Conceptual framework
The stochastic frontier analysis (SFA) has been extensively used in the agriculture sector to measure the TE [29]. The SFA was preferred in many empirical studies due to its ability to manage the effect of data noise and inefficiency [33].Footnote 1 However, the limitation of the original SFA developed by Aigner et al. [34] and Meeusen, van Den Broeck [35] is the failure to address the issue of technological heterogeneity [33]. To address this limitation, economics literature has suggested a meta-frontier framework that allows the estimation and comparison of TEs of farms operating under different production technologies [36]. In this study, the production technology referred to consist of seed technologies.
The concept of meta-frontier was initially proposed by Hayami [37]. In this concept, he assumed that farmers have potential access to different production technologies, and various factors determine the adoption of such technologies. Later, Hayami, Ruttan [38] made a further assumption that the same production function can describe technical possibilities available to farmers classified into different groups. Battese, Rao [39] extended the meta-frontier approach and introduced a SMF production function that allows the estimation of TEs of firms classified into different groups. This method was further extended by Battese et al. [40] and O'Donnell et al. [41] to be a two-stage procedure for the estimation of meta-frontier. The first stage of this approach uses the SFA model to estimate the individual frontiers for each group. In the second stage, a mathematical programming technique, such as Data Envelopment Analysis (DEA) is used to estimate the meta-frontier by combining the group-specific stochastic production functions [22]. However, a potential limitation of this method lies in the estimation of the second stage, in which the meta-frontier estimators lack desirable statistical properties due to the deterministic nature of mathematical programming [22]. In addition, mathematical programming methods cannot separate the idiosyncratic shocks from the model; hence, the estimation results are susceptible to random noise [2, 22].
Due to the limitation of the second stage in the SMF developed by Battese et al. [40] and O'Donnell et al. [41], Huang et al. [21] proposed a SMF model that uses SFA to estimate meta-frontier parameters in the second stage rather than mathematical programming methods. Huang et al. [21] contend that the SFA approach can address the issue of statistical noise and provides desirable statistical properties of the parameters in the estimation of meta-frontier. Moreover, the SMF approach of Huang et al. [21] uses the maximum likelihood estimation (MLE), which allows the computation of statistical inferences without the need to depend on simulations or bootstrapping methods [22]. Huang et al. [21] assert that the SMF approach can estimate the technology gap ratios (TGRs) directly.
Since the SMF approach provides appropriate statistical properties for inference and can isolate TGRs from random shocks which allows TGR to be expressed as a function of exogenous environmental variables [2], we use the SMF approach developed by Huang et al. [21] in this study.
Stochastic meta-frontier specification
Following Huang et al. [21], the SMF model used to estimate the TE of farms adopting different technologies, is specified in a two-step procedure. In the first step, individual group frontiers are estimated, while in the second step, stochastic frontier methods are applied to estimate the meta-frontier production function. The stochastic production function for the \(i\mathrm{th}\) farmer is expressed as
$$Y_{ji} = f^{j} \left( {X_{ji} ;\beta^{j} } \right)e^{{V_{ji} - U_{ji} }} ,\quad i = 1, 2, \ldots ,N_{j} ;\;\;j = 1, 2, \ldots , J$$
(1)
where \({Y}_{ji}\) represents the output of the \(i\mathrm{th}\) farmer in the \(j\mathrm{th}\) group; \({X}_{ji}\) represents a vector of inputs of the \(i\mathrm{th}\) farmer in the \(j\mathrm{th}\) group; \({\beta }^{j}\) is a vector of unknown parameters to be estimated for the \(j\mathrm{th}\) group. Moreover, \({V}_{ji}\) is a random error component that represents the statistical noise and is assumed to be identically and independently distributed with mean as 0 and constant variance as \(\left[{V}_{ji} \sim N\left(0,{\sigma }_{v}^{j2}\right)\right]\). The non-negative random errors \({U}_{ji}\) represent the technical inefficiency of the \({i}^{th}\) farmer in the \({j}^{th}\) group. \({U}_{ji}\) follows a truncated normal distribution and is defined as \({U}_{ji} \sim {N}^{+}\left({\delta }_{j}{Z}_{ji},{\sigma }^{j2}\right)\), where \({Z}_{ji}\) represents a vector of exogenous variables associated with the technical inefficiency of the \(i\mathrm{th}\) farmer in the \(j\mathrm{th}\) group; \({\delta }_{j}\) is a vector of unknown parameters to be estimated.
The TE of the \(i\mathrm{th}\) farmer in \(j\mathrm{th}\) group is specified as the ratio of actual output to the maximum output possible:
$${TE}_{i}^{j}=\frac{{Y}_{ji}}{{f}^{j}\left({X}_{ji}\right){e}^{{V}_{ji}}}={e}^{-{U}_{ji}}.$$
(2)
As suggested by Huang et al. [21], the meta-frontier production function \({f}^{M}\left({X}_{ji}\right)\) that envelops the three group-specific frontiers \({f}^{j}\left({X}_{ji}\right)\) is defined by the following relationship:
$$ f^{j} \left( {X_{ji} } \right) = f^{M} \left( {X_{ji} } \right)e^{{ - U_{ji}^{M} }} ,\;\forall j,i $$
(3)
where \({U}_{ji}^{M}\ge 0\), suggesting that \({f}^{M}\left(\bullet \right)\ge {f}^{j}\left(\bullet \right)\) and the TGR which is computed as the ratio of the frontier production function for the \(j\mathrm{th}\) group relative to the meta-frontier:
$${TGR}_{i}^{j}=\frac{{f}^{j}\left({X}_{ji}\right)}{{f}^{M}\left({X}_{ji}\right)}={e}^{-{U}_{ji}^{M}}\le 1$$
(4)
Following Huang et al. [21], for given levels of inputs, the observed output of the \(i\mathrm{th}\) farmer with respect to the meta-frontier function, adjusted for the corresponding random error is defined as:
$$\frac{{Y}_{ji}}{{f}^{M}\left({X}_{ji}\right)}={TGR}_{i}^{j}\times {TE}_{i}^{j}\times {e}^{{V}_{ji}}.$$
(5)
In the above, it is important to note that though \({TGR}_{i}^{j}\) and \({TE}_{i}^{j}\) are bounded between 0 and 1, the meta-frontier function does not certainly envelope all farmers’ observed outputs due to random noise [22]. Indeed, the unrestricted ratio given in Eq. (5) differentiates the use of the SFA method with mathematical programming to model the meta-frontier. Accordingly, given that the random noise component is obtained from the SFA, the TE of the \(i\mathrm{th}\) farmer with respect to the meta-frontier, is expressed as:
$${MTE}_{ji}=\frac{{Y}_{ji}}{{f}^{M}\left({X}_{ji}\right){e}^{{V}_{ji}}}={TGR}_{i}^{j}\times {TE}_{i}^{j}.$$
(6)
The meta-frontier production function in Eq. (3) was specified using mathematical programming methods proposed by O'Donnell et al. [41]. However, due to the limitations of this approach highlighted earlier, we can reformulate Eq. (3) using the methodology proposed by Huang et al. [21]:
$$\mathrm{ln }{f}^{j}\left({X}_{ji}\right)=\mathrm{ln }{f}^{M}\left({X}_{ji}\right)-{U}_{ji}^{M}.$$
(7)
Here, the individual group frontier \({f}^{j}\left({X}_{ji}\right)\) is unobservable. Still its values can be estimated from the first step, and as the fitted values of \({f}^{j}\left({X}_{ji}\right)\) differ from the actual frontier, Eq. (7) can be redefined as:
$$\mathrm{ln }{\widehat{f}}^{j}\left({X}_{ji}\right)=\mathrm{ln }{f}^{M}\left({X}_{ji}\right)-{U}_{ji}^{M}+{V}_{ji}^{M},$$
(8)
where \({V}_{ji}^{M}\) denotes a symmetric noise that represents the deviation between the fitted values \({\widehat{f}}^{j}\left({X}_{ji}\right)\) and the true frontier \({f}^{j}\left({X}_{ji}\right)\). That is,
$${V}_{ji}^{M}=\mathrm{ln} {\widehat{f}}^{j}\left({X}_{ji}\right)-{f}^{j}\left({X}_{ji}\right),$$
(9)
where the error term \({V}_{ji}^{M}\) is typically distributed as \(N\left(0,{\sigma }_{v}^{M2}\right)\) while the non-negative technology gap component \({U}_{ji}^{M}\ge 0\) is assumed to be distributed as truncated normal, i.e., \({U}_{ji}^{M} \sim {N}^{+}\left({\mu }^{M}\left({Z}_{ji}\right),{\sigma }^{M2}\right)\), where \({Z}_{ji}\) represents the production environment variable vector.
Equation (8) holds the resemblance with the conventional stochastic frontier model, and in consequence, \(\mathrm{ln }{f}^{M}\left({X}_{ji}\right)+{V}_{ji}^{M}\) is referred to as the SMF model. Usually, this model is performed using the maximum likelihood estimation, and the parameter estimates are consistent and asymptotically normally distributed [22]. The SMF model allows for the estimated group-specific frontier to be larger than or equal to the meta-frontier due to statistical noise. However, the meta-frontier should be larger than or equal to the group-specific frontier, i.e., \({f}^{M}\left({X}_{ji}\right)\ge {f}^{j}\left({X}_{ji}\right)\) [22]. Therefore, the estimated TGR is calculated as:
$${T\widehat{G}R}_{i}^{j}=\widehat{E}\left({e}^{-{U}_{ji}^{M}}|{\widehat{\varepsilon }}_{ji}^{M}\right)\le 1,$$
(10)
where \({\widehat{\varepsilon }}_{ji}^{M}=\mathrm{ln} {\widehat{f}}^{j}\left({X}_{ji}\right)-\mathrm{ln }{\widehat{f}}^{M}\left({X}_{ji}\right)\) is the estimated composite residual of Eq. (8).
Functional form specification
Cobb–Douglas and translog functional forms are the most commonly applied in efficiency analysis. In this study, we use the Cobb–Douglas functional form to estimate both group-specific stochastic frontier and stochastic meta-frontier parameters. The choice of Cobb–Douglas is based on the results of the likelihood ratio test. The value of the likelihood ratio test statistic is about 6.21 leading to the acceptance of the null hypothesis stating that Cobb–Douglas functional form is the appropriate representation of the data relative to the translog functional form.Footnote 2 The Cobb–Douglas functional form is specified as follows:
$$\mathrm{ln }\left({Y}_{i}\right)={\beta }_{0}+{\sum }_{j=1}^{4}{\beta }_{j}\mathrm{ln }{X}_{ji}+\left({v}_{i}-{u}_{i}\right)$$
(11)
where \({Y}_{i}\) represents the maize output of the \(i\mathrm{th}\) farmer; \({X}_{ji}\) represents the quantity of the \(j\mathrm{th}\) input used by the \(i\mathrm{th}\) farmer; \(\beta \) denotes a vector of unknown parameters to be estimated; \({v}_{i}\) is the random error term and \({u}_{i}\) is the non-negative inefficient term. Following Battese, Coelli [43], the inefficiency effects model can be written as:
$${\mu }_{i}={\delta }_{0}+\sum_{j=1}^{10}{\delta }_{j}{Z}_{ji},$$
(12)
where \({Z}_{ji}\) denotes a vector of farmer-specific variables that might influence the TE.