Q Open , 2024, 4 , qoae017 https://doi.org/10.1093/qopen/qoae017 Advance access publication date: 23 July 2024 Article Closing the gaps in experimental and observational crop response estimates: a bayesian approach Maxwell Mkondiwa 1 , ∗, Terrance M. Hurley 2 and Philip G. Pardey2 1 International Maize and Wheat Improvement Center (CIMMYT), National Agricultural Science Centre (NASC) Complex, 110012 New Delhi, Delhi, India 2 Department of Applied Economics, University of Minnesota-Twin Cities, 1994 Buford Avenue, Saint Paul, MN, USA ∗Corresponding author: Tel: + 91 7428669054; E-mail: m.mkondiwa@cgiar.org Received: March 1, 2024. Accepted: June 28, 2024 Abstract A stylized fact of African agriculture is that crop responses to inorganic fertilizer application derived from experimental studies are often substantially greater than those from observational studies (e.g. surveys and administrative data). Recent debates on relative costs and benefits of expensive farm input subsidy programs in Africa, have raised the importance of reconciling these estimates. Beyond mean response differences, this paper argues for including parameter uncert aint y and heterogeneit y arising from variations in soil types, environmental conditions, and management practices. We use a Bayesian approach that combines information from experimental and observational data to model uncertainty and heterogeneity in crop yield responses. Using nationally representative experimental, survey, and administrative datasets from Malawi, we find that: (1) crop responses are low in observational data, (2) there are large spatial heterogeneities, and (3) based on sensitivity analysis, ignoring parameter uncert aint y and spatial heterogeneity in crop responses can lead to questionable policy prescriptions. Keywords: Bayesian analysis, Crop responses, Malawi, Yield gaps. JEL code: Q12; C11 T 1 A g v h o b a a © E C d D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 Augu he best fertilizer on any farm is the footsteps of the owner. (Taken from Scott 1998, p. 284, attributed to Confucius) . Introduction stylized fact of African agriculture is that experimentally derived crop responses to inor- anic fertilizer application are often substantially greater than those obtained from obser- ational studies (e.g. using farm survey or official administrative data). There is also a long istory of description of this yield gap, which can be reduced to the presence or absence f positive or negative confounding factors such as biologically optimal crop management y researchers versus biologically sub-optimal (albeit possibly optimal bio-economic) man- gement by farmers; smaller, more uniform, plot sizes used in experiments versus larger, nd heterogeneous plot sizes used by farmers; biases or spatial inconsistencies in site or The Author(s) 2024. Published by Oxford University in association with European Agricultural and Applied conomics Publications Foundation. This is an Open Access article distributed under the terms of the Creative ommons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, istribution, and reproduction in any medium, provided the original work is properly cited. st 2024 https://doi.org/10.1093/qopen/qoae017 https://orcid.org/0000-0003-0008-9095 https://orcid.org/0000-0003-2135-7570 mailto:m.mkondiwa@cgiar.org https://creativecommons.org/licenses/by/4.0/ 2 Mkondiwa et al. s 2 n v a f a t q d a p m y D h B a B e p r o u w t i c W u p a r o o a r p c s ( c D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 ample selection of scientific versus farmer plots; and observer bias (see, e.g. Snapp et al. 014 , Coe et al. 2016 ; Benson et al. 2021 ). In addition to these factors, sample sizes (or the umber of replicates) also differ, with farm surveys that can span thousands of households ersus experiments that often include a few hundred sites at most. According to Bullock nd Bullock (2000 : 97), ‘…the simple fact is that most agronomic experiments are not run or enough years and enough locations to obtain many different observations of weather nd possible field characteristics’. Because a field experiment at a few locations cannot cap- ure all this nuance in variation, the representativeness of agronomic experiments is often uestioned because crop response estimates and recommendations derived from them are ifferent from what is experienced under farmers’ conditions. The aim of this study is to pply a Bayesian approach that combines experimental and observational evidence thereby roviding estimates for making recommendations on fertilizer use that take into account ultiple sources of information. Economists have had long-standing debates on both the causes and solutions related to ield gaps.1 Some of the early work on this topic includes Davidson and Martin (1965) , avidson et al. (1967) , Anderson and Dillon (1968) , and Anderson (1992) . Such debates ave continued in contemporary agricultural policy considerations (see Benson et al. 2021 , enson et al. 2024 ). This study contributes to this prior literature by using experimental nd observational evidence in Malawi to characterize the crop response gaps and applies ayesian linear and hierarchical models to combine the estimates from observational and xperimental studies. The discrepancy in experimental versus commercial yield response can have profound olicy implications. For example, a study by Jayne et al. (2015) using observational crop esponses of the social benefits versus costs of the Malawian farm input subsidy program— ne of the largest targeted national farm input subsidy programs in Africa—found it to be nduly costly.2 In direct contrast, using experimental crop response data, Chirwa & Dor- ard (2013) found the program to be economically beneficial relative to its costs. According o Jayne et al. (2015) , the use efficiency of the nitrogen applied to maize is perhaps the most mportant factor determining the benefits of the Malawi farm input subsidy program. The rux of their case largely hinges on the following yield response relativities: ‘These (crop response) estimates (3.4–9.9 kg of maize output per unit of fertilizer applied per ha) are based on farm survey data and not researcher-influenced plots, and they reflect the range of management practices and production constraints found within Malawi’s smallholder farm sector… Unfortunately, Dorward and Chirwa (2015) maize response estimates of 16–18 kg are derived from researcher-influenced farm trials undertaken in the late 1990s with participants who were largely master farmers’ (Jayne et al., 2015 : 746) hile noting the challenges of reconciling these crop responses, Arndt et al. (2016) eval- ated the subsidy program using crop responses ranging from 11.8 to 18.5 kg of maize er ha per kg of nitrogen fertilizer. They settled on this range of responses, more or less rbitrarily, to cover reported rates from the observational and experimental evidence they eviewed 3 . Arndt and co-authors further comment that reconciling the experimental and bservational crop responses remains an important and unresolved problem. The difficulty f fully reconciling the estimates from these multiple sources of data, which were collected t different time periods in different locations, with different varieties and using different esearch methods, is exacerbated by the reliance on mean response comparisons that com- letely ignore the substantive spatial and temporal heterogeneity in these responses. These oncerns remain especially in recent literature which questions the effectiveness of fertilizer ubsidies given the low yields, persisting food insecurity, and non-compensatory price ratios e.g. Benson et al. 2024 ). Given these challenges, policy making is usually left to guesswork regarding the true rop responses and a reliance on arbitrary approaches to re-adjusting experimental yield Closing the gaps in experimental and observational crop response estimates 3 r m a t a a b r r t t f m M t a s e i a c r t t r a w c T d d t p t r a e I t d s r b l w e r c t b D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 esponses to better reflect farmer conditions. In this paper, we apply a simple and replicable ethod of bringing all these subjective judgements into a formal estimation framework. The pproach is based on the Bayesian paradigm of combining prior information and observa- ional data. We use a Bayesian hierarchical model to incorporate both parameter uncertainty nd heterogeneity in crop response functions and fertilizer recommendations. The Bayesian pproach of combining different evidence on the same phenomenon has recently been used y Fessler & Kasy (2018) to combine predictions of labor demand and wage inequality de- ived from economic theory and empirically derived estimates, byMeager (2019) to combine esults from various randomized control trials of micro-credit interventions across coun- ries, and by Rosas et al. (2018) to impose duality theory restrictions based on experimental rial data to assess market level crop yield responses to prices in the USA. In macroeconomic orecasting, the idea of combining evidence using the Bayesian paradigm has been imple- ented in Dynamic Stochastic General Equilibrium Modeling as well as New Keynesian acroeconomics. It has also been used in the economics of education literature to combine eacher value added measures that are precise but biased with alternative measures based on dmission lotteries for students that are unbiased but imprecise (Angrist et al. 2017 ). Other tudies focus on using machine learning and quasi-experimental approaches to adjust the stimates from either observational or experimental studies (Bernard et al. 2024 ). This paper contributes to the crop responses literature by exploring the possibility of mproving soil fertility recommendations through the careful combination of experimental nd observational crop response evidence, while also taking into account parameter un- ertainty and heterogeneity. Specifically, the study analyzes the effect of observational crop esponses when conditioned on prior (experimental) crop responses. The applications from his modeling approach are many, especially given a lack of directly comparable experimen- al data over time due to changes in experimental designs. Using the Bayesian approach, esearchers can simply use previous estimates as priors in a new analysis. Similarly, in many gronomic research projects scientists are asked to conduct household surveys prior to or hile conducting experiments as part of learning the environment. With this approach, they an formally use the household survey estimates as priors in their experimental analysis. he key rationale is that Bayesian estimates weight the estimate from the present and prior ata using an inverse of the variance parameters so that the uncertainty of the parameters etermine whether the prior or present data dominate. We specifically incorporate parame- er uncertainty in single output and multi-output crop response function estimation, which rovides a more complete description of the crop response parameters. Specifically, we con- ribute to the on-going debates on the use of experimental versus observational mean crop esponses by showing that using the mean response function in combination with arbitrary djustments may result in suboptimal policy prescriptions in most cases because the inher- nt unobserved heterogeneity within and across farms requires site-specific optimization. nstead, researchers are likely better off using the entire distribution of the parameters (as his distribution contains more information than the mean), which entails comparing the istributions of benefit-cost ratios and profits obtained from the different alternatives being tudied. Although endogeneity issues from measurement error, simultaneity, and omitted variables equire close consideration when estimating crop response functions (and such concerns edevil all prior crop response assessments that use observational data), the use of district- evel fixed effects allows comparisons of within district differences of the sources of evidence, hile accounting for uncertainty of parameter estimates, and heterogeneity of crop response stimates. Furthermore, this paper follows a partial identification strategy to test if the crop esponse parameter is observationally equivalent under various prior specifications. Debates on whether crop response estimates are low or high in Malawi and other African ountries are difficult if not impossible to resolve when uncertainty and heterogeneity of he estimates across time and space is ignored. Most importantly, the results we obtain elow show that even with an extremely high prior mean yield response (e.g. 30 kg/ha of 4 Mkondiwa et al. m 1 o f T 2 p o a s r ( a a t B S o 2 2 T l H p f o i w o a i e p m t D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 aize output per kg of nitrogen (N) fertilizer applied) and level of precision (e.g. a value of 0, which is equivalence to a variance of 0.1), the posterior crop response estimates using bservational data can only go as high as 20 kg of maize output for a unit of nitrogen (N) ertilizer applied. In addition, the lowest is around 2 kg of maize output per kg of fertilizer. his implies that there is a 95 per cent probability that the mean crop responses are between and 20 kg of maize output per kilogram of N fertilizer applied.4 Further analysis in the aper shows that there is huge spatial heterogeneity in the crop responses, which should be f importance in policy design because some of the districts are non-responsive to fertilizer pplication. This evidence therefore suggests that resolving policy debates that depend on crop re- ponses should consider variances and heterogeneity in these responses. In summary, the esults illustrate that Malawian maize yield responses are generally low and highly variable over time and space). This underscores the need for evidence-based targeting of locations nd beneficiaries if farm input subsidy programs such as that presently operating in Malawi re to constitute a cost-effective public policy and be profitable for smallholder farmers. The rest of the paper is structured as follows. We present next the model focusing on the heoretical profitability analysis model and the empirical econometric approach that uses ayesian analysis. In Section 3 , we present the data sources and descriptive statistics. In ection 4 , we present the results and discussion. We finally conclude and provide a discussion f limitations and future research in Section 5 . . Model .1 Theoretical model he standard neoclassical approach to production economics on crop response to inputs ike fertilizer is a primal approach based on deterministic profit maximization. Following artley (1983) , the deterministic conditional neoclassical model of fertilizer usage and out- ut response, given that a positive area of land has been allocated to crop j, assumes that armers maximize profits with respect to all variable input levels associated with the area f land ai j which is usually normalized to a unit hectare. The profit associated with crop j n each plot i is defined as πi j = pi j yi j −wi j xi j − FCi j subject to: yi j = fi j ( xi j , ai j , zi j ; θi j ) , (1) here πi j is the profit per unit (hectare) for each plot i and crop j. pi j and wi j are prices f crop outputs and fertilizer respectively. yi j is the crop specific yield (kg/ha) and yi j = fi j (xi j , ai j , zi j ; θi j ) describes the production technology where xi j is the quantity of fertilizer pplied (kg/ha), ai j represents area under crop j in plot i , zi j represents the quantity of other nputs like labor, and θi j represents the set of relevant response parameters that are usually stimated from the data. FCi j represents fixed costs. Under assumptions of twice continuously differentiability, convexity of the production ossibilities set, strict concavity of the objective function, the economic condition for opti- ality is ∂πi j ∂xi j = pi j ∂ fi j ( xi j ∗, ai j , zi j ; θi j ) ∂xi j −wi j = 0 ∂ fi j ( xi j ∗, ai j , zi j ; θi j ) ∂xi j = wi j pi j . (2) Using the implicit function theorem or assuming conventional functional forms for fi j (xi j ∗, ai j , zi j ; θi j ) , it is easy to find the optimal xi j ∗ and this approach has been used ex- ensively in practice to make fertilizer use recommendations. Closing the gaps in experimental and observational crop response estimates 5 θ t t w T s c s h s r t s c f t t s c a s i i t i i u d e w i p o t i r o s G D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 There are fundamental flaws using the neoclassical production model. First, the parameter i j , which essentially drives the optimality as well as heterogeneity across farms, is assumed o be known and certain such that it is usually not included in the optimization. However, hese parameters are rarely if ever known (either in an agronomical or statistical sense), hich implies that economic decisions made on the basis of this assumption are suspect. he fundamental problem in agricultural settings is that the crop is typically grown on oils with an ‘inherent soil fertility gradient,’ which implies that yields even under heavily ontrolled environments will be uncertain because of the unobserved heterogeneity in the oil even a few centimeters apart (Zingore et al. 2007 ).5 In a statistical sense, θi j is usually a set of unknown parameters, about which farmers may ave some prior information based on the performance of the same or different crops under imilar or different input regimes. In conventional theory, there is no provision to incorpo- ate this prior. Second, this conventional approach does not provide any direction as to what ype of data would be required to estimate the production relation fi j (xi j ∗, ai j , zi j ; θi j ) . A re- earcher can conduct experiments to decipher some xi j ∗, but no known experimental design an comprehensively investigate the effect of each of the xi j on yield, while also controlling or all other effects in xi j and zi j . Occasionally, multi-factorial experiments are conducted o (partially) address this challenge. Another line of research uses farm surveys to analyze he observable determinants of yield. Under this approach, the farmer has already made a et of input and crop management choices depending on their observed and unobserved cir- umstances. Using these different approaches result in different estimates of θi j . This study pplies an extensive Bayesian analysis to investigate combined estimates of θi j that are con- istent with theory and the practical challenges (and the relative prices) faced by farmers. To make comparisons across different scenarios we use first order stochastic dominance, n particular ‘posterior stochastic dominance.’ Thus, different information sources are be- ng combined probabilistically and stochastic dominance is being used to compare among hem. Stochastic dominance is normally defined with respect to stochastic outcomes, which n the case of this study are profits. The study therefore concentrates on whether the fertil- zer response parameters dominate each other across the entire measured range of fertilizer se when the prior is updated with additional information. Definition 1 below provides a escription of posterior stochastic dominance. Definition 1: First Order Posterior Stochastic Dominance- Let F( π ) and G( π ) be two cumulative distributions of outcomes (for example profits) based on different experimental priors. Drawing on Levy’s (2016 : 56) definition, the distribution of outcomes F( π ) will first order stochastically dominate the distribution of outcomes G( π ) if and only if F( π ) is less than or equal to G( π ) for every π and there is at least one π for which a strong inequality holds. Using the definition of stochastic dominance and interpretation of posterior parameter stimates as consisting of a prior and a data-based likelihood, two important claims follow hen interpreting the prior scenarios. The first claim, based in the mean responses, is that f the mean for a prior is greater than the mean of the likelihood holding the variance or recision parameter constant, then the resulting posterior parameter is greater than the mean f the likelihood. Second, if the variance for a prior is greater (i.e. has lower precision) than he variance of the likelihood assuming the same mean, then the variance for the posterior s less than the variance of the likelihood. The stochastic dominance ordering is therefore an empirical question that depends on the elative magnitudes of the prior mean and precision versus the mean and the uncertainty f the likelihood. To illustrate the concept of stochastic dominance in comparing the prior cenarios, Fig. 1 demonstrates three hypothetical cumulative distribution functions; F(.), (.) and Q(.). In the figure, F(.) first order stochastically dominates G(.) since F(.) < G(.) 6 Mkondiwa et al. Figure 1 Stochastic dominance of hypothetical posterior outcomes given experimental priors a b 2 T B o t m o ( 2 A O ( l f w [ e D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 cross the entire measured range of profits. Higher order levels of stochastic dominance can e used to compare F(.) and Q(.) or G(.) and Q(.) (see Levy 2016 ). .2. Empirical models o incorporate the facets of the theory above, we use two estimation strategies, namely a ayesian linear model and a Bayesian hierarchical model. These models allow estimation f the crop response parameter which is key to the stochastic dominance comparisons in he theory. All the models are quadratic in the crop response parameter.6 The choice was ade so that the results of the paper are comparable with most of the experimental and bservational estimates hitherto reported for Malawi, which have used this functional form see, e.g. Harou et al. 2017 ). .2.1 Estimating equation: Bayesian linear model Bayesian linear model is used to estimate the ray production function (see nline Appendix B). The Bayesian linear model is equivalent to the ordinary least squares OLSs) regression model when a non-informative prior (e.g. zero mean and an arbitrarily arge variance such as 10,000) is used. The Bayesian linear model for the ray production unction approach is ˜ yi = β0 + β1 xi j + β2 x2 i j + αzi j + J−1 ∑ j=1 ξλi j + εi j , (3) here ˜ yi is the output norm (hereafter referred to as total output index) defined as ˜ y = ∑ p j=1 y 2 j ] 0 . 5 , and y j is the yield (kg/ha) of crop j. When mono-cropped maize is consid- red, the total output index is equivalent to maize yield ;xi j , x2 i j , zi j and λ are vectors of https://academic.oup.com/qopen/article-lookup/doi/10.1093/qopen/qoae017#supplementary-data Closing the gaps in experimental and observational crop response estimates 7 n ( m β a t a e h M M y e y e ( r 2 T c n f h e b n ( t v c k n w t a t e t g A V i ( w ( D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 itrogen (N) fertilizer use (kg/ha), squared N fertilizer use, other explanatory variables like seed use (kg/ha), rainfall etc.), and angular crop output coordinates (representing crop ix), respectively (see Online Appendix B for details). The corresponding parameters are; 1 , β2 , α, and ξ . Without loss of generality, we use the matrix notation for β to represent ll the parameters and X the design matrix for all variables in the model in the derivations hat follow. The disturbance term, ε has a multivariate normal distribution with mean 0 nd covariance matrix σ 2 I, where I is an identity matrix, i.e. ε∼iid N(0 , σ 2 I ) . In Bayesian conometric terminology, the variances σ 2 can be written as precision estimates, h where = σ−2 (Carlin and Louis 2009 ). 7 Though exact sampling from the posterior is possible, the model was estimated using arkov Chain Monte Carlo using Gibbs Sampling. All the models were run with 11,000 CMC iterations with 1,000 used as burn in and the remaining 10,000 for posterior anal- sis. Non-informative priors (0 prior mean and 0.001 prior precision) for the parameter stimates were assumed in the set of models except where stated for the sensitivity anal- ses. The trace plots for key variables (fertilizer use) in both linear and hierarchical lin- ar models showed convergence at small number of runs as expected for linear models see Online Appendix Fig. A2 for some of the trace plots. The rest are available upon equest). .2.2. Heterogeneity in crop responses: Bayesian hierarchical model he enormous heterogeneity in Malawi’s smallholder farming systems implies that even rop response parameters that capture the uncertainty in associated model parameters may ot be sufficient to characterize the different biophysical and socioeconomic circumstances aced by farmers. To address heterogeneity in the crop responses we deployed a Bayesian ierarchical modeling approach. According to Carlin and Louis (2009) , a hierarchical mod- ling approach allows for a more explicit assessment of the heterogeneity both within and etween groups. This modeling approach has been used extensively in statistics and eco- omics literature to model heterogeneity among individuals. For example, Cabrini et al. 2010) uses the Bayesian hierarchical approach to estimate market performance expecta- ions (e.g. prospective prices) of individuals working in agricultural market advisory ser- ices. Chib & Carlin (1999) and Allenby & Rossi (1998) show how the hierarchical model an help in generating consumer and household specific parameters that are useful for mar- eters of consumer products. Following Chib & Carlin (1999) , consider the normal hierarchical model in matrix otation, ˜ yi = Xi β + Wi bi + εi , (4) here each group i has ki observations. The term ‘group’ is being used generally here so hat any type of heterogeneity may be considered. For instance, a group may constitute location (region/district/village/agroecological zone), household, soil type or poverty sta- us. Xi is ki × p design matrix of p covariates. β is a corresponding p × 1 vector of fixed ffects. Wi is ki × q design matrix. bi is q × 1 vector of subject-specific means and enable he model to capture marginal dependence among the observations on the i th group. The roup-specific random effects follow: bi ∼ Nq (0 , Vb ) . And the errors: εi ∼ N(0 , σ 2 Iki ) . ssuming standard conjugate priors, β ∼ Np (μβ, Vβ ) and σ 2 ∼ InvGamma (nu, 1 δ ) and b ∼ InvWishart (r, rR ) where r is set to the number of parameters in the model and R s a diagonal matrix with values along the diagonal equal to the number of parameters Chapman and Feit 2015 ). In the estimation, we have used the MCMChregress function hich implements the Gibbs sampling algorithm based on algorithm 2 in Chib and Carlin 1999) . https://academic.oup.com/qopen/article-lookup/doi/10.1093/qopen/qoae017#supplementary-data https://academic.oup.com/qopen/article-lookup/doi/10.1093/qopen/qoae017#supplementary-data 8 Mkondiwa et al. Table 1. Fertilizer treatments tested in 1995/96 and 1997/98. Treatment Nutrients Fertilizer Name code Nitrogen (kg/ha) Phosphate (kg/ha) Sulphur (kg/ha) Basal (50 kg per ha) Top dressing (50 kg/ha) A 0 0 0 0 0 B 35 0 0 0 1.5Urea C 35 10 2 1 (23:21:0 + 4S) 1 Urea D 69 21 4 2 (23:21:0 + 4S) 2 Urea E 92 21 4 2 (23:21:0 + 4S) 3 Urea F 96 40 0 1.75DAP 3.5Urea Note : The nitrogen (phosphorus and sulphur) rates were computed based on the major nutrients composed in each of the basal and top dressing fertilizer. Consider for example for treatment C which required applying one 50 kg bag of NPK or 23:21:0 + 4S and one 50 kg of Urea. NPK has 23 per cent of its composition in nitrogen while Urea has 46 per cent of it composition in nitrogen. The total nitrogen applied for the C treatment is therefore N = 0 . 23∗50 + 50∗0 . 46 = 35 . 3 3 T u a n b s 3 T i a t e c a i f f c o w ( w f t t t t p a o D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 . Data sources and descriptive statistics .1 Data sources he study uses both experimental and surveyed fertilizer response data for maize. In partic- lar, the paper uses evidence from a) the fertilizer verification experimental data collected nd analyzed by the Malawi Maize Productivity Task Force in 1995/6–1997/8,8 and b) the ationally representative Third Integrated Household Survey data, which were collected etween 2010 and 2011 (and reflect production decisions for the 2008 and 2009 farming easons) across all National Statistical Organization enumeration areas in Malawi. .1.1 Experimental data he study uses geo-referenced on-farm experimental data for the 1995/96 and 1997/8 grow- ng seasons. The trials were carried out as experiments run on farmers’ fields under the uspices of the Malawi Maize Productivity Task Force consisting of national and interna- ional experts. More than 1,500 trials were successfully implemented to evaluate six differ- nt inorganic fertilizer packages for hybrid maize grown by smallholders across the whole ountry (Government of Malawi 1997 ). The distribution of successful trials was unbalanced cross the sites/regions and seasons due to statistical and administrative reasons. As reported n Table 1 , all six treatments (A, B, C, D, E, and F) were tested in the 1995/96 trials, while our (A, C, D, and E) were tested in the 1997/98 trials. The structuring of treatments in the ertilizer trials suggests that the crop yield consequences of nitrogen and phosphorus may be onfounding. That noted, agronomic studies on fertilizer use in Malawi (e.g. Government f Malawi 1997 ) have argued that nitrogen is the most limiting macro-nutrient, and as such e focus on nitrogen responses. In each of the two seasons, two hybrid maize varieties were planted; Malawi Hybrid 17 MH17) was planted in upland sites with historically good rainfall conditions, and MH18 as supplied for trials in lowland areas and at those upland sites in rain-shadow areas. A ew sites also tested composite varieties. The soil texture was recorded for each plot for each reatment plot per year, and a standard protocol was followed across all locations to ensure imely weeding, pest management, and other agronomic management activities. According o Benson (1999 : 12), one notable feature of the standardized protocol was to conduct the rials on farmer’s field that had not received fertilizer or been planted to legumes in the revious two years. The plot size was 6.3 m by 9 m, consisting of seven ridges spaced 90 cm part. The net harvest plot size was five full ridge lengths, or 1/247 ha (0.00405 ha). Table 2 includes descriptive statistics for the fertilizer trials in the two seasons. For each f the treatments, yields in 1995/96 were relatively higher than those obtained in 1997/98, Closing the gaps in experimental and observational crop response estimates 9 Table 2. Descriptive statistics of yields under different fertilizer treatments during 1995/96 and 1997/98 sea- sons . Season Treatment Mean Median Min Max Std. dev CV(Std. dev/Mean)*100 1995/96 A 1,410.47 1,261.18 0.00 7,245.40 873 .26 61.91 1995/96 B 2,182.90 2,028.86 0.00 6,854.25 989 .95 45.35 1995/96 C 2,358.06 2,284.75 182.78 8,577.87 985 .00 41.77 1995/96 D 2,881.76 2,833.09 310.73 9,029.33 1020 .50 35.41 1995/96 E 3,147.30 3,107.26 219.34 8,407.88 1086 .49 34.52 1995/96 F 2,946.88 2,924.48 274.17 7,018.75 1079 .05 36.62 1997/98 A 1,124.05 968.73 0.00 5,117.84 710 .77 63.23 1997/98 C 1,996.54 1,919.19 109.67 5,940.35 927 .44 46.45 1997/98 D 2,523.04 2,467.53 91.39 6,762.86 1029 .17 40.79 1997/98 E 2,914.52 2,833.09 237.61 7,402.59 1157 .81 39.73 Note : Total number of trials is 1,677 for 1995/96 and 1,407 for 1997/98. r a l y r t w c u t 3 T i 2 T i p G fi g t c b i B a fi c c t 3 A i D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 eflecting less favorable weather during the 1997/98 season. In terms of treatments, the verage yields were highest in treatment E, while the nil N treatment (Treatment A) had the owest mean yield, which is expected considering that nitrogen fertilization is considered ield increasing, at least when moving from little or no N. A notable feature of the data summarized in this table, is the large variation in yield esponses across each of the treatments. For both seasons, the coefficient of variation for he nil fertilizer treatment (A) are highest, with the lowest variation observed in treatments ith the highest amount of nitrogen fertilizer applied (E and F). The variability observed an be attributed to interactions between fertilizer application and many other observed and nobserved factors including location, weather and topography. In this study, we explore he importance of understanding variability attributable to location effects. .1.2 Household survey data he household survey was conducted by the Malawi National Statistical Office (NSO) n collaboration with the World Bank’s Living Standards Measurement Survey (LSMS) in 010. It is a nationally representative sample survey covering about 10,000 households.9 he data are analyzed at the crop-plot level to distinguish between input crop responses n single crop versus multi-crop farming systems. The observations pertain to rainy season lots that were owned and/or cultivated by the farm household and that were subject to lobal Positioning System (GPS)-based land area measurement. The data files were merged rst using the available plot geo-codes and then using household geo-variables (e.g. lon- itude, latitude, and distance to road). The merging was done in a way that made sure hat all the households in the final sample had consistent and identifiable household geo- oordinates. The geo-referenced data allow for the analysis of both agronomic and farmer ehavioral responses. The use of this spatially explicit plot level data therefore implies that t is possible to estimate a structural model of multi-crop production enterprises (Fezzi and ateman 2011 ). All plots not grown with either maize or a legume were excluded from the nalysis. In the final data used for the analysis, there are 19,692 plot-crop observations for ve key crops: maize, groundnuts, beans, pigeon peas, and soybeans. This represents 70 per ent of the plot crop observations in the data. These are the major crops for Malawi (ac- ounting for 70 per cent of the country’s total cropped area in 2009–2013, Johnson 2016 ) hat are also featured in the integrated soil fertility management literature. .1.3 Administrative data dministrative data were compiled from annual production estimates included in the Min- stry of Agriculture and Food Security annual statistical bulletin for the period 1983–2015. 10 Mkondiwa et al. Table 3. Descriptive statistics for selected dependent and independent variables ( n = 19,692). Variables Unit Mean Standard deviation Dependent variables Euclidean norm of the yields kg/ha 1,275 .39 1,886 .39 Maize yield kg/ha 763 .22 1,264 .7 Groundnut yield kg/ha 163 .6 1,122 .86 Bean yield kg/ha 45 .98 397 .13 Pigeon pea yield kg/ha 107 .49 501 .61 Soybean yield kg/ha 25 .63 481 .75 Maize dummy Proportion 0 .89 0 .31 Groundnut dummy Proportion 0 .35 0 .48 Bean dummy Proportion 0 .28 0 .45 Pigeon pea dummy Proportion 0 .28 0 .45 Soybean dummy Proportion 0 .29 0 .45 Key independent variables Total inorganic fertilizer applied kg/ha 162 .75 205 .98 Organic fertilizer use (Yes = 1) Proportion 0 .12 0 .33 Inorganic fertilizer use (Yes = 1) Proportion 0 .69 0 .46 Total N applied Kg/ha 51 .37 65 .08 Note : The Euclidean norm of the crop output vector y is computed by ˜ y = [ ∑ p j=1 y 2 j ] 0 . 5 . T a T d s h 3 T f f m p u t i p A d v M r s c s t i D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 hese data are reported at the district level and consist of the total hectarage and production nd average yield for each crop (i.e. maize, groundnuts, beans, pigeon peas, and soybeans). his source does not report any fertilizer use data by district, and thus these administrative ata were only used in calculating cross-district differentials in crop yield performance, a patial dimension of yield gaps.10 For maize, the data has varietal (local, composite, and ybrid) specific yield, hectarage and production information. .2 Descriptive statistics able 3 presents selected descriptive statistics for the various variables characterizing the arm households and plots. Table 3 shows that almost 90 per cent of the plots in the sample were planted with maize ollowed by groundnuts (35 per cent). The maize yields are within the range reported in ost microeconomic studies. Inorganic fertilizer was used on almost 70 per cent of the lots, while only 12 per cent of the plots received organic fertilizers. The average fertilizer se is about 162 kg/ha (corresponding to 51.37 kg N/ha), which is around the applica- ion rate reported for Malawi in other microeconomic studies.11 This figure is higher than n other sub-Saharan African countries possibly because farmers are cultivating very small lots on especially small farms in the context of a generous farm input subsidy program.12 dditional descriptive statistics are presented in Online Appendix Table A1 in the appen- ices. On average, the plots are 0.79 km away from the homestead, though with a huge ariation across the sample (ranging from 0 to 10 km). The average plot size is 0.44 ha. ost farmers perceive that their plots are either good (45 per cent) or fair (43 per cent) in esponse to a question about the perceived soil quality. Most of the plots (59 per cent) have oils that are loam (i.e. between sand and clay) which are considered good soils for crop ultivation. The majority of the households (75 per cent) are male-headed with an average household ize of 4.8 people. About 76 per cent of the household heads have had no formal educa- ion. Almost 46 per cent of these households are classified as poor, with average household ncomes less than MK 37,002 per person per year based on the formal definition of the https://academic.oup.com/qopen/article-lookup/doi/10.1093/qopen/qoae017#supplementary-data Closing the gaps in experimental and observational crop response estimates 11 M r s w 3 T t i T t g t s s o m a e y t e p a d m m 4 4 T e t n m s l p f e m w r m p D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 alawi NSO. Most of the households live in remote rural areas, about 9 km from a main oad and 37 km from the nearest trading center. Online Appendix Fig. A1 in the appendices hows the number of plots planted with each of the crops. Most of the plots are planted ith a pure stand of maize followed by a pigeon pea-maize intercrop. .3 Experimental and observational yield gaps he challenge of combining observational and experimental evidence is that it is unlikely hat one will find directly comparable treatments, that is, yield responses obtained using sim- lar amounts (and types) of fertilizer grown in the same weather events and similar soil types. he nationally representative datasets available are almost 20 years apart (experiments in he 1990s and surveys in the 2010s). To demonstrate that experimental-observational yield aps existed in the 1990s when the experiments were being conducted, we compared the dis- rict averages from the plot level experimental data with the corresponding hybrid varietal- pecific administrative data for each of the two seasons, 1995/96 and 1997/98. Figure 2 hows scatterplots of district level averages of experimental hybrid maize yields for each f the fertilizer treatments (see Table 2 ) and the corresponding district averages of hybrid aize yields from administrative data in each of the respective seasons.13 There are six plots for the 1995/96 agricultural season and four plots for the 1997/98 gricultural season, with each of the plots representing the fertilizer treatments in the xperimental evidence.14 The rays indicate the ratio of experimental to observational ields. Across all the treatments, experimental yields are more than two times higher than he corresponding farm yields reported in the administrative database. As expected, the xperimental-observational yield gaps increase as the amount of nitrogen applied in the ex- erimental data increases. According to the Government of Malawi (1999 ), the 1997/98 gricultural season was a bad maize-growing year in that some districts experienced rought. This is especially evident in Fig. 2 b where the yield gaps for the no-fertilizer treat- ent are much lower. This highlights that the gap between observational and experimental aize yields are affected by environmental and climate conditions. . Results and discussion .1 Overview of the existing maize crop response literature for Malawi he research on crop responses in Malawi dates back to at least the 1960s. Blackie M. t al (1998) reported estimates of experimental maize responses in studies conducted from he 1960s to 1998 ranging from 23.1 to 34 kg of maize per unit of additional applied itrogen. Table 4 below taken from Arndt et al. (2016 : supplemental material) reports the icroeconomic evidence on the marginal returns to fertilizer use for selected types of maize eed. The mean maize responses range from 2.8 to 15 kg/ha for observational studies, much ower than the 23–34 kg/ha range reported in the experimental research. Informed by this rior evidence, below we use maize yield responses in the range of 0–30 kg/ha of maize or an additional kg of applied nitrogen as priors by which to anchor the assessment of stimates in prior studies. In all the prior published assessments of both the experimental and observational aize yield response estimates for Malawi and sub-Saharan Africa, only mean responses ere reported, absent any measures of the associated variation or uncertainty in these eported responses.15 However, to compare across studies and to make sense of these ean crop response parameter estimates, one cannot ignore the associated measures of recision. https://academic.oup.com/qopen/article-lookup/doi/10.1093/qopen/qoae017#supplementary-data https://academic.oup.com/qopen/article-lookup/doi/10.1093/qopen/qoae017#supplementary-data 12 Mkondiwa et al. Figure 2. District level hybrid maize yields from experimental and administrative data for 1997/98 agricultural season. D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 Closing the gaps in experimental and observational crop response estimates 13 Ta b le 4 . M ar gi na l r et ur ns to ni tr og en fe rt ili ze r us e, by m ai ze se ed va rie ty . D or w ar d et al . ( 20 08 ) (S ur ve y of lit er at ur e) H ar ou et al . ( 20 17 ) (M al aw i fi el d tr ia ls ) C hi bw an a et al . (2 01 2) (M al aw i F IS P) R ic ke r- G ilb er t et al . (2 01 1) (M al aw i F IS P) R ic ke r- G ilb er t an d Ja yn e (2 01 1) (M al aw i F IS P) K ilo gr am of m ai ze yi el d fo r an ad di ti on al un it of ni tr og en L oc al va ri et ie s 10 –1 2 12 .0 C om po si te s 15 H yb ri ds 18 –2 0 A ll im pr ov ed va ri et ie s 9. 6 A ll m ai ze se ed 15 24 –3 2 C on te m po ra ne ou s ef fe ct 6. 1 E nd ur in g ef fe ct 11 .7 M ea su re d at th e 10 th pe rc en ti le 2. 8 M ea su re d at th e m ed ia n 7. 6 M ea su re d at th e m ea n 9. 0 M ea su re d at th e 90 th pe rc en ti le 9. 7 So ur ce : A da pt ed fr om A rn dt et al . ( 20 16 ) . D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 14 Mkondiwa et al. Table 5. Experimental and observational maize response function to nitrogen. Experimental Observational Parameter 2.50% Mean 50% 97.50% 2.50% Mean 50% 97.50% (Intercept) 1026.8 1281.63 1281.98 1535.99 181.52 1104.10 1104 2040.73 N fertilizer amount 23.29 25.41 25.41 27.6 10.21 11.09 11.09 11.99 N fertilizer squared −0.1 −0.09 −0.09 −0.07 −0.01 −0.01 −0.01 −0.01 Marginal Effect at N = 55 kg/ha 19.13 20.58 20.58 22.05 9.78 10.56 10.56 11.36 Note : Controls and district fixed effects are included in all specifications. For details, see Online Appendix Tables A2 and A3 in the appendices. The marginal effects are calculated as β1 + β2 N̄ , where β1 and β2 are estimated coefficients and N̄ is the average nitrogen fertilizer evaluated at N = 55 kg/ha. 4 I a A w p l n t o t a O a o G e a p 1 t d 4 T w u d T s m n D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 .2 Experimental, observational and Bayesian crop responses n this section, we present the results from a Bayesian linear model (with results that re the same as using an OLSs on equation 3 ). The set of results (see details in Online ppendix Tables A2 and A3) show the ray production functions for maize intercropped ith either groundnuts, beans, pigeon peas, or soybeans. The variables of interest in the roduction functions include N fertilizer, N fertilizer squared and coordinate angles, the atter representing the crop output mix. The coefficients for the polar coordinate angles are egative for all maize-legume combinations (see Table A3 in the appendices). This implies hat an increase in the output mix reduces the total output index, meaning that the total utput index is lower when maize is intercropped with a particular legume. We estimate that he mono-cropped maize response to N fertilizer application is about 10.56 kg/ha per kg of pplied nitrogen, with a 95 per cent credible interval of 9.78–11.36 kg/ha (see Table 5 and nline Appendix Table A2). The experimental maize responses are about two times higher t 20.58 kg/ha per kg of applied nitrogen, consistent with finding of Anderson (1992) who bserved that ‘There is a systematic overstatement of the extent of responsiveness of crops to applied fertilizer in Africa, relative to what is achievable under most farm conditions. The extent of overstatement is of the order of a factor of, say, two in terms of incremental response ratios.’ Anderson (1992 : 393). iven these results, we can combine the experimental coefficient and the observational co- fficient by simply using the experimental estimate (25.41) and its standard deviation (0.41) s the prior in a regression using the observation data. Figure A3 shows that the resulting osterior distribution of the N coefficient (i.e. median: 12.01, 95 per cent credible interval: 1.53, 12.50) is still closer to the distribution of N responses derived from the observa- ional estimates (i.e. median: 11.09, 95 per cent credible interval: 10.21, 11.99) that the istribution derived from the experimental results. .3 Bayesian analysis with sensitivity testing he foregoing analysis documents the crop response gaps and the hybrid crop responses hen using particular experimental and observational data. It is justifiable to question the se of experimental data that were collected almost two decades before the observational ata. A lot of biophysical factors (including varieties and soil quality) may have changed. herefore, the following set of results uses a range of alternative priors that span the plau- ible range drawing on evidence gleaned from the prior published literature. In particular, the Bayesian sensitivity results indicate changes in Bayesian estimates of the aize yield responses given changes in the prior distributions of crop responses at 55 kg of itrogen per ha. The sensitivity checks are in the changes to the prior on N fertilizer use on https://academic.oup.com/qopen/article-lookup/doi/10.1093/qopen/qoae017#supplementary-data https://academic.oup.com/qopen/article-lookup/doi/10.1093/qopen/qoae017#supplementary-data https://academic.oup.com/qopen/article-lookup/doi/10.1093/qopen/qoae017#supplementary-data https://academic.oup.com/qopen/article-lookup/doi/10.1093/qopen/qoae017#supplementary-data https://academic.oup.com/qopen/article-lookup/doi/10.1093/qopen/qoae017#supplementary-data https://academic.oup.com/qopen/article-lookup/doi/10.1093/qopen/qoae017#supplementary-data Closing the gaps in experimental and observational crop response estimates 15 Table 6. Sensitivity analysis to various priors on the mean and precision. Marginal effects of N fertilizer use Row number Prior precision Prior mean 2.50% 50% 97.50% 1 Baseline: 0.001 0 9 .78 10 .56 11 .36 2 0 .1 0 4 .62 5 .27 5 .92 3 0 .1 6 4 .69 5 .34 5 .98 4 0 .1 12 4 .76 5 .41 6 .05 5 0 .1 18 4 .83 5 .48 6 .12 6 0 .1 24 4 .90 5 .55 6 .19 7 0 .1 30 4 .97 5 .61 6 .26 8 1 0 4 .16 4 .78 5 .39 9 1 6 4 .78 5 .40 6 .01 10 1 12 5 .41 6 .02 6 .64 11 1 18 6 .03 6 .65 7 .26 12 1 24 6 .65 7 .27 7 .88 13 1 30 7 .28 7 .90 8 .51 14 10 0 2 .02 2 .47 2 .91 15 10 6 5 .25 5 .69 6 .13 16 10 12 8 .47 8 .92 9 .36 17 10 18 11 .74 12 .19 12 .64 18 10 24 15 .09 15 .55 16 .01 19 10 30 18 .57 19 .04 19 .51 Note: The prior means for all the controls including N squared term were set to 0 and prior precision was set to 0.1 (so as to make the prior proper for the calculation of the marginal likelihood needed for Bayes Factor). Marginal effects are calculated as β1 + β2 N̄ where β1 and β2 are coefficients for N and N squared terms and N̄ is the average nitrogen fertilizer evaluated at 55 kg/ha. t r e t o m r f ( s o e N ( e t 5 t t e c m p t D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 he mono-cropped maize response function. We considered a range of experimental crop esponse estimates reported for Malawi as summarized by Arndt et al. (2016) and Snapp t al. (2014) to assess if incorporating these priors leads to revisions in the crop responses hat would warrant a change in the recommendations.16 We considered six mean prior levels f the crop response coefficient; specifically values of 0, 6, 12, 18, 24, and 30. A directed search for variance parameters across prior literature revealed that the esti- ates vary as well. For example, using different econometric specifications of a quadratic esponse function as we do, the standard error for the maize response to nitrogen ranges rom about 0.3 to 0.5 in Harou et al. (2017) . Using a quadratic production function, Darko 2016 : 92) estimated standard errors of the crop responses ranging from 1.6 to 3.2. In this tudy, we therefore consider three precision levels: 0.1, 1, and 10 corresponding to variances f 10, 1, and 0.1, respectively. Table 6 shows crop response quantiles for 18 different mod- ls for the various plausible mean and variance priors for the parameter corresponding to fertilizer use. It is important to note that these are based on calculating marginal effects which we also call crop response) not the direct coefficient of the N fertilizer. Marginal ffects are calculated as β1 + β2 N̄ where β1 and β2 are coefficients for N and N squared erms, and N̄ is the average nitrogen fertilizer rate at which the effect is evaluated at (i.e. 5 kg N/ha). The computed crop responses ( Table 6 : row 2–row 13) are largely invariant to changes in he prior mean when the prior precision falls in the 0.1 and 1 range, but are indeed sensitive o the choice of prior means when a higher precision (10) is assumed. This is revealed by the xtent of the overlapping 95 per cent credible intervals when the different prior means are ompared across the same low prior precision level (e.g. compare 5.92 upper quantile for 0 ean prior and 0.1 prior precision with 4.97 lower quantile for 30 mean prior and 0.1 prior recision). In terms of the effect of differences in the precision of the prior estimates, the able shows that when crop response posteriors at different prior precision are compared 16 Mkondiwa et al. Figure 3. Posterior stochastic dominance at different prior mean values [prior precision = 0.1 and 10]. a r t p a p 0 p 4 T t c d g l d o b T m D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 cross the same prior mean, the pattern of crop response posterior results is fuzzy. The eason for this is that the prior precision of one parameter also affects the off-diagonal erms in the variance-covariance matrix, such that depending on the associations across all arameters in the regression, the Bayesian crop responses may not be an intuitive weighted verage. In Bayesian terminology, this is called the effect of nuisance parameters. This also artly explains why the posterior estimates in the baseline case (with 0 prior mean and .001 for all parameters) are different from the sensitivity analyses (0 prior mean and 0.1 rior precision for all other variables except N and N squared terms). .3.1 Mean crop response scenarios he scenarios in Table 6 are illustrated graphically in Fig. 3 below using cumulative dis- ribution functions (cdfs) of the crop response parameters subject to different priors. The omparisons of the cdfs can be interpreted as the posterior stochastic dominance. This is one by first holding the precision constant while varying the mean of the prior for the nitro- en coefficient ( Fig. 3 a and b). In Fig. 3 a, as expected, holding precision constant at the same ow level, the model with the higher prior mean (30 maize kg/ha per kg of N) stochastically ominates all the other models. However, the differences between the posterior distributions f mean responses are small. At a higher precision level, the ordering of the mean response distributions is maintained, ut now the differences between the posterior distributions are quite pronounced ( Fig. 3 b). his implies that achieving lower variance (high precision) may be necessary in the develop- ent of recommendations as also suggested by agronomic research (Vanlauwe et al. 2016 Closing the gaps in experimental and observational crop response estimates 17 Figure 4. Posterior stochastic dominance at different prior precision values [prior mean = 30]. a t i i r 4 I m a d m d o H l 4 I p p D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 nd Coe et al. 2016 ). These results show that comparing mean experimental and observa- ional estimates without considering the variation in responses around the mean can result n final combined recommendations that are a reflection of the scenario in Fig. 3 a or that n Fig. 3 b. Either way, the posterior responses are less than the experimental mean crop esponses that are used to develop fertilizer application recommendations in Malawi. .3.2 Precision of crop response scenarios n Fig. 4 a and b, we show the cases where the precision priors are varied, while the prior ean is fixed at some value. As expected from stochastic dominance, the higher the precision ssumed for the prior, the more likely it is that the posterior will be similar to the prior istribution. Since the prior mean is 0 in Fig. 4 a, the posterior parameter estimates for the odel with a highly informative prior (i.e. a high precision prior = 10) is stochastically ominated by the ones with a weakly informative or effectively non informative priors (1 r 0.1). A contrasting case is presented in Fig. 4 b where a prior mean of 30 is assumed. ere, the model with a high precise prior mean stochastically dominates the models with ow(er) precision. .3.3 Combined mean and precision scenario n the preceding stochastic dominance results, a clear and consistent ordering of either the rior mean or prior precision was assumed. However, what of the case that has a higher rior mean but a lower precision relative to the opposite case (i.e. lower prior mean higher 18 Mkondiwa et al. Figure 5. Posterior stochastic dominance at different experimental priors for mean and precision. p c p l & f w t c r r 1 9 c o q e o 4 B m t l m s m fi a e i D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 recision)? This is where Bayesian stochastic dominance becomes useful. Let’s consider a ase where the prior means and variances are different. In Fig. 5 , a low mean-high variance rior leads to posterior parameter estimates that stochastically dominate the high mean- ow variance results. This implies that the debate between Jayne et al. (2015) and Dorward Chirwa (2015) on whether a lower or higher mean crop yield response is appropriate or assessing the economic veracity of Malawi’s farm input subsidy program is problematic hen the precision (or variance) of the mean estimates are ignored. When uncertainty is incorporated it is the case that for assumed precision levels of 0.1 o 1—which are typical of the precision levels in observational research—, the posterior rop response estimates that are likely relevant for commercial agriculture (observational) ange from 4 to 9 kg of maize output for a unit of fertilizer per ha when the prior mean esponses levels for experimental trials range from 0 to 30. While for precision levels of –10 which are prevalent in experimental research, the posterior crop estimates range from to 19 for prior mean levels from 0 to 30. Based on our evidence, the likely fertilizer rop responses for Malawian agriculture are low and highly variable. Thus, any claims f substantial crop responses to fertilizer application in Malawian maize production are uestionable. Therefore, when evaluating the efficacy of policies that depend on empirical stimates of crop responses, it would be advisable to err on the conservative side (and draw n all the plausible evidence about the mean responses and variations around this mean). .4 Heterogeneity in crop responses: Bayesian hierarchical model results eyond the question of variations around the mean crop responses, the crop response gaps ay also be due to differences in locations where each of the studies were conducted within he country. We therefore need to understand the heterogeneity in the crop responses across ocations. There are two extremes in the way heterogeneity is typically handled in econo- etric analysis. Most studies pool all the data and generate a single response parameter, as- uming a homogenous response for the whole sample. At the other end of the spectrum, one ay consider estimating the response parameters with specific (additive and multiplicative) xed effects for individual cohorts of the data (e.g. individual districts), but this is gener- lly inefficient due to data limitations. A Bayesian hierarchical modeling framework is an fficient (i.e. in terms of degrees of freedom) middle ground, which allows estimation of ndividual specific parameters as random parameters. Closing the gaps in experimental and observational crop response estimates 19 Figure 6. Uncert aint y and heterogeneit y in crop response parameter on fertilizer use (kg/ha). ( l a t r 1 t a r e t d u r d p o i s t D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 Figure 6 shows the density plots of parameter uncertainty (panel A) and heterogeneity panel B). Panel A is based on crop response parameter for the draws from the Bayesian inear model, and illustrates the parameter uncertainty under the maintained assumption of spatially invariant response function. Panel B is based on a random parameter specifica- ion of the district-specific crop response parameter in a hierarchical Bayesian model, and epresents the district-level heterogeneity in crop responses. The results indicate that the model-based parameter uncertainty (ranging from about 8 to 4 additional kgs of maize for additional unit of fertilizer) plotted in Panel A is smaller than he district-to-district heterogeneity plotted in Panel B ( −40–30 additional kgs of maize for dditional unit of fertilizer). The negative responses imply that soils are not conditionally esponsive to fertilizer application in these locations. While it is uncommon for agricultural conomists and agronomists concerned with average responses to report negative responses, his can occur when the fertilizer applied scorches the seed, especially in relatively dry con- itions (Vanlauwe et al. 2011 ). The findings on district-to-district heterogeneity may seem nrealistic when compared with previous analyses that assumed spatial homogeneity in esponses. This result is nonetheless consistent with agronomic research that addresses in- ividual plot heterogeneity. For example, a study by Vanlauwe et al. (2016) compared em- irical distributions (heterogeneity) to model based distributions (measuring uncertainty) f crop responses from agronomic trial data related to maize in Western Kenya and beans n Eastern Rwanda. They concluded that model based distributions provide better preci- ion in the extremes than empirical curves, but that model based distributions depend on he assumption that the model is unbiased. In terms of developing targeted crop response 20 Mkondiwa et al. Figure 7. Scatterplot of hierarchical crop response coefficient by district from experimental and survey evidence. Note: The x and y axes correspond to the hierarchical coefficient for linear term in a quadratic crop response function not the marginal effect. s a a a r g a q q q f a i s r p p e s D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 upport, the heterogeneous model may be more appropriate as it can help identify districts nd plots that are non-responsive. Based on the observational data used in our analysis, the districts of Machinga, Nsanje, nd Chikwawa, for example, appear to be non-responsiveness to fertilizer application (see lso Fig. 7 ). This is in line with experimental evidence (Government of Malawi 1997 ) that eports lower crop responses in the shire valley districts (Nsanje and Chikwawa). This sug- ests a future research strategy that proceeds by answering two questions: (1) will maize in given field respond to fertilizer; (2) if so, what is the optimum fertilizer rate? Answering uestion 2 is more difficult than answering question 1. At a minimum, being able to answer uestion 1 is really impactful. The hierarchical Bayesian model allows one to answer both uestions in that we can identify unresponsive districts and the magnitude of the response or the responsive districts. Figure 7 shows the scatterplot of the district level linear crop response parameters in hierarchical model for experimental and survey data. For almost seven districts (specif- cally, Chikwawa, Mulanje, Machinga, Rumphi, Mangochi, Phalombe, and Mulanje), the oils are not responsive to fertilizer application based on the observational evidence but are esponsive in the experimental evidence. As a research matter, this implies that there still more we need to learn about the bio- hysical and socio-economic aspects that distinguish these districts (i.e. water holding ca- acity or timing of fertilizer application). In terms of policy, it implies that well targeted xtension services are required so that farmers do not waste fertilizer on unresponsive oils. Closing the gaps in experimental and observational crop response estimates 21 5 5 T t r e q e a t a m c t O g e o v m d p a t B n s s h 5 T a g o a t i h s p p f n a a d D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 . Limitations, future research, and conclusion .1. Limitations and future research here are still several remaining limitations in addressing modeling challenges of parame- er uncertainty, heterogeneity and disparate information sources in the estimation of crop esponses to fertilizer application. The first limitation is that given the weaknesses of both xperimental and observational studies, it is difficult to measure the quality of the subse- uent posterior evidence. Recent advances in Bayesian optimization and machine learning, specially mult-fidelity methods (see Frazier 2018 ) hold promise in combining the estimates nd evaluating the quality of such combined estimates. These advanced methods are beyond he scope of this paper. The second limitation is that the scenarios on the effect of changes to the prior on the ssumed posterior parameter estimates is based on the pooled model not the hierarchical odel, which entails that heterogeneity is being treated separately from partial identifi- ation of the distribution of the parameters. This is inevitably the case because estimates reating fertilizer use parameters as random parameters across groups are not available. ther shrinkage models like empirical Bayes modeling and machine learning using ridge re- ression and multi-fidelity methods are potential candidates for future research. In addition, ndogeneity concerns across both the experimental (due to self-selected master farmers) and bservational (management bias and substantial measurement errors) evidence are areas of alid concern that future research could systematically address using quasi-experimental ethods. Finally, the models are not directly linked to any policy parameter like whether to subsi- ize fertilizer, which not only depend on the uncertainty and heterogeneity of crop response arameters, but also on other parameters (e.g. relative profitability of other crops) and the ssociated political economy considerations. While recent literature has demonstrated in- eractive effects of crop management, including weed management (e.g. Burke et al., 2020 , urke et al., 2022 ), and inherent soil quality and degradation (e.g. Berazneva et al. 2023 ), one of these studies attempt to provide whether experimental or observational evidence hould be the basis of such conclusions across heterogeneous environments. Future research hould consider the effect of incorporating multiple sources of information, uncertainty and eterogeneity on a policy decision and the analytical tools used in the paper are appropriate. .2. Conclusion his paper has incorporated three aspects that are often ignored in the crop response liter- ture, namely parameter uncertainty, multiple sources of information, and (spatial) hetero- eneity in the response to fertilizer use. A Bayesian approach is employed to address each f these themes and close the measured gaps in the responses. This is an important goal for gricultural research because of the fairly constant trends of crop output/fertilizer price ra- ios across sub-Saharan Africa, which are indicative of the proposition that long-term trends n fertilizer profitability require improvements in farmer crop response rates. The analysis as shown that using prior knowledge of crop response estimates adds insights to the as- essment of crop responses using observational data. In particular, we find that ignoring the recision parameter when using crop response estimates may lead to inconclusive policy rescriptions. The debates on whether crop responses to fertilizer application are high or low are there- ore questionable when uncertainty that appears to measurably affect the stochastic domi- ance ordering of crop response estimates is ignored. Unless uncertainty is considered, the rguments for or against the use of experimental and observation crop response estimates re inconclusive, thereby leading to questionable policy prescriptions. Moreover, while the ebates have centered on means of crop responses, this paper has shown that both the 22 Mkondiwa et al. m e a v f s a m u c i l s A T A ( I B A t S S D T c o C T E 1 3 4 D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 eans and variances matter in these policy discussions. The results of incorporating het- rogeneity in the estimation by way of using a hierarchical Bayesian modeling approach re quite revealing. We find that the degree of spatial heterogeneity in fertilizer responses aries markedly, with some districts being effectively non-responsive to the application of ertilizer (e.g. Chikwawa), while other districts are highly responsive (e.g. Dedza). These patial differences may be more important to explore for fertilizer policy targeting as they re as wide as national level mean differences between experimental and observational esti- ates. To make progress on reconciling the disparate estimates, we advocate for including ncertainty measures (e.g. standard errors) and specifics on climatic, soil and environmental onditions under which estimates being compared were generated. Methodologically, build- ng on the Bayesian approach we have used, advances in Bayesian optimization and machine earning including multi-fidelity methods hold promise in guiding future experiments and urveys for generating spatially specific and robust recommendations. cknowledgments his paper was prepared with support from the McKnight Foundation and Excellence in gronomy (EiA) Initiative of the CGIAR funded by Bill and Melinda Gates Foundation BMGF). We also acknowledge additional support from the University of Minnesota GEMS nformatics Center, Minnesota Agricultural Experiment Station project MIN-14–171 and MGF funded Cereal Systems Initiative for South Asia (CSISA). The author(s) thank Jeffrey pland and Jeffrey Coulter for extremely helpful comments on this paper. The content of his paper solely reflects the opinions and findings of the author(s). upplementary material upplementary data are available at Q Open online. ata availability statement he main data and code are provided on GitHub: https://github.com/MaxwellMkondiwa/ losing_gaps_bayes. Any other additional data beyond what has been shared will be shared n reasonable request to the corresponding author. onflict of interest he authors declare no conflict of interest. nd Notes See Beddow et al. (2014) for a detailed bio-economic review and evaluation of the yield gap literature. Jayne et al. (2018) reviews subsidy programs for 10 African countries and reports that between 2011 and 2014 the farm input subsidy program in Malawi accounted for 21–44 per cent of the country’s total spending on agriculture. Notice here that some of the crop response gaps are due to semantics. While much of what Jayne et al. (2015) refer to as crop responses are based on total fertilizer applied, Dorward & Chirwa (2015) specifically refer to nitrogen applied. This is a common cause of ostensible differences in yield responses given that agronomic experiments typically report nutrient specific responses, while obser- vational evidence often report crop output responses to the total fertilizer applied. The interpretation holds because we are using a Bayesian credible interval not a frequentist confidence interval. https://academic.oup.com/qopen/article-lookup/doi/10.1093/qopen/qoae017#supplementary-data https://github.com/MaxwellMkondiwa/closing_gaps_bayes Closing the gaps in experimental and observational crop response estimates 23 5 6 7 8 9 1 1 1 1 1 1 1 R A A A A A B B B D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 In reality it is not just soil quality that varies with location. Many other climate, terrain, and physical aspect factors vary by location, often in ways that are imprecisely if at all measured, and variations in these factors also affect crop yields. There is a large body of literature that suggests that choice of a functional form affects the crop responses. The quadratic crop response is used in this paper because it is easy to estimate as com- pared with nonlinear options and also satisfies the basic properties of a production (or yield response) function. By definition, the posterior distribution function p(θ | Y, X ) given crop yields and inputs is derived from a likelihood function L (Y, X| θ ) and a prior distribution function p(θ ) using the Bayes rule as follows: p(θ | Y, X ) = L (Y, X| θ ) p(θ ) p(Y, X ) , where p(Y, X ) is the normalizing constant which is a function of Y and X only, and can be defined by p(Y, X ) = ∫ � L (Y, X| θ ) p(θ ) , where � is the parameter space. This implies that the posterior is proportional to p(θ | X, Y ) ∝ L (X, Y | θ ) p(θ ) . The author is indebted to Dr Todd Benson at IFPRI who participated in the trials and kindly provided the data for the purposes of this study. The data and sampling procedures for the survey can be obtained from the LSMS data repository: https://doi.org/10.48529/w1jq-qh85. 0 The national per capita N fertilizer use in the current survey data is about 51 kg N/ha, while in the period the experiments were conducted (1995/96 and 1997/98), it was about 38 kg of fertilizer per ha (Minot et al. 2000 : p. 50). This implies that the observational application rates were between the 0 and 35 kg N/ha treatments in the experiments. 1 For example, Sheahan & Barrett (2017) report averages of 146 kg/ha (which is equivalent to 53.1 kg/ha of nitrogen) for Malawi, while Komarek et al. (2017) reports 51 kg/ha nitrogen for cen- tral Malawi. The nitrogen application rate reported here, was derived by multiplying 0.23 to basal (23:21:0 + 4S) fertilizer amount applied and 0.46 to top dress (Urea) fertilizer applied, where 0.23 and 0.46 represent the proportion of nitrogen in the fertilizer. 2 The subsidy program targets about 1.5 million farm households representing half of the farm house- holds in Malawi with two 50 kg bags of fertilizer (Arndt et al. 2016 ). 3 MH17 and MH18 were the main improved maize varieties planted during the years of the trial. 4 While the experimental data were parsed into their respective fertilizer treatment cohorts, the same (albeit seasonal and varietal specific) observational data were used in each of the fertilizer cohorts. 5 See a comprehensive review by Jayne & Rashid (2013 : 533, Table 3 ), Snapp et al. (2014) and Benson et al. (2021) . 6 The reported crop response rates are derived from different functional form specifications of the crop response functions (e.g. quadratic, linear), econometric procedures (e.g. ordinary least squares, quantile regression), and using different datasets (e.g. household surveys, demonstration trials). eferences llenby G. M. and Rossi P. E. (1998). ‘Marketing Models of Consumer Heterogeneity.’ Journal of Econo- metrics , 89: 57–78. nderson J. R. (1992). ‘Difficulties in African Agricultural Systems Enhancement? Ten Hypotheses.’ Agri- cultural Systems , 38:387–409. nderson J. R. and Dillon J. R. (1968). ‘Economic Considerations in Response Research’. American Jour- nal of Agricultural Economics , 50:130–42. ngrist J. D. et al. (2017). ‘Leveraging Lotteries for School Value-Added: Testing and Estimation.’ Quar- terly Journal of Economics , 50:871–919. https://doi.org/10.1093/qje/qjx001. rndt C., Pauw K. and Thurlow J. (2016). ‘The Economy-wide Impacts and Risks of Malawi’s Farm Input Subsidy Program.’ American Journal of Agricultural Economics , 98:962–80. http://ajae. oxfordjournals.org/lookup/doi/10.1093/ajae/aav048. arrett C. and Hogset H. (2003). Estimating Multiple-Output Production Functions for the CLASSES Model . Working Paper.m http://crsps.net/wp-content/downloads/BASIS/Inventoried%203. 16/13-2003-8-502.pdf. eddow J. M. et al. (2014). ‘Food Security: Yield Gaps.’ In: Neal, K. Van Alfen (ed.) Encyclopedia of Agri- culture and Food Systems . New York, NY: Elsevier. https://doi.org/10.1016/B978-0-444-52512-3. 00037-1. enson T. (1999). ‘Validating and Strengthening the Area-Specific Fertilizer Recommendations for Hybrid Maize Grown by Malawi Smallholders: A Research Report of the Results of the Nationwide 1997/98 https://doi.org/10.48529/w1jq-qh85 https://doi.org/10.1093/qje/qjx001 http://ajae.oxfordjournals.org/lookup/doi/10.1093/ajae/aav048 http://crsps.net/wp-content/downloads/BASIS/Inventoried%203.16/13-2003-8-502.pdf https://doi.org/10.1016/B978-0-444-52512-3.00037-1 24 Mkondiwa et al. — — B B B B B B B B C C C C C C C C D D D D D D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 Maize Fertilizer Recommendations Demonstration.’ Maize Productivity Task Force, Government of Malawi. —— et al. (2021). Disentangling Food Security from Subsistence Agriculture in Malawi . Washington D.C., USA. International Food Policy Research Institute. https://doi.org/10.2499/9780896294059. Accessed 1 March 2024. —— et al. 2024. Fertilizer Subsidies in Malawi: From Past to Present . Malawi Strategy Support Program, Working paper 44. https://hdl.handle.net/10568/138880. Accessed 1 March 2024. enson T., Mabiso A. and Nankhuni F. (2016). Detailed Crop Suitability Maps and an Agricultural Zona- tion Scheme for Malawi: Spatial Information for Agricultural Planning Purposes . Feed the Future Innovation Lab for Food Security Policy Research Paper, Vol. 17. East Lansing: Michigan State Uni- versity. http://ebrary.ifpri.org/cdm/ref/collection/p15738coll2/id/130494. erazneva J. et al. (2023). ‘Paying for Agricultural Information in Malawi: The Role of Soil Heterogeneity’. Journal of Development Economics , 165:103144. ernard D. R. et al. (2024). How Much Should We Trust Observational Estimates? Accumulat- ing Evidence Using Randomized Controlled Trials with Imperfect Compliance . Working Paper. http://tse-fr.eu/pub/128965. Accessed 1 March 2024. hattacharyya A. and Mandal R. (2016). A Generalized Stochastic Production Frontier Analysis of Tech- nical Efficiency of Rice Farming : A Case Study from Assam, India . Sam Houston State University, Department of Economics and International Business. Working Paper No. 16-03. lackie M. et al. (1998). Malawi: Soil Fertility Issues and Options . A Discussion Paper. https://silo.tips/ download/malawi-soil-fertility-issues-and-options. Accessed 1 March 2024. ullock D. S. and Bullock D. G. (2000). ‘From Agronomic Research to Farm Management Guidelines: A Primer on the Economics of Information and Precision Technology.’ Precision Agriculture , 2:71–101. urke W. J., Snapp S. S. and Jayne T. S. (2020). ‘An In-Depth Examination of Maize Yield Response to Fertilizer in Central Malawi Reveals Low Profits and Too Many Weeds’. Agricultural Economics , 51:923–40. urke W. J., Jayne T. S. and Snapp S. S. (2022). ‘Nitrogen Efficiency by Soil Quality and Management Regimes on Malawi Farms: Can Fertilizer Use Remain Profitable?’ World Development , 152:105792. abrini S. M., Irwin S. H. and Good D. L. (2010). ‘Should Farmers Follow the Recommendations of Market Advisory Services? A Hierarchical Bayesian Approach to Estimation of Expected Performance.’ American Journal of Agricultural Economics , 92:622–37. arlin B. P. and Louis T. A. (2009). Bayesian Methods for Data Analysis Third ., London: Chapman and Hall/CRC Press. hapman C. and Feit E. M. (2015). R for marketing research and analytics . New York, NY: Springer. hib S. and Carlin B. P. (1999). ‘On MCMC Sampling in Hierarchical Longitudinal Models.’ Statistics and Computing , 9:17–26. hib S. (1995). ‘Marginal Likelihood from the Gibbs Output.’ Journal of the American Statistical Asso- ciation , 90:1313–21. hirwa E. and Dorward A. (2013). Agricultural Input Subsidies: The Recent Malawi Experience . Oxford, UK: Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199683529.001.0001. hibwana C., Fisher M. and Shively G. (2012). ‘Cropland Allocation Effects of Agricultural Input Subsi- dies in Malawi’. World Development , 40:124–33. oe R., Njoloma J. and Sinclair F. (2016). ‘To Control or Not to Control: How Do We Learn More About How Agronomic Innovations Perform on Farms?’ Experimental Agriculture , 55:303–9. arko F. A. (2016). ‘Essays on Malawian Agriculture: Micro-Level Welfare Impacts of Agricultural Pro- ductivity; Profitability of Fertilizer Use; and Targeting of Subsidy Programs.’ PhD dissertation, Purude University. avidson B. R. and Martin B. R. (1965). ‘The Relationship Between Yields on Farms and Experiments’. Australian Journal of Agricultural Economics , 9:129–40. avidson B. R., Martin B. R. and Mauldon R. G. (1967). ‘The Application of Experimental Research to Farm Production’. Journal of Farm Economics , 49:900–7. orward A. R. and Chirwa E. W. (2015). ‘How do fertilizer subsidy programs affect total fertilizer use in sub-Saharan Africa? Crowding out, diversion, and benefit/cost assessments.’ Agricultural Economics , 46:739–44. orward A. et al. (2008). Evaluation of the 2006/07 Agricultural Input Subsidy Programme, Malawi. Final Report Submitted to the Ministry of Agriculture and Food Security , Lilongwe: Government of Malawi. https://doi.org/10.2499/9780896294059 https://hdl.handle.net/10568/138880 http://ebrary.ifpri.org/cdm/ref/collection/p15738coll2/id/130494 http://tse-fr.eu/pub/128965 https://silo.tips/download/malawi-soil-fertility-issues-and-options https://doi.org/10.1093/acprof:oso/9780199683529.001.0001 Closing the gaps in experimental and observational crop response estimates 25 D F F F F G G H H H J — — J J J K L L — L M M M D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 uflo E., Kremer M. and Robinson J., 2011. ‘Nudging Farmers to Use Fertilizer: Theory and Experimental Evidence from Kenya.’ American Economic Review , 101:2350–90. http://pubs.aeaweb.org/doi/abs/10. 1257/aer.101.6.2350. essler P. and Kasy M. (2018). ‘How to Use Economic Theory to Improve Estimators: Shrinking Toward Theoretical Restrictions.’ Review of Economics and Statistics , 101:681–98. https://doi.org/10.1162/ rest_a_00795. ezzi C. and Bateman I. J. 2011. ‘Structural Agricultural Land Use Modeling for Spatial Agro- Environmental Policy Analysis.’ American Journal of Agricultural Economics , 93:1168–88. ousekis P. 2002. ‘Distance vs. Ray Functions : An Application to the Inshore Fishery of Greece.’ Marine Resource Economics , 17:251–67. razier P. I. (2018). ‘A Tutorial on Bayesian Optimization’. arXiv . 10.48550/arXiv.1807.02811 overnment of Malawi. (1997). Maize Fertilizer Recommendations Demonstration 1997/98. Lilongwe: Government of Malawi. overnment of Malawi. (1999). Area-Specific Fertilizer Recommendations for Hybrid Maize Grown by Malawian Smallholders: A Manual for Field Assistants. Lilongwe: Government of Malawi. Url: https: //www.cabidigitallibrary.org/doi/pdf/10.5555/20083326757. Accessed 1 March 2024. arou A. P., Liu Y. and Barrett C. B. (2017). ‘Variable Returns to Fertiliser Use and the Geography of Poverty: Experimental and Simulation Evidence from Malawi.’ Journal of African Economies , 26: 342–71. artley M. J. (1983). ‘Econometric Methods for Agricultural Supply under Uncertainty: Fertilizer Use and Crop Response.’ Journal of Mathematical Analysis and Applications , 94:575–601. enningsen G., Henningsen A. and Jensen U. (2015). ‘A Monte Carlo Study on Multiple Output Stochastic Frontiers: A Comparison of Two Approaches.’ Journal of Productivity Analysis , 44:309–20. ayne T. S. et al. (2010). ‘Malawi’s Maize Marketing System.’ Michigan State University. Url: http://fsg. afre.msu.edu/malawi/Malawi_maize_markets_Report_to-DFID-SOAS.pdf. Accessed 1 March 2024. —— et al. (2015). ‘How Do Fertilizer Subsidy Program Affect Total Fertilizer Use In Sub-Saharan Africa? Crowding Out, Diversion, and Benefit/cost.’ Agricultural Economics , 46:745–55. http://doi. wiley.com/10.1111/agec.12190. —— et al. (2018). ‘Review: Taking Stock of Africa’s Second-Generation Agricultural Input Subsidy Programs.’ Food Policy , 75:1–4. ayne T. S. and Rashid S. (2013). ‘Input Subsidy Programs in Sub-Saharan Africa: A Synthesis of Recent Evidence.’ Agricultural Economics , 44:547–62. ohnson M. E., Edelman B. and Kazembe C. (2016). ‘A Farm-Level Perspective of the Policy Challenges for Export Diversification in Malawi: Example of the Oilseeds and Maize Sectors.’ IFPRI Discussion Pa- per No. 01549. http://ebrary.ifpri.org/utils/getfile/collection/p15738coll2/id/130596/filename/130807. pdf. ust R. E., Zilberman D. and Hochman E. (1983). ‘Estimation of Multicrop Production Functions.’ Amer- ican Journal of Agricultural Economics , 65:770–80. omarek A. M. et al. (2017). ‘Agricultural Household Effects of Fertilizer Price Changes for Smallholder Farmers in Central Malawi.’ Agricultural Systems , 154:168–78. evy H. (2016). ‘Stochastic Dominance: Investment Decision Making under Uncertainty.’ 2nd ed, Springer: New York, USA öthgren M. (1997). ‘Generalized Stochastic Frontier Production Models.’ Economics Letters , 57: 255–9. —— (2000). ‘Specification and Estimation of Stochastic Multiple-Output Production and Technical Inefficiency.’ Applied Economics , 32:1533–40. usk J. L. et al. (2002). ‘Empirical Properties of Duality Theory.’ The Australian Journal of Agricultural and Resource Economics, 46:45–68. anski C. F. (2010). ‘Policy Analysis with Incredible Certitude.’ The Economic Journal , 121:F261– F289. eager R. (2019). ‘Understanding the Average Impact of Microcredit Expansions: A Bayesian Hierarchi- cal Analysis of Sevem Randomized Experiments.’ American Economic Journal: Applied Economics , 11:57–91. inot N., Kherallah M. and Berry P. (2000). ‘Fertilizer market reform and the determinants of fertil- izer use in Benin and Malawi.’ International Food Policy Research Institute, Markets and Structural Studies Division , Discussion Paper No. 40. https://www.ifpri.org/publication/fertilizer-market-reform- and-determinants-fertilizer-use-benin-and-malawi. http://pubs.aeaweb.org/doi/abs/10.1257/aer.101.6.2350 https://doi.org/10.1162/rest_a_00795 https://www.cabidigitallibrary.org/doi/pdf/10.5555/20083326757 http://fsg.afre.msu.edu/malawi/Malawi_maize_markets_Report_to-DFID-SOAS.pdf http://doi.wiley.com/10.1111/agec.12190 http://ebrary.ifpri.org/utils/getfile/collection/p15738coll2/id/130596/filename/130807.pdf https://www.ifpri.org/publication/fertilizer-market-reform-and-determinants-fertilizer-use-benin-and-malawi 26 Mkondiwa et al. R R R S S S S V V Z © E C d D ow nloaded from https://academ ic.oup.com /qopen/article/4/2/qoae017/7718810 by C entro Internacional de M ejoram iento de M aiz y Trigo user on 21 August 2024 osas F., Lence S. and Hayes D. J. (2018). ‘Crop Yield Responses to Prices: a Bayesian Approach to Blend Experimental and Market Data.’ European Review of Agricultural Economics , 46: 551–77. https://doi.org/10.1093/erae/jby032. icker-Gilbert J. and Jayne T. (2011). What Are the Enduring Effects of Fertilizer Subsidy Programs on Re- cipient Farm Households? Evidence from Malawi. Staff Paper 2011–09 . Michigan, USA: Department of Agricultural, Food and Resource Economics, Michigan State University. icker-Gilbert J., Jayne T. and Chirwa E. (2011). ‘Subsidies and Crowding Out: A Double Hurdle Model of Fertilizer Demand in Malawi’. American Journal of Agricultural Economics , 93:26–42. cott J. C. (1998). Seeing Like A State: How Certain Schemes To Improve The Human Condition Have Failed . New Haven, USA: Yale University Press. heahan M. and Barrett C. B. (2017). ‘Ten Striking Facts About Agricultural Input Use in Sub-Saharan Africa.’ Food Policy , 67:12–25. napp S. et al. (2014). ‘Maize Yield Response to Nitrogen in Malawi ’s Smallholder Production Sys- tems.’ IFPRI-Malawi Strategy Support Program Working Paper No. 9 , Lilongwe, Malawi. https:// www.ifpri.org/publication/maize-yield-response-nitrogen-malawi%E2%80%99s-smallholder- production-systems. uri T. (2011). ‘Selection and Comparative Advantage in Technology Adoption.’ Econometrica , 79: 159–209. anlauwe B. et al. (2011). ‘Agronomic Use Efficiency of N Fertilizer in Maize-Based Systems in Sub- Saharan Africa within the Context of Integrated Soil Fertility Management.’ Plant and Soil , 339: 35–50. anlauwe B., Coe R. and Giller K. E. (2016). ‘Beyond Averages: New Approaches to Understand Hetero- geneity and Risk of Technology Success or Failure in Smallholder Farming.’ Experimental Agriculture , 55:1–23. ingore S. et al. (2007). ‘Soil Type, Management History and Current Resource Allocation: Three Di- mensions Regulating Variability in Crop Productivity on African Smallholder Farms.’ Fields Crops Research , 101:296–305. https://doi.org/10.1016/j.fcr.2006.12.006. The Author(s) 2024. Published by Oxford University in association with European Agricultural and Applied conomics Publications Foundation. This is an Open Access article distributed under the terms of the Creative ommons Attribution License ( https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, istribution, and reproduction in any medium, provided the original work is properly cited. https://doi.org/10.1093/erae/jby032 https://www.ifpri.org/publication/maize-yield-response-nitrogen-malawi%E2%80%99s-smallholder-production-systems https://doi.org/10.1016/j.fcr.2006.12.006 https://creativecommons.org/licenses/by/4.0/ 1 Introduction 2 Model 2.1 Theoretical model 2.2 Empirical models 3 Data sources and descriptive statistics 3.1 Data sources 3.2 Descriptive statistics 3.3 Experimental and observational yield gaps 4 Results and discussion 4.1 Overview of the existing maize crop response literature for Malawi 4.2 Experimental, observational and Bayesian crop responses 4.3 Bayesian analysis with sensitivity testing 4.4 Heterogeneity in crop responses: Bayesian hierarchical model results 5 Limitations, future research, and conclusion 5.1 Limitations and future research 6 Conclusion Acknowledgments Supplementary material Data availability statement Conflict of interest End Notes References