The study considers panel data of 49 countries in Africa from 2000 to 2019. The minute details of data sources, variable information, individual countries, and regions are discussed in Myeki et al. (2022, 2023). However, the overall summary of variables is presented in Table 1 below. It is clear from the table that the study consisted of a single output, the gross value of agricultural output from crops, livestock, and aquaculture, $1,000 at constant 2015 prices. Three input variables were considered: land, labor, and capital. Land is the quality of adjusted agricultural area measured per 1,000 hectares of “rainfed equivalent cropland.” Labor is the number of economically active adults (male and female) who are employed in agriculture, measured per 1,000 persons. While capital is the net capital stock measured in $1,000, taking 2015 as the base. Compared to 2000 to 2004, output grew by an average of 19% in 2005 to 2009, followed by 41% and 61% in subsequent periods. Land grew at an average of 9%, 21%, and 34% over the reported periods, respectively. For labor, the average growth was 11%, 17% and 24% while capital had 19%, 47%, and 83%, respectively. Much of the growth in both output and inputs happened during 2015 to 2019. As can be seen in Table 1, the gross output averaged $5,274,339 (Std. dev = 9,676,909), ranging from 30 024 in Sao Tome and Principe in 2015 to 5,930,633 in Nigeria in 2019. The summary of inputs shows that land averaged 5,808 hectares (8,740) with a range from 37.39 ha in Djibouti in 2006 to 62,459 ha in Nigeria in 2019. On average, 3,241 people were employed in agriculture, ranging from 11.33 in Sao Tome and Principe in 2002 to 21,298 in Nigeria in 2011. On the other hand, capital had an average of 5,971 (17,604) with a range from 4.83 in Djibouti in 2000 to 188,491 in Nigeria in 2019.

It employs the GTRE model by Colombi et al. (2014), Kumbhakar et al. (2014), and Tsionas and Kumbhakar (2014) to estimate transient and persistent inefficiency in the agricultural sector for the continent. The model is expressed as follows:

where y_it is the dependent variable defined as the logarithm for agricultural output of nation i in period t, α₀ is the intercept, f(x_it, t; β) is the production technology, including a time trend t to account for technological change, x_it is a vector of inputs (in logs), and β is a vector of unknown parameters to be estimated. Additionally, the μ_i represents time-invariant firm-specific latent (unobserved) heterogeneity, η_i denotes the persistent (time-invariant) technical inefficiency, u_it captures transient (time-varying) technical inefficiency, and v_it is a random noise term. The composite error term for the model described in Equation (1) is ε_it = μ_i−h_i+v_it−u_it with μi~N(0,σμ2); vit~N(0,σv2); ηi~N+(0,ση2), and; uit~N+(0,σu2). Numerous stochastic frontier models are embedded in the GTRE model defined in Equation (1). For instance, dropping both μ_i and η_i leads to a “pooled” model proposed by Aigner et al. (1977), described as follows:

The v_it is a random variable assumed to be identically and independently distributed iid with zero mean and variance σv2(vit~N(0,σv2)), assumed to capture the effect of random noise; and u_it is the time-varying non-negative half-normal inefficiency random variable, which is assumed to be iid with zero mean and variance σu2(uit~N+(0,σu2)). Discarding η_i and u_it from the same model, described by Equation (1), yields the standard panel stochastic frontier model with random effects (RE) developed by Pitt and Lee (1981), expressed in Equation (3). The error specification of this equation is ε_it = v_it−h_i, distributed as vit~N(0,σv2) and μi~N+(0,σμ2)

On the other hand, removing η_i alone from Equation (1) leads to Greene (2005a,b)'s true random effect (TRE) model in Equation 4, depending on whether μ_i is correlated with the error term (μi~N+(0,σμ2), vit~N(0,σv2) and; uit~N+(0,σu2)

The study adopted an approach of Filippini and Hunt (2015) and Alberini and Filippini (2018) where the GTRE model and three traditional models described in Equations (2–4) are estimated separately and compared to each other for robustness. Moreover, we also employed the translog functional form based on log-likelihood ratio test results (see Table 2), leading to rejection of the Cobb-Douglass functional form. Thus, the final model is expressed as follows:

where ln y_it represents the agricultural output of individual nation i in period t while α₀ is the intercept. x_jt represents the three key inputs used to derive agricultural output, and they include land, labor, and capital. β_jt is a vector of unknown parameters to be estimated. The μ_i represents time-invariant firm-specific latent (unobserved) heterogeneity, η_i denotes the persistent (time-invariant) technical inefficiency, u_it captures transient (time-varying) technical inefficiency, and v_it is a random noise term. Models described in Equations (1–5) were implemented using STATA 17 software.

For confirmatory purposes, we also estimated the GTRE model using Bayesian SFA analysis, where the different sub-regions of Africa were used as shifters of efficiency. Although extended to include panel data analysis by Koop (1994), the Bayesian framework is novel to Van den Broeck et al. (1994). The advantages of the framework include the ability to minimize the loss associated with an estimation error; permit probability statements about unknown parameters, hypotheses, and models; provide exact finite-sample results for most estimation problems; and allow for the ease of incorporating restrictions that allow for a formal treatment of parameters and model uncertainty. Let θ = (θ₁, …, θ_k) denote the unknown parameters of the GTRE model to be estimates p(θ)≡p(β, σ_w, σ_h, σ_v, σ_u) denotes the probability density function (pdf) of prior information for parameters while L(y, X∣θ) is the likelihood function. The posterior distribution follows from Bayes's theorem as follows:

where p(θ∣y, X) is the posterior pdf and ∝ denotes “is proportional to.” Put simply, the posterior pdf is proportional to the likelihood function multiplied by the prior pdf. Thus, the posterior distribution includes all the information on the parameters contained in the prior and the data. The prior pdf p(θ) can be non-informative or informative. The complex models that do not allow inference by analytical methods, implementing the Bayesian approach, require the use of iterative Markov Chain Monte Carlo (MCMC) algorithms such as the Metropolis-Hastings and Gibbs Sampling, which focus on sampling from the conditional distribution for blocks of the parameter vector. The Gibbs Sampler algorithm used by Koop et al. (1992) is particularly useful for problems involving latent variables, such as SFA models, and is commonly used in the literature. The researcher can write their own MCMC algorithms or just specify the model but use the Bayesian inference using Gibbs Sampling (BUGS) software, such as WinBUGS or JAGS, to handle the MCMC sampling. In our study, we used JAGS in R through the “APEAR” package to implement the final BGTRE model, which is expressed below.

where i = 1,...., n denotes the i-th country, and t = 1. x_jt is a vector of j-th used inputs normalized by the means. The three inputs used in agricultural output are land, labor, and capital. D_l(l = 1, 2, 3) are regional dummy variables, denoting East Africa, North Africa, West Africa, and Southern Africa, all of which were included for the comparisons for regional performance in terms of transient and persistent technical inefficiency. The α and β are unknown parameters to be estimated. Again, the μ_i represents time-invariant firm-specific latent (unobserved) heterogeneity, η_i denotes the persistent (time-invariant) technical inefficiency, u_it captures transient (time-varying) technical inefficiency, and v_it is a random noise term. We also computed partial output elasticities with respect to inputs with the purpose of examining the sensitivity of output change when a change in inputs occurs. The equation to derive partial elasticities is as follows:

where β^j,β^jj and β^jk are the parameters of the BGTRE model estimated in Equation (7). The returns to scale for this model are computed by summing up the partial output elasticities with respect to inputs, as follows:

A series of diagnostic tests were performed to ascertain the most appropriate models, and the results are shown in Table 2. Our first null hypothesis was concerned with testing for the presence of multicollinearity. Since the Variance Inflation Factor (VIF) for the three variables is <10, the null hypothesis is rejected, implying that the independent variables are independent of one another. Using the log-likelihood ratio (LR) test, the findings (test statistic = 959.96; critical = 9.50 and df = 1) show that the null hypothesis stating that inefficiency is not present was rejected, thus indicating the presence of inefficiency and the fact that we cannot use the ordinary least squares. Additionally, the panel effect was found to be significant across the dataset, indicating that heterogeneity is significant. This also suggests that we cannot use the pooled model. The assumption of truncated normal distribution for the inefficiency term was rejected using the Breusch-Pagan (BP) test, which suggests that a half-normal distribution is appropriate. This result implies that we could not test if inefficiency varies over time because such a test requires a truncated normal distribution.

Moreover, testing between fixed and random effects models using the Hausman test showed a p-value of zero, indicating that random effects were appropriate. The results in Table 2 further reveal that the translog stochastic frontier model is preferred over the Cobb-Douglas functional form. Although not presented in the table, we also found that variables were stationary while using the Hadri-LM test. Next, we present the results for the translog stochastic frontier models.

Table 3 presents the parameter estimates for pooled, RE, TRE, and GTRE models. The first three models are nested in the latter. Nonetheless, the coefficient lnt is positive for all estimated models, indicating that agriculture in Africa has experienced technological progress over the study period. The findings suggest that Africa is on the right track in terms of adopting technology to improve agricultural efficiency and productivity. The variables land, labor, and capital are all significant at the 1% level for the pooled and TRE models. This finding was expected because improvement on: (i) land through the application of better fertilizer, (ii) labor through training and education, and (iii) capital through investment in farm machinery and equipment are all highly likely to improve efficiency. Surprisingly, the coefficient for variable capital for RE and GTRE is negative and insignificant, indicating a lack of empirical evidence for the relationship between output and capital for the two models. The coefficient for η is 0.35% and significant in GTRE, suggesting that persistent inefficiency is dominant compared to transient inefficiency (0.07%). This implies that over the study period, agriculture in Africa had an average persistent technical efficiency of 65% and a transient technical efficiency of 93%. The heterogeneity and persistent inefficiency are significant, indicating that the GTRE model is appropriate.

The estimates for the Bayesian stochastic frontier model with translog specification are shown in Table 4. The Gibbs sampler was run for one chain, with a burn-in of 200,000 iterations, 10,000 retained draws, and a thinning to every 40th draw in order to reduce the level of autocorrelation of the chain. This table clearly shows that a percentage increase in any of the inputs leads to an increase in agricultural productivity in Africa. For instance, with each unit increase in agricultural land allocated in Africa, output increases by 0.67%, and this increase falls within the 95% posterior probability interval (0.60%, 0.74%). A unit increase in labor leads to a 0.14% increase in output. In addition, this increase falls in a 0.09% to 0.20% interval with 95% posterior probability. On the other hand, the coefficient for t is 0.02 and positive, indicating technological progress over the study period. These findings are significant, as shown by the same sign for the values of the confidence interval. However, the agricultural output has been more responsive to changes in land compared to labor. On the other hand, a unit increase in capital is associated with a 0.04% increase in output, and this falls in the 0.02% to 0.07% interval with 95% posterior probability. These findings correspond to those discussed earlier in Table 2 and suggest that production behaves well and meets the monotonicity assumption. The results also imply that countries can continue to increase their input levels until any increase in input yields no additional output. The results further reveal that the Monte Carlo (MC) error is lower than the corresponding posterior standard deviation, suggesting convergence of the model and accuracy of estimates. However, if evidence of significance was present, the result would suggest that East Africa is 0.47% more productive compared to Central Africa, while North Africa, Southern Africa, and West Africa are more productive by 0.29%, 0.26%, and 0.10%, respectively. The coefficient for gamma (γ) is significant and positive, indicating that 0.81% of the variation in agricultural output among the African nations is attributed to the variations in technical efficiency. The value of lambda (λ), gives information about the inefficiency level of the agriculture sector. Put in simple terms, it shows by how much an African country has fallen short of the total output. Similarly to Table 2, the persistent technical inefficiency (σ_ui) is dominant at 0.10%, while transient inefficiency (σ_uit) was at 0.08%.

With respect to the results of Equation (9), the value of returns to scale for BGTRE is 0.86, indicating that agricultural production in Africa achieved a decreasing return to scale over the study period. This value is slightly higher by 0.07 compared to that of the GTRE model at 0.79. It suggests that the proportion of output is less than the desired increased inputs during the production process. We found that agricultural output is more responsive to land and labor use. More details of elasticity estimates for each country are presented in Appendix Table 6.

In this section, we present the results for various efficiencies, beginning with those models discussed in Table 3. The pooled model yielded an average efficiency (transient efficiency) score of 0.69%, ranging from 0.22% to 0.93%, while RE had an average (persistent efficiency) score of 0.50% with a range from 0.17% to 0.98%. The TRE model produced an average efficiency score of 0.89, implying room for improving efficiency by 11% without changing the input and current technology. The main model (GTRE) yielded averages of 0.73% and 0.93% for persistent and transient efficiencies, respectively, leading to an overall average efficiency score of 0.68%. This suggests that persistent efficiency has less contribution to the overall efficiency of the agricultural sector in Africa over the study period (see Figure 1). The efficiency estimates of the Bayesian stochastic frontier reveal an overall average efficiency of 0.84%, ranging from 0.52% to 0.91%. However, there is a slight difference between transient and persistent efficiency, showing that transient efficiency dominates by 1%. Nonetheless, further details of efficiency scores obtained from BGTRE for each sampled African nation over the study period are presented in Appendix Table 5. The results for heterogeneity in Appendix Table 7 show that Cameroon, Ghana, South Africa, Egypt, Cote d'Ivoire, Nigeria, Malawi, and Benin have significantly large agricultural outputs. On the other hand, Libya, Niger, Cape Verde, Sao Tome, and Principe, Comoros, Equatorial Guinea and Djibouti had significantly low agricultural output.