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.

[1]  G. Chow,et al.  Best Linear Unbiased Interpolation, Distribution, and Extrapolation of Time Series by Related Series , 1971 .

[2]  James R. Schmidt.  A General Framework for Interpolation, Distribution, and Extrapolation of a Time Series by Related Series , 1986 .

[3]  Perry A. Scheinok,et al.  Spectral Analysis with Randomly Missed Observations: The Binomial Case , 1965 .

[4]  Helmut Lütkepohl,et al.  Linear transformations of vector ARMA processes , 1984 .

[5]  Filippo Moauro,et al.  Temporal Disaggregation Using Multivariate Structural Time Series Models , 2005 .

[6]  Roque B Fernandez,et al.  A Methodological Note on the Estimation of Time Series , 1981 .

[7]  Daniel Peña,et al.  Missing observations in ARIMA models: Skipping approach versus additive outlier approach , 1999 .

[8]  Wilfried R. Vanhonacker,et al.  Estimating dynamic response models when the data are subject to different temporal aggregation , 1990 .

[9]  Thomas F. Walker,et al.  Co-Movement of Australian State Business Cycles , 2007 .

[10]  Richard H. Jones,et al.  SPECTRAL ANALYSIS WITH REGULARLY MISSED OBSERVATIONS , 1962 .

[11]  Benchmarking techniques in the Spanish Quarterly National Accounts , 2006 .

[12]  T. Liu,et al.  Monthly Estimates of Certain National Product Components, 1946-49 , 1951 .

[13]  F. Palm,et al.  Missing observations in the dynamic regression model , 1984 .

[14]  Rob J Hyndman,et al.  25 years of time series forecasting , 2006 .

[15]  A. A. Weiss.  Systematic sampling and temporal aggregation in time series models , 1984 .

[16]  Victor Ginsburgh,et al.  A further note on the derivation of quarterly figures consistent with annual data , 1973 .

[17]  Estimating Monthly GDP In A General Kalman Filter Framework: Evidence From Switzerland , 1999 .

[18]  S. T. Buckland,et al.  An Introduction to the Bootstrap. , 1994 .

[19]  P. D. Jong.  Smoothing and Interpolation with the State-Space Model , 1989 .

[20]  H. Doran,et al.  Prediction of Missing Observations in the Time Series of an Economic Variable , 1974 .

[21]  Cheng Hsiao,et al.  Linear regression using both temporally aggregated and temporally disaggregated data , 1979 .

[22]  Tommaso Proietti.  Distribution and interpolation revisited: a structural approach , 1998 .

[23]  J. Angus.  Forecasting, Structural Time Series and the Kalman Filter , 1992 .

[24]  Miguel Jerez,et al.  Modelling and forecasting time series sampled at different frequencies , 2009 .

[25]  Wolfgang Müller,et al.  Autoregressive Approaches to Disaggregation of Time Series Data , 1971 .

[26]  Abdelwahed Trabelsi,et al.  Benchmarking of Economic Time Series , 1987 .

[27]  Clélia M. Toloi,et al.  SPECTRAL ANALYSIS FOR AMPLITUDE-MODULATED TIME SERIES , 1993 .

[28]  Desagregación conjunta de series anuales: perturbaciones AR(1) multivariante , 2000 .

[29]  Frank T. Denton,et al.  Adjustment of Monthly or Quarterly Series to Annual Totals: An Approach Based on Quadratic Minimization , 1971 .

[30]  Estela Bee Dagum,et al.  Benchmarking time series with autocorrelated survey errors , 1994 .

[31]  William D. Clinger,et al.  On Unequally Spaced Time Points in Time Series , 1976 .

[32]  A. Maravall,et al.  Estimation, Prediction, and Interpolation for Nonstationary Series with the Kalman Filter , 1994 .

[33]  Andrej Petrovic,et al.  Temporal disaggregation of economic time series : towards a dynamic extension , 2003 .

[34]  Santiago Rodr,et al.  Methods for quarterly disaggregation without indicators; a comparative study using simulation , 2003 .

[35]  E. G. Drettakis.  Missing Data in Econometric Estimation , 1973 .

[36]  Daniel O. Stram,et al.  Disaggregation of Time Series Models , 1990 .

[37]  Víctor M. Guerrero,et al.  A recursive ARIMA-based procedure for disaggregating a time series variable using concurrent data , 1995 .

[38]  G. Gudmundsson.  Estimation of Continuous Flows from Observed Aggregates , 2001 .

[39]  Ka Ho Wu,et al.  Comparison of Benchmarking Methods with and without a Survey Error Model , 2007 .

[40]  P. Robinson,et al.  Parametric estimators for stationary time series with missing observations , 1981, Advances in Applied Probability.

[41]  Mohamed Alosh,et al.  A dynamic linear model approach for disaggregating time series data , 1989 .

[42]  Marcel G. Dagenais,et al.  The use of incomplete observations in multiple regression analysis: A generalized least squares approach , 1973 .

[43]  D. Stram,et al.  TEMPORAL AGGREGATION IN THE ARIMA PROCESS , 1986 .

[44]  P. Bloomfield.  Spectral Analysis with Randomly Missing Observations , 1970 .

[45]  Tommaso Proietti,et al.  Dynamic factor analysis with non‐linear temporal aggregation constraints , 2006 .

[46]  Nicola Rossi.  A Note on the Estimation of Disaggregate Time Series When the Aggregate Is Known , 1982 .

[47]  Franz C. Palm,et al.  The construction and use of approximations for missing quarterly observations : A model-based approach , 1985 .

[48]  A. S. Louter,et al.  Estimating Quarterly Values of Annually Known Variables in Quarterly Relationships , 1976 .

[49]  Denis Conniffe.  Small-Sample Properties of Estimators of Regression Coefficients Given a Common Pattern of Missing Data , 1983 .

[50]  Disaggregation of annual flow data with multiplicative trends , 1999 .

[51]  J.M.C. Santos Silva,et al.  The Chow-Lin method using dynamic models , 2001 .

[52]  A. Timmermann,et al.  Economic Forecasting , 2007 .

[53]  J. Pavía,et al.  On Distributing Quarterly National Growth among Regions , 2008 .

[54]  Tommaso Proietti,et al.  Temporal Disaggregation by State Space Methods: Dynamic Regression Methods Revisited , 2006 .

[55]  Víctor M. Guerrero.  Monthly Disaggregation of a Quarterly Time Series and Forecasts of Its Unobservable Monthly Values , 2003 .

[56]  P. Robinson,et al.  Estimation of Time Series Models in the Presence of Missing Data , 1981 .

[57]  Emanuel Parzen,et al.  ON SPECTRAL ANALYSIS WITH MISSING OBSERVATIONS AND AMPLITUDE MODULATION , 1962 .

[58]  James Durbin,et al.  Benchmarking by State Space Models , 1997 .

[59]  Gian Luigi Mazzi,et al.  Introduction to advances in business cycle analysis and forecasting , 2010 .

[60]  Cheng Hsiao,et al.  Missing data and maximum likelihood estimation , 1980 .

[61]  Marcel G. Dagenais,et al.  Incomplete observations and simultaneous-equations models , 1976 .

[62]  F. Palm,et al.  Approximations for Missing Quarterly Observations: A Model-Based Approach , 1986 .

[63]  Stephen G. Hall,et al.  Creating high‐frequency national accounts with state‐space modelling: a Monte Carlo experiment , 2001 .

[64]  J. Pavía,et al.  EstimatingJ (>1) quarterly time series in fulfilling annual and quarterly constraints , 1999 .

[65]  William W. S. Wei,et al.  Effect of systematic sampling on arima models , 1981 .

[66]  P. Bloomfield.  An exponential model for the spectrum of a scalar time series , 1973 .

[67]  Abdelwahed Trabelsi,et al.  A Polynomial Method for Temporal Disaggregation of Multivariate Time Series , 2007, Commun. Stat. Simul. Comput..

[68]  Wai-Sum Chan,et al.  Disaggregation of annual time‐series data to quarterly figures: A comparative study , 1993 .

[69]  J. Boot,et al.  Further Methods of Derivation of Quarterly Figures from Annual Data , 1967 .

[70]  Tommaso Di Fonzo,et al.  Benchmarking Systems of Seasonally Adjusted Time Series , 2005 .

[71]  J. Sargan,et al.  Missing Data in an Autoregressive Model , 1974 .

[72]  G. Chow,et al.  Best Linear Unbiased Estimation of Missing Observations in an Economic Time Series , 1976 .

[73]  M. Friedman.  The Interpolation of Time Series by Related Series , 1962 .

[74]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[75]  J. Jacobs.  "Dividing by 4": A feasible quarterly forecasting method? , 1994 .

[76]  Tommaso Di Fonzo.  Constrained retropolation of high-frequency data using related series: A simple dynamic model approach , 2003 .

[77]  Massimiliano Marcellino,et al.  Interpolation and Backdating with a Large Information Set , 2003 .

[78]  T. Abeysinghe,et al.  Best linear unbiased disaggregation of annual GDP to quarterly figures: the case of Malaysia , 1998 .

[79]  Estela Bee Dagum,et al.  Benchmarking, Temporal Distribution, and Reconciliation Methods for Time Series , 2006 .

[80]  Estela Bee Dagum,et al.  A Unified View of Signal Extraction, Benchmarking, Interpolation and Extrapolation of Time Series , 1998 .

[81]  Robert B. Litterman.  A random walk, Markov model for the distribution of time series , 1983 .

[82]  W. Dunsmuir,et al.  ESTIMATION FOR STATIONARY TIME SERIES WHEN DATA ARE IRREGULARLY SPACED OR MISSING , 1981 .

[83]  Víctor M. Guerrero,et al.  Temporal and contemporaneous disaggregation of multiple economic time series , 1999 .

[84]  H. Werner.  To the temporal aggregation in discrete dynamical systems , 1982 .

[85]  R. Kohn,et al.  Estimation, Prediction, and Interpolation for ARIMA Models with Missing Data , 1986 .

[86]  Two Results Useful for Implementing Litterman's Procedure for Interpolating a Time Series , 1986 .

[87]  J. Sandee,et al.  Derivation of Quarterly Figures from Annual Data , 1964 .

[88]  Baoline Chen,et al.  An Empirical Comparison of Methods for Temporal Disaggregation at the National Accounts , 2007 .

[89]  Tommaso Proietti,et al.  Temporal disaggregation and the adjustment or quarterly national accounts for seasonal and calendar effects , 2008 .

[90]  A Monthly Indicator of GDP , 1997 .

[91]  Theo Nijman,et al.  Efficiency gains due to using missing data procedures in regression models , 1988 .

[92]  Bournay,et al.  Réflexions sur la méthode d'élaboration des comptes trimestriels , 1979 .

[93]  Martin Weale.  Interpolation using a dynamic regression model: specification and Monte Carlo properties , 1997 .

[94]  Richard H. Jones,et al.  Maximum Likelihood Fitting of ARMA Models to Time Series With Missing Observations , 1980 .

[95]  E. Parzen.  Mathematical Considerations in the Estimation of Spectra , 1961 .

[96]  Arnold Zellner,et al.  A Study of Some Aspects of Temporal Aggregation Problems in Econometric Analyses , 1971 .

[97]  William W. S. Wei,et al.  Some Consequences of Temporal Aggregation in Seasonal Time Series Models , 1979 .

[98]  Adriaan Bloem,et al.  Quarterly National Accounts Manual: Concepts, Data Sources, and Compilation , 2001 .

[99]  Tommaso Proietti,et al.  Multivariate temporal disaggregation with cross-sectional constraints , 2011 .

[100]  Víctor M. Guerrero,et al.  Temporal disaggregation of time series : an ARIMA-based approach , 1990 .

[101]  Takeshi Amemiya,et al.  The Effect of Aggregation on Prediction in the Autoregressive Model , 1972 .

[102]  M. Marcellino.  Pooling‐Based Data Interpolation and Backdating , 2005 .

[103]  Abdelwahed Trabelsi,et al.  Bench-marking time series with reliable bench-marks , 1990 .

[104]  Alain Monfort,et al.  On the Problem of Missing Data in Linear Models , 1981 .

[105]  Luis Eduardo Vila Lladosa,et al.  On the performance of the Chow-Lin procedure for quarterly interpolation of annual data: Some Monte-Carlo analysis , 2003 .

[106]  David M. Aadland.  Distribution and interpolation using transformed data , 2000 .

[107]  Andrew Harvey,et al.  Estimating Missing Observations in Economic Time Series , 1984 .

[108]  Tommaso Di Fonzo,et al.  The Estimation of M Disaggregate Time Series When Contemporaneous and Temporal Aggregates Are Known , 1990 .

[109]  Franz C. Palm,et al.  Predictive accuracy gain from disaggregate sampling in ARIMA models , 1990 .

[110]  R. Kohn,et al.  Estimation, Filtering, and Smoothing in State Space Models with Incompletely Specified Initial Conditions , 1985 .

[111]  G. Rajaguru,et al.  Quarterly Real GDP Estimates for China and ASEAN4 with a Forecast Evaluation , 2004 .

[112]  Franz C. Palm,et al.  Consistent estimation of regression models with incompletely observed exogenous variables , 1987 .

[113]  Klaus L. P. Vasconcellos,et al.  Aggregation and Disaggregation of Structural Time Series Models , 1999 .

[114]  John J. Seater,et al.  Temporal Aggregation and Economic Time Series , 1995 .