Abstract
Another category is the ad hoc use of disaggregate information. For example, to produce short-term forecasts of economic activity, City commentators and academic researchers keep an eye on the monthly movements in retail sales or trade, or manufacturing output, believed to be the best available proxy for broader measures of demand and output. Salazar et al. [1994] argue those procedures may, at best, be using existing information inefficiently and, at worse, misleading.
Both economists and statisticians had an interest in the subject during the
1960s, but examples of work can be traced back to the 1930s.
The basic idea can be formalized as follows.
Suppose a series 
is observed at 
regularly spaced periods, but higher frequency measurements are needed.
Information on several,or one related series to is available,
with periodicity 
. Denote by 
a set of
p-related variables to , observed in sub-period u=1,...,K of period t=1,...,N. Therefore, defines a 
matrix with typical column 
i=1,..,p. Assume the hypothetical 
vector
and the related variables follow
where 
is the 
vector of parameters,
is a 
stationary vector of error terms, such that E 
and covariance E 
. Assume, in addition, that a time-invariant
constraint links and , so
where the 
weights are known a-priori. Therefore,
write
where 
and covariance matrix
. Note in equation (3) that is
observed, and 
contains the observed values of the
p-related series at the same frequency as . Taking into account
the constraints (2) we can estimate and by solving
The whole procedure, however, depends on the specification of the model for 
in equation (1). A complete generalization of
Chow and Lin's model, and other extensions, can be found in Salazar
et al. [ibid.] Alternatively, equation (1) can be recast in
state-space form, using the Kalman filter and the fixed point smoother to
estimate and , respectively. Jones [1980] applied
this methodology for ARMA models with missing observations; Harvey and
Pierse [1983], Harvey [1989] and Gomez and Maravall [1994] provide some
useful extensions.
This paper proposes to redefine equation (1) by
where the form of 
is unknown, but can be
recovered from the data. Estimation of (5) is made by a 2-layer Neural
Network model, performing a wavelet decomposition over and
approximating 
to the known values of under a suitable criterion. The learning algorithm ensures in
both models that, at each stage, the linear adding-up constraint given
by (2) is met.
The architecture is a direct adaptation of the Radial Basis Function [RBF] Network, where the RBF activation is substituted by a [discrete] wavelet kernel; seminal papers in this direction are Zhang and Benveniste [1992] and Pati and Krishnaprasad [1993].
Two modeling strategies are considered. First, every neuron on the hidden layer implements a wavelet decomposition using a wavelet basis constructed from the data. This type of architecture is similar to that proposed by Zhang and Benveniste [ibid.] Instead of using a fixed lattice of translation and dilation parameters, these are adaptively determined to form a 'library' of wavelets. The problem then amounts to select the 'best' wavelets from the obtained library, and the estimation of the wavelet coefficients.
The alternative model performs a wavelet decomposition for each of the
p vectors in , selecting for every vector an
appropriate basis. By separating the input space,
and therefore
where 
denotes a 'family' of
wavelets obtained from the mother function 
, for
each i vector in and 
stacks each of the
r=1,...,R decompositions, the contribution by each series may be better
localized.
An application to the estimation of monthly components of Gross Domestic
Product for the United Kingdom is considered, from quarterly aggregates. The
estimates are readily comparable with the estimates in Salazar et al. [op. cit.], and performance measures for each model are analyzed.
Conclusions follow.