A non-equilibrium formulation of food security resilience
AA non-equilibrium formulation of food-security resilience
Matteo Smerlak and Bapu Vaitla
2, 3, ∗ Perimeter Institute for Theoretical Physics,31 Caroline St. N., Waterloo ON N2L 2Y5, Canada Feinstein International Center, Tufts University,114 Curtis Street, Somerville MA 02144, USA T.H. Chan School of Public Health, Harvard University,677 Huntington Avenue, Boston MA 02115, USA (Dated: November 13, 2018)Resilience, the ability to recover from adverse events (“shocks”), is of fundamental im-portance to food security. This is especially true in poor countries, where basic needs arefrequently threatened by economic, environmental, and health shocks. An empirically soundformalization of the concept of food security resilience, however, is lacking. Here we intro-duce a general framework for quantifying resilience based on a simple definition: a unit isresilient if (a) its long-term food security trend is not deteriorating and (b) the effects ofshocks on this trend do not persist over time. Our approach can be applied to any food se-curity variable for which high-frequency time-series data is available, can accommodate anyunit of analysis (e.g., individuals, households, countries), and is especially useful in rapidlychanging contexts wherein standard equilibrium-based economic models are ineffective. Weillustrate our method with an analysis of per capita kilocalorie availability for 161 countriesbetween 1961 and 2011. We find that resilient countries are not necessarily those that arecharacterized by high levels or less volatile fluctuations of kilocalorie intake. Accordingly,food security policies and programs will need to be tailored not only to welfare levels at anyone time, but also to long-run welfare dynamics. ∗ To whom correspondence should be addressed. E-mail: [email protected] a r X i v : . [ q -f i n . E C ] J un I. INTRODUCTION
Ahost of shocks—political conflicts, economic recessions, natural disasters, and epidemicdiseases—continually threatens food security, especially in the developing world. Resilience, theability to recover quickly from shocks, is thus of major interest to social scientific researchers andpolicymakers worldwide.The formalization of resilience, however, is not straightforward. Studies in economics, ecology,engineering, and psychology offer a diversity of approaches [BC14, FS13, FCW + un desirable equilibrium state.The problem with this approach is twofold. First, economies and social structures, especially inpoor rural areas, experience rapid, unpredictable change, and it is not clear that equilibrium-basedmodels appropriately capture this process. Second, even if “real” food security equilibrium statesexist, they are very difficult to identify.In this paper, we introduce a statistical definition of food security resilience that, when high-frequency time series data are available, provides a non-equilibrium alternative. Our approach isbased on the analysis of autocorrelation: the strength of association between past, present, andfuture states of a dynamical system. Strong autocorrelation is termed “persistence,” a concept com-monly used in fields as diverse as physics [RFR94], climatology [KBBH + + + Category Term Definition
Fundamentalproperties Level k t Value of a food security variableTrend g Slope of a variable; the mean increment E (∆ t ) over the time seriesVolatility σ Quantification of mean fluctuation size in the time seriesPersistence π Association between present and past increments in the time seriesKey derivedmetrics Resilience Anti-persistence of shock effects ( π < g ≥ σ < | g | ) given g ≥ both persistence of the effects of shocks, as measured by a chosen food security variable, and thelong-term trend of that variable.More broadly, we suggest that persistence and trend are two of the four fundamental statisticalproperties relevant to the analysis of food security time-series data. The other two properties, level and volatility , are commonly analyzed in international development research. Levels of agiven food security variable are evaluated with reference to a benchmark value (e.g., the level ofcaloric intake relative to need) or relative to the levels of comparable households, countries, orother actors. Volatility can be meaningfully interpreted by evaluating variance in the food securityseries relative to the slope of the long-term trend. We suggest that low volatility, also in the contextof a non-deteriorating long-term trend, defines an actor as “resistant” to shocks. Resilience andresistance are thus distinct but complementary concepts; the former is derived from trend andpersistence, the latter from trend and volatility (Table I).Fig. 1 illustrates these statistical features using per capita kilocalorie availability data fromNamibia, Malawi, and Peru over the years 1961-2011. Note that the various trajectories in Fig.1 have different implications for policies and programs. For example, livelihood support interven-tions, such as disaster insurance and irrigation schemes, may be needed to build resilience andpromote rapid recovery in Namibia and Malawi. Meanwhile, safety net interventions to strengthenresistance, such as subsidized food sales or food aid, might be appropriate to prevent food insecurityin Peru. II. NON-EQUILIBRIUM RESILIENCE
As noted earlier, the chief advantage of our conceptualization of resilience, compared to otherapproaches, is that it does not necessitate the identification of “equilibrium states” or “basins
Namibia Malawi Peru P e r C ap i t a K c a l A v a il ab ili t y FIG. 1. Resilience and resistance with respect to per capita kilocalorie availability in Namibia, Malawi, andPeru between 1961 and 2011. Namibia exhibits a slightly declining long-term trend and high persistence ofthe effects of shocks; neither of the criteria for resilience is met. In contrast, the trend is neutral to positivefor the Malawi and Peru cases. Malawi exhibits strong persistence throughout the time series. Slopes oftime increments tend to be autocorrelated—rising for a decade, falling for two decades, and then risingagain—and so the country is not considered resilient. The Peru series, unlike the other two countries, isvery anti-persistent. The anti-persistence is seen in the relatively rapid reversals of slope; the effects ofshocks do not last. Because the Peru series is both upward trending and anti-persistent, we label it resilient.All three countries, however, are volatile, and so not resistant. of attraction” of food security variables [BC14]. Dynamic economic models often focus on therelationship of current welfare to a theoretical long-run equilibrium state. In ecology, a “basinof attraction” is a state space within which systems are thought to maintain their fundamentalidentity, and ecological resilience is often defined as the ability of a system to stay within such aspace [Hol73]. Recent works have transferred this notion to microeconomic settings, with resiliencedefined as the capacity to remain, in the face of shocks, in a non-poor basin of attraction [BC14].Ciss´e and Barrett [CB16], for example, measure resilience by evaluating the probability of attaininga well-being threshold over a given time horizon.The challenge with equilibrium-based analysis is to accurately make inferences from observabledata about the structure of welfare dynamics. Barrett and Carter [BC13] list some of the mostimportant issues complicating such inferences: unstable equilibria can be difficult to identify insmall samples, because their very instability makes observations around these points rare; in clustersampling designs, homogeneity within clusters can mask equilibrium points because of regressiontowards the mean; and the structure of the underlying production function mapping capital inputsto outputs, and thus to well-being, may be shifting over time. More generally, estimating non-linear production functions (particularly cubic polynomials, the minimal specification for S-shapedpoverty trap functions) in the presence of stochastic errors requires distributional assumptionswhich may be difficult to test empirically, even with fine-grained panel data.There are also conceptual difficulties with defining resilience with reference to a welfare thresh-old. First, such a metric is intrinsically unstable in the face of shocks: by definition, the probabilityto become poor increases, and thus resilience declines, after major asset losses, diseases, etc. Thisapproach results in a characterization of resilience that includes both intrinsic properties of actorsand exogenous conditions. Another issue with equilibrium approaches is the assumption that allactors follow the same (stochastic) dynamics: two actors with the same welfare level at a giventime, must, on average, experience the same level of welfare in the future. To reconcile this assump-tion with observed differences in welfare trajectories requires the identification and measurementof a large number of control variables. This may be impossible, particularly if the structures ofproduction functions vary across actors based on unobserved attributes (e.g., risk aversion).In summary, movement away from an equilibrium state depends on knowing the values thatdefine that equilibrium. If one cannot know whether the pre-shock and post-shock values of awelfare variable are in equilibrium or are unstable, one cannot judge the degree to which thesubsequent trajectory should be interpreted as resilient. The approach we advance in this paperdoes not solve this problem, but rather avoids it: we link resilience not to a target food securitylevel or target speed of recovery, but rather evaluate it as the ability of an actor to disassociatefrom adverse past events while maintaining an improving or neutral long-term food security trend.This method requires high-frequency time series data; when such data is not available, alternativemethods such as those developed by Ciss´e and Barrett [CB16] are preferable.
III. THE STRUCTURE OF WELFARE TRAJECTORIES
We now formalize the above approach. Let k t be a food security variable, measured at sufficientlyhigh frequency across a set of actors. As a rule, we cannot assume that k t is a stationary timeseries: evolving policy, technology, etc., as well as endogenous effects, are likely to shift k t overtime. To account for these non-equilibrium dynamics, we consider not k t directly, but rather itsincrement ∆ t = k t − k t − , as our dependent variable. Our approach to the analysis of food securitytrajectories is then based on two assumptions: ( i ) that the increments ∆ t are (weakly) stationary,and ( ii ) that they can be modeled by an autoregressive moving average (ARMA) ( p, q ) process ofthe form ∆ t = g + p (cid:88) i =1 β i (∆ t − i − g ) + q (cid:88) j =1 θ j (cid:15) t − j + (cid:15) t , (1)where (cid:15) t are independent, identically distributed shocks with standard deviation σ . (1) modelsan actor’s food security change at time t as a linear function of its past changes and of randomexogenous influences. In this model, the β and θ parameters measure, respectively, the marginaleffect of average past increments and the marginal effect of average past shocks on the currentincrement. The larger these parameters, the more autocorrelated the food security trajectory.When fitting the data to estimate the parameters ( g, β, θ, σ ) of this model, parsimonious orders( p, q ) for the ARMA process can be identified using a standard Bayesian criterion such as theAkaike information criterion (AIC). In particular, low orders will be favored when the number ofdata points is small. Within the framework of (1), food security trajectories can be analyzed in terms of four funda-mental dynamical properties:1. The level of the food security variable k t itself. We choose to measure level in the subsequentanalysis by the mean of k t over the entire time series for each actor.2. The trend g of the variable, i.e., the mean increment E (∆ t ). We consider an actor with apositive (negative) trend as improving ( deteriorating ); when the trend is not significantlydifferent from zero, we say that the time series is neutral .3. The volatility of the increment time series ∆ t . We distinguish between the absolute volatility,the standard deviation σ and the relative volatility ρ = σ | g | − . (2)The ρ metric compares the value of σ to the long-term trend g . We consider that a timeseries is not volatile in relative terms when shocks are too small to invert the trend, i.e., when ρ <
0. This condition requires time series with faster growth trends to face proportionallylarger shocks in order to be considered volatile. Note that volatility contains inseparableinformation on both intrinsic shock magnitude and actor response to shocks. A rule of thumb requires a minimum of 50 points for ARMA modeling of empirical data to be meaningful [Cha16],though 100 points would be recommended by many analysts.
4. The persistence of shocks as captured by the increments ∆ t , relative to their average value g .As noted earlier, persistence quantifies the extent to which present increments are partiallydetermined by past increments, i.e., the extent to which the increments retain a memoryof their lagged values. Complete information about such serial associations is provided bythe autocorrelation function γ ( s ), with γ ( s ) the correlation coefficient between the laggedincrement ∆ t − s and the present increment ∆ t . In many cases, however, a more compactmeasure of persistence is desirable. Within the ARMA framework of (1), β i represents thepartial correlation coefficient between the present excess increment (∆ t − g ) and its laggedvalue (∆ t − i − g ), while θ j is the partial correlation coefficient between (∆ t − g ) and pastshocks (cid:15) t − j . This suggests the following simple definition for overall persistence: π = p (cid:88) i =1 β i + q (cid:88) j =1 θ j . (3)We call an actor’s response to shocks anti-persistent if π < random if π = 0, and persistent if π >
0. As noted before, the condition that π < t are quickly reversed—shocksdo not last. A time series exhibiting this property is more predictable in the long run thanone for which shocks have persistent effects ( π > a priori and taken into account whenfitting the welfare variation trajectory with an ARMA model. In particular, persistence can beoverestimated when such breaks are present in the data [Per06]; an actor with a trend that isbest approximated by a spline or other nonlinear function will be misinterpreted by the ARMAmodel in (1) as one with a linear trend but large, long-lasting deviations away from this trend.Second, in most applications the order ( p, q ) will be small ( p + q ≤ p = q = 0, the food security trajectory cannot be distinguished from a randomwalk with drift (a “stochastic trend” or “unit root”); when ( p, q ) = (0 ,
1) and θ = −
1, thetrajectory consists of a linear trend with independent shocks (a “deterministic trend”). Third,the constant 1 / ρ is to some extent arbitrary: its value is contingent upon Though among the simplest definitions available, the particular definition of persistence given in (3) is not the onlyone possible. Alternatives in the literature include the power spectrum at zero frequency, variance ratios [Coc88],mean reversion [DM10], half-life, approximate entropy [Pin91], and, for series with long-range autocorrelation,Hurst exponents and detrended fluctuation analysis [PBH + a quantification of Prob( g + (cid:15) t < − g ) (cid:28)
1. Here we use the common “one sigma” definition.Fourth, while ARMA modeling provides a solid, well-understood framework for understandingserial correlations, more general formulations (e.g., autoregressive fractionally integrated movingaverage (ARFIMA) models or autoregressive conditionally heteroskedastic (ARCH) models, non-constant trends) can be used if required, provided the number of data points is large enough to allowfor such refinements. Fifth, trend, volatility, and persistence can also be defined non-parametrically,e.g., from the empirical mean, standard deviation, and autocorrelation coefficients of the incrementseries ∆ t . These alternative definitions can be used as consistency checks. Finally, note thatshocks themselves do not need to be observed in order to quantify resilience; time series data onfood security outcomes is sufficient. Given the difficulty of measuring the inherent magnitude ofshocks, this is an important feature of our approach.With these observations in mind, we propose the following formal definition of food securityresilience: an actor is resilient if the long-term food security trend is not deteriorating ( g ≥ ) andfood security increments are negatively correlated in time, i.e., exhibit anti-persistence ( π <
0; seeFig. 9). Secondarily, we can think of an actor as resistant to shocks if the long-term food securitytrend is not deteriorating ( g ≥
0) and the fluctuations in increments do not indicate volatility( ρ <
0; see Fig. 10.) These definitions mathematically express our suggestion that a desirable foodsecurity trajectory is one which can resist and recover from shocks along a generally non-negativetrend.
IV. AN ANALYSIS OF COUNTRY KILOCALORIE AVAILABILITY
We now illustrate these concepts with a real-world dataset: annual per capita kilocalorie (kcal)availability between 1961 and 2011 for 161 countries in the world, taken from the FAOSTATdatabase [Foo15]. Note that our approach can accommodate any quantitative variable at any scale.We choose national-level kcal availability because this indicator is one of the most common proxiesfor food security. However, other welfare variables for which high-frequency data is available,including those at household level, could also be used.The kcal dataset is constructed using food balance sheets that add domestic agricultural/livestockproduction to imports, and then subtract exports, livestock feed, seed, and losses during storage Note that the shocks (cid:15) t need not be normally distributed in general. Rather, the skewness S and kurtosis K of thefit residuals should be thought of as providing additional information about the underlying welfare dynamics. Anactor with positive residual skewness (such as Montenegro, S = 2 .
75) and another with negative residual skewness(such as Croatia, S = − .
02) for instance, have qualitatively different trajectories. The former undergoes morefrequent negative shocks; when positive shocks to growth occur, however, they tend to have a larger magnitude.Analyzing departures from normality can thus be part of a finer-grained analysis of welfare trajectories. and transport. The annual net totals are converted to kilocalories and then divided by the coun-try’s population in each year to obtain per capita kcal availability. Note that this figure doesnot capture the often pronounced distributional inequalities within countries. In addition, kcalconsumption is only one aspect of food security, which also encompasses nutrient intake, foodsafety, cultural preferences, and other dimensions [VCM15]. At best, per capita kcal availabilityfigures can be thought of a coarse upper bound estimate of a single aspect of country-level foodsecurity.We first observe that kcal availability k t is far from being stationary (Fig. 5). Most countriesincreased their kcal availability level in the last five decades, although fifteen countries had lowerlevels in 2011 than they did in 1961. These declines occurred despite robust growth in the globalaverage over this period; the entire series has a strong upward trend, broken only temporarily bya five-year interval corresponding to the breakup of the Soviet bloc in the late 1980s and early1990s. An important feature of the global distribution of kcal availability between 1961 and 2011is convergence: while the 1961 kcal distribution was bimodal, the 2011 distribution is unimodal(Fig. 6). However, the distributions of k t remain significantly positively skewed and platykurticat all times, indicating that global inequalities in food availability remain high. Using the Dickey-Fuller F test, a unit root can be excluded at the 1% level for only six countries (Belgium, Ecuador,India, Panama, Sweden, and Switzerland), indicating that k t series are usually not consistent withtrend-stationary processes. This is in itself a notable finding: a shock to kcal growth usually haslong-lasting impact on the levels of k t —that is, k t does not merely revert to its original trajectory.No significant multi-year cycles were observed.Next we performed the regression (1) with AIC-selected orders ( p, q ) after checking that theincrement time series ∆ t in (1) are stationary (in the Dickey-Fuller sense). Figs 2-3 show theperformance of all 161 countries in our dataset between 1961-2011 for the volatility parameter ρ and the persistence parameter π ; similar maps for mean kilocalorie levels and long-term kilocalorietrends are given in the SI (Fig. 7-8). Fig. 8 shows marked geographical differences in the meansize of annual increments. With respect to developing world regions, the performances of WestAfrica, the Middle East, and Central America (as well as Brazil in South America) are notable,in addition to the well-known success stories of East and Southeast Asia. We see relatively worseperformance in much of Central and Southern Africa, as well as South Asia.The patterns of volatility and persistence are more unexpected. By the ρ criterion—having anaverage fluctuation size that does not threaten inversion of the long-term trend—only four countriesin the world are not volatile: Egypt, China, Algeria, and Brazil (2). This structural strength is0 Volatility relative to growth ( σ / 2|g|)Extremely volatile ( ∞ , 6]Very volatile (6, 3]Moderately volatile (3, 2]Mildly volatile (2, 1]Not volatile (1, - ∞ )Data insufficient to calculate series FIG. 2. Volatility of per capita daily kilocalorie availability, based on annual data 1961-2011. Trendinversions g (cid:55)→ − g are unlikely if the mean fluctuation σ is less than twice the size of the long-term growthtrend ( σ < | g | ). All four countries labeled in green—Egypt, China, Algeria, and Brazil—also have improvinglong-term trends, and so are considered resistant. mostly due to steeply positive long-term trends: Egypt, China, and Algeria have the highest g inthe sample, with an increment of around 30 kcal/year, and Brazil is not far behind at 22 kcal/year.These four countries are thus also resistant. However, most countries in the world, especially thosewith very flat trends (e.g., Russia, Argentina, countries in Eastern Europe) and/or high-magnitudefluctuations (Central and Southern Africa), frequently experience serious shocks.In the presence of strong persistence of the effects of shocks, such high volatility can disruptthe long-term growth trajectory. We see, however, a slightly more positive global picture whenlooking at the persistence parameter π : only a handful of countries hold on to past trends. Thetrajectory of most of the world’s countries, in fact, is best characterized as a random walk; theirkilocalorie availability is neither dragged down by nor recovers from shocks. Note that this not anunambiguously positive characteristic—the experience of shocks in these countries delays return topre-shock levels and trends, relative to anti-persistent countries.No country in the entire sample is both resilient and resistant to shocks. However, 36 countries—all of those labeled in green in Fig. 3, with the exception of Chad and Madagascar—are resilient.Within this set are many of the least developed countries in the world, including (in order ofdecreasing g ) Benin, Lesotho, Mozambique, Bangladesh, and Liberia. Given that many other1 Persistence parameter π Very shock persistent (0.5, 1]Shock persistent (0, 0.5]Random walk [0]Shock anti-persistent [0.5, 0)Very shock anti-persistent [-1, -0.5)Data insufficient to calculate series
FIG. 3. Persistence of per capita daily kilocalorie availability, based on annual data 1961-2011. π > π = 0 (gray) suggests a random walk; π < Allcountries colored green except Chad and Madagascar, both of which have a declining long-term kilocalorie trend, are resilient. countries at similar levels of development do not bounce back well from shocks, investigating thedeterminants of resilience is an important direction for future research.
V. IMPLICATIONS FOR POLICY AND RESEARCH
Three key policy and research implications emerge from this analysis. First, while previousworks have examined the dimensionality of food security in terms of what different variables mea-sure [Bar10, MVC14, VCM15], we suggest that multiple interpretations of a single dynamic variableare possible. Assessing food security change over time requires (at least) an investigation of 1) levelsattained, 2) the trend of growth or decline, 3) the volatility of outcomes, and 4) persistence of theeffects of shocks. From subsets of these, the qualities of resilience and resistance can be identified.Surprisingly, the four properties above are independent of each other, which points to the need tothink of them as complements, not substitutes, in the diagnosis of food security trajectories (Fig.4). This latter point is seen by comparing the ten best and worst performing countries in the kcalsample, with respect to each property (Table II).Second, more research on the determinants of variation in persistence and volatility, and by2 ����� � - ���� * ����� ������ - ���� ���������� σ - ����� ����� - ����� ����������� π FIG. 4. Independence of the four fundamental properties of food security trajectories for the countrykilocalorie availability dataset. Corresponding insets are Spearman correlation coefficients. Except for aweak correlation between level and trend, all variables are uncorrelated at p¡0.1. extension resilience and resistance, is critical. We fit a simple linear model to test the hypothesisthat trade openness is positively correlated to kilocalorie volatility and negatively correlated topersistence. Using mean per capita income, literacy rate, and democratization as controls, wefind that trade openness is not significantly associated with either volatility or persistence. GDPper capita is negatively associated with persistence, but the magnitude of the effect is small.These same variables, however, do better in predicting mean kilocalorie levels (especially) andtrends (Table III in SI), again suggesting that distinct forces are driving the various properties offood security trajectories. More detailed work, including on the sub-country level, is needed; theapproach outlined in this paper can be applied at any scale, including at the household level, wheremuch of the key research on the determinants of food security is done.Third, we note that a set of 18 countries are both volatile and exhibit persistence with respectto the effects of shocks. A subset of these—Angola, Cambodia, El Salvador, Iraq, Malawi, Mexico,Namibia, and Sierra Leone—have large segments of the population with low kilocalorie intake.Livelihoods and health are likely to be severely impacted by shocks in these areas, and recovery islikely to be protracted. From a resilience perspective, these are priority countries for international We also fit a larger model including conflict deaths, oil revenue, and transport infrastructure, but none of these vari-ables improved model performance, as evaluated either by overall goodness-of-fit or the magnitude and significanceof individual parameters. In contrast, a country that is structurally anti-persistent will tend to revert back to the trend more predictably. Similarly, the more volatile thefluctuations, the more difficult it will be to predict the sign of the future trend. Data limitationsmay complicate country-level analysis of food security, but household- and individual-level timeseries can help illuminate these issues.
ACKNOWLEDGMENTS
Research at the Perimeter Institute is supported in part by the Government of Canada throughIndustry Canada and by the Province of Ontario through the Ministry of Research and Innova-tion. Thanks to Chris Golden, Robert Gustafson, E. Toby Kiers, Janet Kim, Erwin Knippenberg,William Masters, Dan Maxwell, Mark Constas, Beth Pringle, Ben Rice, Elizabeth Stites, PatrickWebb, and Ahmed Youssef for valuable feedback on the manuscript. High persistence may be desirable in the short-term during periods of rapid positive growth. However, the detrend-ing procedure and construction of the ARMA model requires that persistence during periods of positive growth ismirrored by persistence during periods of decline over the extent of the time series examined. The implications ofthis trade-off for very long-term food security (i.e., longer than the time series in question) is unclear. Overall, highpersistence generally indicates a structural weakness in the ability to be free from the impact of past fluctuations. [Bar10] CB Barrett. Measuring food insecurity. Science , 327(5967):825–828, February 2010.[BC13] CB Barrett and MR Carter. The economics of poverty traps and persistent poverty: empirical andpolicy implications.
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Appendix A: Per capita kilocalorie database
See file kcal.csv for per capita kilocalorie availability database. Values for Belgium 1961-1999and Luxembourg 1961-1999 are taken from combined Belgium-Luxembourg data. Values for CzechRepublic and Slovakia 1961-1990 are taken from combined Czechoslovakia data. Values for Serbiaand Montenegro 1991-2005 are taken from combined Serbia-Montenegro data. Values for Bosnia-Herzegovina, Croatia, Macedonia, Montenegro, Serbia, and Slovenia 1961-1990 are taken fromYugoslav SFR data. Values for Belarus, Estonia, Latvia, Lithuania, Russia, and the Ukraine1961-1990 are taken from USSR data.
Appendix B: Statistical properties database
See file results.csv for statistical properties of country time series, including candidate ARMAmodels and the corresponding AIC scores.7
Appendix C: Global distribution of kcal availability C r o ss - s e c t i ona l a v e r age Global kcal availability C r o ss - s e c t i ona l S D C r o ss - s e c t i ona l a v e r age Global kcal availability C r o ss - s e c t i ona l S D FIG. 5. Left: Evolution of the global average (blue) and standard deviation (red) in country-level kcalavailability for 1961 - 2011, indicating both growth and convergence during this period (with an exceptionaround 1990). Right: Evolution of the sample skewness (blue) and sample kurtosis (red) for the samedataset, indicating non-normality of the global distribution of kcal availability. nu m be r o f c oun t r i e s Distribution of kcal availability ( ) nu m be r o f c oun t r i e s Distribution of kcal availability ( ) FIG. 6. Convergence in food security: the distribution of kcal availability was bimodal in 1961 but becameunimodal by 2011. This is consistent with a decrease in the cross-sectional standard deviation, see Fig. 5above. Appendix D: kcal availability levels and trends
Mean annual level k t FIG. 7. Levels of per capita daily kilocalorie availability. Mean of all annual values for each country between1961-2011 is shown.
Mean annual increment g [-18,0](0,5](5,10](10,15](15,34]Data insufficient to calculate series FIG. 8. Trends of per capita daily kilocalorie availability. Mean of all annual increments in each countrybetween 1961-2011 is shown. Appendix E: Countries rankingDecreasing Level Decreasing Trend Increasing Rel. Volatility Increasing Persistence p -value for autocorrelation of ∆ t according to the Ljung-Box test(***Significant at p < .
01; **significant at p < .
05; *significant at p < . Appendix F: Cross-country regressions
We fit the simple linear model ρ, π = a + b ∗ ( trade ) + m (cid:88) k =1 c i X i + (cid:15) i (F1)to test the hypothesis that trade openness ( trade ) is positively correlated to volatility relative totrend ( ρ ) and negatively correlated to persistence ( π ), given a vector of control variables X ....X m representing economic resources, human capital stocks, and political participation (see Table III).We also include the same determinants in models predicting mean kcal level and growth rate g .In all models, errors are independent and identically distributed with mean zero and standarddeviation σ ; error variances are heteroskedastic, and Huber-White standard errors are used inestimation. We also specify the same models using the subset of developing countries (as classifiedby the United Nations in 2012) only (Table IV); results are similar. kcal (level) g (trend) ρ (volatility relative to trend) π (persistence)trade -0.382 (0.908) -0.046** (.018) -0.083 (0.068) -0.000 (0.001)gdppc 0.010*** (0.002) -0.000 (.000) 0.000 (0.000) -0.000* (0.000)literacy 1006.058*** (185.671) 4.862 (6.395) -64.301 (59.835) 0.145 (0.112)polity 16.185** (6.206) -0.509*** (0.161) -0.248 (0.617) -0.009* (0.004)constant 1596.485 (128.69) 9.360 (5.204) 37.440 (36.119) -0.124 (0.088) r TABLE III. Results of all country models predicting level, trend, volatility relative to trend, and persistenceof per capita daily kcal availability. Standard errors in parentheses. ***Significant at p¡0.01; **significantat p¡0.05; *significant at p¡0.1. Sources: trade openness (sum of exports and imports of goods and servicesmeasured as a share of gross domestic product) [Wor16]; GDP per capita, literacy [FIT15]; polity10 (degreeof democracy and autocracy) [MGJ13]. kcal (level) g (trend) ρ (relative volatility) π (persistence)trade -0.072 (1.151) -0.039** (.022) 0.039 (0.089) -0.000 (0.001)gdppc 0.017*** (0.003) -0.000 (.000) -0.000 (0.000) -0.000 (0.000)literacy 700.963*** (177.437) 8.193 (6.852) -62.530 (59.597) 0.071 (0.119)polity 15.620** (7.379) -0.500*** (0.183) -0.881(1.093) -0.010** (0.004)constant 1785.826 (124.801) 7.158 (5.442) 30.299 (33.334) -0.105 (0.089) r TABLE IV. Results of developing country only models predicting level, trend, relative volatility, and per-sistence of per capita daily kcal availability. Standard errors in parentheses. ***Significant at p < . p < .
05; *significant at p < .
1. Same sources as Table III. Appendix G: Illustration of resilience and resistance with synthetic data
Below we illustrate our concepts of resilience and resistance using synthetic data (300 realiza-tions) generated with different ARIMA(1,1,0) model with identical trends but varying autoregres-sive coefficient β (Fig. 9) or absolute volatility σ (Fig. 10). - - - l e v e l k t anti - persistent ( π = - / ) - - - random walk ( π = ) - - - persistent ( π = / ) - - - - v a r i a t i on Δ t - - - - - - - - FIG. 9. Resilient and non-resilient trajectories, constructed with synthetic data. The actor in the leftcolumn, with π < π >
0, exhibits persistent, and thus non-resilient, behavior. The actor in the middle column correspondsto the marginal case of random walk with drift. - - l e v e l k t not vulnerable ( ρ = / ) - - l e v e l k t marginally vulnerable ( ρ = ) - - l e v e l k t vulnerable ( ρ = - / ) - - - v a r i a t i on Δ t - - - v a r i a t i on Δ t - - - v a r i a t i on Δ t FIG. 10. Resistant vs. volatile trajectories. The actor in the left column, with ρ ≥ g , while the orange line gives theopposite trend − gg