Scientific Research

An Academic Publisher

**A Survey of Methods to Interpolate, Distribute and Extra- polate Time Series** ()

This survey provides an overview with a broad coverage of the literature on methods for temporal disaggregation and benchmarking. Dozens of methods, procedures and algorithms have been proposed in the statistical and economic literature to solve the problem of transforming a low-frequency series into a high-frequency one. This paper classifies and reviews the procedures, provides interesting discussion on the history of the methodological development in this literature and permits to identify the assets and drawbacks of each method, to comprehend the current state of art on the subject and to identify the topics in need of further development. It would be useful for readers who are interested in the techniques but are not yet familiar with the literature and also for researchers who would like to keep up with the recent developments in this area. After reading the article the reader should have a good understanding of the most important approaches, their shortcomings and advantages, and be able to make an informed judgment on which methods are most suitable for his or her purpose. Interested readers, however, will not find much detail of the methods reviewed. Due to the broadness of the subjects and the large number of studies being referenced, it is provided some general assessments on the methods revised without great detailed analysis. This review article could serve as a brief introduction to the literature on temporal disaggregation.

Keywords

Share and Cite:

J. Pavía-Miralles, "A Survey of Methods to Interpolate, Distribute and Extra- polate Time Series,"

*Journal of Service Science and Management*, Vol. 3 No. 4, 2010, pp. 449-463. doi: 10.4236/jssm.2010.34051.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | J. G. de Gooijer and R. J. Hyndman, “25 Years of Time Series Forecasting,” International Journal of Forecasting, Vol. 22, No. 3, 2006, pp. 443-473. |

[2] | M. Marcellino and G. L. Mazzi, “Introduction to Advances in Business Cycle Analysis and Forecasting,” Journal of Forecasting, Vol. 29, No. 1-2, 2010, pp. 1-5. |

[3] | J. Casals, M. Jerez and S. Sotoca, “Modelling and Forecasting Time Series Sampled at Different Frequencies,” Journal of Forecasting, Vol. 28, No. 4, 2008, pp. 316-341. |

[4] | A. Zellner and C. Montmarquette, “A Study of Some Aspects of Temporal Aggregation Problems in Econo- metric Analyses,” The Review of Economic and Statistics, Vol. 53, No. 4, 1971, pp. 335-342. |

[5] | H. Lütkepohl, “Linear Transformations of Vector ARMA Processes,” Journal of Econometrics, Vol. 4, No. 3, 1984, pp. 283-293. |

[6] | Th. Nijman and F. C. Palm, “Series Temporelles Incom- pletes en Modelisation Macroeconomiques,” Cahiers Du Seminaire d’Econometrie, Vol. 29, No. 1, 1985, pp. 141- 168. |

[7] | Th. Nijman and F. C. Palm “Efficiency Gains due to Missing Data Procedures in Regression Models,” Statististical Papers, Vol. 29, 1988, pp. 249-256. |

[8] | Th. Nijman and F. C. Palm, “Consistent Estimation of Regression Models with Incompletely Observed Exogenous Variables,” The Annals of Economics and Statististics, Vol. 12, 1988, pp. 151-175. |

[9] | Th. Nijman and F. C. Palm, “Predictive Accuracy Gain From Disaggregate Sampling in ARIMA Models,” Journal of Business and Economic Statistics, Vol. 8, No. 4, 1990, 189-196. |

[10] | F. C. Palm and Th. Nijman, “Linear Regression Using both Temporally Aggregated and Temporally Disaggregated Data,” Journal of Econometrics, Vol. 19, No. 2-3, 1982, pp. 333-343. |

[11] | A. A. Weiss, “Systematic Sampling and Temporal Aggregation in Time Series Models,” Journal of Econometrics, Vol. 26, No. 3, 1984, 271-281. |

[12] | OECD, “Sources and Methods Used by the OECD Member Countries, Quarterly National Accounts,” Paris, OE- CD Publications, 1996. |

[13] | J. M. Pavía-Miralles and B. Cabrer-Borrás, “On Estima- ting Contemporaneous Quarterly Regional GDP,” Journal of Forecasting, Vol. 26, No. 3, 2007, pp. 155-177. |

[14] | T. DiFonzo and R. Filosa, “Methods of Estimation of Quarterly National Account Series: A Comparison,” unpublished, [Journee Franco-Italianne de Comptabilite Nationale (Journee de Stadistique), Lausanne, 1987, pp. 1-69. |

[15] | J. M. Pavía-Miralles, “La Problemática de Trimestra- lización de Series Anuales,” Valencia, Universidad de Valencia, 1997. |

[16] | Eurostat, “Handbook of Quarterly National Accounts,” Luxembourg, European Commission, 1999. |

[17] | E. B. Dagum and P. A. Cholette, “Benchmarking, Tem- poral Distribution and Reconciliation Methods for Time Series,” New York, Springer Verlag, 2006. |

[18] | J. M. Pavía and B. Cabrer, “On Distributing Quarterly National Growth among Regions,” Environment and Planning A, Vol. 40, No. 10, 2008, pp. 2453-2468. |

[19] | A. M. Bloem, R. J. Dippelsman and N. Maehle, “Quarterly National Accounts Manual. Concepts, Data Sources, and Compilation,” Washington D.C., International Mone- tary Fund, 2001. |

[20] | J. H. C. Lisman and J. Sandee, “Derivation of Quarterly Figures from Annual Data,” Applied Statistics, Vol. 13, No. 2, 1964, pp. 87-90. |

[21] | S. Zani, “Sui Criteri di Calcolo Dei Valori Trimestrali di Tendenza Degli Aggregati della Contabilitá Nazionale,” Studi e Ricerche, Vol. VII, 1970, pp. 287-349. |

[22] | C. Greco, “Alcune Considerazioni Sui Criteri di Calcolo di Valori Trimestrali di Tendenza di Serie Storiche Annuali,” Annali della Facoltà di Economia e Commercio, Vol. 4, 1979, pp. 135-155. |

[23] | H. Glejser, “Une Méthode d’Evaluation de Donnés Mensuelles à Partir d’Indices Trimestriels ou Annuels,” Cahiers Economiques de Bruxelles, No. 29, 1966, pp. 45- 54. |

[24] | C. Almon, “The Craft of Economic Modeling,” Ginn Press, Boston, 1988. |

[25] | L. Hedhili and A. Trabelsi, “A Polynomial Method for Temporal Disaggregation of Multivariate Time Series,” Luxemburg, Office for Official Publications of the European Communities, 2005. |

[26] | L. Zaier and A. Trabelsi, “Polynomial Method for Tem- poral Disaggregation of Multivariate Time Series,” Communications in Statistics-Simulation and Computation, Vol. 36, No. 3, 2007, pp. 741-759. |

[27] | J. C. G. Boot, W. Feibes and J. H. Lisman, “Further Methods of Derivation of Quarterly Figures from Annual Data,” Applied Statistics, Vol. 16, No. 1, 1967, pp. 65-75. |

[28] | V. A. Ginsburgh, “A Further Note on the Derivation of Quarterly Figures Consistent with Annual Data,” Applied Statistics, Vol. 22, No. 3, 1973, pp. 368-374. |

[29] | K. J. Cohen, M. Müller and M. W. Padberg, “Autoregressive Approaches to Disaggregation of Time Series Data,” Applied Statistics, Vol. 20, No. 2, 1971, pp. 119-129. |

[30] | J. M. Pavía, B. Cabrer and J. M. Felip, “Estimación del VAB Trimestral No Agrario de la Comunidad Valen- ciana,” Valencia, Generalitat Valenciana, 2000. |

[31] | H. E. Doran, “Prediction of Missing Observations in the Time Series of an Economic Variable,” Journal of the American Statistical Association, Vol. 69, No. 346, 1974, pp. 546- 554. |

[32] | D. O. Stram and W. W. S. Wei, “Temporal Aggregation in the ARIMA Process,” Journal of Time Series Analysis, Vol. 7, No. 4, 1986, pp. 279-292. |

[33] | G. C. Chow and A. Lin, “Best Linear Unbiased Estimation of Missing Observations in an Economic Time Series,” Journal of the American Statistical Association, Vol. 71, No. 355, 1976, pp. 719-721. |

[34] | S. Rodríguez-Feijoo, A. Rodríguez-Caro and D. Dávila- Quintana, “Methods for Quarterly Disaggregation without Indicators: A Comparative Study Using Simulation,” Computational Statistics and Data Analysis, 2003, Vol. 43, No. 1, pp. 63–78. |

[35] | B. Chen, “An Empirical Comparison of Methods for Temporal Disaggregation at the National Accounts,” 2007. http://www.fcsm.gov/07papers/Chen.V-A.pdf |

[36] | F. T. Denton, “Adjustment of Monthly or Quarterly Series to Annuals Totals: An Approach Based on Quadratic Minimization,” Journal of the American Statistical Association, Vol. 66, No. 333, 1971, pp. 99-102. |

[37] | W. W. S. Wei and D. O. Stram, “Disaggregation of Time Series Models,” Journal of the Royal Statististical Society, Ser. B, Vol. 52, No. 3, 1990, pp. 453-467. |

[38] | P. Nasse, “Le Système des Comptes Nationaux Trime- strels,” Annales de L’Inssée, Vol. 14, 1973, pp. 127-161. |

[39] | C. G. Chang and T. C. Liu, “Monthly Estimates of Certain National Product Components, 1946-49,” The Review of Economics and Statistics, Vol. 33, No. 3, 1951, pp. 219-227. |

[40] | J. Bournay and G. Laroque, “Réflexions sur le Méthode d’élaboration des Comptes Trimestriels,” Annales de L’Insée, Vol. 36, 1979, pp. 3-29. |

[41] | G. C. Chow and A. Lin, “Best Linear Unbiased Interpola- tion, Distribution, and Extrapolation of Time Series By Related Series,” The Review of Economics and Statistics, Vol. 53, No. 4, 1971, pp. 372-375. |

[42] | M. Friedman, “The Interpolation of Time Series by Related Series,” Journal of the American Statistical Association, Vol. 57, No. 300, 1962, pp. 729-757. |

[43] | V. L. Bassie, “Economic Forecasting,” New York, Mc Graw-Hill, 1958, pp. 653-661. |

[44] | ISCO, “L’Aggiustamento delle Stime nei Conti Economici Trimestrali,” Rassegna del Valori Interni dell’Istituto, Vol. 5, 1965, pp. 47-52. |

[45] | OECD, “La Comptabilité Nationale Trimestrelle,” Series Etudes Economiques, Vol. 21, 1966. |

[46] | ISTAT, “I Conti Economici Trimestrali dell’Italia 1970-1982,” Supplemento al Bollettino Mensile di Statistica, Vol. 12, 1983. |

[47] | G. Vangrevelinghe, “L’Evolution à Court Terme de la Consommation des Ménages: Connaisance, Analyse et Prévision,” Etudes et Conjoncture, Vol. 9, 1966, pp. 54- 102. |

[48] | T. DiFonzo, “Temporal Disaggregation of Economic Time Series: Towards a Dynamic Extension,” Luxembourg, Office for Official Publications of the European Communities, 2003. |

[49] | J. Somermeyer, R. Jansen and J. Louter, “Estimating Qu- arterly Values of Annually Know Variables in Quarterly Relationships,” Journal of the American Statistical Association, Vol. 71, No. 355, 1976, pp. 588-595. |

[50] | P. A. Cholette, “Adjusting Sub-Anual Series to Yearly Benchmarks,” Survey Methodology, Vol. 10, No. 1, 1984, pp. 35-49. |

[51] | S. C. Hillmer and A. Trabelsi, “Benchmarking of Economic Time Series,” Journal of the American Statistical Association, Vol. 82, No. 400, 1987, pp. 1064-1071. |

[52] | A. Trabelsi and S. C. Hillmer, “Bench-marking Time Series with Reliable Bench-Marks,” Applied Statistics, Vol. 39, No. 3, 1990, pp. 367-379. |

[53] | P. A. Cholette and E. B. Dagum, “Benchmarking Time Series With Autocorrelated Survey Errors,” International Statistical Review, Vol. 62, No. 3, 1994, pp. 365-377. |

[54] | T. DiFonzo, “Temporal Disaggregation of System of Time Series When the Aggregates is Known,” Luxembourg, Office for Official Publications of the European Communities, 2003 [INSEE-Eurostat Quarterly National Accounts Workshop, Paris-Bercy, R. Barcellan and G. L. Mazzi, Eds., December 1994, pp. 63-78)]. |

[55] | T. DiFonzo and M. Marini, “Benchmarking Systems of Seasonally Adjusted Time Series,” Journal of Business Cycle Measurement and Analysis, Vol. 2, No. 1, 2005, pp. 84-123. |

[56] | Z. G. Chen and K. H. Wu, “Comparison of Benchmarking Methods with and without a Survey Error Model,” International Statistical Review, Vol. 74, No. 3, 2006, pp. 285-304. |

[57] | E. B. Dagum, P. A. Cholette and Z. G. Chen, “A Unified View of Signal Extraction, Interpolation, Benchmarking, and Extrapolation of Time Series,” International Statistical Review, Vol. 66, No. 3, 1998, pp. 245-269. |

[58] | V. M. Guerrero, “Monthly Disaggregation of a Quarterly Time Series and Forecasts of Its Observable Monthly Values,” Journal of Official Statistics, Vol. 19, No. 3, 2003, pp. 215-235. |

[59] | V. M. Guerrero, “Temporal Disaggregation of Time Series: An ARIMA-Based Approach,” International Statistical Review, Vol. 58, No. 1, 1990, pp. 29-46. |

[60] | V. M. Guerrero and J. Martínez, “A Recursive ARIMA-Based Procedure for Disaggregating a Time Series Variable Using Concurrent Data,” Test, Vol. 4, No. 2, 1995, pp. 359-376. |

[61] | V. M. Guerrero and F. H. Nieto, “Temporal and Contemporaneous Disaggregation of Multiple Economic Time Series,” Test, Vol. 8, No. 2 1999, 459-489. |

[62] | D. M. Aadland, “Distribution and Interpolation using Transformed Data,” Journal of Applied Statistics, Vol. 27, No. 2, 2000, pp. 141-156. |

[63] | M. Pinheiro and C. Coimbra, “Distribution and Extrapolation of Time Series by Related Series Using Logarithms and Smoothing Penalties,” Economia, Vol. 17, October 1993, pp. 359-374. |

[64] | T. Proietti, “Distribution and Interpolation Revisited: A Structural Approach,” Statistica, Vol. 58, No. 47, 1998, pp. 411-432. |

[65] | T. DiFonzo, “Temporal Disaggregation Using Related Series: Log-Transformation and Dynamic Extensions,” Rivista Internazionale di Scienze Economiche e Commerciali, Vol. 50, No. 2, 2003, pp. 371-400. |

[66] | R. B. Fernández, “A Methodological Note on the Estimation of Time Series,” The Review of Economics and Statistics, Vol. 63, No. 3, 1981, pp. 471-478. |

[67] | W. R. Vanhonacker, “Estimating Dynamic Response Mo- dels when the Data are Subject to Different Temporal Aggregation,” Marketing Letters, Vol. 1, No. 2, 1990, pp. 125-137. |

[68] | J. Jacobs, “‘Dividing by 4’: A Feasible Quarterly Forecasting Method?” CCSO Series 22. Groningen: Center for Cyclical and Structural Research, 2004. http://ww- w.eco.rug.nl/ccso/CCSO series/ccso22.pdf |

[69] | E. G. Drettakis, “Missing Data in Econometric Estimation,” Review of Economic Studies, Vol. 40, No. 4, 1973, pp. 537-552. |

[70] | J. D. Sargan and E. G. Drettakis, “Missing Data in an Autoregressive Model,” International Economic Review, Vol. 15, No. 1, 1974, pp. 39-59. |

[71] | M. G. Dagenais, “The Use of Incomplete Observations in Multiple Regression Analysis: A Generalized Least Squa- res Approach,” Journal of Econometrics, Vol. 1, No. 4, 1973, pp.317-328. |

[72] | M. G. Dagenais, “Incomplete Observations and Simultaneous Equations Models,” Journal of Econometrics, Vol. 4, No. 3, 1976, pp. 231-241. |

[73] | A. P. Dempster, N. M. Laird and D. B. Rubin, “Maximun Likelihood from Incomplete Data via the EM Algorithm,” Journal of the Royal Statistical Society, Ser. B, Vol. 39, No. 1, 1977, pp. 1-38. |

[74] | C. Hsiao, “Linear Regression Using Both Temporally Aggregated and Temporally Disaggregated Data,” Journal of Econometrics, Vol. 10, No. 2, 1979, pp. 243-252. |

[75] | C. Hsiao, “Missing Data and Maximum Likelihood Estimation,” Economics Letters, Vol. 6, No. 3, 1980, pp. 249-253. |

[76] | C. Gourieroux and A. Monfort, “On the Problem of Missing Data in Linear Models,” Review of Economic Studies, Vol. 48, No. 4, 1981, pp. 579-586. |

[77] | F. C. Palm and Th. Nijman, “Missing Observations in the Dynamic Regression Model,” Econometrica, Vol. 52, No. 6, 1984, pp. 1415-1435. |

[78] | D. Conniffe, “Small-Sample Properties of Estimators of Regression Coefficients Given a Common Pattern of Missing Data,” Review of Economic Studies, Vol. 50, No. 1, 1983, pp. 111-120. |

[79] | Th. Nijman and F. C. Palm, “The Construction and Use of Approximations for Missing Quarterly Observations: A Model Approach,” Journal of Business and Economic Statistics, Vol. 4, No. 1, 1986, pp. 47-58. |

[80] | J. M. C. Santos Silva and F. N. Cardoso, “The Chow-Lin Method Using Dynamic Models,” Economic Modelling, Vol. 18, No. 2, 2001, pp. 269-280. |

[81] | S. Gregoir, “Propositions pour une Désagrégation Tem- porelle Basée sur des Modèles Dynamiques Simples,” Luxembourg, Office for Official Publications of the European Communities, 2003. |

[82] | INE, “Contabilidad Nacional Trimestral de Espa?a. Metodología y Serie Trimestral 1970-1992,” Madrid, Instituto Nacional de Estadística, 1993. |

[83] | ISTAT, “Principali Caratteristiche della Correzione per i Giorni Lavorativi dei Conti Economici Trimestrali,” Rome, ISTAT, 2003. |

[84] | INSEE, “Methodology of French Quarterly National Accounts,” 2004. http://www.insee.fr/en/indicateur/cnat_ trim/ methodologie.htm |

[85] | T. Abeysinghe and C. Lee, “Best Linear Unbiased Dis- aggregation of Annual GDP to Quarterly Figures: The Case of Malaysia,” Journal of Forecasting, Vol. 17, No. 7, 1998, pp. 527-537. |

[86] | T. Abeysinghe and G. Rajaguru, “Quarterly Real GDP Estimates for China and ASEAN4 with a Forecast Evaluation,” Journal of Forecasting, Vol. 23, No. 6, 2004, pp. 33-37. |

[87] | J. M. Pavía and B. Cabrer, “Estimación Congruente de Contabilidades Trimestrales Regionales: Una Aplica- ción,” Investigación Económica, Vol. 62, No. 21, 2003, pp. 119-141. |

[88] | D. Norman, and T. Walker, “Co-movement of Australian State Business Cycles,” Australian Economic Papers, Vol. 46, No. 4, 2007, pp. 360-374. |

[89] | L. R. Acosta, J. L. Cortigiani and M. B. Diéguez, “Trimes- tralización de Series Económicas Anuales,” Buenos Aires, Banco Central de la República Argentina, 1977. |

[90] | J. Cavero, H. Fernández-Abascal, I. Gómez, C. Lorenzo, B. Rodríguez, J. L. Rojo and J. A. Sanz, “Hacia un Modelo Trimestral de Predicción de la Economía Castellano- Leonesa. El Modelo Hispalink CyL,” Cuadernos Ara- goneses de Economía, Vol. 4, No. 2, 1994, pp. 317-343. |

[91] | IGE, “Contabilidade Trimestral de Galicia. Metodoloxía e Series Históricas 1980-1991,” Santiago de Compostela, Instituto Galego de Estadística, 1997. |

[92] | L. Barbone, G. Bodo and I. Visco, “Costi e Profitti in Senso Stretto: un’Analisi du Serie Trimestrali, 1970- 1980,” Bolletino della Banca d’Italia, Vol. 36, 1981, pp. 465-510. |

[93] | E. Quilis, “Benchmarking Techniques in the Spanish Quarterly National Accounts,” Luxembourg, Office for Official Publications of the European Communities, 2005. |

[94] | J. R. Schmidt, “A General Framework for Interpolation, Distribution and Extrapolation of Time Series by Related Series,” In: Regional Econometric Modelling, Boston, Kluwer Nighoff Pub, 1986, pp. 181-194. |

[95] | J. M. Pavía, L. E. Vila and R. Escuder, “On the Perfor- mance of the Chow-Lin Procedure for Quarterly Interpolation of Annual Data: Some Monte-Carlo Analysis,” Spanish Economic Review, Vol. 5, No. 4, 2003, pp. 291- 305. |

[96] | R. B. Litterman, “A Random Walk, Markov Model for Distribution of Time Series,” Journal of Business and Economic Statistics, Vol. 1, 1983, pp. 169-173. |

[97] | P. Nelson and G. Gould, “The Stochastic Properties of the Income Velocity of Money,” American Economic Review, Vol. 64, No. 3, 1974, pp. 405-418. |

[98] | R. B. Fernández, “Expectativas Adaptativas vs. Expecta- tivas Racionales en la Determinación de la Inflación y el Empleo,” Cuadernos de Economía, Vol. 13, No. 40, 1976, pp. 37-58. |

[99] | J. L. Silver, “Two Results Useful for Implementing Litterman’s Procedure for Interpolating a Time Series,” Journal of Business and Economic Statistics, Vol. 4, No. 1, 1986, pp. 129-130. |

[100] | W. Chan, “Disaggregation of Annual Time-Series Data to Quarterly Figures: A Comparative Study,” Journal of Forecasting, Vol. 12, No. 8, 1993, pp. 677-688. |

[101] | E. L. Salazar, R. J. Smith and R. Weale, “Interpolation using a Dynamic Regression Model: Specification and Monte Carlo Properties,” National Institute of Economic and Social Research, No. 126, 1997. |

[102] | E. L. Salazar, R. J. Smith and R. Weale, “A Monthly Indicator of GDP” National Institute of Economic Review, No. 161, 1997, pp. 84-89. |

[103] | T. DiFonzo, “Constrained Retropolation of Highfrequen- cy Data Using Related Series: A Simple Dynamic Model Approach,” Statistical Methods and Applications, Vol. 12, No. 1, 2003, pp. 109-119. |

[104] | E. Quilis, “Desagregación Temporal Mediante Modelos Dinámicos: El Método de Santos Silva y Cardoso,” Boletín Trimestral de Coyuntura, No. 88, 2003, pp. 1-11. |

[105] | A. Abad and E. Quilis, “Software to Perform Temporal Disaggregation of Economic Time Series,” Luxemburg, Office for Official Publications of the European Communities, 2005. |

[106] | T. DiFonzo, “The Estimation of M Disaggregate Time Series when Contemporaneous and Temporal Aggregates are Known,” The Review of Economics and Statistics, Vol. 72, No. 1, 1990, pp. 178-182. |

[107] | E. Quilis, “A MATLAB Library of Temporal Disaggrega- tion Methods: Summary,” Madrid, Instituto Nacional de Estadística, 2002. |

[108] | N. Rossi, “A Note on the Estimation of Disaggregate Time Series when the Aggregate is Known,” The Review of Economics and Statistics, Vol. 64, No. 4, 1982, pp. 695-696. |

[109] | B. Cabrer and J. M. Pavía, “Estimating J(>1) Quarterly Time Series in Fulfilling Annual and Quarterly Constraints,” International Advances in Economic Research, Vol. 5, No. 3, 1999, pp. 339-350. |

[110] | J. M. Pavía-Miralles, “Desagregación Conjunta de Series Anuales: Perturbaciones AR(1) Multivariante,” Investig- aciones Económicas, Vol. 24, No. 3, 2000, 727-737. |

[111] | A. C. Harvey “Forecasting, Structural Time Series and the Kalman Filter,” Cambridge, Cambridge University Press, 1989. |

[112] | R. H. Jones, “Maximum Likelihood Fitting of ARMA Models to Time Series with Missing Observations,” Technometrics, Vol. 22, No. 3, 1980, pp. 389-395. |

[113] | A. C. Harvey and R. G. Pierse, “Estimating Missing Observations in Economic Time Series,” Journal of the American Statistical Association, Vol. 79, No. 385, 1984, 125-131. |

[114] | C. F. Ansley and R. Kohn, “Estimating, Filtering and Smoothing in State Space Models with Incompletely Specified Initial Conditions,” Annals of Statistics, Vol. 13, No. 4, 1985, pp. 1286-1316. |

[115] | R. Kohn and C. F. Ansley, “Estimation, Prediction, and Interpolation for ARIMA Models with Missing Data,” Journal of the American Statistical Association, Vol. 81, No. 385, 1986, pp. 751-761. |

[116] | M. Al-Osh, “A Dynamic Linear Model Approach for Disaggregating Time Series Data,” Journal of Forecasting, Vol. 8, No. 2, 1989, pp. 85-96. |

[117] | V. Gómez and A. Maravall, “Estimation, Prediction and Interpolation for Nonstationary Series with the Kalman Filter,” Journal of the American Statistical Association, Vol. 89, No. 426, 1994, pp. 611-624. |

[118] | P. De Jong, “Smoothing and Interpolation with the State-Space Model,” Journal of the American Statistical Association, Vol. 84, No. 408, 1989, pp. 1085-1088. |

[119] | J. Durbin and B. Quenneville, “Benchmarking by State Space Models,” International Statistical Review, Vol. 65, No. 1, 1997, pp. 23-48. |

[120] | V. Gómez, A. Maravall and D. Pe?a, “Missing Observations in ARIMA Models. Skipping Strategy versus Additive Outlier Approach,” Journal of Econometrics, Vol. 88, No. 3, 1999, pp. 341-363. |

[121] | G. Gudmundsson, “Disaggregation of Annual Flow Data with Multiplicative Trends,” Journal of Forecasting, Vol. 18, No. 1, 1999, pp. 33-37. |

[122] | N. A. Cuche and M. K. Hess, “Estimating Monthly GDP in a General Kalman Filter Framework: Evidence from Switzerland,” Economic and Financial Modelling, Vol. 7, No. Winter, 2000, pp. 1-37. |

[123] | H. Liu and S. G. Hall, “Creating High-Frequency National Accounts with State-Space Modelling: A Monte Carlo Experiment,” Journal of Forecasting, Vol. 20, No. 6, 2001, pp. 441-449. |

[124] | T. Amemiya and R. Y. Wu, “The Effect of Aggregation on Prediction in the Autoregressive Model,” Journal of the American Statistical Association, Vol. 67, No. 339, 1972, pp. 628-632. |

[125] | W. W. S. Wei, “Some Consequences of Temporal Aggre- gation in Seasonal Time Series Models,” In: Seasonal Analysis of Economic Time Series,” A. Zellner, Ed., Washington DC, Government Printing Office, 1978, pp. 433-448. |

[126] | W. W. S. Wei, “Effect of Systematic Sampling on ARIMA Models,” Communications in Statistics A, Vol. 10, No. 23, 1981, pp. 2389-2398. |

[127] | R. J. Rossana and J. J. Seater, “Temporal Aggregation and Economic Time Series,” Journal of Business and Economic Statistics, Vol. 13, No. 4, 1995, pp. 441-451. |

[128] | H. J. Werner, “On the Temporal Aggregation in Discrete Dynamical Systems,” in System Modeling and Optimatization, R. F. Drenick and F. Kozin, Eds., New York: Springer-Verlag, 1982, pp. 819-825. |

[129] | L. K. Hotta and K. L. Vasconcellos, “Aggregation and Disaggregation of Structural Time Series Models,” Journal of Time Series Analysis, Vol. 20, No. 2, 1999, pp. 155-171. |

[130] | F. Moauro and G. Savio, “Temporal Disaggregation Using Multivariate Structural Time Series Models,” The Econometrics Journal, Vol. 8, No. 2, 2005, pp. 214-234. |

[131] | T. Proietti, “Temporal Disaggregation by State Space Methods: Dynamic Regression Methods Revisited,” The Econometrics Journal, Vol. 9, No. 3, 2006, pp. 357-372. |

[132] | R. H. Jones, “Spectral Analysis with Regularly Missed Observations,” Annals of Mathematical Statistics, Vol. 33, No. 2, 1962, pp. 455-461. |

[133] | E. Parzen, “Mathematical Considerations in the Estima- tion of Spectra,” Technometrics, Vol. 3, No. 2, 1961, pp. 167-190. |

[134] | E. Parzen, “On Spectral Analysis with Missing Observa- tions and Amplitude Modulation,” Sankhy? A, Vol. 25, No. 4, 1963, pp. 383-392. |

[135] | P. A. Scheinok, “Spectral Analysis with Randomly Missed Observations: The Binomial Case,” Annals of Mathe- matical Statistics, Vol. 36, No. 3, 1965, pp. 971-977. |

[136] | P. Bloomfield, “Spectral Analysis with Randomly Missing Observations,” Journal of the Royal Statistical Society, Series B, Vol. 32, No. 3, 1970, pp. 369-380. |

[137] | P. Bloomfield, “An Exponential Model for the Spectrum of a Scalar Time Series,” Biometrika, Vol. 60, No. 2, 1973, pp. 217-226. |

[138] | C. M. C. Toloi and P. A. Morettin, “Spectral Analysis for Amplitude-Modulated Time Series,” Journal of Time Series Analysis, Vol. 14, No. 4, 1993, pp. 409-432. |

[139] | W. Dunsmuir, “Estimation for Stationary Time Series When Data are Irregularly Spaced or Missing,” In: D. F. Findley Eds., Applied Time Series Analysis II, New York: Academic Press, 1981, pp. 609-649. |

[140] | W. Dunsmuir and P. M. Robinson, “Parametric Estimators for Stationary Time Series with Missing Observations,” Advances in Applied Probability, Vol. 13, No. 1, 1981, pp. 129-146. |

[141] | W. Dunsmuir and P. M. Robinson, “Estimation of Time Series Models in the Presence of Missing Data,” Journal of the American Statistical Association, Vol. 76, No. 375, 1981, pp. 456-467. |

[142] | W. Clinger and J. W. VanNess, “On Unequally Spaced Time Points in Time Series,” Annals of Statistics, Vol. 4, No. 4, 1976, pp. 736-745. |

[143] | G. Gudmundsson, “Estimation of Continuous Flows from Observed Aggregates,” Journal of the Royal Statistical Society, Ser. D, Vol. 50, No. 3, 2001, pp. 285-293. |

[144] | M. Marcellino, “Pooling-Based Data Interpolation and Backdating,” Journal of Time Series Analysis, Vol. 28, No. 1, 2007, pp. 53-71. |

[145] | T. Proietti, “Multivariate Temporal Disaggregation with Cross-sectional Constraints,” Journal of Applied Statistics, 2010, in-Press. DOI: 10.1080/02664763.2010.505952. |

[146] | E. Angelini, J. Henry and M. Marcellino, “Interpolation and Backdating with a Large Information Set,” Journal of Economic Dynamics and Control, Vol. 30, No. 12, 2006, pp. 2693-2724. |

[147] | T. Proietti and F. Moauro, “Dynamic Factor Analysis with Nonlinear Temporal Aggregation Constraints,” Applied Statistics, Vol. 55, No. 2, 2006, pp. 281-300. |

[148] | T. Proietti, and F. Moauro, “Temporal Disaggregation and the Adjustment of Quarterly National Accounts for Seasonal and Calendar Effects,” Journal of Official Statistics, Vol. 24, No. 1, 2008, pp. 115-132. |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.