Grouped Functional Time Series Forecasting: An Application to Age-Specific Mortality Rates

ABSTRACT Age-specific mortality rates are often disaggregated by different attributes, such as sex, state, and ethnicity. Forecasting age-specific mortality rates at the national and sub-national levels plays an important role in developing social policy. However, independent forecasts at the sub-national levels may not add up to the forecasts at the national level. To address this issue, we consider reconciling forecasts of age-specific mortality rates, extending the methods of Hyndman et al. in 2011 to functional time series, where age is considered as a continuum. The grouped functional time series methods are used to produce point forecasts of mortality rates that are aggregated appropriately across different disaggregation factors. For evaluating forecast uncertainty, we propose a bootstrap method for reconciling interval forecasts. Using the regional age-specific mortality rates in Japan, obtained from the Japanese Mortality Database, we investigate the one- to ten-step-ahead point and interval forecast accuracies between the independent and grouped functional time series forecasting methods. The proposed methods are shown to be useful for reconciling forecasts of age-specific mortality rates at the national and sub-national levels. They also enjoy improved forecast accuracy averaged over different disaggregation factors. Supplementary materials for the article are available online.

[1]  Han Lin Shang,et al.  Methods for Scalar‐on‐Function Regression , 2017, International statistical review = Revue internationale de statistique.

[2]  Claudia Klüppelberg,et al.  An innovations algorithm for the prediction of functional linear processes , 2016, J. Multivar. Anal..

[3]  H. Waldorf.  Aging Gracefully. , 2017, Journal of drugs in dermatology : JDD.

[4]  Han Lin Shang,et al.  Functional time series forecasting with dynamic updating: An application to intraday particulate matter concentration , 2016, 1608.07029.

[5]  Han Lin Shang,et al.  Mortality and life expectancy forecasting for a group of populations in developed countries: A multilevel functional data method , 2016, 1606.05067.

[6]  Rob J. Hyndman,et al.  Fast computation of reconciled forecasts for hierarchical and grouped time series , 2016, Comput. Stat. Data Anal..

[7]  Yu-Ting Chen,et al.  Multivariate functional linear regression and prediction , 2016, J. Multivar. Anal..

[8]  J. Klepsch,et al.  Prediction of functional ARMA processes with an application to traffic data , 2016, 1603.02049.

[9]  Jj Allaire,et al.  Web Application Framework for R , 2016 .

[10]  M. Sherris,et al.  Causes-of-Death Mortality: What Do We Know on Their Dependence? , 2015 .

[11]  W. Härdle,et al.  Stochastic Population Analysis: A Functional Data Approach , 2015 .

[12]  David S. Matteson,et al.  A Bayesian Multivariate Functional Dynamic Linear Model , 2014, 1411.0764.

[13]  H. Shang.  A survey of functional principal component analysis , 2014 .

[14]  Piotr Kokoszka,et al.  Testing stationarity of functional time series , 2014 .

[15]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[16]  Dominik Liebl,et al.  Modeling and forecasting electricity spot prices: A functional data perspective , 2013, 1310.1628.

[17]  M. Kogan,et al.  All-Cause and Cause-Specific Mortality among US Youth: Socioeconomic and Rural–Urban Disparities and International Patterns , 2013, Journal of Urban Health.

[18]  Rob J Hyndman,et al.  Coherent Mortality Forecasting: The Product-Ratio Method With Functional Time Series Models , 2013, Demography.

[19]  Jeng-Min Chiou,et al.  Dynamical functional prediction and classification, with application to traffic flow prediction , 2012, 1301.2399.

[20]  Siegfried Hörmann,et al.  On the Prediction of Stationary Functional Time Series , 2012, 1208.2892.

[21]  Piotr Kokoszka,et al.  Functional prediction of intraday cumulative returns , 2012 .

[22]  Siegfried Hörmann,et al.  Functional Time Series , 2012 .

[23]  Hans-Georg Müller,et al.  Functional Data Analysis , 2016 .

[24]  David E. Tyler,et al.  Robust functional principal components: A projection-pursuit approach , 2011, 1203.2027.

[25]  Rob J. Hyndman,et al.  Optimal combination forecasts for hierarchical time series , 2011, Comput. Stat. Data Anal..

[26]  Maria Russolillo,et al.  The mortality of the Italian population: Smoothing techniques on the Lee-Carter Model , 2011, 1108.0766.

[27]  Rob J. Hyndman,et al.  Nonparametric time series forecasting with dynamic updating , 2011, Math. Comput. Simul..

[28]  Nan Li,et al.  Coherent mortality forecasts for a group of populations: An extension of the lee-carter method , 2005, Demography.

[29]  Harihar Sahoo,et al.  Population Decline and Ageing in Japan — The Social Consequences. , 2011 .

[30]  P. Kokoszka,et al.  Weakly dependent functional data , 2010, 1010.0792.

[31]  Rob J Hyndman,et al.  Rainbow Plots, Bagplots, and Boxplots for Functional Data , 2010 .

[32]  T. Gneiting.  Making and Evaluating Point Forecasts , 2009, 0912.0902.

[33]  Han Lin Shang,et al.  Forecasting functional time series , 2009 .

[34]  Haipeng Shen.  On Modeling and Forecasting Time Series of Smooth Curves , 2009, Technometrics.

[35]  D. Swanson,et al.  Demographic Forecasting , 2009 .

[36]  Erniel B. Barrios,et al.  Principal components analysis of nonstationary time series data , 2009, Stat. Comput..

[37]  H. Müller,et al.  Modeling Hazard Rates as Functional Data for the Analysis of Cohort Lifetables and Mortality Forecasting , 2009 .

[38]  Rob J Hyndman,et al.  Automatic Time Series Forecasting: The forecast Package for R , 2008 .

[39]  Philippe Vieu,et al.  Nonparametric time series prediction: A semi-functional partial linear modeling , 2008 .

[40]  Daniel Peña,et al.  Measuring the Advantages of Multivariate vs. Univariate Forecasts , 2007 .

[41]  Ricardo Fraiman,et al.  Robust estimation and classification for functional data via projection-based depth notions , 2007, Comput. Stat..

[42]  Stergios B. Fotopoulos,et al.  Modeling Financial Time Series With S–PLUS , 2007, Technometrics.

[43]  Rob J. Hyndman,et al.  Robust forecasting of mortality and fertility rates: A functional data approach , 2007, Comput. Stat. Data Anal..

[44]  Christophe Croux,et al.  Robust Forecasting with Exponential and Holt-Winters Smoothing , 2007 .

[45]  D. Willcox,et al.  Aging gracefully: a retrospective analysis of functional status in Okinawan centenarians. , 2007, The American journal of geriatric psychiatry : official journal of the American Association for Geriatric Psychiatry.

[46]  A. Raftery,et al.  Strictly Proper Scoring Rules, Prediction, and Estimation , 2007 .

[47]  Thomas L. Burr,et al.  Modeling Financial Time Series With S—Plus , 2007, Technometrics.

[48]  Ricardo Fraiman,et al.  On the use of the bootstrap for estimating functions with functional data , 2006, Comput. Stat. Data Anal..

[49]  Peter Hall,et al.  Assessing the finite dimensionality of functional data , 2006 .

[50]  H. Müller,et al.  Functional Data Analysis for Sparse Longitudinal Data , 2005 .

[51]  Hans-Georg Ller,et al.  Functional Modelling and Classification of Longitudinal Data. , 2005 .

[52]  B. Willcox,et al.  Successful aging: Secrets of Okinawan longevity , 2004 .

[53]  L. Hantrais.  Population decline and ageing , 2004, Family Policy Matters.

[54]  Jiahui Wang,et al.  Modeling Financial Time Series with S-PLUS® , 2003 .

[55]  L. Berkman,et al.  Socioeconomic status and mortality among the elderly: findings from four US communities. , 2002, American journal of epidemiology.

[56]  R. Fraiman,et al.  Trimmed means for functional data , 2001 .

[57]  Philippe C. Besse,et al.  Autoregressive Forecasting of Some Functional Climatic Variations , 2000 .

[58]  A. Zellner,et al.  A Note on Aggregation, Disaggregation and Forecasting Performance , 2000 .

[59]  K. Hadri.  Testing The Null Hypothesis Of Stationarity Against The Alternative Of A Unit Root In Panel Data With Serially Correlated Errors , 1999 .

[60]  Kenneth B. Kahn.  Revisiting Top-Down versus Bottom-Up Forecasting , 1998 .

[61]  A. McQuarrie,et al.  Regression and Time Series Model Selection , 1998 .

[62]  James O. Ramsay,et al.  Functional Data Analysis , 2005 .

[63]  Alan D. Lopez,et al.  Alternative projections of mortality and disability by cause 1990–2020: Global Burden of Disease Study , 1997, The Lancet.

[64]  国立社会保障・人口問題研究所.  国立社会保障・人口問題研究所 : National institute of population and social security research : 研究所案内 , 1997 .

[65]  J. Sefton,et al.  Reconciliation of national income and expenditure : balanced estimates of national income for the United Kingdom, 1920-1990 , 1995 .

[66]  Byron J. Dangerfield,et al.  Top-down or bottom-up: Aggregate versus disaggregate extrapolations , 1992 .

[67]  P. Phillips,et al.  Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? , 1992 .

[68]  B. Silverman,et al.  Estimating the mean and covariance structure nonparametrically when the data are curves , 1991 .

[69]  Clifford M. Hurvich,et al.  Regression and time series model selection in small samples , 1989 .

[70]  John S. Morris,et al.  Top-down versus bottom-up forecasting strategies , 1988 .

[71]  T. Ishii,et al.  INFLUENCE OF MAJOR HISTOCOMPATIBILITY COMPLEX REGION GENES ON HUMAN LONGEVITY AMONG OKINAWAN-JAPANESE CENTENARIANS AND NONAGENARIANS , 1987, The Lancet.

[72]  Ronald W. Wolff,et al.  Aggregation and Proration in Forecasting , 1979 .

[73]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[74]  H. Akaike.  A new look at the statistical model identification , 1974 .