J. Service Science & Management, 2010, 449-463
doi:10.4236/jssm.2010.34051 Published Online December 2010 (http://www.scirp.org/journal/jssm)
A Survey of Methods to Interpolate, Distribute and
Extrapolate Time Series*
Jose Manuel Pavía-Miralles
Department of Applied Economics, Universitat de Valencia, Valencia, Spain.
Received July 22nd, 2010; revised September 10th, 2010; accepted October 18th, 2010.
ABSTRACT
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 lit-
erature to solve the problem of transforming a low-freq uency 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 litera-
ture 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 rea ders, however, will not find much detail of th e 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 d isaggregation.
Keywords: Adjustment, Benchmarking, High Frequency, Quarterly Accounts, Reconciliation, Temporal Disaggregation
1. Introduction
Time series modeling is commonly used by private and
public economic and financial analysts in order to fore-
cast future values, analyze the properties of a series,
characterize its salient features and monitoring the cur-
rent status of the economy [1,2]. However, despite social
and economic life having quickened and becoming more
turbulent, it is not unusual that some relevant variables
are not available with the desired timing and frequency.
The extra costs associated with collecting data more fre-
quently, some practical limitations to obtain certain
variables with a higher regularity along with delays in
the process of handling and gathering more frequent re-
cords deprive analysts, more often than desired, of the
valuable help that high frequency data would provide to
perform closer and more accurate short-term analyses. In
fact, when temporally aggregated data are used to model,
study and investigate the relationship among variables,
individuals, and/or entities, it is qu ite possible that a dis-
torted view of parameters’ values, lag structures and sea-
sonal components could be reached [3] and that, as a
consequence, poor models and/or forecasts could be ob-
tained and wrong decisions taken [4]. In fact, in econo-
metric modeling, when some of the relevant series are
only available at lower frequencies, obvious improve-
ments in model selection, efficiency in parameters’ esti-
mates and prediction quality are usually obtained if pre-
viously the frequency of the low-frequency time series is
increased increased [5-11].
It is not surprising, therefore, that great quantities of
methods, procedures and algorithms have been proposed
in the literature in order to increase the frequency of
some critical variables, especially in the economic area
where it would facilitate a lower delay and a more pre-
cise analysis of the condition that an economy or com-
pany is experiencing at a given time, making easier to
anticipate changes and to react to them.
Obviously, business and economics are not the only
fields where it would be useful. Other areas also use
these approaches and take advantage of these techniques
to improve the quality of their analysis. This paper,
however, will focus on the contributions made and used
* Supported by the Spanish MICINN project CSO2009-11246.

A Survey of Methods to Interpolate, Distribute and Extrapolate Time Series 450
within economics; since the problems faced by short-
term analysts and quarterly account producers have been
the main catalysts for enlargement and improvement of
most of the procedures proposed in the literature. For
instance, they have been historically linked to the need of
solving problems related to the production of coherent
quarterly national and regional accounts see, e.g., [12]
and [13]. Actually, it is quite probable that some of the
future fruitful developments in this sub ject come to solve
the new challenges posed in these areas. Additionally to
the methods specifically proposed to estimate quarterly
and monthly accounts, another significant number of
methods suggested to estimate missing observations have
been also adapted to this issue. Thus, classifying the
large variety of methods proposed emerges as a neces-
sary requirement in order to perform a systematic and
ordered study of the numerous alternatives suggested.
Furthermore, due to differing algorithms can give rise to
series with different cyclical, seasonal and stochastic
properties - even though they have been derived from the
same research field using the same basic information
[14], a classification would be in itself a proper tool for a
suitable selection of the technique.
Depending on the criteria adopted alternative classify-
cations could be reached. This paper will follow the ba-
sic division proposed in [15,16], which divide the tem-
poral disaggregation problems according to the basic
information available for estimation. Although other di-
visions have been suggested in the literature, as for in-
stance in [17] who divide the literature according to the
nature of the problems, no categorization avoids the
problem of deciding where to place some procedures or
methods (which could belong to different groups and
whose location turns out as extremely complicated) and
it is the belief of the author that the former approach cla-
rifies and makes simpler the exposition. Furthermore, to
make quicker and easier to follow the paper the mathe-
matical expressions have been omitted using a narrative
style to inform about the logic of each procedure.
Previously to introduce any categorization, the different
problems that can appear when disaggregating a time se-
ries must be set up. The next section is devoted to it. In
particular, the rest of the paper is organized as follows.
Section 2 introduces the different types of problems faced
on this framework. Section 3 exposes the criteria followed
to classify the different alternatives and details the main
characteristics of each group. Sections 4 to 7 present the
methods and section 8 discusses and concludes.
2. Interpolation, Distribution and Extrapo-
lation Problems
In general, inside this framework and depending on the
kind of variable handled (either, flow or stock) two dif-
ferent main problems can emerge: the distribution prob-
lem and the interpolation problem. On the one hand, the
distribution problem appears when the observed values
of a flow low-frequency (LF) series of length N must be
distributed among kN values (where k is the number of
sub-periods in which each period of low-frequency is
divided: For instance, k = 3 if a monthly series must be
estimated from a quarterly observed series, k = 4 if
quarterly estimates for yearly data are desired and k = 12
if monthly data are required for an annually observed
series), such that the temporal sum of the estimated
high-frequency (HF) series fits the values of the LF se-
ries. It is, if y = [y1,...,yN‘ represents the (Nx1) vector of
observed LF values and z = [z1,...,zT‘ the (T1) vector of
missing HF values, with T = kN; the vectors y and z can
be related by y = Bz, with B = IN  u, where IN is the
identity matrix of order N,
 stands for Kronecker’s
product and u is a (1k) vector of ones. On the other
hand, the interpolation problem consists in generating a
HF series with the values of the new series being the
same as the ones of the LF series for those temporal
moments where the latter is observed. It is u is equal to
either a vector (1k) of zeroes except for the first or the
last component that is one, depending on the corre-
sponding high-frequency point time where the LF series
is observed.
In both cases, when estimates are extended out of the
period covered by the low-frequency series, the problem
is called extrapolation. Extrapolation is used, therefore,
to forecast values of the high-frequency series when no
temporal constraints from short series are available; al-
though, nevertheless, in some cases (especially in multi-
variate contexts), other different forms of constraints can
exist. In these cases, the matrix B is replaced by a proper
design matrix [18].
Furthermore, related to distribution problems, they can
be found benchmarking and balancing problems [17]
which are mainly used in management and by official
statistical agencies to adjust the values of a HF series of
ballpark figures (usually obtained employing sample
techniques) to a more accurate LF series and other tem-
poral disaggregation problems where the temporal ag-
gregation function is different from the sum function.
Anyway, despite the great quantity of procedures for
temporal disaggregation of time series being proposed in
the literature, the fulfillment of the constraints derived
from the observed LF series is the norm in the subject.
3. Criteria of Classification
As has been stated on the introduction many different
criteria could be proposed to divide the great amount of
proposals that can be found in this framework. A first
division could arise attending to the plane from which

A Survey of Methods to Interpolate, Distribute and Extrapolate Time Series451
the problem is faced, either the frequency or the temporal
plane. This division, however, is not well-balanced, since
the temporal perspectiv e has b een by larg e more popu lar.
On the one hand, the procedures that deal with the prob-
lem from a temporal perspective will be analyzed in Sec-
tions 4, 5, and 6. On the other hand, the methods that try
to solve the problem from the frequency domain will be
introduced in Section 7.
Another possible criterion of division attends to the
use or not of related series, usually called indicators.
Economic events tend to made visible in different ways
and to affect many dimensions. The economic series are
therefore correlated variables that do not evolve in an
isolate way. Consequently, it is not unusual that some
variables available in HF could display similar fluctua-
tions than those (expected) for the target series. Some
methods try to take advantage of this fact to temporally
distribute the target series. Thus, the use or not of indi-
cators has been considered as another criterion of classi-
fication. The procedures which deal with the series in an
isolated way and compute the missing data of the disag-
gregated HF series from the temporal plane taking into
account only the information given by the objective se-
ries are presented in Section 4.
Complementary to the previous methods appear the set
of procedures which exploit the economic relationships
between indicators and objective series. This group,
composed for an extent and varied collection of methods
widely used, has an enormous success. As example, it
could be cited that the use of procedures based on indi-
cators is the rule among the governmental statistic agen-
cies that use indirect methods to estimate quarterly and
monthly accounts [16,19]. These procedures are pre-
sented in Section 5.
Finally, a last group has been considered: the Kalman
filter approaches. The methods that use the state space
representation to estimate the non available values have
been grouped in Section 6. The great flexibility that of-
fers the representation of the temporal processes in the
state space and the enormous possibilities that these rep-
resentations present to properly deal with log-trans- for-
mations and dynamic approximations justify its own sec-
tion, despite all the approaches of this section being clas-
sifiable in the previous two sections.
4. Non-Indicators Algorithms
Different approaches and strategies have been employed
without using related variables. The first methods pro-
posed were developed without any theoretical justifica-
tion for the elaboration of quarterly (or monthly) national
accounts as mere instruments. They were quite me-
chanical and generated the HF series imposing some a
priori desired conditions. Gradually, nevertheless, new
methods-theoretically founded on the Autoregressive
Integrated Moving Average (ARIMA) representation of
the series to be disaggregated-were progressively ap-
pearing, introducing more flexibility in the process.
The design of these primary methods, nevertheless,
was already in those first days influenced for the need of
solving one issue that appears recurrently in the subject
and whose solution must tackle every method suggested
to temporal disaggregate a time series: the spurious step
problem. To prevent series with undesired discontinuities
from one year to the next, the pioneers proposed to esti-
mate quarterly values using figures from several con-
secutive years. The disaggregation methods proposed by
[20-22] were devised to estimate the quarterly series
corresponding to year t as a weighted average of the an-
nual values of periods t-1, t and t+1. They estimate the
quarterly series through a fix weight structure. The dif-
ference among these methods lies on the election of the
weight matrix. [20] calculated the weight matrix by re-
quiring that the estimated series verify some a priori
“interesting” properties. [21] assumed that the curve of
the quarterly estimates is located upon a second degree
polynomial that passes by the origin. And [22] extended
[21]’s proposal to polynomials with other degrees. Fur-
thermore, following pioneers’ ideas, [23] expanded [20]
to the case of distributing quarterly or annual series into
monthly ones and later [24] provided, in an econometric
computer package G, a method to convert annual figures
into quarterly series, assuming that a cubic polynomial is
fitted to each successive set of two points of the low-
frequency series. All these methods, however, are uni-
variate and it was necessary to wait two more decades to
find inside this approach a solution for the multivariate
problem. Just recently, [25,26] have extended, for both
stock and flow variables, [24]’s univariate polynomial
method to the multivariate case.
A different approach was proposed in [27] who, using
also an ad-hoc mathematical procedure, build the quar-
terly series by solving an optimization problem. In par-
ticular, their method builds the quarterly series as solu-
tion of the minimization of the sum of squares of either
the first or the second differences of the (unknown) con-
secutive quarterly values, under the condition that the
annual aggregation of the estimated series adds up the
available annual figures. Although [27] algorithms most-
ly reduced the subjective charge of the preceding meth-
ods, their way of solving the problem of spurious steps
was still a bit subjective and therefore it was not free of
criticism. On the particular, it could be consulted, among
others [6,14,15,28] .
In the same way that polynomial procedures were ge-
neralized, [27] approach was also extended in both flexi-
bility and in the number of series to be handled. [29]
extended [27] work introducing flexibility in the process

A Survey of Methods to Interpolate, Distribute and Extrapolate Time Series 452
in a double way. On one hand, they dealt with any pair of
possible combination of high and low frequencies. And,
on the other hand, they considered the minimization of
the sum of the squared of the ith differences between
successive sub-period values (not only first and second
differences). The multivariate extension, nevertheless,
was introduced later, in [30].
Additionally to the abovementioned ad-hoc mathe-
matical algorithms, many other methods could be also
classified within this group of techniques. Among them,
it will be highlighted [31 ] and [32]. [31] assumed th at the
target series is observed in its higher frequency in a part
of the sample period (generally at the end of the sample)
and proposed to use this subsample to estimate the
non-available values employing the temporal character-
istics of the series obtained from it. This strategy, how-
ever, as proved by [33], yields inefficient estimates. [32],
on the other hand, proposed to obtain the target series
minimizing a squared function defined by the inverse
covariance matrix associated to the quarterly stationary
ARMA(p,q) process obtained taking differences on the
non-stationary ARIMA(p,d,q) one (an ARIMA process
with autoregressive order p, integrated order d, and
moving average order q). In particular, they suggested
adjusting an ARIMA model to the low-frequency series,
selecting a model for the quarterly values compatible
with the annual model and minimizing the dth differ-
ences of the objective series by the loss function using
the annual series as constraint. Unfortunately, according
to [34], [32] method only performs well when the series
are long enough to permit a proper estimation of the
ARIMA process. [32], however, made possible to reas-
sess [27]. They showed [27]’s algorithm being equiva-
lent to use their procedure assuming that the series to be
estimated follow either a temporal integrated process of
order one or two (i.e., I(1) or I(2)).
To know the advantages and disadvantages of many of
the methods introduced in this section and decide which
to use under what circumstances, it could be consulted
[34,35]. They performed a simulation and a real-data
exercise in which [20,21,27,36]in its variant without
indicators [32,37] and Lagrange polynomial interpolators
were compared.
5. Based Method Indicators
Procedures based on related variables have been the most
popular and the most widely used and successful. Thus, a
great number of procedures can be found within this
category. Comparing with the algorithms not-using indi-
cators, related variable procedures have been assigned
two main principal advantages: (i) they present better
foundations in the construction hypothesis (which can
comparatively affect the results validation); and, (ii) they
make use of relevant economic and statistical informa-
tion, being more efficient. Although, in return, the re-
sulting estimates depend crucially on the indicators cho-
sen and, as [38] observed, they hide an implicit hypothe-
sis according to which the LF relationship is accepted to
be maintained in the HF.
The first drawback implies that a special care should
be taken in selecting indicators. This problem, however,
far from being closed has been opened during decades.
Already in 1951, [39] tried to establish some criteria the
indicators should fulfill, and this issue has been repeat-
edly treated again and again [38,40,12,30]. It, however,
does not yield that some universal accepted sound crite-
ria have been proposed. Indeed, although the use of in-
dicators to predict the probable evolution of key series
throughout the quarters of a year is the rule among the
countries that estimate quarterly accounts by indirect
methods there are no apparent fixed criteria to select
them. A summary of the indicators used by the different
national statistics agencies can be found in [12] and a list
of features that a variable should fulfill to be selected as
indicator is pointed out in [13].
In order to better manage the large quantity of proce-
dures classified in this category, this section has been
divided in three subsections. The first subsection is de-
voted to those procedures, called benchmarking tech-
niques, which given an initial estimated HF series adjust
their values using some penalty function in order to ful-
fill the LF constraints. Subsection 2 presents the proce-
dures that take advantage of econometric models to ap-
proximate the incompletely observed variables. Some
techniques which use dynamic regression models in the
identification of the relationship linking the series to be
estimated and the (set of) related time series have been
also included in this subsection. Finally, subsection three
shows those methods, named optimal procedures, which
jointly estimate both parameters and HF series combin-
ing target LF series and HF indicators and incorporating
the LF constraints in the process of estimation, basically
[41] and its extensions.
5.1. Benchmarking/Adjusting Algorithms
As a rule, benchmarking and adjusting methods are
composed of two stages. In the first step an initial ap-
proximation of the objective series is obtained. In the
second step the first estimates are adjusted by imposing
the constraints derived from the available and more reli-
able annual series. The initial estimates are reached using
either sample procedures or some kind of relationship
among indicators and target series. Among those options
that use related variables to obtain initial app roximations
both non-correlated and correlated strategies could be
found. The non-correlated proposals (the former from an

A Survey of Methods to Interpolate, Distribute and Extrapolate Time Series453
historical perspective) do not explicitly take into account
the existing correlation between target series and indica-
tors and rapidly lost analysts’ favor. [42] can be con-
sulted for a wide summary of these algorithms. The cor-
relation strategies, on the other hand, usually assume a
lineal relationship between the objective series and the
indicators from which an initial HF series is obtained.
Once the initial approximation is available, it is adjusted
to make it congruent with the observed LF series. The
discrepancies between both LF series (the observed se-
ries and the series obtained by aggregation of the initial
estimates) are then removed.
A great quantity of benchmarking procedures can be
found in the literature. Bassie [43] propo sed to distribute
annual discrepancies by a structure of fixed weights cal-
culated taking into account the discrepancies corre-
sponding to two successive years and assuming that the
weights function follows a third degree polynomial. [43]
has been historically applied to series of the French and
Italian economy [44-46] and currently Finland and Den-
mark use variants of this method to adjust their quarterly
GDP series [12]. However, it lacks of theoretical justifi-
cation and, according to [43], the method spawns series
with irregularities and cyclical components different to
the initial approximations when the annual discrepancies
are too big [12].
Vangrevelinghe [47] planne d a differe nt appro ach. H is
proposal (primary suggested to estimate the French
quarterly familiar consumption series) consists of (1)
applying [20] to both objective annual series and indica-
tor annual series to obtain, respectively, an initial ap-
proximation and a control series, to then (2) modifying
the initial estimate by aggregating the discrepancies be-
tween 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 generaliz-
ing [88] by allowing the weight structure to be different
for each quarter and year, with the weight structure ob-
tained, using annual co nstraints, 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 sophis-
ticated 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 with out too much interven-
tion by the producer;” and where “the statistical proce-
dures involved should be run in an accessible and well
known, possibly us er friendly, and well sound ed softwar e
program, interfacing with other relevant instruments
typically used by data producers (i.e. seasonal adjust-
ment, forecasting, identification of regression mod-
els,…)”.
Denton [36] suggested adjusting the initial estimates
minimizing a loss function defined by a square form.
Therefore, the choice of the symmetrical matrix deter-
mining the specific square form of the loss function is
the crucial element in [36]. Denton concentrated on the
solutions obtained minimizing the hth differences be-
tween the to-be-estimated series and the initial approxi-
mations and found [27] as a particular case of his algo-
rithm. Later on, [50] proposed a slight modification to
this function family in order to avoid dependence on the
initial conditions. The main extensions of [36] neverthe-
less were reached by [51-55] who made more flexible the
algorithm and extended it to the multivariate case.
Hillmer and Trabelsi [51,52] worked on the problem
of adjusting a univariate HF series using data obtained
from different sampling sources, and found [36] and [50]
as particular cases of their proposal. In particular, they
relaxed the requirements about the LF series p ermitting it
to be observed with error; although, as compensation,
they had to suppose known the temporal structure of the
errors caused by sampling the LF series. When bench-
marks are observed without error, the problem trans-
forms into minimizing the discrepancies between the
initial estimates and the LF series according to a loss
function of the square form type [52]. In these circum-
stances, they showed that the method of minimizing the
hth differences proposed by [36] and [50] implies to im-
plicitly admit: (1) that the rate between the variances of
the observation errors and the ARIMA model errors of
the initial approximation tends to zero; and, (2) that the
observation erro rs follow an I(h) process with eith er null
initial conditions, in [36], or with the initial values of the
series of observation errors begin in a remote past, in
[50].
In sample survey most time series data come from re-
peated surveys whose sample designs usually generate
autocorrelated errors and heteroscedasticity. Thus, [53]
introduced a regression model to take into account it ex-
plicitly and showed that the gain in efficiency of using a
more complex model varies with the ARMA model as-
sumed for the survey errors. In this line, [56] showed,
through a simulation exercise and assuming that the sur-
vey error series follows an AR(1) process, that [53] and
[57] have great advantages over [36] and that they are
robust to misspecification of the survey error model. The
multivariate extension of [36], on the other hand, was

A Survey of Methods to Interpolate, Distribute and Extrapolate Time Series 454
introduced under a general accounting constraint system
in [54] and [55]. They assumed a set of linear relation-
ships among target variables and indicators from which
initial estimates are obtained, to then, applying the
movement preservation principle of Denton approach
subject to the whole set of contemporaneous and tempo-
ral aggregation relationships, reach estimates of all the
series verifying all the constraints.
Although [36] and also [55] do not require any reli-
ability measurement of survey error series to be applied,
their need in many other [36]’s extensions [51] led [58]
to propose an alternative approach. In particular, to
overcome some arbitrariness in the choice of the stochas-
tic structure of the high frequency disturbances, [59] and
[60] developed a new adjustment procedure assuming
that the initial approximation and the objective series
share the same ARIMA model. They combined an
ARIMA-based approach with the use of high frequency
related series in a regression model to obtain the Best
Linear Unbiased Estimate (BLUE) of the objective series
verifying LF constraints. This approach permits an au-
tomatic (which takes a recursive form in [60]) ‘revision’
of the estimates with each new observation, what intro-
duces an important difference with the other procedures
where the estimates obtained for the periods relatively
far away from the last period of the sample are in prac-
tice ‘fixed’. On the other hand, the multivariate and ex-
trapolation extensions of Guerrero’s approach were
likewise provided by Guerrero and colleagues [58,61].
[61] suggested a procedure for estimating unobserved
values of multiple time series whose temporal and con-
temporaneous aggregates are known using vector auto-
regressive models; while [58] proposed a recursive ap-
proach to estimate current disaggregated values of the
series and a method to predict future disaggregated val-
ues. In these cases, nevertheless, it should be noted that
even though the problem can be cast using a state-space
formulation, the Kalman filter approach cannot be ap-
plied directly in these circumstances since the usual as-
sumptions underlying Kalman filtering are not fulfilled.
When dealing with economic variables, it is not un-
common to use logarithms or other transformations of
original data (for example, most time series become sta-
tionary after applying first differences to their logarithms)
to achieve better models of time series due to, as [141]
showed, “…the failure to account for data transforma-
tions may lead to serious errors in estimation”. A very
interesting variant in this framework therefore emerges
when log-transformations are taken. The problem of
dealing with log-transformed variables in the temporal
distribution framework was first considered by [63], and
later treated, among others, in [64] and [1]. However,
due to the logarithmic transformation not being additive,
the LF aggregation constraint cannot be directly applied .
When only log-transformation are taken, [64] proposed
to obtain initial estimates applying the exponential func-
tion to the approximations reached using a linear rela-
tionship between log-transformation of the target series
and indicators, to then in a second step adopt [36] algo-
rithm to get the final values; although according to [47]
this last step could be unnecessary as “the disaggrega ted
estimates present only negligible discrepancies with the
observed aggregated values.” On the other hand, when
the linear relationship is expressed in terms of rate of
change of the target variable (i.e., using the logarithmic
difference), [47] and [65] suggested to obtain initial es-
timates for the non-transformed values of the objective
variable using [66] and further benchmark (using Den-
ton’s formula) for either flow or index variable to exactly
fulfill the temporal aggregation constraints.
5.2. Econometric Model Approaches
Economic theory stands for functional relationships
among variables. The econometric models express those
relations by means of equations. Models based on annual
data conceal higher frequency information and are not
considered sufficiently informative to policy makers.
Building quarterly and monthly macro-econometric
models is therefore imperative. Sometimes, the fre-
quency of the variables taking part in the model is not
homogeneous and expressing the model in the lower
common frequency almost never offers an acceptable
approximation, due to, as [67,68] showed, it is preferable
to estimate the missing observations simultaneously with
the econometric model rather than previously interpo-
lated the unavailable values to directly handle the high-
frequency equations. Thus, putting the model in the de-
sired frequency and use the same model, not only to es-
timate the unknown parameters but also to estimate the
non-observed values of the target series, represents in
general a good alternative.
There are a lot of econometric models susceptible of
being formulated. And therefore, many strategies may be
adopted to estimate the missing observations [7,10,
57,69-79].
Drettakis [69] formulated a multiequation dynamic
model about the United Kingdom economy with one of
the endogenous variables observed only annually for a
part of the sample and obtained estimates for the pa-
rameters and the unobserved values by Maximum Like-
lihood (ML) with complete information. [70] extended
[69] to the case in which the number of unobserved se-
ries is higher than one and introduced an improvement to
reduce the computational charges of the estimation pro-
cedure. The use of ML was also followed in [74], [75]
and [10]. As example, [10] derived the ML estimator

A Survey of Methods to Interpolate, Distribute and Extrapolate Time Series455
when data are subject to different temporal aggregations
and compared its sample variance with those obtaining
after applying to [74,75] estimators Generalized Least
Squares (GLS) and Ordinary Least Squares (OLS). On
the other hand, GLS estimators were employed by
[71,76,78] for models with missing observations in the
exogenous variables and, therefore, probably with a het-
eroscedastic and serially correlated disturbance term.
In the extension to dynamic regression models, the
ML approach was again used in Palm and Nijman’s
works. [79] considered a simultaneous equations model,
not completely specified, about the Dutch labor market
with some variables only annually observed and pro-
posed to obtain initial estimates for thos e variables using
the univariate quarterly ARIMA process derived from
the observed annual series. These initial estimates were
used to estimate the model parameters by ML. [77] stud-
ied the problem of p arameters identification and [79] and
[9] the estimation one. To estimate the parameters they
proposed two alternatives based on ML. The first one
consisted of building the likelihood function from the
forecast errors, using the Kalman filter. The second al-
ternative consisted of applying the EM algorithm adapted
to incompletes samples. This adaptation was developed
in a wide and long paper by [73]. [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 esti-
mator [17]. Additionally, although other temporal disag-
gregation 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 ex-
tensions 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.
5.3. Optimal Procedures
Optimal methods get their name from the estimation
strategy they adopt. Such pr ocedures directly incorporate
the restrictions derived from the observed annual series
into the estimation pro cess to jointly obtain the BLUE of
both parameters and quarterly series. To do that, a linear
relationship between target series and indicators is usu-
ally assumed. This group of methods is one of the most
widely used and in fact its root proposal, [41], h as served
as basis for many statistical agencies [16,82, 83,84] and
analysts [85,86,87,88] to quarterly distribute annual ac-
counts and to provide flash estimates of quarterly growth,
among other tasks.
There are many links between benchmarking and op-
timal procedure; however, according to [14], “(1) …
compared to optimal methods, adjustment methods make
an inefficient (and sometimes, biased) use of the indica-
tors; (2) the various methods have a different capability
of providing statistically efficient extrapolation”. In con-
trast, the solution of optimal methods crucially depends
on the correlations structure assumed for the disturbances
of the linear relationship. In fact, many optimal proposals
are only different in that point. All of them, nevertheless,
pursue to avoid spurious steps in the estimated series.
Friedman [42] was the first one in applying this ap-
proach. He solved the stock variable case obtaining the
BLUE of both coefficients and objective series. Never-
theless, [41] were who solved, using a common notation,
the interpolation, distribution, and extrapolation prob-
lems and wrote the paper probably most influential and
cited in this subject. They focused on the case of con-
verting a quarterly series into a monthly one and as-
sumed an AR (1) hypothesis for the disturbances in order
to avoid unjustified discontinuities in the HF estimated
series. Under this hypothesis, the covariance matrix is
governed by the autoregressive coefficient of order one
of the HF disturbance series, which is unknown. Hence,
to apply the method it ha s to be pr ev ious ly estimated. [41]
suggested exploiting the functional relationship between
autoregressive coefficients of order one of the LF and the
HF errors to estimate it. Specifically, they proposed an
iterative procedure to estimate the monthly AR (1) coef-
ficient from the rate between elements (1, 2) and (1, 1) of
the quarterly error covariance matrix.
The [41] strategy of relating the autoregressive coeffi-
cients of order one of the high and low error series,
however, cannot be completely generalized to any pa ir of
frequencies [89] and consequently several other strata-
gems were followed to solve the issue. In line with [41],
[55] obtained for the annual-quarterly case a function
between the two au toregressive coefficients. The relation
reached by [55], unfortunately, only has unique solution
for non-negative annual autoregressive coefficients. De-
spite it, [90,91] took advantage of such a relation to sug-
gest two iterative procedures to handle the Chow-Lin
method with AR (1) errors in the quarterly-annual case.
[90] even provided a solution to apply the method when
an initial negative estimate of the autoregressive coeffi-
cient is obtained. Althoug h, to handle the problem of the
sign, [40] had already proposed to estimate the autore-
gressive coefficient through a two-step algorithm in
which, in the first step, the element (1, 3) of the covari-
ance matrix of the annual errors is used to determinate
the sign of the autoregressive coefficient. In addition to
the above possibilities, strategies based on the maximum
likelihood (with the hypothesis of normality for the er-
rors) have been also tried. Examples of this approach can

A Survey of Methods to Interpolate, Distribute and Extrapolate Time Series 456
be found in [46,92,93].
Despite the AR (1) temporal error structure being the
most extensively analyzed, other disturbance structures
has been proposed. Among the stationary ones, [94] held
MA (1), AR (2), AR (4), and a mix between AR (1) and
AR (4) as reasonable possibilities to deal with the an-
nual-quarterly case. These complexities, however, were
probed as unnecessary in [95], who through a Monte
Carlo experiment showed that, although disturbances
actually follow other stationary structures, assuming an
AR (1) hypothesis on the disturbance term does not sig-
nificantly influence the quality of the estimates. In regard
to the extensions towards no stationary structures, [66]
and [96] can be cited. On the one hand, [66], based on
[97,98] results, recommended to use [36] and showed
that such an app roach to the problem is equ ivalent to use
[41] with a random walk hypothesis for the errors. And,
on the other hand, [96] studied the problem of monthly
disaggregating a quarterly series and extended [41] for
the case in which the residual series followed a Markov
random walk. [96], however, did not solve the problem
of estimating the parameter of the Markov process for
small samples. Fortunately, [95] found a solution to this
problem and extended [99] to the case of annual series
and quarterly in dicators.
Despite [55] abovementioned words about the superi-
ority of optimal methods over adjustment procedures, all
the previous methods can be, in fact, obtained as solu-
tions of a quadratic-linear optimization problem [63],
where the metric matrix that defines the loss function is
the inverse of the high-fr equency covariance error matrix.
Theoretically, therefore, other structures for the distur-
bances could be easily managed. Thus, for example, in
line with [37], the HF covariance error matrix could be
estimated from the ARIMA structure of the LF covari-
ance matrix. Despite it, low AR order models are still
systematically chosen in practice due to (1) the covari-
ance 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).
The estimates obtained following [41] are, however,
only completely satisfactory wh en the temporal aggrega-
tion constraint is linear and there are no lagged depend-
ent variables in the regression. Thus, to improve accu-
racy of estimates taking into account dynamics specifica-
tions, usually en countered in app lied econometrics works,
several authors [80,81,101,102] have proposed to gener-
alize [41] (including [66] and [96]) by the use of linear
dynamic models, which permit to perform temporal dis-
aggregation problems providing more robust results in a
broad range of circumstances. In particular, [80] follow-
ing the works initiated by [81,101,102] which were very
difficult to implement in a computer program proposed
an extension of [41] that is particularly adequate when
the series used are stationary or co-integrated [103].
They besides solve the features of the estimation of the
first low-frequency period and produce disaggregated
estimates and standard errors in a straightforward way. In
[104] a MATLAB library to perform it is provided. This
library completed the MATLAB libraries, also detailed
in [105], that to run [27,36,41,54,66,96,106] offers the
Spanish Statistical Official Institute (INE) free of charge
[107]. On the other hand, empirical applications of these
procedures could be consulted in [49,65], where fur-
thermore a panoramic revision of [80]’s procedure is
offered.
The [41] approach and its abovementioned extensions
are all univariate, thus to handle problems with more
than J (>1) series to-be-estimated, multivariate exten-
sions are required. In this situations, apart from the
low-frequency temporal constraints, some additional
cross-section, transversal or contemporaneous aggregates
among the HF target series are usually available. To deal
with this issue, different procedures [41] have been pro-
posed in the literature. [108] was the first who faced this
problem. [108] assumed that the contemporaneous HF
aggregate of the J series is known and proposed to apply
an estimation procedure in two steps. In the first step, he
suggested applying [41], in an isolated way, to each one
of the J series imposing only the corresponding LF con-
straint and assuming white noise residu als. In the second
step, he proposed to run again [41] procedure, imposing
as constraint the observed contemporaneous aggregated
series under a white noise error vector, to simultaneously
estimate the J series using as indicators the series esti-
mated in the first step. This strategy, however, as [106]
pointed out, does not guarantee the fulfillment of the
temporal restrictions.
[102], attending to [108] limitation, generalized the
[41] estimator and got the BLUE of the J series, fulfilling
simultaneously the temporal and the transversal restric-
tions. Similar to [41], [106] again obtained that the esti-
mated series crucially depend on the structure assumed
for the disturbances. Nevertheless, he only offered a
practical solution under the hypothesis of errors tempo-
rally uncorrelated. That hypothesis unfortunately is in-
adequate due to it can produce spurious steps in the es-
timated series. In order to solve it, [109] and [110] in-
troduced a structure for the disturbances in which each
one of the J error series follow either an AR (1) process
or a random walk with shocks only contemporaneously

A Survey of Methods to Interpolate, Distribute and Extrapolate Time Series457
correlated. [110], additionally, extended the estimator
obtained in [106] to situations with more general con-
temporaneous aggregations and provided an algorithm to
run such so complex disturbance structure in empirical
works. Algorithm that is unnecessary under [54]’s sim-
plification, which proposes assuming a multivariate ran-
dom walk structure for the error vector. Finally, the mul-
tivariate extrapolation issue was introduced in [13] ex-
tending [10 9] pr o posal .
6. The Kalman Filter Strategies
In the study of time series, one of the approaches is to
consider the series as a realization of a stochastic process
with a particular model generator (e.g., an ARIMA proc-
ess), which depends on some parameters. In order to
predict how the series will behave in a future or to re-
build the series estimating the missing observation it is
necessary to know the model parameters. The Kalman
filter permits to take advantage of the tempor al sequence
of the series to implement through a set of mathematical
equations a predictor-corrector type estimator, which is
optimal in the sense th at it minimizes the estimated error
covariance when some presumed conditions are met. In
particular, it is an efficient recursive filter that estimates
the state of a dynamic system from a series of incomplete
and noisy measurements. This approach appears very
promising within the temporal disaggregation problem
due to its great versatility. Moreover, for economists, it
presents the additional advantage of making possible that
both unadjusted and seasonally adjusted series can be
simultaneously estimated.
Among the different approaches to approximate the
population parameters of the data generating process it
stands out ML. The likelihood function of the stochastic
process can be calculated in a relatively simple and very
operative way by the Kalman filter. The density of the
process, under a Gaussian distribution assumption for the
series, can be easily derived from the forecast errors.
Prediction errors can be computed in a straightforward
way by representing the process in the state space, and
the Kalman filter can then be used. In general, the pio-
neers methods based on the representation in the state
space supposed an ARIMA process for the objective
series and computed the likelihood of the process
through the Kalman filter by employing the smooth
point-fixed algorithm (see, e.g., [111] for details in both
univariate and multivariate cases) to estimate the not
available values.
Despite the representation of a temporal process in the
state space not being unique, the majority of the propos-
als to adapt Kalman filter to manage missing observa-
tions can be reduced to th e one proposed by [112]. [112]
suggested building the likelihood function excluding the
prediction errors associated to those temporal points
where no observation exist and proposed to use forecasts
obtained in the previous instant to go on running the
Kalman filter equations. Among others, this pattern was
followed by [111-117]. Additionally to [112] approach,
other approaches can be found. [118] developed a new
filter and some smooth algorithms which allow interpo-
lating the non observed values with simpler computa-
tional and analytical expressions. [119] used state-space
models to adjust a monthly series ob tained from a survey
to an annual benchmark. And, [120] followed the strat-
egy of estimating missing observations considering them
as outliers, while [121] introduced a prescribed multipli-
cative trend in the problem of quarterly disaggregating
an annual flow series using its state-space representation.
Jones [112], pioneer in the estimation of missing ob-
servations from the state space representation, treated the
case of a stock variable which is assumed to follow a
stationary ARMA process. Later on, [113], also dealing
with stationary series, extended [112] proposal for the
case of flow variables. Besides, they adapted the algo-
rithm to the case in which the target series follows a re-
gression model with stationary residuals and dealt with
the problem of working with logarithms of the variable.
What’s more, [113] also extended the procedure to the
case of stock variables following non stationary ARIMA
processes; although in this case, they compelled the tar-
get variable being available in HF for a large enough
sample subperiod. In the non stationary case, however,
when [113]’s hypothesis is not verified, building the li-
kelihood of the process becomes difficult. Problems in
converting the process into stationary and in defining the
initial conditions arise. In order to solve it, [114] pro-
posed to consider a diffuse initial distribution in the
pre-sample, while [115] suggested transforming the ob-
servations in order to define the likelihood of the process.
[115]’s transformation made possible to generalize the
previous results (including those reached by [113]), al-
though at the cost of destroying the sequence of the se-
ries, altering both smoothing and filtering algorithms.
Fortunately, [117] went be yond this difficu lty and solved
it making possible to use the classical tools to deal with
the problem of non stationary processes whatever the
structure of the missing observations. However, althoug h
[115] and [117] extended the issue to the treatment of
regression models with non stationary residuals (allow-
ing related variables to be included in this framework),
they did not deal with the case of flow variables in an
explicit way. [1 16] was who handled such a pr oblem and
extended the solu tion to non stationary flow series. [116],
moreover, suggested using the Kalman filter for the re-
cursive estimate of the non-observed values as a tool to
overcome the problem of the change of the estimates due

A Survey of Methods to Interpolate, Distribute and Extrapolate Time Series 458
to the increasing of the available sample. In this line,
[122] used information contained in related series to es-
timate the monthly Swiss GDP from the quarterly series,
while [123] estimated a monthly US GDP series from
quarterly values after testing several state-space repre-
sentations to, through a MonteCarlo experiment, identify
which variant of the model gives the best estimates. They
found the more simple representations did almost as well
as more complex ones.
Most of above proposals, however, consider the tem-
poral structure (the ARIMA process) of the objective
series known. In practice, however, it is unknown and it
is required to specify the orders of the process to deal
with it. In order to solve it, some strategies have been
followed. Some attempts have tried to infer the process
of a HF series from the observed process of the LF one
[6,60,116]; while many other studies have concentrated
on analyzing the effect of aggregation over a HF process
(e.g., among others, [32], [124-126], and, more recently,
[127] and [3]) and on studying its effect over stock vari-
ables observed in fixed step times (among others, [128],
or [11]). Fortunately, the necessary and sufficient condi-
tions under which the aggregate and/or disaggregate se-
ries can be expressed by the same class of model was
derived by [129].
Both multivariate and dynamic extensions have been
also tackled from this framework, although they are just
incipient. On the one hand, the multivariate approach
started by [111] was contin ued in [130], who suggested a
multivariate seemingly unrelated time series equations
model to estimate the HF series using the Kalman filter
when several constraints exits. The framework they pro-
posed is flexible enough to allow for almost any kind of
temporal disaggregation problems of both raw and sea-
sonally adjusted time series. On the other hand, [131]
offered a dynamic extension providing, among others
contributions, a systematic treatment of [96], which per-
mits to explain the d ifficulties commonly encountered in
practice when est imating [96] ’s model.
7. Frequency Domain Approaches
A great amount of energy has been devoted to deal with
the matter from the temporal perspective. Similarly, great
efforts have been also devoted from the frequency do-
main, although they have had less successful and have
done less fruits. In this approach, the greatest efforts
have been invested on estimating the spectral density
function or spectrum of the series, the main tool of a
temporal process in the frequency plane. The estimation
of the spectrum of the series has been undertaken from
both angles: the parametric and the non-parametric per-
spective.
Both Jones [132] and Parzen [133,134] were pioneers
in the study of missing observations from the frequency
domain. They analyzed the problem under a systematic
scheme for the observed (and therefore also for the un-
observed) values. [132], one of the pioneers in studying
the problem of estimating the spectrum, treated the case
of estimating the spectral function of a stock stationary
series sampled systematically. This problem was also
faced by [134] who introduced the term of amplitude
modulation, the key element in which later spectral de-
velopments were based on in their search for solutions.
The amplitude modulation defines itself as a zeros and
ones series in the sample period. The value of the ampli-
tude modulation is one in those periods where the series
is observed, whereas it is zero in case of not being ob-
served.
Different schemes for the amplitude modulation have
been considered in the literature. [135] studied the case
in which the amplitude modulation followed a Bernoulli
random scheme. This random scheme was extended to
others by [136] and [137]. More recen tly, [138] obtained
estimators of the spectral function for three types of
modulation sequences: determinist, random and corre-
lated random. On the other hand, [139], [140] and [141]
followed a different approach; they assumed an ARIMA
process and estimated its parameters with the help of the
spectral approximation to the likelihood func tion.
Although the great majority of patterns for the missing
observations can apparently be treated from the fre-
quency domain, not all of them have a solution. This fact
is a consequence of the impossibility of completely esti-
mating the autocovariances of the process in many prac-
tical situations. In this sense, [142] studied the situations
in which it is possible to estimate all the auto cov arian ces.
On the particular, it must be remembered [139] words:
“… (the estimators) are asymptotically efficient when
compared to the Gaussian maximum likelihood estimate
if the proportion of missing data is asymptotically negli-
gible.”. Hence, the problem of disaggregating an annual
time series in quarterly figures is one of those that do not
still have a satisfactory solution from this perspective.
Likewise, the efforts made to emplo y the sp ectral tools to
estimate the missing values using the information given
by a group of related variables have required so many
restrictive hypotheses that its use has not been advisable
until now. Nevertheless, from a related approach, [143]
have made some advances proposing a method to esti-
mate (under some restrictive hypothesis and in a con-
tinuous way) a flow variable.
8. Discussion
The large list of the reference section clearly shows that
a vast quantity of methods, procedures, and algorithms
has been proposed in the literature to d eal with the prob-

A Survey of Methods to Interpolate, Distribute and Extrapolate Time Series459
lem of transforming a low-frequency series (either an-
nual or quarterly) into a high-frequency one (either quar-
terly or monthly). The first proposals, which built series
using pure mathematical ad-hoc procedures, were pro-
gressively overcame and gradually strategies based on
related indicators were gaining researchers’ preferences,
with the [41] method, and all its extensions, highlighting.
Likewise, interesting solutions have been also suggested
representing the series into the state-space and using the
Kalman filter to handle the underlying dynamic system.
In fact, in my opinion, the great flexibility of this strat-
egy makes it a proper tool to deal with the future chal-
lenges to appear in th e subject and to hand le situations of
non-systematic missing observations (non-treated in this
paper). The advances made from the frequency domain
however do not seem encouraging. None of the ap-
proaches, nevertheless, should be discarded rapidly due
to, according to [144], pooling estimates obtained from
different procedures can improve the quality of the dis-
aggregated series.
An analysis of the historical evolution of the topic,
nevertheless, points towards dynamic regression models
and techniques using formulations in terms of unob-
served component models/structural time series and the
Kalman filter as the research lines that will hold a
pre-eminent position in the future. On the one hand, the
extension of the subject to deal with multivariate dy-
namic models is still waiting to be tackled; and, on the
other hand, the state-space methodology offers the gen-
erality that is required to address a variety of inferential
issues that have not been dealt with previously. In this
sense, both approaches could be combined in order to
solve one of the main open problems currently posed in
the area: to jointly estimate some high-frequency series
of rates when the low-frequency series of rates, some
transversal constraints and several related variables are
available. For example, it could be used to solve the
problem of distributing among the regions of a country
the national quarterly rate of growth, when the annual
series of regional growth rates are known and several
high-frequency regional indicators are available and,
moreover, both the regional and sector structure of
weights change quarterly and/or annually. In this line,
Prioreti’s [145] recent paper is a first step.
In addition, a new emerging approach that is taking
into account the more recent developments in applied
statistical literature (including data mining, dynamic
common component analyses, time series models envi-
ronment and Bayesian modeling) and that takes advan-
tage of the continuous advances in computer hardware
and software (making u se of large dataset available) will
likely turn up in the future as a main line in the subject.
Indeed, as [146] point out: “Existing methods … are ei-
ther univariate or based on a very limited number of
series, due to data and computin g constraints …until the
recent past. Nowadays large datasets are readily avail-
able, and models with hund reds of parameters are easily
estimated”. In this line, [147] dealt with a dynamic facto r
model using the Kalman filter to perform an index of
coincident US economic indicators; [146] modeled a
large datasets with a factor model and developed an in-
terpolation procedure which clearly improves univariate
approaches that exploits the estimated factors as sum-
mary of all the available information; and, [148] pro-
posed a bivariate basic structural model that permits to
carry out simultaneously the seasonal and calendar ad-
justment and the temporal disaggregation.
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