Table 2. Hypothesis test for the parameters of the frontier model of environmental inefficiency for the farms studied.

Null Hypothesis	Log (likelihood)	${\chi }_{0.95}^2-\mathit{\text{valor}}$ †	Statistics Test	Decision
${\mathrm{{\rm H}}}_0=\gamma ={\delta }_0=\dots {\delta }_4=0$	78.77	9.48	14.05 *	Rechaza H 0
${\mathrm{{\rm H}}}_0=\gamma =0$	-1.93	7.85	36.29 *	Rechaza H 0
${\mathrm{{\rm H}}}_0={\delta }_1={\delta }_2={\delta }_3=0$	71.44	7.85	14.05 *	Rechaza H 0

^† The likelihood-ratio statistical test, λ= -2{log [likelihood (H0)] - [log [likelihood (H1)]} has about an x-distribution with estimators equal to the number of estimators assumed to be zero in the null hypothesis, H ₀; subsequently H ₁ is true. If the estimator, γ, is zero, then the variances in the inefficiency effects are zero and so the model reduces to the traditional mean response function. In this case, the estimators, δ0 and δ1, are not defined. Henceforth, the critical value for the statistical test for this second hypothesis was obtained from the χ ₁ ² distribution.

^*One asterisk in the estimate of the statistical test indicated that it exceeds the 95th percentile for the corresponding Chi-square distribution (χ ²) and consequently the null hypothesis was rejected.