Estimating Quarterly Values of Annually Known Variables in Quarterly Relationships

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

“…His proposal (primary suggested to estimate the French quarterly familiar consumption series) consists of (1) applying [20] to both objective annual series and indicator annual series to obtain, respectively, an initial approximation and a control series, to then (2) modifying the initial estimate by aggregating the discrepancies between the observed quarterly indicator and the control series, using as scale factor the Ordinary Least Squares (OLS) estimator of the linear observed annual model. Later, minimal variations of [46] were proposed by [28] and [49]. [28] suggested obtaining the initial estimates using [27], instead of [20], and [48] proposed generalizing [88] by allowing the weight structure to be different for each quarter and year, with the weight structure obtained, using annual constraints, from a linear model.…”

“…One of the most successful methods in the area (not only among benchmarking procedures) is the approach proposed by [36] in 1971. The great attractiveness of methods such as [36] -and also [41] -among analysts and statistical agencies [17], despite using more sophisticated procedures generally yielding better estimates [13], can be explained because as [49] pointed out short-term analysis in general and quarterly accounts in particular need disaggregation techniques being "…flexible enough to allow for a variety of time series to be treated easily, rapidly and without too much interven-tion by the producer;" and where "the statistical procedures involved should be run in an accessible and well known, possibly user friendly, and well sounded software program, interfacing with other relevant instruments typically used by data producers (i.e. seasonal adjustment, forecasting, identification of regression models,…)".…”

“…[57], on the other hand, presented a general dynamic stochastic regression model, which permits to deal with the most common short-term data treatment (including interpolation, benchmarking, extrapolation and smoothing), and showed that the GLS estimator is the minimum variance linear unbiased estimator [17]. Additionally, although other temporal disaggregation procedures based on dynamic models have been proposed [49,80,81], they will be considered in the next subsection as they can be observed as dynamic extensions of [41]. It must be noted however that they might be also placed on the previous subsection due to they could be seen under the perspective of the classical two-step approach of adjusting algorithms.…”

“…Thus, for example, in line with [37], the HF covariance error matrix could be estimated from the ARIMA structure of the LF covariance matrix. Despite it, low AR order models are still systematically chosen in practice due to (1) the covariance matrix of the HF disturbances cannot be, in general, uniquely identified from the LF ones and (2) the typical sample sizes occurring in economics usually provide poor LF error matrix estimates [64,49]. In fact, the Monte Carlo evidence presented in [100] showed that this approach would likely perform comparatively badly when the LF sample size is lower than 40 (a really non infrequent size in economics).…”

Missing Observations in the Dynamic Regression Model

“…A more sophisticated smoothing method has been proposed and applied by Somermeyer et al (1976). A review of various other methods of interpolation is given by Gelauff and Harkema (1977).…”