Disaggregation of Time Series Models

Abstract

SUMMARY We develop a model disaggregation method to derive a disaggregate model from a given aggregate model, which is then used to perform data disaggregation. Since a time series model and its autocovariance structure are closely related, we approach the problem by exploring the possibility of estimating the autocovariance structure for the unobserved disaggregated series from the available autocovariances of an aggregate model. Let the time series aggregates be the non‐overlapping sums of m consecutive disaggregated observations. Given an aggregate autoregressive integrated moving average ARIMA(p, d, r) model with r ≤ p + d + 1, assume that there is no hidden periodicity of order m. It is shown that, if m is odd or if m is even but all the real roots of the autoregressive polynomial of the given aggregate model are positive, then there exists a disaggregate model whose autocovariances can be uniquely derived from the autocovariances of the given aggregate model. Both non‐seasonal and seasonal models are discussed. Empirical examples are presented to illustrate the method.

A quarterly post-Second World War real GDP series for New Zealand

“…4. The disturbance term assumptions are: AR(1) for Chow and Lin (1971); I(1) for Ferna´ndez (1981); ARIMA (1,1,0) for Litterman (1983), and ARIMA(p, d, q) for Wei and Stram (1990). 5.…”

“…2 These methods can be dated back to the adjustment procedures advocated by Denton (1971) and the still widely used 'static' method due to Chow and Lin (1971). 3 Other forms of static model, which essentially treat dynamic behaviour through different assumptions about the disturbance term, have been offered by Ferna´ndez (1981), Litterman (1983) and Wei and Stram (1990). 4 Univariate dynamic approaches have been advocated by Di Fonzo (2003), Gregoir (2002), Salazar et al (1994), and Santos Silva and Cardoso (2001); while multivariate dynamic state-space form (SSF) models, accounting for missing observations and utilising an augmented Kalman filter, have been put forward by Harvey (1989, ss.…”

A Benchmarking Approach to Temporal Disaggregation of Economic Time Series by Related Series

“…Under some assumptions (Wei and Stram, 1990), we can say that e t (the temporal disaggregation of E t follows the autoregressive moving average ARMA model:…”

“…The first provides procedures which estimate the disaggregated series on the basis of information derived only from aggregated series (e.g., Almon, 1988;Cohen et al, 1971;Lisman and Sandee, 1964;Stram and Wei, 1986;Wei and Stram, 1990), whereas the second type uses a disaggregation scheme based on additional information available from other related series (e.g., Bassie, 1958;Chow and Lin, 1971;Fernandez, 1981;Friedman, 1962;Guerrero, 1990;Litterman, 1983;Vangrevelinghe, 1966). We can also distinguish between non-model based methods (e.g., Almon, 1988;Lisman and Sandee, 1964) and model based method (e.g., Harvey and Pierse, 1984;Stram and Wei, 1986;Wei and Stram, 1990). Temporal disaggregation methods based on additional information available from related series lead to disaggregated figures inconsistent with the aggregated figures (e.g., for each year, the annual figure should agree with the total of quarterly figures).…”

Optimization models for degrouping population data

“…The disaggregation of a general chronological series-such as the annual gross domestic product (GDP) of a country-a problem closely related to the one posed above, has been dealt with extensively (see Lisman and Sandee 1964;Cohen et al 1971;Denton 1971;Wei and Stram 1990;Guerrero 2003;Pavía-Miralles 2010;Proietti 2011). The expansion of an abridged series of mortality data given in age groups has also received some attention.…”