Causal Inference on Networks under Continuous Treatment Interference

This paper presents a methodology to draw causal inference in a non-experimental setting subject to network interference. Specifically, we develop a generalized propensity score-based estimator that allows us to estimate both direct and spillover effects of a continuous treatment, which spreads through weighted and directed edges of a network. To showcase this methodology, we investigate whether and how spillover effects shape the optimal level of policy interventions in agricultural markets. Our results show that, in this context, neglecting interference may lead to a downward bias when assessing policy effectiveness.



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1 Introduction

Policies often involve interventions that spillover across units and generate indirect effects. These effects, which occur when an agent’s treatments indirectly affect other agents’ outcomes through different types of interference, are pervasive in many economic and social contexts (Cox, 1958). Understanding the mechanism of interference is therefore crucial for the optimal design of an intervention, because it allows to leverage or reduce spillover effects and improve the overall policy effectiveness (Moffitt, 2001). This is, for example, the case when contrasting criminal involvement (Glaeser et al., 1996), improving immigrants’ access to labor market (Beaman, 2012), providing financial education (Bursztyn et al., 2014; Cai et al., 2015), and designing health programs (Miguel et al., 2004), managerial incentive systems (Bandiera et al., 2009), or retirement plans (Duflo and Saez, 2003).111Or also, encouraging schooling attendance (Lalive et al., 2009) and responding to trade restrictions (Giordani et al., 2016).

Broad-based policy experimentation represents the ideal framework to develop a rigorous evaluation of an intervention (Athey & Imbens, 2017), and a large array of cleverly designed experiments have been developed to deal with the issue of interference (see Baird et al., 2018, for a recent review).222 The theoretical foundations often employed by this literature have been laid by Hudgens & Halloran (2008), and further developed by Liu & Hudgens (2013), Tchetgen Tchetgen & VanderWeele (2012), and Baird et al. (2018), who approached the estimation of spillover effects using two-stage randomization designs. In a later refinement, Leung (2019) provided a framework to account for interference when conveyed by a single social network. However, more than often the only viable option to assess a policy effectiveness is through the analysis of observational data. In these cases, the existing impact evaluation methodologies (e.g., the propensity score matching (PSM), Rosenbaum & Rubin, 1983; Dehejia et al., 2002), which rely on the assumption that an agent’s treatment does not spill over on to other agents,333This is also called Individualistic Treatment Response (ITR) assumption (Manski, 2013), and combined with the unique treatment assumption is referred to as the Stable Unit Treatment Value Assumption (SUTVA) (Rubin, 1980). largely neglect the existence of interference.

In this paper, we present one of the first efforts to evaluate policy interventions in the presence of interference using observational data. Our proposed methodology aims to: i) produce unbiased estimates of the policy impact by correcting for the bias resulting from both treatment selection and interference, and ii) quantify both direct and spillover effects of the treatment.

We provide a general formalization of the issue under the potential-outcomes framework, by developing a joint propensity score (JPS) for the estimation of treatment and spillover effects in observational data. In doing so, we assume that spillover effects are mediated through a network structure (Jackson et al., 2017). This allows us to identify how the agent’s treatment has also an effect on another agent, track the pattern of policy diffusion in the network444This is also the case of Banerjee et al. (2013), who employed social network data to identify best practices for the diffusion of a microfinance program in India., and conceive each agent as subject to two treatments, namely: the individual treatment and the network treatment. The latter is the agent’s exposure to the treatments of neighborhood agents. Causal estimation of these effects is then obtained by balancing individual and network characteristics across agents under different levels of the individual and network treatments. The foundation of this approach rests upon the work by Forastiere et al. (2020), who were among the first to tackle the issue of interference in observational network data and causal inference. With respect to that study, a key novelty of this paper is represented by the source of treatment considered. While Forastiere et al. (2020) proposes a method to study binary treatments, here we consider a continuous treatment at both individual and network level. Moreover, our method allows to model different degree of exposures to spillover effects, by considering both symmetric or asymmetric network connections, and those characterized by heterogeneous intensities.

The methodology developed in this paper is illustrated through the investigation of policy effectiveness in agricultural markets. Policy interventions in these markets provide an ideal case-study for the analysis interference for two main reasons. First, these interventions are not randomly distributed, leading to a potential treatment selection bias. Second, they can significantly interfere with one another because of the emerging interconnections of the agri-food international networks (Johnson & Noguera, 2017; Balié et al., 2018). In this context, our method allows to assess the impact of a policy while correcting for potential biases resulting from both treatment selection and interference. In addition, it allows to model the non-discrete nature of the treatment, accounting for the heterogeneous intensity of policy interventions, which differ from country to country and over time. The potential spillover effects of national policies in agricultural markets have represented a matter of interest for academics and policy makers since the 2008 food crisis, when riots erupted in many developing countries, and it is currently highly debated during the COVID-19 crisis, as some countries are resorting to restrict their exports in order to ensure adequate domestic supplies.555As reported by the International Food Policy Research Institute (IFPRI) (Glauber et al., 2020): on March 2020 Russia, the world’s second largest exporter of cereals, has already implemented a ban on its export of processed grains. Soon after, Kazakhstan, Thailand, and Cambodia followed suit and Vietnam, the world’s third largest exporter of rice, put a moratorium on new export contracts. Contributing to this emerging debate, we show that agricultural policies have indirect effects which significantly interfere with those implemented by commercial partners.

The contribution of this paper is threefold. First, by introducing a new framework to deal with interference in observational studies, it contributes to the vast literature on impact evaluation techniques (for a review, see Sacerdote, 2014; Athey & Imbens, 2017). Second, it proposes an innovative approach to properly frame the role of network structures in causal inference, which is still very much debated in both economic and network analysis (Doreian, 2001; Arpino et al., 2017). Third, it provides new insights into the optimal design of agricultural policies in the presence of non-negligible spillover effects.

The rest of this paper is organized as follows. In Section 2 we describe the JPS estimator. In Section 3, we present an empirical application of this method to investigate spillover effects in agricultural markets. Finally, Section 4 concludes and draws policy implications.

2 Methodology

In this section we introduce the notation and main assumptions of our analytical setting. First, we assume that agents are embedded in a network; that is, the agents are represented by nodes and the diffusion channel through which the treatments interfere with one another, mediating spillover effects, is identified by a link. In other words, i and j are connected through a link if treatment on an agent has also an effect on the other agent. This network can be represented by the adjacency matrix , with element being a continuous value on the realm of positive real numbers registering exposure of agent to treatment of agent .666Observe that in this framework, connections can be defined both by a social or a spatial criterion.

Then, let the network be composed of nodes. For each node , we define a partition of the set of nodes as . The set is referred to as the neighborhood of agent , which has cardinality and contains all nodes directly connected to . The number of neighbors is referred to as the degree centrality of agent for consistency with the literature on social networks (see e.g., Jackson (2010)). Similarly, the set contains all nodes other than that are not in .

We now denote by the observed outcome for agent , and by Y

the corresponding vector. We assume that

is subject to two treatments. The former is the individual treatment for agent , denoted by the continuous variable , for which the corresponding vector is Z. The latter is the neighborhood treatment, denoted by , which is the individual treatment received by other agents in and to which is indirectly exposed. For each unit , the object defines the following partitions of the treatment and outcome vectors: and .

Finally, defines the vector of individual-level covariates for agent ; for instance, ’s economic and social characteristics. Similarly, denotes the vector of neighborhood covariates for agent . This may include two types of ’s neighborhood-level covariates: i) variables representing the structure of the neighborhood (e.g. the degree , the centrality, the reciprocity, the topology, etc.), and ii) variables representing the composition of the neighborhood (i.e., aggregational characteristics summarizing individual attributes of nodes ).777Specifically, this function takes the form of , where is a -matrix collecting all the neighbors’ individual covariates and is a function summarizing the matrix into a vector of dimension . and are then combined into the vector composed by covariates, which represents the set of all exogenous pre-treatment variables for agent .

2.1 The Stable Unit Treatment on Neighborhood Value Assumption (SUTNVA)

In this section, we discuss how interference is modelled in our setting. In impact evaluation methods, it is standard to assume that agent’s potential outcome (i.e., ) depends only on the agent’s own treatment; that is, the individual treatment Z. This assumption, combined with the unique treatment assumption, is referred to as the Stable Unit Treatment Value Assumption (SUTVA) (Rubin, 1980). However, under interference, SUTVA does not hold, since agents are also exposed to the treatment received by their neighborhood; that is, the neighborhood treatment . Formally, this violation of the SUTVA requires the introduction of a new more general assumption, which takes into account interference within the neighborhood. Accordingly, in what follows, we propose some adjustments to the main elements constituting the SUTVA.

The first component of the SUTVA is the so-called consistency assumption (Rubin, 1980, 1986). Specifically, the outcome observed at node depends on the entire treatment assignment vector Z and can be written as . As pointed out by Rubin (1986), this potential outcome is only well defined if the following assumption holds:

Assumption 1 (No Multiple Versions of Treatment (Consistency))

The mechanism used to assign treatments does not matter and assigning the treatments in a different way does not constitute a different treatment.

The second component of the SUTVA is the assumption that the potential outcome of unit i only depends on the individual treatment , which rules out the presence of interference between units. In our setting, the assumption of no interference is replaced with a new local interference assumption within the neighborhood:

Assumption 2 (First-Order Interference)

The potential outcome of agent depends on the individual treatment (i.e., ) and the treatment received by connected agents (i.e., ):

Assumption 2 states that interference acts only within the immediate neighborhood; that is, agent is exposed to spillover effects generated from the treatment of direct connections in the network. Building on this idea, we can formalize the dependence of agent i’s outcome from the treatments received by neighboring agents through a specific summarizing function , defined by Aronow & Samii (2017) as exposure mapping function. Specifically, this function allows us to: i) map agent’s exposure to the treatment that spillover across the network, and ii) identify the area of the network (i.e., the neighborhood) from which spillovers originate (e.g. friends or commercial partners). In doing so, it can be used to disentangle the effect of the individual treatment from that resulting from the exposure to the treatment received by other agents located in the neighborhood. By means of this function, we are able to introduce the following assumption:

Assumption 3 (First-Order Interference with Exposure Mapping)

Given a function , and such that , the following equality holds:

In combination with Assumption 1, this is referred to as Stable Unit Treatment on Neighborhood Value Assumption (SUTNVA) (Forastiere et al., 2020).

2.2 Potential Outcomes and Neighborhood Interference

We now define the form of the neighborhood treatment , and the potential outcomes of the agents . Therefore, we make use of the exposure mapping function introduced in the previous section.

The level of treatment received by agent from her neighbors, , can be expressed through the exposure mapping function as . This implies that can be either univariate or multivariate. In this context, and without loss of generality, we consider it to be:


where is the entry of matrix recording the level of interaction between and . A desirable feature of Assumption 3, with defined as in (1), is that it allows for the existence of heterogeneous network effects. This implies that the level of first-order interference depends on the position of the agent in the network, and one can take into consideration both direction and intensity of the link between agents.

From this definition of the neighborhood treatment, and by virtue of Assumption 3, we are also able to define the potential outcomes of the agents as a joint treatment of the form ; henceforth, . Specifically, this expression represents the potential outcome of node under treatment , whereby agent is exposed to the neighborhood treatment through connected agents.888Here, we take a perspective that is sometimes referred to as ‘super-population’; that is, the potential outcomes are fixed quantities and expectations are simple averages of these outcomes in the large, near-infinite, super-population of cardinality (Imbens & Rubin, 2015).

2.3 Causal Estimands

Given the continuous individual treatment and the bivariate joint treatment, the potential outcome of unit can be seen as a dose-response function. Accordingly, we define the marginal mean of the potential outcome , for each value of and , as the average dose-response function (aDRF), denoted by . Formally, let


which can be marginalized to get the univariate average dose-response functions


where and are the observed marginal distributions of the neighborhood and individual treatments. The univariate average dose-response functions allow to define direct effects of the treatment as any comparison , or as the first derivative of the average dose-response function . Similarly, spillover effects can be defined as the difference between the average potential outcome corresponding to two different levels of the neighborhood treatment and : , or as the first derivative of the average dose-response function .

2.4 Unconfoundedness of the Joint Treatment

To draw causal inference, the standard literature relies on the unconfoundedness assumption, which implies that the treatment can be considered to be randomly assigned after accounting for agents’ differences in a fixed set of exogenous pre-treatment characteristics (Rubin, 1990). However, in the presence of interference, we require that both the individual and the neighborhood treatments should be unconfounded conditional on covariates. The extension of unconfoundedness to neighborhood treatment is a crucial element of this study because only when both treatments are considered as random is any potential confounding effect offset. Formally, this leads to the identification of the dose-response function based on the unconfoudedness assumption of the joint treatment, which is the conditional independence between the joint treatment and potential outcomes:

Assumption 4 (Unconfoudedness of the Joint Treatment)

Conditional on the vector of covariates , the potential outcome is independent of the level of the treatments and :

Assumption 4 states that for agents with the same values of covariates , the distribution of a potential outcome does not depend on the actual treatments and that each agent receives. Conditional independence of essentially posits an exogeneity assumption of the joint treatment and it rules out the presence of unmeasured factors affecting the potential outcome of an agent and alternatively their own treatment or the treatment received by their neighbors (see Forastiere et al. (2020) for a detailed discussion on the plausibility of this assumption).

2.5 Joint Propensity Score-based Estimator

To get an unbiased estimate of both the treatment and the spillover effects, we now present the JPS-based estimator. This is modelled to correct for interference by balancing individual and neighborhood covariates across agents under different levels of individual and neighborhood treatments.

Formally, we define a joint propensity score

as the probability of being subject to direct treatment

and being exposed to a weighted average of the treatments of the agent’s connections equal to , given characteristics :


where denotes the probability of neighborhood treatment at level conditional on a specific value of the individual treatment and on the vector of covariates (i.e., ), which we refer to as the neighborhood propensity score. Similarly, denotes the probability of individual treatment at level conditional on covariates (i.e., ), which we refer to as the individual propensity score.

By definition, the individual and neighborhood propensity scores are two joint balancing scores; that is, . This means that within strata with the same value of and

, the joint probability distribution of the individual treatment

and the neighborhood treatment does not depend on the value of . It is then straightforward to show that, given the Assumption 4 and the balancing property of the propensity scores, the assignment to the joint treatment is unconfounded given both the individual and the neighborhood propensity scores (i.e., ). This result implies that any bias associated with differences in the distribution of covariates across groups with different treatment levels can be removed by adjusting for both propensity scores.999A provides a detailed discussion of this result. As an adjustment method, we use an extended version of the model-based generalized propensity score approach (GPS) introduced by Hirano & Imbens (2004). Thus, given the factorization of the joint propensity score into the product of the individual propensity score and neighborhood propensity score (4), we are able to use a generalized propensity score approach on both propensity scores to adjust for confounding covariates .

As already mentioned, this approach builds on Forastiere et al. (2020), who deal with a binary individual treatment and use a subclassification method to adjust for the individual propensity score and (within each stratum) the model-based GPS approach to adjust for the neighborhood propensity score. With respect to this method, our estimator is modified by replacing the subclassification on the propensity score of the binary treatment with a second generalized propensity score for continuous treatment.

2.6 Estimation Procedure

In what follows, we outline the estimating procedure for the average dose-response function .

Consider the following general models for the individual treatment Z, the neighborhood treatment G, and the outcome Y:


According to the models in (5), (6), and (7), the estimation procedure requires the following steps:

  1. Estimate the parameters and of the models for the individual treatment in (5) and for the neighborhood treatment in (6);

  2. Use the estimated parameters in Step 1 to predict for each unit the actual individual propensity score and the actual neighborhood propensity score ; that is, the probabilities for unit i to receive the individual treatment and being exposed to the neighborhood treatment , where and are the values that were actually observed;

  3. Estimate the parameters of the outcome model in (7) by using the observed data , and the predicted propensity scores and ;

  4. For each level of the joint treatment , predict for each unit the individual and the neighborhood propensity scores corresponding to that level of the treatment (i.e., and

    ) and use these predicted values to impute the potential outcome


  5. To estimate the average dose-response function , for each level of the joint treatment take the average of the potential outcomes over all units

Standard errors and 95% confidence intervals can be derived using bootstrap methods (Efron, 1979; Forastiere et al., 2020).

3 Empirical Application

In this section, we illustrate how our methodology works in practice through an application to the agricultural markets in the presence of spillover effects. The use of our (generalized) propensity score-based estimator to this context is justified by two main reasons. First, policy interventions in these markets are not random; rather, they are driven by a series of macro-economic factors such as the country’s level of development and agroclimatic conditions, among others. In such cases, our method allows us to assess the impact of a policy while correcting for potential biases resulting from both treatment selection and interference. Second, the intensity of policy interventions is highly heterogeneous – differing from country to country and over time – and our framework provides the means to model the non-discrete nature of this treatment.

The agricultural sector represents an ideal case study because it has been subjected to some of the most heavy-handed governmental interventions over the last century (Anderson et al., 2010). Despite the successful conclusion of the Uruguay Round Agreement on Agriculture (URAA) in 1994, which helped to reduce distortions, many countries still prefer to regulate agricultural markets and subsidize farmers (Carter and Steinbach, 2018). The reason for this is that preventing losses in a sector with a significant presence (in terms of output, employment, etc.) may loom large in the government’s objective function (Trefler, 1993; Freund and Ozden, 2008).101010For instance, in Africa the agriculture sector still generates about 25% of the gross domestic product (GDP), or 50% if we look at the broader agribusiness sector, and involves roughly 65% of the local population (Balié et al., 2018).

An additional element of interest, is the dramatic level of interconnection reached by agricultural markets, which provides an optimal environment to study interference. The emergence of the so-called agri-food global value chains (GVCs) (Johnson & Noguera, 2017; Balié et al., 2018) has increased the probability of spillover effects generated by national policy interventions111111Giordani et al. (2016) document the existence of a "multiplier price effect" that can take place when countries impose restricting trade measures. (Gouel, 2016; Bayramoglu et al., 2018; Beckman et al., 2018; Fajgelbaum et al., 2020), thereby challenging the way in which policy makers establish their policies.

Finally, while there is an intense debate over the effects that openness to trade in the primary sector has had on consumers’ welfare (e.g., on income during a famine in India (Burgess and Donaldson, 2010), on health in Mexico (Giuntella et al., 2020) and on labor market distortions (Tombe, 2015)), to the best of our knowledge few efforts have been made to analyse the effects on food security (Magrini et al., 2017; Allcott et al., 2019) and none of them takes spillover effects into account. Our empirical application aims at complementing this literature by providing the first investigation on whether and how agricultural policy interventions in a particular country impact not only food security in the country itself, but also spill over on to commercial partners.

3.1 Data

The empirical analysis requires four sets of data: i) the treatment variable measuring the intensity of policy interventions, ii) a set of observed country’s characteristics, iii) an outcome measuring food security, and iv) a network describing the trade relationships.

Following Anderson and Nelgen (2012a, b), here we assess the policy intensity using the Nominal Rate of Assistance (NRA). Used in the guise of treatment, this is an estimate of direct government policy intervention, as it measures the percentage by which these policies have raised (lowered) gross returns to farmers above (below) what they would have been without the government’s intervention. In other words, the NRA is the percentage by which the domestic producer price is above (or below, if negative) the border price of a like product, net of transportation and trade margins. Therefore, NRA is pivotal to testing our methodology in that it is a continuous measure accounting for both traditional policy instruments (e.g., tariffs, export subsidies, and import quotas) and the additional measures untamed by the URAA (e.g., trade remedies).121212It is worth mentioning, however, that trade policies such as export bans and import tariffs account for 60% of the NRA at the global level (Anderson et al., 2013). For ease of interpretation, we shift the support of the treatment (NRA + 1), which is known as Nominal Assistance Coefficient (NAC). Therefore, for any given country, a NAC>1 signals the presence of policies supporting the agricultural producers (and a farm-gate price above the border price) while a NAC<1 indicates a disincentive (i.e., taxation) for the agricultural sector. Figure 1 shows that while the richest countries are decreasing their policy support to farmers, developing countries are increasingly switching from taxing agricultural production to applying protectionist measures, often exceeding the level of support provided by OECD countries (Swinnen et al., 2012).131313Anderson and Nelgen (2012a, b) provide a detailed explanation of the method used to develop the NRA and of the interpretation of NAC.

Figure 1: NAC values in OECD (red) and Non-OECD (blue) countries from 1990 to 2010

The characteristics selected to explain the policy intensity (i.e., set ii) are borrowed from the agricultural and trade policy literature (Anderson et al., 2013; Magrini et al., 2017). Specifically, we consider: real per capita GDP and total population as a proxy of the country demand and size, respectively; per-capita arable land and the agricultural total factor productivity growth index to assess the country’s relative agricultural comparative advantage; the ratio of food imports to total exports, net food exports, and absolute (positive and negative) percentage deviations from the trend in international food prices as a measure of country’s access to, dependence from, and position in the global market; and the international food price volatility index to capture country’s response to changes in price levels. Finally, we include a dummy to capture the effect of the food crisis of 2007-08, and a set of regional dummies to control for unobservable characteristics of African, Asian, European-transition, Latin American, and high-income countries.

Food security (i.e., set iii) is primarily measured as the level of food availability, that is the supply of food commodities in kilo-calories per person. According to the guidelines of the Committee on World Food Security (CFS, 2009), consumers are better off when this measure is maximized.141414Note that supply figures do not include consumption-level waste (i.e. that wasted at retail, restaurant and household levels), and therefore represents food available for consumption at the retail level, rather than actual food intake. As an example, Unites States shows around 3700 per capita calorie supply in 2000, while Kenya only 2000 kcal. Alternatively, as also suggested by the CFS (2009), we analyze food utilization as proxied by the prevalence of anemia among children aged under five.151515Consequently, consumers are better off when the measure of food utilization is minimized.

Finally, the network (i.e., set iv) is built using the value of bilateral agri-food trade in each given year (see Figure 2). The time window considered goes from 1990 to 2010. Furthermore, it is worth emphasizing that Magrini et al. (2017), to avoid the risk of interference, exclude the countries most likely to generate or be affected by spillover effects from the analysis (i.e., the top global exporters and importers), namely: the United States, Germany, France, Italy, Spain, the Netherlands, Belgium, China, Brazil, Canada, Japan and the UK. In contrast, we keep these countries because it is our interest to account for interference.161616In a robustness check, we also exclude from our analysis these countries. Consequently, we have a sample of 74 countries (see Figure 6). Summary statistics are reported in Table 2. For additional information on data sources, see Table 3 in B.

Figure 2: International Trade Network in 2010. Only top five incoming flows are displayed for OECD (red) and non-OECD (blue) countries

3.2 Model Setup

The general framework presented in Section 2 allows us to model the spillover effects using different measures. For the purpose of this empirical application, we assume that the extent to which the intensity of a policy of country affects country depends on the value of its bilateral agri-food exports, normalized by the average world trade value. Consequently, equation (1) becomes:


In addition, we leverage the information provided by the observation of the network structure over time, to obtain the square adjacency matrix , where the generic element if and : i) refer to the same country, or ii) have no trade relationships at time , or iii) indicate observations at different points in time.171717Note that this requires independence between observations across years, which is a common assumption in most spatial applications (Anselin et al., 2008).

Following the estimating procedure for the average dose-response function of Section 2.6

, in this empirical application we first apply a zero-skewness Box-Cox transformation

(Box and Cox, 1964) to the vector of treatment, ,181818Note that is chosen so that the skewness of the transformed variable is zero. and assume the following normal model for :


where contains the the set of covariates relative to agent (i.e., ).191919The implicit assumption is that self-selection mechanisms are only driven by the individual characteristics of the agent and not by the neighborhood characteristics (i.e., is not considered). We will relax this assumption in Section (3.4). Similarly, we assume the following model for the neighborhood treatment



follows a normal distribution with mean

and variance

. Finally, we postulate a normal model for the outcome given the propensity scores:


where is the sum of cubic polynomials and their interactions. We also include in an interaction term between the country NAC and the network NAC. This allows the direct effect of national policies to vary depending on the policies implemented in partner countries, and the spillover effects to vary depending on the country NAC.

3.3 Results

Table 1 reports the estimated parameters of the models for the individual treatment (Direct NAC) and the neighborhood treatment (Network NAC). The set of covariates used in these models, i.e, , are those described in Section 3.1.

(1) (2)
Direct NAC Network NAC
(Eq. 5) (Eq. 6)
real pc GDP 0.038 (0.007) 0.686 (0.068)
pc arable land 0.040 (0.006) 0.066 (0.055)
population 0.013 (0.004) 0.392 (0.037)
agricultural productivity 0.001 (0.0003) 0.004 (0.003)
food import/total exports 0.023 (0.008) 0.360 (0.071)
net exports 0.017 (0.002) 0.096 (0.017)
positive deviation food price 0.144 (0.143) 1.842 (1.300)
negative deviation food price 0.236 (0.132) 2.378 (1.208)
food price volatility 2.843 (1.102) 19.944 (10.065)
food crisis 0.030 (0.018) 0.118 (0.160)
Z 1.139 (0.208)
Constant 0.430 (0.080) 9.787 (0.732)
Observations 930 930
R 0.527 0.420
Adjusted R 0.519 0.410
Residual Std. Error 0.126 (df = 915) 1.150 (df = 914)
F Statistic 72.728 (df = 14; 915) 44.066 (df = 15; 914)
Regional dummies Yes Yes
  • Notes: Significance levels: * p 0.1; ** p0.05; *** p0.01. Real pc GDP, pc arable land and population variables are in log and one year lagged. Agricultural productivity, food import/total exports, net exports, positive deviation food price and negative deviation food price variables are one year lagged (Source: FAOSTAT, WDI, USDA).

Table 1: Individual and neighborhood propensity scores

Propensity scores are instrumental to fit the outcome model. Nonetheless, a number of relevant insights can be obtained by the analysis of the estimated parameters presented in this table. Take for instance column 1, which indicates the correlation between the intensity of direct policy interventions and pre-treatment country characteristics. A country is more likely to strengthen the support to its agricultural sector when the local demand expands, due to an increase in per capita GDP or in the population. In contrast, a country tends to increase the taxation of revenues when it has a comparative advantage in the agricultural sector, as measured by per capita arable land, or when it heavily relies on imports, as proxied by the variables food import over total exports and net food exports. This also happens when positive and negative deviations of international prices from their trend and high level of food price volatility occur, suggesting that when prices spike, governments tend to reduce the support to their domestic markets by imposing restrictions on exports and lowering protection on imports. This is also confirmed by the negative and significant coefficient on food crisis.

Now consider column 2, which describes instead the correlation between the country characteristics and the (weighted average) intensity of policies implemented by its commercial partners. We observe that partner countries tend to provide a high support to their own agricultural sector when the country features a large local demand or it increases its reliance on imports. In contrast, they reduce the level of support when trading with partners who experience price volatility. Finally, we observe that the direct NAC is negatively correlated with the network NAC, which suggests that, country characteristics being held constant, the higher the level of support in the country the lower will be that of partner countries.

Following Steps 2 and 3 of the methodology, we predict the individual and neighborhood propensity scores. These are then used to estimate the conditional expectation of the outcome given by the model (7).202020Since we make use of nonlinear functions of the individual and network NAC, model (7) implies a nonlinear functional form on the direct and network NAC. The estimated coefficients are reported in Table 5 in the B. We then obtain the dose-response functions by following Steps 4 and 5. That is, we first predict the probability of observing each pair of values of the direct NAC () and network NAC (), and we then use the individual and neighborhood propensity scores to predict the country-level outcomes corresponding to . Finally, we obtain the dose-response function by averaging these potential outcomes across all countries.

Figure 3 reports the marginal aDRF for the food availability outcome when neglecting interference (i.e., when we do not include either the network NAC or the neighborhood propensity score in the outcome model). The figure shows that the highest level of food availability is registered when NAC is about 48% (i.e., when governments provide a limited support to the price received by their agricultural producers). Moreover, marginal benefits are also obtained when NAC values range from 0.9 to 1.48, that is when the response function is increasing. In contrast, negative effects are produced in cases of strong incentives (i.e., NAC higher than 1.48) or high taxation (i.e., NAC lower than 0.9). This suggests that: i) in line with Anderson et al. (2013), taxing agricultural producers to obtain additional resources to be invested in more dynamic sectors comes at a cost of lower food availability212121Anderson et al. (2013) shows that taxation affects both producers and consumers. For producers, it reduces both profits and incentives to respond to market signals. For consumers, if taxation discourages farming activity, then it can negatively affect both demand for farm labor and wages for unskilled workers in farm and non-farm jobs.; and ii) a strong support of the primary sector may result in a protection of inefficient domestic producers or crop varieties (Tombe, 2015).

Figure 3: Dose-response function of direct NAC on food availability (log scale) w/o interference

Similarly, Figure 4 displays the marginal aDRF (left-hand panel) and (right-hand panel) when interference is taken into account. By comparing this response function with that of Figure 3 (and the relative coefficients of the two outcome models (Tables 5 columns 1 and 2), we obtain meaningful information on the extent and direction of the bias when neglecting spillover effects.

Specifically, we observe that when ignoring interference, the impact of national policies () is overestimated by about 30%. The left-hand panel of Figure 4, which represents the aDRF of the direct NAC when interference is taken into account and marginalized over, shows that the highest benefit in terms of food supply is registered when NAC value is equal to 1.78. This suggests that domestic policies may require additional efforts if they are to be effective in an interconnected world.

In addition, the dose-response estimator allows us to assess the spillover effects of policy interventions in partner countries. The right-hand panel of Figure 4, representing the average dose-response function of the network NAC , shows that as a result of the emergence of agri-food GVCs, it is crucial to take into account commercial partner policies when determining the optimal level of a domestic intervention because they can either boost or counteract the effect of local measures. Specifically, high levels of domestic food availability are reached when trading partners provide incentives to their own agricultural producers, as shown by the increasing aDRF. This is straightforward bearing in mind that producer support may boost exports and therefore food availability in the importing country i.

Figure 4: Marginal dose-response function of direct NAC (left-hand) and marginal dose-response function of network NAC (right-hand) on food availability (log scale) with interference

The effect of the correlation between domestic and foreign policies — as mediated by the trade network — is clearer when we look at Figure 5, which represents the bivariate aDRF . Even when governments are able to maximize their objective functions and reach the highest level possible of welfare, the intensity of policies implemented in partner countries may still push the supply of food far from the desired level.

Figure 5: Average dose-response function of direct NAC and network NAC on food availability (log scale) with interference

3.4 Robustness Checks

In Section 2 we show that when the selection of neighborhood treatment is driven by the characteristics of both the agent and the network, unconfoundedness may be achieved through a decomposition of the vector into two subvectors encompassing individual and neighborhood chracteristics. While in the previous analysis we made use of only the former type of covariates, in this section we introduce a specific network-level variable, namely the weighted average real per capita GDP of the network . Table 6 in B reports the estimated parameters of the models for the individual and the neighborhood treatments when including this additional covariate. The results are in line with our baseline specification and further show that network real per capita GDP is negatively correlated with the direct NAC and positively associated with the network NAC.222222The goodness of fit of the Network NAC model clearly improves when including this additional covariate, as it explains most of the variability of the variable, the network NAC. Moreover, we find that the outcome model still confirms our main results (Table 7, B), although with wider confidence intervals for both aDRFs and a lower maximum point for (1.61) (Figure 7).

Further, we check whether our results are robust to sample composition. First, we re-run our baseline model excluding the main global exporters and importers, namely the United States, Germany, France, Italy, Spain, the Netherlands, Belgium, China, Brazil, Canada, Japan and the UK, to check whether these are indeed the countries more responsible for spillover effects. As expected, the results reported in Tables 8, 9 in B and in Figure 8 show no point of maximum in the considered NAC range and a no longer increasing aDRF of the network NAC (right-hand panel). This corroborates our main findings, whereby agricultural policies implemented by large global players have a strong influence over the food security of their trade partners and, accordingly, cannot be ignored when assessing the effectiveness of policies.

Finally, we make use of an alternative proxy for food security, i.e. food utilization. Measured as the prevalence of anemia among children aged under five, consumers are better off when this measure is minimized. Table 2 in B reports the coefficients of the four outcome models so far analyzed. A low level of intervention (i.e., when NAC is about 1.4) is still conducive of high consumer welfare (Figure 9). But again this optimal level of support is underestimated when considering interference, as the minimum point moves from 1.4 to 1.6 (Figure 10). Similar results are also obtained when we introduce the neighborhood-level characteristic, and when excluding top global exporters and importers from the analysis, as reported in Figure 12 and Figure 13, respectively.

4 Conclusions

Causal inference in observational studies has often neglected the presence of interference, which has proven to be pervasive in many economic and social contexts. By developing a JPS estimator extended to the case of continuous treatment, this paper provides a methodology to evaluate policies when spillover effects matter. Specifically, we develop a generalized propensity score-base estimator that corrects for the bias resulting from both treatment selection and continuous treatment interference, balancing individual and neighborhood covariates across units under different levels of individual treatment and of exposure to neighbors’ treatment.

The empirical relevance of our methodology is illustrated through the assessment of the effects of agricultural policies on food security. On the one hand, our results show that policy interventions matter and they have a non-linear impact on food security. In particular, both a local excessive taxation and support for the primary sector are detrimental for food availability. On the other hand, the average direct effect estimated neglecting interference underestimates the optimal level of producers’ support.

The correlation between local and foreign policies — as mediated by the trade network — points to new directions of research and it may provide interesting insights to assess the indirect effects of policy changes. This is, for instance, the case of the Single Farm Payment implemented in 2003 under the Common Agricultural Policy of the European Union (EU), which consisted in detaching farmers’ income payments from the production of specific crops to increase the flexibility of farm decision making and ultimately reduce the level of EU intervention. It is also the case of the currently restrictions policy measures implemented by some countries in order to ensure adequate domestic supplies and shield their consumers from price volatility during the highly debated COVID-19 crisis. The framework provided in this paper might contribute to an assessment of the indirect consequences of these policies on partner countries.

However, even though this paper makes a concrete step forward in causal inference studies accounting for network structures, a serious limitation lies in the assumption that limits spillover effects to first-order network neighbors. While this is common in the literature, the plausible presence of higher-order interference might bias the estimates of both direct and first-order spillover effects. Finally, the bootstrap procedure that we employ in our empirical application relies on an independent sampling strategy with replacement, which is only appropriate if the analyst can rule out the presence of a residual correlation between potential outcomes of partner countries after conditioning for covariates. Clustering at the geographical level or by employing a community detection algorithm (see Forastiere et al. (2019)) would be a promising avenue for future research.


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Appendix A Balance Check

In Section 2.5 we described the balancing property of the propensity scores; that is, the fact that the covariates are balanced across levels of the joint treatment within strata defined by the values of both propensity scores and . As long as the estimated propensity scores satisfy this property, the proposed adjustment method ensures unbiased estimates of the causal estimands of interest. Therefore, this balancing property can be employed to empirically assess the adequacy of the estimated propensity scores.

With a binary treatment, this balance check is usually conducted by comparing the distribution of covariates between treated and control units within strata defined by the propensity scores (Rosenbaum & Rubin, 1983). With a continuous treatment, Hirano & Imbens (2004) propose to first divide the levels of the treatment into intervals and, within these, stratify individuals into groups according to the median values (of the corresponding interval) of the generalized propensity score. Then, it is possible to test whether the observed covariates are balanced within these GPS strata. Unfortunately, in our framework each unit is affected by two different continuous treatments and a similar approach seems to be unfeasible.

However, we may implement a regression-based approach as in Flores et al. (2012). We first check the balancing property of both the individual and neighborhood propensity scores. This is done by regressing each covariate on the two treatments with and without the generalized propensity scores and then comparing the coefficients of the treatments. A likelihood ratio (LR) test can compare the two models and check whether conditional on the GPS, the covariates have little explanatory power. If so, they are sufficiently balanced by the GPS. Finally, to check for the balancing property of each propensity score, we split the balance check into two steps. The first step checks the balancing property of the individual propensity score , through a comparison of models for the individual treatment given the individual propensity score with and without covariates. In the second step, we check the balancing property of the neighborhood propensity score by comparing a model for the neighborhood treatment given the individual treatment and the neighborhood propensity score with and without covariates.

Appendix B Figures and Tables

Figure 6: Map of the World. Countries included in the sample are indicated by the color grey.
Variable Mean St. Dev. Max Min
Outcomes food availability 2,772.66 453.82 3,522 2,046
food utilization 36.08 22.15 79.00 10.50
Covariates real pc GDP 13,883.05 18,126.58 91,617.28 302.13
pc arable land 0.32 0.39 2.81 0.03
population (/100) 680,600.84 1,585,192.15 12,309,806.91 3,174.14
agriculture productivity 113.28 15.97 180.44 83.47
food import/total exports 0.13 0.18 1.94 0.01
net exports 1.66 2.38 24.61 0.00
positive deviation food price 0.01 0.03 0.15 0.00
negative deviation food price 0.05 0.04 0.12 0.00
food price volatility 0.01 0.00 0.02 0.00
food crisis 0.10 0.29 1.00 0.00
Treatment NAC (Z) 1.14 0.26 2.21 0.77
Network Trade value 0.92 1.50 10.27 0.00
Table 2: Summary statistics of outcomes, covariates, treatment and network variables
Variable Description Source
Outcomes food availability food supply in kcal/capita/day FAO - Food Balance Sheets
food utilization prevalence of anemia among children World Bank - World Development Indicators
– percentage of – children under 5
Covariates real pc GDP real per capita GDP (2005 Int. dollar per person) World Bank - World Development Indicators
pc arable land per capita arable land (hectares per person) World Bank - World Development Indicators
population population (in thousands) World Bank - World Development Indicators
agriculture productivity Agricultural Total Factor Productivity (TFP) United States Department of Agriculture
growth index (base year 1961=100) - Economic Research Service
food import/total exports food imports over total exports FAOSTAT
net exports net food exports FAOSTAT
positive deviation food price Deviation of international food prices from trend (%) World Bank - GEM Commodity Price Data
negative deviation food price Deviation of international food prices from trend (%) World Bank - GEM Commodity Price Data
food price volatility international food price volatility FAOSTAT
regional group dummies African Developing Countries (Group 1); World Bank dataset (Anderson and Nelgen, 2012b)
Asian Developing Countries (Group 2);
Latin American Developing Countries (Group 3);
European Transition Economies(Group4);
High-income Countries (Group 5)
food crisis Food crisis dummy (1 if year 2007 and 2008) Authors calculation
NAC NAC= NRA + 1. Nominal Rates of Assistance (NRA): World Bank dataset (Anderson and Nelgen, 2012b)
Value of production-weighted average NRA all (primary)
Agriculture, total for covered and non-covered and
non-product-specific assistance
Network trade value value of agricultural and food bilateral exports FAOSTAT
Table 3: Variables and data sources
Country Mean St. Dev. Max Min Country Mean St. Dev. Max Min
ARG 0.87 0.11 1.00 0.70 KEN 1.01 0.09 1.16 0.69
AUS 1.02 0.02 1.06 1.00 KOR 2.28 0.35 2.78 1.52
AUT 1.42 0.23 1.82 1.07 LKA 1.03 0.11 1.19 0.85
BEL 1.30 0.12 1.54 1.10 LTU 1.16 0.26 1.65 0.55
BEN 0.99 0.02 1.01 0.93 LVA 1.18 0.27 1.73 0.55
BFA 0.98 0.03 1.02 0.90 MAR 1.65 0.15 1.98 1.45
BGD 0.93 0.12 1.06 0.63 MDG 1.00 0.10 1.24 0.90
BGR 0.95 0.14 1.18 0.64 MEX 1.14 0.13 1.41 0.85
BRA 1.01 0.09 1.11 0.80 MLI 0.98 0.02 1.02 0.94
CAN 1.21 0.09 1.43 1.08 MOZ 1.04 0.06 1.24 0.95
CHE 2.77 0.83 4.32 1.48 MYS 0.99 0.05 1.06 0.87
CHL 1.07 0.03 1.10 1.01 NGA 1.00 0.08 1.22 0.87
CHN 1.03 0.13 1.27 0.71 NIC 0.90 0.08 1.05 0.73
CIV 0.76 0.05 0.82 0.68 NLD 1.39 0.17 1.62 1.08
CMR 0.99 0.01 1.01 0.96 NOR 2.68 0.61 3.67 1.63
COL 1.17 0.10 1.34 0.96 NZL 1.02 0.01 1.06 1.00
CZE 1.21 0.12 1.48 1.07 PAK 0.96 0.08 1.12 0.78
DEU 1.39 0.20 1.79 1.07 PHL 1.19 0.14 1.41 0.87
DNK 1.34 0.18 1.70 1.06 POL 1.18 0.14 1.60 0.98
DOM 1.07 0.15 1.43 0.76 PRT 1.27 0.11 1.44 1.08
ECU 0.93 0.14 1.22 0.70 RUS 1.14 0.19 1.42 0.55
EGY 0.97 0.07 1.10 0.84 SDN 0.82 0.29 1.47 0.31
ESP 1.28 0.14 1.56 1.08 SEN 0.98 0.12 1.23 0.83
EST 1.10 0.18 1.41 0.62 SVK 1.24 0.12 1.43 1.07
ETH 0.86 0.19 1.27 0.50 SVN 1.56 0.29 2.06 1.09
FIN 1.58 0.47 2.54 1.07 SWE 1.46 0.30 2.13 1.07
FRA 1.37 0.22 1.88 1.06 TCD 0.99 0.01 1.01 0.96
GBR 1.42 0.21 1.88 1.09 TGO 0.98 0.02 1.00 0.93
GHA 0.98 0.03 1.05 0.92 THA 1.00 0.06 1.14 0.90
HUN 1.20 0.12 1.45 1.07 TUR 1.25 0.11 1.43 1.01
IDN 1.01 0.11 1.20 0.78 TZA 0.85 0.15 1.12 0.50
IND 1.08 0.11 1.26 0.88 UGA 0.96 0.08 1.02 0.76
IRL 1.57 0.26 2.05 1.08 UKR 0.93 0.15 1.13 0.54
ISL 2.76 0.86 4.94 1.62 USA 1.12 0.04 1.18 1.04
ITA 1.29 0.15 1.57 1.07 ZAF 1.05 0.07 1.21 0.93
JPN 2.08 0.25 2.72 1.68 ZMB 0.94 0.00 0.94 0.94
VNM 1.13 0.15 1.32 0.88
Notes: Country names are denoted using ISO3 Code.
Table 4: NAC distribution by country
(1) (2)
Food Availability Food Availability
(w/o interference) (with interference)
z 2.042 (0.813) 0.523 (0.664)
0.970 (0.591) 0.057 (0.486)
0.095 (0.136) 0.087 (0.113)
0.246 (0.078) 0.132 (0.061)
0.046 (0.046) 0.039 (0.036)
0.010 (0.008) 0.006 (0.007)
0.145 (0.019) 0.028 (0.016)
0.107 (0.028)
0.021 (0.006)
0.002 (0.0005)
0.265 (0.772))
1.181 (3.792)
3.095 (5.902)
0.192 (0.047)
0.036 (0.013)
Constant 6.834 (0.342) 7.545 (0.281)
Observations 930 930
R 0.358 0.617
Adjusted R 0.353 0.610
Residual Std. Error 0.134 (df = 922) 0.104 (df = 914)
F Statistic 73.333 (df = 7; 922) 98.015 (df = 15; 914)
  • Notes: Significance levels: * p 0.1; ** p0.05; *** p0.01.

Table 5: Outcome models
(1) (2)
Direct NAC Network NAC
(Eq. 5) (Eq. 6)
real pc GDP 0.047 (0.008) 0.038 (0.011)
pc arable land 0.040 (0.006) 0.043 (0.009)
population 0.019 (0.004) 0.005 (0.006)
agricultural productivity 0.001 (0.0003) 0.002 (0.0005)
food import/total exports 0.016 (0.008) 0.028 (0.011)
net exports 0.015 (0.002) 0.006 (0.003)
positive deviation food price 0.160 (0.141) 0.593 (0.207)
negative deviation food price 0.262 (0.131) 0.383 (0.193)
food volatility 3.034 (1.091) 3.933 (1.604)
food crisis 0.030 (0.017) 0.061 (0.025)
Network real pc GDP 0.002 (0.0005) 0.132 (0.001)
Z 0.198 (0.034)
Constant 0.598 (0.087) 0.188 (0.127)
Observations 930 930
R 0.538 0.985
Adjusted R 0.530 0.985
Residual Std. Error 0.125 (df = 914) 0.183 (df = 913)
F Statistic 70.849 (df = 15; 914) 3,829.031 (df = 16; 913)
Regional dummies Yes Yes
  • Notes: Significance levels: * p 0.1; ** p0.05; *** p0.01. Real pc GDP, pc arable land, population variables and network real pc GDP are in log and one year lagged. Agricultural productivity, food import/total exports, net exports, positive deviation food price and negative deviation food price variables are one year lagged (Source: FAOSTAT, WDI, USDA).

Table 6: Individual and neighborhood propensity scores with neighborhood-level covariates
Food Availability
z 1.338 (0.718)
0.560 (0.525)
0.032 (0.122)
0.134 (0.064)
0.031 (0.038)
0.006 (0.007)
0.123 (0.018)
0.187 (0.023)
0.025 (0.005)
0.002 (0.0004)
0.061 (0.126)
0.051 (0.099)
0.019 (0.025)
0.007 (0.007)
0.068 (0.014)
Constant 7.174 (0.311)
Observations 930
R 0.542
Adjusted R 0.535
Residual Std. Error 0.113 (df = 914)
F Statistic 72.185 (df = 15; 914)
  • Notes: Significance levels: * p 0.1; ** p0.05; *** p0.01.

Table 7: Outcome model with neighborhood-level covariates
Figure 7: Marginal dose-response function of direct NAC (left) and marginal dose-response function of network NAC (right) on food availability (log scale) with neighborhood-level covariates
(1) (2)
Direct NAC Network NAC
(Eq. 5) (Eq. 6)
real pc GDP 0.024 (0.008) 0.537 (0.041)
pc arable land 0.031 (0.007) 0.374 (0.036)
population 0.010 (0.005) 0.097 (0.025)
agricultural productivity 0.0001 (0.0004) 0.009 (0.002)
food import/total exports 0.022 (0.008) 0.100 (0.041)
net exports 0.015 (0.002) 0.087 (0.009)
positive deviation food price 0.063 (0.164) 0.012 (0.792)
negative deviation food price 0.158 (0.152) 0.835 (0.732)
food volatility 3.660 (1.272) 11.673 (6.160)
food crisis 0.029 (0.021) 0.069 (0.099)
Z 0.734 (0.125)
Constant 0.335 (0.104) 4.777 (0.509)
Observations 777 777
R 0.374 0.635
Adjusted R 0.362 0.628
Residual Std. Error 0.130 (df = 762) 0.628 (df = 761)
F Statistic 32.460 (df = 14; 762) 88.249 (df = 15; 761)
  • Notes: Significance levels: * p 0.1; ** p0.05; *** p0.01. Real pc GDP, pc arable land and population variables are in log and one year lagged. Agricultural productivity, food import/total exports, net exports, positive deviation food price and negative deviation food price variables are one year lagged (Source: FAOSTAT, WDI, USDA).

Table 8: Individual and neighborhood propensity scores excluding main exporter and importer countries
Food Availability
z 0.130 (0.858)
0.302 (0.629)
0.137 (0.144)
0.104 (0.077)
0.021 (0.048)
0.005 (0.009)
0.096 (0.021)
0.137 (0.050)
0.040 (0.018)
0.007 (0.003)
0.257 (0.385)
1.539 (1.089)
2.359 (0.966)
0.196 (0.033)
0.104 (0.030)
Constant 7.749 (0.359)
Observations 777
R 0.435
Adjusted R 0.424
Residual Std. Error 0.118 (df = 761)
F Statistic 39.076 (df = 15; 761)
  • Notes: Significance levels: * p 0.1; ** p0.05; *** p0.01.

Table 9: Outcome model excluding main exporter and importer countries
Figure 8: Marginal dose-response function of direct NAC (left) and marginal dose-response function of network NAC (right) on food availability (log scale) excluding main exporter and importer countries
(1) (2) (3) (4)
Food Utilization Food Utilization Food Utilization Food Utilization
(w/o interference) (with interference) (with neighborhood-level (excluding main
covariates) exp imp countries)
z 4.768 (2.975) 1.119 (2.428) 1.285 (2.479) 2.305 (2.620)
2.154 (2.193) 1.858 (1.807) 0.088 (1.844) 1.031 (1.965)
0.107 (0.510) 0.670 (0.424) 0.202 (0.432) 0.070 (0.462)
0.428 (0.285) 0.026 (0.222) 0.111 (0.224) 0.563 (0.255)
0.092 (0.176) 0.263 (0.136) 0.304 (0.138) 0.214 (0.167)
0.015 (0.033) 0.035 (0.026) 0.039 (0.026) 0.033 (0.033)
0.795 (0.074) 0.504 (0.063) 0.671 (0.065) 0.865 (0.075)
0.810 (0.098) 0.957 (0.088) 0.207 (0.154)
0.149 (0.025) 0.135 (0.020) 0.068 (0.048)
0.012 (0.002) 0.010 (0.002) 0.001 (0.007)
4.660 (2.680) 0.125 (0.305) 0.324 (1.439)
37.926 (14.551) 0.063 (0.231) 8.812 (4.342)
62.926 (24.362) 0.030 (0.054) 12.310 (4.173)
0.376 (0.172) 0.010 (0.020) 0.822 (0.123
0.225 (0.046) 0.321 (0.046) 0.442 (0.091)
Constant 6.390 (1.235) 3.813 (1.023) 4.772 (1.044) 4.785 (1.093)
Observations 952 952 952 786
R 0.442 0.673 0.665 0.564
Adjusted R 0.438 0.668 0.650 0.555
Residual Std. Error 0.501 (df = 944) 0.385 (df = 936) 0.395 (df = 936) 0.387 (df = 770)
F Statistic 106.934 (df = 7; 944) 128.644 (df = 15; 936) 118.692 (df = 15; 936) 66.375 (df = 15; 770)
  • Notes: Significance levels: * p 0.1; ** p0.05; *** p0.01.

Table 10: Food utilization outcome models
Figure 9: Dose-response function of direct NAC on food utilization (log scale) w/o interference
Figure 10: Marginal dose-response function of direct NAC (left) and marginal dose-response function of network NAC (right) on food utilization (log scale) with interference
Figure 11: Average dose-response function of direct NAC and network NAC on food utilization (log scale)
Figure 12: Marginal dose-response function of direct NAC (left) and marginal dose-response function of network NAC (right) on food utilization (log scale) with neighborhood-level covariates
Figure 13: Marginal dose-response function of direct NAC (left) and marginal dose-response function of network NAC (right) on food utilization (log scale) excluding main exporter and importer countries