WPS8117
Policy Research Working Paper 8117
On the Predictability of Growth
Matthieu Cristelli
Andrea Tacchella
Masud Cader
Kirstin Roster
Luciano Pietronero
IFC Country Economics & Engagement
June 2017
Policy Research Working Paper 8117
Abstract
A country’s productive structure and competitiveness are a selection mechanism (the Selective Predictability Scheme),
harbingers of growth. Growth is a dynamic process based defining future growth trajectories for similar countries, and
on capabilities that are difficult to define and measure across compares projected long-term, five-year forecasts with tra-
countries. This paper uses a global measure of fitness (or ditional methods used by the International Monetary Fund.
complexity-weighted diversity of production) as a method The analysis finds that production structure is a good long-
to explore a country’s relative growth potential. The analysis term predictor of growth, with prediction performance
finds that there are two types of growth, predictable or lam- falling off for countries not yet in the laminar classification.
inar, and unpredictable. This classification is used to create
This paper is a product of the IFC Country Economics & Engagement. It is part of a larger effort by the World Bank to
provide open access to its research and make a contribution to development policy discussions around the world. Policy
Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at
mcader@ifc.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
On the Predictability of Growth
How Industrial Structure Predicts Long-Term Growth
Matthieu Cristelli, Andrea Tacchella, Masud Cader, Kirstin Roster,
Luciano Pietronero
JEL classification: O11, O14, O47, O50, O57, E17, F14, F43, F47, C14, C40
Keywords: growth projections, long-term forecast, growth predictability, growth potential,
economic complexity, productive structure, country competitiveness, capabilities, diversification.
1. Introduction
1.1 How to select the past to model future growth: A long-standing
question and a concrete answer
Growth is the aggregate result of uncountable interactions among economic actors
occurring at different temporal and geographical scales and its modeling still represents
one of the main challenges of economics. Modeling the conditions leading countries to
increase competitiveness and enabling country growth,1 in fact, has a pivotal role. On one
hand the understanding of the growth process allows to provide the suitable tools to
design the most effective and inclusive economic policies with the ultimate goal to foster
broad economic growth. On the other hand, economic practice is rooted in the concept
of expectation and consistent modeling is essential to provide consistent forecasts.
In this work we aim to show that countries’ productive structures, and more specifically
cross-country differences in productive structure, are a good mid-long term predictor of
economic performance. Specifically, on the basis of a recently proposed complexity
weighted production diversity measure - fitness [Tacchella, 2012] - we can define the
fitness-GDP per capita plane which allows to select in the past the closest comparators
for the country we aim to forecast.
The definition of this new space carries two pivotal conceptual points. Our forecasting
scheme is, conceptually speaking, rooted in what is called, with the jargon of dynamical
systems, the method of the analogues [Lorenz, 1969]. Let us suppose we want to
forecast the evolution trajectory of an event of this system a number of periods ahead and
we do not know the rules of evolution of the system (i.e., the underlying equations).
Provided some degree of stationarity of the system2 and provided a way to measure the
distance between events in their space of evolution, we can devise, in principle, an
alternative approach to obtain a data-driven assessment of this projection: use the past to
model the future. We can look for the closest events to the event we observed in the past
and use their evolution to model the projection we want to assess. However, this
approach is often unfeasible because it faces the so-called curse of dimensionality
[Bellman, 1957; Cecconi et al., 2012]: either the system has an extremely low number of
dimensions (i.e., 2-3) or we need an exponentially growing past statistics to be
successful. The latter condition is usually hardly achievable, especially for economic
data.
We need therefore an extra step to model the economic future with the economic past in
a scientific and successful way because the development of countries is the result of
uncountable interactions and factors at different levels. Is there an economic level which
comprehensively and inherently is the result of all these factors? The fitness (and in
general the branch of economics named economic complexity) [Hausmann et al. 2007;
Hidalgo et al., 2009; Tacchella et al., 2012] aims to be a framework to provide a positive
1
In this work we refer to growth on the long run. Throughout the text we will refer to it simply as
growth.
2
The stationarity is a subtle point which is hardly assessable. However, we can pragmatically deal
with this point the other way around. Until this strategy works, we can argue that, to some extent,
stationarity is at work and tracking the forecasting power of the strategy in time allows to have
insights into this feature.
1
and concrete answer to this challenge. The level encoding all the factors and their
interaction is identified as the economic output of a country evaluated on a competitive
basis. In details the cross-country differences of the productive capacities represent the
arena to define an aggregate measure of countries’ competitiveness. In the literature,
several attempts have been proposed to decode this competitiveness, see for instance
[Lall, 2001, Hausmann, 2007, Hidalgo, 2009, Tacchella, 2012, Cristelli, 2013] and there
are examples of attempts to build an effective low dimensional space in which to embed
and predict countries’ dynamics, using a competitiveness dimension together with a
measure of GDP [McArthur, 2001]. The dynamical part has been treated with linear
regression models in order to predict GDP growth [Hidalgo, 2009; Podobnik, 2012]. Such
approaches have been shown to be flawed because of the insufficient quality of the
competitiveness measures used [Lall, 2001; Cristelli 2013]. Here we show that a much
more effective low dimensional space can be built by using fitness together with per
capita GDP.3 The validity of such choice can be appreciated a-posteriori, by observing
an emerging predictable dynamical structure4 (we refer to the next sections for further
details on this aspect).
To clarify even more, we can interpret the practical implementation of our scheme, named
the Selective Predictability Scheme (SPS hereinafter) with the jargon of the kernel
regression [Nadaraya, 1964; Watson, 1964]. We want to stress that this identification is
only formal and it is made to explain the practical SPS specifications. The kernel
regression identification alone would not allow why SPS is a scientific framework to model
the future with the past to be appreciated.
SPS works as a two-dimension kernel regression, where the two dimensions are the
fitness and the GDP per capita. The kernel i) selects those past events which are in the
neighborhood of the event we want to forecast, i.e., the set of the closest past we
previously mentioned and ii) averages the k-period ahead displacements of those
selected events. This average displacement provides the projection for the evolution
trajectory of both the country competitiveness (the fitness) and the country GDP per
capita (the growth).
The key point and the novelty of this approach is that, in order to build a reliable predictor
from past events, we need to properly evaluate the neighbors of the future development
we want to forecast. The wealth dimension alone is not enough to filter out these
neighbors as two countries may have achieved the same level of GDP per capita for
extremely heterogeneous factors but simultaneously we cannot directly use all these
discriminating factors because this task is made unfeasible by the curse of
dimensionality. The fitness wants to be a synthetic measure to include in a feasible way
those factors and their (complex) interaction by assessing their emerging complexity from
the differences of the productive structure across countries.
The forecast scheme we devise is extremely parsimonious in terms of model complexity
but, still, it performs comparably (and in some regimes outperforms) with the IMF’s 5-year
projections. In addition, the scheme allows us to define an empirical measure for the
predictability of a country’s economic growth depending on the stage of its development,
which allows to group countries into predictable and unpredictable types, with such
groupings evolving in time. The groupings derived from our scheme can be used to
separate IMF projections as well: we find that the prediction performances are higher for
3
We will use GDP per capita PPP in current USD.
4
We instead refer to [Tacchella et al., 2012] for a technical discussion on why GDPpc and not
GDP is the proper counterpart for the Fitness dimension.
2
predictable type of events also in this case. This underpins that the predictability
dimension quantifies a fundamental feature of a country’s economic status, and is not a
concept arising only in our forecasting scheme.
It is also worth to stress that in this work we are dealing with a forecasting problem, i.e.
the country productive structure is a predictor for long-term development, and we are not
addressing whether countries’ productive structure causes growth or vice versa (i.e.
causation problem). This is a crucial difference to settle in the perspective of the
identification problem [Fisher, 1966] which only applies to the latter class of problems.
We refer to [Kleinberg, 2015] for a general discussion on this aspect and on the
importance of prediction issues in the policy perspective.
The effort to reconcile these approaches with standard economic theory to provide them
an economic foundation is beyond the scope of this work. The present paper has the sole
goal to discuss how economic modeling and economic practice can benefit from such
types of approaches and how they can enhance and complement long-term schemes for
growth forecasting by rooting in a data-driven framework the modeling of the selection of
growth comparatives.
1.2 Paper organization
The paper is organized as follows:
1. In the remainder of this introductive section, we discuss how the economic complexity
framework relates to growth forecasting and modeling, we deepen the discussion
concerning its conceptual grounding and the economic implications and we
summarize the main result of the paper.
2. In section 2, we detail the general concepts backing the Fitness dimension as a proxy
for the country competitiveness. We also briefly discuss the data sets used in this
work.
3. In the section 3, we present the main SPS results, namely:
i. We compare SPS performances with WEO-IMF projections and three limited
intelligence univariate models;
ii. We discuss the meaning of the economic dimension Predictability that we
define within SPS.
4. In Section 4, we present two case studies, Thailand and Ghana, in order to concretely
illustrate how the Predictability dimension reflects a higher/lower degree of coherence of
the comparator events’ trajectories – the past events we select to be the closest to the
event we want to forecast.
5. In Section 5, we provide a detailed technical description of the mathematical
specifications of SPS.
6. Section 6 is devoted to a general discussion of how SPS can provide and enhance
present economic practice.
1.3 How economic complexity relates to growth modeling
Starting from Adam Smith, economics has witnessed several attempts to tackle economic
growth modeling which fall into two major groups: empirical data-driven approaches and
theoretical ones. The former are heterogeneous in terms of methods but they all share a
common guideline: they try to model growth from the knowledge of the past. Turning
towards theoretical approaches, limiting our attention to the last decades and proceeding
in a non-exhaustive way, the forefather of modern theoretical approaches to model
growth is represented by the Solow model [Solow, 1956] where the aggregate output of
3
an economy is modeled in terms of capital and labor to model productivity. Successive
attempts have tried to increasingly describe economic dynamics as an endogenous
process. The original Solow model is indeed a purely exogenous model. As examples of
this tendency, in [Romer, 1986] Romer proposes to endogenize innovation while in
[Aghion, 1990] Aghion and Howitt propose a model to endogenize the Schumpeterian
concept of creative destruction.
Growth and in general aggregate economic output are, as mentioned, the results of
complex and heterogeneous interactions occurring at different scales. However, the
economic mainstream tends to neglect the possible effects of these complex interactions.
For instance, variety of productions, of inputs, of technology etc. are usually reconciled
within de facto representative product, input, technology, firm, etc. models. In such a way
modeling underestimates the role of behaviors that can emerge only at aggregate scales
(meso and macro) but cannot be directly related to features of the representative
actors/dimensions. The last decade has witnessed the build-up of the awareness that the
heterogeneity and complexity of these interactions must be included in the description of
economic aggregates features in order to properly deal with their dynamics.
Coherently with this new vision, a number of works have underpinned that countries’
productive structure [Hausmann, 2007] and competitiveness are harbingers of growth.
Country productive structure, defined on a competitive basis, is proposed as a proxy that
carries information about the capabilities owned by a country. Capabilities are country
endowments whose combinations define what a country is able to produce and compete
on. Thus competitiveness and growth turn out to be a dynamic process based on
capabilities’ dynamics: the more capabilities a country acquires, the more likely new
competitive production variety will be accessible. 5 However, differently from a
representative capability framework, the extreme heterogeneity of these endowments
makes a direct tackling unfeasible as they are both difficult to define and measure across
countries as we have previously mentioned in section 1.1. The stand-off can be overtaken
by reversing the usual way to proceed between endowments and economic output, not
anymore from the former to the latter but vice-versa: cross-country differences of
production structure can be leveraged to assess a country’s relative competitiveness.6
Seminal works in this direction are the indexes proposed in [Lall, 2001] and in
[Hausmann, 2007]. A further step is represented by [Hidalgo, 2009] which, differently
from previous approaches, explicitly accounts for the heterogeneous networked nature of
the production structure. However, as discussed in [Tacchella, 2012] and [Cristelli, 2013],
more sophisticated non-linear mathematical specifications are required in order to
properly define an index of complexity weighted diversity of production (Fitness) which is
consistent with the statistical features of the production network. For a detailed discussion
we refer to [Cristelli, 2013] and Section 2.
1.4 More on the curse of dimensionality
A direct selection of the past in order to model the future is unfortunately a problem which
is intractable as the state of an economic system, as mentioned previously, is specified,
conceptually speaking, by a very large set of endowments and, in practice, by thousands
of economic indicators. The time evolution of this system, i.e. the development
trajectories, is the result of the interaction among these economic dimensions. These
5
It can be shown that the production variety gain (i.e. production diversification) as a function of
the number of capabilities is an exponential.
6
Further details on the conceptual grounding of this approach are provided in the following
sections.
4
dimensions are known under several names in the economic literature with slightly
different meaning according to the specific field: determinants of growth in empirical
studies, capabilities in trade-related works and endowments for the theory of economic
growth.
The evolution of a country’s economic development creates a trajectory which technically
can be compared to other countries which went through a similar trajectory. While
intuitively appealing, this type of comparative thinking is affected by the curse of
dimensionality [Bellman, 1957; Cecconi et al., 2012] – i.e., as you measure economic
performance in more ways, it becomes difficult to state anything definitive about the
similarity of different economies without unrealistically long histories of countries’
evolution. Provided we cannot forge longer histories of countries for obvious reasons, a
natural way to deal with such a scenario is then to define a suitable procedure to reduce
the dimensionality of the problem in order to pinpoint the driving dimensions.
Economic literature has proposed a number of strategies to implement such
dimensionality reduction. However, most of these strategies share a common feature:
they all try to reduce the dimensionality of the space by building an index or any form of
compact description as a combination of the endowments (or capabilities, we will use the
terms as synonyms throughout the paper). Among the most popular there are direct
(linear) combinations of economic variables either serving as regressors [Barro, 1991] or
as pillars to build new economic indexes such as the Global Competitiveness Index
[WEF, 2016]. More refined approaches treat the problem of the identification of relevant
features in a large data set as a statistical optimization problem, treated with techniques
ranging from Principal Component Analysis [Jolliffe, 2002], to Self-Organizing Maps
[Kohonen, 1982], to other kinds of shallow [Mikolov, 2013] and deep [Bengio, 2007]
neural networks.
1.5 Reversing the approach: From economic output to endowments,
and the fitness-GDPpc plane
The approach of New Development Economics (i.e. economic complexity) is to reverse
this perspective and adopt a different strategy in order to perform the dimension
reduction of the space embedding economic dynamics. We do not go from endowments
to final output anymore, the information flow goes exactly in the opposite direction.
Namely it is the set of produced products that informs on the capabilities of a country and
its potential competitiveness in a compact description.
This dimension reduction achieved by reversing the way we proceed between economic
output and endowments belongs to the complexity paradigm, specifically, the economic
complexity paradigm.7 The complexity paradigm provides a natural way of thinking to
properly deal with adaptive, competitive and heterogeneous systems. In particular, this
paradigm provides the playground to define and model the macroscopic and aggregated
description of those systems where the interaction heterogeneity among their actors,
attributes and activities is a non-negligible element. In our economic setting, actors read
as economic actors, specifically countries, attributes as endowments (or capabilities) and
activities as economic output, specifically exported products.
7
Finding an economic level encoding the relevant endowments and defining a decoding
framework to end up with a few new indicators (i.e. few dimensions) is, conceptually speaking, not
equivalent to find the ‘right’ variable (for example to feed a regression model) out of hundreds
indicators.
5
SPS roots can be therefore traced back in a number of efforts which could be
summarized in the fact that what you export matters [Hausmann et al. 2007; Hidalgo et
al., 2009; Tacchella et al., 2012]. The fundamental idea backing these approaches is that
the differences in capabilities among countries drive and determine the differences in the
export basket of countries. More specifically the differences in capabilities are the drivers
of qualitative more than quantitative differences in the export basket of countries. Thus
the specific diversity of the set of exported products comprehensively encodes all those
dimensions which can drive economic competitiveness.
Fitness is then both a measure of the relative uniqueness of a country’s production as well
as the diversity of its production capabilities. By taking these two aspects into account
we can define a fitness measure, and the competitiveness dynamics of country
development which yields insights on development traps, selecting comparator countries,
and projected growth.
To define this fitness measure, we follow the mathematical specifications proposed in
[Tacchella et al., 2012] and in [Cristelli et al., 2013]. These specifications are consistently
defined with respect to the constraints set by the statistical features of the export basket
diversity differently from previous attempts in this field [Cristelli et al., 2013] (we refer to
Section 2 for further details).
The resulting 2-dimensional space, the fitness-GDPpc plane, is the space in which we
select the right comparatives to model growth projections. We find indeed that the notion
of closeness is highly effective to select those comparatives in the past which are
informative on the future we want to estimate. This backs and strengthens the
effectiveness of our dimension reduction scheme. Moreover, the regularity of the
economic evolution in this plane turns to be a proxy for the predictability of the economic
growth. On this account it introduces the concept of heterogeneity of the growth
predictability and the predictability dimension, which we discuss in Section 5. This
dimension is proven to be effective not only for SPS projections but also for other sources
of long-term growth projections, such as the ones provided by the IMF. Growth
predictability as measured by the SPS is then a standalone piece of information which
provides an assessment of the confidence of the projected growth rates underpinning a
real feature of economic systems specific to the stage of development of a country and
regardless of the source of the growth projections.
1.6 Summary of the main results
The comparison with traditional methods used by the IMF shows that the SPS model and
in general a country’s productive structure is a good and parsimonious predictor of
growth in the long run.8 This work bears three major results:
2. Production structure as measured by the Fitness dimension can be used to
model growth over long term horizons (5+ years) by defining, together with an
intensive measure of wealth, an effective dynamical space for the SPS. SPS defines an
operative and generalizable procedure to select comparator countries and trajectories
to model future GDP growth projections.
8
It is worth noticing that the specifications here proposed for the SPS are the minimal ones. This
implies that there exists a significant space of development for the model, although its forecasting
accuracy in the present version is already is already comparable or even higher than model
embedding significantly more intelligence.
6
3. The SPS provides an accuracy comparable to IMF forecasts, while being
characterized by a much lower model complexity. Here we compare our results
with the available long-term GDP growth projections available through the World
Economic Outlook by the IMF with three different accuracy metrics. SPS is a unique
parameter-free model, while IMF models have specifications and assumptions tailored
to single nations.
4. The SPS allows definition of a novel economic dimension which measures the
predicability of economic growth (and of development at large). This dimension is
shown to be applicable to IMF projections as well, defining a country segmentation for
which the predictive accuracy of both IMF and SPS projections is a-posteriori shown
to be significantly improved.
The implications for economic practice are threefold: i) the Fitness dimension and New
Development Economics (i.e. Economic Complexity) define a model which couples
simplicity and parsimony in terms of assumptions and whose performances are
comparable with the state of the art on the same time horizon (5 years); ii) the SPS
provides growth estimates on time horizons even longer than the state of the art and
keeps a comparable level of reliability of the one achieved in a 5-year time horizon; iii) the
framework offers a scientific foundation for selecting the proper past comparatives to
provide a forecast of the future and simultaneously for assessing the expected accuracy
of such forecast. This predictability assessment provides a criterion to assess the
confidence of growth estimates that mostly depends on the status of the country itself, as
identified by its Fitness and per capita GDP (GDPpc).
7
2. Measuring country competitiveness: The Fitness of
countries
Fitness synthetically measures differences in country competitiveness from cross-country
output differences, specifically from differences in terms of number and complexity (or
sophistication) of classes of products.
The complex and heterogeneous structure of the network of interactions between
economic actors and economic activities/products plays a special role in order to
underpin the main features of economic outputs (even at aggregate level such as the
Gross Domestic Product, GDP). The features of this country specific network are the
results of latent economic dimensions we have to deal with in order to segment the past
in relation to our final aim to find the appropriate comparators to model the future. This is
traditionally rephrased and illustrated by stating that the economic output can be seen as
the result of country specific multipartite (for the sake of simplicity, tripartite) networks
where the three composing layers are the economic actors (here countries), the
capabilities and the economic activities (or products) as shown in Fig. 1. This picture
clarifies the special role played by the network due to economic actors and economic
activities/products as it turns out to be the bipartite projection of this tripartite structure
which encodes information we want to access.
It also follows that the mathematical specification defining the measure of country
competitiveness becomes a crucial point, far from being a simple exemplification of the
narrative. They must conversely be driven by the features of the structure of the bipartite
structure of actors and activities which, at country scale, becomes the bipartite network of
Fig. 1: Cross-country endowment (blue diamonds) differences define the relative
competitiveness of countries. Cross-country output differences in terms of number and nature
of products synthetically encode this relative competitiveness because the bipartite network
defined by countries (green squares) and economic output and activities – in this specific
example, the products (red circles) – is the projection of the tripartite network countries-
endowments-products. This tripartite modeling is intractable via a direct approach being the
result of heterogeneous economic dimensions and hardly measurable interactions.
Conversely bilateral trade data are a robust proxy for the bipartite network country-product.
8
countries-products. Conceptually speaking, the measure of intangible features here
discussed is similar to the effort of a class of methods in network science9 which aims at
inferring features of the network nodes starting from the node connections (i.e. from the
topology). However, the analogy holds only at a conceptual level as the bipartite nature of
the export network and the features of this network itself call for specifications which are
not simply an extension of a linear ranking algorithm to the bipartite case as proposed in
[Hidalgo et al., 2009].
We follow the commonly accepted procedure to build a proxy for the bipartite structure of
countries-products starting from country trade flows, i.e. export [Hidalgo et al, 2009]. We
refer to Appendix B.1 for the motivations of this choice.
As mentioned, we are interested in qualitative differences (i.e. what) rather than
quantitative (i.e. volumes) differences. This implies the definition of a filter to make the raw
export volumes binary quantities. We again leverage a standard and widely accepted
indicator to perform this task: the Revealed Competitive Advantage (RCA) index
proposed originally by Balassa in [Balassa, 1965]. We end up with a binary matrix whose
entries Mcp are 1 if the country c has a revealed comparative advantage in product p
larger than 1, they are 0 otherwise (see Appendix B.1 for the specifications of the RCA
matrix and further details about the RCA interpretation).
The visualization of the matrix M (see Fig. 2) reveals a clear statistical feature of
nestedness. Adapting the ecological definition of a nested system to economics, an
economic system is said to be nested when specialist (non-diversified) countries tend to
produce a subset of products which is also made by generalist (diversified) countries; on
the other hand only generalist (ubiquitous) products are produced by specialist countries.
Nestedness is thought to be one of the principal signatures of complex ecosystems,
where actors compete for finite resources.10 The nested structure sets strong constraints
on the mathematical specifications of the algorithm which intends to measure country
competitiveness.
2.1 Mathematical specifications of the Fitness dimension
The nested structure of the country-product bipartite network as shown in Fig. 2 sets non-
trivial constraints on the amount of information carried by the network edges in order to
assess country competitiveness. Let us consider four specific cases to illustrate why
different links of the country-product network carry different information:
i. a product is exported by a largely diversified country;
ii. a product is exported by a poorly diversified country;
iii. a country exports a highly nonexclusive product;
iv. a country exports a highly exclusive product.
It turns out that statements i) and iii) carry little information in order to determine the level
of sophistication (complexity hereinafter) of a product and the country competitiveness
(Fitness) respectively. Diversified countries (e.g. Germany) being competitive on a wide
9
The eigenvector centrality measure and following evolution, the PageRank [Page et al, 1999] is
likely the most known in this class of approaches.
10
In support of this, similar nested structures are also observed in ecological systems. Ecological
and economic systems have strong analogies as they are both adaptive, evolutionary systems with
limited resources. We refer to [Dominguer-Garcia et al, 2015] where the Fitness and Complexity
algorithm has been shown to be insightful to assess and rank the importance of species in terms of
the ecosystem stability.
9
range of products carries little information on their complexity. The limit case of a country
exporting all products (even though this limit cannot practically occur due to RCA) would
not provide information at all on product complexity. Similarly, statement iii) provides little
information to assess the country fitness as non-exclusive products tend to be produced
by all countries. Again, if all countries exported a product (even though RCA prevents to
observe this limit case), this product would not convey information at all about the Fitness
of countries.
Conversely both statements ii) and iv) have a leading role in assessing product
complexity and country fitness respectively. Statement ii) is conveying crucial information
that that product is produced in a country which is able to be competitive only on non-
exclusive products and statement iv) underpins a country able to export a product which
only very few diversified countries are able to compete on.
Fig. 2: The nested structure of the binary bipartite network defined by countries and
exported products. Rows represent country export baskets and columns instead
specify products. A dark orange dot means that Mcp =1 while light orange means
Mcp =0. Rows and columns are rearranged according to Fitness and Complexity
dimensions defined in Section 2.2. A system is said to be nested when specialized
actors (i.e. countries) tend to produce only a set of products which is also made by
generalist or diversified countries. Source [Cristelli et al., 2013].
To sum up, the nested structure of the bipartite network countries-products, on one hand,
calls for a leading role of the diversity of a country production to underpin its
competitiveness. On the other hand, nestedness also implies that the complexity of a
product must be non-linearly related to this competitiveness dimension (the Fitness) of
countries and specifically this relation must be dominated by the less fit exporter. In other
words, the Complexity cannot be a simple average of the Fitness of its exporters.
By combining these two arguments, we obtain the algorithm firstly proposed in [Tacchella
et al., 2012] which consists in self-consistent non-linear coupled equations for the Fitness
of countries - the competitiveness dimension for countries we were looking for - and the
Complexity of products.11 In Eq. 1 we report the specifications of this iterative scheme.
11
We refer to [Cristelli et al., 2013] for a detailed description of the differences of these
specifications with previous attempts.
10
(1)
The fixed point of this map12 operatively defines the measure for country competitiveness
(i.e., the Fitness dimension) and the product Complexity. In Eq. 1, Fc(n) and Qp(n) are
respectively the Fitness of a country and the Complexity of a product at n-th iteration of
the algorithm and Mcp are the entries of the previously defined binary matrix M. The
symbol ∙ denotes the average, in the case of Fc(n), over all Fitness values and, in the
case of Qp(n) over all complexities values. We remind the reader that the binary C × P
matrix M defines the topology of the bipartite network whose nodes are countries and
exported products (C and P are, respectively, the number of countries and the number of
products).
It is worth noticing that, at each step of the algorithm, the Fitness is the diversity of a
country weighted by the Complexity of products while the Complexity of products is the
harmonic mean of the Fitness of the countries exporting that product up to the
normalization factor. On one hand this implies, consistently with the nested structure of
the matrix M, that the lowest Fitness value among those countries exporting a product is
an upper bound for the Complexity of a product. On the other hand, it also means that the
larger the number of countries exporting a product, the lower will be the Complexity.
In [Cristelli et al., 2013; Pugliese et al., 2014; Wu et al., 2016] the reader can find
extended discussion on the convergence properties of Eq. 1.
2.3 Data sets
The International Trade Statistics Database is made publicly available by United Nations
and accessible via UN Comtrade website.13 In this specific work we will use a dataset
derived from UN Comtrade raw data: the BACI data set released by CEPII. The BACI
dataset is the result of reconciliation procedure of the UN Comtrade which essentially
fixes the inconsistencies between import and export flows, see [Gaulier et al., 2010] for
details.
The latest release of the BACI data set spans the period 1995-2014. This data set
provides all yearly trade volumes, expressed in current USD, between pairs of countries
broken down at the product level. Products are available up to the 6-digit level of the
Harmonized System classification. In this specific work, we will use products aggregated
12
The fixed point of a mathematical map f is simply an element of the domain of the function which
is mapped into itself. In formula given the function y=f(x), x=p is said to be a fixed point of f if and
only if f(p)=p. As a concrete example, economic equilibria are usually fixed points. In this specific
case, the fixed point of the Fitness and Complexity map is said to be an attractive fixed point as
there is a non-empty subset of the domain of the map for which the iterated sequence x, f(x),
f(f(x)), f(f(f(x))), … converges to the fixed point. Furthermore, for this specific map we numerically
show that the attractive fixed point is also unique in the subdomain of the map which is
economically meaningful (see [Cristelli et al, 2013]).
13
https://comtrade.un.org
11
at 4-digit level (HS2007). At 4-digit level, there are approximately 1,150 products
exported by at least one country (we discard those products for which all trade flows are
zero).
Eventually we filter out countries according to a population and total export volume
threshold. This procedure selects approximately 140 countries which account for more
than 95% of world GDP.
The source of all the remaining economic dimensions used in this work is World Bank
Open Data platform,14 except when explicitly stated otherwise.
14
http://data.worldbank.org
12
3 Results: Testing SPS performances and measuring
growth predictability
3.1 SPS model in a nutshell
As mentioned in the previous sections, SPS is a scheme providing a set of criteria to
select examples from the past to model future growth estimates and in particular, it
provides an operative answer to which past should be used to forecast which future.
The evolution of a country’s fitness over time defines a trajectory in the fitness-GDP plane.
Let’s call this the development trajectory. The challenge is to then define the space such
that similar trajectories represent comparable development events, which form the basis
of similar development futures. This is the role of the SPS.
Country economic states are defined in the Fitness-GDPc plane by 2x1 vectors
representing the logarithm of the Fitness and the logarithm of the GDPpc PPP. We the
define the set of comparators for an economic state we want to forecast D-period ahead
as the set defined by the countries which were in the past in the neighborhood of this
economic state in the Fitness-GDPpc plane (we refer again to Section 5 for all the details
and conditions to filter out comparators in order to avoid biases or country over
representations). Provided this set, we average the D-period ahead trajectories of the
comparators, the starting point is the first year a comparator country is found in the
neighborhood of the event to forecast. Let us make an example, we want to forecast the
growth of Vietnam from 2014 to 2019 and, for instance, we find that Mexico in 1995 was in
the neighborhood of Vietnam in 2014 and Indonesia in 2005. Mexico’s trajectory from
1995-2000 and Indonesia’s one from 2005 to 2010 will be averaged in order to model the
evolution of Vietnam in the period 2014-2019. The comparator set yields an average
displacement in the Fitness-GDPpc plane which represents our projections for the event
we want to forecast. This implies that SPS provides both a forecast for the GDPpc and the
Fitness since SPS provides an assessment of the D-period ahead position in the Fitness-
GDPpc of the country to forecast. For the sake of comparison with other source of GDPpc
projection, we convert this result into an annualized growth rate.
3.2 Overview of IMF’s growth modeling
In order to stress the huge parsimony of the SPS in terms of parameters, variables and
assumptions, we provide a brief overview of the modeling underlying IMF’s growth
projections as conceptual benchmark for the remainder of this section. The IMF publishes
GDP growth projections in the annual World Economic Outlook. There is no global unified
methodology: the projections are computed country by country and are subsequently
homogenized and aggregated through a multi-step process with feedbacks among
different teams, schematized in Fig. 3 Reading from the IMF website15:
“The IMF’s World Economic Outlook uses a “bottom-up” approach in producing its
forecasts; that is, country teams within the IMF generate projections for individual
countries. These are then aggregated, and through a series of iterations where the
aggregates feed back into individual countries’ forecasts, forecasts converge to the
projections reported in the WEO.
Because forecasts are made by the individual country teams, the methodology can vary
from country to country and series to series depending on many factors.”
15 https://www.imf.org/external/pubs/ft/weo/faq.htm#q1g
13
Fig. 3: A scheme representing broadly the IMF methodology for GDP forecasting. (Source: IMF
website)
The IMF projections are based on a number of precise assumptions, as stated in [IMF,
2016]. Those assumptions range from exchange rates, to a precise estimate of the oil
prices in 2016 and 2017, to interbank rates for several different currencies. On top of
these global assumptions there are country-specific assumptions on the continuity of
national fiscal and monetary policies.
This results in a very hard to grasp global picture of the models used by the IMF. It is not
directly possible to estimate the number of parameters and assumptions used for the
forecasting as well as the aggregation procedure since they are inherently country
specific.
3.3 Other benchmark models
We also compare our predictive power against three simple country specific univariate
time series models. Simple zero/limited intelligence models have been shown to have a
non-negligible predictive power in the long run (see for instance [Pritchett et al, 2014] and
[Kraay et al., 1999]) and they represent an important benchmark.
Model 1: we use, as a predictor of the annualized D-period ahead growth rate, the
annualized growth rate observed in the last D-period, in formula:
!,!,!!! = !,!!!,!
where gc,t,s denotes the annualized growth rate of the GDPpc of country c in the period of
time from t to s.
Model 2: we model the evolution process for the logarithm of GDPpc of country c at time t
(denoted with yc,t) as a country specific AR(1) process:
!,! = ! + ! !,!!!
Model 3: as for model 2 we use a country specific AR(1) process for the evolution
process of the logarithm of GDPpc of country c at time t (denoted with yc,t) and we
include a trend term linearly related to the time:
!,! = ! + ! + ! !,!!!
14
Model 1 represents the simplest specifications for the argument proposed in [Pritchett et
al, 2014] while model 2 and 3 are the simplest specifications for the country specific
models proposed in [Kraay et al., 1999].16
3.4 Comparison of SPS with IMF’s projections and other benchmark
models
In order to assess the accuracy of the growth projections we compare the annualized
forecasted growth rates with the corresponding realized annualized growth rates over the
same time window. Hereinafter we will refer to annualized growth rates as simply growth
rates for the sake of simplicity. We benchmark SPS and IMF projections accuracy using
three metrics:
• Pearson Correlation (PC) coefficient between projected growth and actual growth
rates;
• Mean Absolute Error (MAE): the average of absolute errors, where the error is
defined as the difference between forecasted rates giproj and actual growth rates giactu,
in formula
!
1 !"#$
= ! − !!"#$
! !!
• Root Mean Squared Error (RMSE): the square root of the average squared errors,
where the error is defined as the difference between forecasted rates giproj and actual
growth rates giactu, in formula
!
! !
1 !"#$ !
= ! − !!"#$
! !!
While the last two metrics are similar, the RMSE gives a greater importance to large
errors. In this way we can also detect the nature of the errors and distinguish scenarios
which are on average similar but show differences in terms of the behavior of the outliers
of the event distribution. In the strict sense, correlation is not a performance indicator. It
rather provides directional information of the error dispersion, we might have an almost
vanishing MAE and simultaneously zero correlation or perfect correlation and a non-zero
MAE.
IMF projections are provided in the World Economic Outlook on a yearly basis and
historical data are publicly available on WEO website.17 Long term projections (4-5 years)
are available only after 2007. We therefore test IMF performances on three time windows
2008-2013, 2009-2014, 2010-2015.
16
In [Hidalgo et al., 2009] and in [Hausmann et al, 2014] the authors relate complexity index ECI to
growth and propose a regressive model which, iterated forward, can be leveraged to forecast
growth. In the web page http://atlas.cid.harvard.edu/rankings/growth-predictions/ the authors
release their 10 years forecast from 2004 to 2014. This implies that a limited comparison would be,
in principle possible, for the period 2004-2014 and 2005-2015 but unfortunately they do not specify
whether the forecasted growth refers to nominal or real GDP (or GDPpc). We cannot therefore
determine which is the proper actual growth to benchmark the projections based on this model. For
these reasons we decided not to include these projections in our comparison.
In addition, a direct comparison with the present scheme cannot be performed as the dynamics in
the ECI-GDPpc plane is observed essentially chaotic everywhere.
17
http://www.imf.org/external/ns/cs.aspx?id=28
15
SPS estimates are available over six periods: 2005-2010, 2006-2011, 2007-2012, 2008-
2013, 2009-2014, 2010-2015. In both cases, we compare the growth rate projections of
GDPpc PPP.
For the model 2 and model 3, we estimate the parameters using the available data up to
time t-1 where t represents the starting time of the period we want to forecast. We
measure the performance over the same six periods used for SPS. In order to have a
setting as similar as SPS, we train models 2 and 3 using data from 1995 only.
Let us first consider a setting in which all available countries are considered (‘All’). In
Table 1 we show the results of the comparison for the accuracy of SPS and IMF growth
projections on a 5 year-time horizon in terms of correlation. 18 In all the tables two
specifications for SPS are provided. The main difference between these two
specifications is that ‘SPS + trend’, differently from ‘SPS’, explicitly accounts for an
autoregressive term for past growth trends.19 In section 5 we discuss the motivation of the
two specifications and in next section we show that they are substantially equivalent in
the predictable regime reinforcing the validity of the selected comparators. In terms of
directional information, we find that the IMF performs approximately 10% better than SPS.
‘All’
SPS SPS + trend IMF
PC 0.30 0.37 0.42
(p-value) (<0.00001) (<0.00001) (<0.00001)
N. Obs. 758 758 384
Table 1: Comparison of SPS (two specifications) and IMF performances with respect
to Pearson correlation. We color in green the scheme with higher correlation.
Conversely, as shown in Table 2, we observe a reversed scenario when we consider the
accuracy of the projections. Both SPS specifications outperform IMF in terms of MAE and
RMSE. Moreover, the difference of the typical error size between IMF and SPS is higher,
in relative terms, for RMSE than MAE (SPS is up to 15% and 3% more accurate than IMF
on the basis of RMSE and MAE respectively). This has a twofold implication: i) both SPS
specifications are more accurate in terms of magnitude of errors and ii) they make smaller
extreme errors, i.e. the dispersion of the forecast error distribution is smaller for SPS than
for IMF.
It is worth noticing that, although Fitness is an export and manufacturing driven
dimension, the statistical features of SPS’ errors are marginally dependent on GDP sector
composition. For instance, SPS approximately performs on average the same error for
countries driven by manufacturing or by services and the same finding applies as a
function of the GDP share due to exports. We refer to Appendix A.2 for extensive
discussions on this point.
18
For each row of the provided tables we highlight in green the method with the best performance.
For MAE and RMSE the lower the value, the higher is the forecasting power. This means that
MAE=0 (RMSE=0) correspond to an error-free forecast. For correlation (PC) the information
provided is only directional and higher or slower correlation does not imply necessarily higher or
slower accuracy in terms of projection errors.
19
In this way, we can deal with the known empirical facts that past growth rates, explain
approximately 10% of future growth rates variance [Pritchett et al, 2014].
16
‘All’
SPS SPS + IMF Model 1 Model 2 Model 3
trend
MAE 2.01 1.97 2.03 2.71 4.62 5.42
(CI) (1.91,2.11) (1.87,2.07) (1.85,2.22) (2.52,2.89) (4.26,5.03) (5.02,5.87)
Accuracy gain % (ref. IMF) 1.0% 3.0% 0.0% -33% -129% -167%
RMSE 2.64 2.60 3.05 3.79 7.07 8.10
(CI) (2.49,2.80) (2.45,2.75) (2.64,3.48) (3.50,4.09) (6.20,8.03) (6.87,9.45)
Accuracy gain % (ref. IMF) 13.4% 14.8% 0.0% -26% -131% -166%
N. Obs. 758 758 384 763 760 760
Table 2: Comparison of SPS (two specifications), IMF, Model 1-2-3 performances with respect to
MAE and RMSE. The confidence interval CI is estimated via bootstrapping and values reported
correspond to 95% CL. We color in green the scheme with the highest accuracy.
In table 2 we also show the predictive power for Model 1-2-3 and, although Model 1
performs surprisingly well, their accuracy is significantly lower than SPS and IMF
projections. We can conclude that SPS and IMF provide a significant improvement of the
accuracy when compared with county specific limited intelligence models.
In summary, SPS and IMF behave overall similarly on the basis of the three metrics but:
• SPS uses less heterogeneous data than IMF and is more parsimonious in terms of
parameters and still achieves similar performance
• SPS performs slightly better than IMF in both RMSE and MAE terms.
In next section we discuss how SPS can extract a further dimension, the Predictability,
and how it can be used to refine the IMF forecast.
3.5 The predictability of growth
The dynamics in the Fitness-GDPpc allows defining a dimension, the Predictability
hereinafter, which essentially measures how regular the flow of the economic evolution is
(see Section 5 for the mathematical specifications). As shown in Figs. 6, 8 and 9, the two
regimes (P and UnP) emerge. Dividing the countries on the basis of this variable
illustrates if there are systematic differences of the SPS performances on the two subsets.
As a first test, we say that a country belongs to the predictable regime if it has ln(f) > -1
and to the unpredictable regime otherwise (see Section 5 for a discussion of this
threshold). As shown from the first two rows of the last column of Table 3 and 4,
approximately 60% of the original events fall into the predictable regime and the
remaining 40% in the unpredictable regime.
3.5.1 Segmenting countries on the basis of Predictability: SPS cases
Let us first consider the two SPS specifications in order to discuss whether the
Predictability defines a non-trivial segmentation of countries. If Predictability dimension is
informative on the degree of growth predictability, we should observe that accuracy of ‘P’
regime is higher than the one of the overall case (‘All’) and that the ‘All’ case has a higher
accuracy than the one measured for countries belonging to the ‘UnP’ regime.
17
‘All’ ‘P’ – Laminar/Predictable ‘UnP’ – Chaotic/Unpredictable
SPS SPS + trend SPS SPS + trend SPS SPS + trend
PC 0.30 0.37 0.41 0.38 0.21 0.42
(p-value) (<0.00001) (<0.00001) (<0.00001) (<0.00001) (0.00012) (<0.00001)
N. Obs. 758 758 429 429 329 329
Table 3: Comparison of SPS (two specifications) with respect to Pearson
correlation for countries belonging to the Predictable regime (ln(Fitness) > -1) and
to the Unpredictable regime (ln(Fitness) < -1). We report in columns ‘All’ the
results of Table 1 for the sake of comparison.
In Table 3 we report SPS comparison in terms of correlation. For SPS without trend
specifications the expected ordering of the correlation magnitude is observed: ‘P’ is more
correlated than ‘All’ case and ‘All’ case is more correlated than ‘UnP’ regime. We observe
a 35% correlation gain for SPS without trend (see Section 5). In the case SPS + trend, the
correlation is instead substantially left unchanged (as we will see in Table 4 the same
behavior is observed for MAE and RMSE: the relative accuracy gain is higher for SPS
than for SPS + trend). We discuss in Section 3.5.3 the conceptual implications for the
nature of the comparative events of this different behavior.
‘All’ ‘P’ – Laminar/Predictable ‘UnP’ – Chaotic/Unpredictable
SPS SPS + trend SPS SPS + trend SPS SPS + trend
MAE 2.01 1.97 1.81 1.94 2.26 2.08
(CI) (1.91,2.11) (1.87,2.07) (1.69,1.93) (1.81,2.06) (2.09,2.45) (1.92,2.25)
Accuracy gain 0.0% 0.0% 10.0% 1.5% -12.4% -5.6%
% (ref. ‘All’)
RMSE 2.64 2.60 2.33 2.51 3.00 2.75
(CI) (2.49,2.80) (2.45,2.75) (2.19,2.58) (2.35,2.67) (2.72,3.28) (2.50,3.01)
Accuracy gain 0.0% 0.0% 11.7% 3.5% -13.6% -5.8%
% (ref. ‘All’)
N. Obs. 758 758 429 429 329 329
Table 4: Comparison of SPS (two specifications) performances with respect to
MAE and RMSE. The confidence interval CI is estimated via bootstrapping and
values reported corresponds to the interval defining 95% CL. We color in green
the scheme with the highest accuracy. We report in columns ‘All’ the results of
Table 2 for the sake of comparison.
In Table 4 we show for both SPS specifications the comparison in terms of forecasting
accuracy. Filtering countries on the basis of the splitting ‘P’ and ‘UnP’ yields a significant
improvement of SPS forecasting accuracy when we consider MAE and RMSE metrics. In
detail, the accuracy for predictable countries ‘P’ is larger than accuracy for ‘P’ and ‘UnP’
countries combined together (i.e. ‘All’ columns): from 1.5% to 10% more accurate for
MAE and from 3.5% to about 12% for RMSE. ‘All’ case yields results which are from 5.6%
to 12.4% more accurate than ‘UnP’ countries for MAE and from 5.8% to 13.6% for RMSE.
We therefore find that ‘P’ accuracy is always higher than ‘All’ case accuracy and the
accuracy of ‘UnP’ is always lower than the accuracy of the ‘All’ case.
The direct comparison of ‘UnP’ and ‘P’ shows that ‘P’ regime is up to 20% more accurate
than ‘UnP’ regime for MAE and up to 22% for RMSE.
18
3.5.2 Why predictability matters for IMF projections
Let us replicate the analysis of the previous section for IMF projections in order to discuss
how the predictability dimension can be leveraged to reduce forecasting errors and
enrich the information provided by IMF projections. As shown in Table 5 correlation is left
unchanged by the segmentation of countries as in the scenario SPS + trend.
‘All’ ‘P’ – Laminar/Predictable ‘UnP’ – Chaotic/Unpredictable
IMF IMF IMF
PC 0.42 0.42 0.41
(p-value) (<0.00001) (<0.00001) (<0.00001)
N. Obs. 384 213 171
Table 5: Comparison of IMF with respect to Pearson correlation for countries
belonging to the Predictable regime (ln(Fitness) > -1) and to the Unpredictable
regime (ln(Fitness) < -1). We report in columns ‘All’ the results of Table 1 for the
sake of comparison.
In terms of accuracy, the ‘P’/‘UnP’ country segmentation surprisingly delivers results
which are similar to the scenario we observe for SPS. IMF projections restricted on ‘P’
regime are more accurate than the case considering all countries and the ‘All’ case is
more accurate than the ‘UnP’ regime. ‘P’ regime for IMF is 19% and 33% more accurate
than ‘UnP’ in terms of MAE and RMSE respectively.
‘All’ ‘P’ – Laminar/Predictable ‘UnP’ – Chaotic/Unpredictable
IMF IMF IMF
MAE 2.03 1.87 2.22
(CI) (1.85,2.22) (1.70,2.06) (1.87,2.58)
Accuracy gain % 0.0% 7.9% -9.4%
(ref. ‘All’)
RMSE 3.05 2.46 3.66
(CI) (2.64,3.48) (2.22,2.68) (2.88,4.37)
Accuracy gain % 0.0% 19.3% -20%
(ref. ‘All’)
N. Obs. 384 213 171
Table 6: Comparison of IMF performances with respect to MAE and RMSE. The
confidence interval CI is estimated via bootstrapping and values reported
corresponds to the interval defining 95% CL. We report in columns ‘All’ the
results of Table 2 for the sake of comparison.
Accuracy increase is then observed not only for SPS but also for IMF. While for SPS this
observation is coherent with the definition of Predictability, this is a priori unexpected for
IMF. Even more surprisingly, the performance gain is, in relative terms, even higher for
IMF than for SPS (both cases) in terms of RMSE (IMF gain is about 20% while SPS gain is
at most about 12%). Filtering countries on the basis of Predictability therefore reduces the
average size of IMF projection errors and it significantly reduces the number of
projections producing large errors. This means that, even if IMF growth modeling is
completely different, the Predictability dimension is capturing a real feature of the
countries (and consequently of the economic regime) pooled in those areas of the plane
where the Predictability is found to be high.
19
As a general comment, while SPS and IMF perform similarly when all available countries
are considered (IMF outperforms SPS in terms of correlation but SPS outperforms IMF in
terms of MAE and RSME), on this reduced set of predictable countries SPS (the
specification without the trend component) matches or outperforms IMF accuracy in all
cases.
In Appendix A.1 we provide a table summarizing all the results provided in Table 1-6. In
Appendix A.5 we show the results of Table 6 for Model 1-2-3. For the three univariate
models, we observe much more limited relative differences between the three scenarios
‘P’, ‘UnP’ and ‘All countries’. This finding is unsurprising because the three univariate
models leverage the past of the country itself which might be very far – i.e. in a different
regime - in the fitness-GDPpc plane from the trajectory we want to forecast. For these
reasons, we do not expect that the ‘P’ and ‘UnP’ classification should affect the
forecasting performance as for SPS or IMF.
3.5.3 Conceptual implications of the Predictability for SPS comparators and IMF
projections
Let us now discuss the different behavior of SPS and SPS + trend when we restrict our
analysis to the regime ‘P’ and the implications of this on the nature of the comparators
SPS select. For the sake of clarity of this section, the difference between the two
specifications consists in the fact that SPS does not directly account for an
autoregressive component in the growth estimates (i.e. in the set of past comparators of a
country we exclude those provided by the country itself) while ‘SPS + trend’
specifications explicitly account for an autoregressive term for past growth trends as
discussed in Section 5 where we provide further details for both specifications.
By comparing SPS and SPS + trends performance gains as in Tables 3 and 4, we find
that filtering countries by predictability improves in relative terms SPS accuracy more than
‘SPS + trend’ one and makes SPS outperforming ‘SPS with trend’ and consequently
reverses the ranking observed in Table 1 and 2. By coupling these observations with the
specifications’ differences, we can conclude that in the predictable regime, the past trend
component is very similar for all comparatives. They are not only good comparatives for
the future trajectory but they are also good comparators in terms of past behavior. In
other words, by selecting predictable countries we are selecting countries in a regime
where there exists a typical future trajectory of evolution, a typical past growth trend and
the comparators selected by SPS provide good estimates of both aspects.
Conversely in the other regime and in the case of with available countries, although the
closeness in the plane is still informative, this closeness is only partial and the trend
components measured by the autoregressive term of country past growth is not
represented as well as in the predictable regime by comparatives’ dynamics. For this
reason, in the ‘UnP’ regime the growth trend complements much better the simple SPS
forecasting, providing a greater improvement to all the metrics considered with respect to
the ‘P’ case.
The evidence that for countries in the regime ‘P’ the closeness is concretely pinpointing a
closeness of economic states, points in the direction of underlying typical trajectories of
development in this regime and the fitness-GDPpc plane and SPS are revealing those
patterns. This perspective then explains why filtering countries by predictability also
increases IMF performances. SPS is signaling that trajectories of growth exist, are less
heterogeneous than in ‘UnP’. Therefore the signal-to-noise ratio is more favorable for all
modeling including IMF ones and not only for SPS.
20
3.6 SPS accuracy beyond 5 years
‘Growth is devilishly hard to predict’ is the opening of a 2016 editorial that appeared in
the Economist’s columns [The Economist, 2016] and the scarcity of systematic and cross-
country sources of long term growth projections witnesses the inherent challenges of this
task. SPS intends to lay the foundation of a playground to provide growth insights beyond
the 5-year time horizon. The possibility within SPS to naturally discuss longer time horizon
crucially relies on the model’s parameter parsimony and on the extreme homogeneity of
the leveraged sources of data: a proxy of the production structure. This permits to avoid
entering a number of macroeconomic estimates underlying more traditional growth
modeling scheme (e.g. IMF). These estimates are crucial to capture short term variability
of growth but become an intrinsic limit for increasing time horizon: the more there are
parameters/variables to estimate, the higher the chance of errors and chains of error for
composite estimates. Conversely if there are stages or regimes of growth where countries
undergo (on average) recurrent low dimensional patterns of growth on the long run - and
our analysis does support the existence of these typical trajectories - the knowledge of
these trajectories becomes increasingly the driving strategy to set up modeling of future
growth variability.
It is not by chance that the SPS playground is defined in a logarithmic space which is
coherent with the idea of smoothing out country specific growth profiles in order to first
map these stage specific long-term trends. Even though this perspective sound as
alternative to standard modeling, this narrative is instead synergic with standard
indicators and with the body of knowledge concerning feedbacks between
macroeconomic dimensions and growth. SPS sets a hierarchical framework to discuss
growth modeling on the long term by disentangling intrinsic growth trajectories due to the
development stage of a country and country specific growth profile originated by cross
section differences of their macroeconomic dimensions. Cross section differences are
here intended at the same point of closeness in the fitness-GDPpc plane and not at the
same point of time as they are usually intended.
Let us discuss SPS results for time windows longer than 5 years. We limit our analysis in
this section to the simplest specifications of SPS model without considering the
SPS+trend specifications.20 In order to ensure as much as possible a ceteris paribus
comparison among different time horizons, we restrict the performance analysis of SPS
on those projections with starting year in the time interval 2000-2004. The projections of a
country for a specific starting year are retained only if they are available for all time
horizons we consider, otherwise we discard it.21 This implies that all the performance
indicators we compute in the following are assessed on the same number of
observations.
Panels of Fig. 4 and Fig. 5 illustrate SPS projections’ performances in terms of PC and
MAE, RMSE respectively as a function of the time horizon D. All metrics suggest that SPS
performances tend to increase (diminishing MAE and RSME) for increasing time
windows. As an example, SPS on a 9-year time horizon is about 25% more accurate than
SPS with 5-year specifications in terms of MAE and approximately 30% more accurate in
20
The main constraints of extending SPS on longer time horizon are set by a decreasing statistics
of comparatives to train SPS. To model growth projections on a 5 years time horizon we need
comparative trajectories of 5 years length and therefore we can select comparators with a starting
point up to 2010, for 7 years horizon up to 2008 and for 10 years only up to 2005.
21
we discuss in Appendix A.4 the performance of SPS as a function of the parameter defining the
size of the neighborhood of an event in the Fitness-GDPpc plane in order to select its comparators.
In this subsection the value for this parameter is the same we use throughout Section 3.
21
terms of RMSE. Similar patterns are observed restricting the analysis to countries in the
‘P’ regime, we simply have downward shifted patterns for MAE and RMSE and upward for
PC (there are 329 observations in this case).
Fig. 4: Correlation between projected and actual growth rates for projection with
initial point in the interval 2000-2004 as a function of the forecasting time horizon
expressed in years. CI is estimated via bootstrapping. All points are estimated with
622 observations.
Fig. 5: MAE (left panel) and RMSE (right panel) of SPS projected growth rates for
projection with initial point in the interval 2000-2004 as a function of the forecasting
time horizon expressed in years. CI is estimated via bootstrapping. All points are
estimated with 622 observations.
3.7 Summary of SPS key findings
To sum up, this section provides several key findings:
• The SPS allows to introduce the concept of heterogeneity of the degree of
predictability of an economic system. Countries are economic systems whose
predictability depends on the development stage itself. This finding is also
consistent with the observation that the probability of occurrences of wars,
conflicts, and all major sources of economic volatility are dependent on the
development maturity of a country.
22
• The concept of predictability naturally arises from this scheme, and it can be used
to assess the confidence of growth projections, regardless of the modeling used,
as in the case of IMF. This dimension is then used to enhance the value carried by
those projections by segmenting countries on the basis of the estimate reliability.
• The closeness of comparators in the predictable regime points in the direction of
concrete closeness of economic states, in other words, in the direction of the
existence of typical and less heterogeneous trajectories of development with
respect to other regimes. It means that a predictable regime selects countries
which offer more favorable conditions to modeling, including the IMF one.
• SPS modeling is more parsimonious than IMF in terms of parameters and
leverages less heterogeneous economic indicators. SPS and IMF perform in a
comparable way when all countries are considered and SPS outperforms IMF in
the predictable regime.
23
4. Case Scenarios
This section illustrates SPS results for 2019 for Thailand and Ghana, two countries which
fall into the predictable and unpredictable regimes respectively. Considering these cases
shows the importance of the Predictability dimension as a tool to estimate the forecasting
power of growth models such as SPS.
As shown in section 3.5.1, SPS forecasts are very accurate for countries like Thailand,
where growth dynamics follow predictable pathways laid out by comparator countries.
These predictable countries are located in the red areas of the Fitness-GDPpc plane (see
Fig. 6). 22 Between 2014 and 2019, Thailand is expected to follow the well-bounded
pathway of comparators like Denmark or Portugal in the late nineties, and increase its per
capita GDP by 5.4 annually (see Table 7).
Growth is more difficult to estimate for countries in the unpredictable (blue) areas of the
Fitness-GDP plane, since per capita GDP is influenced by numerous exogenous
economic indicators whose dynamics are difficult to model even conceptually. Ghana, for
example, belongs to the set of unpredictable countries, where models like SPS or IMF’s
forecast are less accurate in determining how a country’s income will change. While
Ghana’s per capita GDP is also expected to increase by over 5 percent annually, its
comparators follow very heterogeneous trajectories. They do not move in a similar
direction across the Fitness-GDPpc plane as is the case with Thailand (Figure 6).
Because of this, Ghana’s anticipated rise in per capita GDP in 2019 is less certain.
Fig. 6: Ghana’s per capita GDP is more difficult to predict than that of Thailand,
which is illustrated by their position in the Fitness-GDPpc plane. Thailand’s
comparators follow a fairly uniform trajectory, while Ghana’s comparators are more
heterogeneous.
22
We refer to Section 5.1.2 (Step 3) for the specifications and a technical discussion of the
predictability dimension.
24
Thailand Ghana
Per capita GDP (PPP), Prediction for 2019 $20,523 $5,274
Per capita GDP (PPP) in 2014 $15,776 $4,102
Predicted CAGR in per capita GDP 5.40% 5.15%
Predictability 1.1 0.5
Table 7: Forecasts and predictability for Thailand and Ghana
While the expected growth rates are similar for Thailand and Ghana, the countries are
differentiated by the Predictability dimension. Thailand’s predictability has improved from
0.6 in 1995 to 1.1 in 2014 (see figure 7). Following its expected development trajectory
(into the darker red area of the Fitness-GDPpc plane) will further increase its predictability
and may place it on par with some of its current comparators, such as Hungary or
Malaysia. Ghana displays less certain behavior: The trajectories of its comparator set are
quite heterogeneous and predictability is low. The progression of Ghana’s trajectory will
in turn determine the predictability of any future forecasts. It is unclear whether Ghana’s
per capita GDP will become more or less difficult to estimate. Ghana currently sits at the
periphery between the unpredictable and predictable zones of the fitness-GDPpc plane.
Depending on changes in its Fitness and per capita GDP, Ghana can move to a more
(red) or less (blue) predictable position. The volatility of Ghana’s historical predictability
(figure 7) illustrates that such switches between more and less predictable regimes are
not unprecedented. Countries in different predictability regime are structurally different
as, for instance, shown in Table 8 where we report the typical dependence on natural
resources of Thailand and Ghana’s comparators as a percentage of GDP. Thailand’s
comparators are countries with a marginal dependence on low complexity primary
products as natural resources while Ghana’s comparators are economies strongly driven
by natural resources (on average 10% of their GDPs).
Thailand’s Ghana’s
comparators comparators
Total natural resource rents (% of GDP) Average: 1.9 Average: 11.9
Median: 0.8 Median: 10.0
Fitness Average: 2.1 Average: 0.1
Median: 1.9 Median: 0.1
Table 8: Forecasts and predictability for Thailand and Ghana
For the set of unpredictable economies, it is important to consider country-specific
exogenous determinants of growth. Ghana, for example, is reliant on natural resource
extraction, and changes in commodity prices can affect its per capita GDP. The
Predictability dimension can benefit both the SPS and IMF models, as it confirms the
predictive power of growth forecasts for laminar countries and highlights where predictive
limitations exist. It enables a clear segmentation of predictable and unpredictable
countries without requiring specifications for the countless economic indicators that
influence country dynamics (see sections 1.1 and 1.4). It also captures predictability
changes over time, as countries move between predictable and unpredictable regimes.
In this way, Predictability provides information on which countries can be modeled
accurately, and which trajectories are likely influenced by the volatility of exogenous
dynamics.
25
Fig. 7: Predictability of development trajectories for Ghana and Thailand.
Thailand’s per capita GDP is increasingly easier to estimate, while Ghana’s
predictability has been volatile. The negative peak in 2011 corresponds to the fact
that Ghana enters for a short period in the blue area in the middle of the heatmap
discussed in Fig. 6. This region is characterized by an extremely low coherence of
the distribution of the comparators.
26
5. Methods: SPS specifications
In this section, we discuss the SPS mathematical specifications and how the Fitness-
GDPpc plane is used to select the comparatives modeling the future we want to estimate.
5.1 Mathematical specifications of SPS
5.1.1 The Fitness-GDPpc plane and the economic dynamics in the plane
Fitness is a measure of country competitiveness and competitiveness is a driver of the
differences in growth profiles. More competitive economies are expected to grow faster
and consistently than less competitive economies.
The direct comparison defines the Fitness-GDPpc plane [Cristelli et al., 2015], in which it
is possible to track simultaneously the evolution of an economic system in terms of
endowment competitiveness and per capita monetary performance. The Fitness-GDPpc
plane is defined by the natural logarithm of the Fitness and of GDPpc, in such way we
can encompass the dynamics of countries over several orders of magnitude (GDPpc
ranges approximately from 102 USD to 105 USD) while Fitness approximately ranges from
0 to 10.
In Fig. 8 we report the local 1-year average displacement of countries in this plane and
we visualize the results as a vector field. In detail, we split the Fitness-GDPpc plane
according to a grid and we average all 1-year displacements belonging to a box. A 1-
year displacement is said to belong to a box if the starting point of this displacement
belongs to the box.
This procedure then compares the evolution of economic states (the position of a country
in the Fitness-GDPpc plane) which may be a priori far in time but are close in the plane,
as an example we find that nowadays frontier African countries are potentially on the
verge of a sustained industrialization. E.g. Kenya and Uganda are in an economic state in
this plane which is close to the position of Vietnam in the early 1990s or, going back
further, to the Republic of Korea in the 1960s and 1970s.
The grey band in Fig. 8 is an estimate of the expected level of GDPpc of a country
provided its level of Fitness. We refer to Appendix A.3 for the mathematical details of this
estimate. Countries below the line have a per capita wealth lower than what is expected
from Fitness. Conversely for those above, Fitness level does not account for the whole
GDPpc. Despite a highly tempting strategy, the signed distance from the Fitness-GDPpc
is not trivially correlated with the future growth of countries as witnessed by the smoothed
dynamics of Fig. 8.
27
Fig. 8: the heterogeneity of the economic development in the Fitness-GDP
plane. The plane is divided into square boxes and arrows are the local 1-
year average displacement of countries in each box. Grey band is an
estimate of he expected level of GDPpc provided the level of Fitness, see
Appendix A.3 for the details of this estimate.
The comparison of economic states allowed by the Fitness dimension yields non-trivial
results as witnessed by the fact that the dynamics of countries shows different degree of
smoothness. On the right part of Fig. 8 the flow of evolution tends to be laminar (or
predictable) while in left bottom corner and in left top part the emerging dynamics is
much less regular. This leads us to observe that, on average and in the long-term, we
expect to find region more predictable than others and where using the behavior of the
past will be informative on the future evolution of countries in a similar area of the plane.
In this sense the Fitness-GDPpc plane will act as a feature selector in the region where
the flow is laminar and regular where close countries tend to evolve on average in a
similar way. This heterogeneity of regimes, on one hand, is the main reason why a
regressive approach will fail in attempting to forecast growth. Different regions have
substantially different dependences on Fitness (and GDPpc) and why instead a non-
parametric approach is more fruitful in this context as described in the next section.
Furthermore, if we use for the same comparison the raw diversification of countries (using
the notation of Section 2 the diversification of country c would be ∑p Mcp) instead of
Fitness, this would produce a pattern resembling an incoherent or random motion in the
diversification-GDPpc plane. This strengthens the non-trivial ordering provided by
Fitness. It holds back the concept of closeness induced by this economic dimension as
highly informative and effective at reducing the dimensionality embedding the economic
dynamics.
28
5.1.2 SPS: Selecting the right comparatives and definition of the Predictability
The mathematical specifications of the SPS can be described in terms of a 3-step
process:
1. Selection of the candidates to be comparatives in the Fitness-GDPpc plane;
2. Modeling of growth projections as a result of the average development trajectories
derived from the previously selected comparatives;
3. Measuring the Predictability of the region the previously selected comparatives
belong to as a result of a generalized signal-to-noise ratio.
Provided these specifications, SPS specifications be identified as a two dimensional non-
parametric regression (also known as kernel regression [Nadaraya, 1964; Watson, 1964])
where the selected trajectories of the comparatives are playing the role of the sample
statistics and the density we want to estimate is calculated only in those points centered
at the points in the fitness-GDPpc plane corresponding to the events we want to project.
The conceptual basis of the scheme to select comparatives can be found in [Cristelli et
al., 2015]. The specific version of the scheme here proposed is a tailored version of the
concepts of [Cristelli et al., 2015] to deliver growth projections. SPS as discussed in
[Cristelli et al., 2015] is aimed at investigating general features of the flow of the economic
dynamics. In this sense, SPS in [Cristelli et al., 2015] can be thought of as a discrete
Eulerian specification of the economic flow field. Here instead, as we are interested in
modeling the growth projections of specific events, i.e. a country growth over a given time
window, we develop a sort of Lagrangian specification of the economic flow dynamics.23
Step 1: Comparatives selection
We define the event ec,t as the economic state of the country c at time t. Considering that
our events lie in the Fitness-GDPpc, ec,t is a 2x1 vector representing the logarithm of
Fitness and the logarithm GDPpc at time t of country c, in formula ec,t ≡ (fc,t,yc,t) where for
the sake of notation simplicity, in this section we will refer to the logarithm of fitness as
simply f and to the logarithm of GDPpc as y.
Definition: the set of comparatives of a reference event ec,t0 =(fc,t0,yc,t0) is the set composed
of all those events ec’,t in fitness-GDPpc plane for which it holds:
• || ec’,t -ect,0 || < r where ||.|| denotes the Euclidean distance (i.e. L2 norm)
• c ≠c’
• t1) and a loss (P<1) of
predictive resolution.
In order to visualize the behavior of this dimension, we perform a kernel regression in the
Fitness-GDP plane with a Gaussian kernel as shown in Fig. 9 (top panel). The pattern
reveals a strong degree of heterogeneity of the Predictability, the color scheme is
logarithmic and the above defined ratio covers almost two decades.
The threshold ln(Fitness) = -1 we use to define predictable ‘P’ and unpredictable ‘UnP’
countries is approximately the median (50th percentile) of the Predictability marginalized
along Fitness axis as shown in Fig. 9 (bottom panel). The threshold corresponds
approximately to log10(P) = -0.25, namely a predictability of 0.5 which means that the
dispersion of the comparators’ evolution is approximately twice the dispersion of the
originating point of the comparators’ trajectories.
5.2 SPS + trend: Accounting for the autoregressive growth component
The exclusion of the self-contribution of a country as a comparative on one hand avoids
biases, especially in those areas of the fitness-GDPpc where the density of comparatives
is very low. The exclusion is indeed substantially negligible for a large set of
comparatives. An example conversely for which the self-contribution would lead to a
growth projection strongly biased by the past is the case of China, where the number of
comparatives is in the range of 5 to 10. The scheme would hardly detect the smooth
slowdown of China growth rates from double digit rates to current values in the range 6-
8%.
However, neglecting totally this autoregressive term means to discard a variable which
we know accounts for approximately 10% of the variance of the quantity we want to
project. The preferred way to deal with such issue would be to add a third dimension to
our scheme, the past growth rates and perform the comparative search in this 3-
dimensional space. A point in this new space would be a three-dimensional array
specified by fitness, GDPpc and past value of growth of the GPDpc. The neighborhood of
an event would be then a sphere of radius r and formally the specifications to implement
SPS would be similar. However, in such a setup we would deal again with the curse of
dimensionality and, as a general trend, we would have much smaller statistics to estimate
growth projection. The number of countries for which we would be able to yield a growth
estimate would be drastically reduced.
With the priority being the delivery of growth estimates for as many countries as possible,
we propose an alternate scenario to reconcile SPS with an explicit term accounting for the
autoregressive component of growth.
We propose a simple model, SPS + trend, in the form of a regressive scheme as follows:
!"!!!"#$% = !"! + ! !"#$% !! + ! !"#$% !!
where gSPS is the estimated growth rate as described in the previous section, gtrend 1y and
gtrend 2y are the growth rates one and two years before the starting time of the period we
want to project. As long as a term including recent past trend is provided in the model,
we find substantially similar results. The proposed specifications are the ones which best
perform (the difference among the different specifications are marginal). ! and ! are
trained on the time window 1995-2005 via a standard OLS optimization. We refer to
Section 3.5.2 for the motivation for which the trend term is not needed for modeling
growth estimates of countries in the predictable regime.
33
6. Discussion and Perspectives
The ability to assess a consistent measure of the competitiveness of a country’s
productive structure proves to be a crucial element when approaching the complex task
of predicting growth. Such measures are traditionally built with a bottom-up approach, i.e.
by an informed aggregation of a large number of indicators. This way of measuring
competitiveness suffers from many shortcomings, most importantly the heterogeneity of
such indicators across different countries and the objective difficulty of defining suitable
“rules of sum” that allow to effectively synthesize such indicators into a coherent measure
of competitiveness.
In this work, we point out how reversing this approach allows the definition of a measure
of competitiveness, the fitness, that is extremely effective in capturing the growth potential
of countries within a simple forecasting model. The Fitness is defined as a measure of the
outcome of a country’s productive structure in terms of diversity and complexity of
produced products. This definition provides the advantage of i) relying on a single global
dataset of world trade, which is consistent across countries and widely available, and ii)
automatically incorporate the “rules of sum” of capabilities by looking directly at the
outcome that these capabilities allow.
Being an intensive measure of competitiveness, the Fitness is naturally compared with
per capita GDP. Countries display a remarkably peculiar dynamical behavior when
analyzed with respect to fitness and GDPpc time series, that allow us to define two
distinct regimes of growth, a predictable - or laminar - regime, and a unpredictable - or
chaotic - regime. In the laminar regime, we observe that countries within a specific region
of the fitness-GDPpc plane tend to have similar growth patterns. This allows us to define
the Selective Predictability Scheme (SPS), that uses past trends of selected comparator
countries, that have been in the same area of the Fitness-GDPpc plane in the past, to
predict future growth.
The main advantages of the SPS are three:
• It naturally provides an accurate measure of the forecasts, that is defined as the
average predictability of past trajectories in the neighborhood of the country’s
present position in the fitness-GDPpc plane;
• Such accuracy is not a mere property of the SPS method, but rather an estimate of
the actual “predictability” of a country’s economy, that implicitly affects the
accuracy of other forecasting models as well (e.g. IMF);
• It achieves results in terms of forecasting errors that are comparable with very
complex, multi-parameter, country-specific models by using one global model,
virtually parameter-free, to forecast every country in the data set.
An approach like the SPS is able to describe and forecast effectively medium-long term
growth patterns, as it is based on a fundamental measure of competitiveness, that drives
countries over a several-year time horizon. For this reason, it must be complemented with
other, more fine grained methods when there is need to forecast on shorter time horizons,
or to incorporate informed knowledge about relevant industrial and monetary policies,
and international and local scenarios.
34
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Appendix A: Results
Appendix A.1: Comparison SPS vs IMF accuracy (full
table)
We provide a table summarizing all the results provided in Table 1-6 of Section 3.
PC MAE RMSE Observ.
All P UnP All P UnP All P UnP All P UnP
SPS 0.30 0.42 0.21 2.01 1.81 2.26 2.64 2.33 3.00 758 429 329
SPS + Trend 0.37 0.38 0.42 1.97 1.94 2.08 2.60 2.51 2.75 758 429 329
IMF 0.42 0.42 0.41 2.03 1.87 2.22 3.05 2.46 3.66 384 213 171
Table A.1: Comparison of SPS and IMF performances with respect to three metrics:
Pearson Correlation (PC), Mean Absolute Error (MAE) and Root Means Square
Error (RMSE). For SPS we propose the results of two specifications of the scheme,
the latter explicitly accounting for past growth trends. We color in green in each
column the model which best performs. Considering all available countries, columns
‘All’, in terms of correlation IMF slightly outperforms SPS while in terms of average
errors SPS outperforms IMF. When we consider countries only from the predictable
regions as measured by the Fitness-income plane (columns ‘P’), performances tend
to be improved both for IMF and SPS. Results in columns ‘UnP’ correspond to
select only country from the low predicability region.
Appendix A.2: Projection accuracy vs GDP sector
composition
SPS relies on Fitness and Fitness is an export driven measure of cross-country diversity
relying on trade data accounting mainly for manufacturing. Here we show that these
features do not introduce any significant bias: SPS’ projection accuracy is essentially
independent on the specific country GDP sector composition. We consider six economic
indicators for each country27: services, manufacturing, industry, agriculture and natural
resources rents; all indicators are provided as a percentage of GDP. We also include the
export share to address whether there are systematic differences between countries
driven by domestic market or by export.
We compare projection error distribution and MAE as defined in Section 3 but we
segment countries in four groups for each variable. These four groups are defined by the
sector composition quartile a country belongs to. Let us consider for instance
manufacturing, countries in the first quartile of manufacturing GDP share are those
countries whose share is between 0 and 25th percentile of the manufacturing GDP share
distribution. Those in the second quartile have shares between the 25th and 50th percentile
27
Source: World Bank Data platform http://data.worldbank.org
38
and so on for the third and fourth quartiles. The fourth quartile is composed of countries
which are dominated by a specific sector (or are export driven in the case of export
share). Conversely the first quartile is composed of countries for which the sector (or the
export) is less relevant. For instance, in the time windows we test SPS on, manufacturing
accounts on average for 6.9% of GDP for countries in the first quartile and for 22.3% in
the fourth quartile. We measure and compare MAE and error distribution for each quartile
of each variable. MAE error bars are estimated via bootstrapping while error distributions
are compared via a box plot. As shown in the panels in the following, we conclude that
SPS accuracy is marginally dependent on the country sector GDP composition and on
the size of the domestic market.
As expected for the first quartile in the case of manufacturing and industry, we observe
larger errors as Fitness is essentially a blind proxy being those sectors negligible. The
remaining quartiles show instead a higher and similar accuracy.
For agriculture, accuracy is slightly decreasing as a function of the quartile and this can
be explained in terms of an increasing exposure to the exogenous volatility of food
commodities. For services, we have the same pattern observed for manufacturing and
industry, larger errors for countries in the first quartile and constant profile for the
remaining ones.
Considering natural resources rents, we have a pattern similar to Agriculture and, again, it
can be interpreted as a signal of countries with an increasing exposure to the volatility
due to commodities.
A slightly upward and expected trend of the accuracy (i.e downward MAE) is observed
as a function of the GDP share due to export.
A2.1 Manufacturing
Manufacturing Average share
(percentiles) (% GDP)
0-25 6.9
25-50 12.3
50-75 16.1
75-100 22.3
Fig. A.2.1. boxplot comparison of the SPS error distribution (left panel) and mean
absolute error (right panel) as a function of manufacturing GDP share quartile.
39
A2.2 Industry
Industry Average share
(percentiles) (% GDP)
0-25 18.9
25-50 26.0
50-75 31.3
75-100 44.9
Fig. A.2.2: boxplot comparison of the SPS error distribution (left panel) and mean
absolute error (right panel) as a function of industry GDP share quartile.
A2.3 Agriculture
Agriculture Average share
(percentiles) (% GDP)
0-25 1.9
25-50 6.2
50-75 15.4
75-100 32.8
Fig. A.2.3: boxplot comparison of the SPS error distribution (left panel) and mean
absolute error (right panel) as a function of agriculture GDP share quartile.
40
A2.4 Services
Services Average share
(percentiles) (% GDP)
0-25 38.5
25-50 51.4
50-75 60.9
75-100 71.5
Fig. A.2.4: boxplot comparison of the SPS error distribution (left panel) and mean
absolute error (right panel) as a function of services GDP share quartile.
A2.5 Natural resources rents
Natural Resources Average share
(percentiles) (% GDP)
0-25 0.5
25-50 3.1
50-75 9.4
75-100 31.7
Fig. A.2.5: boxplot comparison of the SPS error distribution (left panel) and mean
absolute error (right panel) as a function of natural resources rents GDP share quartile.
41
A2.6 Export (goods and services)
Goods&Services Average share
Exports (percentiles) (% GDP)
0-25 18.0
25-50 29.7
50-75 43.1
75-100 69.8
Fig. A.2.6: boxplot comparison of the SPS error distribution (left panel) and mean
absolute error (right panel) as a function of export GDP share quartile (goods and
services).
Appendix A.3: Fitness-GDPpc relationship
In order to estimate the expected level of GDPpc provided the level of Fitness of a
country, we estimate the parameters of a linear relationship between the logarithm of the
Fitness and of GDPpc which minimizes the weighted Euclidian distance from the scatter
plot points as follows:
The weights wc are the share of a country's GDP with respect to world GDP.
Appendix A.4: Dependence of SPS results on r
We replicate the analysis of Section 3.4 in order to test the variability of SPS’ results as a
function of the parameter r setting the size of the neighborhood to select comparators. In
order to make a ceteris paribus comparison as in Section 3.4 we must further restrict the
set of countries we retain for the analysis as we have to enforce that countries must be
42
available for all time horizons and all values of the parameter r. We show the results in
Fig. A.4.1 and Fig. A.4.2. In terms of accuracy, results profile is essentially the same
regardless of the value of r while increasing values of r appears to marginally enhance
the directional correlation. All results shown in this paper are obtained setting r=0.6, a
value belonging to the range in which SPS’ results are essentially independent on the
value of this parameter.
Fig. A.4.1: Correlation between projected and actual growth rates for time windows
with starting point in the interval 2000-2004 as a function of the forecasting time
horizon expressed in years and of the parameter r. CI is estimated via
bootstrapping. All points are estimated with 600 observations. Increasing value of r
slightly increases correlation.
Fig. A.4.2: MAE (left panel) and RMSE (right panel) of SPS projected growth rates
for time windows with starting point in the interval 2000-2004 as a function of the
forecasting time horizon expressed in years and the parameter r. CI is estimated
via bootstrapping. All points are estimated with 600 observations. Accuracy turns
out to be independent on r in the range investigated.
43
Appendix A.5: Performances of the reference models in
the ‘P’ and ‘UnP’ regime
‘All’ ‘P’ – Laminar/Predictable ‘UnP’ – Chaotic/Unpredictable
Model 1 Model 1 Model 1
MAE 2.71 2.83 2.57
(CI) (2.52,2.89) (2.59,3.07) (2.28,2.88)
Accuracy gain % 0.0% -4.4% 5.1%
(ref. ‘All’)
RMSE 3.79 3.76 3.82
(CI) (3.49,4.08) (3.45,4.08) (3.28,34.35)
Accuracy gain % 0.0% 0.8% -0.8%
(ref. ‘All’)
N. Obs. 763 424 339
‘All’ ‘P’ – Laminar/Predictable ‘UnP’ – Chaotic/Unpredictable
Model 2 Model 2 Model 2
MAE 4.62 4.57 4.71
(CI) (4.25,5.02) (4.11,5.06) (4.12,5.35)
Accuracy gain % 0.0% 1.1% -1.9%
(ref. ‘All’)
RMSE 7.07 6.76 7.44
(CI) (6.20,8.03) (5.90,7.65) (5.81,9.19)
Accuracy gain % 0.0% 4.4% -5.2%
(ref. ‘All’)
N. Obs. 760 424 336
‘All’ ‘P’ – Laminar/Predictable ‘UnP’ – Chaotic/Unpredictable
Model 3 Model 3 Model 3
MAE 5.42 5.63 5.17
(CI) (5.02,5.88) (5.13,6.19) (4.51,5.92)
Accuracy gain % 0.0% -3.9% 4.6%
(ref. ‘All’)
RMSE 8.10 7.88 8.36
(CI) (6.87,9.44) (6.67,9.35) (6.28,10.8)
Accuracy gain % 0.0% 4.1% -4.8%
(ref. ‘All’)
N. Obs. 760 424 336
44
Appendix B: Methods
Appendix B.1: Exports flows and Specifications of the
Revealed Comparative Advantage
Exports are economic outputs endowed with four non-trivial features which are not shared
by internal production: i) they are the results of forces shaped by international
competition, ii) export data are standardized and homogeneous cross country and cross
sector, as a result of the harmonization of customs offices, iii) they are available up to a
disaggregate level which is deep enough to pinpoint the heterogeneous structure of
productive networks and iv) they are consistently available starting from the 1960s.
The RCA is a non-linear filter which compares two shares: the share of the product with
respect to the country export basket and the share of this product with respect to the total
volume (in this case with respect to the world GDP due to export). RCA achieves the non-
trivial result of filtering out the trivial correlation between country sizes and export
volumes. RCA is indeed a relative and multi-scale threshold rather than a flat thresholding
procedure.
The entries of the RCA matrix are defined as follows:
!"
!! !"!
!" =
!! !!!
!!!! !!!!
where qcp is the export of country c of product p expressed in current USD. We define the
binary country-product matrix M whose entries Mcp are defined as follows:
1 !" ≥ 1
!" =
0 ℎ
where RCAcp are the entries of the RCA matrix. The matrix M straightforwardly defines the
topology of the bipartite country-product network: an edge exists between a country and
a product if (and only if) the corresponding entry of the matrix M is 1.
45