Long-memory process and aggregation of AR(1) stochastic processes: A new characterization
aa r X i v : . [ m a t h . S T ] A ug Long-memory process and aggregation of AR(1)stochastic processes: A new characterization
Bernard Candelpergher a , Michel Miniconi b and Florian Pelgrin c Abstract
Contemporaneous aggregation of individual AR(1) random processes might lead todifferent properties of the limit aggregated time series, in particular, long memory(Granger, 1980). We provide a new characterization of the series of autoregressivecoefficients, which is defined from the Wold representation of the limit of the ag-gregate stochastic process, in the presence of long-memory features. Especially theinfinite autoregressive stochastic process defined by the almost sure representationof the aggregate process has a unit root in the presence of the long-memory prop-erty. Finally we discuss some examples using some well-known probability densityfunctions of the autoregressive random parameter in the aggregation literature.JEL Classification Code: C2, C13.Keywords: Autoregressive process, Aggregation, Heterogeneity, Complex variableanalysis. a University of Nice Sophia-Antipolis, Laboratoire Jean-Alexandre Dieudonn´e. E-mail: [email protected]. b University of Nice Sophia-Antipolis, Laboratoire Jean-Alexandre Dieudonn´e. E-mail: [email protected]. c EDHEC Business School, E-mail: fl[email protected] paper has benefited from very useful comments from and discussions with St´ephane Gregoir. Theusual disclaimer applies. Introduction
Aggregation is a critical and widely acknowledged issue in the empirical and theoreticalliterature in economics and other fields. Especially, since the contributions of Grangerand Morris (1976) and Granger (1980), it is well-known that the contemporaneous ag-gregation of individual random AR(1) stochastic processes might lead to long memorymodels. Notably, Granger (1980) considers the case of a Beta-distribution for the randomautoregressive parameter and thus points out that the long memory property dependson the behavior of the density of the random autoregressive parameter near unity andthat the common and idiosyncratic components might exhibit a different degree of longmemory. In particular, Zaffaroni (2004) generalizes these results by studying the limitof the aggregate process with a quite flexible (semi-parametric) assumption regardingthe behavior near unity of the probability density function of the random autoregressivecoefficient and makes clear the asymptotics of both the common and idiosyncratic partsof the aggregated process. Among others results, Zaffaroni (2004) shows formally thatthe more concentrated is the distribution of the random autoregressive coefficient nearthe unit, the stronger is the long-memory property of the limit aggregated process. . Fol-lowing these contributions, we study the aggregation of heterogenous individual AR(1)stochastic processes that leads to the long-memory property. In contrast to the litera-ture, we focus on the infinite autoregressive representation of the limit of the aggregateprocess, and especially the sum of the autoregressive coefficients, rather than the usualinfinite moving representation.Indeed the use of the (equivalent) infinite autoregressive representation of the limit of theaggregated process might be meaningful in various contexts of the aggregation (panel)literature. As pointed out by Lewbel (1994), the autoregressive representation is insight-ful among others when estimating the (macro) aggregate dynamics with unobservableindividual series in the presence of unobserved parameter heterogeneity, when identifyingand estimating certain distributional features of the micro parameters from aggregate re- In the sequel, a stochastic process is said to have a long memory property if its autocovariancefunction is not summable (Beran, 1994; Beran et al., 2013). For a recent survey, see Leipus et al. (2013). Obviously, if the limit of the aggregate process is represented al-most surely by a short memory process, the infinite autoregressive representation andespecially the determination of its persistence (e.g., through the sum of the autoregres-sive coefficients) easily obtains using standard results of time series analysis (Brockwelland Davis, 2002). In contrast, if the limit of the aggregate process is represented almostsurely by a long memory process, the convergence and thus the determination of theseries of autoregressive coefficients is challenging since the series of moving average co-efficients is no longer absolutely summable. Such a characterization might be necessaryfor instance in the case of the estimation of the aggregate dynamics with unobservableindividual series in which finite parameter approximation for the infinite lag distributionis required—the autoregressive representation of the limit aggregated process displays aninfinite parametrization whereas the econometrician has only finitely many observationsand thus finite-parameter approximations might lead to near observational aggregateprocesses with different statistical properties as long as the sum of the autoregressivecoefficients is not correctly identified. In this respect, our paper tackles this issue andproposes a new characterization of the autoregressive representation of limit long-memoryaggregate processes that results from individual random AR(1) stochastic processes.Notably aggregation of individual random AR(1) processes is analyzed in this paper underthe assumptions that some common factors exist, the (positive) random autoregressivecoefficient takes values in [0 ,
1) with probability distribution µ , and all of the noncentralmoments exist. In particular, the series of all of the noncentral moments might be eitherabsolutely summable or divergent and the limit of the aggregated stochastic process sat-isfies almost surely both an infinite moving average and autoregressive representations.Within this framework, we show that the sum of the autoregressive coefficients of thestochastic process defined from the limit in L of the aggregate process, say P k ≥ a k , mightequal one (respectively, less than one) when the limit aggregate process has the longmemory property (respectively, short memory property). Say differently, the divergenceof the series of noncentral moments is fully equivalent to the presence of a unit root in See also Pesaran (2003), Pesaran and Chudik (2014), Jondeau and Pelgrin (2014a,b). r → − a ( r ) of the(ordinary) generating function of the sequence of the autoregressive coefficients does notinsure its equality to P k ≥ a k or even the convergence of this sum. On the other hand, wecannot apply a standard Hardy-Littlewood Tauberian theorem since, to the best of ourknowledge, there is no general proof of the positiveness of the autoregressive coefficientsfor all probability distributions µ defined on [0 , L but does not belong to L .The rest of the paper is organized as follows. In Section 2, we discuss the main assump-tions regarding the individual random AR(1) stochastic processes and then we derivethe limit aggregate process. In Section 3, we provide the two main results of our paper,namely the determination of the sum of the autoregressive parameters and the charac-terization of the stochastic process defined by the limit in L of the aggregated process.In Section 4, we assume that the distribution of the random autoregressive coefficient issubsequently a Beta distribution (of the first kind), a uniform distribution and a randompolynomial density function. In so doing, we apply the results of Section 3 and generalizessome results in the literature. Proofs are gathered in the Appendix. In this section, we first discuss the assumptions regarding the individual AR(1) processes.Then we derive the limit aggregate process.4 .1 Individual dynamics
Consider the individual random AR(1) model for i = 1 , · · · , N and t ∈ Z : x i,t = ϕ i x i,t − + v i,t , (1)where ϕ i denotes an individual-specific (random) parameter drawn from a fixed randomvariable ϕ and v i,t is an error term that can decomposed into a common component, ǫ t ,and an idiosyncratic (individual-specific) component, η i,t : v i,t = ǫ t + η i,t . (2)The macro variable results from the aggregation of micro-units, with the use of time-invariant nonrandom weights W N = ( w , · · · , w N ) ′ , with N P i =1 w i = 1, so that the aggregatevariable is defined as the weighted average of the micro-units X N,t = N P i =1 w i x i,t . The fol-lowing assumptions hold: Assumption 1: ϕ a fixed random variable with probability distribution µ the sup-port of which is in [0 , Assumption 2:
The moments u k = E ( ϕ k ) exist for all integer k ≥ Assumption 3: ǫ t and η i,t are white noise processes with means of zero and variance of σ ǫ and σ η , respectively; ǫ t and η i,t are mutually orthogonal at any lag and lead. Assumption 4:
Realizations of ϕ are independent of those of ǫ t . Assumption 5: As N → ∞ , k W N k = O (cid:0) N − / (cid:1) and w i / k W N k = O (cid:0) N − / (cid:1) for all i ∈ N .Before discussing our main assumptions, we introduce some notations. Let H x de-note the Hilbert space generated by all of the random variables that compose the panel Alternatively one might assume that t ∈ N and consider an asymptotically stationary limit aggregateprocess (Gon¸calves and Gourieroux, 1994). Such dynamics have been used in economics (among others) to represent consumption expendituresacross households (Lewbel, 1994), consumer price inflation across subindices (Altissimo et al., 2009), realexchange rates across sectors (Imbs et al., 2005), or real marginal cost across industries (Imbs et al.,2011). x i,t ) i =1 , ··· ,N,t ∈ Z , H x i the Hilbert space generated by the stochastic process ( x i,t ) t ∈ Z , H x,t and H x i ,t the corresponding subspaces of H x and H x i up to time t . Assumption 1 outlinesthat ϕ is a random variable with distribution on the support [0 ,
1) such that ϕ ∈ T t ⊕ i H x i ,t . This assumption is consistent with many parametric specifications of the cross-sectionaldistribution of ϕ . We only rule out situations in which some individual processes are notalmost surely (asymptotically) stationary, i.e. P ( | ϕ | ≥ > B ( p, q ). In this case, the representation ofthe (limit) aggregate process does depend on the mass distribution of the Beta distri-bution around unity: the smaller is q , the larger is the mass of the distribution aroundunity (Zaffaroni, 2004). In contrast, imposing the condition 0 ≤ ϕ ≤ c < c will guarantee that there are no individual unit root parameters that woulddominate at the aggregate level (Zaffaroni, 2004) and that the limit aggregate process (as N → ∞ ) displays short memory with an exponentially decaying autocorrelation func-tion. Assumption 2 insures that noncentral moments u k = E ( ϕ k ) of any (nondegenerate)random variable ϕ , defined on [0 , > u ≥ · · · ≥ u k ≥ , ∀ k ≥ , and u k → k → ∞ .Assumption 3 defines the statistical properties of the two components of the error term,the common shock (factor) and the idiosyncratic shock. Several points are worth com-menting. First, without loss of generality, one might assume that the stochastic process( v i,t ) i =1 , ··· ,N,t ∈ Z is weakly linearly exchangeable (Aldous, 1980), i.e. ( v i,t ) i =1 , ··· ,N,t ∈ Z isa purely non deterministic (regular) covariance stationary process and the covariancestructure is invariant by translation with respect to the time index and invariant bypermutation with respect to the unit index. Second, the stochastic process ( ǫ t ) t ∈ Z is as-sumed to be known and given. Given that H x = ⊕ i H x i and thus H x,t = ⊕ i H x i ,t , one has ǫ t ∈ H x,t T ( H x,t − ) ⊥ . Third, taking that H x i = H ǫ ⊕ H x i − ǫ , the idiosyncratic stochasticprocess is such that η i,t ∈ H x i − ǫ,t T ( H x i − ǫ,t − ) ⊥ . Fourth, the assumption that the vari- Note that the set ( − ,
0) has been excluded from the support of ϕ for simplicity’s sake. Without altering our main results but at the expense of further assumptions, (i) the common errorterm at time t might be multiplied by the realization of a (scaling) random variable κ to introduce someform of heteroscedasticity, (ii) the stochastic process ( ǫ t ) t ∈ Z might be assumed to be unknown (usinga two-step procedure), and (iii) multiple independent common error terms might be introduced in thespecification (Zaffaroni, 2004). . Assumption 4 avoids any correlation between theerror terms and ϕ . Assumption 5 is a granularity condition, which insures that the weightsused to define the aggregate process are not dominated by a few of the cross-sectionalunits (Gabaix, 2011; Pesaran and Chudik, 2014). The empirical cross-sectional moments of ϕ are ˜ E N (cid:0) ϕ k (cid:1) = N P i =1 w i ϕ ki , ∀ k ≥
1. For sake ofsimplicity and without loss of generalization with respect to Assumption 4, we assumethat w i = 1 /N for all i . Consequently, as N → ∞ , ˜ E N (cid:0) ϕ k (cid:1) a.s. → u k .Using Eqs. (1)–(2), the exact aggregate dynamics can be written as: N Y j =1 (1 − ϕ j L ) X N,t = 1 N N X i =1 Y j = i (1 − ϕ j L ) v i,t (3)or equivalently X N,t = 1 N N X i =1 (1 − ϕ i L ) − ǫ t + 1 N N X i =1 (1 − ϕ i L ) − η i,t (4) The condition is that the maximum eigenvalue of the covariance matrix of η t = ( η ,t , · · · , η N,t ) ′ remains bounded as the cross-section dimension increases. Our results extend to the case of (time-varying) stochastic weights. Such an extension requires atleast that the weights be distributed independently from the stochastic process defining the randomvariable. Put differently, it is an ARMA(N,N-1) in the absence of common roots in the individual processes(Granger and Morris, 1976). L is the lag operator ( z t − = Lz t ). Taking Eq. (4), we can characterize theasymptotic behavior of both the idiosyncratic component and the common component.This is done in the following proposition. Results are known but are reported here forsake of completeness. Proposition 1
Suppose that Assumptions 1–5 hold. Given the disaggregate model de-fined in Eqs. (1)–(2), the limit in L of the aggregated process as N → ∞ satisfies (almostsurely) the two equivalent representations: X t = ∞ X k =0 u k ǫ t − k (MA form) , (5) X t = ∞ X k =1 a k X t − k + ǫ t (AR form) , (6) where X N,t L → X t and ˜ E N (cid:0) ϕ k (cid:1) a.s. → u k = E (cid:0) ϕ k (cid:1) as N → ∞ . The sequence { a k , k ≥ } where a k = E [ A k ] satisfies the recurrence relation : A = ϕ , A k +1 = ( A k − a k ) ϕ (7)Proof: Gon¸calves and Gouri´eroux (1988), Lewbel (1994).Several points are worth commenting. First, as shown by Robinson (1978), with theexception of a degenerate distribution for ϕ (e.g., Dirac distribution), the dynamics ofthe limit aggregate process is richer than the individual dynamics because of the non-ergodicity of the individual random AR(1) process. Second, using the infinite movingaverage representation and the positiveness of the moments, the (limit) aggregate pro-cess displays short memory if P ∞ k =0 u k < + ∞ whereas it has a long-memory propertyif P ∞ k =0 u k = + ∞ . It is worth noting that (i) the (limit) aggregate process is in L but not necessarily in L , and (ii) the sum of the autoregressive coefficients is (abso-lutely) convergent if the aggregate process has short memory (since the spectral repre-sentation is unique). Third, the central moments of the cross-sectional distribution of ϕ can be easily obtained from the infinite autoregressive representation of the aggregateprocess. This is useful when considering the standard disaggregation problem in statis-8ics. For instance, the first four cross-sectional moments are E [ ϕ ] = a , V [ ϕ ] = a , S [ ϕ ] = ( a − a a ) / ( a ) / , and K [ ϕ ] = ( a − a a + a a + a ) / ( a ) . Fourth, Equa-tion (6) shows that aggregation leads to an infinite autoregressive model for X t (seeRobinson, 1978, Lewbel, 1994). Notably, using Eq. (7), the autoregressive parameters a k are nonlinear transformations of the noncentral moments of ϕ and satisfy the followingnon-homogenous difference equations (for k ≥ a k +1 ≡ E [ A k +1 ] = u k +1 − k X r =1 a r u k − r +1 a = E [ ϕ ] . Fifth, the persistence of the stochastic process defined by the limit of the aggregatedprocess can be defined as the sum of the autoregressive coefficients, which is denoted a (1) = P ∞ k =1 a k . More specifically, if the limit of the aggregate process belongs to L andthus ( X t ) t ∈ Z is a short-memory stochastic process, then it is straightforward to show that a (1) = (cid:18) E (cid:20) − ϕ (cid:21)(cid:19) − E (cid:20) ϕ − ϕ (cid:21) . or equivalently (1 − a (1)) − = E (cid:20) − ϕ (cid:21) . In particular, the limit of the aggregate stochastic process has short memory if and onlyif E h − ϕ i < ∞ (Robinson, 1978; Gon¸calves and Gourieroux, 1994). In the spirit ofZaffaroni (2004), a sufficient condition on the probability density function h of ϕ forthe short memory property is that there exists α ∈ (0 ,
1) and a constant C such thatlim x → − h ( x )(1 − x ) − α = C . Sixth, we can define the generating function of the a k terms as follows.Taking the recurrence relation, one can write formally X k ≥ A k z k = zϕ − zϕ − X k ≥ a k z k ! X k ≥ a k z k = X k ≥ u k z k − X k ≥ a k z k X k ≥ u k z k . Let m denote m ( z ) = X k ≥ u k z k , we obtain the formal generating function of the a k terms a ( z ) ≡ X k ≥ a k z k = m ( z )1 + m ( z ) , (8)and thus the mapping between the generating function of the infinite autoregressive lagpolynomial and the one of the infinite moving average representation. As explained inSection 3, Eq. (8) is fundamental since our proof is intimately related to the function m m . In this section, we show that the sum of the autoregressive coefficients equals one in thepresence of a (limit) aggregate process in L but not in L . In so doing, we emphasize thatcomplex analysis is required to obtain the convergence of the series P k ≥ a k . Then weprovide a characterization of the (limit) aggregate long-memory process in the presenceof individual random AR(1) processes. m Before showing the convergence of the series of autoregressive parameters, we need anintermediate result regarding the m function, and especially to make clear the relationshipbetween the sum of the moving average coefficients P k ≥ u k and m ( r ) when r → − .Proposition 2 clarifies this link and turns to be extremely useful when characterizing10he infinite sum of the autoregressive coefficients of a (limit) aggregate long-memoryprocess. Proposition 2 ∞ P k =1 u k = + ∞ if and only if lim r → − m ( r ) = + ∞ Proof : See Appendix A. a ( z ) = m ( z )1 + m ( z ) and the convergence of P k ≥ a k One key issue to study the convergence of the autoregressive coefficients is that the ex-istence of the limit, lim r → − a ( r ), does not insure its equality to P k ≥ a k or even theconvergence of this sum. Supplementary
Tauberian conditions are needed for this kindof results (Hardy, 1949; Titchmarsh, 1939; Korevaar, 2004). In particular, the conver-gence of this series has to be studied by making use of complex analysis, especially in thecase of a (limit) aggregate long-memory process.All of the proofs related to Theorem 1 are gathered in Appendix B. To summarize,the proof of the convergence of the autoregressive coefficients proceeds as follows. Wefirst define m as an analytic function of the complex variable z within the open discaround 0 of radius one, D (0 , m ( z ) as an integral on [0 ,
1) with parameter z ∈ D (0 , D (0 , \{ } (see Lemma 1 in Appendix B). Then a second lemma (see Lemma 2 in Appendix B)proves that the function a ( z ) = m ( z )1+ m ( z ) is well defined in the disc D (0 ,
1) (i.e., the de-nominator does not vanish) and then it can be extended to the whole closed disc D (0 , P u k :if P u k < ∞ the function m is continuous in the closed disc, so is a ; and if P u k = ∞ then | m ( z ) | → ∞ when z → D (0 , a ( z ) → z → D (0 , a is univalent and provides the Note obviously that ∞ P k =0 u k = + ∞ if lim r → − m ( r ) = + ∞ . For instance, consider the power series defined for | r | < h ( r ) = ∞ P k =0 ( − k r k . Then h ( r ) = r andlim r → − h ( r ) = . However ∞ P ( − k is not convergent. P a n (Theorem 1). Theorem 1
Let { a k , k ≥ } denote the sequence defined in Proposition 1. The series ∞ P k =1 a k is convergent and + ∞ X k =1 a k = lim r → − m ( r )1 + m ( r )Proof : See Appendix B.Taking Proposition 2 and Theorem 1, it is then straightforward to determine the sum ofthe autoregressive coefficients. Proposition 3
Let { a k , k ≥ } denote the sequence defined in Proposition 1. The sumof the autoregressive coefficients, ∞ P k =1 a k , equals one if and only if lim r → − m ( r ) = + ∞ orequivalently if and only if ∞ P k =1 u k = + ∞ . Three points are worth commenting. First, if the limit aggregate process is a second-order stationary process (Proposition 1) and the series of its moving average coefficientsis absolutely summable, then the sum of the autoregressive coefficients is less than one.Notably, this result obtains with classical time series results whereas there is a need ofcomplex analysis when the series of the moving average coefficients is not absolutelysummable and one studies what happens at the pole z = 1. A second and related point isthat the behavior of the series of the autoregressive coefficients depends on whether thelimit second-order stationary aggregate process belongs to L or not. Consequently, asexplained below, this provides a new characterization of a (limit) aggregate long-memoryprocess as a result of the aggregation of random AR(1) processes. Third, as stated inCorollary 1, the function Φ( z ) = 1 − a ( z ) admits only one zero on D (0 , Corollary 1
Let Φ( z ) = 1 − a ( z ) . Then z = 1 is a zero of the function Φ , which isdefined on D (0 , , if and only if ∞ P k =1 u k = + ∞ . X N,t ) t ∈ Z admits a unit root whereas the aggregate pro-cess is weakly stationary. There is a one-to-one relationship between the long memoryproperty and the presence of a unit root of the infinite autoregressive limit aggregateprocess. Taking Theorem 1 and Proposition 3, we are now in a position to provide a new charac-terization of a (limit) aggregate long-memory process that results from the aggregationof individual random AR(1) processes.
Theorem 2
Suppose that Assumptions 1 and 2 hold true. Let ( X t , t ∈ Z ) denote along-memory process with the following Wold decomposition X t = ∞ X k =0 u k ǫ t − k where the u k terms are nonnegative and ∞ P k =0 u k = ∞ . Then the a j terms of the equivalentinfinite autoregressive representation, X t = ∞ X k =1 a k X t − k + ǫ t , which are defined from a k +1 = u k +1 − P kr =1 a r u k − r +1 and a = u , satisfy P ∞ k =1 a k = 1 . Several points are worth discussing. First, the (limit) aggregate long-memory processdoes not belong to the class of ARFIMA or self-similar stochastic processes. In partic-ular, it is not possible to express the moving average coefficients of a fractional ARIMAprocess (by binomial expansion) such that they match the moving average weights of the A continuous-time stochastic process ( Y t ) is said to be self-similar with self-similarity paramter H , iffor any sequence of time points t , · · · , t k and any positive constant a , c − H ( Y at , · · · , Y at k ) has the samedistribution as ( Y t , · · · , Y t k ). H (Beran, 1994; Beran et al., 2013). Second, the factthat the sum of the autoregressive coefficients of the (limit) aggregate process is equalto one is also consistent with the standard definition of long-memory processes (Beran,1994; Beran et al., 2013), i.e. there exists a real number α ∈ (0 ,
1) and a constant c ρ such that lim k →∞ ρ ( k ) c ρ k α = 1 where ρ ( k ) = γ X ( h ) /γ X (0) (with γ X ( h ) = E (cid:2) ϕ h / (1 − ϕ ) (cid:3) ) is theautocorrelation of order k of the (limit) aggregate stochastic process ( X t ).Third, as pointed out by Beran (1994), observing long-range dependence in an aggre-gate time series (e.g., at the macro-level) does not necessarily mean that this is dueto the genuine occurrence of long memory in the individual series (micro-level). Thismight be induced artificially by aggregation. Say differently, identifying the source oflong memory would require to look carefully at the behavior of the possibly unobservableindividual series. Fourth, Theorem 2 has some implications, which are beyond the scopeof this paper, regarding some aggregation problems often encountered in the theoreticaland empirical aggregation research (Pesaran and Chudik, 2014), namely the derivationof the macro dynamics from heterogenous individual dynamics and the identification andestimation of certain distributional features of the micro parameters from aggregate rela-tions (disaggregation problem). For instance, in the case of the disaggregation problemwhen individual series are not available, since the autoregressive specification in Proposi-tion 1 displays an infinite parametrization and the econometrician has only finitely manyobservations, one might proceed with a finite-parameter approximation for the infinitelag distribution (Sims, 1971, 1972; Faust and Lepper, 1997; P¨otscher, 2002) and mightaccount possibly for the constraint a (1) = 1. Notably, finite-parameter approximationsmight lead to near observational aggregate processes with different statistical propertiesand thus to the incorrect identification of distributional features of the micro parame-ters. Other applications might concern, among others, the estimation of the aggregate See Lewbel (1994), and Jondeau and Pelgrin ( 2014b). The convergence of the estimates of the approximation is not sufficient to guarantee the conver-gence of some functions of those parameters—pointwise convergence does not imply (locally) uniformconvergence. L that admits a Wold decomposition with decreasing and nonnega-tive moment moving average coefficients and does not belong to L displays a unit root.This comes from the formal identity between the generating functions of ( a k ) and ( u k ), P k ≥ a k z k = P k ≥ u k z k − P k ≥ a k z k P k ≥ u k z k . In this section, we review three examples, namely the Beta B ( p, q ) distribution (of thefirst kind), the uniform distribution with p = q = 1, and the random polynomial density. Beta distribution
Following Morris and Granger (1976), Granger (1980) and Gon¸calvesand Gouri´eroux (1994), we assume that ϕ is Beta-distributed. B ( p, q ; x ) = Γ( p + q )Γ( p )Γ( q ) x p − (1 − x ) q − [0 , ( x ) , p > , q > . In this respect, to the best of our knowledge, Proposition 4 provides a new characterizationof the series of autoregressive coefficients (i.e., the persistence of the (limit) aggregateprocess ( X t )). Proposition 4
Suppose that ϕ is Beta-distributed (of the first kind). Given the disag-gregate model defined in Eqs. (1)–(2), the series of the autoregressive coefficients of thelimit aggregate process defined in Proposition 1 is given by:- If q > , then ∞ P k =1 a k = pp + q − ; Note that Gon¸calves and Gouri´eroux (1994) study extensively the aggregation of individual AR(1)processes in which ϕ is Beta-distributed (after an homothety) and provide a discussion regarding aggre-gate long-memory processes. However they do not consider the sum of the autoregressive coefficients. If q ≤ , then ∞ P k =1 a k = 1 . Proposition 4 can be shown as follows. Taking the integral form of m ( r ) we have1 + m ( r ) = Γ( p + q )Γ( p )Γ( q ) Z x p − (1 − x ) q − − rx dx, and thus (by the monotone convergence theorem)lim r → − (1 + m ( r )) = Γ( p + q )Γ( p )Γ( q ) Z x p − (1 − x ) q − dx. Consequently, this is a convergent integral if and only if q >
1. In this case,lim r → − (1 + m ( r )) = Γ( p + q )Γ( p )Γ( q ) Γ( p )Γ( q − p + q −
1) = 1 + pq − + ∞ X k =1 a k = lim r → − m ( r )1 + m ( r ) = pp + q − . On the other hand, it follows that lim r → − (1 + m ( r )) = + ∞ if and only if q ≤
1, and P + ∞ k =1 a k = 1 (Theorem 1). Uniform distribution
We now assume that p = q = 1 such that the random variable ϕ is uniformly distributed over the interval [0 ,
1) and non central moments are given by u k = k +1 . Proposition 5
Suppose that ϕ is uniformly distributed over the interval [0; 1) . Giventhe disaggregate model defined in Eqs. (1)–(2), the autoregressive coefficients of the limitaggregate process defined in Proposition 1 are given by: a k = | I k | k ! where I k = R x ( x − · · · ( x − k +1) dx has the same sign as ( − k − for k ≥ . Moreover, P ∞ k =1 a k = 1 . Proposition 5 can be shown by using either Theorem 1 or a new lemma provided in Ap-pendix C. Notably, the coefficients of the series a ( z ) = m ( z ) / (1 + m ( z )) can be computed16s follows. First, the generating moment series is1 + m ( z ) = X k ≥ u k z k = X k ≥ z k k + 1 = − log(1 − z ) z where | z | < a ( z ) = m ( z )1 + m ( z ) = 1 + z log(1 − z ) . Second the expression of z log(1 − z ) is derived by a power series development of the function a . Indeed, using the notation ψ ( z ) = log(1 − z ), one has Z e xψ ( z ) dx = e ψ ( z ) − ψ ( z ) = − z log(1 − z )where the power series development of e xψ ( z ) = (1 − z ) x is defined to be:(1 − z ) x = X n ≥ (cid:18) xn (cid:19) ( − n z n = 1 + X n ≥ ( − n n ! z n x ( x − · · · ( x − n + 1) . Since this series converges absolutely for | z | < x in [0 , Z e xψ ( z ) dx = Z (1 − z ) x dx = 1 + X n ≥ ( − n n ! z n I n where I n = R x ( x − · · · ( x − n + 1) dx for n ≥
1. Finally, Z e xψ ( z ) dx = − z log(1 − z ) = 1 − X k ≥ k ! z k | I k | and thus a ( z ) = X k ≥ | I k | k ! z k . Several points are worth commenting. On the one hand, the non-negativeness of theautoregressive coefficients for the uniform distribution allows for the use of a standardHardy-Littlewood Tauberian result and thus might not require Theorem 1. Howeverthe non-negativeness of the autoregressive coefficients is not proved for all probabilitydistributions µ with support [0 , Random polynomial density
We consider as a last example the case of the poly-nomial aggregated AR(1) model. More specifically, we suppose that ϕ has a continuousdistribution over [0 ,
1] that can be represented by a polynomial of degree d ≥ f ( ϕ ) = d X s =0 c s ϕ s [0 , ( ϕ )where P ds =0 c s s +1 = 1 (the density has to be integrated to one) and f is non-negative in[0 , f (1) = 0 or f (1) > f (1) = c + c + · · · + c d is the value of the density at x = 1. Proposition 6
Suppose that ϕ has a polynomial density function of order d . Given thedisaggregate model defined in Eqs. (1)–(2), the series of the autoregressive coefficients ofthe limit aggregate process defined in Proposition 1 is given by:- If f (1) = 0 , then a (1) = ∞ X k =1 a k = 1 − d − P n =0 n P k =0 c n − k n +1 ; - If f (1) > , then ∞ P k =1 a k = 1 . d , the non-central moments u k = E (cid:2) ϕ k (cid:3) , k ≥
0, are given by: E (cid:2) ϕ k (cid:3) = d X s =0 c s s + k + 1 , and thus the generating moment series,1 + m ( z ) = X k ≥ u k z k = ∞ X k =0 (cid:16) d X s =0 c s s + k + 1 (cid:17) z k , is convergent at least for | z | <
1. Therefore, one needs to study the convergence of theseries 1 + m (1) = ∞ X k =0 d X s =0 c s s + k + 1 = lim K →∞ K X k =0 d X s =0 c s s + k + 1The terms of the double sum S ( K ) = P Kk =0 P ds =0 c s s + k +1 form an array with K + 1 rowsand d + 1 columns. Without loss of generality, we can suppose K > d . Let n = s + k , wehave n − d ≤ k ≤ n . Taking diagonal sums along the lines s + k = 0 , , . . . , d + K of thearray, one can write S ( K ) = S ( K ) + S ( K ) + S ( K ) with S ( K ) = d X n =0 n X k =0 c n − k n + 1 = c + c + c · · · + c + c + · · · + c d d + 1 S ( K ) = K X n = d +1 n X k = n − d c n − k n + 1 = K X n = d +1 n + 1 n X k = n − d c n − k = f (1) K X n = d +1 n + 1and S ( K ) = d + K X n = K +1 n X k = n − d c n − k n + 1 = d + K X n = K +1 n + 1 n X k = n − d c n − k = c + c + · · · + c d K + 2 + c + · · · + c d K + 3 + · · · + c d K + D + 1 . The first sum, S ( K ), is finite and independent of K . The second sum, S ( K ), is clearlyconvergent as K → ∞ if and only if f (1) = 0. Finally, the third sum, S ( K ), is finiteand its limit is zero when K approaches infinity. In this respect, the generating moment19eries 1 + m (1) converges if and only if f (1) = 0. In this case its limit is S = S ( K ) with S ( K ) = c + c + c · · · + c + c + · · · + c d d + 1In contrast, when f (1) > P k ≥ u k diverges and thus the sum of the au-toregressive coefficients equals one. Consequently, the (limit) aggregate process displayslong-range dependance.In this respect, Proposition 6 provides formally the sufficient condition discussed byChong (2006), i.e. f (1) > c = 1, we end up with the uniform distribution (with f (1) > ax (1 − x ) for a = 0 and x ∈ [0 , f (1) = 0 and the long-runpersistence is given by a (1). In contrast, if the probability density function has a supporton [0; 1], the aggregation process leads to a generalized integrated process (Lin, 1991;Granger and Ding, 1996). As a final remark, it is worth emphasizing that any distribu-tion such that the generating function of the sequence of the moments is not convergentleads to a long-memory process characterized by Theorem 2. In this paper, we study the aggregation of individual random AR(1) processes underthe assumptions that some common factors exist, the (positive) random autoregressivecoefficient takes values in [0 ,
1) with probability distribution µ , and all of the noncen-tral moments exist. Notably we show by making use of complex analysis that sum ofthe autoregressive coefficients equals one in the presence of limit aggregate long mem-ory processes: the divergence of the series of noncentral moments is fully equivalent tothe presence of a unit root in the autoregressive representation of the stochastic processdefined by the limit of the aggregate process. We then illustrate our results using someprominent examples of distribution for the aggregation of random autoregressive AR(1)20rocesses. This provides some new insights that might deserve some empirical applica-tions and some theoretical developments, as for instance the disaggregation problem.21 ppendix: Proofs Appendix A
Proof of Proposition 2: Taking that the sequence of the moments u k = E ( ϕ k ) = Z [0 , x k dµ ( x )is positive and decreasing, one has lim k → + ∞ u k = 0 (by monotone convergence appliedto the sequence ( x k ) k ≥ and thus the radius of convergence of the series P k ≥ u k z k is atleast 1.Let ( r n ) an increasing sequence in [0 ,
1) such that lim n → + ∞ r n = 1. For x ∈ [0 ,
1) thefunctions f n ( x ) = r n x − r n x are positive therefore by Fatou’s Lemma we getlim inf n m ( r n ) = lim inf n Z [0 , f n dµ ( x ) ≥ Z [0 , lim inf n f n dµ ( x ) = Z [0 , x − x dµ ( x ) = + ∞ X k =0 u k Thus if P u k is divergent then m ( r ) → + ∞ when r → − . Moreover by the AbelTheorem (Titchmarsh, 1939, pp.9-10) if P u k is convergent then lim r → − m ( r ) is finiteand lim r → − m ( r ) = P u k . The equivalence then follows. Appendix B
In Appendix B, we provide the proof of Theorem 1. In so doing, we proceed with threelemmas. Notably we first define m as an analytic function of the complex variable z inthe disc D (0 ,
1) and rewrite m ( z ) as an integral on [0 ,
1) with parameter z ∈ D (0 , D (0 , \{ } . ThenLemma 2 proves that the function a ( z ) = m ( z )1+ m ( z ) is well defined in the disc D (0 ,
1) (i.e.,the denominator does not vanish) so that it can be extended to the whole closed disc D (0 , P u k : if P u k < ∞ the function m is continuous in the closed disc, so is a ; and if P u k = ∞ then | m ( z ) | → ∞ when z → D (0 , a ( z ) → z → (0 , a is shown to be univalentand provides the use of a Tauberian condition. Then Theorem 1 is proven.Let m the function defined in the open disc D (0 ,
1) = { z ∈ C with | z | < } by m ( z ) = + ∞ X n =1 u n z n This function is analytic. Moreover, by a classical theorem, on the boundary C (0 ,
1) = { z ∈ C with | z | = 1 } the series P n ≥ u n e int is convergent for all t ∈ ]0 , π [ and by Abel’stheorem : m ( e it ) = + ∞ X n =1 u n e int = lim r → − + ∞ X n =1 u n r n e int Thus the function m is defined in D (0 , \{ } = { z ∈ C with | z | ≤ , z = 1 } . Bypositivity we have + ∞ X n =1 E ( ϕ n ) r n = E ( + ∞ X n =1 ϕ n r n )for all 0 ≤ r <
1. Thus for r ∈ [0 ,
1) we get m ( r ) = + ∞ X n =1 u n r n = E ( ϕr − ϕr ) = Z [0 , rx − rx dµ ( x ) Lemma 1
For z ∈ D (0 , \{ } m ( z ) = Z [0 , zx − zx dµ ( x )Proof: First we prove that the function z R [0 , zx − zx dµ ( x ) is analytic in the open disc D (0 , D (0 , \{ } and thus is defined.Let K denote a compact set in D (0 , \{ } . This compact is included in D (0 , \ D (1 , ε )23ith ε > z ∈ K we get zx ∈ D (0 , \ D (1 , ε ) for all x ∈ [0 , | − zx | ≥ ε. Therefore for all z ∈ K and x ∈ ]0 , (cid:12)(cid:12)(cid:12)(cid:12) zx − zx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε x. This proves the continuity of the function z R [0 , zx − zx dµ ( x ) over K . Moreover, byanalyticity of z zx − zx for all x ∈ [0 ,
1) and the previous boundness condition, thefunction z R [0 , zx − zx dµ ( x ) is analytic in D (0 , m ( r ), r ∈ [0 , m ( z ) = Z [0 , zx − zx dµ ( x )for all z ∈ D (0 , z = e it with t = 0 m ( e it ) = lim r → − + ∞ X n =1 u n r n e int = lim r → − Z [0 , re it x − re it x dµ ( x ) = Z [0 , e it x − e it x dµ ( x ) (cid:3) Extension by continuity of the function m m Lemma 2
The function m m can be extended to a continuous function over D (0 , . Proof: The function 1 + m doesn’t vanish in D (0 ,
1) because1 + m ( z ) = Z [0 , − zx dµ ( x )and if z = a + ib with a + b ≤ ax < x ∈ [0 ,
1) soRe(1 + m ( z )) = Z [0 , − ax (1 − ax ) + b x dµ ( x ) > m m is therefore defined and continuous in D (0 , \{ } and analytic in D (0 , z = 1, we need to consider two cases :1. If P u k is convergent then the series P u k z k is normally convergent in D (0 ,
1) andthe function m is continuous in D (0 ,
1) and consequently so is m m .2. If P u k is divergent it is sufficient to prove that | m ( z ) | → + ∞ when z →
1. Indeed m ( z )1 + m ( z ) = 11 + m ( z ) → m m by 1 at the point z = 1.We now show that | m ( z ) | → + ∞ when z → . In so doing, consider a sequence of points z k = a k + ib k with a k → , b k → a k + b k ≤
1. ThenRe( m ( z k )) = − Z [0 , − a k x (1 − a k x ) + b k x dµ ( x ) . As a k x ≤ k and all x ∈ [0 ,
1) the functions f k : x − a k x (1 − a k x ) + b k x are positive in [0 , k Z [0 , f k dµ ( x ) ≥ Z [0 , lim inf k f k dµ ( x )where lim inf k f k ( x ) = lim f k ( x ) = 11 − x . It is therefore sufficient to remark that if P u k is divergent then Z [0 , − x dµ ( x ) ≥ Z [0 , x − x dµ ( x ) = + ∞ X n =0 u n = + ∞ . So for all sequence ( z k ) converging to 1 in the disc, we have Re ( m ( z k )) → + ∞ and thus | m ( z k ) | → + ∞ . (cid:3) Convergence of an analytic function D (0 , , which is continuous on D (0 , emma 3 provides the use of a Tauberian condition, which turns to be crucial forthe proof of Theorem 1. Lemma 3
Let f ( z ) = P k ≥ b k z k an analytic function on D (0 , , continuous on D (0 , . If f is injective on D (0 , then the series P k ≥ b k is convergent. Proof: We proceed in two steps. First, we prove that1 n n X k =1 k | b k | → . On the one hand, the function f is analytic on D (0 ,
1) thus the image U = f ( D (0 , C and U is included in the compact set f ( D (0 , f is continuous on D (0 , . On the other hand, the function f being injective on D (0 ,
1) we have f ′ ( z ) = 0 forall z ∈ D (0 ,
1) thus f is a C diffeomorphism between D (0 ,
1) and U . By the change ofvariables formula we get λ ( U ) = Z D (0 , | f ′ ( x + iy ) | dxdy and with the use of polar coordinates we get the finite Lebesgue measure of U as the sumof the series λ ( U ) = π + ∞ X n =1 n | b n | The convergence of this last series now implies that n P nk =1 k | b k | → N ≥ n > N write1 n n X k =1 k | b k | = 1 n N X k =1 k | b k | + 1 n n X k = N +1 k | b k | . n n X k =1 k | b k | ≤ n N X k =1 k | b k | + 1 n ( n X k = N +1 k ) / ( n X k = N +1 k | b k | ) / ≤ n N X k =1 k | b k | + 1 n ( n X k = N +1 k | b k | ) / and thus lim sup n → + ∞ n n X k =1 k | b k | ≤ n ( n X k = N +1 k | b k | ) / . Since n ( P nk = N +1 k | b k | ) / → n P nk =1 k | b k | → Lemma 3 . Indeedlet t n = n P nk =1 kb k (with t = 0). We have b n = ( t n − t n − ) + n t n − . The series P n ≥ ( t n − t n − ) is convergent because N X n =1 ( t n − t n − ) = t N → , and thus the series P n ≥ ( t n − t n − ) is Abel summable. Since the series P n ≥ b n is alsoAbel summable by continuity of f , we get the Abel-summability of the series P n ≥ n t n − .Finally, since n (cid:0) n t n − (cid:1) = t n − → , it follows from the classical Tauber’s theorem that the series P n ≥ n t n − (and thus P n ≥ b n ) is convergent. (cid:3) Proof of Theorem 1
Using
Lemma 2 , the function f = a = m m is defined and continuous on D (0 ,
1) andanalytic in D (0 , Lemma 3 that it remains to prove that f is injectivein D (0 , m ( z )1 + m ( z ) = m ( z )1 + m ( z ) ⇔ m ( z ) = m ( z ) ,
27t is sufficient to prove the injectivity of the function m on D (0 , D (0 , r ) for all 0 ≤ r < m is not injective on D (0 ,
1) thenthere exists z = z such that m ( z ) = m ( z ) ; so m is not injective on D (0 , r ) where r > max( | z | , | z | ).For all 0 ≤ r < m is analytic on D (0 , r ) and continuous on D (0 , r ). UsingDarboux’s theorem (Burckel, 1979, p. 310), we could then establish the injectivity of m on D (0 , r ) by showing that m is injective on the circle of radius r with center 0. Indeed,let ϕ ( t ) = Re ( m ( re it )) = Z [0 , rx cos( t ) − r x − rx cos( t ) + r x dµ ( x ) . We have ϕ ′ ( t ) = sin t Z [0 , − rx (1 − r x )(1 − xr cos t + r x ) dµ ( x ) . We see that the function ϕ is decreasing on ]0 , π [ et increasing on ] π, π [. It is symmetricacross π : we have ϕ ( t ) = ϕ (2 π − t ). Therefore the only points t = t with ϕ ( t ) = ϕ ( t )are the pairs ( t, π − t ) with t ∈ [0 , π [. Moreover we haveIm( m ( re it )) = Z [0 , xr sin( t )1 − xr cos( t ) + r x dµ ( x )hence Im( m ( re it )) = − Im( m ( re i (2 π − t ) ))Since we have sin( t ) Z [0 , xr − xr cos( t ) + r x dµ ( x ) > t ∈ [0 , π ), we can’t have Im( m ( re it )) = Im( m ( re i (2 π − t ) ))Therefore t m ( re it ) is injective. By the Abel Theorem the sum of the series P a n isequal to the limit lim r → a ( r ) where r ∈ [0 , (cid:3) eferences [1] Aldous, 1980, Sminaire de Probabilit´es, Ecole d’´et´e de Saint-Flour, Springer Verlag.[2] Altissimo, F., Mojon, B., Zaffaroni, P., 2009. Can Aggregation Explain the Persistenceof Inflation? Journal of Monetary Economics 56, 231–241.[3] Beran, J., 1994, Statistics for long-memory processes , Monograph on statistics andapplied probability 61, Chapman and Hall.[4] Beran, J., Feng, Y., Ghosh, S. and R. Kulik, 2013.
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