Nº 503 ISSN A note on Chambers s long memory and aggregation in macroeconomic time series. Leonardo Rocha Souza

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1 Nº 53 ISSN A oe o Chambers s log memor ad aggregaio i macroecoomic ime series Leoardo Rocha Souza Seembro de 3

2 A Noe o Chambers s Log Memor ad Aggregaio i Macroecoomic Time Series Leoardo Rocha Souza Graduae School of Ecoomics, Geulio Vargas Foudaio. Sepember 3 Absrac Chambers (1998) eplores he ieracio bewee log memor ad aggregaio. For coiuous-ime processes, he aes he aliasig effec io accou whe sudig emporal aggregaio. For discree-ime processes, however, he seems o fail o do so. This oe gives he specral desi fucio of emporall aggregaed log memor discree-ime processes i ligh of he aliasig effec. The resuls are differe from hose i Chambers (1998) ad are suppored b a small simulaio eercise. As a resul, he order of aggregaio ma o be ivaria o emporal aggregaio, specificall if d is egaive ad he aggregaio is of he soc pe. JEL classificaio: C14, C, C43 Kewords: Temporal Aggregaio, Log Memor, Aliasig Correspodig auhor: Leoardo Rocha Souza leors@fgv.br Phoe: Fa: Address: EPGE Fudação Geúlio Vargas. Praia de Boafogo, 19, 11 adar, Rio de Jaeiro CEP 5-9 Brazil The auhor would lie o ha FAPERJ for he fiacial suppor, ad Marcelo Ferades for helpful commes o previous versios of his wor.

3 1 - Iroducio Chambers (1998) ivesigaes he specral desi fucio of soc ad flow aggregaed log memor processes, as well as coiuous-ime log memor processes observed a discreeime iervals ad cross-secioall aggregaed log memor processes. For he emporal aggregaio of discree-ime processes, however, he does o ae io accou he aliasig effec as he does i he case of he coiuous-ime processes. This oe derives he specral desi fucio of aggregaed soc ad flow log memor processes i ligh of he Aliasig Theorem. The resulig formulae are differe from hose i Chambers (1998), ad a simulaio sud brigs evidece i m favor. The Aliasig Theorem is adaped o he case i which a discree-ime process is observed a a slower samplig rae. Oe of he mai resuls of Chambers (1998), amel ha he iegraio order is ivaria o emporal aggregaio, sill holds i mos cases, he ecepio beig aggregaio of soc processes wih a egaive order of iegraio. O he oher had, he secod esable implicaio of he heor (Chambers 1998, Secio.5) ma be seriousl impaired b he bias icurred b emporal aggregaio i soc aggregaes, as repored b Souza ad Smih (). The e secio coais m specific pois, while Secio 3 provides some discussio o he resuls. Accouig of he aliasig effec I modif slighl he oaio used b Chambers (1998), paricularl i ha I defie he properies i he ime ui of he origial samplig frequec, whereas he uses he ime ui of he aggregaes. I use as he level of aggregaio, isead of p, so as o avoid cofusio wih he AR order p. Also, I op for savig oaio i some equaios. Defiiio 1: The emporall aggregaed variable Y is observed as follows: 1a) If X is a soc variable, he Y = X, = 1,, T. 1b) If X is a flow variable, he = X j = j Y B X, = 1,, T. j= j= The differece bewee Defiiio 1a) ad 1b) is ha a movig average filer (1 + B + + B -1 ) is applied o X i he case of a flow variable before sip-samplig, while he soc variable is simpl sip-sampled..1 Resuls from Theorem 1 of Chambers Chambers (1998) wors wih he followig frequec-domai defiiio of log memor. X is a log memor process if is specral desi fucio saisfies 1

4 d (1) f ( λ) ~ c λ as λ, for some < c < ad.5 < d <.5, where λ is he frequec. This defiiio implies ha f(λ) has a zero or a pole a λ = respecivel if d < or d >. Auoregressive fracioall iegraed movig average (ARFIMA) models, iroduced b Hosig (1981) ad Grager ad Joeu (198), are able o reproduce he behavior described b (1). X follows a ARFIMA(p,d,q) model if Φ d ( B)(1 B) X = Θ ( B) ε, where ε is a mea-zero, cosa variace (σ ε ) whie oise process, B is he bacward shif operaor such ha BX = X -1, ad Φ(B)=1-φ 1 B- -φ p B p ad Θ(B)=1+θ 1 B+ +θ q B q are he auoregressive ad movig-average polomials, respecivel. If he roos of Φ(B) are ouside he ui circle, he process is saioar ad if he roos of Θ(B) are ouside he ui circle he process is iverible. The specral desi fucio of saioar ARFIMA processes is give b: iλ σ Θ ε λ d ( e ) i () f ( λ) = 1 e, < λ π, π iλ Φ( e ) where i = -1. Wih hese resuls, Chambers argues ha, if X has he Wold represeaio (1 B) X ( δ + d ) = h= ρ ε h h = ρ( B) ε, where ρ = 1, ρ <, ε is a whie oise sequece h=1 h wih variace σ ε, -.5 < d <.5, ad δ is a ieger umber; he he aggregaed process Y has he followig specral desi: σ δ ε iλ / ( + d ) iλ / (3) f Y ( λ) = 1 e ρ( e ), < λ π, π if X is a soc variable; ad σ λ δ + ε λ λ (4) λ = ρ 1 i / ( d ) i / i / f Y ( ) 1 e ( e ) e, < λ π, π = if X is a flow variable. I shall poi ou here ha a specral desi is well defied ol if he process is secod-order saioar, which is o ae io accou i (3-4), as δ is allowed o ae o posiive ieger values, characerizig o-saioari. However, his is a mior comme i his oe, he major oe referrig o he aliasig effec eplaied e.. Aliasig effec Temporal aggregaio as defied icludes a some par he ac of sip-samplig. This causes he aliasig pheomeo, well ow i he sigal processig lieraure for coiuous-

5 ime processes observed a discree-ime iervals. The lieraure seems o give lile heed, however, o he fac ha he same causes for he aliasig o appear whe observig a coiuous process a discree-ime are also prese i he ac of sip-samplig. I fac, a umber of sigal processig ad ime series boos (e.g. Koopmas 1974, Bloomfield 1976, Priesle 1981, Oppeheim ad Schafer 1989, Hamilo 1994) eplai he aliasig effec ol as a pheomeo which arises whe observig coiuous-ime processes a discree iervals. However, he eplaaio of his pheomeo ad he derivaio of is effecs whe observig a discree process a a lower samplig rae are almos ideical. The differece lies i ha he specral desi fucio of discree-ime processes is defied ol over he rage (-π, π] while ha of coiuous-ime processes is defied over he real lie R. A iuiive eplaaio of he aliasig pheomeo occurrig i discree processes observed a a lower samplig frequec is he followig. Whe he samplig frequec is lower ha ha of he uderlig process b a facor, a compoe wih frequec ω i he origial process will have (omial) frequec λ = ω i he ewl sampled series, possibl fallig ouside (-π, π]. Aleraivel, he frequec ierval (-π, π] for λ i he specrum of he aggregaed process is equivale o (-π/, π/] for ω i he origial process. Clearl some frequecies of he origial process will o be direcl observed i he aggregaed process (ad herefore will o appear i is specrum), for he will complee more ha a eire ccle bewee wo subseque observaios, sice heir respecive periods are smaller ha he samplig period. Isead, compoes wih hese frequecies will have a appare (lower) frequec i he aggregaed process, differe from he real frequec. All frequecies uder he same appare frequec will be observed ogeher. This is, loosel speaig, he aliasig effec ad is equivale o foldig he specrum imes io he ierval (-π/, π/]. The aliasig effec arisig from aggregaig discree-ime soc processes is give as par of he followig heorem: Theorem 1: Le X be a covariace saioar discree-ime process wih specral desi fucio f (ω) ad Y he correspodig aggregaed process. The specral desi fucio of Y, f (λ), is give b: 1a) If is a odd umber: 1 λ jπ λ jπ (5) f ( λ) = g, + f +, < λ π ; j= 3

6 1b) If is a eve umber: (6) 1 λ jπ λ jπ g, + f +, < λ j= 1 f ( λ) =, 1 1 λ jπ λ jπ g, + f +, < λ π j= where g(,ω) = 1 if he aggregaio is of he soc pe ad g(,ω) = si ( θ / ) πf ( ω) = lim if he aggregaio is of he flow pe. θ ω si ( θ / ) The pure aliasig effec appears i he aggregaio of soc variables, whe g(,ω) = 1, bu i also appears i he aggregaio of flow variables, mied wih oher effecs iroduced b he movig average filer (1 + B + + B -1 ). The fucio F (ω) is he Fejer erel (deails i Priesle, 1981) ad is periodic wih period equal o π. For ω resriced o he ierval (-π, π], i has he highes pea a he frequec zero (far higher ha he subsidiar peas) ad zeros a frequecies ha are ozero muliples of π/ (Nquis frequec) as show i Figure 1 for = 6. Figure 1 illusraes clearl ha afer applig a movig average filer (1+B+ +B -1 ), he low frequecies predomiae. Furhermore, he Fejer erel firs derivaive is zero a all (zero ad ozero) muliples of he Nquis frequec ad he ozero muliples will be folded io he frequec zero afer a furher sip-samplig, so ha he aliasig effec is offse i he aggregaio of he flow pe a lower frequecies. The mai par of he proof of Theorem 1 is adaped from he proof of he Aliasig Theorem for coiuous-ime processes observed a discree-ime iervals, easil foud i he Specral Aalsis boos, e.g. Priesle (1981) ad Oppeheim ad Schafer (1989)..3 Specrum of aggregaed log memor processes Havig saed Theorem 1, i is sraighforward o calculae he specrum of emporal aggregaes of fracioall iegraed processes usig equaio (). Corollar 1: Le X have he Wold represeaio X d = (1 B) ρ ( B) ε where ρ = 1, h=1 ρ h <, ε is a whie oise sequece wih variace σ ε, -.5 < d <.5. The, for π < λ π, he specrum of he aggregaed variable Y is give b: 1a) If X is a soc variable: 4

7 (7) 1 σ + = ( ) / d ( + ) / ε i λ j π i λ j π f ( ) 1 ( ) λ e ρ e ; π = 1b) If X is a flow variable: (8) f ( λ) = σ j= j 1 + i( λ jπ ) / d i( λ+ j π ) / 1 ( ) (( + ) / ) ε e ρ e F λ j π. The chage i he summaio idices is uderae o uif he resuls for eve ad odd values of, ad does o affec he resuls because of he periodici displaed b he epoeial of imagiar umbers..4 Simulaio I his subsecio a small simulaio is carried ou o compare he specral desi fucio give i Corollar 1 for aggregaed log memor processes wih ha give b Theorem 1 of Chambers (1998) as displaed i equaios (3-4). Figure shows he specral fucio derived i his oe for aggregaed soc ARFIMA processes ad he oe derived i Chambers (1998), ogeher wih he periodogram ordiaes averaged across 1 realizaios of he process. The X ais shows he idices j = 1,,, T/ represeig he Fourier frequecies jπ/t. Figure 3 does he same as Figure, bu for flow processes. The processes are aggregaed from ARFIMA(,.3,), wih = 3; ARFIMA(1,.3,) wih φ =.8 ad = 4; ARFIMA(,.3,1) wih θ = -.8 ad = 4; ARFIMA(1,.3,1) wih φ = -.4, θ = -.8 ad = 3. The aggregaed series legh is 51 observaios ad he error variace is ae as σ ε = 1. As we ca see for boh soc ad flow aggregaed ARFIMA processes he averaged periodogram ordiaes (dos) are scaered aroud he solid lie, which represes he formulae (7-8) derived i his oe. The formulae derived i Chambers (1998) ad displaed i (3-4), represeed b a dashed lie, ields values somewha differe from he observed i he simulaio eperime. 3 Discussio This oe aims a dispuig some of he resuls derived i Chambers (1998), specificall he specrum of emporal aggregaes of discree-ime log memor processes. The mai poi is ha he aliasig effec was o ae io accou. I provide here he correspodig specrum i ligh of he aliasig effec. A small simulaio provides evidece i m favor. As a resul, wo of he implicaios of Chambers (1998) resuls mus be reviewed. Firs, ha he iegraio order remais cosa afer emporal aggregaio. This is rue for he 5

8 aggregaio of flow variables, as he movig average filer (1 + B + + B -1 ) assigs weighs equal o zero o he frequecies which will alias righ oo he frequec zero ad ver small for hose aliasig o is vicii. For aggregaio of soc variables, if d is posiive, he ubouded eerg i he specrum for low frequecies i he origial process domiaes he eerg comig from aliases of hese low frequecies i he aggregaed process (if his eerg is bouded), ad a codiio similar o (1) sill holds wih same parameer d. However, if aggregaig a soc variable wih egaive d, he codiio (1), ha implies a zero i he frequec zero, will surel be desroed, uless i he ver uliel case where he specrum i he frequecies which will alias o he frequec zero is zero. A curious oe is ha, if log memor is defied i he ime domai as a hperbolical deca of he auocorrelaios, d 1 (9) ρ ~ c as, emporal aggregaio of eiher pe (soc or flow) is o able o desro his proper, ad o eve o chage he iegraio order d. This resul, however, is of lesser pracical imporace because a egaive d is rarel observed empiricall, bu arises frequel from overdifferecig a process. As praciioers usuall aggregae series before differecig hem ad o oherwise, i is uliel ha a (soc) process wih egaive d will be aggregaed. The secod implicaio o be reviewed is he secod esable hpohesis of Chambers (1998, Secio.5). He argues ha he order of iegraio should be he same whe esimaed from differe frequecies of he daa. This is rue for flow variables bu o for soc oes. Eve hough he aggregaio of soc variables reais he specrum behavior i a small eighborhood of zero if d >, he aggregaio of flow variables do i i a far wider eighborhood, irrespecive of he sig of d. These differe frequec-domai behaviors of soc ad flow variables will disicl affec he (semiparameric) esimaio of aggregaed fracioall differeced processes based o he low-frequec periodogram ordiaes. If he process is a flow variable, less bias is liel o be iduced b aggregaio, while if i is a soc variable, i is liel ha aggregaio will icur some bias. I paricular, if d <, he aggregaio of soc fracioall iegraed processes will desro codiio (1). However, for posiive d, whe he sample size icreases he bias eds o disappear as a arrower bad of frequecies is used for esimaio. These cojecures are largel cosise wih Moe Carlo resuls from Souza ad Smih (, 3). 6

9 Appedi Proof of Theorem 1 The specrum of a covariace saioar discree-ime process X is defied b: 1 f e iω ( ω) = γ, -π < ω π (A1) π = where γ is he -h order auocovariace of X. As he auocovariace fucio of real valued processes is a eve fucio, (A1) reduces o: 1 f ( ω) = γ cosω, -π < ω π. (A) π = The specrum f (ω) is he defied as a Fourier cosie series whose coefficies are he auocovariaces of X. As cos ω, =, 1,,..., is a complee orhogoal se over he ierval ( π, π] for eve fucios (ad he specral desi is a eve fucio) he relaio give b (A) is equivale o: π γ = f ( ω)cos ω d = cos ω df ( ), =, ± 1, ±,... (A3) π where F (ω) is he specral disribuio fucio of X. Cosider firs he aggregaio of soc variables. The auocovariaces of Y = X are give hus b: π γ = γ = f ( ω)cos ω d = cos ω df ( ), =, ± 1, ±,... (A4) π Firs ae he simpler case where is a odd umber. The iegral i (A4) ca be spli io: j= ( j+ 1) π / ( j 1) π / π / γ = cos ω df ( ) = cos( ω + jπ ) df ( + jπ / ) (A5) j= / Sice cos(a + jπ) = cos(a), where j is a ieger umber, (A5) rewries o: π / γ = cos( ω) df ( + jπ / ) (A6) j= / Maig λ = ω where λ is he frequec measured i he ime ui of Y, we have: π 1 γ = cos( λ) df ( λ / + jπ / ) (A7) j= However, b (A3) we ca wrie he -h order auovariace of Y as: π γ = f ( λ)cos λ dλ = cos λ df ( λ), =, ± 1, ±,... (A8) π 7

10 The fac ha cos λ, =, 1,,..., is a complee orhogoal se over he ierval ( π, π] for eve fucios, ogeher wih (A7) ad (A8) impl (5) wih g(,ω) = 1. Now if is a eve umber (A5) rewries o: γ + = 1 j= j= 1 / π / cos( ω + jπ ) df ( cos( ω + jπ ) df ( + jπ / ) + (A9) + jπ / ) ad he res of he proof (for soc processes) follows as i he case is odd. Now cosider he aggregaio of flow variables. Le Z = B j= 1 j X be he overlappig aggregaed process of X. The movig average represeaio of (1 + B + + B -1 ) sraighforwardl gives he followig relaioship bewee he specra of Z ad X : 1 ( ) ( ) f z ω = f ω e = iω, -π < ω π. The muliplicaive erm 1 i e ω is equivale o = si ( θ / ) lim = π. F ( ω) θ ω. This laer resul ca be easil verified (see, for eample, si ( θ / ) Bloomfield 1976, p. 51). Now, he aggregaed process Y is obaied from Z hrough a aggregaio of he soc pe, ad he relaioship bewee he specra of Y ad X is ha give i Theorem 1. The proof is complee. 8

11 Refereces Bloomfield, P., 1976, Fourier Aalsis of Time Series: A Iroducio (Wile, New Yor). Chambers, M. J., 1998, Log memor ad aggregaio i macroecoomic ime series, Ieraioal Ecoomic Review 39, Grager, C. W. G. ad R. Joeu, 198, A iroducio o log memor ime series models ad fracioal differecig, Joural of Time Series Aalsis 1, Hamilo, J. D., 1994, Time Series Aalsis (Priceo Uiversi Press, New Jerse). Hosig, J., 1981, Fracioal differecig, Biomeria 68, 1, Koopmas, L.H., 1974, The Specral Aalsis of Time Series (Academic Press, New Yor). Oppeheim, A. V. ad R. W. Schafer, 1989, Discree-Time Sigal Processig, (Preice-Hall, New Jerse). Priesle, M. B., 1981, Specral Aalsis ad Time Series (Academic Press, Lodo). Souza, L. R. ad J. Smih,, Bias i he memor parameer for differe samplig raes, Ieraioal Joural of Forecasig 18, Souza, L. R. ad J. Smih, 3, Effecs of emporal aggregaio o esimaes ad forecass of fracioall iegraed processes: A Moe Carlo sud, Ieraioal Joural of Forecasig, forhcomig. 9

12 Figure 1: The Fejer erel for = 6, resriced o (-π, π] Figure : Chambers s (1998) ad his oe s heoreical specral fucios for aggregaed soc ARFIMA processes, respecivel he dashed ad he coiuous lies. The dos correspod o periodogram ordiaes averaged across 1 realizaios of he processes. 1

13 Figure 3: Chambers s (1998) ad his oe s heoreical specral fucios for aggregaed flow ARFIMA processes, respecivel he dashed ad he coiuous lies. The dos correspod o periodogram averaged across 1 realizaios of he processes. 11

14 ENSAIOS ECONÔMICOS DA EPGE 456. A CONTRACTIVE METHOD FOR COMPUTING THE STATIONARY SOLUTION OF THE EULER EQUATION - Wilfredo L. Maldoado; Humbero Moreira Seembro de 14 págs TRADE LIBERALIZATION AND THE EVOLUTION OF SKILL EARNINGS DIFFERENTIALS IN BRAZIL - Gusavo Gozaga; Naércio Meezes Filho; Crisia Terra Seembro de 31 págs DESEMPENHO DE ESTIMADORES DE VOLATILIDADE NA BOLSA DE VALORES DE SÃO PAULO - Berardo de Sá Moa; Marcelo Ferades Ouubro de 37 págs FOREIGN FUNDING TO AN EMERGING MARKET: THE MONETARY PREMIUM THEORY AND THE BRAZILIAN CASE, Carlos Hamilo V. Araújo; Reao G. Flores Jr. Ouubro de 46 págs. 46. REFORMA PREVIDENCIÁRIA: EM BUSCA DE INCENTIVOS PARA ATRAIR O TRABALHADOR AUTÔNOMO - Samaha Taam Dar; Marcelo Côres Neri; Flavio Meezes Novembro de 8 págs DECENT WORK AND THE INFORMAL SECTOR IN BRAZIL Marcelo Côres Neri Novembro de 115 págs. 46. POLÍTICA DE COTAS E INCLUSÃO TRABALHISTA DAS PESSOAS COM DEFICIÊNCIA - Marcelo Côres Neri; Aleadre Pio de Carvalho; Hessia Guilhermo Cosilla Novembro de 67 págs SELETIVIDADE E MEDIDAS DE QUALIDADE DA EDUCAÇÃO BRASILEIRA Marcelo Côres Neri; Aleadre Pio de Carvalho Novembro de 331 págs BRAZILIAN MACROECONOMICS WITH A HUMAN FACE: METROPOLITAN CRISIS, POVERTY AND SOCIAL TARGETS Marcelo Côres Neri Novembro de 61 págs POBREZA, ATIVOS E SAÚDE NO BRASIL - Marcelo Côres Neri; Wager L. Soares Dezembro de 9 págs INFLAÇÃO E FLEXIBILIDADE SALARIAL - Marcelo Côres Neri; Maurício Piheiro Dezembro de 16 págs DISTRIBUTIVE EFFECTTS OF BRAZILIAN STRUCTURAL REFORMS - Marcelo Côres Neri; José Márcio Camargo Dezembro de 38 págs O TEMPO DAS CRIANÇAS - Marcelo Côres Neri; Daiela Cosa Dezembro de 14 págs EMPLOYMENT AND PRODUCTIVITY IN BRAZIL IN THE NINETIES - José Márcio Camargo; Marcelo Côres Neri; Maurício Corez Reis Dezembro de 3 págs. 47. THE ALIASING EFFECT, THE FEJER KERNEL AND TEMPORALLY AGGREGATED LONG MEMORY PROCESSES - Leoardo R. Souza Jaeiro de 3 3 págs.

15 471. CUSTO DE CICLO ECONÔMICO NO BRASIL EM UM MODELO COM RESTRIÇÃO A CRÉDITO - Bárbara Vascocelos Boavisa da Cuha; Pedro Cavalcai Ferreira Jaeiro de 3 1 págs. 47. THE COSTS OF EDUCATION, LONGEVITY AND THE POVERTY OF NATIONS - Pedro Cavalcai Ferreira; Samuel de Abreu Pessoa Jaeiro de 3 31 págs A GENERALIZATION OF JUDD S METHOD OF OUT-STEADY-STATE COMPARISONS IN PERFECT FORESIGHT MODELS - Paulo Barelli; Samuel de Abreu Pessoa Fevereiro de 3 7 págs AS LEIS DA FALÊNCIA: UMA ABORDAGEM ECONÔMICA - Aloísio Pessoa de Araújo Fevereiro de 3 5 págs THE LONG-RUN ECONOMIC IMPACT OF AIDS - Pedro Cavalcai G. Ferreira; Samuel de Abreu Pessoa Fevereiro de 3 3 págs A MONETARY MECHANISM FOR SHARING CAPITAL: DIAMOND AND DYBVIG MEET KIYOTAKI AND WRIGHT Ricardo de O. Cavalcai Fevereiro de 3 16 págs INADA CONDITIONS IMPLY THAT PRODUCTION FUNCTION MUST BE ASYMPTOTICALLY COBB-DOUGLAS - Paulo Barelli; Samuel de Abreu Pessoa Março de 3 4 págs TEMPORAL AGGREGATION AND BANDWIDTH SELECTION IN ESTIMATING LONG MEMORY - Leoardo R. Souza - Março de 3 19 págs A NOTE ON COLE AND STOCKMAN - Paulo Barelli; Samuel de Abreu Pessoa Abril de 3 8 págs. 48. A HIPÓTESE DAS EXPECTATIVAS NA ESTRUTURA A TERMO DE JUROS NO BRASIL: UMA APLICAÇÃO DE MODELOS DE VALOR PRESENTE - Aleadre Maia Correia Lima; João Vicor Issler Maio de 3 3 págs ON THE WELFARE COSTS OF BUSINESS CYCLES IN THE TH CENTURY - João Vicor Issler; Afoso Arios de Mello Fraco; Osmai Teieira de Carvalho Guillé Maio de 3 9 págs. 48. RETORNOS ANORMAIS E ESTRATÉGIAS CONTRÁRIAS - Marco Aoio Boomo; Ivaa Dall Agol Juho de 3 7 págs EVOLUÇÃO DA PRODUTIVIDADE TOTAL DOS FATORES NA ECONOMIA BRASILEIRA: UMA ANÁLISE COMPARATIVA - Vicor Gomes; Samuel de Abreu Pessoa;Ferado A. Veloso Juho de 3 45 págs MIGRAÇÃO, SELEÇÃO E DIFERENÇAS REGIONAIS DE RENDA NO BRASIL - Eesor da Rosa dos Saos Juior; Naércio Meezes Filho; Pedro Cavalcai Ferreira Juho de 3 3 págs THE RISK PREMIUM ON BRAZILIAN GOVERNMENT DEBT, Adré Soares Loureiro; Ferado de Holada Barbosa - Juho de 3 16 págs FORECASTING ELECTRICITY DEMAND USING GENERALIZED LONG MEMORY - Lacir Jorge Soares; Leoardo Rocha Souza Juho de 3 págs.

16 487. USING IRREGULARLY SPACED RETURNS TO ESTIMATE MULTI-FACTOR MODELS: APPLICATION TO BRAZILIAN EQUITY DATA - Álvaro Veiga; Leoardo Rocha Souza Juho de 3 6 págs BOUNDS FOR THE PROBABILITY DISTRIBUTION FUNCTION OF THE LINEAR ACD PROCESS Marcelo Ferades Julho de 3 1 págs CONVEX COMBINATIONS OF LONG MEMORY ESTIMATES FROM DIFFERENT SAMPLING RATES - Leoardo R. Souza; Jerem Smih; Reialdo C. Souza Julho de 3 págs. 49. IDADE, INCAPACIDADE E A INFLAÇÃO DO NÚMERO DE PESSOAS COM DEFICIÊNCIA - Marcelo Neri ; Wager Soares Julho de 3 54 págs FORECASTING ELECTRICITY LOAD DEMAND: ANALYSIS OF THE 1 RATIONING PERIOD IN BRAZIL - Leoardo Rocha Souza; Lacir Jorge Soares Julho de 3 7 págs. 49. THE MISSING LINK: USING THE NBER RECESSION INDICATOR TO CONSTRUCT COINCIDENT AND LEADING INDICES OF ECONOMIC ACTIVITY - JoãoVicor Issler; Farshid Vahid Agoso de 3 6 págs REAL EXCHANGE RATE MISALIGNMENTS - Maria Crisia T. Terra; Frederico Esrella Careiro Valladares Agoso de 3 6 págs ELASTICITY OF SUBSTITUTION BETWEEN CAPITAL AND LABOR: A PANEL DATA APPROACH - Samuel de Abreu Pessoa ; Silvia Maos Pessoa; Rafael Rob Agoso de 3 3 págs A EXPERIÊNCIA DE CRESCIMENTO DAS ECONOMIAS DE MERCADO NOS ÚLTIMOS 4 ANOS Samuel de Abreu Pessoa Agoso de 3 págs NORMALITY UNDER UNCERTAINTY Carlos Eugêio E. da Cosa Seembro de 3 8 págs RISK SHARING AND THE HOUSEHOLD COLLECTIVE MODEL - Carlos Eugêio E. da Cosa Seembro de 3 15 págs REDISTRIBUTION WITH UNOBSERVED 'EX-ANTE' CHOICES - Carlos Eugêio E. da Cosa Seembro de 3 3 págs OPTIMAL TAXATION WITH GRADUAL LEARNING OF TYPES - Carlos Eugêio E. da Cosa Seembro de 3 6 págs. 5. AVALIANDO PESQUISADORES E DEPARTAMENTOS DE ECONOMIA NO BRASIL A PARTIR DE CITAÇÕES INTERNACIONAIS - João Vicor Issler; Rachel Couo Ferreira Seembro de 3 9 págs. 51. A FAMILY OF AUTOREGRESSIVE CONDITIONAL DURATION MODELS - Marcelo Ferades; Joachim Grammig Seembro de 3 37 págs. 5. NONPARAMETRIC SPECIFICATION TESTS FOR CONDITIONAL DURATION MODELS - Marcelo Ferades; Joachim Grammig Seembro de 3 4 págs.

17 53. A NOTE ON CHAMBERS S LONG MEMORY AND AGGREGATION IN MACROECONOMIC TIME SERIES Leoardo Rocha Souza Seembro de 3 11págs. 54. ON CHOICE OF TECHNIQUE IN THE ROBINSON-SOLOW-SRINIVASAN MODEL - M. Ali Kha Seembro de 3 34 págs.