sus ainabili y
A icle
The P obabili y Dis ibu ion o Wo ldwide Fo es A eas
Ra ael González-Val 1,2


Ci a ion: González-Val, R.
The P obabili y Dis ibu ion
o Wo ldwide Fo es A eas.
Sus ainabili y 2021,13, 1361.
h ps://doi.o g/10.3390/su13031361
Academic Edi o : Sil es e Ga cia
de Jalon
Recei ed: 15 Decembe 2020
Accep ed: 19 Janua y 2021
Published: 28 Janua y 2021
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2021 by he au ho .
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
1Facul ad de Economía y Emp esa, Campus Pa aíso, Uni e sidad de Za agoza, 50005 Za agoza, Spain;
a aelg@uniza .es
2Ins i u d’Economia de Ba celona (IEB), Facul a d’Economia i Emp esa, Uni e si a de Ba celona,
08034 Ba celona, Spain
Abs ac :
This pape analyses he p obabili y dis ibu ion o wo ldwide o es a eas. We ind mode a e
suppo o a Pa e o- ype dis ibu ion (powe law) using FAO da a om 1990 o 2015. Powe laws
a e common ea u es o many complex sys ems in na u e. A powe law is a plausible model o
he wo ld p obabili y dis ibu ion o o es a eas in all examined yea s, al hough he log-no mal
dis ibu ion is a plausible al e na i e model ha canno be ejec ed. The andom g ow h o o es
a eas could gene a e a powe law o log-no mal dis ibu ion. We s udy he change in o es co e age
using pa ame ic and non-pa ame ic me hods. We iden i ied a sligh con e gence o o es a eas o e
he ime e iewed; howe e , andom o es a ea g ow h canno be ejec ed o mos o he dis ibu ion
o o es a eas. The e o e, ou esul s gi e suppo o heo e ical models o s ochas ic o es g ow h.
Keywo ds:
o es s; FAO da a; p obabili y dis ibu ion; powe law; Pa e o dis ibu ion; log-no mal
dis ibu ion; exponen ial dis ibu ion; a e o change; s ochas ic o es g ow h
1. In oduc ion
A cu en eme ging en i onmen al conce n is he loss o o es a ea in many de eloped
coun ies. Economic and popula ion g ow h equi es inc easing amoun s o esou ces (such
as land and imbe ). I hese esou ces a e no enewed, o egene a ion is no adequa e,
one migh expec a g adual deple ion o esou ces o e ime.
A e se e al decades o wo ldwide de o es a ion, ecen da a epo s good news.
In ecen yea s, he cu en a es o de o es a ion ha e diminished in many coun ies. The
Global Fo es Resou ces Assessmen (FRA) 2020, elabo a ed on by he Food and Ag icul u e
O ganiza ion (FAO) o he Uni ed Na ions [
1
], highligh s ha “ he a e o ne o es loss
dec eased subs an ially o e he pe iod 1990–2020 due o a educ ion in de o es a ion in
some coun ies, plus inc eases in o es a ea in o he s h ough a o es a ion and he na u al
expansion o o es s.”
This changing end om a dec ease o an expansion in o es s was de ined as a o es
ansi ion by Ma he [
2
] and can be exp essed in e ms o he en i onmen al Kuzne s
cu e [
3
]. Empi ical e idence suppo ing he o es ansi ion is inc easing (as epo ed
in [
4
–
7
]). Ne e heless, mos o hese s udies a e case s udies, and he a e o a coun y’s
o es a ea depends on coun y- and a ea-speci ic idiosync a ic ac o s [
8
]. These ac o s
include anspo cos s and ade issues, changes in land use, mig a ions om u al o
u ban a eas, ag icul u al sec o p oduc i i y (which educes he p essu e on a able land),
ene gy di e si ica ion (which educes ene gy dependence on wood uel), clima e change,
and changes o he mindse o indi iduals who a e becoming inc easingly conce ned abou
he p ese a ion o na u e.
Ra he han ocusing on a pa icula case o s udy, ou app oach he e is global. We
aimed o analyse he p obabili y dis ibu ion o wo ldwide o es a eas and o sea ch
o consis en s a is ical pa e ns in o es -a ea- equencies. This pape con ibu es o
he li e a u e in se e al ways. Fi s , using FAO yea ly da a om 1990 o 2015, we will
desc ibe he a ia ion in he equency dis ibu ion o wo ldwide o es a eas o e ime.
Sus ainabili y 2021,13, 1361. h ps://doi.o g/10.3390/su13031361 h ps://www.mdpi.com/jou nal/sus ainabili y
Sus ainabili y 2021,13, 1361 2 o 19
Ou benchma k model is a powe law. Mandelb o in oduced he idea ha one o he
main cha ac e is ics o “na u e” was ha i possessed so-called “scaling laws” (o “powe
laws”) [
9
], which is a p ope y ela ed o he ac al s uc u e o na u e, wi h he wo d
“na u e” designa ing bo h physical geog aphy and human geog aphy [
10
]. The e o e,
powe laws a e common ea u es o many complex sys ems and a e applied in s udies
on a ied en i onmen al- ela ed phenomena, such as he in ensi y o ea hquakes [
11
,
12
],
losses caused by loods [
13
], p ecipi a ion [
14
], o es i es [
15
] and he size dis ibu ion o
na ional ca bon dioxide emissions [
16
]. They ha e also been applied in human geog aphy
o analyse ci y and coun y size dis ibu ions [17,18].
Using he me hod o Clause e al. [
19
], we ind ha he powe law is a plausible i in
all yea s, bu he e is a plausible al e na i e model—namely, he log-no mal dis ibu ion.
I he p obabili y dis ibu ion o o es a eas ollows a powe law, he ela ionship be ween
magni ude and equency could be sa is ac o ily i ed by a s aigh dec easing line wi h
a nega i e slope. This s iking empi ical egula i y could ha e impo an empi ical, he-
o e ical, and policy implica ions. Howe e , o ou knowledge, his issue has emained
unexplo ed om ei he a heo e ical o an empi ical poin o iew.
Secondly, exploi ing ou da a’s yea ly empo al dimension, we s udy he beha iou
o he a es o o es a ea changes. In his case, we es whe he he e is any ela ionship
be ween a coun y’s ini ial o es a ea and i s a e o change: do la ge o es a eas show
highe a es o change, o on he con a y, a e hei de o es a ion a es highe ? In pa icula ,
we empi ically es andom (o s ochas ic) o es a ea g ow h (i.e., he change in o es
co e age is independen o he ini ial o es a ea), which could gene a e bo h Pa e o and
log-no mal dis ibu ions. Al hough we ind e idence o a sligh con e gence in o es a eas
o e he pe iod conside ed, ou esul s suppo andom g ow h in o es a eas om 1990
o 2015 o mos o he dis ibu ion o o es a eas, indica ing ha no sys ema ic pa e n o
change can be iden i ied a a coun y le el.
The pape is o ganised as ollows: Sec ion 2in oduces he ma e ials and me hods
used. Sec ion 3shows he esul s, con aining he s a is ical analysis o he p obabili y
dis ibu ion o wo ldwide o es a eas and he analysis o i s e olu ion o e ime. Las ly,
Sec ion 4discusses he main esul s and concludes.
2. Ma e ials and Me hods
2.1. Da a
Fo es s da a come om he FAO s a is ics (FAOSTAT) conce ning o es land by
coun y. Al hough o es s do no con o m o geopoli ical bounda ies ( o example, he
Amazon ain o es spans nine coun ies), i is common o assess o es a ea changes by
coun y. As o es s a e managed by coun ies, any changes in o es a eas can e lec
coun y-le el o es policies.
FAO de ines o es land as “land spanning mo e han 0.5 hec a es wi h ees highe
han 5 me es and a canopy co e o mo e han 10 pe cen , o ees able o each hese
h esholds in si u.” Chen e al. highligh he limi a ions p esen ed by FAO’s concep o
o es land [
20
]. The main one is ha he e m “ o es ” in he FRA epo s ep esen s
a ype o land use a he han physical ees compa ed o he sa elli e-based land co e
da ase s. Thus, an a ea can be classi ied as a o es i i is egis e ed as “ o es ” land use,
e en i he e is no ee. Da a a e collec ed om FAO membe coun ies h ough he annual
FAO ques ionnai e on land use, i iga ion, and ag icul u al p ac ices. The ques ionnai e
design p ocess in ol ed use s, na ional co esponden s, and expe s om a ious echnical
backg ounds. S a ing om he Global Fo es Resou ce Assessmen [
21
–
25
] da a on o es
a ea o he yea s 1990, 2000, 2005, 2010 and 2015, FAO p o ides annual da a ia linea
in e pola ion o ob ain a comple e ime se ies om 1990 o 2015, which is he pe iod
conside ed in his s udy. Thus, we can es ima e he yea -by-yea s a is ical dis ibu ion.
Table 1shows he sample sizes o each yea and he desc ip i e s a is ics. The numbe
o obse a ions ep esen s he numbe o coun ies included in he sample, which sligh ly
inc eased o e ime. Fo es land is epo ed in 1000 hec a es. Values o he a e age o es
Sus ainabili y 2021,13, 1361 3 o 19
a ea and i s s anda d de ia ion a e qui e pe sis en , bu a sligh dec ease in bo h s a is ics
can be obse ed o e ime. This dec ease is obse ed in he i s hal o he 1990s, and
in he las se e al decades, he mean o es a ea has s abilised abo e 18 million hec a es.
The maximum alue co esponds o he Russian Fede a ion ( o me USSR in 1990 and
1991) in all yea s, while he minimum o es a ea is always eco ded a he Fa oe Islands.
Ou sample includes all coun ies wi h no size es ic ion bu , al hough he numbe o
coun ies in he sample is high ( anges om 196 in 1990 o 223 in 2015), we acknowledge
ha he FAO da a includes only a pa ial lis o coun ies because i does no p o ide
in o ma ion on all coun ies wo ldwide because some coun ies a e non- ep esen ed in he
FAO Regula P og amme. Mo eo e , hese da a could be biased owa ds some a eas on he
plane (and some o es ypes) as no ed in FAO epo s using hese da a o assess global
o es s. The e o e, ou esul s a e es ic ed o a subse o he plane . Ne e heless, in 2015
mos da a we e epo ed om coun ies hemsel es (a o al o 155 epo s)–coun ies ha
con ain 98.8% o he wo ld’s o es s acco ding o he FAO da a.
Table 1. Fo es land: desc ip i e s a is ics by yea .
Yea Obse a ions
(Coun ies) Mean Fo es Land S anda d
De ia ion Minimum Maximum
1990 196 21,062.6 81,025.61 0.083 849,424.4
1991 199 20,708.56 80,373.09 0.083 849,563.7
1992 217 18,957.31 75,040.84 0.083 809,013.6
1993 219 18,751 74,635.55 0.083 809,045.5
1994 219 18,717.82 74,556.07 0.083 809,077.3
1995 219 18,684.64 74,477.48 0.083 809,109.2
1996 219 18,651.46 74,399.77 0.083 809,141.1
1997 219 18,618.28 74,322.94 0.083 809,172.9
1998 219 18,585.09 74,247 0.083 809,204.8
1999 219 18,551.91 74,171.95 0.083 809,236.6
2000 220 18,434.55 73,938.98 0.083 809,268.5
2001 220 18,413.77 73,869.62 0.083 809,172.8
2002 220 18,392.99 73,801.48 0.083 809,077.1
2003 220 18,372.21 73,734.57 0.083 808,981.4
2004 220 18,351.43 73,668.89 0.083 808,885.7
2005 220 18,330.65 73,604.43 0.083 808,790
2006 221 18,232.26 73,473.26 0.083 810,059.1
2007 221 18,216.81 73,499.95 0.083 811,328.3
2008 221 18,201.36 73,527.28 0.083 812,597.4
2009 221 18,185.91 73,555.25 0.083 813,866.5
2010 221 18,170.47 73,583.87 0.083 815,135.6
2011 221 18,155.5 73,567.33 0.083 815,094.6
2012 223 18,098.42 73,222.85 0.083 815,053.6
2013 223 18,082.80 73,207.06 0.083 815,012.6
2014 223 18,067.19 73,191.62 0.083 814,971.5
2015 223 18,051.57 73,176.52 0.083 814,930.5
No e: Uni : 1000 ha. Sou ce: FAO Fo es Resou ce Assessmen s, FAOSTAT.
The quali y o da a also imp o ed o e ime. Keenan e al. ound ha es ima es o abou
60% o global o es a ea in 2015 we e epo ed o be based on da a o he highes quali y
(e.g., using emo e sensing da a), while his igu e was 57% in 1990 [
26
]. This imp o emen
in da a quali y is especially signi ican in ce ain a eas, such as he opical coun ies [
27
],
al hough se e al me hodological issues pe sis . Na u al a ia ion in o es g ow h and es ima-
ion e o s in o es in en o ies can cause unce ain y in o es g ow h and de elopmen al
p edic ions [
28
]. E en pixel-based compa isons o land co e maps buil using emo e sensing
da a can e eal spa ial disag eemen and unce ain y [
29
]. This p oblem usually a ises in he
measu emen o na u al esou ces, and se e al echnical me hods ha e been de eloped o
add ess his issue; o some eal-li e case s udies o unce ain y on sus ainabili y see [
30
–
35
].
Focusing on o es land da ase s, Chen e al. compa e i e global land co e da ase s and
Sus ainabili y 2021,13, 1361 4 o 19
Global Fo es Resou ces Assessmen s o e eal unce ain ies in he global o es changes in he
ea ly 21s cen u y, inding ha hese da ase s displayed subs an ial di e gences in o al a ea,
spa ial dis ibu ion, la i udinal p o ile, and annual a ea change [
20
]. Ne e heless, Chen e al.
conclude ha an inconsis en de ini ion o o es a eas is no he majo ac o d i ing he
inconsis encies in he o e all global o es a ea change, and acknowledge ha he FRA epo s
a e he mos comp ehensi e o es assessmen da ase s, which a e widely used o o es
condi ions popula iza ions, policy guidance, and land co e da a accu acy alida ions [20].
2.2. Powe Laws and Cu e Fi ing
Le
S
deno e he o es a ea (measu ed in hec a es) by coun y. I o es a ea is
dis ibu ed acco ding o a powe law, also known as a Pa e o dis ibu ion, he densi y
unc ion is p(S) = a−1
SS
S−a
∀S≥S, and he complemen a y cumula i e densi y
unc ion P(S)is P(S) = S
S−a+1
∀S≥S, in which a>0 is he Pa e o exponen
(o he scaling pa ame e ), and
S
is he numbe o o es hec a es in he coun y a he
unca ion poin , which is he lowe bound o he powe law beha iou .
Taking na u al loga i hms, we ob ain a linea speci ica ion:
ln R=ln A−aln S+u, (1)
whe e
u
ep esen s a s anda d andom e o (
E(u)=
0 and
Va (u)=σ2
) and
ln A
is
a cons an . The g ea e he coe icien
ˆ
a
, he mo e homogeneous o es a eas a e ac oss
coun ies. Simila ly, a small pa ame e (less han 1) indica es a hea y- ailed dis ibu ion.
F om Equa ion (1), i seems easy o es ima e he Pa e o exponen because i is jus he
slope o a line i ed by O dina y Leas Squa es (OLS). Howe e , his eg ession analysis
used commonly in he li e a u e can p esen some p oblems [
36
]. The main one is ha he
Maximum Likelihood (ML) es ima o is mo e e icien i he unde lying s ochas ic p ocess
is eally a Pa e o dis ibu ion [
37
,
38
]. Fu he mo e, bo h [
37
] and [
19
] highligh ha he
OLS es ima es o he Pa e o exponen a e subjec o sys ema ic and po en ially la ge e o s.
Finally, his p ocedu e is s ongly biased in small sample sizes [39].
The e o e, o o e come hese limi a ions we use we use an inno a i e me hod p o-
posed by Clause e al. o es ima e powe laws, based on he (ML) es ima o o he Pa e o
exponen [19]:
ˆ
a=1+n n
∑
i=1
ln Si
S!,∀Si≥S,
whe e
n
is he numbe o da a poin s. Sample size is an impo an issue o he ML
es ima ion. The ML es ima o ’s p ope ies a e consis ency, no mali y, and e iciency, bu
only when he sample size app oaches in ini y. In he ield o powe laws es ima ion, [38]
showed ha he a iance o he es ima es ob ained wi h he ML es ima o is no ably
lowe han ha o he es ima es using a linea i on he i s i e bins in he equency
dis ibu ion. In ac , he ML es ima o has been shown ma hema ically o be he minimum
a iance unbiased es ima o [
40
]. The e o e, he ML es ima o has a lowe a iance han
any o he unbiased es ima o o all possible alues o he scaling pa ame e . This makes
he ML es ima o he mos accu a e and obus me hod o es ima ing he powe law
scaling pa ame e [
41
]. The e o e, e en i he sample size is low, ML is less biased han
o he es ima o s.
Clause e al. [
19
] p opose an i e a i e me hod o es ima e he adequa e unca ion
poin (
S
). The exponen
a
is es ima ed o each
Si≥S
using he ML es ima o (boo s apped
s anda d e o s a e calcula ed wi h 1000 eplica ions). Then, he Kolmogo o –Smi no (KS)
s a is ic is compu ed o he da a and he i ed model. The
S
lowe bound ha is inally
chosen co esponds o he alue o Si o which he KS s a is ic is he smalles .
Sus ainabili y 2021,13, 1361 5 o 19
Clause e al. [
19
] p oposed se e al goodness-o - i es s (al e na i e s a is ical es s o
check whe he a a iable is Pa e o dis ibu ed a e a ailable; o ins ance, see [
42
]). In he
same way as B zezinski, we used a semi-pa ame ic boo s ap app oach [
43
]. This p o-
cedu e is based on he i e a i e calcula ion o he KS s a is ic o 500 boo s ap da ase
eplica ions. This me hod samples om obse ed da a and checks how o en he esul -
ing syn he ic dis ibu ions i he ac ual da a as poo ly as he ML-es ima ed powe law.
Thus, he null hypo hesis is he powe law beha iou o he o iginal sample o
Si≥S
.
Ne e heless, his es has an unusual in e p e a ion because we can always i a powe
law ega dless o he ue dis ibu ion om which ou da a we e d awn. Clause e al. [
19
]
ecommend he conse a i e choice ha he powe law is uled ou i he p- alue is below
0.1: “ ha is, i is uled ou i he e is a p obabili y o 1 in 10 o less ha we would me ely
by chance ge da a ha ag ee as poo ly wi h he model as he da a we ha e”. The e o e,
his p ocedu e only allows us o conclude whe he he powe law is a plausible i o he
da a. Finally, we compa e he linea powe law i wi h he i p o ided by o he non-linea
s anda d s a is ical dis ibu ions, he log-no mal and exponen ial dis ibu ions. The densi y
unc ions o hese dis ibu ions a e
p(S) = 1
1−e cln S−µ
σS√2πσ2e−(ln S−µ)2
2σ2 o he log −no mal and p(S) = e−λSλeλS
o he exponen ial. We use Vuong’s model selec ion es o make bila e al compa isons
be ween he powe law and he o he dis ibu ions. Al hough he e a e speci ic es s
designed o compa e he i p o ided by a powe law and a log-no mal dis ibu ion [
44
],
Vuong’s es allows o he compa ison be ween any wo dis ibu ions. The es is based
on he no malised log-likelihood a io; he null hypo hesis is ha he wo dis ibu ions
a e equally a om he ue dis ibu ion, while he al e na i e is ha one o he es
dis ibu ions is close o he ue dis ibu ion. High p- alues indica e ha one model canno
be a ou ed o e he o he , while low alues indica e ha one o he wo dis ibu ions
p o ides a be e i o he ue dis ibu ion. I he null hypo hesis is ejec ed, he sign o
he no malised log-likelihood a io indica es which one o he wo compa ed dis ibu ions
is close o he empi ical da a.
2.3. Pa ame ic and Non-Pa ame ic Empi ical Models o Change in Fo es Co e age
Again, le
Si
be he o es a ea (measu ed in hec a es) o he coun y
i
a ime
and
le
Changei
be i s loga i hmic change in o es co e age; hen
Changei =ln Si −ln Si −1
.
One possible issue ela ed o he change in o es co e age is ha i migh change simply
because he coun y’s a ea changed o e ime. To check whe he his scena io could be
he case, we use coun y a ea da a om FAO o compu e he change a e o coun y’s a ea.
Mos o he a es a e ze o since coun ies’ bounda ies usually do no change o e ime.
Only in 187 cases (3.4% o he o al), we ob ained a non-ze o a e o change in land a ea.
We hen calcula e he co ela ion be ween he a e o change in o es co e age and he a e
o change in he coun y a ea in he same yea , when he la e is non-ze o: Spea man’s ho
= 0.0038. Fu he mo e, we also un a es in which he null hypo hesis is ha bo h a es a e
independen , and he p- alue o he es is 0.9614. The e o e, we can conclude ha changes
in o es co e age a e no signi ican ly d i en by changes in coun y a ea.
Nex , we de ine
gi
as he no malized a e o change (by sub ac ing he con empo a y
mean and di iding by he s anda d de ia ion in he ele an yea ). Ra es o change a e
no malised because we a e conside ing a panel o a es o change om di e en yea s.
The hypo hesis we aim o es is he andom (o s ochas ic) g ow h o o es a eas, ha is,
whe he he a e o change o he o es a eas is independen o i s ini ial a ea. Fi s , we
conside he ollowing pa ame ic model o change in o es co e age:
gi =µ+β1ln Si −1+β2(ln Si −1)2+φj+δ +ui , (2)

Sus ainabili y 2021,13, 1361 6 o 19
whe e
φj
a e coun y ixed e ec s,
δ
a e ime ixed e ec s, and
ui
is he esidual e m,
which we assume o be iden ically and independen ly dis ibu ed o all coun ies, wi h
E(ui )=
0 and
Va (ui )=σ2∀i
,
. The speci ica ion includes a squa e e m o he ini ial
o es a ea (a quad a ic unc ion) o cap u e non-linea i y in he ela ionship be ween
change in o es a ea and ini ial size. No e ha Equa ion (2) could be easily ex ended o
include any o he a iables ha can ha e an impo an in luence on he g ow h o o es s,
om human ac ions o clima e change [
45
,
46
]. Ne e heless, as ou main in e es is o es s
he andom (o s ochas ic) g ow h o o es a eas and no o analyse he di e en d i e s o
change in o es co e age, he model in Equa ion (2) does no include addi ional eg esso s
and he
βj
a e he key coe icien s, cap u ing he e ec o he ini ial o es a ea on he a e
o change.
Howe e , al hough he
βj
coe icien s help o de ec non-linea i ies, a pa ame ic
eg ession is no necessa ily he bes way o add ess such non-linea ela ionships. Some
au ho s [
47
] ha e highligh ed he ad an ages o he non-pa ame ic app oach o e he
s anda d pa ame ic one. Mainly, non-pa ame ic me hods do no impose any s uc u e on
unde lying ela ionships ha may be non-linea and may change o e ime (no need o
es ic he ela ionship o being s a iona y).
The e o e, we also pe o m a non-pa ame ic analysis using ke nel eg essions [
48
].
This consis s o aking he ollowing speci ica ion:
gi=m(si)+εi,
whe e
gi
is again he no malized a e o change (by sub ac ing he con empo a y mean
and di iding by he s anda d de ia ion in he ele an yea ) and siis he loga i hm o he
i h coun y’s o es a ea
(si=ln Si)
. Ins ead o making assump ions abou he unc ional
ela ionship
m
,
ˆ
m(s)
is es ima ed as a local mean a ound poin
s
and is smoo hed using
a ke nel, which is a symme ical, weigh ed, and con inuous unc ion in
s
. The ke nel used
is an Epanechniko , and he bandwid h is se using Sil e man’s ule o humb. Thus,
his non-pa ame ic es ima e allows he a e o change o a y wi h he ini ial o es a ea
o e he en i e dis ibu ion. We un he ke nel eg ession o each pe iod and o a pool
om 1990 o 2015, using he Nada aya–Wa son me hod o es ima e ˆ
m(s). As a obus ness
check, we e-es ima ed he ke nel eg ession using he LOcally WEigh ed Sca e plo
Smoo hing (LOWESS) algo i hm ins ead o he Nada aya–Wa son me hod and iden i ied
simila esul s ( hese esul s a e a ailable om he au ho upon eques ). As he a es
o change a e no malised, i he change was independen o he ini ial o es a ea, he
non-pa ame ic es ima e would be a s aigh line on he ze o alue and alues di e en
om ze o would in ol e de ia ions om he mean: signi ican highe - han-ze o alues
would indica e di e gence (wi h la ge o es a eas showing a es o change highe han
hose o he smalle ones), while nega i e es ima es would poin o con e gence wi h he
la ges o es a eas showing a es o change lowe han hose o he smalle uni s.
3. Resul s
3.1. The P obabili y Dis ibu ion o Wo ldwide Fo es A eas
We use he yea ly FAO da ase o es ima e he p obabili y dis ibu ion o wo ldwide
o es a eas by yea om 1990 o 2015 by i ing a powe law o each pe iod o ou yea ly
sample o coun ies. Da a in e pola ion could gene a e some doub s abou he obus ness
o he annual da a se . Fu he mo e, one possible conce n wi h ou analysis migh be
ha he change in he lis o coun ies migh b ing abou bias o ou esul s. Thus, as a
obus ness check, in Appendix A, we e-es ima e he main esul s using only he FRA
pe iodical da a, conside ing a ixed lis o coun ies.
Figu e 1shows he esul s o ou selec ed yea s: 1990, 2000, 2010, and 2015 ( he
esul s o all o he yea s a e a ailable om he au ho upon eques ). The da a, plo ed as
a complemen a y cumula i e dis ibu ion unc ion (CCDF), a e i ed by a powe law, and
i s exponen is es ima ed using he ML es ima o . Fo illus a i e pu poses, he log-no mal
dis ibu ion is also i ed o he da a by ML ( he blue do ed line). The op imal lowe
Sus ainabili y 2021,13, 1361 7 o 19
bound o bo h dis ibu ions is es ima ed using he me hod p o ided by Clause e al.’s [
19
]
me hod. The black line indica es he powe law beha iou o he uppe ail dis ibu ion.
Sus ainabili y 2021, 13, x FOR PEER REVIEW 7 o 20
s. The ke nel used is an Epanechniko , and he bandwid h is se using Sil e man’s ule
o humb. Thus, his non-pa ame ic es ima e allows he a e o change o a y wi h he
ini ial o es a ea o e he en i e dis ibu ion. We un he ke nel eg ession o each pe iod
and o a pool om 1990 o 2015, using he Nada aya–Wa son me hod o es ima e
()
ˆ
ms.
As a obus ness check, we e-es ima ed he ke nel eg ession using he LOcally WEigh ed
Sca e plo Smoo hing (LOWESS) algo i hm ins ead o he Nada aya–Wa son me hod
and iden i ied simila esul s ( hese esul s a e a ailable om he au ho upon eques ).
As he a es o change a e no malised, i he change was independen o he ini ial o es
a ea, he non-pa ame ic es ima e would be a s aigh line on he ze o alue and alues
di e en om ze o would in ol e de ia ions om he mean: signi ican highe - han-ze o
alues would indica e di e gence (wi h la ge o es a eas showing a es o change highe
han hose o he smalle ones), while nega i e es ima es would poin o con e gence wi h
he la ges o es a eas showing a es o change lowe han hose o he smalle uni s.
3. Resul s
3.1. The P obabili y Dis ibu ion o Wo ldwide Fo es A eas
We use he yea ly FAO da ase o es ima e he p obabili y dis ibu ion o wo ldwide
o es a eas by yea om 1990 o 2015 by i ing a powe law o each pe iod o ou yea ly
sample o coun ies. Da a in e pola ion could gene a e some doub s abou he obus ness
o he annual da a se . Fu he mo e, one possible conce n wi h ou analysis migh be ha
he change in he lis o coun ies migh b ing abou bias o ou esul s. Thus, as a obus -
ness check, in Appendix A, we e-es ima e he main esul s using only he FRA pe iodical
da a, conside ing a ixed lis o coun ies.
Figu e 1 shows he esul s o ou selec ed yea s: 1990, 2000, 2010, and 2015 ( he
esul s o all o he yea s a e a ailable om he au ho upon eques ). The da a, plo ed
as a complemen a y cumula i e dis ibu ion unc ion (CCDF), a e i ed by a powe law,
and i s exponen is es ima ed using he ML es ima o . Fo illus a i e pu poses, he log-
no mal dis ibu ion is also i ed o he da a by ML ( he blue do ed line). The op imal
lowe bound o bo h dis ibu ions is es ima ed using he me hod p o ided by Clause e
al.’s [19] me hod. The black line indica es he powe law beha iou o he uppe ail dis-
ibu ion.
Sus ainabili y 2021, 13, x FOR PEER REVIEW 8 o 20
Figu e 1. The p obabili y dis ibu ion o o es a eas. No es: FAOSTAT da a, FAO Fo es Resou ce Assessmen s. The da a
a e plo ed as a complemen a y cumula i e dis ibu ion unc ion (CCDF),
()
P SS≥.
The es ima ed Pa e o exponen is qui e pe sis en o e ime wi h a alue a ound 1.8
ac oss all yea s (see Table 2). No only is he scaling pa ame e pe sis en , he h eshold
es ima e is also consis en o e ime wi h alues a ound 8000 in mos yea s. Thus, he
powe law ail s a s om o es a eas ≥ 8,000,000 hec a es, including an a e age numbe
o 61 coun ies by yea . No e ha his op imal h eshold iden i ies he poin o he o es
a ea dis ibu ion a which he da a’s powe law beha iou s a s. We i he di e en dis-
ibu ions o he uppe ail, which means ha ou analysis ocuses only on he coun ies
wi h he la ges o es a eas, and al hough he h eshold and he size o he uppe ail may
a y be ween yea s (no d ama ically, as shown in Table 2), he e a e ew a ia ions in he
sample o coun ies included in he uppe ail. The e o e, al hough some coun ies en e
he sample o FAO da a o e ime (see Table 1), his a ia ion in he se o coun ies should
no cause a signi ican change in ou esul s o he uppe ail dis ibu ion because usually
hey a e small coun ies. Ne e heless, Appendix A demons a es ha ou esul s a e o-
bus ega dless o new coun y en ies in he sample.
Table 2. Powe law i .
Da a Lowe
Bound Pa e o Exponen Powe Law
Tes
Powe Law s.
Log-No mal
Powe Law s.
Exponen ial
S ˆ
a S anda d E o p-Value p-Value p-Value
1990 8201 1.839 0.107 0.544 0.654 0.003
1991 7962 1.834 0.105 0.582 0.627 0.003
1992 7746 1.843 0.105 0.614 0.666 0.002
1993 7613 1.833 0.103 0.708 0.618 0.002
1994 7694 1.829 0.104 0.692 0.590 0.002
1995 7899 1.835 0.105 0.684 0.611 0.002
1996 7822 1.829 0.104 0.752 0.581 0.003
1997 7745 1.824 0.104 0.776 0.551 0.003
1998 7668 1.818 0.103 0.820 0.522 0.003
1999 8224 1.827 0.107 0.798 0.546 0.004
2000 8032 1.839 0.107 0.764 0.610 0.002
Figu e 1.
The p obabili y dis ibu ion o o es a eas. No es: FAOSTAT da a, FAO Fo es Resou ce Assessmen s. The da a
a e plo ed as a complemen a y cumula i e dis ibu ion unc ion (CCDF), P (S≥S).
The es ima ed Pa e o exponen is qui e pe sis en o e ime wi h a alue a ound 1.8
ac oss all yea s (see Table 2). No only is he scaling pa ame e pe sis en , he h eshold
es ima e is also consis en o e ime wi h alues a ound 8000 in mos yea s. Thus, he
powe law ail s a s om o es a eas
≥
8,000,000 hec a es, including an a e age numbe
o 61 coun ies by yea . No e ha his op imal h eshold iden i ies he poin o he o es
a ea dis ibu ion a which he da a’s powe law beha iou s a s. We i he di e en
dis ibu ions o he uppe ail, which means ha ou analysis ocuses only on he coun ies
wi h he la ges o es a eas, and al hough he h eshold and he size o he uppe ail may
a y be ween yea s (no d ama ically, as shown in Table 2), he e a e ew a ia ions in
he sample o coun ies included in he uppe ail. The e o e, al hough some coun ies
en e he sample o FAO da a o e ime (see Table 1), his a ia ion in he se o coun ies
should no cause a signi ican change in ou esul s o he uppe ail dis ibu ion because
usually hey a e small coun ies. Ne e heless, Appendix Ademons a es ha ou esul s
a e obus ega dless o new coun y en ies in he sample.
Sus ainabili y 2021,13, 1361 8 o 19
Table 2. Powe law i .
Da a Lowe Bound Pa e o Exponen Powe Law Tes Powe Law s.
Log-No mal
Powe Law s.
Exponen ial
S^
aS anda d E o p-Value p-Value p-Value
1990 8201 1.839 0.107 0.544 0.654 0.003
1991 7962 1.834 0.105 0.582 0.627 0.003
1992 7746 1.843 0.105 0.614 0.666 0.002
1993 7613 1.833 0.103 0.708 0.618 0.002
1994 7694 1.829 0.104 0.692 0.590 0.002
1995 7899 1.835 0.105 0.684 0.611 0.002
1996 7822 1.829 0.104 0.752 0.581 0.003
1997 7745 1.824 0.104 0.776 0.551 0.003
1998 7668 1.818 0.103 0.820 0.522 0.003
1999 8224 1.827 0.107 0.798 0.546 0.004
2000 8032 1.839 0.107 0.764 0.610 0.002
2001 7958 1.834 0.106 0.800 0.582 0.003
2002 7884 1.828 0.105 0.796 0.555 0.003
2003 8174 1.841 0.108 0.798 0.607 0.002
2004 8171 1.842 0.108 0.800 0.611 0.002
2005 8168 1.843 0.108 0.818 0.615 0.002
2006 8456 1.855 0.110 0.782 0.667 0.002
2007 8475 1.857 0.111 0.752 0.679 0.002
2008 8495 1.860 0.111 0.716 0.693 0.002
2009 8144 1.845 0.108 0.766 0.624 0.002
2010 8138 1.846 0.108 0.824 0.629 0.002
2011 8136 1.847 0.108 0.802 0.637 0.002
2012 9136 1.879 0.115 0.794 0.739 0.002
2013 8594 1.850 0.111 0.836 0.603 0.002
2014 8614 1.852 0.111 0.878 0.615 0.002
2015 8634 1.855 0.111 0.908 0.628 0.002
No es: The lowe bound and he Pa e o exponen a e es ima ed using Clause e al.’s [
19
] me hodology. The powe law es is a goodness-
o - i es . H
0
is ha he e is powe law beha iou o
Si≥S
. The powe law e sus log-no mal es is Vuong’s model selec ion es , based
on he no malized log-likelihood a io: H
0
is ha bo h dis ibu ions a e equally a om he ue dis ibu ion while H
A
is ha one o he
es dis ibu ions is close o he ue dis ibu ion.
The powe law appea s o p o ide a good desc ip ion o he dis ibu ion beha iou .
In con as , he i o he log-no mal dis ibu ion does no seem o be isually appealing,
especially o he highe obse a ions. Ne e heless, isual me hods can lead o inaccu a e
conclusions, especially a he uppe ail because o la ge luc ua ions in he empi ical
dis ibu ion [
49
], so nex we conduc s a is ical es s on he goodness o i . Table 2shows
he esul s o he es s. The p- alues o he es a e always highe han 0.1, con i ming ha
he powe law is a plausible app oxima ion o he da a’s eal beha iou , as we canno ejec
he powe law in any case.
Resul s in Table 2show ha he log-no mal dis ibu ion is a plausible al e na i e
o he powe law ha we canno ejec ( o un he es we used he same lowe bound,
he es ima ed alue co esponding o he powe law). In con as , while he exponen ial
dis ibu ion is clea ly ejec ed wi h low p- alues o he es and a posi i e and la ge
alue (no shown o size es ic ions, bu a ailable om he au ho upon eques ) o he
no malized log-likelihood a io o all yea s. The e o e, using he e minology desc ibed by
Clause e al.’s [
19
], we ob ain mode a e suppo o he powe law beha iou o he c oss-
coun y p obabili y dis ibu ion o o es -a ea- equencies: he powe law is a plausible i ,
bu he e is also a plausible al e na i e.
3.2. An Analysis o Change in Fo es Co e age
The abo e esul s sugges wha can be conside ed as a snapsho o he p obabili y
dis ibu ion o wo ldwide o es a eas om 1990 o 2015. Fo each yea , we es ima ed he
Pa e o exponen and conduc ed a goodness-o - i es o indica e he plausibili y o a powe
Sus ainabili y 2021,13, 1361 9 o 19
law model. Fu he mo e, ou es ima es e ealed ha he exponen o he powe law o
he uppe - ail dis ibu ion emained s able h oughou he conside ed pe iod. Howe e ,
his inding does imply ha he dis ibu ion o wo ldwide o es a eas emains s a ic.
To illus a e his poin , Figu e 2shows he empi ical densi y unc ions o he i s and
las pe iods in ou sample (1990 and 2015), which was es ima ed using adap i e ke nels.
Con a y o he p e ious analysis whe e we ocused on he uppe - ail dis ibu ion beha iou ,
all obse a ions a e conside ed; he a e age es ima ed h eshold was 8145 (see Table 2),
which in loga i hmic e ms co esponds o a alue oughly equal o 9. Al hough he shape
o he empi ical dis ibu ion is qui e simila in bo h pe iods, we can obse e a loss o densi y
in bo h he uppe and lowe ails and, as a esul , an inc ease in densi y in he cen al alues.
The e o e, in 2015 we ind ha he empi ical dis ibu ion o o es a ea co e age was sligh ly
mo e e en han in ea lie yea s.
Sus ainabili y 2021, 13, x FOR PEER REVIEW 10 o 20
he cen al alues. The e o e, in 2015 we ind ha he empi ical dis ibu ion o o es a ea
co e age was sligh ly mo e e en han in ea lie yea s.
Figu e 2. Empi ical densi y unc ions o o es a ea co e ages.
Economic li e a u e on he dis ibu ion o inancial asse s [50], i m size [51], and ci y
size [52] usually concludes ha a Pa e o- ype dis ibu ion is gene a ed by a andom
g ow h p ocess (in he i m and ci y size li e a u e, his hypo hesis is called Gib a ’s law).
Fu he mo e, o he plausible, al e na i e models ha canno be ejec ed in he p e ious
empi ical analysis, such as he log-no mal dis ibu ion, can also gene a e andom a es o
change in o es a eas. The hypo hesis ha is usually es ed s a es ha he a e o change
o he a iable is independen o i s ini ial size (i.e., he unde lying g ow h model is a
mul iplica i e p ocess).
In ecology and bioeconomics he common app oach is also o ea changes in he
esou ce popula ion as a andom a iable [53]. The bioeconomics li e a u e ypically as-
sumes ha he s anda d de ia ion is p opo ional o he esou ce popula ion and com-
bines his wi h a mean g ow h componen ollowing a B ownian mo ion ha can be geo-
me ic [54] o logis ic [55]. The e o e, ou analysis o he a es o change can be conside ed
as a es o he models o s ochas ic o es g ow h.
We ca y ou a dynamic analysis o he change in o es a eas using he pa ame ic
and non-pa ame ic me hods, as he FAO da ase enables us o calcula e he yea ly a es
o change in o es a eas by coun y. Table 3 shows he esul s o he OLS es ima ion o
he pa ame ic model o Equa ion (2). The i s column co esponds o a simple bi a ia e
eg ession, which indica es a nega i e and signi ican impac o he ini ial o es a ea on
he change in o es co e age. In column 2, we add coun y and yea ixed e ec s o con-
ol o empo al shocks and unobse ed cha ac e is ics ha can a y a a coun y le el.
The es ima ed coe icien o he ini ial o es a ea is no signi ican , al hough i emains
nega i e. Finally, in column 3 we use he ull speci ica ion, including bo h he log-le el o
ini ial o es a ea and i s squa e e m, and he coun y and ime ixed e ec s. The es ima ed
coe icien s a e signi ican ly di e en om ze o, wi h 1
ˆ0
β
< and 2
ˆ0
β
>, which implies
a U-shaped ela ionship be ween change in o es co e age and ini ial o es a ea, poin ing
o a obus non-linea ela ionship be ween bo h a iables.
0.00 0.05 0.10 0.15
Densi y
-5 0 5 10 15
Fo es land (ln scale)
1990 2015
Figu e 2. Empi ical densi y unc ions o o es a ea co e ages.
Economic li e a u e on he dis ibu ion o inancial asse s [
50
], i m size [
51
], and ci y
size [
52
] usually concludes ha a Pa e o- ype dis ibu ion is gene a ed by a andom g ow h
p ocess (in he i m and ci y size li e a u e, his hypo hesis is called Gib a ’s law). Fu he mo e,
o he plausible, al e na i e models ha canno be ejec ed in he p e ious empi ical analysis,
such as he log-no mal dis ibu ion, can also gene a e andom a es o change in o es
a eas. The hypo hesis ha is usually es ed s a es ha he a e o change o he a iable is
independen o i s ini ial size (i.e., he unde lying g ow h model is a mul iplica i e p ocess).
In ecology and bioeconomics he common app oach is also o ea changes in he e-
sou ce popula ion as a andom a iable [
53
]. The bioeconomics li e a u e ypically assumes
ha he s anda d de ia ion is p opo ional o he esou ce popula ion and combines his
wi h a mean g ow h componen ollowing a B ownian mo ion ha can be geome ic [
54
] o
logis ic [
55
]. The e o e, ou analysis o he a es o change can be conside ed as a es o he
models o s ochas ic o es g ow h.
We ca y ou a dynamic analysis o he change in o es a eas using he pa ame ic
and non-pa ame ic me hods, as he FAO da ase enables us o calcula e he yea ly a es
o change in o es a eas by coun y. Table 3shows he esul s o he OLS es ima ion o
he pa ame ic model o Equa ion (2). The i s column co esponds o a simple bi a ia e
eg ession, which indica es a nega i e and signi ican impac o he ini ial o es a ea on
he change in o es co e age. In column 2, we add coun y and yea ixed e ec s o
con ol o empo al shocks and unobse ed cha ac e is ics ha can a y a a coun y le el.
Sus ainabili y 2021,13, 1361 16 o 19
Finally, Figu e A3 shows he non-pa ame ic esul s o a pool wi h all a es o change
be ween wo consecu i e pe iods; now, he e a e 756 o es a ea– a e o change pai s.
G aph (a) in Figu e A3 shows he ke nel eg ession o he a e o change o he pool. The
es ima ed mean a e dec eases wi h he ini ial o es land, bu he es ima ed alues a e
smoo he han hose p esen ed in Figu e 4a. G aph (b) in Figu e A3 displays he s ochas ic
ke nel es ima ion o he dis ibu ion o no malised a es o change, condi ional on he
dis ibu ion o ini ial o es a eas a he same da e, showing a e y simila plo o ha shown
in Figu e 4b. Again, mos o he bi a ia e densi y is concen a ed a ound he ze o alue.
O e all, esul s in his Appendix using a balanced sample o coun ies and excluding
in e pola ed alues o o es a eas a e qui e simila o hose ob ained in p e ious sec ions
conside ing he ull sample o coun ies by yea p o ided by FAO. The e o e, we con i m
ha ou esul s do no appea o ha e been d i en by changes in he sample size o
in e pola ion issues.
Sus ainabili y 2021, 13, x FOR PEER REVIEW 18 o 20
(a) Ke nel es ima e o he a e o change in o es a eas
(b) S ochas ic ke nel
Figu e A3. Change in o es co e age om 1990 o 2015; includes 756 obse a ions (FRA da a and a ixed sample o coun-
ies).
Re e ences
1. FAO. Global Fo es Resou ces Assessmen 2020—Key Findings; UN Food and Ag icul u e O ganiza ion: Rome, I aly, 2020. A aila-
ble online: h p://www. ao.o g/documen s/ca d/en/c/CA8753EN (accessed on 8 July 2020).
2. Ma he , A.S. The o es ansi ion. A ea 1992, 24, 367–379.
3. P a , A.S.P.; Walke , R. Regional in e dependence and o es “ ansi ions”: Subs i u e de o es a ion limi s he ele ance o local
e e sals. Land Use Policy 2010, 27, 119–129.
4. B ay, D.B.; Klepeis, P. De o es a ion, Fo es T ansi ions, and Ins i u ions o Sus ainabili y in Sou heas e n Mexico, 1900–2000.
En i on. His . 2005, 11, 195–223.
5. Fa ley, K.A. Pa hways o o es ansi ion: Local case s udies om he Ecuado ian Andes. J. La . Am. Geog . 2010, 9, 7–26.
6. F aye , J.; Mülle , D.; Sun, Z.; Mun oe, D.K.; Xu, J. P ocesses Unde lying 50 Yea s o Local Fo es -Co e Change in Yunnan,
China. Fo es s 2014, 5, 3257–3273.
-0.5 0.0 0.5 1.0
Ra e o Change
-5 0 5 10 15
Fo es Land (ln scale)
Pool 1990-2015
Figu e A3.
Change in o es co e age om 1990 o 2015; includes 756 obse a ions (FRA da a and a ixed sample o coun ies).

Sus ainabili y 2021,13, 1361 17 o 19
Re e ences
1.
FAO. Global Fo es Resou ces Assessmen 2020—Key Findings; UN Food and Ag icul u e O ganiza ion: Rome, I aly, 2020; A ailable
online: h p://www. ao.o g/documen s/ca d/en/c/CA8753EN (accessed on 8 July 2020).
2. Ma he , A.S. The o es ansi ion. A ea 1992,24, 367–379.
3.
P a , A.S.P.; Walke , R. Regional in e dependence and o es “ ansi ions”: Subs i u e de o es a ion limi s he ele ance o local
e e sals. Land Use Policy 2010,27, 119–129. [C ossRe ]
4.
B ay, D.B.; Klepeis, P. De o es a ion, Fo es T ansi ions, and Ins i u ions o Sus ainabili y in Sou heas e n Mexico, 1900–2000.
En i on. His . 2005,11, 195–223. [C ossRe ]
5.
Fa ley, K.A. Pa hways o o es ansi ion: Local case s udies om he Ecuado ian Andes. J. La . Am. Geog .
2010
,9, 7–26.
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