Stochastic Demographic Dynamics,

Long-memory and Economic Growth∗

Tapas Mishraa and Alexia Prskawetzb

a International Institute for Applied Systems Analysis (IIASA)A-2361 Laxenburg, Austriae-mail: mishra@iiasa.ac.at

b Institute for Mathematical Methods in EconomicsVienna University of Technology &Vienna Institute of Demography

Austrian Academy of Science, 1040 Vienna, Austria

This version: March, 2008

Abstract

This paper exploits for the first time the temporal dynamic features of de-mographic system in economic growth model. Standard practice in economicgrowth models has been to treat population growth as either constant, exoge-nous, and/or stationary to ensure that long-run equilibrium characteristics of aneconomy is not affected by non-mean converging demographic shocks. Rather,the stand-in practice has been to allow (endogenous) technological change orhuman capital to play crucial role in determining long-term equilibrium. Weargue in this paper against the stationary assumption of demographic system(broadly the population growth) and show empirically that population growthfollows a long-memory process. Using a stochastic version of the Solow-Swanmodel, we demonstrate analytically as well as with simulation example the ef-fect of long-memory demographic shocks on economic growth. The generalconclusion is that stochastic demographic system has had discernible impact oneconomic growth in the past four decades and that the source of long-memory in

∗This research was completed when the first author was visiting Vienna University of Technologyas a Marie Curie RTN student. The research is (partly) financed by the European Commissionunder the RTN Grant project No. HPRN-CT-000234-2002. Any opinion, findings, and conclusionsor recommendations expressed in this material are those of the authors and do not necessarily reflectthe views of the European Commission.

1

economic growth is due to stochastic demographic system. Finally, highly non-linear relationship is observed between long-memory demographic system andgrowth volatility among four sets of geographically clustered economies, viz.,Asia, Europe, Africa and Latin America & Offshore countries, during 1960-2000.JEL Classification: C14, C31, E61, J11, O47Key words : economy-demography interaction model, long-memory, economicgrowth, stochastic demography.

2

1 Introduction

In the last three decades since Dickey and Fuller (1979), economic growth literature (boththeoretical as well as empirical) has experienced interesting turning points, specifically whilesynchronizing the theoretical and empirical demand of time’s explicit role in model formu-lation. A quick survey into theoretical growth literature would reveal that incorporating theconsistent temporal dynamic features of interacting variables in the system did not alwaysfit into their agenda. ‘Time’ as a broad concept, often gets appended over ‘span’ in, for in-stance, (dynamic) overlapping generations type model (in terms of two/three generations).Similarly, in standard Solow-Swan or endogenous growth models, time is not allowed to playexplicit role. Empirically, one would come across the problems of non-constant parameter,shifts and changes which would most likely affect the long-term equilibrium. That is be-cause, the economy as a system evolves over time and it is highly likely that during evolutionsome amount of shocks would have crept into the system and that such shocks (endogenousor exogenous) will exert long-term effects. To a large extent, time has often found ‘dormant’not ‘dominant’ role in theoretical growth literature. An important determinant of growththrough which time was expected to have played pivotal role is population growth, or morebroadly the demographic system. A common feature in most growth models (classical ormodern) is that population growth is constant, exogenous and stationary. Such an assump-tion has so far helped delivering steady state equilibrium characteristics of the economy.One can classify this kind of assumption of population as a perfect foresight/complete in-formation model - that is an economy would behave correctly and would lead to expectedchanges in future pattern only when population growth is assumed to be stationary. Puttingotherwise, the demographic system is unrealistically assumed not to be subject to environ-mental and behavioral changes and therefore any stochastic shock that might arise in thissystem and could have long-term destabilizing effect, is prima facie eliminated.

However, the recent resurrected interests in the theoretical and empirical treatmentof population growth problem point to the important connection between economic growthand stochastic shocks in demographic variables (e.g., Azomahou, Diebolt and Mishra, 2007;Mishra, 2006). Wider attention is now laid on the consequences of demographic dynam-ics, for instance, age-structure variation on long-term economic growth and planning (e.g.,Boucekkine et al., 2002). The centrality of stochastic demographic shocks in the building-blocks of growth models has conventionally been downplayed and has often been assumedto exert only short-run mean converging effect on economic growth. However, realisticdemographic and economic situations reveal a different story altogether.1 For about twocenturies since Malthus (1798), the possibility of long-term mean non-convergent effect ofdemographic shocks on economic growth has hardly been given a thorough treatment. Asa result the evolutionary characteristics of demographic system, particularly, the impact oftemporal shocks on the interacting system was often sidelined.

It is well known that any system that exists always evolves over time. During evo-lution, the system is subjected to shocks of varying magnitudes, the impact of which maybe carried long-time in the future. The persistent shock may or may not converge to thelong-run mean depending on the complexity of interaction of the system with others anddepending in part on the strength of innovation shocks. Not surprisingly, a shock’s ubiq-uity in any economic system calls forth for a more tractable and prudent treatment of itsconstituents. Despite its all pervasiveness, conventional economic thinking persuaded us to

1Historical evidence is replete with the fact that the greatest of human sufferings come morethrough our unwillingness to admit a bleak future due to our own action and lack of proper visionthan by our inability to solve a particular problem.

3

the point that shocks are transitory in nature; long-run equilibrium is assured after a fewinitial spurts which might occur due to some demand side disturbances in the economy.Productivity shocks might tend to exert long-term impacts in the economy, but the conven-tional wisdom treated this as being an impossibility. Economic thinking has evolved andby now we recognize that shocks, however small in magnitude it may be, cannot be ignoredwhen it accumulates in the system. Granger and Joyeux (1980) proved that shocks of smallmagnitudes, under certain distributional condition and upon accumulation over a long timecan give rise to ‘long-memory characteristics in the sense that the shock persisting in theeconomy will take very long time (even may not) to return to its steady state values.

As noted earlier an important assumption in the growth theoretic literature is thatpopulation growth is exogenous - implying either constancy and/or stationarity2 of popu-lation in the model. A stationary population in this case implies that environmental andbehavioral factors are held constant such that immigration and migration do not occur, thegrowth rate is zero, an equal number of individuals are present in each cohort and thatthe probability of dying in each cohort is the same. Similarly, a stable population is onein which rates of age-specific mortality and fertility are constant for a particular period oftime. When a stationary model is used for stable population, the predicted life expectancyis more a measure of fertility than mortality. This is because stationary models cannotcompensate for the rate of natural increase in stable population. Since environmental andbehavioral factors do account for a large of fluctuations/uncertainty in population growth,there is ample reason to assume that a stochastic shock arising from either behavioral rela-tion of demography with economic system or due to external environmental changes - arecorrelated. Therefore, non-stationary (of certain degree) demographic system is a rathernatural characterization than a stationary system.

Although demographic variables, like population growth, do not exhibit frequent fluc-tuations as is the case for most macroeconomic aggregates, the underlying mechanism thatdescribes demographic process is exceedingly complex, characteristically non-linear and mayresult in a pattern which exhibits chaotical dynamics (Prskawetz and Feichtinger, 1995).This is not surprising given that demographic process experiences many endogenous shiftsover time due to the interaction mechanism between demography and the economy and thefeedback effect following them. These typical features are however infrequently studied inthe theoretical literature where temporal variation of the demographic variables come intothe picture. A clear delineation between the two therefore is warranted.

In view of the above, this paper exploits for the first time the temporal dynamicfeatures of demographic system in economic growth model. We argue in this paper againstthe stationary assumption of demographic system (broadly the population growth) and showempirically that population growth follows a long-memory process. In particular we showthat the long-run mean and variance of output growth is a function of stochastic demographicshocks. From standard growth theoretic perspective, we build a stochastic version of Solow-Swan model where stochasticity is shown to arise in the demographic process. In this contextwe show that since demography and economic growth are inextricably linked, a shock to thedemographic system will be directly captured by the economic system as well. Second, therole of stochastic shocks explaining economic growth fluctuations is considered. The buildingblock of this idea is that while the behavior of stochastic shocks are studied in a regressionas a product of the analysis contained in error term, here the shocks enter the regression as adependent variable and also as an explanatory variable. Therefore, the impact of exogenous

2The statistical definition of stationarity, which we will be defined in the following section isbased on the characteristics of lack of correlation between current and remote past values of avariable.

4

system (which we consider here as random and with long-run impact on the economy) in theregression equation help us explain the interaction mechanism of stochastic shocks and theeconomic system. Taking the long-memory estimates of total population (estimated for theperiod 1960-2000), we empirically illustrate with cross-sectional regression of four groups ofcountries (Asia, Africa, Europe, Latin America and Offshore) that stochastic demographicshocks have had significant impact on economic growth and vice versa.

The rest of the chapter is planned as follows. Section 2 describes the source ofstochasticity in demographic system. Section 3 provides a theoretical framework explicatingthe demographic-economic growth linkage with long-memory dynamics. Empirical constructand illustration are provided in Section 4. Finally, Section 5 concludes with our findingswith a discussion of the results and implications for development objectives.

2 Source of stochasticity

Among several reasons investigating why demographic system might exhibit stochasticity,Shaffer’s (1987) proposition is of interest to our purpose here. Shaffer (1987) argues that de-mographic stochasticity is caused by chance realizations of individual probabilities of deathand reproduction in a (finite) population. Moreover, demographic stochasticity is causedby environmental stochaticity from a nearly continuous series of small or moderate pertur-bations that similarly affect the birth rates of individuals within each age in a population.Additionally, catastrophes are large environmental perturbations that produce sudden ma-jor reductions in population size. Whether in terms of simple environmental or behavioralchanges or large scale environmental changes (in terms of catastrophes), demographic sys-tem by and large is often subject to continuous perturbations. Sometimes the effect of theseperturbations go unnoticed, however, as it happens, these small disturbances contribute tolarge scale environmental and demographic changes in the long-run.

Demographic system does not evolve independent of economic and environmentalsystems. The continuous interaction between demographic, economic and environmentalfactors thrives on continuous feedback effects from one system to the other and that rendersthe relation highly non-linear (Azomahou and Mishra, 2007). Interesting to note that theevolutionary mechanism of economic and demographic systems are different. To a reasonableextent it can be said that unless certain demographic standards are met (say minimumpopulation with standard replacement rate), the internal dynamics of economic system willbe severely upset. For instance, given the speed of demographic growth, specifically in termsof age-structured population growth, declining fertility and mortality, increase in educatedmass, and faster population aging, economic functioning must take recourse to cognizablepolicy changes so as to restore balance for a ‘sustainable demography-economic’ growth inthe longer run. The natural occurrence of demographic and economic system interactionprovides reason to stress that any endogenous shift occurring in one system would havelong-term consequences for the other. This convention has been stressed, nonetheless, inmost of the population literature.

Interestingly, in spite of differences in evolution structure of different systems, mostof them share common properties. For instance, initial modeling strategies of macroeco-nomic/financial time series were in line with random walk, that is they are cyclic and non-periodic. However, recent strategy stresses that most macroeconomic time series resembleneither random walk nor white noise, suggesting that some comprise or hybrid between therandom walk and its integral may be useful. In a similar vein, for over centuries we haveobserved a cyclical behavior of demographic system, captured in terms of demographic tran-sition. It has also been observed that the demographic states are repetitive after long years,

5

so that gives rise to a kind of long-swing behavior with history dependence property of thesystem. Intuitively, this implies that a particular kind of demographic state tend to settlefor some period of time due to a specific interaction nature of demography-economic state.After saturation, a new demographic state emerges which owes its course due to innovationand development. Once that also gets saturated, the demography-economic system tendto return to old equilibrium path. Typically, this has been summarized in terms of demo-graphic transition and multi-transition demographic states in demographic literature. Forour current exposition purpose, this feature is important as the period and state of transitionthat generally persists for long time before beginning or settling to another, very frequentlyencounters small or large continuous perturbations within that cycle. Technically, it can bedescribed as follows.

At time τ , denote the state of the demographic system as Dτ . The elements of Dτ

comprises of points (assuming it to be infinite) pi described at each point in time, t. Wehave in mind that τ consists of broader time span where t forms the space of τ . Now, ourconjecture is during the transition of demographic state, Dτ from sayDτ1 to Dτ2 , continuousperturbations might have occurred at different t interpoints. Over the span τ1 to τ2, theperturbations accumulate and over time the sum of perturbations are likely to produce anon-mean convergent distribution of the system. While this is a natural possibility, extantgrowth and demography theories assumed this to be stationary, in the sense that shocks getsmoothed out and summed perturbations always tend to converge in the long-run. Fromnon-stationary time series and empirical macroeconomic and growth literature, we now knowthat this is not true. In the strict sense, this is rather an unrealistic characterization of anevolving system. General observation of a seemingly smooth aggregate time series may infact underline many hidden non-linear and non-stationary dynamics. For demographic statetransition, this is often the case. The transition probabilities (from one state to the other)are very naturally non-constant, because assumption of constancy of transition probabilitiesover time precludes consideration of factors which may affect economic movements and itsimpact on demographic system through changes in transition probabilities.

The source of stochasticity in the demographic system which naturally lends to strongpersistence of the system (in the sense of high dependence of observations across distant timepoints) emerge from varied sources, as described above. The basic source, as we discussed, isfrom non-stationary transition of the system from one state to the other. During transitionsome ‘trend-shift’ type of changes, such as catastrophes might occur. Apart from thispossible natural (exogenous) changes, a continuous perturbation always occur which mostlyemerge from the endogenous interaction mechanism with economic and environment system.Together it implies that constancy and stationarity of demographic system in conventionalgrowth modeling needs a revisit and that a new growth theoretic mechanism needs to bedesigned with persistent demographic shocks with non-stationary feature where stationarydemographic system is one of the limiting cases.

3 Model

To demonstrate how stochastic demographic system affects long term economic growth, weapproach the problem in two steps. First, it is shown that k-period output growth (where kis time period in future) is a function of stochastic long-memory demographic system. Highermemory implies slower convergence to long-run mean. Therefore, the greater the degree ofstochastic demographic shocks the slower would be the system’s and its interacting agent’s(the economy) convergence to long-run mean level (defined under steady state values). Thesecond approach is based on the building of a stochastic version of Solow-Swan model with

6

long-run demographic characteristics. For the purpose of elicitation, we define below theconcept of long-memory in population and economic growth and depict the consequences ofstochasticity in the demography-economic system.

3.1 Long-memory in population growth

Population growth (nt) is defined as the difference in fertility (Ft) and mortality (Dt) rateswhile accounting for net migration rate (Nmt) in the economy. This is written as

nt = (Ft −Dt) +Nmt (1)

where for simplicity assume that Nmt = 0.3 Most growth models still assume that thedemographic system basically is stable (or broadly defined under stationarity). Even if it isoften assumed it is still unknown whether the demographic system (with continuous inter-action with the aggregate economy) will tend to converge to a stable long-run equilibriumlevel. Indeed, while stability (or stationary) assumption of demographic system is an appar-ent possibility, it may not be the only possibility. Rather stationary demographic systemcan be a limiting case of a more general data generation process where shocks of variedmagnitude could exhibit different degrees of convergence to the long-run mean. One of theinteresting properties of such process is that the shock converges rather slowly and thatthe slow convergence have different impacts on economic growth. The standard Markovianassumption of demographic shocks in growth models downplays the possibility of non-meanconvergent or very slow convergent effect of shock on growth thus directly allowing the modelto acquire built-in uncertainty which is not desirable for forecasting. Moreover, looking atthe recent demographic trends and the impact of endogenous demographic shifts (possiblyleading to chaotic dynamics in economic growth, Prskawetz and Feichtinger, 1995), the sta-bility/stationarity assumption ignores much of the demographic dynamics and its role ineconomic growth, now and in the future.

Stochastic shocks, thus which can possibly be present in the demographic systemexhibit various persistence properties depending on how fast shocks taper-off over time.The slow or fast decay of stochastic demographic shock (e.g., in the population growth) hasimportant implications for economic growth: volatile demographic system induce volatilityin output (both over time and across space). To know how it works, we define below long-memory in population growth.

Definition 1 Denote d as the integration parameter lying on the real line, k as the laglength. Now, suppose that nt is a process with autocovariance function γ(k) ∼ C(k)k2d−1 ask → ∞, C(k) 6= 0, where k defines the lag between current and distant observations. Thennt is a long-memory process if the autocovariance function decays slowly to the mean valueover time.

Now let’s define the fractional integrated autoregressive (AR) moving average (MA)process (ARFIMA (p,d,q)) for p AR order and q MA order along with fractional d for ntwith/without feedback effect from the economy. Two cases are distinguished. In the puredemographic model, dynamics of population growth (nt) is determined by its autoregressive

3The assumption of zero migration rate helps us distinguish between pure population growthshocks and external shocks such as migration variation. In any case, we presuppose that positivemigration effects may at times net out or add-in to the persistent shocks the magnitude of whichcan vary depending on the size of migration and seasonality.

7

and moving average structure such that nt at time t is led by its own evolutionary charac-teristics and by the evolution of some stochastic shocks. In this setting, no feedback effectaccrues from economy to demography and the converse, but the dynamics is governed byexogenous growth generating mechanism. Interaction model (as described below), however,may contain terms which explain structural dynamics of nt even while being explained byARMA features. The models are presented as:

• Pure demographic model :

(1 − L)dΦ(L)(nt − µ0) = µ1 + θ(L)ut (2)

• Interaction model :

(1 − L)dΦ(L)(nt − µ0) = µ1 + βxt + θ(L)ut (3)

where ut ∼ iid(0, σ2u); Φ(L) = (1 − φ1L − · · · − φpL

p): AR(p); θ(L) = (1 + θ1L + · · · +θqL

q): MA(q). Furthermore, µ0 and µ1 are intercepts which affect the demographic systemdifferently (due to the way they enter the system). xt is the vector of explanatory variableswhich may include lagged dependent variable and others with a possibly distributed lagstructure. Formally, (1 − L)d can be described by power series expansion mechanism:

(1 − L)d =

∞∑

j=0

(−1)j(

d(d− 1)(d− 2) . . . (d− j + 1

j!

)

(4)

where d(d−1)(d−2)...(d−j+1j! is the binomial coefficient which is defined for any real number

d and non-negative integer j. The most intuitive exposition of (1 − L)d for a time seriesis via their infinite order moving average (MA) or autoregressive (AR) representations. Inthis instance, expressing MA(∞) of (1 − L)d for the time series would mean that we havean expression:

∑

∞

j=0 hjLj, where h0 = 1 and

hj =−dΓ(j − d)

Γ(1 − d)Γ(j + 1)=j − d− 1

jhj−1, j ≥ 1. (5)

It may be noted that the AR and MA representations of fractionally differenced seriesillustrate the central properties of fractional process, particularly long-range dependence,which is the focus of this paper. Allowing d to lie on the real line renders a flexible mechanismto display varied shock convergence properties. For instance, with d = 0 in (1 − L)d = ǫtwithout AR and MA components, that is with a fractional Gaussian process, the systemexhibits ‘short memory’ because the autocorrelations in this case is summable and decayfairly rapidly so that a shock has only a temporary effect completely disappearing in thelong run. Long memory and persistence is observed for d > 0. In this case, the shock affectsthe historical trajectory of the series. However, greater is the magnitude of d, stronger is thememory and greater is shock persistence. For d ∈ (0, 0.5), the series is covariance stationaryand the autocorrelations take much longer time to taper-off. When d ∈ [0.5, 1), the series isa mean reverting long-memory and non-stationary process. This implies even though remoteshocks affect the present value of the series, this will tend to the value of its mean in thelong run. For −1/2 < d < 0 the process is known to be fractionally over-differenced. In this

8

case, there is still short memory with summable autocovariances, but the autocovariancesequence sums to 0 over (−∞, +∞). For d < −1/2 the series is covariance stationary butnot invertible. And finally, when d ≥ 1 the series is nonstationary and exhibits ‘perfectmemory’ or ‘infinite memory’. There is no unconditional mean defined for the series in thiscase. The process defined by this value of d is non-stationary and non-mean reverting. Inthis case, the mean of the series has no measured impact on the future values of the process.Important to note that for 0.5 ≤ d < 1, there is no variance, so the existence of the meanwould need to be established in each case. There is a median, however. So this case may bedescribed by ’median reversion’. The results are summarised in Table 1.

Table 1: Fractional components and their interpretation

d Interpretation0 : Short-memory population growth, log population is I (1)1 : Non-stationary population growth, log population is I (2)

< 0, 0.5 > : Long-memory population growth, log population is I (d+1)

3.2 Long-memory in output

To know how stochastic demographic system may induce volatility in economic growth, wewould like to show that the conditional mean and variance of k−period aggregate output isa function of stochastic memory of demographic system.

To show this, assume a simple economic-demography growth model (EDM):

yt = γnt−1 + ηt (6)

where ηt ∼ (0, σ2η). This model provides us with simple explanation that past population

growth impacts output at time t because it takes time for the economy to feel the effect ofpopulation rise (which is presented in terms of net resource users). Similarly, by adding 1on each side of this equation, we get a relation that implies output at period t+ 1 dependson population growth at period t. Thus, there is a feedback effect from economy to thedemographic system and the converse.

Proposition 1 Under the assumption of feedback effect between economy and demographicsystem and given the EDM model as above, long memory in output growth, yt, can berepresented by the long memory in the demographic system.

Proof of proposition 1

Two mechanisms are explored: First, we explore the link between output growth (yt) andaggregate population growth (nt) and second, with age-structured population.

Method 1: Aggregate population

Let nt in EDM model (Equation 6) follow an ARFIMA(p,d,q) process:

(1 − φ1L− φ2L2 − · · · − φpL

p)(1 − L)dnt= (1 + θ1L+ θ2L

2 + · · · + θqLq)ǫt

(7)

9

with usual definitions: E[ηtǫs] = σ2τǫ if t = s, 0, otherwise. We assume φ(L) 6= 0 for

z ≤ 1. Re-write (7) as: φ(L)−1(1 − L)−dθ(L)ǫt. Now, denote ω(L) = φ(L)−1, whereω(L) =

∑

∞

i=0 ωiLi and use the identity ω(L)φ(L) = 1 to find the unknown coefficients re-

cursively:

ω0 = 1,ω1 = φ1ω0,ω2 = φ1ω1 + φ2ω0 and so,ωi = φ1ωi−1 + · · · + φpωi−p for i = p, p+ 1, · · · .

Further, using Binomial expansion of (1−L)d, we have (1−L)−d =∑

∞

i=0(d+j−1)···(d+1)d

i! Li.Multiplying (1 − L)−d and φ(L)−1, we get

(1 − L)−dφ(L)−1 =

∞∑

j=0

zjLj (8)

wherezj = 1 if j = 0,

zj = ω0(d+j−1)···(d+1)d

j! +

ω1(d+j−2)···(d+1)d

(j−1)! + · · · + ωj−1d+ ωj , otherwise.

And finally, for j ≥ 0, describeψj = zj + zj−1θ1 + · · · zj−qθqwith z−1 = · · · = z−q = 0.

Denote the cumulative k−period output, yt at time t as Y(k)t and the MA(∞) representation

of yt as

yt =∞∑

j=0

ψjǫt−j. (9)

To know the effect of stochastic demographic shocks on aggregate output, we utilize EDMand MA(∞) representations such that:

Y(k)t =

∑kl=1 yt+l = γ.

∑kl=1

∑

∞

j=0 ψjǫt−1−j+l +∑kl=1 ηt+l

Denoting ζ(k)i ≡ ψi + ψi−1 + · · · + ψi−(k−1),

we can writeY

(k)t = γ.

∑

∞

i=0 ζ(k)i ǫt−1−j+l +

∑kl=1 ηt+l

The conditional expectation of Y(k)t then equals:

E

[

Y(k)t

]

= γ.

∞∑

i=0

ζ(k)i ǫt−1−j+l (10)

and the conditional variance of k−period cumulative output is:

V art

(

Y(k)t − E

[

Y(k)t

])

= γ2.∑k

l=1

(

ζ(k)k−l

)2

σ2ǫ + γ.

∑(k)k−l σǫη + σ2

η

(11)

10

Expressed in terms of ζ, aggregate output is a function of stochastic memory compo-nent both in mean and variance, thus completing the proof of long memory in output dueto long-memory in aggregate population. �

Method 2: Components of population

Here we make use of Granger’s (1980) aggregation theorem which states that if compo-nents of an aggregate follow certain distributional structure (for example, beta distribution),then aggregation of them exhibits a long-memory pattern. In our case, let Vt denote thevector of population of different age structure, viz., Vt = ((Age0− 14)t, ..., (Age65+)t). Themodel is

yt = β0 + β1Age(0 − 14)t + β2Age(14 − 29)t+...+ β5Age(65+)t + ǫt

(12)

A compact expression of the above equation is

yt = β0 +∑

i

βiVit + ǫt (13)

where i refers to population of age group 0 − 14, 15 − 29, ..., 65+ respectively. ǫt ∼ I(0). Ifcomponent of population is assumed to be integrated of order I(d) and contains memorystructure, then Vit can be described by (1−L)dVit which has the following shock expansionmechanism:

Vit = Vi + 1 + dVi,t−1 −d(d−1)

2! Vi,t−2 + ...− (−1)j d(d−1)...(d−j+1)j! Vi,t−j + ..., (14)

with Vi = 0 and d ≥ 0. Noting that (1 − L)d =∑

∞

j=0 hjLj, where hj indicates

declining weights, we can denote this as impulse-response coefficient of Lj. Hamilton (1994)proves that for large j, hj ∼ (j+1)d−1 with d < 1. Given the demographic-economic relation,the growth of output, yt is represented as a function of the impulse-response coefficient hj .Applying aggregation theorem, under certain distributional assumption of component ofpopulation, the aggregate population follows long-memory. Moreover, when each populationcomponent can be expressed in terms of impulse response coefficient, then output as afunction of population component is also expressed by impulse-response, the magnitude andlength of which depends on the estimate of d. On a different note, it can be proved thatpopulation components affect economic growth via a designed ‘transfer function’ mechanism.Under stability assumption of the function, possible stochastic memory in the componentsof population also result in stochastic memory in aggregate output. �

3.2.1 A Stochastic Solow Model

In this section we provide a theoretical construct expounding the relation between long mem-ory and output growth. We use a stochastic version of Solow-Swan model where populationgrowth in the model, instead of being constant, is assumed to be stochastic so that dynamicsof population growth can determine the dynamics of output in the economy. Drawing onthe intuition and construct of long-memory population growth described in the preceding

11

section, we allow population in Solow-Swan model to follow a long-memory data genera-tion process (DGP). The economy is assumed to be closed. The production function of therepresentative agent is given a Cobb-Douglas type:

Yt = AKαt N

1−αt (15)

where 0 < α < 1, Yt is output at time t, Kt is capital input at t. Labor input, Nt governedby the growth of population, nt so that

Nt = (1 + nt)Nt−1 (16)

where population growth, nt, in our system is assumed to follow a long-memory data gen-erating process which evolves as

(1 − L)dΦ(L)nt = Θ(L)ǫt (17)

L is the lag operator as defined before and

(1 − L)d =∞∑

j=0

Γ(j − d)

Γ(j + 1)Γ(−d)Lj (18)

Φ(L) = (1+φ1L+ ...+φpLp) and Θ(L) = (1−θ1L− ...−θqL

q) are AR and MA polynomialsrespectively. Moreover, the investment, It and capital stock equations are described as

Kt+1 = (1 − δ)Kt + It (19)

In the above equation, capital stock is assumed to decline at a constant rate of δ (0 < δ < 1)per period. Given that s is the fraction of Y to be invested, then

It = sYt (20)

Consumption is defined according to

Ct = (1 − s)Yt (21)

Proposition 2 Given a production function of Solow-Swan type where population growthfollows a long-memory data generating process (DGP) (Equations 8 - 12), the output growthin the economy will also follow a long-memory DGP. Long-run convergence of output willbe determined depending on the degree of memory of stochastic population shocks.

Proof of proposition 2

The immediate effect of long-memory population growth on economy’s long-termoutput, consumption and investment growth can be observed by plugging the long-memoryDGP of nt in the production, capital, and consumption equations. Assuming that ψ(L) =(1−θ1L−...−θqL

q)(1+φ1L+...+φpLp) ≡ 14 in equation 17 and substituting it in equation 16 and then in equation

15, we obtain

Yt = AKαt [(1 + (1 − L)−dψ(L)ǫt)Nt−1)]

1−α (22)

4This assumption is not binding but assumed for simplicity.

12

The output per capita, yt = (Yt/Nt) in this case is a function of sequence of shocks, thusregulating the ’efficiency unit of output’ by the stability of shocks. Moreover, since (1−L)d

can be represented by impulse-response mechanism, viz.,∑

∞

j=0(j + 1)d−1, inducting this inequation 22 then depicts

Yt = AKαt [(1 +

∞∑

j=0

(j + 1)1−dψ(L)ǫt)Nt−1)]1−α (23)

Assuming the effect of technology, A, to be constant on Yt, or by assuming thatgrowth in A is caused by population pressure, a unit shock in nt in equation 23 can exhibithow Yt responds to it. Nevertheless, it is clear that depending on the magnitude of d,the behaviour of Nt can determine the nature of output growth in the economy. Now,since consumption and investment are a function of output, the persistence of shocks inoutput, consumption and investment growth in the economy. Denoting, aggregate outputand aggregate consumption at T as QT and CT it can be shown that

∑Tt=1 Yt = f(K,n(d)),

and∑T

t=1 Ct = f(Y, n(d)) where n(d) denotes long-memory population growth.The steady-state growth of output and investment can be derived from the above

characterizations of stochastic output, consumption and investment equations. Note that aswe gradually increase the value of d from 0 till 1, i.e., from stationarity (short-memory) tonon-stationary long-memory, Yt’s response (in equation 16) to stochastic shocks due to highmagnitude of memory will vary widely. The precise effect of long-memory population shockon output is demonstrated in Figure 1. The number of years are indicated on the horizontalaxis and response of output in the vertical axis. It may be observed from Figure 1 that aswe increase the value of d from 0.1 till 0.8, i.e., from stationarity to high non-stationarity,the response of output to such variation also increases over time, viz., from a slow responseto a very steep response as the economy progresses. �

13

Figure 1: Long-memory effect on Output

4 Empirical analysis

4.1 Estimation of d

To provide evidence of fractionally integrated population growth with non-mean convergentshock dynamics, it is important to provide an efficient and unbiased estimation of d. Wefollow this up in the spectral domain and adopt Kim and Phillips’ (2000) modified log-periodogram regression method.

Given the formulation of fractionally integrated population growth process: (1 −L)dnt = ǫtif population growth is governed by the mean of the process nt = nt + ǫt, thenthe spectral density of nt is given by

fn(λ) =σ2ǫ

2π(24)

However, in the presence of persistence of stochastic shocks and continuous pertur-bations, the spectrum is governed by the stochastic memory in the demographic systemequation such that

ln[In(λζ)] = −2dln|1 − eiλζ | + ln(fu(λζ)) + ηj (25)

where the periodogram ordinates (left hand side of the equation, ln[In(λζ)], of popu-lation growth is regressed over the spectral representation of the error term, fu(λζ) and thetransformation of (1 − L)d in the frequency domain |1 − eiλζ |. The ordinates are evaluatedat the fundamental frequencies ζ = 1, ..., ν.

14

Phillips (1999) notes that Eq. 25 is a moment condition and not a data generatingmechanism, and the analysis of this regression estimator is complicated while characterisingthe asymptotic behavior of the discrete fourier transform (dft) FP (λζ) which is centralto determining the properties of the regression residual ηj . A detailed description of thismethod is presented in the Appendix.

A practical problem is the choice of ν, the number of periodogram ordinates to beused in the regression. Geweke and Porter-Hudak (GPH, 1983) suggests that the optimalν = Tα where α = 1/2 and T is the sample size. The choice involves a tradeoff that maybe described as follows. The smaller the bandwidth, the less likely the estimate of d iscontaminated by higher frequency dynamics, i.e., the short-memory. However, at the sametime smaller bandwidth leads to smaller sample size and less reliable estimates. As in thecase of GPH method, the smaller value of α (as in ν = Tα) implies the smaller numberof harmonic ordinates (i.e., the smaller bandwidth) will be used for the estimation of d.Generally, in empirical analysis, preference is given to increasing the value of α to checkfor the consistency of the estimate of d although simulation experiments can confirm thevalidity of the selection. For our purpose, we have used α = 0.60 through α = 0.80 toestimate d. We choose5 α = 0.7 based on a Monte Carlo simulation experiment (see tablebelow) where we have minimum bias for that bandwidth. Davidson’s (2005) TSM softwareis used to carry out the simulation experiment which is built for the GPH model (assumingthat the simulation results will not drastically change if we had used MLPR).

4.2 Empirical framework

From dicussions in Section 3 it is apparent that stochastic demographic shocks have sig-nificant effect on economic growth. To supplement the analytical results, in this sectionwe investigate empirically if and how demographic variability and stochastic demographiclong-memory (is caused by or) has caused growth volatility. For the purpose of illustra-tion we have divided the developed and developing countries into four regional groups, viz.,Asia, Africa, Europe, and Latin America and Offshore countries. Our idea is to perform across-section regression over the period 1960-2003 and investigate how output and demo-graphic system responded to each other over four decades when elements of stochasticityand volatility were considered. The basic framework of the model is as follows:

di,(t,T ) = Γ1(log(GDPi,(t,T )), Zi,(t,T )) + ǫit (26)

log(GDPi,(t,T )) = Γ2(di,(t,T ), Zi,(t,T )) + ψ0it (27)

σ(GDPi,(t,T )) = Γ2(di,(t,T ), Zi,(t,T )) + ψ1it (28)

σ(ngri,(t,T )) = Γ2(σ(GDPi,(t,T )), Zi,(t,T )) + ψ2it (29)

where t = (1, ..., T ), di,(t,T ) is the estimate of long-memory population growth during the pe-

riod (t, ..., T ) for ith country, Zi,(t,T ) represent a vector of other demographic variables, suchas average schooling rate, life expectancy at birth and population density, etc. ǫit and ψit∼ (0, σ2), σ for GDP and ngr are the standard deviation of output and population respec-tively. σ is an indicator of volatility. GDP growth is calculated by taking the log differences

5The estimates of d for other bandwidth are available with the authors though we have notreported in the main text due to space limitation.

15

between period t and T . Equations 26 and 27 describe the relation between long-memorydemographic shocks and economic growth for which causality runs from either direction.Equations 28 and 29 present volatility equations for GDP and population respectively alongwith other regressors summarized in Z variable. These equations can be precisely presentedas instability equations. σ can appear both as dependent and explanatory variables in theregression. For instance, average output growth rate (over the sample span) can be regressedon σ of output growth (which measure instability) and σ of other variables such as humancapital and population density. Note that ǫit ∼ (0, σ2

ǫ ), ψ0it ∼ (0, σ2ψ0), ψ1it ∼ (0, σ2

ψ1),

ψ2it ∼ (0, σ2ψ2) and that they have zero covariance between them.

4.3 Data characteristics

The empirical illustration in this paper is based on a cross-section regression of averageper capita output growth on long memory estimates of total population, growth and demo-graphic instability, population density, and human capital for the four set of geographicallyclustered countries over the period 1960-2003. The country groups are Asia, Africa, Eu-rope, and Latin America and Offshore (See Table 2 for the list of countries in each group).The motivation to subdivide the countries into four groups is that we would like to run across-section regression where cross-sectional units share common shocks (for instance, withrespect to demographic and socio-economic-geographic characteristics). With such a geo-graphically structured country groups, we aim to present a clear relational structure betweeneconomic growth and stochastic demographic system.

Table 2: List of countries.

Asia Africa Europe LatAmericaoffshore

Bahrain Benin Austria ArgentinaBangladesh Burki Faso Belgium AustraliaCambodia Cameroon Bulgaria Bolivia

China CentAfrRepub Denmark BrazilHong Kong Chad Finland Canada

India Cte d’Ivoire France ChileIndonesia Egypt Germany Colombia

Iran Gabon Greece Costa RicaJapan Ghana Hungary CubaJordan Guinea Ireland Dom.Republic

Malaysia Kenya Italy EcuadorNepal Madagascar Netherlands El Salvador

Pakistan Malawi Norway GuatemalaPhilippines Mali Poland HondurasSingapore Mauritius Portugal MexicoS.Korea Morocco Romania New Zealand

Sri Lanka Mozambique Spain PanamaSyria Namibia Sweden Paraguay

Thailand Niger Switzerand PeruTurkey Nigeria UK USA

Vietnam South Africa UruguayTogo

UgandaZambia

ZimbabweN = 21 N = 25 N = 20 N = 21

16

To investigate if volatility in demographic system cause (or is caused by) volatilityin economic growth, we have calculated standard deviation of population growth and percapita GDP growth over 1960-2003 period. World Bank data is used for age-specific andtotal population, and population density. For per capita income at purchasing power parity,we have used Maddison data which is in 1995 US dollar. Educational attainment data arebased on IIASA-VID database (see Lutz et al. 2007a, 2007b).6 Most importantly, thisdata set allows a cross-classification of education data by age groups (in age intervals offive years), and thus allows us to obtain estimates of the full demographic distribution ofeducational attainment. The data on long-memory demographic system concerns estimationof d of aggregate and age structured population growth for different countries. Employingmodified log periodogram regression method of Kim and Phillips (2000), we estimated d forthe four country groups. The estimates have been obtained from Azomahou, Diebolt andMishra (2007).

4.4 Results

The effect of stochastic demographic shocks on economic growth (and the converse) is de-picted by performing both non-parametric and parametric regression. Since there is no apriori reason to impose a specific functional form on how economic growth and demographicshocks affect each other, we have performed a nonparametric regression to check the truefunctional form of their impact.

The centrality of our investigation in fact lies in delineating the relational structureand discerning the effect between economic growth instability (measured by σygr) and long-memory population shocks. Azomahou and Mishra (2007) employed nonparametric panelregression method to study the relation between population age-dynamics and economicgrowth of OECD and non-OECD countries. The authors found significant evidence of non-linearity between age-structure and economic growth over four decades. For the purposeof our present investigation we have performed kernel regression of σygr on d for aggregatepopulation growth for the cross-section of sub-structured country groups . The principalmotivation of this regression is to study if and how stochastic demographic shocks haveinfluenced economic growth instability over the past four decades. Nadayara-Watson non-parametric regression has been employed on the standard deviation of countries’ average percapita GDP growth over four decades on long-memory estimates of aggregate populationgrowth. The converse effect (i.e., regression of long memory aggregate population growthon average GDP growth) has also been tested. The bandwidth for regression is given by:S.n−1/5 where S is the sample standard deviation of the explanatory variable and n is thesample size. The nonparametric plots for each regression are presented in Figures 2-5. Ex-amination of these figures confirms our prediction that long-memory demographic shocksimparted non-linear and instability propelling impact on economic growth, a result whichhas interesting policy implications.

Simple cross-section regression (correcting for heteroscedastic errors) have also beenperformed (equations 26-29) for four country groups. From Table 2 it is evident that ag-gregate population growth had negative impact on economic growth of Asia and Africa,

6Some specific characteristics of the educational attainment data are in order. This humancapital data set was produced in a joint effort by the International Institute for Applied SystemsAnalysis (IIASA) and the Vienna Institute of Demography (VID) and improves enormously onpreviously available data on education in several respects. In contrast to most earlier attemptsto improve data quality, which were concentrated on raising more empirical information or usingeconomic perpetual inventory methods and interpolation.

17

Latin America and Offshore countries and positive but insignificant effect for Europe. Thefindings are closer to empirical expectation: aggregate population growth exerted growthretarding impact for developing countries and growth-enhancing effect in developed coun-tries. Table 2 report results from cross-section regression for the described models and fourcountry groups. The results for each country groups are sub-divided into two parts: resultsfrom average GDP per capita growth rate as dependent variable are reported first, while thesecond set of results are based on growth instability indicator (σygr) as the dependent vari-able. The idea is to study how growth and instability (volatility) are explained by stochasticdemographic factors along with other policy variables.

In general we would expect an empirically negative relation between aggregate popu-lation growth and economic growth. It is also expected that instability in economic growthwould retard growth and thus a negative relation can be gauged between average outputgrowth and σ for output growth. Similarly, given the analytical discussions in the precedingsection, stochastic population growth shocks (represented by estimates of d for total popu-lation) would render growth retarding effect in the past years. We examine the empiricalregularity for each premise in case of the defined four country groups.

A closer look into Table 2 reveals that population growth (ngr) in the past four decadeshad significant negative impact on average GDP per capita growth in Asia, Africa, and LatinAmerica and Offshore countries, but positive effect in Europe. Even though ngr does notreflect the distinct characteristics of the growth of its components, the sign of the aggregateeffect indicates however that, a net rise in population number in developing countries hadgrowth-retarding effect due to heavy diversion of resources for consumption and maintenanceof growing population. For developed countries, positive population growth has been seen ascontributing to long-run growth in view of the faster ageing in these economies. Populationdensity also appears to have rendered negative impact on average economic growth in Asia,Africa, Latin America and Offshore countries during 1960-2003. From growth theory weknow that population density is supposed to exert growth enhancing effect as this could betaken as a proxy of technical change. However, it is important to remember that the growththeoretic assertion is valid under the high density of educated mass where resource and ideageneration forms the centrality in growth. Uneducated mass density is always detrimentalto growth and this is what is reflected in our results of negative effect of population densityon growth over last four decades.

An important aspect of Table 2 concerns the explanation of growth instability dueto volatile demographic system. As indicated before, σygr and σngr have been calculatedfor four groups of countries which indicate growth instability and demographic instabilityrespectively. The second part of regression for each country group in Table 2 concernswith the regression of σygr on σngr and other variables. Without exception, we find thatdemographic instability has had significant effect on economic growth instability for allcountry groups during 1960-2003. It may be noted that the data were converted into naturallogarithmic terms. So the parameter estimates are indicative of elasticity measure. Thesignificant positive sign of σngr for each country group reflects the fact that response ofgrowth volatility to demographic shocks is positive, i.e., as demographic volatility increasesgrowth volatility increases along with, however, the magnitude of response differ widelyacross country groups.

Now, when we analyze the effect of long-memory population (d) shocks on economicgrowth, we find interesting pattern. Aggregate population shocks (d-totpop) had variedimpacts on economic growth for the country groups. Asia experienced negative impactof aggregate population shocks, while Africa experienced positive effect for d-totpop, butnegative age-structured population growth shocks (d014, d1564, d65p. For Europe and

18

Table 3: Cross-section regression results: (Sample 1960-2003)

Dependent variable: Average per capita GDP growth: 1960-2003

ngr d-tpop σygr σngr dens H1529 H3049 H5064 d014 d1564 d65p Const R2

ASIA: Dependent var - Avg GDP percapita growth

AS1 −1.321(0.278)

- - - - - - - - - - 5.677(0.803)

0.322

AS2 - −1.640(1.08)

- - −1.531(0.317)

- - - - - - 7.50(1.217)

0.385

AS3 - −1.520(1.071)

−0.092(0.212)

- −1.390(0.488)

- - - - - - 7.549(1.400)

0.40

AS4 - −1.169(1.092)

- −1.751(1.052)

−0.892(0.491)

0.611(0.312)

- - −0.684(0.821)

2.915(2.114)

−0.103(0.840)

2.982(2.595)

0.69

ASIA: Dependent Var - σygr

AS5 - - - 3.893(1.068)

- - - - - - - 2.728(0.696)

0.35

AS6 - - - 3.574(1.113)

0.631(0.574)

- - - −2.283(1.333)

−2.952(1.454)

−0.189(1.00)

6.875(2.408)

0.62

AFRICA: Dependent var - Avg GDP percapita growth

AF1 −1.023(0.664)

- - - - - – - - - - 3.322(1.640)

0.056

AF2 - 2.997(0.856)

- - −1.172(0.430)

- - - - - - 1.602(1.147)

0.40

AF3 - 2.871(0.807)

−.044(0.99)

- −1.113(0.413)

- - - - - - 1.8469(1.575)

0.41

AF4 - 1.441(0.941)

- 0.452(0.860)

−1.343(0.496)

0.076(0.283)

- - −1.343(1.288)

1.303(1.082)

1.020(0.953)

2.161(1.214)

0.54

AFRICA: Dependent var - σygr

AF5 - - - 2.692(1.815)

- - - - - - - 5.620(1.061)

0.02

AF6 - - - −1.384(3.208)

2.564(1.223)

- - - −8.433(4.788)

3.092(2.434)

−1.101(1.132)

7.726(4.089)

0.21

EUROPE: Avg GDP percapita growth

EU1 0.627(0.702)

- - - - - - - - - - 2.444(0.287)

0.014

EU2 - 0.256(0.689)

- - 0.367(0.622)

- - - - - - 2.352(0.602)

0.385

EU3 - 0.318(0.692)

0.187(0.088)

- 0.494(0.628)

- - - - - - 1.666(0.656)

0.15

EUROPE: Dependent var - σygr

EU5 - - - 4.847(1.976)

- - - - - - - 1.478(0.616)

0.37

EU6 - - - 4.492(1.602)

−0.625(0.793)

- - - −1.750(1.330)

−0.834(0.950)

1.985(1.070)

2.460(2.459)

0.62

LATIN AMERICA & OFFSHORE: Avg GDP percapita growth

LA1 −0.586(0.291)

- - - - - - - - - - 2.678(0.595)

0.17

LA2 - 0.103(0.566)

- - −0.563(0.297)

- - - - - - 2.556(0.802)

0.16

LA3 - 0.108(0.545)

−0.115(0.97)

- −0.602(0.301)

- - - - - - 3.098(0.981)

0.20

LA4 −11.520(0.282)

- - 0.116(0.620)

11.017(4.109)

- - - −0.780(0.503)

2.669(1.073)

0.212(0.602)

0.231(1.904)

0.43

LATIN AMERICA & OFFSHORE:: Dependent var - σygr

LA5 - - - 1.226(1.636)

- - - - - - - 3.573(0.750)

0.021

LA6 17.086(8.265)

- - 1.716(1.753)

−17.296(8.133)

- - - - -()

3.694(1.180)

0.10

Note: Bracketed values are robust standard errors.

Latin America and Offshore countries, although d-totpop appears to be positive, but theyare not significant implying no measurable effect of aggregate population shocks on economicgrowth. This could be because the volatility of demographic system is already captured byσ term in the estimation and thus demographic volatility could well proxy for long-memorypopulation shocks. However, to eke out individual distinctions, both volatility and long-memory variables have been included in the estimation. Further, it may be noted thatthe relation between economic growth and long-memory demographic system could be aresult of some reduced-form equation describing demography-economic system, as is thecase in our stochastic Solow-Swan model. Additional variables in the regression are humancapital for age-structured population. For different country groups, we do not find significant

19

evidence of the effect of age-specific human capital on economic growth, although we findsome positive and significant evidence for Asia (for the age group: 1529 represented by thevariable hcap1529). It is possible that cross-sectional regression under aggregate variablesetting may not reflect the true nature of relationship among dependent and explanatoryvariables. However, the results presented here served our purpose for the explanation ofstochastic/volatile demographic system on economic growth.

5 Conclusion

Due to evolutionary characteristics of the demographic and demographic-economic system,certain stochastic shocks do not always attain long-run convergence. While convergentshocks in any system ensures (long-run) stability, the lack of it is renders trouble for growthplanning and development. This paper took the lead to fully exploit the time series (orevolutionary) dynamic features of the demographic system and analytically showed howa possible stochastic shock could induce non-mean convergent growth shocks in the long-run. We stressed that dynamic temporal features of demographic system should be fullytaken into account while performing long-term economic growth planning. In particular,we found that variation in the degree of persistence also induce a variation in the responseof output in the economy. Stochastic version of Solow-Swan model was considered wherewe introduced long-memory in the population growth which so far has been treated asexogenous and stationary in conventional growth models. A possible source of stochasticity(the long memory) is shown in this paper to come emanate from population growth. Thusthe paper contributed to the growth literature by accommodating historical characteristicsof demographic shocks - big or small random variations accumulated over time - in explaininggrowth variations.

Our empirical strategy followed a deviation from the traditional practice in thatstochastic shocks appear as dependent variable and then entered as explanatory variable inthe regression. We showed that stochastic demographic shocks accumulated in the last fourdecades can be significantly explained by economic growth variations of developing countrieswhile the latter is also significantly explained by stochastic population shocks both in thedeveloped and developing countries during the last 40 years. This provides direct evidenceof the debated issue on how and why stochastic population shocks need to be accounted foras historical contingencies in economic growth model of developed and developing countries.The implication of this model is straightforward: that conventional growth accounting neednot necessarily avoid stochastic variations that arise in demographic processes. They mayno longer be treated as exogenous in the model, rather while treating them endogenouslycan reflect more about the growth pattern and importance of imperfect institutional set-upin the concerned economy.

20

References

Azomahou, T. and T. Mishra (2007), “Age Dynamics and Economic Growth: An Analysisin a Nonparametric Setting”, Economics Letters, (in press).

Azomahou, T., C. Diebolt and T. Mishra (2007), “Spatial persistence of demographic shocksand economic growth”, Journal of Macroeconomics (in press).

Boucekkine, R., D. de la Croix, and O. Licandro (2002), “Vintage Human Capital, Demo-graphic Trends, and Endogenous Growth,” Journal of Economic Theory, 104: 340-375.

Dickey, D.A. and W.A. Fuller (1979), “Distribution of Estimators for Autoregressive TimeSeries with a Unit Root,” Journal of Amercian Statistical Association, 74: 427-431.

Davidson, J. (2005), Time Series Modeling Version 4.15,http://www.timeseriesmodeling.com/.

Granger, C.W.J. (1980), “Long Memory Relationships and the Aggregation of DynamicModels,” Journal of Econometrics, 14, 227-238.

Granger, C. W. G. and R. Joyeux (1980), “An introduction to long memory time seriesmodels and fractional differencing,” Journal of Time Series Analysis, 1, 15-29.

Geweke, J. and S. Porter-Hudak (1983), “The Estimation and Application of Long MemoryTime Series Models,” Journal of Time Series Analysis, 221-238.

Hamilton, J.D. (1994), Time Series Analysis, Princeton University Press.

Kim, C.S. and P.C.B. Phillips, (2000), “Modified Log Periodogram Regression,” WorkingPaper, Yale University.

Lau, S-H. P. (1999), “I(0) In, Integration and Contegration Out: Time Series Properties ofEndogenous Growth Models,” Journal of Econometrics, 93, 1-24.

Lutz, W., A. Goujon, and A. Wils (2007a), “The population dynamics of human capi-tal accumulation”, in Population Aging, Human Capital Accumulation and ProductivityGrowth, edited by A. Prskawetz, D. Bloom, and W. Lutz. A special issue of Populationand Development Review, Vol. 33. New York: Population Council.

Lutz, W., A. Goujon, S. K.C., and W. Sanderson (2007b), Reconstruction of Populations byAge, Sex and Level of Educational Attainment for 120 Countries for 1970-2000, IIASA In-terim Report IR-07-002. Laxenburg, Austria: International Institute for Applied SystemsAnalysis.

Mishra, T. (2006), Dynamics of Demographic Changes and Economic Development, Doctoraldissertation, Catholic University of Louvain, Belgium. Mimeo.

Phillips, P.C.B.(1999), “Discrete Fourier Transforms of Fractional Processes,” Unpublishedworking paper No. 1243, Cowles Foundation for Research in Economics, Yale University.http://cowles.econ.yale.edu/P/cd/d12a/d1243.pdf

Prskawetz, A. and G. Feichtinger (1995), “Endogenous Population Growth may ImplyChaos,” Journal of Population Economics, 8(1):59-80.

21

Shaffer, M.L. (1987), “Minimum Viable Populations: Coping with Uncertainty,” Pages 69-86 in M.E. Soule, ed. Viable Populations for Conservation, Cambridge UniversityPress, New York.

22

Figure 2: Nadaraya-Watson nonparametric regression of growth instability on long-memory aggregate population: Asia

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4

stdevygr

Nadaraya-Watson Regression on dtotpop

23

Figure 3: Nadaraya-Watson nonparametric regression of growth instability on long-memory aggregate population: Africa

2

4

6

8

10

12

14

16

18

20

0 0.2 0.4 0.6 0.8 1 1.2 1.4

stdevygr

Nadaraya-Watson Regression on dtotpop

24

Figure 4: Nadaraya-Watson nonparametric regression of growth instability on long-memory aggregate population: Europe

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

stdevygr

Nadaraya-Watson Regression on dtotpop

25

Figure 5: Nadaraya-Watson nonparametric regression of growth instability on long-memory aggregate population: Latin America and Offshore

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4

stdevygr

Nadaraya-Watson Regression on dtotpop

26

Appendix

6 d estimates from Modified Log Periodogram regres-

sion

Table 4: Long memory population parameter estimates, AfricaCountries Aggregate Age 0-14 Age 15-64 Age 65+Algeria 0.702 1.001 0.714 0.983Angola 0.337 0.692 0.350 0.598Benin 0.717 0.993 0.667 1.057Botswa 1.172 0.886 1.156 0.583Burki Faso 0.867 1.099 0.469 1.060Cameroon 0.842 1.183 0.727 0.990Cape Verde 1.395 0.838 0.721 0.325Central African Republic 0.555 1.304 0.786 0.759Chad 0.309 0.532 0.507 0.648Congo Republic 0.976 1.063 0.991 0.415Cote d’Ivoire 1.030 1.186 1.075 1.026Egypt 0.952 0.925 0.826 1.177Gabon 0.767 0.938 0.886 0.497Gambia 0.912 1.323 0.808 0.203Ghana 0.480 0.461 0.840 0.780Guinea 0.411 0.682 0.526 0.494Kenya 0.734 1.223 0.959 1.042Lesotho 1.268 0.747 1.136 0.418Liberia 0.196 1.154 1.087 0.900Madagascar 0.612 1.143 0.709 0.751Malawi 0.645 1.098 0.944 0.843Mali 0.877 0.925 0.707 1.015Mauritius 1.049 1.148 1.410 0.378Morocco 0.819 0.997 0.824 1.247Mozambique 0.160 0.923 0.736 0.772Namibia 0.724 1.054 0.822 -0.121Niger 0.755 1.053 0.786 0.604Nigeria 0.703 1.229 1.024 0.869Senegal 0.581 1.170 0.869 0.866Sierra Leone 0.137 0.813 0.551 0.759South Africa 1.005 1.066 0.981 0.832Sudan 0.573 1.091 0.908 0.586Tanzania 0.503 1.298 1.001 0.872Togo 0.201 0.896 0.749 0.898Tunisia 0.228 0.778 0.844 1.061Uganda 1.021 1.127 1.124 1.142Zaire 0.374 0.877 0.930 1.060Zimbabwe 0.774 0.708 1.273 0.738

27

Table 5: Long memory population parameter estimates, AsiaCountries Aggregate Age 0-14 Age 15-64 Age 65+Afghanistan 0.590 0.864 0.927 0.876Bahrain 0.630 0.557 1.049 -0.312Bangladesh 0.642 0.849 0.710 1.066Burma 0.750 0.924 1.078 0.942Cambodia 1.145 0.943 1.005 0.892China 0.416 0.905 1.007 0.971Hong Kong 0.334 0.746 0.933 0.932India 0.702 1.076 1.065 1.215Indonesia 1.067 1.154 1.062 0.631Iran 0.976 0.901 0.927 0.871Iraq 0.095 1.099 0.939 0.878Israel 0.135 1.147 1.383 1.085Japan 0.953 1.345 0.822 0.988Jordan 0.188 1.318 0.818 0.699Kuwait 0.305 1.216 1.020 0.000Laos 0.834 0.920 0.742 1.205Lebanon 1.133 0.902 1.003 0.432Malaysia 0.909 0.871 0.984 0.807Nepal 0.583 1.201 0.681 0.552North Korea 1.104 1.025 1.199 0.617Oman 1.338 0.873 0.712 0.031Pakistan 0.708 1.209 0.839 0.672Philippines 1.047 1.006 1.039 1.041Qatar 0.724 0.927 1.189 -0.345Singapore 0.870 1.015 1.096 0.946South Korea 1.054 1.255 1.112 0.426Sri Lanka 0.734 1.107 1.428 0.768Syria 0.997 0.948 0.718 1.002Thailand 0.846 1.123 1.349 0.848Turkey 1.170 1.413 1.028 1.238United Arab Emirates 1.283 1.084 1.186 0.497Vietnam 1.213 0.579 0.760 0.832Yemen 1.167 1.152 0.890 1.114

28

Table 6: Long memory population parameter estimates, EuropeCountries Aggregate Age 0-14 Age 15-64 Age 65+Albania 0.366 0.724 1.305 0.842Austria 0.582 1.210 1.047 1.096Belgium 0.857 1.164 1.422 1.066Bulgaria 0.743 0.638 0.947 1.035Denmark 0.853 1.018 0.842 1.191Finland 0.714 1.097 0.888 1.469France 0.553 1.014 1.176 0.938Germany 0.844 1.277 1.034 1.043Greece 0.762 1.059 0.981 1.093Hungary 0.599 0.926 1.087 1.121Ireland 0.919 1.266 0.924 0.873Italy 0.705 1.090 0.885 0.852Netherlands 0.549 1.043 1.141 1.385Norway 1.145 1.379 1.120 1.243Poland 1.156 0.891 1.254 1.397Portugal 0.479 1.300 1.115 1.059Romania 0.596 0.674 1.106 1.205Spain 1.209 1.164 1.035 0.766Sweden 0.687 1.062 1.103 1.107Switzerland 0.670 1.260 1.179 1.159United Kingdom 0.895 1.080 0.992 1.034

Table 7: Long memory population parameter estimates, Latin America & OffshoreCountries Aggregate Age 0-14 Age 15-64 Age 65+Argentina 0.889 0.934 1.075 1.173Australia 0.428 1.089 1.021 1.157Bolivia 0.168 1.229 0.856 0.590Brazil 0.873 1.019 1.201 1.269Canada 0.790 1.041 1.318 1.270Chile 0.494 0.886 1.163 1.110Colombia 1.204 1.026 1.169 1.181Costa Rica 0.279 0.500 1.188 0.601Cuba 0.623 1.109 1.252 1.029Dominican Republic 0.838 0.964 1.230 0.702Ecuador 1.178 1.076 1.021 0.856El Salvador 0.647 1.364 1.079 0.677Guatemala 0.445 1.248 0.891 0.575Honduras 0.471 1.247 0.957 0.572Jamaica 0.518 1.020 0.855 1.059Mexico 1.266 0.854 1.047 1.137New Zealand 1.032 1.179 1.168 1.301Nicaragua 0.358 1.119 0.872 0.424Panama 0.735 0.996 1.123 0.745Paraguay 0.851 1.237 0.826 1.335Peru 1.235 0.989 1.107 1.255Puerto Rico 0.334 1.083 1.140 1.103Trinidad and Tobago 0.703 0.654 1.117 0.581United States 0.962 1.275 1.413 0.760Uruguay 1.216 0.910 0.900 0.951Venezuela 0.714 1.018 1.184 1.096

29