = ~ z2 + z3)`. Use `NULL` for no endogeneity (the model is then a plain spatial SF with fixed effects). An endogenous variable may appear in the frontier, in the scaling, or both.
`scaling`	One-sided formula of the variables entering $h_{it}=\exp(z_{it}'\phi)$, e.g. `~ x2 + q2`. There is no intercept (it is absorbed into $\sigma_u$). Needs at least one time-varying term.
`mu`	`FALSE` (default) gives a half-normal inefficiency ($\mu=0$); `TRUE` estimates the truncation point $\mu$ (truncated-normal).
`control`	List passed to `stats::optim` (`maxit`, `reltol`).

5. What is returned, and the available methods

spefa() returns an object of class "spefa". The estimated parameter vector contains the frontier slopes $\beta$, the instrument slopes $\delta_j$, the scaling coefficients $\phi$, the reduced-form variances $\sigma^2_{\varepsilon_j}$, the endogeneity correlations $\rho_j$, $\sigma^2_v$, $\sigma^2_u$, (optionally $\mu$) and the spatial parameter $\lambda$.

Method	Returns
`summary(fit)`	Coefficient table with standard errors, $z$-statistics and $p$-values; log-likelihood, AIC, BIC; convergence code; $\lambda$ and the endogeneity correlations $\rho$.
`coef(fit)`, `vcov(fit)`	Point estimates and the (delta-method) covariance matrix.
`logLik(fit)`, `AIC(fit)`, `BIC(fit)`, `nobs(fit)`	Standard fit statistics.
`efficiency(fit, spatial = TRUE)`	Conditional-mean (Jondrow et al., 1982) inefficiency, optionally rescaled by the spatial multiplier $(I-\lambda W)^{-1}$. Returns the $N\times T$ inefficiency matrix `u`, the efficiency matrix `eff = exp(-u)`, and the per-unit `Eu_i`.
`impacts(fit)`	Direct, indirect and total marginal effects (LeSage and Pace, 2009) of each frontier regressor through $(I-\lambda W)^{-1}$.

Standard errors come from the Hessian at the optimum (computed on the unconstrained scale) propagated to the natural parameters by the delta method.

6. A worked example

A small demo panel (spefademo, $N = 40$, $T = 12$) ships with the package and is used here. The data-generating values are $\beta=(0.5,0.5)$, $\delta=(1,1)$, $\phi=(0.3,0.3)$, $\sigma^2_\varepsilon=0.09$, $\sigma^2_v=0.04$, $\rho=(0.5,0.5)$, $\sigma^2_u=0.09$, $\lambda=0.5$; q1 and q2 are endogenous, q1 enters the frontier and q2 the scaling.

library(spefa)
data(spefademo)

fit <- spefa(y ~ x1 + q1, data = spefademo, index = c("id", "time"),
             W = spefaW, endogenous = list(q1 = ~ z1, q2 = ~ z2),
             scaling = ~ x2 + q2)

summary(fit)

Indicative output from a larger sample ($N=150$, to show parameter recovery; the bundled $N=40$ panel gives the same point estimates with wider standard errors):

               Estimate Std.Error  z value  Pr(>|z|)
x1              0.5002   0.0027    182.2    <2e-16 ***
q1              0.5007   0.0032    155.2    <2e-16 ***
delta.q1.z1     0.9991   0.0056    179.8    <2e-16 ***
delta.q2.z2     1.0029   0.0054    186.6    <2e-16 ***
x2              0.2995   0.0120     24.9    <2e-16 ***
q2              0.3051   0.0120     25.5    <2e-16 ***
rho.q1          0.4763   0.0134     35.5    <2e-16 ***
rho.q2          0.5095   0.0130     39.2    <2e-16 ***
sigma2_v        0.0391   0.0010     38.5    <2e-16 ***
sigma2_u        0.0879   0.0162      5.4    5e-08  ***
lambda          0.5027   0.0044    114.5    <2e-16 ***

Spatial parameter lambda = 0.5027
Endogeneity correlations rho (q1, q2): 0.476, 0.509

impacts(fit)            # direct / indirect / total marginal effects
ef <- efficiency(fit)   # spatially-corrected technical efficiency
summary(as.vector(ef$eff))

7. Reading the results

Endogeneity. A $\rho_j$ statistically different from zero signals endogeneity of variable $j$; the $z$-test in the coefficient table is a direct test of $H_0:\rho_j=0$. When all $\rho_j=0$ the control-function correction vanishes and the estimator collapses to the exogenous case.
Spatial dependence. $\lambda$ measures the strength of spillovers; the impacts() table splits each regressor’s effect into a direct (own-unit) and an indirect (spillover) component, with total $\approx \beta_k/(1-\lambda)$ under row-normalised W.
Efficiency. efficiency(fit)$eff gives unit-by-period technical efficiency in $(0,1]$; with spatial = TRUE the scores incorporate neighbours’ inefficiency through $(I-\lambda W)^{-1}$.

8. Practical notes and identification

The panel must be balanced, and the rows/columns of W must follow the sorted unit id used in data.
The inefficiency parameters $\phi$ and $\sigma_u$ are identified only when the scaling variables vary over time (otherwise first differencing removes the inefficiency) and when the inefficiency is non-negligible. If the inefficiency is very small, expect $\phi$ and $\sigma_u$ to be weakly identified.
The endogeneity correction maintains the usual control-function assumptions: joint normality of the structural and reduced-form errors, and inefficiency independent of the regressors.
The likelihood evaluates the reduced-form (instrument) density on the data and counts the Gaussian normalising constant once per unit; the truncated-normal terms use the log-CDF for numerical stability.

References

Giannini, M. (2025). A spatial stochastic frontier model with fixed effects and endogenous environmental variables. Spatial Economic Analysis, 20(3), 420–441.

Jondrow, J., Lovell, C. A. K., Materov, I. S., & Schmidt, P. (1982). On the estimation of technical inefficiency in the stochastic frontier production function model. Journal of Econometrics, 19(2–3), 233–238.

Kutlu, L., Tran, K. C., & Tsionas, M. G. (2020). A spatial stochastic frontier model with endogenous frontier and environmental variables. European Journal of Operational Research, 286(1), 389–399.

LeSage, J., & Pace, R. K. (2009). Introduction to Spatial Econometrics. CRC Press.

Wang, H.-J., & Ho, C.-W. (2010). Estimating fixed-effect panel stochastic frontier models by model transformation. Journal of Econometrics, 157(2), 286–296.

spefa: spatial stochastic frontier with fixed effects and endogeneity

Massimo Giannini (University of Rome Tor Vergata, Department of Enterprise Engineering)

2026-05-21

1. What this package estimates

2. Installation

3. The data you supply

4. The `spefa()` function and its options

5. What is returned, and the available methods

6. A worked example

7. Reading the results

8. Practical notes and identification

References

spefa: spatial stochastic frontier with fixed effects and endogeneity

Massimo Giannini (University of Rome Tor Vergata, Department of Enterprise Engineering)

2026-05-21

1. What this package estimates

2. Installation

3. The data you supply

4. The spefa() function and its options

5. What is returned, and the available methods

6. A worked example

7. Reading the results

8. Practical notes and identification

References

4. The `spefa()` function and its options