An assessment of the global food security index

Abstract

Several measures of food insecurity, whether at the household or at the national level, have been introduced during the past two or three decades. Some concentrate on the determinants of food security while other emphasize more the consequences of food insecurity. The main focus of this paper is on the food security indicators introduced by the Economist Intelligence Unit (EIU), the Global Food Security Index (GFSI). The paper has two goals. It first checks whether the set of weights selected by the panel of experts of the EIU plays a crucial role in the ranking of countries by level of food security. Then it examines to what extent the ranking of countries given by the GFSI is sensitive to the list of indicators selected. The empirical analysis conducted, based on statistical techniques such as principal components and efficiency analysis, led us to conclude that both the weights selected and the choice of indicators give a reasonable ranking of countries by level of their food security.

Appendices

Appendix 1: Methodology: the various approaches used

1.1 Principal Component Analysis

Principal components analysis (PCA) posits an underlying structure relating the indicator variables to a set of latent factors:

$$ {x}_{1i}={\sum}_{k=1}^K{b}_{1k}{y}_{ki},\dots .,{x}_{hi}={\sum}_{k=1}^K{b}_{hk}{y}_{ki},\dots, {x}_{Ki}={\sum}_{k=1}^K{b}_{Kk}{y}_{ki} $$

(1)

where x_hi refers to the value of variable h in country i; y_ki is the value of principal component k for the i-th country; and the b’s are coefficients. The coefficients are estimated and the system is inverted to derive

$$ {y}_{fi}={\sum}_{k=1}^K{a}_{fk}{x}_{fi} $$

(2)

for each principal component f.

The first principal component accounts for as much of the variability in the data as possible. Each other principal component has the highest variance possible under the constraint that it is orthogonal to the other components. The principal components are in fact the eigenvectors of the covariance matrix of the original variables. The first principal component therefore gives an index providing maximum discrimination between the countries, the variables varying the most across countries being given a larger weight.

1.2 Efficiency Analysis

1.2.1 On Efficiency analysis and the distance function

Efficiency analysis is largely based on the concepts of duality and input distance functions in production theory.

Let x_i = (x_1i, ..., x_ji, ..., x_ki) denote the vector of levels of food security observed in the various k domains of food security for country i and let y_i denote the overall level of food security for country i. A country’s performance, as far as food security is concerned, may hence be represented by the pair (x_i, y_i), i =1, …I where I is the total number of countries.

A food security index FS can then be estimated using a Malmquist input quantity index:

$$ FS\left(y,{x}^s,{x}^t\right)={D}_{input}\left(y,{x}^s\right)/{D}_{input}\left(y,{x}^t\right) $$

(3)

where x^s and x^t are two different "food security inputs" vectors and D_input is an input distance function. The idea behind the Malmquist index is to provide a reference set against which to judge the relative magnitudes of the two vectors of "food security inputs".

To be more precise, let L(y) represent the input set of all input vectors x that can produce the output vector y. That is,

$$ L(y)=\left\{x:x\ \mathrm{can}\ \mathrm{produce}\ y\right\}. $$

(4)

The input distance function D_input (x,y) involves then (in a way similar to that in which this concept is used in consumption theory) the scaling of the input vector and will be defined as

$$ {D}_{input}\ \left(x,y\right)=\operatorname{Max}\ \left\{\rho :\left(x/\rho \right)\in L(y)\right\} $$

(5)

It may be proven that

• the input distance function is increasing in x and decreasing in y;

• it is linearly homogeneous in x;

• if x belongs to the input set of y (i.e. x∈L(y)) then D_in(x,y) ≥ 1;

• the input distance function is equal to unity if x belongs to the “frontier” of the input set (the isoquant of y).

1.2.2 Estimation procedures

Corrected ordinary least squares (COLS)

Let us take as a simple illustration the case of a Cobb-Douglas production function. Let ln y_i be the logarithm of the output of firm i, i =1 to I, and x_i a (N+1) row vector, whose first element is equal to one and the others are the logarithms of the N inputs used by the firm. We may then write that

$$ \mathit{\ln}\ \left({y}_i\right)={x}_i\cdotp \beta -{u}_i\kern0.5em ,\kern0.5em i=1,\dots, I. $$

(6)

One of the methods allowing the estimation of this output-oriented Farrell measure of technical efficiency T_i (see Farrell 1957) is to use an algorithm proposed by Richmond (1974) which has become known as corrected ordinary least squares (COLS). This method starts by using ordinary least squares to derive the (unbiased) estimators of the slope parameters. Here β is a (N+1) column vector of parameters to be estimated and u a non-negative random variable, representing the technical inefficiency in production of firm i.,

The ratio of the observed output of firm i to its potential output will then give a measure of its technical efficiency T_i so that

$$ {T}_i={y}_i/\exp \left({x}_i\cdotp \beta \right)=\exp \left({x}_i\cdotp \beta -{u}_i\right)/\exp \left({x}_i\cdotp \beta \right)=\exp\ \left(-{u}_i\right) $$

(7)

Then in a second stage the (negatively biased) OLS estimator of the intercept parameter β₀ is adjusted up by the value of the greatest negative residual so that the new residuals have all become non-negative. Naturally the mean of the observations does not lie any more on the estimated function: the latter has become in fact an upward bound to the observations.

The stochastic production frontier

One of the main criticisms of the COLS method is that it ignores the possible influence of measurement errors and other sources of noise. All the deviations from the frontier have been assumed to be a consequence of technical inefficiency. Aigner et al. (1977) and Meeusen and van den Broeck (1977) independently suggested an alternative approach called the stochastic production frontier method in which an additional random error v is added to the non-negative random variable u.

$$ \ln\ \left({y}_i\right)={x}_i\cdotp \beta +{v}_i-{u}_i $$

(8)

The random error v is supposed to take into account factors such as the weather, the luck, etc. v is assumed to be i.i.d. normal random variables with mean zero and constant variance σ_v², independent of u, the latter being taken generally to be i.i.d. exponential or half-normal random variables. In the latter case where u is assumed to be i.i.d truncations (at zero) of a normal variable N(0,σ), Battese and Corra (1977) suggested to proceed as follows. Calling σ_s² the sum (σ² + σ_v²), they defined the parameter γ = (σ²/σ_s²) —so that γ has a value between zero and one— and showed that the log-likelihood function could be expressed as

$$ \ln (L)=-\left(N/2\right)\ln \left(\pi /2\right)-\left(N/2\right)\ln \left({\sigma}_s^2\right)+{\sum}_{i=1}^I\left[1-\varPhi \left({z}_i\right)\right]-\left[1/\left(2{\sigma}_s^2\right)\right]{\sum}_{i=1}^I{\left(\ln {y}_i-{x}_i\beta \right)}^2 $$

(9)

where z_i = [(ln y_i - x_i·β)/σ_s]·√(γ/(1-γ)) and Φ(·) is the distribution function of the standard normal random variable.

The Maximum Likelihood estimates of β, σ_s² and γ are obtained by finding the maximum of the log-likelihood function defined previously where this function is estimated for various values of γ between zero and one. More details on this estimation procedure is available in programs such as FRONTIER (Coelli 1992) or LIMDEP (Green 1992). The same methods (COLS and Maximum Likelihood) may naturally be also applied when estimating distance functions.

COLS or the stochastic production frontier have been applied in various areas in Economics. Ramos and Silber (2005) used, for example, this approach to study the dimension of human development, Deutsch and Silber (2005) to measure multi-dimensional poverty and Deutsch et al. (2013) to analyze the performance of pupils taking the PISA (Program for International Student Assessment) test (for more information on the PISA test, see, OECD 2012).

Data envelopment analysis (DEA)

Data Envelopment Analysis (DEA) (see, for example, Coelli et al. 2005) is a linear programming methodology to measure the efficiency of multiple decision-making units (DMUs) when the production process includes several inputs and outputs. It has been used by economists to measure the efficiency of production units. Whereas in traditional statistical analyses the performance of producers is compared to that of the average producer, DEA compares each producer (decision making unit in the language of Charnes et al.’s 1978, path breaking paper) with the best one. The idea is to derive a production frontier but this non-parametric approach does not assume a particular functional form for the frontier. The function is in fact determined by the most efficient producers. The idea behind such a frontier is that one assumes that if a firm, using specific levels of inputs, is able to produce a certain output level, another firm of similar scale should be able to reach similar results. In other words if a given producer P1 manages to produce Y(P1) units of output with X(P1) inputs, other producers should also have such a capacity, if they are efficient. The same is true, say, when one considers a producer P2 which produces Y(P2) units of output with X(P2) inputs, etc…As a consequence these producers P1, P2,…, P(N), assuming N producers, can be theoretically “amalgamated” to obtain a kind of virtual producer using a combination of the inputs of the various producers to produce a combination of the outputs they produce. The goal of DEA then turns out to be a search for the best virtual producer for each actual producer. If this virtual producer produces more output than the actual producer with the amount of inputs the latter uses or uses less inputs to produce the output of the actual producer, the latter is assumed to be inefficient.

Data Envelopment Analysis has been applied in various areas in Economics. Rabar (2017) reviews the application of data envelopment analysis to studies that empirically explore socio-economic efficiency of OECD (Organization for Economic Cooperation and Development) member countries.

The Anderson et al. (2011) approach: A nonparametric empirical method for deriving food security rankings, based also on data envelopment analysis

This approach has the advantage of not requiring the selection of a particular aggregation function or some weighting scheme. It involves the calculation of a lower bound on the distance function measure of food security. Assume that there are m indicators measuring different aspects of food security for each of the observations (countries) in our dataset. Let x_i refer to the i^th observation and let X be the m by n matrix of all these observations. Let now FS denote a food security function that aggregates the variables associated with an observation into a single scalar measure. The following two assumptions about FS are made. The first one is monotonicity, which is defined as FS(x) ≥ FS(y) if x ≥ y. Monotonicity hence implies that food security does not fall with an increase in one of the indicators. The second assumption is that of quasi-concavity which we define as FS(x) = FS(y) ≤ FS(αx + (1 − α)y) with 0 ≤ α ≤ 1. Quasi-concavity therefore implies that for a given distribution of the indicators, food security is (weakly) increasing when there is an inequality reducing reallocation between the observations.

The distance measure function used by Anderson et al. (2011) is then defined as

$$ {\displaystyle \begin{array}{l}D\left(F{S}_{x_j}\right)= Mi{n}_D\Big\{D\left(F{S}_{x_j}\right)\ \mathrm{with}\ D>0,{x}_j\in X, FS\ \mathrm{satisfying}\ \mathrm{monotonicity}\ \mathrm{and}\ \mathrm{quasi}-\\ {}\mathrm{concavity}\Big\}\end{array}} $$

(10)

Let now LCH(X) refer to the lower convex hull of the data. Then, if the reference level is that of the country in the dataset with the lowest level of food security, we may write that

$$ D\left({FS}_{x_j}\right)={\mathit{\operatorname{Min}}}_D\left\{D\left({FS}_{x_j}\right)\ \mathrm{with}\ D>0\ \mathrm{and}\ D\left({FS}_{x_j}\right)\in LCH(X)\right\} $$

(11)

Let also UMH(X)denote the upper monotone hull of the data.We may then write, if the reference level is that of the country in the dataset with highest level of food security, that

$$ D\left({FS}_{x_j}\right)={\mathit{\operatorname{Min}}}_D\left\{D\left({FS}_{x_j}\right)\ \mathrm{with}\ D>0\ \mathrm{and}\ D\left({FS}_{x_j}\right)\in UMH(X)\right\} $$

(12)

These distance measures correspond therefore to the minimum amount by which one need to scale each observation so that they share an equal ranking with the worse-off or best observations. As stressed by Anderson et al. (2008) “they represent lower bounds on these measures for any and all ways of choosing to weigh the various indicators, as long as the weighting formula is monotone and quasi-concave”.

To apply the approach of Anderson et al. (2011) we used a program originally written by T.W. Leo for their paper.

Combining principal components analysis and data envelopment analysis

In the empirical section of the present paper, we also combine principal components analysis and data envelopment analysis. We proceeded as follows.

In a first stage we applied PCA to each of the domains of food security distinguished by The Economist, namely affordability, availability and quality and safety. After deriving the first principal components x_AFFi, x_AViand x_QSi in each of these three domains for each country i, we transformed these components in such a way that the transformed components would always be positive. This transformation is required because in PCA the value of each component will be negative for some of the countries. We therefore defined these transformed components as

$$ {y}_{AFFi}=\frac{\left({x}_{AFFi}-\mathit{\operatorname{Min}}\left\{{x}_{AFFi}\right\}\right)}{\left(\mathit{\operatorname{Max}}\left\{{x}_{AFFi}\right\}-\mathit{\operatorname{Min}}\left\{{x}_{AFFi}\right\}\right)} $$

(13)

$$ {y}_{AVi}=\frac{\left({x}_{AVi}-\mathit{\operatorname{Min}}\left\{{x}_{AVi}\right\}\right)}{\left(\mathit{\operatorname{Max}}\left\{{x}_{AVi}\right\}-\mathit{\operatorname{Min}}\left\{{x}_{AVi}\right\}\right)} $$

(14)

$$ {y}_{QSi}=\frac{\left({x}_{QSi}-\mathit{\operatorname{Min}}\left\{{x}_{QSi}\right\}\right)}{\left(\mathit{\operatorname{Max}}\left\{{x}_{QSi}\right\}-\mathit{\operatorname{Min}}\left\{{x}_{QSi}\right\}\right)} $$

(15)

Then, in a second stage, we apply a methodology originally proposed by Lovell et al. (1994). The idea is as follows.

Let FS_i refer to the level of food security achieved in country i. Then a country’s endowment of inputs and food security achievement may be written as (y_AFFi, y_AVi, y_QSi, FS_i), i = 1 to I , I being the number of countries. The relative level of food security in country i, RFS_i , may then be estimated using a Malmquist input quantity index by writing

$$ {\displaystyle \begin{array}{l} RF{S}_i\left(F{S}^{ref},{y}_{AFF i},{y}_{AV i},{y}_{QS i},{y}_{AFF}^{ref},{y}_{AV}^{ref},{y}_{QS}^{ref}\kern0.50em \right)\\ {}={D}_{inputs}\left(F{S}^{ref},{y}_{AFF i},{y}_{AV i},{y}_{QS i}\right)/{D}_{inputs}\left(F{S}^{ref},{y}_{AFF}^{ref},{y}_{AV}^{ref},{y}_{QS}^{ref}\right)\end{array}} $$

(16)

where ${FS}^{ref},{y}_{AFF}^{ref},{y}_{AV}^{ref}\ \mathrm{and}\ {y}_{QS}^{ref}$ are respectively the level of food security, of affordability, availability and quality and safety in some hypothetic country of reference and D_inputs is an input distance function.

The idea behind the Malmquist index is to provide a reference set against which to judge the relative magnitudes of the two input vectors. This reference set is some isoquant L(FS^ref) and the radially further away from this isoquant the point corresponding to the inputs y_AFFi, y_AVi and y_QSi is, the higher the relative level of food security, because these inputs must be reduced more to reach the isoquant of reference L(FS^ref).

The problem however is that such a Malmquist index is generally a function of this reference isoquant. To solve this issue, Lovell et al. (1994) suggested to proceed as follows (their analysis referred to individuals but we applied their ideas to countries). Treat all countries equally and assume that each country has the same level of food security, namely one unit of food security. Let e refer to such a level of food security so that the isoquant of reference may be written as (e) . This isoquant clearly bounds the input vectors from below because countries whose input vector locates them on L(e) have clearly the lowest possible level of food security and their relative level of food security will be equal to 1. Countries with larger input vectors will then have higher levels of food security and their relative level of food security will be higher than 1.

How can we estimate such a distance function? Lovell et al. (1994) proposed to define a (N − 1) dimensional vector z, N being the number of inputs. In our case N=3 and we define z as z_i = (z_AVi,  z_QSi) with z_AVi = (y_AVi/y_AFFi) and z_QSi = (y_QSi/y_AFFi). We can then define an input distance function D_inputs(e, z_i ) as

$$ {D}_{inputs}\left(e,{z}_i\kern0.5em \right)=\left(1/{y}_{AFFi}\right){D}_{inputs}\left(e,{y}_{AFFi},{y}_{AVi},{y}_{QSi}\kern0.5em \right) $$

(17)

and, since D_inputs(e, y_AFFi, y_AVi, y_QSi ) ≥ 1, we may conclude that (1/y_AFFi) ≤ D_inputs(e, z_i ).

This implies that we may also write that

$$ \left(1/{y}_{AFFi}\right)={D}_{inputs}\left(e,{z}_i\kern0.5em \right)\ \mathit{\exp}\left(\upvarepsilon \right) $$

(18)

with ε ≤ 0.

If we now assume that D_inputs(e, z_i ) has a translog functional form, we may write that

$$ \mathit{\ln}\left(1/{y}_{AFFi}\right)={\upalpha}_0+{\upalpha}_1\mathit{\ln}{z}_{AVi}+{\upalpha}_2\mathit{\ln}{z}_{QSi}+\left(1/2\right){\upalpha}_{12}\mathit{\ln}{z}_{AVi}\mathit{\ln}{z}_{QSi}+{\upvarepsilon}_i $$

(19)

Estimates of the coefficients α₀, α₁, α₂ and α₁₂ may be obtained using corrected ordinary least squares (see, Coelli et al. 1998, for more details on this technique) while the input distance function D_inputs(FS^ref, y_AFFi, y_AVi, y_QSi) for each country i will be obtained via the transformation D_inputs(FS^ref, y_AFFi, y_AVi, y_QSi) =  exp (Max{ε_i} − ε_i).

This distance will, by definition, be greater than or equal to one (since its logarithm will be non-negative) and will hence indicate by how much a country’s inputs must be scaled down in order to reach the isoquant of reference. Such a procedure thus guarantees that all input vectors lie on or above the isoquant of reference L(e). The relative level of food security for country i will then be obtained by dividing D_inputs(e, z_i ) by the minimum observed distance value, which by definition equals 1.

Another possibility is to use (10) to write that

$$ \mathit{\ln}\left(1/{y}_{AFFi}\right)={\upalpha}_0+{\upalpha}_1\mathit{\ln}{z}_{AVi}+{\upalpha}_2\mathit{\ln}{z}_{QSi}+\left(1/2\right){\upalpha}_{12}\mathit{\ln}{z}_{AVi}\mathit{\ln}{z}_{QSi}+{\upvarepsilon}_i $$

(20)

$$ \left(1/{y}_{AFFi}\right)=f\left({z}_{AVi},{z}_{QSi}\right) $$

(21)

that is, to consider the variable (1/y_AFFi) as an output produced with the two inputs z_AVi and z_QSi and estimate the lower bound of the isoquants defining this production function via Data Envelopment Analysis, assuming evidently constant returns to scale.

This is what we did and in Section 4 we present empirical results derived from such an approach, using the program DEAP. This DEAP program, freely available, was written by Tim Coelli and is used to construct DEA frontiers for the calculation of technical and cost efficiencies and also for the calculation of Malmquist TFP (Total Factor Productivity) Indices.

This two stages analysis has been used in the past. In a paper on social exclusion in Macedonia, Deutsch et al. (2012) used in a first stage correspondence analysis to reduce the number of variables and in a second stage the stochastic production frontier approach The same approach has been used by Deutsch et al. (2013) who used the PISA data to estimate an education production function. In another paper on school externalities and scholastic performance, Deutsch et al. (2019) used correspondence analysis in a first stage and data envelopment analysis in the second stage.

Appendix 2: List of variables taken into account by the database of The Economist in the measurement of food security

1.1 AFFORDABILITY

• Food consumption as a share of household expenditure

• Proportion of population under global poverty line

• Gross domestic product per capita (at PPP)

• Agricultural import tariffs

• Presence of food safety net programmers

• Access to financing for farmers

1.2 AVAILABILITY

• Sufficiency of supply

• Average food supply

• Dependency on chronic food aid

• Public expenditure on agricultural R&D

• Agricultural infrastructure

• Existence of adequate crop storage facilities

• Road infrastructure

• Port infrastructure

• Volatility of agricultural production

• Political stability risk

• Corruption

• Urban absorption capacity

• Food loss

1.3 QUALITY AND SAFETY

• Diet diversification

• Nutritional standards

• National dietary guideline

• National nutrition plan or strategy

• Nutrition monitoring and surveillance

• Micronutrient availability

• Dietary availability of vitamin A

• Dietary availability of animal iron

• Dietary availability of vegetal iron

• Protein quality

• Food safety

• Agency to ensure the safety and health of food

• Percentage of population with access to potable water

• Presence of formal grocery sector

Appendix 3: List of variables taken into account by the FAO to measure food security

1.1 Availability

• Average dietary energy supply adequacy

• Average value of food production

• Share of dietary energy supply derived from cereals, roots and tubers

• Average protein supply

• Average supply of protein of animal origin

1.2 Access

• Percent of paved roads over total roads

• Road density

• Rail lines density

• Gross domestic product per capita (in purchasing power equivalent)

• Average dietary energy supply adequacy

• Average value of food production

• Share of dietary energy supply derived from cereals, roots and tubers

• Average protein supply

• Average supply of protein of animal origin

• Domestic food price index

• Prevalence of undernourishment

• Share of food expenditure of the poor

• Depth of the food deficit

• Prevalence of food inadequacy

1.3 Stability

• Cereal import dependency ratio

• Percent of arable land equipped for irrigation

• Value of food imports over total merchandise exports

• Political stability and absence of violence/terrorism

• Domestic food price volatility

• Per capita food production variability

• Per capita food supply variability

1.4 Utilization

• Access to improved water sources

• Access to improved sanitation facilities

• Percentage of children under 5 years of age affected by wasting

• Percentage of children under 5 years of age who are stunted

• Percentage of children under 5 years of age who are underweight

• Percentage of adults who are underweight

• Prevalence of anemia among pregnant women

• Prevalence of anemia among children under 5 years of age

• Prevalence of vitamin A deficiency in the population

• Prevalence of iodine deficiency

Appendix 4

Table 7 Rankings of all the 105 countries for the various approaches