ENDOGENEITY IN STOCHASTIC PRODUCTION FRONTIER WITH ONE AND TWO-STEP MODELS: AN APPLICATION WITH MUNICIPAL DATA FROM THE BRAZILIAN AGRICULTURAL CENSUS

Oliveira, Kessys Lorrânya Peralta de; Andrade, Bernardo Borba de; Silva e Souza, Geraldo da; Castro, Bruno Soares de

doi:10.1590/0101-7438.2022.042.00243504

Kessys Lorrânya Peralta de Oliveira ^**Corresponding author

Federal Institute of Rondônia (IFRO), BR 435, KM 63, 76993-000, Colorado do Oeste, RO, Brazil - E-mail: kessys.oliveira@ifro.edu.br

https://orcid.org/0000-0002-9321-7756

Bernardo Borba de Andrade

Department of Statistics, University of Brasília (UnB), Computer Science and Statistics Building, Darcy Ribeiro University Campus, 70910-900, Brasília, DF, Brazil - E-mail: bbandrade@unb.br

https://orcid.org/0000-0003-4688-9733

Geraldo da Silva e Souza

Department of Statistics, University of Brasília (UnB), Computer Science and Statistics Building, Darcy Ribeiro University Campus, 70910-900, Brasília, DF, Brazil - E-mail: bbandrade@unb.br

Brazilian Agricultural Research Corporation (Embrapa/Sede), Av. W3 Norte, s/n, 70770-901, Brasília, DF, Brazil - E-mail: souzag832@gmail.com

https://orcid.org/0000-0002-6697-5383

Bruno Soares de Castro

Federal University of Rondônia (UNIR), R. Rio Amazonas, 351, 76900-726, Ji-Paraná, RO, Brazil - E-mail: bruno.castro@unir.br

https://orcid.org/0000-0002-9711-3401

*Corresponding author

Approach	Distribution of u	ln L	AIC	BIC	Cor(y, $\hat{y}$ )	Bias	RMSE
FIML	Half-normal	-874.8	1827.5	2081.4	0.8792	0.19	1.135
	Exponential	-1330.6	2739.3	2993.2	0.8769	0.32	1.138
	Truncated normal	-432.2	944.5	1204.9	0.7315	2.87	2.895
LIML	Half-normal	-892.1	1814.3	1911.9	0.8792	0.20	1.126
	Exponential	-1284.7	2599.3	2697.0	0.8772	0.26	1.199
	Truncated normal	-435.7	903.5	1007.6	0.7636	2.88	2.901

Variable	Coefficient	Standard error	z	P-value	Lower limit	Upper limit
Frontier
constant	2.477	0.125	19.852	0.000	2.233	2.722
ln(land)	0.110	0.014	7.639	0.000	0.082	0.139
ln(labor)	0.206	0.011	17.945	0.000	0.184	0.229
ln(techinputs)	0.509	0.018	28.977	0.000	0.475	0.544
region_north	0.066	0.055	1.201	0.230	-0.042	0.174
region_northeast	0.129	0.053	2.425	0.015	0.025	0.233
region_southeast	0.242	0.046	5.211	0.000	0.151	0.333
region_south	0.335	0.048	7.003	0.000	0.241	0.429
ln( $σ_{u}^{2}$ )
constant	5.637	0.201	28.040	0.000	5.243	6.031
techassist	0.931	0.525	1.772	0.076	-0.099	1.961
financing	-2.931	0.579	-5.061	0.000	-4.067	-1.796
gini	-9.938	0.299	-33.267	0.000	-10.523	-9.352
ln( $σ_{w}^{2}$ )
constant	-1.078	0.023	-47.806	0.000	-1.122	-1.034
η _techassist
constant	0.890	0.079	11.194	0.000	0.734	1.045
η _financing
constant	0.200	0.081	2.484	0.013	0.042	0.358
$σ_{w}^{2}$	0.340	0.008	44.359	0.000	0.325	0.355

Variable	Coefficient	Standard error	z	P-value	Lower limit	Upper limit
Frontier
constant	2.552	0.126	20.306	0.000	2.306	2.799
ln(land)	0.111	0.015	7.521	0.000	0.082	0.140
ln(labor)	0.202	0.012	16.813	0.000	0.178	0.226
ln(techinputs)	0.507	0.018	28.133	0.000	0.472	0.543
region_north	0.062	0.055	1.123	0.261	-0.046	0.170
region_northeast	0.131	0.053	2.455	0.014	0.026	0.235
region_southeast	0.231	0.046	4.995	0.000	0.140	0.321
region_south	0.319	0.047	6.736	0.000	0.226	0.412
ln( $σ_{u}^{2}$ )
constant	6.337	0.594	10.662	0.000	5.172	7.502
techassist	-0.182	0.578	-0.315	0.753	-1.314	0.950
financing	-2.215	0.617	-3.592	0.000	-3.423	-1.006
gini	-10.441	0.841	-12.420	0.000	-12.089	-8.794
ln( $σ_{w}^{2}$ )
constant	-1.065	0.023	-45.429	0.000	-1.111	-1.019
η _techassist
constant	0.664	0.080	8.335	0.000	0.508	0.820
η _financing
constant	0.238	0.080	2.986	0.003	0.082	0.395
$σ_{w}^{2}$	0.345	0.008	43.938	0.000	0.329	0.360

Approach	Wald	Likelihood ratio	P-value
FIML	201	342	0.000
LIML	149	208	0.000

Production factor	Coefficient	Standard error	t	P-value	Lower limit	Upper limit
FIML
Labor	0.25	0.01	18.30	0.00	0.22	0.28
Land	0.13	0.02	7.69	0.00	0.10	0.17
Capital	0.62	0.02	31.01	0.00	0.58	0.66
Returns to scale	0.83	0.01
LIML
Labor	0.25	0.01	18.13	0.00	0.22	0.27
Land	0.14	0.02	7.91	0.00	0.10	0.17
Capital	0.62	0.02	31.47	0.00	0.58	0.66
Returns to scale	0.82	0.01

y_{i} = x_{1 i}^{⊤} β + v_{i} - u_{i}, x_{i} = Z_{i} δ + ε_{i}, [\begin{matrix} {\tilde{ε}}_{i} \\ v_{i} \end{matrix}] \equiv [\begin{matrix} Ω^{- \frac{1}{2}} ε_{i} \\ v_{i} \end{matrix}] ~ N ([\begin{matrix} 0 \\ 0 \end{matrix}], [\begin{matrix} I_{p} & σ_{v i} ρ \\ σ_{v i} ρ^{⊤} & σ_{v i}^{2} \end{matrix}]) .

\ln L_{y | x} (θ) \propto \sum_{i = 1}^{n} \{- \frac{1}{2} \ln σ_{i}^{2} + \ln Φ (- \frac{e_{i} λ_{i}}{σ_{i}}) - \frac{e_{i}^{2}}{2 σ_{i}^{2}}\}, \ln L_{y | x} (θ) \propto \sum_{i = 1}^{n} \{- \frac{1}{2} \ln σ_{u i}^{2} + \frac{σ_{w i}^{2}}{2 σ_{u i}^{2}} + \ln Φ (\frac{- e_{i} - \frac{σ_{w i}^{2}}{σ_{u i}}}{σ_{w i}}) + \frac{e_{i}}{σ_{u i}}\}, \ln L_{y | x} (θ) \propto \sum_{i = 1}^{n} \{- \frac{1}{2} \ln σ^{2} - \ln Φ (\frac{μ_{i}}{σ_{u}}) + \ln Φ (\frac{μ_{i}}{σ λ} - \frac{e_{i} λ}{σ}) - \frac{1}{2} {(\frac{e_{i} + μ_{i}}{σ})}^{2}\}, \ln L_{x} (θ) = \sum_{i = 1}^{n} \{\frac{- p \ln (2 π) - \ln (| Ω |) - ε_{i}^{⊤} Ω^{- 1} ε_{i}}{2}\}, e_{i} = y_{i} - x_{1 i}^{⊤} β - \frac{σ_{w i}}{σ_{c w}} η^{⊤} (x_{i} - Z_{i} δ), ε_{i} = x_{i} - Z_{i} δ, σ_{i}^{2} = σ_{u i}^{2} + σ_{w i}^{2}, λ_{i} = \frac{σ_{u i}}{σ_{w i}}, μ_{i} \neq 0 \Leftrightarrow u_{i} ~ ℕ^{+} (μ_{i}, σ_{u}^{2}),

U (δ) = \sum_{i = 1}^{n} Z_{i}^{⊤} ε_{i} Ω^{- 1} - \sum_{i = 1}^{n} Z_{i}^{⊤} \{\frac{σ_{w i}}{σ_{c w}} [\frac{e_{i}}{σ_{i}^{2}} + \frac{λ_{i}}{σ_{i}} A_{i}]\} η^{⊤}, U (β) = \sum_{i = 1}^{n} x_{1 i}^{⊤} \{\frac{e_{i}}{σ_{i}^{2}} + \frac{λ_{i}}{σ_{i}} A_{i}\}, U (η) = \sum_{i = 1}^{n} ε_{i}^{⊤} \{\frac{σ_{w i}}{σ_{c w}} [\frac{e_{i}}{σ_{i}^{2}} + \frac{λ_{i}}{σ_{i}} A_{i}]\}, U (φ_{u}) = \sum_{i = 1}^{n} x_{2 i}^{⊤} \{\frac{1}{2 σ_{i}^{2}} [\frac{e_{i}^{2}}{σ_{i}^{2}} - \frac{e_{i}}{λ_{i} σ_{i}} A_{i} - 1]\} σ_{u i}^{2}, U (φ_{w}) = \sum_{i = 1}^{n} x_{3 i}^{⊤} \{\frac{1}{2 σ_{i}^{2}} [\frac{e_{i}^{2}}{σ_{i}^{2}} + \frac{e_{i} λ_{i}}{σ_{i}} A_{i} (2 + λ_{i}^{2}) - 1]\} σ_{w i}^{2} + + \sum_{i = 1}^{n} {\tilde{x}}_{3 i}^{⊤} \{\frac{1}{2} η^{⊤} ε_{i} \frac{σ_{w i}}{σ_{c w}} [\frac{e_{i}}{σ_{i}^{2}} + \frac{λ_{i}}{σ_{i}} A_{i}]\},

U (δ) = \sum_{i = 1}^{n} Z_{i}^{⊤} ε_{i} Ω^{- 1} - \sum_{i = 1}^{n} Z_{i}^{⊤} \{\frac{σ_{w i}}{σ_{c w}} [\frac{1}{σ_{w i}} B_{i} - \frac{1}{σ_{u i}}]\} η^{⊤}, U (η) = \sum_{i = 1}^{n} ε_{i}^{⊤} \{\frac{σ_{w i}}{σ_{c w}} [\frac{1}{σ_{w i}} B_{i} - \frac{1}{σ_{u i}}]\}, U (β) = \sum_{i = 1}^{n} x_{1 i}^{⊤} \{\frac{1}{σ_{w i}} B_{i} - \frac{1}{σ_{u i}}\}, U (φ_{u}) = \sum_{i = 1}^{n} x_{2 i}^{⊤} \{\frac{1}{2 σ_{u i}^{2}} [\frac{σ_{w i}}{σ_{u i}} B_{i} - \frac{σ_{w i}^{2}}{σ_{u i}^{2}} - \frac{e_{i}}{σ_{u i}} - 1]\} σ_{u i}^{2}, U (φ_{w}) = \sum_{i = 1}^{n} x_{3 i}^{⊤} \{\frac{1}{2 σ_{u i}^{2}} + \frac{1}{2 σ_{w i}} B_{i} [\frac{e_{i}}{σ_{w i}^{2}} - \frac{1}{σ_{u i}}]\} σ_{w i}^{2} + + \sum_{i = 1}^{n} {\tilde{x}}_{3 i}^{⊤} \{\frac{1}{2} η^{⊤} ε_{i} \frac{σ_{w i}}{σ_{c w}} [\frac{1}{σ_{w i}} B_{i} - \frac{1}{σ_{u i}}]\},

U (δ) = \sum_{i = 1}^{n} Z_{i}^{⊤} ε_{i} Ω^{- 1} - \sum_{i = 1}^{n} Z_{i}^{⊤} \{\frac{σ_{w}}{σ_{c w}} [\frac{e_{i} + μ_{i}}{σ^{2}} + \frac{λ}{σ} D_{i}]\} η^{⊤}, U (η) = \sum_{i = 1}^{n} ε_{i}^{⊤} \{\frac{σ_{w}}{σ_{c w}} [\frac{e_{i} + μ_{i}}{σ^{2}} + \frac{λ}{σ} D_{i}]\}, U (β) = \sum_{i = 1}^{n} x_{1 i}^{⊤} \{\frac{e_{i} + μ_{i}}{σ^{2}} + \frac{λ}{σ} D_{i}\}, U (τ) = \sum_{i = 1}^{n} x_{2 i}^{⊤} \{\frac{1}{λ σ} D_{i} - \frac{\sqrt{λ^{- 2} + 1}}{σ} C_{i} - \frac{e_{i} + μ_{i}}{σ^{2}}\}, U (φ_{u}) = \sum_{i = 1}^{n} \{\frac{1}{2 σ_{u}^{2}} \frac{μ_{i}}{σ_{u}} C_{i} + \frac{1}{2 σ^{2}} [\frac{(e_{i} + μ_{i})^{2}}{σ^{2}} - \frac{1}{λ σ} D_{i} (2 μ_{i} + e_{i} + \frac{μ_{i}}{λ^{2}}) - 1]\} σ_{u}^{2}, U (φ_{w}) = \sum_{i = 1}^{n} \{\frac{1}{2 σ^{2}} [\frac{(e_{i} + μ_{i})^{2}}{σ^{2}} + \frac{λ}{σ} D_{i} (μ_{i} + 2 e_{i} + e_{i} λ^{2}) - 1]\} σ_{w}^{2},

V_{1} = (q \times q) Asymptotic variance matrix of {\hat{θ}}_{1} based on \ln L_{1} (θ_{1}), V_{2} = (p \times p) Asymptotic variance matrix of {\hat{θ}}_{2} based on \ln L_{2} (θ_{2} | θ_{1}), C = (p \times q) matrix given by E [(\frac{\partial \ln L_{2}}{\partial θ_{2}}) (\frac{\partial \ln L_{2}}{\partial θ_{1}^{⊤}})], R = (p \times q) matrix given by E [(\frac{\partial \ln L_{2}}{\partial θ_{2}}) (\frac{\partial \ln L_{1}}{\partial θ_{1}^{⊤}})] .

{\hat{V}}_{1} = {[- \frac{1}{n} \sum_{i = 1}^{n} (\frac{\partial^{2} \ln f_{1 i}}{\partial {\hat{θ}}_{1} \partial {\hat{θ}}_{1}^{⊤}})]}^{- 1}, \hat{C} = \frac{1}{n} \sum_{i = 1}^{n} (\frac{\partial \ln f_{2 i}}{\partial {\hat{θ}}_{2}}) (\frac{\partial \ln f_{2 i}}{\partial {\hat{θ}}_{1}^{⊤}}), {\hat{V}}_{2} = {[- \frac{1}{n} \sum_{i = 1}^{n} (\frac{\partial^{2} \ln f_{2 i}}{\partial {\hat{θ}}_{2} \partial {\hat{θ}}_{2}^{⊤}})]}^{- 1}, \hat{R} = \frac{1}{n} \sum_{i = 1}^{n} (\frac{\partial \ln f_{2 i}}{\partial {\hat{θ}}_{2}}) (\frac{\partial \ln f_{1 i}}{\partial {\hat{θ}}_{1}^{⊤}}) .

U_{1} (δ) = - 2 \sum_{i = 1}^{n} Z_{i}^{⊤} ε_{i}, U_{2} (δ) = - \sum_{i = 1}^{n} Z_{i}^{⊤} \{\frac{e_{i}}{σ_{i}^{2}} + \frac{λ_{i}}{σ_{i}} A_{i}\} η^{⊤}, U_{2} (η) = \sum_{i = 1}^{n} {\hat{ε}}_{i}^{⊤} \{\frac{e_{i}}{σ_{i}^{2}} + \frac{λ_{i}}{σ_{i}} A_{i}\}, U_{2} (β) = \sum_{i = 1}^{n} x_{1 i}^{⊤} \{\frac{e_{i}}{σ_{i}^{2}} + \frac{λ_{i}}{σ_{i}} A_{i}\}, U_{2} (φ_{u}) = \sum_{i = 1}^{n} x_{2 i}^{⊤} \{\frac{1}{2 σ_{i}^{2}} [\frac{e_{i}^{2}}{σ_{i}^{2}} - \frac{e_{i}}{λ_{i} σ_{i}} A_{i} - 1]\} σ_{u i}^{2}, U_{2} (φ_{w}) = \sum_{i = 1}^{n} x_{3 i}^{⊤} \{\frac{1}{2 σ_{i}^{2}} [\frac{e_{i}^{2}}{σ_{i}^{2}} + \frac{e_{i} λ_{i}}{σ_{i}} (2 + λ_{i}^{2}) A_{i} - 1]\} σ_{w i}^{2},

U_{1} (δ) = - 2 \sum_{i = 1}^{n} Z_{i}^{⊤} ε_{i}, U_{2} (δ) = - \sum_{i = 1}^{n} Z_{i}^{⊤} \{\frac{1}{σ_{w i}} B_{i} - \frac{1}{σ_{u i}}\} η^{⊤}, U_{2} (η) = \sum_{i = 1}^{n} {\hat{ε}}_{i}^{⊤} \{\frac{1}{σ_{w i}} B_{i} - \frac{1}{σ_{u i}}\}, U_{2} (β) = \sum_{i = 1}^{n} x_{1 i}^{⊤} \{\frac{1}{σ_{w i}} B_{i} - \frac{1}{σ_{u i}}\}, U_{2} (φ_{u}) = \sum_{i = 1}^{n} x_{2 i}^{⊤} \{\frac{1}{2 σ_{u i}^{2}} [\frac{σ_{w i}}{σ_{u i}} B_{i} - \frac{σ_{w i}^{2}}{σ_{u i}^{2}} - \frac{e_{i}}{σ_{u i}} - 1]\} σ_{u i}^{2}, U_{2} (φ_{w}) = \sum_{i = 1}^{n} x_{3 i}^{⊤} \{\frac{1}{2 σ_{u i}^{2}} + \frac{1}{2 σ_{w i}} [\frac{e_{i}}{σ_{w i}^{2}} - \frac{1}{σ_{u i}}] B_{i}\} σ_{w i}^{2},

U_{1} (δ) = - 2 \sum_{i = 1}^{n} Z_{i}^{⊤} ε_{i}, U_{2} (δ) = - \sum_{i = 1}^{n} Z_{i}^{⊤} \{\frac{e_{i} + μ_{i}}{σ^{2}} + \frac{λ}{σ} D_{i}\} η^{⊤}, U_{2} (η) = \sum_{i = 1}^{n} {\hat{ε}}_{i}^{⊤} \{\frac{e_{i} + μ_{i}}{σ^{2}} + \frac{λ}{σ} D_{i}\}, U_{2} (β) = \sum_{i = 1}^{n} x_{1 i}^{⊤} \{\frac{e_{i} + μ_{i}}{σ^{2}} + \frac{λ}{σ} D_{i}\}, U_{2} (τ) = \sum_{i = 1}^{n} x_{2 i}^{⊤} \{\frac{1}{λ σ} D_{i} - \frac{\sqrt{λ^{- 2} + 1}}{σ} C_{i} - \frac{e_{i} + μ_{i}}{σ^{2}}\}, U_{2} (φ_{u}) = \sum_{i = 1}^{n} \{\frac{1}{2 σ_{u}^{2}} \frac{μ_{i}}{σ_{u}} C_{i} + \frac{1}{2 σ^{2}} [\frac{(e_{i} + μ_{i})^{2}}{σ^{2}} - \frac{1}{λ σ} D_{i} (2 μ_{i} + e_{i} + \frac{μ_{i}}{λ^{2}}) - 1]\} σ_{u}^{2}, U_{2} (φ_{w}) = \sum_{i = 1}^{n} \{\frac{1}{2 σ^{2}} [\frac{(e_{i} + μ_{i})^{2}}{σ^{2}} + \frac{λ}{σ} D_{i} (μ_{i} + 2 e_{i} + e_{i} λ^{2}) - 1]\} σ_{w}^{2},

\ln (i n c o m e_{i}) = β_{0} + β_{1} \ln (l a n d_{i}) + β_{2} \ln (l a b o r_{i}) + β_{3} \ln (t e c h i n p u t s_{i}) + β_{4} r e g i o n_{n o r t h_{i}} + + β_{5} r e g i o n_{n o r t h e a s t_{i}} + β_{6} r e g i o n_{s o u t h e a s t_{i}} + β_{7} r e g i o n_{s o u t h_{i}} + v_{i} - u_{i}, \ln (σ_{u i}^{2}) = φ_{u 0} + φ_{u 1} t e c h a s s i s t_{i} + φ_{u 2} f i n a n c i n g_{i} + φ_{u 3} g i n i_{i}, \ln (σ_{w i}^{2}) = φ_{w 0}, t e c h a s s i s t_{i} = δ_{0} + δ_{1} \ln (l a n d_{i}) + δ_{2} \ln (l a b o r_{i}) + δ_{3} \ln (t e c h i n p u t s_{i}) + δ_{4} r e g i o n_{n o r t h_{i}} + + δ_{5} r e g i o n_{n o r t h e a s t_{i}} + δ_{6} r e g i o n_{s o u t h e a s t_{i}} + δ_{7} r e g i o n_{s o u t h_{i}} + δ_{8} s o c i a l_{i} + + δ_{9} d e m o g r a p h i c_{i} + δ_{10} e n v i r o n m e n t_{i} + δ_{11} g i n i_{i} + ε_{1 i}, f i n a n c i n g_{i} = γ_{0} + γ_{1} \ln (l a n d_{i}) + γ_{2} \ln (l a b o r_{i}) + γ_{3} \ln (t e c h i n p u t s_{i}) + γ_{4} r e g i o n_{n o r t h_{i}} + + γ_{5} r e g i o n_{n o r t h e a s t_{i}} + γ_{6} r e g i o n_{s o u t h e a s t_{i}} + γ_{7} r e g i o n_{s o u t h_{i}} + γ_{8} s o c i a l_{i} + + γ_{9} d e m o g r a p h i c_{i} + γ_{10} e n v i r o n m e n t_{i} + γ_{11} g i n i_{i} + ε_{2 i},

Variable	Coefficient	Standard error	z	P-value	Lower limit	Upper limit
techassist
constant	-0.260	0.036	-7.277	0.000	-0.331	-0.190
ln(land)	0.005	0.004	1.247	0.212	-0.003	0.013
ln(labor)	0.010	0.003	3.071	0.002	0.004	0.017
ln(techinputs)	0.082	0.005	17.294	0.000	0.073	0.091
region_north	0.048	0.015	3.129	0.002	0.018	0.078
region_northeast	0.048	0.015	3.201	0.001	0.019	0.078
region_southeast	0.071	0.013	5.292	0.000	0.045	0.097
region_south	0.165	0.014	11.542	0.000	0.137	0.194
social	0.406	0.023	17.754	0.000	0.361	0.451
demographic	-0.016	0.028	-0.560	0.575	-0.071	0.040
environment	0.040	0.034	1.180	0.238	-0.027	0.107
gini	-0.579	0.031	-18.884	0.000	-0.639	-0.519
financing
constant	-0.508	0.036	-13.939	0.000	-0.580	-0.437
ln(land)	0.027	0.004	6.831	0.000	0.019	0.035
ln(labor)	-0.005	0.003	-1.381	0.167	-0.011	0.002
ln(techinputs)	0.129	0.005	26.873	0.000	0.120	0.138
region_north	-0.076	0.015	-4.884	0.000	-0.106	-0.045
region_northeast	-0.080	0.015	-5.241	0.000	-0.110	-0.050
region_southeast	-0.057	0.014	-4.223	0.000	-0.084	-0.031
region_south	0.104	0.015	7.168	0.000	0.076	0.133
social	0.156	0.024	6.590	0.000	0.109	0.202
demographic	-0.222	0.029	-7.576	0.000	-0.279	-0.165
environment	-0.398	0.036	-11.173	0.000	-0.467	-0.328
gini	-0.197	0.032	-6.087	0.000	-0.260	-0.133

Variable	Coefficient	Standard error	z	P-value	Lower limit	Upper limit
techassist
constant	-0.293	0.036	-8.148	0.000	-0.363	-0.222
ln(land)	0.003	0.004	0.864	0.388	-0.004	0.011
ln(labor)	0.004	0.003	1.167	0.243	-0.003	0.010
ln(techinputs)	0.079	0.005	16.716	0.000	0.070	0.088
region_north	0.055	0.015	3.598	0.000	0.025	0.085
region_northeast	0.047	0.015	3.124	0.002	0.018	0.077
region_southeast	0.059	0.013	4.419	0.000	0.033	0.085
region_south	0.155	0.014	10.819	0.000	0.127	0.183
social	0.487	0.022	21.740	0.000	0.443	0.531
demographic	-0.003	0.029	-0.109	0.913	-0.060	0.054
environment	-0.018	0.035	-0.510	0.610	-0.086	0.051
gini	-0.426	0.029	-14.643	0.000	-0.483	-0.369
financing
constant	-0.521	0.036	-14.313	0.000	-0.593	-0.450
ln(land)	0.027	0.004	6.682	0.000	0.019	0.035
ln(labor)	-0.007	0.003	-2.186	0.029	-0.014	-0.001
ln(techinputs)	0.128	0.005	26.669	0.000	0.118	0.137
region_north	-0.073	0.015	-4.698	0.000	-0.103	-0.042
region_northeast	-0.081	0.015	-5.269	0.000	-0.111	-0.051
region_southeast	-0.062	0.014	-4.588	0.000	-0.089	-0.036
region_south	0.100	0.015	6.879	0.000	0.072	0.129
social	0.189	0.023	8.320	0.000	0.145	0.234
demographic	-0.217	0.029	-7.359	0.000	-0.275	-0.159
environment	-0.421	0.035	-11.912	0.000	-0.491	-0.352
gini	-0.134	0.030	-4.527	0.000	-0.192	-0.076

[1] *Corresponding author

Brasil

Brasil

ENDOGENEITY IN STOCHASTIC PRODUCTION FRONTIER WITH ONE AND TWO-STEP MODELS: AN APPLICATION WITH MUNICIPAL DATA FROM THE BRAZILIAN AGRICULTURAL CENSUS

ABSTRACT