FONDATION POUR LES ETUDES ET RECHERCHES SUR LE DEVELOPPEMENT INTERNATIONAL

Measuring macroeconomic volatility
Applications to export revenue data, 1970-2005.

byJoël Cariolle

Working paper n°I14“Innovative indicators” series
March 2012

Measuring macroeconomic volatility: applications toexport revenue data, 1970-2005.
Joë l Cariolle†

Summary

The literature on macroeconomic volatility covers an extremely wide field, as evidenced by thevery broad spectrum of indicators used to grasp it as a phenomenon. In general there seems to
be little discussion about the choice of indicator for macroeconomic volatility, on the grounds
that the different methods, based on stationary series, give rise to scores which are closely
correlated. Although these methods may converge when analysing only the average magnitudeof distribution around a reference value, however, they diverge significantly when one examines
either asymmetry or kurtosis (instances of extreme deviation). This article reviews the existing
literature and sets out the principal methods used for calculating volatility. These methods arecompared and their properties analysed based on export revenue data for 134 countries from1970 to 2005. In particular, a distinction is drawn between measurements of the magnitude ofvolatility and measurements of asymmetry and the incidence of extreme deviations.

† Foundation for International Development Study and Research, 63 Bvd François Mitterrand, 63000 ClermontFerrand, France. I would like to thank Patrick Guillaumont, Michaël Goujon, Jean-Louis Combes and Laurent Wagner
for their practical support and comments.

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TABLE OF CONTENTS
1.

Introduction ........................................................................................................................................... 4
1.1.
Macroeconomic volatility and development. ................................................................................... 4
The costs of volatility .............................................................................................................................................................. 4
Internal economic conditions resulting in greater vulnerability to economic volatility. .............................. 5
The origins of macroeconomic volatility ........................................................................................................................... 7
1.2.
Measures of economic volatility: a variety of indicators. ............................................................. 8
Economic volatility as the standard deviation of the growth rate of a variable .............................................. 9
Economic volatility as the standard deviation of the residual of an econometric regression ..................... 9
Economic volatility as the standard deviation of the cycle isolated by a statistical filter ......................... 10

2.

The question of reference values.................................................................................................. 12
2.1.
Breakdown of economic series ............................................................................................................13
Observation of series behaviour ...................................................................................................................................... 13
Theoretical breakdown of series ..................................................................................................................................... 14
Volatility as a measure of variability, risk or uncertainty? ................................................................................... 15
2.2.
Parametric approach ................................................................................................................................16
Estimate based on a linear deterministic trend ........................................................................................................ 16
Estimate based on a mixed trend .................................................................................................................................... 18
Estimate based on a rolling mixed trend ..................................................................................................................... 20
2.3.
Filtering approach ....................................................................................................................................22

3.

Calculating deviation from the trend.......................................................................................... 26
3.1.
3.2.
3.3.
3.4.

Calculation period for volatility..........................................................................................................26
The magnitude of volatility ..................................................................................................................27
The asymmetry of volatility..................................................................................................................30
Frequency of extreme deviations ........................................................................................................33

Conclusion .................................................................................................................................................... 38
Bibliography ................................................................................................................................................. 39
Appendices ................................................................................................................................................... 43
FIGURES
Figure 1. Export series and spectrum densities for South Korea, Argentina, Venezuela, Kenya,
Ivory Coast and Burundi. ................................................................................................................................... 14
Figure 2. Change in export revenues, simple linear trend. ...................................................................................... 17
Figure 3. Residual of linear trend ............................................................................................................................................ 17
Figure 4. Change in export revenues, mixed trend...................................................................................................... 19
Figure 5. Correlogram of residuals, mixed trend.......................................................................................................... 20
Figure 6. Rolling mixed trend, comparative change and correlogram of residuals, Argentina. ........... 21
Figure 7. Global and rolling mixed trends, comparative change and correlogram of residuals,
Burundi. ....................................................................................................................................................................... 22
Figure 8. Export revenues smoothed by HP filter ........................................................................................................ 25
Figure 9. Magnitude of volatility – deviations (as % of trend) for the last five years around amixed trend ................................................................................................................................................................ 28
Figure 10. Graphical illustration of correlations between magnitude of volatility indicators. .............. 29

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Figure 11 Graphical illustration of a positive and negative asymmetric distribution of identicalmagnitude. ................................................................................................................................................................. 30
Figure 12. Correlation between magnitude and asymmetry of volatility indicators, by reference
value. ............................................................................................................................................................................. 32
Figure 13. Graphic representation of the degree of flattening of a distribution ............................................ 33
Figure 14. Graphical illustration of correlations between magnitude and kurtosis of volatility
indicators, by reference value ........................................................................................................................... 35
Figure 15. Correlation between asymmetry and kurtosis of volatility indicators, by reference
value .............................................................................................................................................................................. 37

TABLES
Table 1: Specification and unit root test on panel data .............................................................................................. 19
Table 2. Correlation between magnitude of volatility indicators. ........................................................................ 28
Table 3. Descriptive statistics of magnitude of volatility indicators (%). .......................................................... 29
Table 4. Correlations of coefficients of asymmetry calculated over the period 1982-2005. ...................... 31
Table 5. Descriptive statistics for kurtosis calculated for the period 1982-2005. ........................................... 34
Table 6. Correlations in kurtosis calculated over the period 1982-2005............................................................. 34
Table 7. Correlations between kurtosis and coefficient of asymmetry for each reference value,
1982-2005........................................................................................................................................................................................... 36

APPENDICES
Appendix A.1 Overview of indicators of volatility and their application in the literature. .................... 44
Appendix A.2. Mixed trends and correlogram of residuals .................................................................................... 46
Appendix A.3. Correlogram of export revenue cycles smoothed by the HP filter ...................................... 48
Appendix A.4 Comparative evolution of deviations (in % of trend values) ................................................... 49

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1. Introduction
The numerous global economic crises of the 20th century have made macroeconomic volatility a key
issue in analysing the determinants of economic growth. The multiplicity of ways in which it affects
the long-term growth potential of economies, its diverse causes and the array of methods by which
it is measured, make economic volatility a complex and multidimensional phenomenon. We
therefore consider the term “volatility” as a generic term, combining all the techniques available formeasuring economic fluctuations.

1.1. Macroeconomic volatility and development.
The literature provides an extensive analysis of the costs and consequences of macroeconomic
volatility. Although the positive relationship between risk and capital yield may, under certain
conditions, explain a positive relationship between economic volatility and growth, most research
agrees that this phenomenon has a negative impact on long-term growth and well-being. Indeed,over the long term, volatility contributes to a reduction in levels of consumption, investment andfactor productivity, to an increase in the volatility and unpredictability of economic policy, and to
deterioration in the institutional environment. The effects on performance are even more marked indeveloping countries, which are often subjected to more significant external shocks but which do
not enjoy the internal conditions that would allow them to absorb them more easily. Hnatkovska
and Loayza (2005), Aizenman and Pinto (2005) and Loayza et al (2007) offer an exhaustive overview
of the consequences of macroeconomic volatility and the factors that cause it.

The costs of volatility
Macroeconomic volatility is a major obstacle to growth. According to estimates produced by
Hnatkovska and Loayza (2005), based on a sample of 79 countries, increasing the average value ofvolatility by the value of its standard deviation results in an average loss of 1.3 points for growth in
GDP over the period 1960-2000, and 2.2 points for the decade 1990-2000. Volatility can, indeed, act
as an obstacle to the key factors in economic and social development.
An early series of research articles examined the impact of macroeconomic volatility on growth fromthe point of view of investment or production factor productivity. Dawe (1996) analyses the effect ofvolatility in exports on investment and growth. He finds both a positive effect of volatility on
investment through an increase in precautionary savings, and a negative effect on growth throughan allocation of capital to sectors with lower yields. Dehn (2000), on the other hand, identifies asignificant and negative impact of shocks in the price of raw materials on investment in developing
countries. Guillaumont, Guillaumont-Jeanneney, and Brun (1999) highlight a negative effect ofvolatility in the rate of investment on growth, based on a decline in average productivity. In asimilar area, Koren and Tenreyro (2006) empirically test a theoretical model where development isaccompanied by increased diversification of inputs into the production system, thus reducing theeffect of volatility in world prices on production factor productivity and therefore on growth.
Finally, Combes and Guillaumont (2002) show that volatility in relation to terms of trade has a

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negative impact on the growth rate of capital productivity. Macroeconomic volatility therefore
seems to be an obstacle to economic growth insofar as it discourages investment decisions, has anegative effect on factor productivity and diverts capital from the most productive sectors.
Other studies analyse the impact of macroeconomic volatility on growth and well-being through its
effect on the quality of economic policy. Easterly et al (1993) show that positive shocks in relation toterms of trade influence the long-term growth path of economies, in part through an improvementin economic policy. Ramey and Ramey (1995) show that the unpredictability of economic policy
caused by volatility in growth rates has a negative effect on the average growth rate of the economy.
Guillaumont and Combes (2002) show that vulnerability to volatility in global prices has a negative
effect on the quality of economic policy and growth. To take two other examples, both Fatas andMihov (2006, 2007) and Afonso and Furceri (2010) have emphasised the negative impact ofvariability in budget policy on growth in both OECD countries and developing nations.
The negative effect of macroeconomic volatility on growth and well-being, based on volatility inpublic and private consumption, has also been examined in a number of studies. Aizenman and
Pinto (2005) and Wolf (2005) point out that in the case of imperfect financial markets, the State andindividual households are unable to protect themselves fully against risks which affect their revenue
and adjust their consumption to the vagaries of economic activity. The result is that volatility driven
by external factors, for example in relation to terms of trade, generates internal volatility in relation
to consumption, particularly in developing countries (Aguiar and Gopinath, 2007; Loayza et al,2007).
Economic policy and consumption therefore appear to be internal channels by whichmacroeconomic volatility is transmitted and may even be magnified, with a concomitant negative
effect on growth and development.

Internal economic conditions resulting in greater vulnerability to economic volatility.
One positive effect of volatility on growth has already been mentioned (Imbs, 2007; Rancière et al,2008; Hnatkovska and Loayza, 2005). This can be explained primarily by the positive correlation
between risk and return on investment projects. Hnatkovska and Loayza (2005) suggest, however,that a positive effect of this kind is dependent on the existence of risk-sharing mechanisms andrespect for rights of ownership, which are in turn supported by a well-developed financial systemand high-quality institutions. A country’s vulnerability to macroeconomic volatility is therefore
driven by a number of handicaps, which are either structural or depend on the level of economic
development. These factors explain why, in general terms, developing countries are more
vulnerable to macroeconomic volatility. Developing countries are more exposed to shocks, and do
not always have the mechanisms or internal conditions in place to enable them to absorb them. The
size of the population, the degree of diversification of the economy and the capacity for operating acountercyclical economic policy, the existence of well-developed financial institutions andinstitutional quality are therefore determining factors in the impact of volatility on growth.
The literature on economic vulnerability has made a significant contribution to our understanding
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of the internal and external conditions of vulnerability to shocks (Guillaumont, 2007, 2009a, 2009b,
2010; Cariolle, 2011; Loayza and Raddatz, 2007; Combes and Guillaumont, 2002). The researchcarried out distinguishes structural factors in relation to vulnerability from more transitory factors
linked to economic policy (or “resilience”). As far as structural factors are concerned, a distinction isdrawn between the magnitude and frequency of shocks (commercial or natural) and exposure tosuch shocks. Factors that affect exposure to shocks (such as the size of the population, the degree of
economic diversification, distance from global markets and geographical isolation) increase thepropensity of economies to suffer shocks and the negative impact of such shocks on growth. A
recent study by Malik and Temple (2009), on the structural determinants of volatility in relation to
growth, suggests a negative relationship between access to global markets and macroeconomic
volatility. According to the authors, countries which are isolated from global markets tend to lackdiversity in terms of exports and experience greater volatility in relation to GDP. As to the resilienceof particular countries, although this is to some extent dependent on structural factors, it is linked
primarily to economic policy and institutions. As a result, development strategies with a focus on
foreign trade, the procyclicality and countercyclicality of economic policy and the quality of
governance and democratic institutions (Rodrik, 1998; 2000) can determine both the magnitude ofvolatility experienced by countries and its effect on their development.
The contribution made by foreign trade to macroeconomic volatility is one of the themes mostcommonly addressed in the literature. Whilst the role played by openness to trade has not beenclearly established (Combes and Guillaumont, 2002), several pieces of research have examined therelationship between a country’s degree of specialisation, its level of development andmacroeconomic volatility. Di Giovanni and Levchenko (2010) study the extent to which openness totrade can result in a specialisation of export sectors which are highly exposed to external shocks,
and thus to increased economic volatility. They show that countries with a low or moderatecomparative advantage in high-risk export sectors diversify their economy in order to attenuate therisk affecting their export revenues. Conversely, countries with a very high level of comparativeadvantage in these sectors tend to specialise in them, and are thus more exposed to volatility inrelation to terms of trade, exports and GDP growth per head. Similarly, Koren and Tenreyro (2006,2007) show that poor countries specialise in a limited number of sectors, with relatively simple
production technologies and a limited range of inputs, and are therefore more vulnerable to shocks
in global prices. Development is therefore supported by diversification into sectors based on more
complex technologies, using a wider variety of inputs, and with less exposure to macroeconomic
volatility. Van der Ploeg and Poelhekke (2009) find that growth volatility, driven by volatility inglobal raw materials prices, is the main determinant of the “natural resource curse”. Having
controlled for growth volatility, they show that supplies of natural resources have a positive andsignificant effect on economic growth. Development is thus generally accompanied by economic
diversification and by specialising in sectors which are less exposed to global volatility. Conversely,
the negative effect of a low level of economic diversification on development is assumed to be
dependent on a high level of exposure to volatility in global prices.

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Other studies have examined the quality of institutions in attenuating or contributing to economic
volatility. Acemoglu et al (2003) showed that poor institutions result in poor governance, which inturn contributes to macroeconomic volatility. Mobarak (2005) finds that democracy reduces
volatility through increased scrutiny by citizens of the management of economic policy. These
results reflect those found by Rodrik (2000), according to which democratic political structuresencourage political consensus around political responses to external shocks. Numerous research
articles have also examined the role of the development of financial markets in transmitting
macroeconomic volatility (Beck et al, 2006; Aghion et al, 2005; Aghion et al, 2004). In general,
financial development therefore tends to attenuate shocks, although it seems to be able to magnify
the effect of shocks of monetary origin on volatility in GDP (Beck et al, 2006). The quality of political,
economic and financial institutions therefore seems to be both a source and a vector ofmacroeconomic volatility.

The origins of macroeconomic volatility
Another set of literature examines the sources of macroeconomic volatility. This research generally
draws a distinction between external forms of volatility (exports, global prices, terms of trade orinternational interest rates) and internal forms (such as economic policy, agricultural production
and natural or climatic disasters). Similarly, it is possible to distinguish between exogenous sourcesof macroeconomic volatility (related to international trade, agricultural production and natural
disasters) and endogenous sources (linked to volatility in economic policy or domestic sociopolitical conditions). Finally, several studies draw a distinction between so-called “normal”
fluctuations and “crisis” fluctuations, the magnitude of which exceeds a particular threshold(Rancière et al., 2008; Hnatkovska and Loayza, 2005).
The literature on the economic vulnerability of developing countries emphasises the significant
contribution made by the magnitude and frequency of external and natural shocks to the structuralvulnerability of developing countries (Guillaumont, 2007, 2009a, 2009b, 2010; Cariolle, 2011; Loayza
and Raddatz, 2007; Combes and Guillaumont, 2002). Research by Mauro and Becker (2006) identifies
the external shocks that cause growth shocks, such as a deterioration in relation to terms of trade
and a sudden halt to the movement of capital. Similarly, Raddatz (2007) examines the sources ofvolatility in GDP in the Least Developed Countries. He sets out an analysis of the total and relative
contribution of external shocks to volatility in GDP based on a breakdown of GDP variance.
Raddatz shows that external shocks (terms of trade, price of primary products, LIBOR anddevelopment aid) have only a marginal effect on the volatility of the growth rate of GDP, whilst
internal factors related to economic policy (level of public deficit, inflation and overvaluation of
exchange rates) make a significant contribution. These results confirm those of Fatas and Mihov(2006, 2007), showing that economic volatility results in part from volatility in budget policy.
Finally, Hnatkovska and Loayza (2005) distinguish between the effect on growth of “normal”
fluctuations in GDP (positive or negative, repeated and on an average scale) and the effect of“crises” measured by falls in GDP over a certain threshold. The simultaneous introduction of thesetwo volatility variables in their regressions shows the important and significant effect of a “crisis”

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level of volatility on growth and the insignificant effect of “normal” volatility. The sameclassification of volatility was used by Rancière et al (2008) and applied to the effect of a financial“systemic risk” on growth. They find a positive relationship between financial crisis and growth,which is explained by the leverage effect of companies’ level of indebtedness on their investments.

The literature on macroeconomic volatility therefore covers something of a wide field, showing thesignificant interest in understanding it and its decisive role on economic performance in relation to
growth. As we will see below, however, there is a diverse range of methods available for measuring
volatility, although there has been no real discussion of the advantages and disadvantages of suchmethods since the research carried out by Gelb (1979) and Tsui (1988).

1.2. Measures of economic volatility: a variety of indicators.
Common definitions of volatility often refer to the notion of disequilibrium. Measuring economic
volatility involves evaluating the deviation between the values of an economic variable and its
equilibrium value. This equilibrium value, or reference value, in turn refers to the existence of apermanent state or trend. In statistical terms, economic volatility is traditionally measured by thesecond (standard deviation) or sometimes a higher moment1 (Rancière et al, 2008), of the
distribution of a variable around its mean or a trend, which then represents the equilibrium value(to which the variable tends to return quickly after deviating in response to a shock). It is frequent
for macroeconomic series (GDP, export revenues, final consumption) to be “non-stationary”, i.e.
they fluctuate around a trend which itself varies over time, or for shocks to make the variabledeviate from its previous tendency over the long term or permanently. It then becomes necessary touse so-called “stationarisation” techniques in order to separate the permanent (or trend) component
from the transitory (or residual) component of the evolution of a series (see section 2.1). The
volatility indicators obtained using these techniques are intended to reflect the effects of the episodic
variations of an economic series around a reference value (mean, deterministic trend or variable
over time). Calculating volatility thus relies on two key questions: that of calculating a reference value– or the choice of a stationarisation method – and measuring fluctuations around said reference value.
A distinction can be drawn between two main families of volatility indicators: on the one hand,those which measure the variability of an economic series, i.e. taking into account all of the transitory
variations of a statistical series, and on the other, those which measure economic uncertainty, or theunpredictability of variations in total variability (Wolf, 2005). We set out below a review of the mainindicators of economic volatility as presented in the literature, noting how reference values andfluctuations have been calculated by various authors. A summary table of these indicators can befound in Appendix A.1.
1 Asymmetry and kurtosis coefficients of financial values are often used as measures of risk in the financial sector. In

economics, only Rancière et al (2008) have, to our knowledge, incorporated the effects of economic volatility into theirresearch.

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Economic volatility as the standard deviation of the growth rate of a variable
Most of the research proposes measuring volatility on the basis of the standard deviation of thegrowth rate of a variable, which assumes that said variable is stationary at first difference. In otherwords, this approach puts forward restrictive hypotheses as to the behaviour of a series without any
prior testing.
Ramey and Ramey (1995), for example, propose studying the effect of economic variability using thestandard deviation of the growth rate of GDP per-capita. Servén (1997) examines the effects ofvolatility on investment in sub-Saharan Africa and uses two measures of macroeconomic volatility,
namely the standard deviation and coefficient of variation of several aggregates2 (terms of trade,
black-market premium, inflation, etc.). Acemoglu et al (2003) study the effect of institutional quality
on macroeconomic volatility and measure the latter using standard deviations of GDP growth rates
and terms of trade. Similarly, Di Giovanni and Levchenko (2010), and Van der Ploeg and Poelhekke(2009) examine the effects of a high level of exposure to external shocks and measuremacroeconomic volatility using the standard deviation of the growth rates of terms of trade, GDP
per inhabitant and exports. Raddatz (2007) also uses measures based on the standard deviation of
the growth rate of several macroeconomic variables (price of primary products, terms of trade, aid
per inhabitant, GDP per inhabitant and LIBOR) to examine the contribution of external shocks to thevolatility of GDP in African countries. It is thus common to apply measures of volatility based on
variance (standard deviation or coefficient of variation) to differentiated series such as GDP, termsof trade, export revenues, prices of goods or international interest rates.

Economic volatility as the standard deviation of the residual of an econometric regression
Other measures of volatility are based on the residual or explanatory power of econometric
regressions. Pritchett (2000) proposes three measures of volatility. The first is based on thecoefficient of determination of a growth-rate regression on a linear temporal trend. The lower thecoefficient of determination, the more the explanatory power of the temporal trend is limited andthe greater the level of volatility. The second measure is based on the difference in growth rates
before and after a break year identified by minimising the sum of the squares of the residuals of aregression on a simple linear trend3. The author also proposes a measure of economic volatility
based on calculating the standard deviation of the residual of a regression of GDP on a mixed
deterministic and stochastic trend, along with Servén (1998), Combes and Guillaumont (2002), andGuillaumont and Chauvet (2007)4. In a similar approach, Lensink and Morrissey (2006) examine the2 See section 3.2 for standard deviation and coefficient of variation formulae.
3 y = a I (t < t*) + b t*I (t < t*) + a *I(t > t*)+ b t*I (t > t*) +e
1
1 1
1
1
11
11
11
t

Where I() is an indicative function, and the break year, t*, is chosen to minimise the sum of the square of residuals, and t* to ≥ 6 and T - t ≥ 6.
4 Amongst other things, Servén (1998) uses a specification similar to that used by Combes and Guillaumont (2002), tomeasure uncertainty, imposing a nullity constraint on the coefficient associated with a second lag.

Page | 9effect on economic growth of the volatility of Foreign Direct Investments (FDI) as measured by thestandard deviation of the residual of a regression of FDI on its three lags and a deterministic trend.
Other authors have tended to concentrate more on the effects of economic uncertainty on
investment and growth. Ramey and Ramey (1995) propose a measure of the uncertainty componentof volatility based on the standard deviation for prediction error against the growth rate (thepredicted value being obtained by a growth rate regression against a quadratic trend, linear trend,
two GDP lags and the initial values of the share of investment in GDP, population and humancapital)5. Servén (1998) and Dehn (2000) in turn examined the effect of economic uncertainty on
investment using several measures of uncertainty generated by price volatility in raw materials andcalculating the conditional standard deviation for prediction error for these series obtained using a
GARCH process (1,1).
Volatility measures based on the residual of econometric regressions thus have the merit of being
based on a less restrictive formalisation of the process underlying the change in the trend of
economic series. Nonetheless, it remains to be seen whether said formalisation allows proper series
stationarisation, and whether the interpretation of the residual is correct (uncertainty orvariability?).

Economic volatility as the standard deviation of the cycle isolated by a statistical filter
Finally, several studies have used the filtered value of a statistical series as a reference value. This
technique can be used to disaggregate a series into trend variations (long term) and cyclicalvariations (short term). This type of volatility indicator is therefore based on cyclical or cycle
fluctuations. The filtering technique is different from the previous two methods insofar as it does
not formulate the behaviour of a series in advance (order of integration, difference-stationarity or
trend-stationarity) and filters series on the basis of their past and future behaviour. Section 2.3
examines this technique in more detail.
Dawe (1996) thus filters export series using a moving average based on five years, i.e. on [t-2;t+2],
and bases his volatility measurement on the average difference between the series observed and thismoving average. Other authors use the Hodrick-Prescott (Hodrick and Prescott, 1997) filter tocalculate their volatility indicator; they include Chauvet and Guillaumont (2007), who use thestandard deviation of the development aid cycle isolated using this filter (HP), i.e. the standard
deviation of the difference between the value smoothed by the filter and the observed value of theiraid variable. Becker and Mauro (2006) propose a shock variable based on the HP filter and identify
decreases in GDP as an event equating to a decline in filtered GDP of over 7%. Hnatkovska and
Loayza (2005) isolate the cyclical fluctuations in GDP series using the Baxter and King filter (1999)
and calculate their standard deviation. Finally, Afonso and Furceri (2010) use the standard deviationof the cyclical component of public spending and tax revenues isolated using HP and Baxter-Kingfilters.
5 The authors regress the following model for the whole of the sample for each country:∆ y it = α 0 + α 1 t + α 2 t 2 + α 3 t 1974 + ϕ dummy 1974 + β y it − 1 + γ y it − 2 + λ I itinitial + δ pop itinitial + θ Khum itinitial + ε itPage | 10

The usual indicators of economic volatility can therefore be distinguished by the method used to
calculate the reference value selected. It is thus possible to draw a distinction between indicators
based on first-difference series variance, indicators based on the variance of the residual of a more
complex model of economic series, and those based on cyclical variance identified by applying
statistical filters. These distinctions become more blurred, however, with regard to the method used
to calculate deviations from the reference value, since the literature limits the analysis of volatility toan analysis of the variance of volatility, i.e. the average magnitude of fluctuations6.
In the following section, we present an analysis of the various methods used for calculating areference value based on export data for the period 1970-2005 and illustrate the differences in theanalysis depending on the method used. In section three, we outline various ways of characterising
the fluctuations of a variable around a reference value, showing that it is possible to quantify
volatility not only by the average magnitude of economic fluctuations, but also in terms of theirasymmetry and the occurrence of extreme variations (or kurtosis).

6 Except for Rancière et al (2008).

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2. The question of reference values
If volatility refers to the notion of disequilibrium, then it must be measured using stationary series.
Most economic aggregates, however, are not “naturally” stationary. It therefore becomes necessary
to calculate a reference value or trend value, around which series will be stationary, which in turnmeans identifying the right method of stationarisation.
Or yt = µt + εt,, a non-stationary process with µt a non-constant term and εt the residual. Stationarising
yt consists of calculating or estimating the trend component µt so that the residual (or cycle) εt meetsthe following conditions7:(1) E(εt) = 0;
(2) V(εt)= σε2 < ∞ for all of t; and(3) Cov(εt ; εt -k) = ρk.
These three conditions require the residual εt to have a zero mean (1), a variance (2) and a finite
autocovariance (3) independent of time. Volatility measures are based on the residual (or cycle) εt,,which therefore reflects only transitory fluctuations. If the series is poorly stationarised, variations
which are attributable to a long-term (or permanent) change in yt may be included in the residual,
thus breaching conditions (1), (2), and (3). The volatility measures based on them would therefore beincorrect, because they do not correspond with the definition given to them. Stationarisation is aprerequisite condition for calculating volatility based on non-stationary level series. Calculating areference value is therefore a fundamental step, since it results in identifying and isolating the trend
or permanent component in the change of an economic variable, from its transitory or stationary
component (Dehn, 2000; Hnatkovska, 2005). To understand this issue in more detail, we firstexamine the theoretical breakdown of change in statistical series. We then present the principles andproperties of the usual methods for calculating reference values, which we apply to export data.
We illustrate our analysis using the annual change in export revenues for 134 (developed anddeveloping) countries over the period 1970-2005 from the World Development Indicators8. The
advantage of using these series is that export volatility is an important aspect of macroeconomic
volatility, which is addressed in depth in the literature on economic volatility. Fluctuations in export
revenues may reflect both the change in domestic (changes in domestic production conditions,natural disasters, etc.) and international economic conditions (volatility in global prices). Within thisframework, we present standard techniques for series stationarisation and calculating the volatilityof exports applicable to a broad range of developed and developing countries over the period 19702005.

7 We are referring here to conditions of “weak” stationarity.
8 http://data.worldbank.org/data-catalog/world-development-indicators

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2.1. Breakdown of economic series
Observation of series behaviour
Figure 1 shows the changes in export revenues in constant dollars (2000) in six different countries(South Korea, Argentina, Venezuela, Kenya, Ivory Coast and Burundi) and the spectrum densities
associated with them. The spectrum of a series is a representation of the contributions of eachfrequency variation9 to the total variation of the series. Observing the spectrum density of a series
then makes it possible to identify whether the change in a series is dominated by variations over alonger period or shorter period. Examining the spectrum is therefore a very useful diagnostic tool if
we wish to represent correctly the dynamics of change in a series. A peak at a given frequency
indicates that a significant proportion of the total variance in a series can be explained by thevariations in said frequency.
Figure 1 shows that the countries represented had a change in exports dominated by variations overa long period, with an increasing trend (excluding Burundi) and a decreasing spectrum density.
Except for South Korea10, a large proportion of the series spectrum is located around variations inperiodicity of around 20 years (with a peak in density at a frequency of around 0.05). It all appears
that variations in average periodicity contribute strongly to total variability for these sample
countries. In particular, variations between 5 and 7 years (with a frequency between 0.1 and 0.3)
seem to contribute substantially to total variability in series from Venezuela, Ivory Coast and
Burundi. This reflects the conclusions reached by Rand and Tarp (2002), and Aguiar and Gopinath(2007), according to whom developing economies experience greater volatility in their rate of
economic growth than developed countries. Finally, an examination of the spectrum for the changeof exports can be used to identify the existence of peaks of density at high frequencies
corresponding to periodicities of around 2-3 years, which suggest that the total variability of theseries used can also be explained by fluctuations over short periods.
Examining the changes in export series for a sample of countries therefore suggests that, althoughvariations in export series over a long period explain most of their variability, fluctuations over a
medium period also play a part. The choice of a reference value is therefore important, since thismakes it possible to distinguish trend variations (long/medium periodicity) from the short-termtransitory variations on which volatility calculations are based. A subsequent theoretical breakdownof economic series provides additional information for understanding changes in statistical series.

9 The calculation for switching from frequency to period is as follows: F=1/T, where F = frequency and T = period.
10South Korea presents a “Granger profile” (Granger, 1966), with most of the power of the spectrum close to a zerofrequency.

Page | 13

Figure 1. Export series and spectrum densities for South Korea, Argentina, Venezuela,
Kenya, Ivory Coast and Burundi.

2004

1.000e+10

year

1970

1970

2004
year

Exports

Burundi

2004

0Exports

Ivory Coast

01.746e+09

ExportsKenya

1.140e+101970

2.000e+08

2004
year

Venezuela

Exports

Argentina

Exports
4.250e+09

1970

3.932e+09

5.086e+09

Exports

South Korea

3.728e+103.615e+102.801e+11

Change in exports (USD, 2000)

1970

year

2004

1970

year

2004
year

Log Periodogram
-2 0
2
4
-6

-4

Log Periodogram
-2 0
2
4
-4

-4

Log Periodogram
-2 0
2
4

Burundi-6-4

Log Periodogram
-2 0
2
4

Ivory Coast

-6

-6

-4

Log Periodogram
-2 0
2
4

Kenya

0.00 0.10 0.20 0.30 0.40 0.50
Frequency
60.00 0.10 0.20 0.30 0.40 0.50
Frequency
660.00 0.10 0.20 0.30 0.40 0.50
Frequency

0.00 0.10 0.20 0.30 0.40 0.50
Frequency

Venezuela

Argentina

-6

-6

-4

Log Periodogram
-2 0
2
4

South Korea

6

6

6

Spectrum densities

0.00 0.10 0.20 0.30 0.40 0.50
Frequency

0.00 0.10 0.20 0.30 0.40 0.50
Frequency

Theoretical breakdown of series
According to Dehn (2000) and Hnatkovska (2005), economic series have a trend or permanent (yPt)component and a cyclical or transitory (yCt) component:
Page | 14

yt = yPt + yCt(1)

The permanent component is made up of a deterministic part (y0+αt), with a temporal trend t11, and astochastic (εPt) part:yPt = y0 + αt + εPt(2)εPt represent stochastic shocks affecting the series trend on a permanent or prolonged basis. By way of
example, productivity shocks or a change in the preferences of economic agents can affect thechange in an economic series over the long term.
The cyclical or transitory component comprises a predictable (yCPt component – associated withstructural factors such as the level of development, foreign exchange system, the size of the country,etc.) and an unpredictable (εCt) component:
yCt = yCPt + εCt(3)εCt represents unpredictable shocks with a temporary effect on the series cycle, such as suddenchanges in international prices of raw materials or unforeseen climatic events.

Volatility as a measure of variability, risk or uncertainty?
As Azeinman and Pinto (2005) suggest, the literature generally sees volatility as associated witheconomic risk or uncertainty12. According to the authors, whilst volatility provides information on
the observed results of a variable, it can also, by extension, provide information on possible results
and thus represent an approximation of the risk associated with it. The volatility indicatorspresented in section 1.2 would then be similar to risk indicators. The authors emphasise, however,that such a measure can overestimate risk by also including predictable fluctuations. Pure risk or
uncertainty would then need to be measured by the residual obtained from a volatility prediction
model, e.g. conditional variance models such as GARCH models (Dehn et al, 2005; Dehn, 2000;
Serven, 1998). Techniques for measuring volatility would therefore be divided into two mainfamilies, namely those which provide measures of the total variability of a series, and those which
provide measures of uncertainty or risk (Wolf, 2005).
Models based on uncertainty indicators, however, such as GARCH models, generally apply to highfrequency economic data (daily, monthly or quarterly price changes, for example), whereas here weare examining volatility in exports, which are reported annually13. As a result, this paper does notaddress uncertainty indicators but instead reviews calculation methods for the reference values usedas the basis for indicators of volatility in terms of variability, i.e. based on the variations of yC in (3).

11Here we are examining the case of a linear deterministic trend, accepting that this may take diverse forms (quadratic andexponential trends, etc.)
12Ignoring the “Knightian” distinction between risk and uncertainty.
13 As in the majority of research on macroeconomic volatility.

Page | 15

2.2. Parametric approach
Most measures of macroeconomic volatility are based on a univariate parametric approach, whichmodels economic series on the basis of past change over a given period. Much of the research set out
in the previous section uses volatility indicators based on residual variance. The most common
techniques are presented below.

Estimate based on a linear deterministic trend
The traditional approach consists of developing a volatility indicator based on an average deviation
around a linear trend. In reality, export series, like all other actual macroeconomic variables (GDP,exports, interest rates, etc.) are series dominated by low frequencies14 (figure 1), which justifies
modelling them using a deterministic trend (linear, polynomial or exponential). In its simplified(linear) form, this technique consists of estimating the following model:

y t = α + βt + ε t(4)Where yt is the variable whose volatility is being measured, α a constant, t a linear trend, and εt azero mean error term. In this case, the reference value is the trend:yˆ t = αˆ + βˆt(5)Deviations from the trend (εt) in principle have no permanent effects on yt. In other terms, these
deviations are assumed to be stationary around the trend and can therefore represent the volatilityof yt. A measure of volatility based on εt thus relies on three key hypotheses: i) that the serieschanges at a constant rate over time, ii) that the long-term change in the series is perfectlypredictable and iii) that all deviations or shocks affecting it are transitory around the trend.
Beveridge and Nelson (1981) highlighted the limitations of an approach of this kind.

14A series whose variations over a long period are those which contribute the most to total variance. See Guay and Saint-

Amant (1997).

Page | 16

Figure 2. Change in export revenues, simple linear trend.
Argentina

002.000e+08

2.000e+104.000e+08

6.000e+08

4.000e+10

Belize

1980

1985

1990

1995

2000

2005

1970

1975

1980

1985

1990

1995

2000

2005

year

Year
Exports (USD, 2000)

Exports (USD, 2000)

Linear trend

Linear trend

To illustrate these limitations Figure 2 shows the actual change in exports and their trend as shownin (7) in Belize between 1980 and 2004, and in Argentina between 1970 and 2005 (Figure 3 shows theresiduals for the estimates produced). Although the change in export revenues in Belize appears at
first sight to be linear, the series observed may, in spite of this, depart from its trend over the longterm. This problem is all the more obvious in Argentina, where the growth in export revenues does
not, at first sight, follow a linear trend. This example illustrates the limitations of this technique forcalculating volatility. Such a specification risks overestimating the importance of shocks by wrongly
including a part of the non-constant trend in the residual.
Figure 3. Residual of linear trend
Argentina

1980

1985

1990

1995
year

2000

2005

-1.00e+10-5.00e+07-5.00e+09

Residus0

Residuals
0

5.00e+09

5.00e+07

1.00e+10

Belize

1970

1975

1980

1985

1990

1995

2000

2005

year

Page | 17

Estimate based on a mixed trend
Up to this point, we have assumed that deviations from the deterministic trend are only transitory.
Equation (2), however, suggests that it is possible that residual variations (εT) are not transitory, andhave a permanent effect on the trend followed by a series. It thus becomes possible to estimate thereference value on the basis of a stochastic trend (a random process below), represented by thefollowing first-order autoregressive AR(1) process,(yt = yt −1 + ε t avec ε t ~ N 0, σ 2)(6)which we can re-write as follows:ty t = y 0 + ∑ε t
1

In this case, the change in y, is determined by a successive history of random shocks, with the resultthat a shock occurring in the past, even the distant past, has the same effect on the series as a shockin the present (long-memory process). The series is then considered to be difference-stationary or thatits trend follows a random path. The series contains a unit root. After differencing, the series can be
rewritten as follows:(∆yt = ε t avec ε t ~ N 0,σ 2)This hypothesis is nevertheless quite strong in the context of examining macroeconomic variables
dominated by low frequencies (Nelson and Kang, 1981). The analysis of the export series and theirspectrum in section 2.1 (see Figure 1) would justify specifying a mixed trend, suggesting theexistence of fluctuations around a trend which is both deterministic and stochastic. Namely theprocess AR(1), which also includes a deterministic trend:

y t = α + βt + δy t −1 + ε t(7)

Our aim is to discover whether the trend as specified in equation (7) makes it possible to stationarise
the export series for our sample of countries correctly. To do so, we calculate the p-value of the unit
root test using Maddala-Wu panel data (based on the Phillips Perron test), carried out in 134
countries, and Fisher statistics on the joint nullity hypothesis for coefficients associated with drift,
trend and lag (table 1).

Page | 18

Table 1: Specification and unit root test on panel data
H0: the series is non-stationary∆y it = α i + β i t + φ1 y it−1 + εit∆∆y it = α i + β i t + φ 2∆y it−1 + εit

Prob>Chi2

F-test

1.000
0.000

47.47
36.61

Countries (Observations): 134(3693).

The results of the tests carried out thus do not justify the rejection of a null hypothesis for unit root
and tend to justify the use of a mixed trend with drift (cf. equation (9)). The non-stationarity
hypothesis is thus rejected once the series is differenced for a second time, and the F-test statistics
lead us to reject the joint nullity hypothesis for the specification coefficients (9). Figure 4 illustrates
the trends obtained for the case of Belize and Argentina.
Figure 4. Change in export revenues, mixed trend.

002.000e+08

2.000e+104.000e+08

4.000e+10

Argentina

6.000e+08

Belize

1970
1980

1985

1990

1995

2000

2005

1975

1980

1985

1990

1995

2000

2005

year

year
Exports (USD, 2000)
Exports (USD, 2000)

Mixed trend

Mixed trend

Figure 5 shows the correlogram for the residuals of equation (7) estimated for Belize and Argentina
respectively. An examination of the correlogram shows white noise, because the correlations
associated with lags are not significantly different from zero15. With regard to these two example
countries and those shown in Appendix A.2, using a mixed trend therefore proves to be more
appropriate to stationarise export revenues.
However, an examination of Figure 4 and Appendix A.2 suggests that this method of estimating
creates a trend whose profile appears to be a slightly smoothed and offset version of the change seenin the exports. This trend therefore seems to reproduce in t the change in exports observed in t-1,
which therefore contributes to an artificial creation of volatility. This phenomenon is all the moremarked in the cases of Argentina (Figure 4), Venezuela, Burundi and Kenya (Appendix A.2).
15The Portmanteau statistics, which are not presented in this article, do not make it possible to reject a null hypothesis for awhite-noise process. They can be supplied to the reader on request.

Page | 19

Figure 5. Correlogram of residuals, mixed trend
Argentina

-0.40

-0.40

-0.20

-0.20

Autocorrelations
0.00
0.20

Autocorrelations
0.00
0.20

0.40

0.40

Belize

1

2

3

4

5

6

1

2

3

Lags

4

5

6

Lags

Bartlett's formula for MA(q) 95% confidence bands

Bartlett's formula for MA(q) 95% confidence bands

Estimate based on a rolling mixed trend
The main problem with a mixed trend as calculated in the previous sub-section is that it is based ona strong hypothesis of constancy over time of the coefficients associated with the trend of the series.
Effectively, this so-called “global” trend is predicted each year for each country based on coefficients
estimated for the whole period of data availability. It also excludes the possibility of a change of
regime in the deterministic and stochastic change of the series concerned. Maddala and Kim (1996)
emphasise that important changes in the deterministic component of the trend taken by a series canlead, wrongly, to a failure to reject the unit root hypothesis. It is therefore possible that the mixed
trend we have estimated attributes to random trend variations a change in the deterministic changeof the series, which may then lead to an overestimate of the magnitude of fluctuations. Althoughtests do exist (e.g. CUSUM, Max Chow) to identify breaks in a trend during unit root tests (Maddala
and Kim, 1996), an alternative and practical solution is to produce a “rolling” estimate of the mixed
trend over a shorter period (Guillaumont, 2007), allowing the coefficients estimated to change fromyear to year and thus reflecting recent changes in the series trend. This “rolling mixed trend” iscalculated for each country and each year based on estimating equation (9) over the period [t; t-k],
rather than over the whole of the period:yˆ tT = αˆ T + βˆ T t + δˆ T y t −1
and

(8)

yt = yˆ tT + ε twhere T is the estimation period for the trend corresponding to (t; t – k).
The results obtained when k = 12 are shown for Argentina in Figure 6 and compared with thoseobtained based on the previous “global” mixed trend. Appendix A.2 shows the predictions andPage | 20

correlograms for residuals obtained using global and rolling mixed trends for other country
examples. We can see that the trend thus obtained is smoother than with the previous model andthat it does not have a “sawtooth” profile resulting from constant parameters over time. Similarly,
an examination of the correlogram in Figure 6 and Figure 7, and the correlograms in the appendix(Appendix A.2), suggests that the residuals resulting from this approach are stationary for thecountry examples chosen.

1970

1980

1990
year

Exports (USD, 2000)
Rolling mixed trend

Global mixed trend

2000

-0.50

0

2.000e+10

Autocorrelations
0.00

0.50

4.000e+10

Figure 6. Rolling mixed trend, comparative change and correlogram of residuals, Argentina.

1

2

3

4

5

6

Lags
Bartlett's formula for MA(q) 95% confidence bands

Nonetheless, this technique presents a number of disadvantages. On the one hand, it reduces thetime coverage of volatility indicators, since the first trend value is only available from t = 1+k. On
the other hand, a bias may appear as the result of a limited trend estimation period.
Estimates based on a rolling mixed trend are not necessarily an ideal solution where there is a breakin the trend of the series. Changes prior to the break may continue to exhibit inertia when the rolling
trend is estimated, once the break has occurred. Moreover, by proposing an estimate of the trend
over a shorter period than the global mixed trend, this technique may include some long-periodicity
fluctuations in the residual as a result of being more sensitive to medium-periodicity fluctuations.
This phenomenon is illustrated in particular by the respective behaviour of two mixed trends
applied to Burundi between 1985 and 1995 in Figure 7. The rolling mixed trend tends tounderestimate the trend for exports to fall relative to the “global” trend over the same period.
The choice of period for calculating the trend is therefore important. In this case we have chosen an
estimation period for the rolling trend of 12 years, in order to highlight the advantages anddisadvantages of this technique compared with the global mixed trend. It could be considered that arolling trend calculated over a period of between 10 and 20 years represents a reasonable basis withregard to the spectrum densities presented previously in Figure 1, since medium-term fluctuations
seem to contribute strongly to the total variability of the series. This choice should be justified basedon a prior examination of series behaviour (unit root tests, graphical examination of series andexamination of their spectrum density) and supported by a study of the literature on the topic.
Page | 21

Global mixed trend

1.000e+08

1.500e+08

Autocorrelations
-0.50
0.00
0.50

2.000e+08

Figure 7. Global and rolling mixed trends, comparative change and correlogram of residuals,
Burundi.

1

2

3

4

5

6

5

6

Lags

1970

1980

1990

2000

year
Exports (USD, 2000)
Rolling mixed trend

Global mixed trend

Rolling mixed trend

Autocorrelations
-0.50
0.00
0.50

50000000

Bartlett's formula for MA(q) 95% confidence bands

1

2

3

4
Retards

Bartlett's formula for MA(q) 95% confidence bands

2.3. Filtering approach
Some research on economic volatility uses a statistical filtering approach to isolate the cyclical andtrend components of changes in the series (Becker and Mauro, 2006; Chauvet and Guillaumont,2007). The standard deviation of the cyclical component then becomes an example of a volatility
indicator. The most popular filter remains the Hodrick-Prescott filter (1997). This can stationarise
potentially integrated series up to order four (King and Rebelo, 1993). The band-pass (BP) filter put
forward by Baxter and King (1999) is also used in the literature on economic fluctuations(Hnatkovska and Loayza, 2005). Although the BP filter maintains the properties of the series moreaccurately, this advantage comes at the price of a loss of observations at the end of the sample16(Baxter and King, 1999). Moreover, when applied to our export data, both techniques give extremely
similar results17. In this article we therefore only present the results obtained with the HodrickPrescott (HP) filter.
The advantage of the filtering method compared with those described above is that it does not
impose a priori any particular (and sometimes arbitrary) form (mixed trend, deterministic, random,etc.) on the behaviour of the series. Moreover, a statistical filtering method enables changes to thetrend over time, which is a definite advantage over the time series-based approach we have

16The BP filter

is a bilateral filter, which requires a minimum number of observations before and after each filteredobservation point in order to increase the precision of the filtering. Baxter and King recommend ignoring the first 12 andlast 12 quarters when filtering quarterly series. For annual series, they recommend ignoring the first three and last three
years of the sample.
17Available to the reader on request.

Page | 22

presented. Hodrick and Prescott break down the change in a series into a non-stationary trend
component (yPt), and a stationary cyclical component (yCt):
yt = yPt + yCt ,T = 1, 2, 3,…, t.
(9)
The HP filter consists of isolating the cyclical component by optimising the following programme inrelation to Yp:
T −1
T
2
p 2
minT ∑ (Yt − Yt ) + λ∑ (∆2Yt p ) {Yt p }t =1  t=1
t= 2

(10)

It is similar to a symmetrical infinite moving average filter. The parameter λ is called a “smoothing
parameter”. The first term of the equation (10) minimises variance in the cyclical component (YCt)
whilst the second term smoothes the change in the trend component. When λ tends towards infinity,
the variance in the growth of the trend component tends towards 0, which implies that the trend
component – or filtered series – is close to a simple linear temporal trend. Conversely, when λ tends
toward 0, the filtered series is close to the original series. Isolated cyclical fluctuations will thus
exhibit a higher (short) frequency (periodicity) the lower the value of the smoothing parameter.
The choice of the value of λ remains arbitrary and there is as yet no unanimity in the literature.
Whilst Hodrick and Prescott advocate a parameter λ equal to 100 for annual data, some studies
suggest a higher value, of between 100 and 400 (Baxter and King, 1999), whilst others prefersignificantly lower values, of between 6 and 10 (Maravall and Del Rio, 2001). Given that the aim of
this article is not to debate the appropriate value for the smoothing parameter, we will compare theresults obtained when it is equal to 100 and 6.5. We will thus obtain deviations compared with along-term trend (λ=100) and a medium-term trend (λ=6.5).
An examination of Figure 8 suggests that use of the HP filter can apply to a large number ofcountries with varied profiles for changes in exports. As predicted, values filtered with a smoothing
parameter of 6.5 are more sensitive to fluctuations in the medium term than those filtered with aparameter of 100. Appendix A.3 shows the correlograms between the cycles of the countries shownin Figure 8. Both these figures thus suggest that the cycles extracted are stationary and not
significantly autocorrelated. The p-values of the Phillips-Perron test below the correlograms in
Appendix A.3 show that we can reject the null hypothesis for first-order integration of the cycle witha confidence interval of 99%, which indicates that the series seems to be correctly stationarised forthese countries with this technique.
The approach nonetheless presents a number of disadvantages. Whilst this method does not impose
a particular functional form on the change in the trend, it does impose, in an ad hoc manner,
weightings identical to the values before and after the filtered value. It thus suggests the hypothesisthat the cyclical component and trend component of economic series are independent. In light of thesignificant amount of research on the effects of macroeconomic volatility on long-term growth,
however, this hypothesis seems restrictive. Moreover, a further defect of the HP filter is that it
becomes unilateral at the end of the sample, thus giving less good results at the start and end of the
Page | 23

data availability period (Van Norden, 2004). Finally and above all, as regards the choice of asmoothing parameter, a low value provokes compression effects: part of the short-periodicity cyclicalvariation can be attributed to the trend. The estimated trend then appears to be highly volatile andthe cyclical component – and therefore volatility – may be underestimated in this case. Conversely,
the choice of a high smoothing parameter provokes leakage effects: some of the long-periodicity
variations can be attributed to the cyclical component. The trend thus appears less volatile and thecyclical component tends to be overestimated. As regards macroeconomic volatility in developing
countries, an examination of the spectra of the export series in Figure 1 suggests the choice of a lowsmoothing parameter, given the significance of medium-term fluctuations. Figure 8 shows agraphical representation of the consequences of choosing a high or low smoothing parameter.
As we have seen in this section, the method of calculation used for the reference value is central tocalculating volatility. If we wish to reflect the effects of transitory variations – for example due toeconomic cycles, certain commercial shocks or climatic events – it is essential to use an adequate
series stationarisation method. In this regard we have set out the two main approaches adopted in
the literature on macroeconomic volatility: the parametric approach and the filtering approach.
Although both these approaches provide acceptable results in relation to stationarisation, the global
mixed trend seems to create volatility artificially: this is less true of the rolling mixed trend and thefiltering method.

Page | 24

02.000e+10

Argentina

1970

1980

1990

2000

year
Exports (USD, 2000)

1.000e+10 2.000e+10 3.000e+10 4.000e+10

4.000e+10

Figure 8. Export revenues smoothed by HP filter
Venezuela

1970

1980

HP 100

Exports (USD, 2000)

0South Korea

1980

1990

2000

year
Exports (USD, 2000)

HP 100Ivory Coast

1970

1980

1990

2000

year

HP 100

Exports (USD, 2000)

HP 100HP 6.52.000e+08 4.000e+08 6.000e+08

HP 6.54.000e+09

2000

HP 6.52.000e+09 4.000e+09 6.000e+09 8.000e+09

1.000e+11 2.000e+11 3.000e+11

HP 6.5

1970

1990
year

Belize02.000e+09

3.000e+09

Kenya

1970

1980

1990

2000

1980

1985

1990

year
Exports (USD, 2000)
HP 6.5

1995

2000

2005

year
HP 100

Exports(USD, 2000)

HP 100HP 6.5

Page | 25

3. Calculating deviation from the trend
Unlike shock variables, which reflect instantaneous fluctuations, measuring volatility involves
calculating a deviation which reflects an accumulation of fluctuations. The deviations presented inthis section consist of summing the deviations around a trend over a given period, so as to highlighta particular aspect of the distribution of a variable (stationary or stationarised): its variance,
asymmetry or degree of kurtosis (or flattening) of the distribution.
Limiting the analysis of volatility only to traditional indicators based on the variance of a variable,
such as standard deviation or the coefficient of variation18, can mask other important aspects ofvolatility, such as the asymmetry of deviations and the occurrence of extreme deviations. In fact,measures such as the standard deviation of deviations, which measure the average magnitude of a
distribution, do not take account of the asymmetry of the reactions of economic agents to positive or
negative shocks. Similarly, they are not able to identify whether the distribution around the trend ischaracterised by frequent shocks on a limited scale, or dominated by infrequent shocks on a large
scale. These characteristics of a distribution are important, however, if, for example, the aim is to
study how volatility affects the behaviour of economic agents. We present below the possible
options for quantifying these phenomena, based on the various moments of distributions aroundthe reference values presented in the previous section.

3.1. Calculation period for volatility
If an indicator of volatility is intended to reflect past experience of shocks, the choice of period overwhich the deviation is calculated is a first step towards calculating said deviation. If, for example,
the aim is to examine the short-term effects of volatility, it is common to calculate an indicatorreflecting fluctuations over the last five or ten years. If the aim is to study the medium/long-term
effects of volatility, a calculation period of over ten years may be needed. Moreover, where thechange in a series presents distinct episodes of volatility over time, it may be relevant to align the
period over which deviations are summed to the approximate period of the episodes of volatility(Wolf, 2005). In fact it is possible to sum over a short period, deviations around a trend which has
itself been estimated over a longer period (see section 2.2). In this case, it will be a matter of studying
the short-term effects of fluctuations around a shorter- or longer-term trend.
In the remainder of the document, we compare the results from different types of volatility
indicators used to analyse the three dimensions (or moments) of the distribution of a variable: thescale of values of a variable, their asymmetry and the occurrence of extreme deviations.

18Their respective formulae are set out at the end of this section.

Page | 26

3.2. The magnitude of volatility
Numerous research articles cited in the list of references examine the consequences anddeterminants of the magnitude of volatility. Several methods are available to quantify this aspect ofvolatility. The most common method consists of calculating the standard deviation of a variable
based on its reference value: y t − yˆ t 1
INS1 = 100 ×∑T t  yˆ t 

2with T=1, …, t.

We compare the deviations with the reference value in order to ensure comparability between theindicators of volatility of a variable whose order of magnitude differs from one country to another.
It is also possible to calculate the average absolute deviation of deviations from the trend:

INS 2 = 100 ×y t − yˆ t
1
∑
T t
yˆ t

with T=1, …, t.

Where y t is the observed value of the series, and ŷ t is the reference value. The question of therespective advantages of standard deviation and absolute average to measure the ‘average
deviation’ of a distribution is an old debate (Gorard, 2005). The preference for standard deviation ishistoric and is related in part to the fact that numerous statistical tests and numerous indicators are
based on it. The preference arose because i) it was shown that in the case of a normal/Gaussian
distribution, standard deviation provides a more efficient estimate of ‘average deviation’ than
average absolute deviation; and ii) it is easier to manipulate in algebraic terms. In the context of an
economic analysis of the effects of the magnitude of volatility, the use of standard deviation isjustified in the case where the effects of volatility increase exponentially with the size of fluctuations.
It has been shown, however, that average absolute deviation is a more efficient measure where there
are measuring errors, or when the distribution is not normal. We illustrate the differences between
these two measures of the magnitude of volatility based on the results obtained for Venezuela and
Kenya shown in Figure 9. Both of these measures are calculated on a rolling annual basis over a
calculation period of five years (t; t-5). Figure 9 suggests that the peaks of volatility corresponding to
the end of the 1970s, 1980s and the early 1990s and 2000s are more heavily weighted when we use
the mean square deviation (INS1).We subsequently prioritise use of the standard deviation (INS1) asa measure of the average magnitude of deviations, given that this method of calculation makes it
possible to distinguish more easily between different episodes of volatility, because of its frequent
use in the literature and because the other measures of volatility presented in the paper are based on
this statistic. We then compare the standard deviations obtained from the reference values presented
in the previous section. This time, deviations are calculated over the period 1982-2005. The results
are shown in tables 2 and 3, and in figure 10.

Page | 27

Figure 9. Magnitude of volatility – deviations (as % of trend) for the last five years around amixed trend

0

5

10

5

15

10

20

15

Kenya

25

Venezuela

1975

1980

1985

1990
year

1995

INS 1

2000

2005

1975

1980

1985

INS 2

1990
year
INS 1

1995

2000

2005

INS 2

Moreover, it appears that measures of the magnitude of volatility calculated on the basis of mixed
trends – global and rolling – and the HP filter are very similar, as suggested by the correlations
shown in table 2 and figure 10. Nonetheless, the magnitude of volatility around the global mixed
trend and the HP100 filter tends on average to be higher than other indicators.
In the same way that using an HP filter with a high smoothing parameter can provoke leakage
effects and result in an overestimate of the magnitude of volatility, a mixed trend may also wrongly
include elements of the trend component in the residual (see sections 2.2 and 2.3). Moreover, a trend
estimated over a long period is less influenced by medium-term variations than a mixed trend
estimated over a shorter period (see, for example, Appendix A.2. Finally, the propensity of thistechnique to create volatility artificially, as we saw in the previous section, contributes to an averageincrease in volatility scores.
Table 2. Correlation between magnitude of volatility indicators.
(1)
Global mixed trend(2)
Rolling mixed trend(3)
HP 6.5(4)
HP 100Volatilities calculated over the period 1982-2005
(1)

1.00

(2)

0.92*

1.00

(3)

0.96*

0.95*

1.00

(4)

0.87*

0.80*

0.87*

1.00

* Significant at 5%. Observations: 134.

Page | 28

Table 3. Descriptive statistics of magnitude of volatility indicators (%).
Volatility Mixed trend

Volatility Rolling Mixed
trend

Volatility HP 6.5

Volatility HP 100

13.6

11.5

9.2

13.3

8.7

7.8

6.1

7.8

Mean
Standard
deviation
Observations: 134.

The values smoothed by the HP filter are calculated on the basis of a symmetric moving-average
process and used to take account of any breaks in the trend. It has been noted that the trend in the
numerous developing countries which make up our sample is more volatile (Rand and Tarp, 2002).
This is why the use of the HP(100) filter is relevant if the aim is to reflect the trend in exports inindustrialised economies, over a longer period and on a larger scale. As with the global mixed trend,
the HP100 filter is therefore less sensitive than other trends to variations over a medium period. This
explains why volatility calculated on the basis of the HP100 filter correlates less closely – though still
at a high level of around 80% – and shows a higher mean than other methods.

SLE
IRN
ALB
KWT
GNB
SOM
NGA UGABDI HTI
DOM
GHA
RWA
COM
SUR
ZAR
TON
STP
SAU
SLB
BFA
DZA
MDG
ARG
GAB
COG
PRY
PNG
NIC
DMA
BGD
AGO
LAO
BGR
LSO
OMN
SLV
TGO
VUTVEN
CMR
SWZ
NER
ZMB
PER
ARE
GUY
ATG
MWI
GMB
BEN
BTN
BWA
SYR
TTO
NPL
BOL
ETH
CAF
ZWE
KNA
LCA
BLZ
CIV
SEN
MUS
URY
MOZ
VCT
MRT
BHR
CRI
CHL
HUN
MLT
ECU
MLI
TUR
FJI
PAN
COL
PRT
NAM
PHL
HND
ZAF
IDN
THA
CYP
CHE
AUT
DEU
EGY
KEN
GRC
ISL
HKG
KOR
BRA
PAK
BRB
GTM
FIN
MYS
MEX
LUX
LKA
ITA
MAR
SWE
BEL
IND
SYC
TUN
DNK
JPN
PRI
AUS
JOR
FRA
CHN
NZL
ESP
NLD
JAM
NOR
MAC
ISR
IRL
USA
GBR
CAN

20

40
60
Rolling mixed trend

80

KIR

TCD
SLE
IRN
KWT
ALB
UGA
HTI
SOM GNB
NGA
BDI
DOM
GHA
RWA
COM
SUR
ZAR
TON
STP
SAU
SLB
BFA
DZA
MDG
ARG
GAB
COG
PRY
NIC
PNG
DMA
BGD
AGO
BGR
LAO
LSO
VEN
OMN
SLV
TGO
CMR
SWZ
NER
VUT
ZMB
PER
ARE
GUY
ATG
MWI
GMB
BEN
BTN
SYR
BWA
TTO
NPL
BOL
ETH
CAF
ZWE
LCA
SEN
BLZ
KNA
CIV
VCT
MUS
URY
MOZ
MRT
BHR
CRI
HUN
MLT
CHL
ECU
TUR
MLI
PRT
FJI
NAM
PAN
COL
PHL
HND
IDN
ZAF
THA
CYP
DEU
CHE
AUT
EGY
ISL
GRC
HKG
PAK
KEN
KOR
BRA
GTM
MYS
BRB
LKA
LUX
MEX
FIN
MAR
BEL
SWE
TUN
IND
ITA
SYC
DNK
JPN
AUS
PRI
FRA
JOR
ESP
CHN
NZL
JAM
NLD
MAC
NOR
IRL
ISR
USA
GBR
CAN

Rolling mixed trend
0 20 40 60 80

0

10

20

HP65

30

40

UGA KWTSLETCD
IRN
ALB
NGA
SOM
GHA
GNB
RWA
DOM
HTI
ARG
COM
AGO
SAU
SUR
SLB
ZAR
TON
STP
GAB
TGO
DZA
LSO
BDI
NIC
PER
NER
COG
BFA
GUY
ZMB
PNG
PRY
SLV
OMN
BGD
BGR
ARE
SWZ
ETH
VEN
SEN
BEN
CMR
BOL
DMA
MWI
VCT
TUR
TTO
SYR
ATG
NAM
EGY
BHR
BWA
ECU
GMB
MDG
IDN
MOZ
CIV
LKA
MRT
NPL
CAF
MLT
MLI
LAO
PRT
DEU
AUT
HUN
URY
BLZ
COL
CRI
LUX
CHE
CHL
HND
ZWE
BTN
CYP
BEL
BRB
MEX
ISL
GRC
GTM
LCA
AUS
MAR
PAK
SWE
KEN
KOR
BRA
FJI
MUS
PAN
KNA VUT
DNK
ITA
MYS
PHL
ZAF
JAM
THA
TUN
FRA
SYC
FIN
HKG
IRL
MAC
ESP
PRI
NZL
IND
NLD
JPN
CHN
JOR
NOR
CAN
GBR
ISR
USA

0

10

20

HP65

30

40

0

50

KIR

50

KIR
TCD
HTI
SLE
SOM ALBUGAIRNGNBGHA
NGA
SUR
DOMKWT
RWA
BGR
COM
SAU
SLB
ZAR
GAB
BDI
BFA
PRY
PNG
VUT
TGO
LSO
COG
STP
NIC
DZA
BTN
MOZ
TON
AGO
PER
SLV
ATG
GUY
ETH
SWZ
NER
OMN
ZMB
MWI
SYR
BWA
GMB
DMA
ARG
BEN
NPL
MDG
MLT
CMR
KNA
ARE
CIV
CAF
VEN
EGY
SEN
PRT
MUS
NAM
ZWE
ZAF
ECU
THA
AUT
VCT
LCA
HND
URY
SYC
HUN
MLI
TTO
BGD
IDN
DEU
CHE
MEX
ISL
CRI
LUX
BLZ
BOL
BEL
PAN
PHL
C
OL
HKG
SWE
DNK
MRT
CHL
BHR
CHN
FIN
GRC
ITA
KOR
MAR
MYS
BRB
BRA
CYP
FRA
GTM
T
FJI
UR
IND
LKA
NLD
PRI
PAK
KEN
IRL
MAC
TUN
ESP
NZL
JPN
NOR
ISR
JOR
AUS
GBR
CAN
JAM
USA

Global mixed trend
020 40 60

Global mixed trend
020 40 60

0

LAO

HP100
0 10 20 30 40 50

KIR

TCD

10

20

HP65

30

40

TCD

50

KIR

SLE
IRN
KWT
ALB
GNB
UGA
HTI
NGA
BDI RWADOM
GHA SOM
COM
ZAR
TON
STP
SAU SUR
SLB
BFA
DZA
MDG
ARG
GAB
COG
NIC
PNG
PRY
BGD
DMA
AGO
BGR
LSO
VEN
OMN
SLV
TGO
CMR
NER
SWZ
VUT
ZMB
PER
ARE
GUY
MWI
ATG
BEN
GMB
SYR
BTN
TTO
B
WA
BOL
NPL
ETH
CAF
ZWE
BLZ
LCA
SEN
CIV
KNA
VCT
URY
MUS
MOZ
BHR
MRT
CRI
CHL
HUN
ECU
MLT
TUR
MLI
FJI
COL
PAN
NAM
PRT
PHL
HND
IDN
ZAF
CYP
CHE
DEU
THA
AUT
EGY
GRC
ISL
PAK
KEN
GTM
BRA
KOR
HKG
LKA
BRB
MYS
FIN
LUX
MEX
MAR
SWE
BEL
TUN
IND
ITA
SYC
AUS
JPN
PRI
DNK
JOR
FRA
NZL
ESP
CHN
JAM
NLD
NOR
MAC
ISR
IRL
USA
GBR
CAN

0

Rolling mixed trend
0 20 40 60 80

Global mixed trend
020 40 60

Figure 10. Graphical illustration of correlations between magnitude of volatility indicators.

10

20

30
HP100

LAO

40

50

KIR

UGA
KWT
SLE TCD
IRN
ALB
NGA
SOM
GHA
GNB
RWA
DOM
HTI
ARG
COM
AGO
SAU
SUR
SLB
ZAR
TON
STP
GAB
DZA
TGO
LSO
BDI
NIC
NER
PER
COG
BFA
ZMB
GUY
PNG
OMN
SLV
PRY
BGD
BGR
ARE
SWZ
VEN
ETH
SEN
BEN
BOL
CMR
DMA
MWI
TUR
TTO
VCT
SYR
ATG
BHR
ECU
NAM
EGY
MDG
GMB
B
WA
IDN
CIV
MOZ
LKA
MRT
CAF
NPL
MLI
MLT
COL
BLZ
CRI
DEU
HUN
URY
AUT
PRT
CHL
LUX
CHE
HND
ZWE
BTN
GTM
CYP
BRB
GRC
BEL
MEX
ISL
LCA
AUS
PAK
KEN
FJI
BRA
MAR
KOR
SWE
P
MUS
AN
KNA VUT
MYS
ITA
DNK
PHL
ZAF
JAM
TUN
FRA
FIN
SYC
THA
NZL
ESP
MAC
IRL
PRI
IND
HKG
JOR
JPN
NLD
CHN
NOR
CAN
GBR
ISR
USA

0

10

20

30
HP100

40

LAO

50

Page | 29

3.3. The asymmetry of volatility
Measures of magnitude do not provide a means of identifying the effects of the asymmetry ofshocks (Wolf, 2005). A study of the effects of the asymmetry of fluctuations is justified, however, inlight of the distinction in agents’ responses to negative or positive shocks (Dercon, 2002; Elbers et al,2007). The coefficient of asymmetry or skewness, describes the macroeconomic volatility profile
facing a country, since it indicates the propensity of countries to suffer negative or positive shocks.
This value is calculated as follows:

Skewness = 100 × y t − yˆ t 
1∑T t  yˆ t 3

1
 y t − yˆ t  ∑
 T t  yˆ t  


2

3/ 2

avec T = 1,..., tA negative skewness indicates that the distribution is weighted towards the left of the reference
value, whilst a positive coefficient indicates a distribution weighted towards the right (see figure11). A symmetrical distribution will have a coefficient whose value is close to zero. A distribution
with a positive (negative) skewness therefore indicates an experience of volatility dominated bypositive (negative) shocks. Moreover, asymmetry increases as shocks (whether positive or negative)
become greater. In other words, skewness also indicates the propensity of a variable to undergo
crises (when negative) or booms (when positive) (Rancière et al, 2008).Although the measures of themagnitude of deviations around a trend presented previously are strongly correlated (see table 2
and figure 10), the asymmetry of distributions of deviations around them can present a significantly
different profile depending on the reference value chosen (see Table 4). Thus, for a given magnitudeof volatility, the experience of shocks of two different countries can present a contrasting asymmetry(see figure 11).
Figure 11 Graphical illustration of a positive and negative
asymmetric distribution of identical magnitude19.

19Source: Wikipedia.fr.

Page | 30

Table 4 shows that the correlations between skewness based on different values are weak, whichsuggests that each of the reference values expresses the asymmetry of shocks differently. The
similarity in the profile of distributions of deviations around the filtered values is confirmed by arelatively strong correlation between their skewness (65%), whilst the skewness obtained from theglobal mixed trend is weakly correlated to the others. The asymmetry of the distribution of resultsaround the mixed trend shows correlations with other, weaker indicators (14% with the skewnessHP(6.5), and 2% with the skewness-HP(100)).
Table 4. Correlations of coefficients of asymmetry calculated over the period 1982-2005.
(1)
skewness (Global mixed
trend)(2)skewness (Rolling mixed
trend)(3)skewness (HP(6.5))(4)skewness (HP(100))

Skewness calculated over the period 1982-2005
(1)

1

(2)

0.23*

1

(3)

0.08*

0.14*

1

(4)

0.29*

0.02

0.65*

1

* Significant at 10%. Sample = 134 countries.

Figure 12 compares indicators of the magnitude of volatility with indicators of the asymmetry ofvolatility for each reference value. The straight-line correlations obtained for each reference value
suggest a fairly weak positive correlation, particularly if certain outliers are excluded. This graphunderlines the fact that for episodes of volatility on a similar magnitude, the asymmetry of shocks
may be opposite, suggesting an entirely distinct experience of shock.
The asymmetry of distributions around the reference value is therefore a separate aspect ofvolatility, which cannot only be assessed by the indicators of magnitude presented in section 3.2.

Page | 31

Figure 12. Correlation between magnitude and asymmetry of volatility indicators, by reference
value.

Correlation = 36%

KIR

80

Rolling mixed trend

Correlation = 23%

60

Global mixed trend

TCD

60

Magnitude

0

20

40

40

SLE
IRN
KWT
ALB
GNB
UGA
HTI BDI
NGA
SOM
DOM
GHA
RWA
COM
SUR TON ZARSAU STPSLBDZABFA
ARG
MDG
GAB
COG
PRY
NIC
PNG
AGO
BGD
BGR
LAO
LSO
OMN
VEN
TGO
SLV ZMBSWZ
NER
PER
VUT BTNCMR DMA
ARE BEN
GUY
MWI
ATG
GMB
SYR
BWA
TTO
ETH
NPL
BOL
CAF
ZWE
CIV
BLZ
KNA
SEN
LCA
URY
VCT
MOZ
MUS
BHR
MRT
CRI
ECU
CHL
MLT
HUN
MLI
TUR
COL
FJI
PAN
NAM
PRT
PHL
HND
IDN
ZAF
EGY
CHE
THA
AUT
GRC
ISL
CYPDEU
KEN
GTM
BRA
PAK
KOR
HKG
BRB
LUX
MEX
MYS
FIN
LKA
BEL
SWE
MARDNKITASYC TUN
IND
AUS
JPN
PRI
JOR
FRA
CHN
ESP
NZL
JAM
NLD
MAC
NOR
ISR
GBR
USA
CAN IRL

020

Magnitude

KIR

-100

0

100

200

300

UGA
KWT
SLE ALB
IRN
SOM
GHA
GNBNGATCD
RWA
HTI
ARG
COM SURSAUAGO
ZAR
TON
GAB
STP
TGO
DZA
BDI SLB
LSO
NIC
BFA
PER
COG
NER
PRY
PNG
GUY
ZMB
OMN
BGD
SLV
BGR
ARE
SWZ
VEN
ETH
SEN
CMR
BEN
MWI
DMA
BOL
VCT
TTO
SYR
TUR
ATGLAO
MDG
BWA
BHR
NAM
ECU
EGY
GMB
MOZ
IDN
NPL
CIV
CAF
MRT
MLT
MLI
BLZ
PRT
CRI
DEU
HUN
URY
BTN
AUT
CHE
COLLKA
HND
CHL
MEX
ZWE
GRC
ISL
BRB
LCA
CYP
GTM
BEL
MUS
PAN
KNA
FJI
SWE
MAR
BRA
PAK
AUS
KOR
KEN
MYS
ZAF
ITA
DNK
PHL
SYC
TUN
FIN
THA
VUT
FRA
JAM
ESP
HKG
IRL
PRI
IND LUX
JOR
MAC
NZL
NLD
JPN
NOR
CHN
CAN
ISR
GBR
USA

-200

-100

0

Asymmetry

200

300

Asymmetry

HP 100

Correlation = 18%

Correlation = 38%
LAO

50

50

HP 6.5
KIR

-200

-100

0
Asymmetry100

TCD
SLE HTIUGAALB SOMIRNNGAGNB GHA
SUR
KWT
RWA
COM
BGR
SAU
SLBBDI ZARGABPRY
BFA
VUT
PNG
TGO
LSO
COG
STP
DZA
NIC
TON
BTN
MOZ
AGO
PER
SLV
ATG
SWZ
GUY
NER
ETH
OMN
ZMB
MWI
SYR
BWA
DMA
GMB
ARG
BEN
KNA
MDG
NPL
CIV
ARE
CAF
VEN
CMR
MLT
EGY
SEN
PRT
ZWE
ECU
ZAF
MUS
LCA
NAM
HND
THA
AUT
URY
VCT
SYC
BGD
HUN
MLI
TTO
IDN
DEU
ISL
CHE
MEX
CRI
LUX
BLZ
BOL
BEL
PAN
COL
PHL
SWE
HKG
CHL
BHR
DNK
FIN
CHNMRT
GRC
KOR
ITA
MYS
MAR
BRB
BRA
FRA
TUR
CYP
GTM
FJI
IND
PRI
LKA
NLD
KEN
PAK
ESP
IRL
MAC
TUN
JPN
NOR
ISR
JOR
AUS
GBR
JAM
CANNZL
USA

DOM

0

10

DOM

30

Magnitude

ALB

20

TCD
HTI
GNB SLEIRNSOM NGAUGA
GHABDI RWA SUR ZAR
COM
SLB
GAB
BFA
COG
STP
AGO
DZA
PNG
SAU
TON
LAO
ARG
LSO
NIC
OMN
NER
BGR
MWIPERVUT
BEN
MDG
TGO
VEN
SLV
SWZ
GUY
ZMB
PRY
BGD
BTN
SYR
CAF
DMA
ATG
CIVPAN
GMB
ETH
ECU
ARE
MOZ
BOL
TTO
KNA
BWA
BHR
CRI
CMR
COL
BLZ
URY
SEN
MLI
ZWE
NPL
ZAF
EGY
NAM
HND
MUS MLT
FJI
VCT
LCA
BRA
GTM
TUR
GRC
CHL
HUN
ISL
FIN
SYC
MEX
KOR
AUT
MRT
CHE
DEU
KENISR
SWE
LUX
PRT
IDN
PHL
JOR
LKA
BRB
MYS
IND
ITA
PAK
CHN
NZL
BEL
PRI
ESP
MAR
DNK
FRA
AUS
JPN
NOR
CYP
TUN
THA
NLD
IRL
MAC
GBR
JAM
HKG
CAN
USA
KWT

40

KIR

40
30

Magnitude

20
10
0

100

DOM

200

-200

-100

0

100

200

300

Asymmetry

Page | 32

3.4. Frequency of extreme deviations
A final dimension of the volatility of a macroeconomic variable is the occurrence of extreme
deviations within a given distribution. This is measured by the fourth moment of the distribution ofobservations around their reference value, namely kurtosis. This value is calculated as a percentageof the reference value, as follows:

Kurtosis = 100 × y t − yˆ t 
1∑T t  yˆ t 

42
1
 y t − yˆ t  



T ∑
yˆ t  
t 



2

avec T = 1,..., tIn the case of a normal distribution,
kurtosis is equal to 3 (or 300% when
expressed as a percentage of thetrend). The kurtosis is a measure ofboth the peakedness and tails’
fatness of a random variable’s
probability distribution relative tothose of a normal distribution. It
indicates the extent to which thenumber of observations close to themean is high compared withobservations away from the mean.
In other words, a high value
indicates that the distribution tends
to be pointed around the mean withthick tails. A low coefficient indicates that the distribution tends to be concentrated around the mean
with thin tails. Figure 13 illustrates three types of flattening of distributions – leptokurtic (kurtosis >3), normal (kurtosis = 3), and platikurtic (kurtosis < 3). This thus gives an indication of thepropensity of a variable to take high values. The risk of kurtosis is seen as a major risk in financeand on this basis is used as a measure of volatility in its own right. In the case of export volatility, it
makes it possible to have an idea of the propensity of certain countries to experience significant
variations in foreign trade in terms of volume or prices. As underlined by Rancière et al (2008), ahigh value of kurtosis should be interpreted with caution since it may both “be generated either byextreme events or by a cluster of observations around the mean that affect the peakedness of the
distribution” (p.386). They find that about one-fifth of their sample of countries exhibiting highkurtosis was affected by observations in the neighbourhood of the distribution centre. One should
hence look carefully at country data and combine such measure of volatility with a measure ofskewness, to ensure that countries actually experienced sharp unusual fluctuations (see figure 15).
Figure 15 also illustrates that such combination can reflect country’s propensity to experience
Figure 13. Graphic representation of the degree offlattening of a distribution

Page | 33

negative or positive external shocks and thus avoid establishing arbitrary thresholds beyond whichcountries experience either a “boom” or a “crisis” (Rancière et al, 2008).
Table 5 shows the descriptive statistics for kurtosis associated with each reference value.
Distributions around reference values are thus, on average, slightly leptokurtic (higher than 300%)
but remain close to the kurtosis value of a normal law. It should be noted that the distributions ofdeviations around HP(100) filtered values have a similar degree of flattening, on average, to those ofdeviations around HP(6.5) filtered values. Distributions around mixed trends show, on average,
more extreme results than distributions around filtered values.
Table 5. Descriptive statistics for kurtosis calculated for the period 1982-2005.
Kurt. (global mixed
trend)

Kurt. (rolling mixed
trend)

Kurt. (HP(6.5))

Kurt. (HP(100))

352.7

367.9

320.0

312.2

155.7

173.0

139.0

146.2

Mean (%)
Standard
deviation
Total sample: 134 countries.

Table 6 shows the correlations between kurtoses obtained by the four reference values. Comparedwith the correlations between dissymmetry coefficients, the correlations between kurtoses areslightly stronger. The degree of occurrence of large-scale positive or negative deviations is logicallyless influenced by the choice of reference value than their asymmetry. The correlations between the
kurtosis of distributions around a rolling mixed trend and those around filtered values, however,are the weakest. The reference values shown in this paper point to different analyses with regard to
the propensity of countries to experience extreme deviations.
Table 6. Correlations in kurtosis calculated over the period 1982-2005(1)
Kurt. (global mixed trend)(2)
Kurt. (rolling mixed trend)(3)
Kurt. (HP(6.5))

(1)

1

(2)

0.39*

1

(3)

0.38*

0.28*

1

(4)

0.49*

0.22*

0.62*

(4)
Kurt. (HP(100))1*Significant at 5%. Sample = 134 countries.

Figure 14 compares indicators of the magnitude of volatility with indicators of the kurtosis ofvolatility and shows a slightly stronger correlation between these two than between indicators of themagnitude and asymmetry of volatility. The average magnitude of deviations thus seems to berelatively independent of the occurrence of extreme values. By way of example, a country such as
the Dominican Republic (abbreviated to “DOM” in the graphs in Figure 14) has a very high kurtosis
coefficient but is subject to volatility on a moderate scale. Conversely, Guinea-Bissau (“GNB” inPage | 34

Figure 14) is a country which, according to the indicators shown here, has experienced volatility ona fairly large scale but has a more platikurtic distribution of observations around the reference
values. The strongest positive correlation between the average magnitude of deviations and their
kurtosis around the HP(100) filter confirms the propensity that this reference value has to isolate
cycles over a longer period, thus explaining a relatively more pointed distribution around thereference value.

Global mixed trend

Rolling mixed trend

Correlation = 29%

Correlation = 20%

KIR

80

60

Figure 14. Graphical illustration of correlations between magnitude and kurtosis of volatility
indicators, by reference value

TCD

60
40

UGA
KWTIRN
SLE
ALB
TCD
NGA
SOM
GHA RWA
GNB
HTI
ARG
COM
AGO
SAU
SUR
SLB
ZAR
TON
STP
GAB
TGO
DZA
BDI
LSO
NIC
BFA
PER
NER
COG
PNG
PRY
GUY
ZMB
OMN
BGD
SLV
BGR
ARE
SWZ
VEN
ETH
SEN
BEN
MWI
CMR
DMA
BOL
VCT
TTO
SYR
TUR
BWA
MDG
BHR
ATGCOLNAM HUN
EGY
ECU
MOZ
GMB
NPL
IDN
CIV
MRT
CAF
LAO
MLT
LKA
MLI
CRI
BLZ
PRT
DEU
BTN
URY
AUT
CHE
HND
CHL
LUX
ISL
GRC
ZWE
MEX
CYP
LCA
GTM
BRB
BEL
KNA
MAR
BRA
PAK
MUS
SWE
PAN
FJI
AUS
KOR
KEN
MYS
ZAF
ITA
DNK
SYC
TUN
THA
FIN
PHL
VUT
JAM
FRA
HKG
PRI
IND
ESP
IRL
MAC
NZL
NLD
JOR
JPN
NOR
CHN
CAN
ISR
GBR
USA

20

UGA
DOM
DMA
LAO

0

IRN
ALB
KWT
GNB
HTI
NGA
SOMBDI
GHA
RWA
COM
SUR
ZAR
TON
STP
SAU
SLB
BFA
DZA
ARG
MDG BGD
GAB
COG
PRY
PNG
NIC
AGO
BGR
LSO
OMN
VEN
TGO
SLV
NER
SWZ
VUT
PERBTN
CMR
ZMB
ARE
GUY
MWI
BEN
ATG
GMB
SYR
BWA
TTO
ETH
NPL
BOL
CAF
ZWE
CIV
BLZ
KNA
SEN
LCAHUN
URY
VCT
MOZ
MUS
BHR
MRT
CRI
ECU
MLT
CHL
MLI
TUR
COL
PAN
FJI
NAM
PRT
PHL
HND
ZAF
IDN
EGY
CHE
THA
AUT
DEU
GRC
CYP
ISL
KEN
HKG
BRA
PAK
KOR
GTM
BRB
LUX
FIN
MYS
LKA
MEX
BEL
SWE
MAR
ITA
IND
SYC
TUN
DNK
AUS
PRI
JPN
FRA
JOR
CHN
ESP
NZL
JAM
NLD
MAC
ISR
IRL
GBR
UNOR
SA
CAN

Magnitude

40

SLE

0

20

Magnitude

KIR

200

400

600

800

1000

0

500

HP filter 100

Correlation = 37%

Correlation = 41%
LAO

50

50

HP filter 6.5
KIR

200

400

600
Peakedness

800

DOM

30
0

BGD

20

DOM

TCD
HTI
SLE
SOM
UGA ALB
IRN
GHA
NGA
GNB
SUR
KWT
RWA
BGR
COM
SAU
SLB
ZAR
GAB
BDI
BFA
PRY
VUT
PNG
TGO
LSO
COG
STP
DZA
NIC
BTN
TON
MOZ
AGO
PER SWZ
SLV
ATG
NER
GUY
ETH
OMN
ZMB
MWI
SYR
BWA
GMB
DMA
ARG
BEN
NPL
CIV
KNA
MDGBGD
CAF
CMR
ARE
VEN
MLT
SEN
EGY
ZWE
PRT
ZAF
ECU
NAM
HND
LCA
VCT
URY
AUT
THA
TTO
MLI
HUN
SYC
ISL
DEU
IDN
MEX
CHE
CRI
BOL
BLZ
LUX
PAN
BEL
COL
PHL
SWE
CHL
MRT
HKG
BHRMUS
FIN
DNK
CHN
GRC
KOR
MYS
ITA
MAR
BRB
BRA
TUR
FRA
FJI
GTM
CYP
IND
LKA
PRI
PAK
KEN
NLD
IRL
ESP
MAC
TUN
NZL
JPN
NOR
ISR
JOR
AUS
GBR
CAN
JAM
USA

10

Magnitude

TCD
HTI
GNB SLEIRN ALB
KWT
SOM
NGA
UGA
RWA
COM
SUR
BDI GHASLBZAR
GAB
BFA
STP
COG
AGO
DZA
VUT
PNG
SAU
TON
LAO
ARG
NER
LSO
OMN
NIC SLV
MWI
BGR
BEN
MDG
TGO
VEN
SWZ
ZMB
GUY
PRY
SYR
BTN
CAF
DMA
CIV
ATG PER
GMB
ETH
ECU
ARE
BOL
MOZ
TTO
KNA
B
WA
CRI
BHR
CMR
COL
PAN
SEN
URY
BLZ
NPL
ZAF
ZWE
MLI
EGY
NAM
HND
FJI
VCT
MUS
BRA
LCA
TUR
CHL
GRCGTM
FIN
MLT
ISL
HUN
KOR
MEX
SYC
AUT
MRT
DEU
C
HE
KEN
PRT
LUX
SWE
IDN
PHL
LKA
BRB
MYS
PAK
ITA
CHN
IND
NZL ISR JOR
ESP
PRI
BEL
MAR
AUS
FRA
JPN
DNK
NOR
TUN
CYP
NLD
THA
IRL
MAC
GBR
JAM
HKG
CAN
USA

40

KIR

40
30

Magnitude

20
10

1500

Peakedness

Peakedness

0

1000

DOM

1000

0

500

1000

1500

Peakedness

Page | 35

Table 7. Correlations between kurtosis and coefficient of asymmetry for each reference value, 1982-2005.
Global mixed trend

Rolling mixed trend

HP (6.5)

HP (100)

Total sample+0.65*+0.51*+0.12*+0.35*

Positive asymmetry
AC>0%+0.85*+0.83*+0.84*+0.91*

Negative asymmetry
AC<0%-0.58*-0.48*-0.70*-0.45*+0.16*+0.33*+0.35*

Weak asymmetry+0.24*
CA =[-100%; 100%]*Significant at 5%. Sample = 134 countries.

Figure 15 compares the scores for asymmetry of volatility with scores for kurtosis of volatility. A Ushaped relationship can be observed between the two measures, whose return point is around azero value for the coefficient of asymmetry. Table 7 shows a very high positive correlation between
these two indicators when the sample is limited to countries with a positive asymmetry for the
distribution of exports, and a significant negative correlation for countries with a negative
asymmetry for the distribution of exports. Moreover, an analysis of these correlations shows that ahigh kurtosis most frequently corresponds to a positive asymmetry for shocks, particularly for
distributions around mixed trends and the HP(100) filter. Conversely, for degrees of asymmetry
between 0 and 100% in absolute values, we see a very moderate positive relationship both ingraphical terms and in terms of coefficients, suggesting that these two aspects of volatility are
relatively independent of each other for these values.
A low skewness in absolute values therefore appears to correspond to small-magnitude, highfrequency variations, whilst a high coefficient of asymmetry seems to correspond to largemagnitude but lower frequency variations. It therefore becomes possible to see a bidimensional
measure of instability in the coefficient of asymmetry of a distribution around its trend, reflecting
both asymmetry and the occurrence of extreme deviations.

Page | 36

Figure 15. Correlation between asymmetry and kurtosis of volatility indicators, by reference
value

200

1000

1500
100

DOM

0

HUN
NAM
RWA
LKA
UGA
CHE
TUR
COL
GHA
ATG
BHR
BEL
ARG
MEX
DZA
BRB
KIR
LUX
IRN
ZAR
Z
MB
SEN
FRA
DNK
DEU
ZWE
MDG
AUT
PER
PHL
JORFJI
GMB
IRL
CAF
ITA
AUS BOL
JAM
SWZ
CHL
ZAF
CMR
OMN
KWT
ESP
GRC
COG
KEN
ISL
SYR
IND
SLV
NLD
MRT
VEN
GUY
PRT
MLT
ARE
PRI
ECU
BFA
SLE
BWA
PAN
SUR
TCD
ALB
GAB
LSO
SWE
NER
SAU
MUS
BDI SLB
TTO
NGA
FIN
URY
HKG
ETH
PAK
BGD
CHN
MLI
MWI
BLZ
EGY
PRY
JPN
BRA
THA
CRI
NOR
CAN
MOZ
LAO
HND
HTI
AGO
KOR
TGO
BGR
USA
GNB
NIC
DMA
STP
SOM
CIV
VCTISR
TUN
GBR
BEN
MAR
GTM
KNA
BTN
COM
MYS
NZL
PNG
SYC
MAC
TON
IDN
NPL
LCA
VUT
CYP

300

-200

-100

0

100

Asymmetry

HP 6.5HP 100

200

300

1500

Asymmetry

KWT
SWZ

HTI

MDG

COL KIR
ECU
MUS
JOR
COM
VUT
GHA
SLV
PER
SYC
KENISRLKA
BDI
ZAR
NIC RWAALBMEX
ATG
SLE
BHR
NZL
ARG
HND
IDN
GUY
KOR
SLB
VEN
HUN
GTM
EGY
MYS
IND
MOZ
MLI
BWA
BFA
OMN
TUR
CYP
BLZ
TCD
MAR
BEN
BRB
CRI
SUR
ZMB
IRN
BGR
AGO
TON
BTN
HKG
CHE
LAO
LSO
MAC
UGA
ARE
BOL
MWI
GRC
JAM
CHL
ISL
GBR
CIV
DNK
PRY
URY
ETH
GNB
PHL
KNA
NAM
MRT
COG
USA
SYR
MLT
TGO
DMA
BEL
ZWE
PRI
THA
STP
SWE
PNG
GAB
CAN
SOM
SEN
NER
VCT
CAF
LUX
AUT
CHN
FIN
LCA
CMR
ZAF
DZA
FRA
JPN
DEU
PAN
PRT
ITA
PAK
BRA
TTO
NPL
AUS
GMB
TUN
FJI
ESP
NOR
SAU
IRL
NGA
NLD

-200

-100

0
Asymmetry

100

200

LAO

1000

BGD

K urtosis

DOM

500

K urtos is0

K urtosis

GHA
BGD
MDG HUN CMR
LUX
CHE
LCA
GTM
SLE
MEX
BTN
HTI
COL
MUS
IRL
PER
BWA
VEN
CRI
VUT
ISL
DEU
KWT ARG
IDNTTO
IRN
DNK
TUR
RWA
KOR
SEN
MRT
DZANAM
ITA
EGY
CAF
ARE
ATG
ALB
LKA
ZAR
ECU OMN
GUY
BOL
BEL
AUT
JPN
ZAF
PRY
KNA
MAR
ISR
SWE
SYC
NICBGR
JOR
COM
BFA
IND
NPL
PAK
SWZ
SLV
JAM
ZWE
BRB
CAN
KIR
ZMB
VCT
CYP
BEN
COG
MOZ
HND
TON
NER
PRT
FRA
THA
BDI
SLB
KEN
GAB
MYS
NOR
PHL
MLI
AGO
ETH
BLZ
TGO
PRI
BHR
USA
PNG
GRC
SYR
CIV
NZL
SUR
BRA
MWI
ESP
CHL
STP
AUS
URY
GMB
CHN
LSO
NLD
HKG
TUN
FIN
GNB
MLT
FJI
MAC
SOM
SAU
GBR
PAN
NGA

500

UGA
DOM
LAO
DMA
TCD

-100

200 400 600 800 10001200

Rolling mixed trend

DOM
KWT
HTI
MUS
SWZ BGDJOR VUT
MDG
GHA
RWA
BHR
IRN
CRI BRB
SYC
MLT
PER
TCD
ECU
ISR
COL
LUX
TON
HUN
VEN
KNA
MAC
GBR
ALB
IND
SLV
EGY
PRI
NIC
BDI
GRC
ARG
COM
THA
CHE
ZAR
BGR
TUN
ARE
SLE
FRA
ATG
CYP
GUY
BEL
IDN
JAM
LKA
LCA
SUR
HND
CMR
LSO
OMN
NZL
AUT
ITA
MAR
HKG
BWA
BTN
URY
BLZ
AGO
SLB
STP
MRT
DEU
DNK
CIV
PHL
CHN
MWI
NLD
ISL
MOZ
ZMB
KOR
NPL
PAN
BEN
ETH
ZAF
AUS
GTM
PNG
NOR
KEN
KIR
PRT
JPN
DMA
ESP
GNB
NAM
NGA
MLI
SYR
CAN
BFATUR
VCT
SAU
GAB
BOL
MEX
PRY
COG
CHL
USA
PAK
TGO
SEN
MYS
FZWE
JI
NER
SWE
DZA
UGA
GMB
BRA
CAF
TTO
FIN
IRL
SOM

0

K urtosis

200

400 600

800 1000

Global mixed trend

-200

-100

0

100

200

300

Asymmetry

Page | 37

Conclusion
The aim of this article is to review the research into macroeconomic volatility along with an analysisof the usual indicators of volatility. The literature on the subject, although very extensive, uses adiverse range of volatility indicators. The techniques used vary in terms of the choice of reference
value and the way in which deviations from the reference value are calculated. Although, in general,
the research is limited to examining the magnitude of volatility, it is possible to extend the analysis
to the effects of the asymmetry of shocks and the occurrence of extreme deviations. In fact, both ofthese aspects are separate dimensions of volatility but largely ignored in the literature on thesubject.
Moreover, we have limited our analysis to indicators designed to reflect the transitory variations –both certain and uncertain – in export revenues. Measuring the uncertainty resulting from thefluctuations of an economic variable requires the use of specific statistical tools, which have not beenaddressed in this paper. We have suggested four possible methods for calculating a reference value.
The first two involve predicting the change in exports based on a mixed trend, including adeterministic and a random component. This method is reliant on selecting an appropriate
functional form. Although observing the behaviour of the series for each country can be timeconsuming when working on panel data, a thorough review of the literature on the order ofintegration of the variable concerned, combined with a spectrum analysis and the use of unit root
tests can provide us with useful information on specifying an appropriate functional form.
Moreover, in our view it is better to opt for a rolling estimated trend than a “global” mixed trend,since it enables the coefficients estimated to vary over time and reflects recent changes in the trendof the series more accurately.
A filtering approach has the advantage of not being based on an a priori formalisation of the change
in the series. It also offers the advantage of being sensitive to breaks in the trend over time.
Problems of leakage or compression can nevertheless introduce/exclude undesirable/relevant
components in order to study the effects of short-term fluctuations. Applied to developing
countries, the choice of a smoothing parameter between 6 and 10 initially appears preferable, since
the trend in these countries seems to fluctuate more significantly.
We have also set out the various possibilities for calculating deviations and the various effects thelatter are assumed to reflect. We have drawn a distinction between the magnitude of volatility –measured by an indicator of average dispersion around the reference value, asymmetry – measured
by the coefficient of asymmetry for the distribution of deviations, and the occurrence of the extreme
values of deviations – measured by the kurtosis of the distribution of deviations. Whilst theindicators for the magnitude of volatility obtained from the various reference values are verystrongly correlated, this is not the case for measures of asymmetry and kurtosis. It thus appears thatevery moment in the distribution of deviations around the reference values expresses a particular
aspect of volatility. The coefficient of asymmetry seems particularly interesting because of its
Page | 38

bidimensional nature. Not only does it express the degree of asymmetry of fluctuations around atrend over a given period, but it also reflects the frequency of extreme deviations.

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Appendices

Page | 43

Appendix A.1 Overview of indicators of volatility and their application in the literature.

Filters

Growth rate/first difference

Indicators

Authors

Phenomenon

Variables concerned (yt)Standard deviation or coefficient of variation of growth rate

Servén (1997), Acemoglu et al.
(2003), Mobarak (2005), Koren andTenreyro (2006), Raddatz (2007), Di
Giovanni and Levchenko (2007),Malik and Temple (2009), Ploeg andPoelhekke (2009)

Variability

GDP/inhabitant, inflation,
terms of trade, actual
exchange rate, black-market
premium, international
interest rate (LIBOR),
development aid/inhabitant,public spending, ratio of
wheat cultivation yields tonational yield

Variance of growth rate over five years

Koren and Tenreyro (2006)

variability

Annual growth rate of work
productivity

Decline in GDP: decrease of more than 1% of the (annual) series log
smoothed by the HP filter (lambda=1000)

Mauro and Becker (2006)

Variability

GDPStandard deviation of the cyclical component, i.e. the standard deviationof the difference between series smoothed by the HP or BK filter andactual series.

Hnatkovska and Loayza (2005);
Guillaumont and Chauvet (2007),
Afonso and Furceri (2010)

Variability

Aid, export revenues, GDP.

Variability

GDP, budget variables(transfers, subsidies, public
spending, tax revenues, etc.)

Page | 44

Average over five years of the ratio of the absolute deviation between theobserved value of export revenues (X) and the value filtered using amoving average process over five years, based on GDP (Y):
j+2

INST

=

X j −
T−2

1
∑
T − 4 j= 3

1
∑ Xk
5 k= j− 2

Dawe (1996)

Variability

Export revenues

Ramey and Ramey (1995)

Uncertainty

Growth rate of GDP

Pritchett (2000), Mobarak (2005)

Variability

Growth rate of GDP per
inhabitant, growth rate ofcapital per worker

Yj∆ y it = α 0 + α 1 t + α 2 t 2 + α 3 t1974 + ϕ dummy 1974 + β y it −1 + γy it − 2 + λ I itinitial + δpop itinitial +εThe standard deviation of the residual it , is seen as a measure ofvolatility reflecting uncertainty. This approach is adopted for the wholeof the sample and for all countries, to produce an estimate of coefficients
specific to each country.

εThe standard deviation of the residual t , obtained using a regression ofyt on a linear trend:
Forecasts/Estimates

yt = α + β t + ε tRolling standard deviation or average absolute deviation of the residual

ε t obtained based on a regression of y on a rolling mixed trend,ty t = α + β t + γ y t −1 + ε tStandard deviation of the error in a regression of FDI over three lags anda temporal trend:

Guillaumont and Combes (2002),
Guillaumont and Chauvet (2007),
Servén (1998)

Lensink and Morrissey (2006)

y t = α + β t + γ 1 y t −1 + γ 2 y t − 2 + γ 3 y t − 3 + ε t

Variability/uncertainty

Development aid, exports,
terms of trade, inflation rate,
relative price of capital,
actual exchange rate, growthrate of GDP.

Uncertainty

FDI/GDP, FDI

Uncertainty

Prices of raw materials,
inflation rate, price of capital
relative to actual exchange
rate

They estimate volatility measures for each country based on thefollowing GARCH(1,1) model:∆ y it = α 0 + α 1 t + α 2 t 2 + β 1 ∆ y it − 1 + β 2 y it − 2 + δ D t + ε it
where t=1…,Tand D the vector of mute quarterly variables.

By imposing the following constraint on conditional varianceσ it2 = γ i , 0 + γ i ,1 ε i2, t − 1 + δ i σ it2 − 1The estimated value of

Dehn (2000), Servén (1998) andAfxentiou and Serletis (2000)

σ̂ it represents the uncertainty of y .
itPage | 45

Global mixed trend

Autocorrelations-0.40-0.20 0.00 0.20 0.40

Venezuela

1

2

3

4

5

6

5

6

5

6

5

6

5

6

5

6

Lags
Bartlett's formula for MA(q) 95% confidence bands

1970

1980

1990

2000

year
Exports (USD, 2000)

Global mixed trend

1

2

3

4
Lags

Rolling mixed trend

1.000e+11 2.000e+11 3.000e+11

Rolling mixed trend

Autocorrelations
-0.50
0.00
0.50

1.000e+10 2.000e+10 3.000e+10 4.000e+10

Appendix A.2. Mixed trends and correlogram of residuals

Bartlett's formula for MA(q) 95% confidence bands

South Korea
Autocorrelations
-0.50
0.00
0.50

Global mixed trend

1

2

3

4
Lags

1970

1980

1990

2000

Year
Exports (USD, 2000)

Global mixed trend

1

2

3

4
Lags

Rolling mixed trend

1.000e+08 1.500e+08 2.000e+08

Rolling mixed trend

Autocorrelations
-0.50
0.00
0.50

0

Bartlett's formula for MA(q) 95% confidence bands

Bartlett's formula for MA(q) 95% confidence bands

Burundi
Autocorrelations
-0.50
0.00
0.50

Global mixed trend

1

2

3

4
Lags

1970

1980

1990

2000

year
Exports (USD, 2000)
Rolling mixed trend

Global mixed trend

Rolling mixed trend

Autocorrelations
-0.50
0.00
0.50

50000000

Bartlett's formula for MA(q) 95% confidence bands

1

2

3

4
Retards

Bartlett's formula for MA(q) 95% confidence bands

Page | 46

4.000e+09

Global mixed trend

3.000e+09

Autocorrelations-0.40-0.20 0.00 0.20 0.40

Kenya

1

2

3

4

5

6

5

6

Lags

1970

1980

1990
year

Exports (USD, 2000)
Rolling mixed trend

Global mixed trend

2000

Rolling mixed trend

Autocorrelations
-0.50
0.00
0.50

2.000e+09

Bartlett's formula for MA(q) 95% confidence bands

1

2

3

4
Lags

Bartlett's formula for MA(q) 95% confidence bands

Page | 47

Appendix A.3. Correlogram of export revenue cycles smoothed by the HP filter
Belize
Autocorrelations (HP 100)
-0.50
0.00
0.50

Autocorrelations (HP 100)
-0.50
0.00
0.50

Argentina

1
2
3
P-value de Mackinnon (lambda=100) = 0,00

4

5

6

1

2

3

Lags

4

5

6

Lags
P-value de MacKinnon (lambda=100)=0,03
P-value de MacKinnon (lambda=6,5)=0,00
Autocorrelations (HP 6,5)
-0.40-0.20 0.00 0.20 0.40

Autocorrelations (HP 6,5)
-0.50
0.00
0.50

P-value de MacKinnnon (lambda=6,5)=0,00

1

2

3

4

5

6

1

2

3

Lags

4

5

6

4

5

6

Lags
Bartlett's formula for MA(q) 95% confidence bands
Bartlett's formula for MA(q) 95% confidence bands

Kenya

Autocorrelations (HP 100)
-0.50
0.00
0.50

Autocorrelations (HP 100)
-0.50
0.00
0.50

Ivory Coast

1

2

3

4

5

6

1

2

3

Lags

Lags
P-value de MacKinnon (lambda=100)=0,01$

P-value de MacKinnon (lambda=100)=0,003

P-value de MacKinnon (lambda=6,5)=0,001
Autocorrelations (HP 6,5)
-0.50
0.00
0.50

Autocorrelations (HP 65)
-0.50
0.00
0.50

P-value de MacKinnon (lambda=6,5)=0,00

1

2

3

4

5

6

1

2

3

Lags

4

5

6

4

5

6

Lags

Bartlett's formula for MA(q) 95% confidence bands

Bartlett's formula for MA(q) 95% confidence bands

Autocorrelations (HP 100)
-0.40 -0.20 0.00 0.20 0.40

Venezuela

Autocorrelations (HP 100)
-0.50
0.00
0.50

South Korea

1

2

3

4

5

6

1

2

3

Lags

Lags
P-value de MacKinnon (lambda=100)=0,004

P-value de MacKinnon (lambda=100)=0,00
P-value de MacKinnon (lambda=6,5)=0,00
Autocorrelations (HP 6,5)
-0.40 -0.20 0.00 0.20 0.40

Autocorrelations (HP 65)
-0.50
0.00
0.50

P-value de MacKinnon (lambda=6,5)=0,00

1

2

3

4

5

6

1

2

Lags
Bartlett's formula for MA(q) 95% confidence bands

3

4

5

6

Lags
Bartlett's formula for MA(q) 95% confidence bands

Page | 48

Appendix A.4 Comparative evolution of deviations (in % of trend values)
Argentina

-40

-.2

-20

0

in % of trend
020

in % of trend
.2

40

60

.4

Belize

1970

1980

Year

1990

2000

1980

1985

1990

year

1995

2000

Rolling mixed trend

Global mixed trend

Rolling mixed trend

Global mixed trend

HP 6.5HP 100HP 6.5HP 100Ivory Coast

-.6

-.2

-.4

-.1

in % of trend
0.1

in % of trend
-.2
0
.2

.2

.4

.3

Burundi

2005

1970

1980

1990

2000

1970

1980

1990

year

2000

year

Rolling mixed trend

Global mixed trend

Rolling mixed trend

Global mixed trend

HP 6.5HP 100HP 6.5HP 100

-.4

-.2

-.2

0

0

in % of trend.2.4

in % of trend
.2
.4

.6

Venezuela.6South Korea

1970

1980

1990

2000

1970

1980

year

1990

2000

year

Global mixed trend

Rolling mixed trend

Global mixed trend

Rolling mixed trend

HP 6.5HP 100HP 6.5HP 100Page | 49