%Paper: ewp-em/9501001
%From: "Leigh Tesfatsion" <S1.TES@ISUMVS.IASTATE.EDU>
%Date: Fri, 13 Jan 95 13:37:16 CST

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                         \begin{flushright}
{\bf ISU Economic Report No.\ 28\/} \\
{\bf Revised 6 January 1994}
                         \end{flushright}
\vspace*{2mm}

                         \begin{flushleft}
{\Large{\bf A Multicriteria Approach to Model \\Specification and
Estimation$^*$}} \\ \vspace*{4 mm}

{\large{\bf Robert Kalaba}} \\
{\bf Departments of Electrical and Biomedical Engineering} \\
{\bf University of Southern California, Los Angeles, CA  90089} \\
\vspace*{4mm}

{\large{\bf Leigh Tesfatsion}} \\
{\bf Department of Economics and Department of Mathematics} \\
{\bf Iowa State University, Ames, IA 50011-1070} \\
\vspace*{4mm}

                              \end{flushleft}

\noindent {\it Abstract:\/} In decision theory, incommensurabilities among
conflicting decision
criteria are typically handled by multicriteria optimization methods such as
Pareto efficiency and mean-variance analysis.  In econometrics and
statistics, where conflicting model criteria replace conflicting decision
criteria, probability assessments are routinely used to transform disparate
model discrepancy terms into apparently commensurable quantities.  This
tactic has both strengths and weaknesses.  On the plus side, it permits the
construction of a single real-valued measure of theory and data
incompatibility in the form of a likelihood function or a posterior
probability distribution. On the minus side, the amalgamation of conceptually
distinct model discrepancy terms into a single real-valued incompatibility
measure can make it difficult to untangle the true source of any diagnosed
model specification problem.  This paper discusses recent theoretical and
empirical work on a multicriteria ``flexible least squares'' (FLS) approach
to model specification and estimation.  The basic FLS objective is to
determine the ``cost-efficient frontier,'' that is, the set of estimates that
are minimally incompatible with a specified set of model criteria.  The
relation of this work to previous work in econometrics, statistics, and
systems science is also clarified.

\vspace*{4mm}

\noindent {\bf Keywords:} Model Specification; Set-Valued Estimation;
Multicriteria Decision Making; Efficiency; Vector-Valued Optimization;
Flexible Least Squares.
\vspace*{2mm}

\vspace*{4mm}

{\footnotesize
\setlength{\baselineskip}{15pt}
\indent  $*$This work was partially supported by NIH Grant No. DK 33729 and
has been presented at meetings of the Econometric Society, the Society for
Economic Dynamics and Control, the Midwest Econometrics Group, and the IC$^2$
Institute.  A preliminary abridged version of this paper appears in the
IC$^2$ proceedings volume \cite{kt5}.  The authors are grateful to the editor
and two anonymous referees for helpful comments.  Please address
correspondence to L.\ Tesfatsion (tesfatsi@iastate.edu).
 }

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% Here is the LaTeX file mc1.tex for Section 1
\section{Introduction}

\indent
\indent
     Why have multicriteria decision making (MCDM) techniques played only a
minor role in econometric and statistical methodology to date?

     On the surface, this minor role is surprising.  Every postulated
theoretical relation is almost surely false.  A cross-sectional function for
household demand may be misspecified as linear rather than nonlinear.
Dynamic relations may be misspecified because, for example, a wealth
accumulation function omits an important variable.  Measurement errors
stemming from imprecise measuring instruments may not be additive, or, even
if additive, they may not be normally distributed.  The important point is
that conceptually distinct types of theoretical relations are false for
conceptually distinct reasons.  Consequently, model specification and
estimation would seem, intrinsically, to be a {\it multicriteria\/} decision
problem.  Any model will typically entail various conceptually distinct types
of model specification error, and a researcher undertaking the estimation of
the model would presumably want {\it each\/} type of error to be small.

     The apparent explanation for the minor MCDM role is that standard
econometric and statistical techniques routinely require researchers to cast
their inference problems in an all-encompassing stochastic framework.  As
will be discussed more carefully in subsequent sections, the actual data
generating process is assumed to be describable by means of some well-defined
probability distribution either objectively, i.e., apart from any observer,
or subjectively, as a coherent reflection of a researcher's beliefs.  Within
this all-encompassing stochastic framework, discrepancy terms arising from
model misspecification are interpreted as random quantitites governed by
joint probability distributions.  The determination of the separate and joint
behavior of the theoretical variables in relation to process observations can
then be analyzed in terms of a likelihood function or a posterior probability
distribution.  The problem of reconciling imperfect theory with observations
is thus transformed into the problem of determining the most probable
parameter values for a stochastic model whose structural form is assumed to
be correctly and completely specified.

      What are the strengths and weaknesses of this standard approach?  On
the plus side, it provides a powerful and elegant way in which to scale and
weigh disparate sources of information.  All discussion of theoretical
variables is conducted in terms of assumed joint probability relations, so
that a common level of abstraction is achieved.  This permits the
construction of a single {\it real-valued\/} measure of incompatibility
(goodness of fit) between theory and observations, e.g., the construction of
a likelihood function.  To use an analogy from decision theory, it is as if
the preferences of decision makers with potentially conflicting objectives
could always be represented in aggregate form by a single real-valued utility
function.

     On the minus side, it forces an inferential study to proceed under the
generally false presumption of correct model specification.  This standard
``null hypothesis'' is to be employed even when a researcher is fully aware
that he has resorted to conventional or otherwise arbitrary probability
assessments for model discrepancy terms.  Residuals (estimates for the model
discrepancy terms) can of course subsequently be subjected to various
diagnostic procedures to check for model misspecification.  Yet the fact
remains that all incompatibilities between theory and observations, whatever
their actual source, are forced to reveal themselves as inconsistencies
between postulated and empirical {\it probability\/} relations; the
cross-sectional, dynamic, or measurement relations tend to be pushed into the
background or lost sight of entirely through various analytical
manipulations.  Untangling the true source of a diagnosed specification
problem can thus be difficult.

     In refs.\ \cite{kt0}-\cite{kt4} the problem of model specification and
estimation is re-examined from a multicriteria perspective.  A framework is
developed which encompasses a broad range of views concerning the appropriate
interpretation and treatment of model discrepancy terms.  On the one hand,
conceptually distinct discrepancy terms can be considered without
amalgamation, as illustrated by the ``flexible least squares'' (FLS)
approach.  The basic FLS objective is to determine the set of estimates that
are ``cost efficient'' in the sense that no other estimates yield uniformly
smaller discrepancy terms.  Alternatively, when appropriate, joint
probability assessments can be used to achieve a complete amalgamation of the
discrepancy terms into a single real-valued measure of theory and data
incompatibility.

     Section 2 illustrates the FLS approach for a time-varying linear
estimation problem in which a researcher is unable or unwilling to provide
probability assessments for model discrepancy terms.  Section 3 contrasts the
FLS handling of this problem with the standard inferential approach in which
probability assessments for discrepancy terms are assumed to be available.  A
variety of FLS simulation studies and empirical applications are reviewed in
Section 4.  A more general multicriteria framework for model specification
and estimation is outlined in Section 5.  Section 6 discusses the
relationship of this multicriteria framework to previous uses of
multicriteria methods in econometrics, statistics, and systems science.
Final remarks are given in Section 7.

.....................................................................
% Here is the LaTeX file mc2.tex for Section 2
\vspace*{2mm}
\section{The FLS Approach: An Illustrative Example}

\indent
\indent
     Suppose scalar observations $y_1,y_2,\ldots ,y_T$ have been obtained on
a process at successive time points $1,2,\ldots ,T$.  The basic estimation
objective is to understand the way in which the process has evolved over the
course of the observation period.

    The state of the process at each time $t$ is described by an $N \times 1$
column vector $x_t$ of unknown process attributes.  For example, for a
time-varying linear regression problem, $x_t$ might simply be a listing of
the time $t$ regression coefficients.  For an economic growth problem, $x_t$
might include stocks of real and financial assets available at time $t$,
together with various structural parameters characterizing the objectives and
constraints faced by firms and households.

     The relationship between the observation $y_t$ and the state vector
$x_t$ at each time t is postulated {\it a priori\/} to be approximately
linear.  In addition, the evolution of the state vector $x_t$---although not
well understood {\it a priori\/}---is postulated to be gradual in the sense
that $x_t$ undergoes at most a small change from one observation time to the
next.  These prior postulates of approximately linear measurement and gradual
state evolution are modelled as follows:
   \vspace*{2mm}

\noindent {\bf Measurement Relations [Approximate Linearity]:}
               \begin{equation}  \label{1}
          y_t  -  h'_{t}x_t ~ \approx ~ 0 ~  , ~  t  =  1,...,T ~,
                    \end{equation}
   \vspace*{1mm}
{\it where $h'_t$ is a $1 \times N$ row vector of known exogenous variables.}


   \vspace*{4mm}

\noindent {\bf Dynamic Relations [Gradual State Evolution]:}
                        \begin{equation} \label{2}
     x_{t+1}  -  x_t ~ \approx ~ {\bf 0}~   , ~  t  =  1,\ldots ,T-1~.
                         \end{equation}

    In accordance with the basic estimation objective, suppose an attempt is
now made to determine all possible estimates ${\bf \hat{X}}_T$ =
$(\hat{x}_1,\ldots ,\hat{x}_T)$ for the state sequence ${\bf X}_T$ =
$(x_1,\ldots ,x_T)$ that are minimally incompatible with the given
theoretical relations (1) and (2), conditional on the given observation
sequence ${\bf Y}_T$ = $(y_1,\ldots ,y_T)$.  The multicriteria nature of this
estimation problem is seen as follows.  Two conceptually distinct types of
discrepancy terms can be associated with each possible state sequence
estimate ${\bf \hat{X}}_T$.  First, the choice of ${\bf \hat{X}}_T$ could
result in nonzero measurement discrepancy terms $y_t - h'_t\hat{x}_t$ in
(\ref{1}).  Second, the choice of ${\bf \hat{X}}_T$  could result in nonzero
dynamic discrepancy terms $\hat{x}_{t+1} - \hat{x}_t$ in (\ref{2}).  In order
to conclude that the theoretical relations (1) and (2) are in reasonable
agreement with the observations, {\it each\/} type of discrepancy would have
to be small in some sense.

     Suppose a measurement cost $c_M({\bf \hat{X}}_T,{\bf Y}_T,T)$ and a
dynamic cost $c_D({\bf \hat{X}}_T,{\bf Y}_T,T)$ are separately assessed for
the two disparate types of discrepancy terms entailed by the choice
of a state sequence estimate ${\bf \hat{X}}_T$.  These costs represent the
degree to which nonzero discrepancy terms are viewed as undesirable.  For
illustration, suppose these costs take the form of sums of squared
discrepancy terms, implying that positive and negative discrepancies are
viewed as equally undesirable.  More precisely, for any given state sequence
estimate ${\bf \hat{X}}_T$, let the measurement cost associated with ${\bf
\hat{X}}_T$ be given by
                      \begin{equation} \label{3}
c_M({\bf \hat{X}}_T,{\bf Y}_T,T) ~ = ~ \sum_{t=1}^{T} [y_t -
                                                    h'_t\hat{x}_t]^2~,
                       \end{equation}
and let the dynamic cost associated with ${\bf \hat{X}}_T$ be given
by
                    \begin{equation}  \label{4}
c_D({\bf \hat{X}}_T,{\bf Y}_T,T)~=~ \sum_{t=1}^{T-1}[\hat{x}_{t+1}-
               \hat{x}_t]'D [\hat{x}_{t+1} - \hat{x}_t] ~ ,
                    \end{equation}
where $D$ is a suitably selected positive definite scaling matrix.%
     \footnote{The scaling matrix $D$ can be specified so that the ``FLS''
estimates obtained below for the state vectors $x_t$ are essentially invariant
to the choice of units for the components of the exogenous vectors $h_t$.
See ref.\ \cite[Footnote 3]{tes} }.

     If the prior beliefs (1) and (2) concerning the measurement and dynamic
relations hold true with absolute equality, then selecting the actual state
sequence ${\bf X}_T$ as the state sequence estimate would result in zero
values for both $c_M$ and $c_D$---the ``ideal'' cost point in the terminology
of Yu \cite[p.\ 67]{yu}.  In all other cases, each potential state sequence
estimate ${\bf \hat{X}}_T$ will entail positive measurement and/or dynamic
costs.  Nevertheless, not all of these state sequence estimates are equally
interesting.  In particular, a state sequence estimate ${\bf \hat{X}}_T$ that
is dominated by another estimate ${\bf X}^*_T$, in the sense that ${\bf
X}^*_T$ yields a lower value for one type of cost without increasing the
value of the other, should presumably be excluded from consideration.

     Attention is therefore focused on the set of undominated state sequence
estimates.  Such estimates are referred to as {\it flexible least squares
(FLS)\/} estimates.  Each FLS estimate shows how the process state vector
could have evolved over time in a manner minimally incompatible with the
prior measurement and dynamic relations (1) and (2).  Without additional
modelling criteria, restricting attention to any proper subset of the FLS
estimates is an arbitrary decision.  Consequently, the FLS approach envisions
the generation and consideration of a representative sample of the FLS
estimates in order to determine the similarities and divergencies displayed
by these potential state sequences.  The similarities might be used to
construct more structured hypotheses regarding the measurement and evolution
of the state vector.  The divergencies reflect the uncertainty inherent in
the problem formulation regarding the true nature of the underlying process.

     Define the {\it cost possibility set\/} to be the collection
                \begin{equation} \label{5}
C(T) ~ = ~ \{ c_D({\bf \hat{X}}_T,{\bf Y}_T,T), c_M({\bf \hat{X}}_T,{\bf
             Y}_T,T) \mid {\bf \hat{X}}_T \in R^{TN} \}
                      \end{equation}
of all possible configurations of dynamic and measurement costs attainable at
time $T$, conditional on the given observation sequence ${\bf Y}_T$.  In
analogy to the usual Pareto-efficient frontier, the {\it cost-efficient
frontier\/} $C^F(T)$ is then defined to be the collection of all undominated
cost vectors $c$ $=$ $(c_D,c_M)$ in $C(T)$, i.e., all cost vectors $c$ in
$C(T)$ for which there exists no other cost vector $c^*$ in $C(T)$ satisfying
$c^*$ $\leq$ $c$ with $c^* \ne c$.  Formally, letting vmin denote vector
minimization,
                   \begin{equation}  \label{6}
                     C^F(T) ~ = ~ \mbox{vmin}\, C(T) ~.
                        \end{equation}
By construction, then, the cost-efficient frontier is the collection of all
cost vectors associated with the FLS state sequence estimates.

     If the $N \times T$ matrix $[h_1,\ldots ,h_T]$ has full rank $N$, the
cost-efficient frontier $C^F(T)$ is a strictly convex curve in the $c_D-c_M$
plane giving the locus of vector-minimal costs attainable at time $T$,
conditional on the given observations.  In particular, as depicted in Figure
1, $C^F(T)$ reveals the measurement cost $c_M$ that must be paid in order to
achieve a zero dynamic cost $c_D$, i.e., time-constant state vector
estimates.
                           \begin{center}
                 {\bf ---Insert Figure 1 About Here---}
                          \end{center}

     Once the FLS estimates and the cost-efficient frontier have been
determined, three different levels of analysis can be used to investigate the
degree to which the theoretical relations (1) and (2) are incompatible with
the observations $y_1,\ldots ,y_T$.

     First, one can determine the efficient attainable trade-off between the
measurement and dynamic costs $c_M$ and $c_D$ at any point $\mu $ along the
cost-efficient frontier, where $\mu$ denotes the slope of the frontier
multiplied by $-1$; i.e., $\mu$ $\equiv$ $-dc_M/dc_D$.  Second, one can
generate the FLS estimates whose cost vectors correspond to a rough grid of
$\mu$-points spanning the frontier.  Each of these FLS estimates yields a
possible time path for the actual state vector, and summary descriptive
statistics (e.g. average value and standard deviation) constructed for these
estimates can be used to indicate the extent to which the state vector
evolves over time.  Finally, the time-paths traced out by the FLS estimates
can be directly examined for evidence of systematic and possibly
idiosyncratic time variations in individual state variables that are
difficult to discern from summary statistical characterizations.  Various
simulation and empirical studies making use of this three-stage FLS analysis
are discussed in Section 4, below.

     In summary, the basic FLS objective is to characterize the set of all
state sequence estimates that achieve vector-minimal incompatibility between
process observations and imperfectly specified theoretical relations,
whatever form these theoretical relations might take.  Although probability
relations can be incorporated along with other types of theoretical relations
(see \cite{kt0,kt4}), they do not play a distinguished role.  Indeed, as
illustrated above, they may be absent altogether.  In contrast, commonly used
statistical estimation techniques such as maximum a posteriori (MAP) and
maximum likelihood estimation are point estimation techniques that attempt
to determine the most probable state sequence estimate for a stochastic model
whose structure is assumed to be correctly and completely specified.  The
crucial distinction between the two approaches lies in the use of probability
theory to transform potentially disparate model discrepancy terms into
apparently commensurable quantities.

     The next section illustrates this distinction by re-formulating the
state estimation problem (1) and (2) in accordance with standard statistical
practice.

.................................................................
% Here is the LaTeX file mc3.tex for Section 3
\section{Standard Approach to the Section 2 Problem}

\indent
\indent
     Suppose scalar observations $y_1,\ldots ,y_T$ obtained on a process are
postulated to be approximately linearly related to a sequence of state
vectors $x_1,\ldots ,x_T$.  The prior measurement relations take the
following form: \vspace*{2mm}

\noindent {\bf Measurement Relations [Approximate Linearity]:}
                  \begin{equation} \label{7}
          y_t ~ = ~ h'_tx_t  +  v_t ~ , ~   t = 1,...,T,
                    \end{equation}
{\it where $x_t$ denotes an $N \times 1$ column vector of unknown state
variables,
$h'_t$ denotes a $1 \times N$ row vector of known exogenous variables, and
$v_t$ denotes a scalar measurement discrepancy term.}

\vspace{2mm}
     If no restrictions are placed on the discrepancy term $v_t$, then
equation (\ref{7}) is simply a defining relation for $v_t$.  That is, $v_t$
is a slack variable, and equation (\ref{7}) is true by definition whether or
not an approximately linear relation exists between $y_t$ and $x_t$ in
actuality.  The slack variable $v_t$ depends on everything affecting $y_t$
that is not captured by the term $h'_tx_t$---that is, everything unknown, or
not presumed to be known, about how $y_t$ might depend on higher order terms
in $x_t$, on missing variables, and so forth.  To give content to the prior
of ``approximately linear measurement,'' the discrepancy term $v_t$ must
further be restricted to be small in some sense.

     Suppose in addition to (\ref{7}) that the state vector $x_t$ is assumed
to evolve gradually over time.  The prior dynamic relations take the following
form:
\vspace*{2mm}

\noindent {\bf Dynamic Relations [Gradual State Evolution]:}~
                  \begin{equation} \label{8}
        x_{t+1} ~ = ~ x_t  +  w_t ~, ~    t  =  1,...,T-1,
                  \end{equation}
{\it where the $N \times 1$ vector $w_t$ denotes a dynamic discrepancy term.}

\vspace{2mm}
     As before, if no restrictions are placed on the discrepancy term $w_t$,
then equation (\ref{8}) simply defines $w_t$ to be a slack variable
incorporating everything unknown, or not presumed to be known, about how the
differenced state vector $[x_{t+1}-x_t]$ depends on higher order terms in
$x_t$, on missing variables, and so forth.  Consequently, as it stands,
equation (\ref{8}) is true regardless of the actual relation between
$x_{t+1}$ and $x_t$.  To give content to the prior of ``gradual state
evolution,'' the discrepancy term $w_t$ must further be restricted to be
small in some sense.

     If no additional theoretical relations are introduced, the
estimation problem described above is simply an alternative representation
for the multicriteria estimation problem outlined in Section 2.  Each
possible estimate for the state sequence $(x_1,\ldots ,x_T)$ entails two
conceptually distinct apple-and-orange types of discrepancy
terms---measurement and dynamic---and a researcher undertaking this
estimation would presumably want {\it each\/} type of discrepancy to be
small.

     However, standard econometric and statistical techniques invariably do
introduce a third type of theoretical relation at this point in the
description of an estimation problem:  namely, probability relations
restricting discrepancy terms.  Consider, for example, the following commonly
assumed relations implying that the measurement and dynamic discrepancy terms
$v_t$ and $w_t$ in (\ref{7}) and (\ref{8}) are random quantities with known
probability density functions (PDF's) governing both their individual and
joint behavior:
   \vspace*{2mm}

\noindent {\bf Probability Relations:}
                 \begin{eqnarray}
\mbox{}   & (v_t)~\mbox{and} ~(w_t) ~  =  ~ \mbox{mutually and serially
          independent processes}; \label{9} \\
\mbox{} & (PDF~\mbox{for} ~v_t) ~ = ~ P_v~,~t = 1,\ldots ,T;  \label{10} \\
\mbox{} & (PDF~\mbox{for} ~ w_t)~ = ~ P_w~,~ t = 1,\ldots ,T-1; \label{11} \\
\mbox{}  & x_1 ~ \mbox{distributed independently of} ~v_t~\mbox{and}
                 ~w_t~\mbox{for each}~ t. \label{12} \\
\mbox{}  & (PDF ~\mbox{for} ~ x_1) ~  =  ~ P_x; \label{13}
                   \end{eqnarray}

     Since (\ref{7}) and (\ref{8}) are still interpreted as equations in the
usual exact mathematical sense, $v_t$ and $w_t$ now appear in these equations
as commensurable ``disturbance terms'' impinging on correctly specified
theoretical relations.  The previous interpretation for $v_t$ and $w_t$ as
apple-and-orange discrepancy terms incorporating everything unknown about the
measurement and dynamic aspects of the process is thus dramatically altered.

     Once the commensurability of the discrepancy terms $w_t$ and $v_t$ is
assumed, a single real-valued measure of theory and data incompatibility can
be constructed.  Specifically, combining the measurement relations (\ref{7})
with the probability relations (\ref{9})-(\ref{13}) permits the derivation of
a probability density function $P({\bf Y}_T\mid {\bf X}_T)$ for the
observation sequence ${\bf Y}_T$ $=$ $(y_1,\ldots ,y_T)$ conditional on the
state sequence ${\bf X_T}$ $=$ $(x_1,\ldots ,x_T)$.  Combining the dynamic
relations (\ref{8}) with the probability relations (\ref{9})-(\ref{13})
permits the derivation of a ``prior'' probability density function $P({\bf
X}_T)$ for ${\bf X}_T$.  The joint probability density function for ${\bf
X}_T$ and ${\bf Y}_T$ then takes the form
                      \begin{equation}  \label{14}
  P({\bf X}_T,{\bf Y}_T)~=~P({\bf Y}_T\mid {\bf X}_T)\cdot P({\bf X}_T).~
                          \end{equation}
The joint probability density function (\ref{14}) elegantly combines the two
distinct sources of theory and data incompatibility---measurement and
dynamic---into a single {\it real-valued\/} measure of incompatibility for
any considered state sequence ${\bf X}_T$.

     As detailed in \cite{wh}, an objective commonly assumed for estimation
problems described by relations of the form (\ref{7})-(\ref{13}) is maximum a
posteriori (MAP) estimation, i.e., the determination of the state sequence
${\bf X}_T$ that maximizes the posterior probability density function $P({\bf
X}_T\mid {\bf Y}_T)$.  Since the observation sequence ${\bf Y}_T$ is assumed
to be given, this objective is equivalent to determining the state sequence
${\bf X}_T$ that maximizes the product of $P({\bf X}_T\mid {\bf Y}_T)$ and
$P({\bf Y}_T)$.  In accordance with Bayesian rules of probability theory,
                    \begin{equation} \label{15}
 P({\bf X}_T\mid {\bf Y}_T)\cdot P({\bf Y}_T)~ = ~P({\bf Y}_T\mid {\bf
                            X}_T)\cdot P({\bf X}_T) ~,
                        \end{equation}
where, as earlier explained, the right-hand expression can be evaluated using
the relations (\ref{7})-(\ref{13}).  Determining a MAP state sequence is thus
equivalent to determining a state sequence which minimizes the real-valued
incompatibility cost function
                     \begin{equation}  \label{16}
 c({\bf X}_T,{\bf Y}_T,T) ~=~ - \log [P({\bf Y}_T\mid {\bf X}_T)P({\bf X}_T)]~.
                     \end{equation}

     In summary, what ultimately has been accomplished by the augmentation of
the measurement and dynamic relations (\ref{7}) and (\ref{8}) with the
probability relations (\ref{9}) through (\ref{13})?  {\it The multicriteria
problem of achieving vector-minimal incompatibility between imperfectly
specified theoretical relations and process observations has been transformed
into the single-criterion problem of determining the most probable state
sequence for a stochastic model whose structure is assumed to be correctly
and completely specified.\/}

     One basic objection to this standard estimation approach is that it
entails an interpretation for the discrepancy terms that is at odds with the
originally specified priors (\ref{1}) and (\ref{2}).  In particular, the
time-trend smoothness prior (\ref{2}) is replaced with the prior of a random
walk, even though these two priors represent different conceptualizations for
the movement of the underlying state vectors.  The time-trend prior (\ref{2})
postulates that successive state vectors evolve gradually from one time
period to the next, a movement that might be captured by a straight line or a
sine wave, for example.  In contrast, the random walk model implies that
``error terms'' are persistently accumulated in successive state vectors,
resulting in a nonstationary process exhibiting jagged discontinuities
between successive state vectors.

     It is sometimes countered that this distinction is unimportant if
the {\it variances\/} of the random walk error terms are anticipated to
be small.  However, as stressed in recent macroeconometric work, e.g., Nelson
and Plosser \cite{np}, the dynamic properties of a time-trend model are
altogether different from the dynamic properties of a random walk model,
however one models the variances of these error terms.  Consequently, ``small
discrepancy terms'' and ``small error term variances'' are not conceptually
interchangeable descriptions.  In particular, for initial diagnostic checks
of poorly understood structures, the probabilistic assumption of ``small
variances'' can be an overly restrictive concept (cf. Ruspini \cite{rus}).

     Another important objection to the standard estimation approach is that
the probability relations (\ref{9})-(\ref{13}) imply that $w_t$ and $v_t$ are
governed by a well-defined joint probability distribution and hence are
cardinally comparable.  For many processes it is hard to maintain this
assumption in a publicly credible way.  For example, the observations
$y_1,\ldots ,y_T$ might be the outcome of a nonreplicable experiment,
implying that probability assessments for the discrepancy terms $w_t$ and
$v_t$ cannot be put to an objective test.  Alternatively, as stressed in
Section 2, the theoretical relations (\ref{7}) and (\ref{8}) might represent
tentatively held conjectures concerning a poorly understood process, or a
linearized set of relations obtained for an analytically intractable
nonlinear process.  In this case it is questionable whether the discrepancy
terms are governed by any meaningful probability relationships.  A researcher
might then have to resort to specifications determined largely by convention
if he is forced to provide a probabilistic characterization for the
discrepancy terms.

     A third objection to the standard estimation approach is that
conceptually distinct discrepancy terms are amalgamated into a single
real-valued incompatibility measure such as (\ref{16}).  This amalgamation
makes it difficult to detect and correctly sort out which aspects of the
model, if any, are seriously misspecified.  There is of course no way to
determine from the single real-valued measure (\ref{16}) that a serious
specification error has occurred, e.g., in the dynamic relations (\ref{8})
rather than the measurement relations (\ref{7}).  In fact, (\ref{16}) is
constructed under the premise that no specification error has occurred, and
there is no way to use it per se to check for any kind of modelling
difficulty.  Rather, subsequent tests must be conducted to check whether the
data appear to be anomalous with respect to the given model specification, or
whether other plausible model specifications exist that make the data appear
less anomalous.

     A further difficulty here, as detailed in ref.\ \cite[section 5.1]{kt4},
is that standard diagnostic procedures force all incompatibilities between
theory and observations to reveal themselves as incompatibilities between
theoretically anticipated probability relations and empirically determined
statistical properties.  For example, suppose the dynamic relations (\ref{8})
are fundamentally misspecified because the true dynamic dependence of
$x_{t+1}$ on $x_t$ is highly nonlinear.  Using standard diagnostic tests on
the dynamic residual terms $\hat{w}_t$ $\equiv$ $[\hat{x}_{t+1}-\hat{x}_t]$,
a researcher would presumably perceive that the properties of these residuals
are at odds with the probability relations assumed for $w_t$ in (\ref{9}) and
(\ref{11}).  The tendency of the researcher might then be to concentrate on
modifying the probability assumptions for $w_t$ to improve statistical
fit---e.g., to replace serial independence with first-order serial
correlation, or to assume that $w_t$ has a time-varying covariance
matrix---rather than to think more carefully about the actual physical or
behavioral relationships connecting $x_{t+1}$ to $x_t$.

      These three objections to the standard estimation
approach---potentially distorted priors, inappropriate and potentially
misleading assumptions of cardinal comparability, and the confounding of
conceptually distinct discrepancy terms---would be of purely academic
interest if treating discrepancy terms as commensurable random disturbance
terms constituted the only way to obtain estimates for unknown process
states. However, Section 2 suggests to the contrary that an alternative
multicriteria treatment of discrepancy terms is also feasible for this
purpose.

...............................................................
% Here is the LaTeX file mc4.tex for Section 4
\vspace*{2mm}
\section{FLS Simulation and Empirical Studies}

\indent
\indent
     In the previous two sections a case is made for the conceptual
desirability of a multicriteria FLS approach to the estimation of process
states for processes whose properties are poorly understood {\it a priori\/}
and hence whose descriptions incorporate potentially significant
specification errors.  Not yet examined, however, is the extent to which the
FLS approach permits the recovery of {\it accurate\/} information about
process states.  The present section briefly reviews a number of simulation
and empirical studies that have addressed this issue.

     Ref.\ \cite{kt2} undertakes an FLS analysis of a time-varying
linear regression problem, a special case of (1) and (2) in which the time
$t$ state vector $x_t$ denotes the vector of time $t$ regression coefficients
and the time $t$ exogenous vector $h_t$ denotes the vector of time $t$
regressor variables.  The basic estimation objective is to determine whether
the regression coefficients have exhibited any systematic time-variation over
the course of the observation period.

     A Fortran program for generating the FLS estimates is provided in ref.\
\cite{kt2}, together with an explanation of the program logic.%
     \footnote{This FLS program for time-varying linear regression has
recently been incorporated into the statistical package SHAZAM; see
\cite{whi}, or email info@shazam.econ.ubc.ca for information.  See also
\cite{kt3} for a more general FLS Fortran program, GFLS, applicable for
systems characterized by approximately linear measurement and dynamic
relations.}
     Various FLS simulation experiments making use of this program are
reported and graphically depicted in \cite{kt2} and \cite{krt1}.  These
experiments demonstrate the ability of the FLS method to track and recover
linear, quadratic, sinusoidal, and elliptical motions in the true underlying
regression coefficients, despite noisy observations, and relying only on
prior measurement and dynamic relations of the form (1) and (2).  Indeed, the
motions are recovered with good qualitative accuracy all along the FLS
frontier.

     For example, experiments were carried out for which the components of
the true two-dimensional coefficient (state) vectors $x_t$ =
$(b_{t1},b_{t2})$ were simulated to be sinusoidal functions of $t$.  The
first component, $b_{t1}$, moved through two complete periods of a sine wave
over the interval of time from $t=1$ to $t=30$, and the second component,
$b_{t2}$, moved through one complete period of a sine wave over this same
time interval.  Each observation $y_t$ was generated in accordance with the
linear regression model $y_t$ = $h'_tx_t + v_t$, where the components of the
regressor vector $h'_t$ were taken to be deterministic cyclic functions of
$t$ and the components of the measurement discrepancy term $v_t$ were
independently generated from a pseudo-random number generator for a normal
distribution $N(0,0.5)$.

     As depicted in Figure 2, the FLS estimates for $b_{t1}$ and $b_{t2}$
closely tracked the true values for these coefficients both qualitatively and
quantitatively at the point $\mu = 1$ along the cost-efficient frontier.  As
$\mu $ was increased from $1$ to $1000$ by powers of ten, the FLS estimates
were pulled steadily inward toward the zero dynamic cost (ordinary least
squares) solution, $(b_{t1},b_{t2}) = (0.03,0.04)$ for $t = 1,\ldots ,30$.
Nevertheless, for each $\mu$, the two-period and one-period sinusoidal
motions of the true coefficients were still reflected.  Thus, sixty
coefficients were recovered from only thirty observations, with good
qualitative accuracy, all along the cost-efficient frontier.
                           \begin{center}
                {\bf ---Insert Figure 2 About Here---}
                          \end{center}

     Although these simulation experiments indicate that the FLS estimates
are able to track smooth motions in the regression coefficients, the
question remains whether discontinuous motions cause the FLS method to fail.
This issue arose in the FLS money demand study \cite{tes}, for the focus of
the study concerned possible step-function breaks in money demand regression
coefficients.  Various simulation experiments were therefore conducted in
which the components of the true regression coefficients were shifted
idiosyncratically at various points in time.  Surprisingly, using only
measurement and smoothness priors analogous to (1) and (2), the FLS estimates
were able to track and recover these step-function shifts with good
qualitative accuracy all along the cost-efficient frontier despite the
absence of any prior knowledge concerning the timing, number, and magnitude
of the shifts.  Indeed, the larger the magnitude of the shifts, the better
the accuracy of the estimates.

     To understand this seeming paradox, consider what happens if an
underlying true linear regression coefficient $\beta_{ti}$ undergoes a single
step-function shift from its current value $b$ to a new value $b'$ at some
time $t=t'$.  If the FLS estimate $\hat{\beta }_{ti}$ for $\beta_{ti}$ is
equal to $b$ for $t < t'$, and if it remains at $b$ over the remainder of the
observation period from $t'$ to $T$ despite the shift in $\beta_{ti}$ at
$t=t'$, then the result is an accumulation of measurement costs over $t'$ to
$T$; and the larger the magnitude of the shift, the larger the accumulation
of measurement costs.  On the other hand, if $\hat{\beta }_{ti}$ were
likewise to shift from $b$ to $b'$ at $t=t'$, the result would be a one-time
dynamic cost but {\it no\/} subsequent accumulation of measurement costs.
Thus, cost-minimization considerations will generally dictate that the FLS
estimates should shift in response to shifts in the underlying true
coefficients as long as the shifts are spaced sufficiently far apart and do
not occur close to the final observation time $T$.

     Given the promising nature of these shift simulation results, the FLS
method was next used in \cite{tes} to undertake an empirical money demand
investigation.  Measurement and dynamic relations analogous to (1) and (2)
were used to model U.S.\ money demand over the volatile period
1959:Q2-1985:Q3.  In particular, no prior information regarding possible
shift times was used in the FLS estimation procedure.  The time paths traced
out by the FLS coefficient (state) estimates were found to exhibit a
clear-cut downward shift in 1974, during the time of the first OPEC oil price
shock, at each tested point along the cost-efficient frontier.  This finding
was in accordance with previous ordinary least squares (OLS) studies of U.S.\
money demand that had investigated the possibility of a 1974 shift in the
money demand regression coefficients using variants of the Chow test and
recursive least squares.

     In addition, however, the FLS results in \cite{tes} also indicated the
presence of systematic idiosyncratic time variations in the regression
coefficients---e.g., a sharp and steady decline in the coefficient for the
inflation rate---which Chow tests and recursive least squares are not
designed to detect.  Moreover, the ``unit root'' nonstationarity problem
reported in these previous OLS money demand studies was seen to disappear
once the FLS coefficient estimates were allowed to exhibit even small amounts
of time variation in accordance with the dynamic smoothness prior (2).

     A number of other empirical FLS studies have recently appeared that
suggest the potential usefulness of FLS as a diagnostic tool.  For example,
Dorfman and Foster \cite{df} use FLS to develop a new measure of productivity
change.  They assume that measurement errors are independently and
identically distributed random variables whereas the coefficients
characterizing the production relation evolve slowly over time in an unknown
deterministic manner.  Under these assumptions they are able to provide a
statistical interpretation for their FLS coefficient estimates and hence also
for their FLS measure of productivity change.

     Dorfman and Foster then apply their FLS productivity measure to U.S.\
agricultural data for the period 1948-1983.  They compare the FLS measure
with two more traditional measures that assume time-constant production
function parameters---total factor productivity, and a measure of technical
change based on the elasticity of production with respect to time.  They find
(pp.\ 286-8) that the FLS measure is more stable than these latter measures
in the sense of having a smaller variance around a constant percentage growth
rate.  Interestingly, the FLS measure also produces considerably lower
estimates of productivity growth than the total factor productivity measure
and generally higher estimates of productivity growth than the elasticity
measure.

    In \cite{lut1}, L\"{u}tkepohl uses FLS to obtain detailed information
on the variability of individual coefficients for a U.S.\ money demand
relation specified in error-correction form.  He shows (Figure 1, p.\ 735)
that all long-run coefficients are relatively stable over the thirty-three
year period 1954-1987, with the least stable being the coefficient on the
short-term interest rate (proxied by the discount rate on 91-day Treasury
bills) and the most stable being the coefficient on transactions volume
(proxied by real GNP).  On the other hand, the short-run coefficients on
rates of change in the interest rate, transactions volume, and the general
price level (proxied by the GNP deflator) are considerably more volatile than
the long-run coefficients over this same period.  He concludes (p.\ 742) that
these FLS findings are consistent with a financial innovations explanation of
money demand instability over this period.

     In a different study, L\"{u}tkepohl and Herwartz \cite{lh} generalize
the FLS time-varying linear regression method developed in \cite{kt2,kt3} by
allowing for anticipated seasonal periodicities as well as for time trends.
They first undertake a study of their generalized FLS algorithm for three
artificially generated time series, each having a seasonal pattern, in which
a variable is linearly dependent on its value in some past time.  In the
first model, the intercept and slope coefficients are both time invariant; in
the second model, these coefficients are both periodic; and in the third
model, the intercept is periodic and undergoes a structural shift in
mid-sample whereas the slope coefficient is time invariant.  They show
(Tables 1 and 2) that their generalized FLS measure is able to detect the
time invariance of the coefficients in the first model and the coefficient
periodicities in the second and third models, as well as the structural shift
for the third model.

     L\"{u}tkepohl and Herwartz then use their generalized FLS measure to
study actual consumption and income time series data for the (West) German
economy, with the goal of detecting specific types of coefficient variations
and identifying any coefficients that appear to be time-invariant.  They
interpret their FLS findings for income as evidence in favor of a model in
which the intercept is periodic and remaining coefficients are time
invariant; and they interpret their FLS findings for consumption as evidence
in favor of a model with time-invariant coefficients for first and
second-order lagged terms but with periodically varying coefficients for the
intercept and higher-order lagged terms.

     Finally, Schneider \cite{sch} carries out an extensive comparative study
between maximum likelihood (EM and scoring) and FLS time-varying linear
regression methods, where the latter is characterized (p.\ 192) as a
descriptive variant of Kalman filtering that constitutes a ``simple but
powerful tool of exploratory data analysis.'' He first applies FLS as a
preliminary descriptive stability test to a standard Goldfeld-type model of
money demand for (West) Germany.  As depicted in his Figures 14.2-14.7 (pp.\
206-208), he concludes (p.\ 211) that only the coefficients for the
short-term and long-term interest rates and the 90-day swap rate exhibit a
distinct time-varying behavior.  In particular, he notes that the behavior of
the short-term interest rate is particularly remarkable: an apparent
stabilization from 1974 onward that coincides with the date when the German
central bank officially switched from an interest-rate target regime to a
money target regime.  [In subsequent discussion (p.\ 212) he notes that an
FLS argument can also be made for a step change in the swap rate in 1974, the
introduction date of flexible exchange rates.]  In support of these
conclusions, he notes that the patterns in the individual paths of the FLS
coefficient estimates persist over a large portion of the cost-efficient
frontier.

     Schneider next checks what paths for the regression coefficients are
picked out by maximum likelihood (ML) when the descriptive dynamic and
measurement costs $c_D$ and $c_M$ are reinterpreted as elements of a
likelihood function generated from a random walk model for the regression
coefficients.  First cautioning that little is known about the sampling
distributions of ML estimators for this time-varying linear regression model,
he concludes (p.\ 212) that the movements in the coefficients for the
short-term interest rate and the swap rate appear to be significantly
identified to be time-varying at a type I error level of 1 percent.  The
estimates for the variances are low, reflecting the fact that the random walk
model spreads out the time variation over the entire sample period.  Although
the FLS-apparent step changes in the short-term interest rate and swap rate
are thus considerably smoothed, the ML-estimated paths for the individual
coefficients nevertheless exhibit the same general features as the
FLS-estimated paths.  This is seen in his Figures 14.8-14.13 (pp.\ 214-217)
depicting two-standard-deviation bands about the means of proxied {\it a
posteriori\/} distributions for the regression coefficients, conditional on
ML estimates for remaining structural parameters.

................................................................
% Here is the LaTeX file mc5.tex for Section 5
\vspace*{2mm}
\section{Generalizations}

\indent
\indent
     In previous sections it is shown how FLS can be used to investigate the
basic incompatibility of theory and data for processes characterized by
approximately linear measurement relations and gradual state evolution.  In
this section we describe a more general multicriteria approach to estimation
developed in \cite{kt4}.  We also suggest how the latter approach might be
recast in the form of a utility maximization problem subject to a budget
constraint.

     Consider a situation in which a sequence ${\bf Y}_T$ $=$
$(y_1,\ldots ,y_T)$ of noisy observations $y_t$ has been obtained on some
process of interest.  The basic objective is to learn about the sequence of
states ${\bf X}_T$ $=$ $(x_1,\ldots ,x_T)$ through which the process has
passed.

     Suppose the degree to which each possible state sequence estimate ${\bf
\hat{X}}_T$ is incompatible with the given observation sequence ${\bf Y}_T$
is measured by a $K$-dimensional vector $c({\bf \hat{X}}_T,{\bf Y}_T,T)$ of
incompatibility costs.  These costs may represent penalties imposed for
failure to satisfy criteria {\it conjectured\/} to be true (theoretical
relations), and also penalties imposed for failure to satisfy criteria {\it
preferred\/} to be true (objectives).  Let $C(T)$ denote the set of all
incompatibility cost vectors $c$ $=$ $c({\bf \hat{X}}_T,{\bf Y}_T,T)$
corresponding to possible state sequence estimates ${\bf \hat{X}}_T$.  The
{\it cost-efficient frontier\/}, denoted by $C^F(T)$, is then defined to be
the collection of undominated cost vectors $c$ in $C(T)$.  That is, a cost
vector $c$ in $C(T)$ is an element of $C^F(T)$ if and only if there exists no
other cost vector $c^*$ in $C(T)$ satisfying $c^* \leq c$ with $c^* \ne c$.

     By construction, the state sequence estimates ${\bf \hat{X}}_T$ whose
cost vectors attain the cost-efficient frontier are characterized by a basic
efficiency property:  For the given observations, no other possible state
sequence estimate yields lower incompatibility cost with respect to each of
the $K$ modelling criteria included in the incompatibility cost vector.  Each
of these state sequence estimates thus represents one possible way the actual
process could have evolved over time in a manner minimally incompatible with
the prior theoretical relations and objectives.

     The basic multicriteria estimation problem can be summarized as follows:
\vspace*{2mm}

\noindent {\bf The Basic Multicriteria Estimation Problem:} {\it Given a
process length $T$, an observation sequence ${\bf Y}_T$, and a vector-valued
incompatibility cost function $c(\cdot ,{\bf Y}_T,T$), determine all possible
state sequence estimates ${\bf \hat{X}}_T$ that vector-minimize the
incompatibility cost $c({\bf \hat{X}}_T,{\bf Y}_T,T)$.  That is, determine
all possible state sequence estimates ${\bf \hat{X}}_T$ whose cost vectors
$c({\bf \hat{X}}_T,{\bf Y}_T,T)$ attain the cost-efficient frontier
$C^F(T)$.}
     \vspace*{2mm}

     The cost-efficient frontier $C^F(T)$ can be obtained by means of a
multicriteria extension of the usual scalar dynamic programming equations.%
     \footnote{General multicriteria dynamic programming algorithms have
previously been developed by a variety of other researchers.  See, for
example, ref.\ \cite{li}.}
     Consider the estimation problem at any intermediate time $t$.  Suppose a
$K$-dimensional vector $c({\bf \hat{X}}_t,{\bf Y}_t,t)$ of incompatibility
costs can be associated with each $t$-length state sequence estimate ${\bf
\hat{X}}_t$ $=$ $(\hat{x}_1,\ldots ,\hat{x}_t)$, conditional on the sequence
of observations ${\bf Y}_t$ $=$ $(y_1,\ldots ,y_t)$.  Let $C(\hat{x}_t,t)$
denote the set of all cost vectors $c({\bf \hat{X}}_t,{\bf Y}_t,t)$
attainable at time t, conditional on the time-$t$ state estimate being
$\hat{x}_t$; and let $C^F(\hat{x}_t,t)$ denote the cost-efficient frontier
for $C(\hat{x}_t,t)$.  Given certain regularity conditions, it is shown in
\cite{kt4} that the state-conditional frontier at any intermediate time $t$
is mapped into a state-conditional frontier at time $t+1$ in accordance with
a vector-valued recurrence relation having the form
                \begin{equation} \label{17}
C^F(\hat{x}_{t+1},t+1)~=~\mbox{vmin} \, ( ~ \bigcup_{\hat{x}_t}\, [
    C^F(\hat{x}_t,t) + \Delta c(\hat{x}_t,\hat{x}_{t+1},y_{t+1},t+1) ]~) ~,
                     \end{equation}
where vmin denotes vector-minimization and $\Delta c(\cdot )$ denotes a
vector of incremental costs associated with the state transition
$(\hat{x}_t,\hat{x}_{t+1})$.  The cost-efficient frontier at the
final time $T$ is then given by
               \begin{equation}  \label{18}
   C^F(T) ~ = ~ \mbox{vmin} \, [ \, \bigcup_{\hat{x}_T} C^F(\hat{x}_T,T)\, ]~.
                   \end{equation}

     Three well-known state estimation algorithms are derived in \cite{kt4}
as single-criterion special cases of the multicriteria recurrence relations
(\ref{17}) and (\ref{18}):  namely, the Kalman filter \cite{kal}, the Viterbi
filter \cite{for,vit}, and the Larson-Peschon filter \cite{lar} for
sequentially generating maximum a posteriori (MAP) probability estimates.  In
addition, an algorithm for sequentially generating the FLS estimates for the
problem discussed in Section 2, above, is derived as a bicriteria special
case of (\ref{17}) and (\ref{18}).

     Finally, it is interesting to note that the basic multicriteria
estimation problem outlined above can be recast as a problem of utility
maximization subject to constraint.  That is, one can include in the cost
vector only those costs corresponding to criteria conjectured to be true
(i.e., theoretical relations), so that the resulting cost-efficient frontier
depicting the feasible efficient trade-offs among model discrepancy terms is
analogous to a ``budget constraint.''  One could then superimpose on this
frontier the indifference curves for a researcher's ``utility function'' that
assigns a utility value to each possible configuration of costs (discrepancy
terms), thus permitting for that researcher the selection of a unique
``best'' model specification along the frontier.  In this way it might be
possible to separate the subjective selection of a model based on properties
preferred by individual researchers from the more objective identification of
model specifications that are efficient with regard to possible trade-offs
among discrepancy terms.

...................................................................
% Here is the LaTeX file mc6.tex for Section 6
\vspace*{2mm}
\section{Relation to Previous Work}

\indent
\indent
     Roughly stated, multicriteria decision making (MCDM) is the study of
decision situations in which one or more agents with potentially conflicting
objectives must somehow decide on the implementation of an action.  Due in
large part to the seminal work of Charnes and Cooper, Yu, Zeleny and others
dating back to the early nineteen sixties, MCDM has now become an established
interdisciplinary field that cuts across the boundary lines separating
operations research, management science, systems science, computer science,
applied mathematics, psychology, and many other disciplines.  See, for
example, refs.\ \cite{dye,klw,ste,yu,zel1,zio}.

     The duality between decision making (control) and estimation (system
identification) for single-criterion optimization problems has been known for
over thirty years (Kalman \cite[p.\ 42]{kal}).  Surprisingly, however, the
interconnections between {\it multicriteria\/} decision making and {\it
multicriteria\/} estimation have yet to be systematically explored.

     Some use of multicriteria methods has of course occurred in statistical
inferential studies.  Multicriteria methods have traditionally been used to
describe the trade-off between Type I and Type II errors.  In addition,
multicriteria methods have been used to describe the trade-off between bias
and variance (fidelity and smoothness) which some estimation procedures
entail.  See, for example, the discussion of ridge trace procedures in Judge
et al. \cite[pp.\ 915-916]{jud}, the discussion of smoothing splines in Wahba
\cite{wah}, and the discussion in Good and Gaskins \cite{goo} of penalized
likelihood methods for the location and probabilistic evaluation of ``bumps''
in estimated probability densities.

     One also finds instances in which researchers have advocated using
multicriteria methods for other types of estimation purposes.
For example, in the systems literature, Benedict and Bordner \cite{ben}
proposed a bicriteria estimation algorithm for a class of radar tracking
problems.  Moreover, various researchers have proposed using bicriteria
methods for handling the dual objectives of system optimization and system
identification which arise for ``dual control'' problems, i.e., for problems
in which an agent is attempting to control a system at the same time he is
attempting to learn about its characteristics.  See, for example, Haimes et
al.\ \cite{hai} and Koussoulas \cite{kou}.  In the MCDM literature, both
Narula and Wellington \cite{nar} and Zeleny \cite[pp.\ 469-471]{zel1} have
proposed the use of multicriteria methods for linear regression analysis.
Also, Charnes and Cooper have developed a ``data envelopment analysis''
method for the estimation of the Pareto-efficient frontier of an
empirically-determined multi-input/multi-output production function.   The
method has been used to classify organizations using the same kinds of inputs
and outputs either as efficient or inefficient; see \cite{bcc}.  Other
potential applications of this method are discussed in Charnes et
al.~\cite{cha} and Seiford and Thrall \cite{sei}.

     In the econometrics literature, Leamer \cite[pp.\ 141-170]{lea1}
introduces the notion of an ``information contract curve'' in the space of
regression coefficients to discuss regression selection strategies in the
case in which only the contours (iso-density surfaces) of the prior
probability density function and the likelihood function are known.
Specifically, the information contract curve is a locus of points giving all
feasible estimates for the regression coefficient vector which are efficient
relative to two potentially conflicting criteria: maximization of the prior
probability density function specified as a contour map; and maximization of
the sample-conditioned likelihood function specified as a contour map.

    In subsequent work (see \cite{lea2}), Leamer proposes a more general
``global sensitivity analysis'' for investigating the sensitivity of
posterior distribution inferences to alternative choices of prior probability
distributions.  A related line of work on ``set-valued filtering'' has been
developed in the systems science and statistics literatures; see, for
example, Stirling and Morrell \cite[Section V.B]{sti} and Morrell \cite{mor}.
These studies argue that unique estimates cannot be inferred from data sets
when, for whatever reason, a data analyst is unable to use probability
assessments to fully scale and weigh disparate sources of information in the
form of a uniquely specified posterior probability distribution.

     Many statisticians, econometricians, and systems scientists are either
unwilling or unable to undertake a complete scalarization of their estimation
problems in the form of a posterior probability distribution.  Nevertheless,
rather than considering the sensitivity of inferences to alternative prior
probability distributions, the majority of these researchers instead rely on
ordinary least squares and maximum likelihood methods for initial estimation
purposes, followed by subsequent diagnostic testing to check for model
misspecification.

     Hendry and Richard \cite{hen} have attempted to systematize the latter
model specification procedure.  They formulate various model design criteria
which they believe to be of particular relevance for econometric modelling.
For any one model, these model design criteria could be construed as
constituting an incompatibility cost vector $c$ in the sense of Section 5,
above.  However, Hendry and Richard do not attempt to determine the
trade-offs among the criteria in accordance with any systematic
multicriteria (vector optimization) procedure.

     Rather, as is standard in the diagnostic testing literature, Hendry and
Richard advocate the sequential application of their model design criteria,
opening themselves to the usual criticism (see, e.g., Judge et al.\
\cite[pp.\ 869-870]{jud}) that the choice of a final model might depend upon
the particular order of application.%
     \footnote{The observation that the ``decision path'' can affect a final
choice is also well known in the MCDM literature; see, e.g., Korhonen et
al. \cite{kmw}.}
     One way to interpret this path-dependence criticism is to note that
Hendry and Richard may simply be ending up at one among many possible
points on a frontier of models that are all equally acceptable (efficient)
relative to their postulated set of criteria.  In other words, assuming that
the various criteria represent an over-identifying set of constraints, a
systematic multicriteria treatment of the modelling problem would necessarily
lead to a {\it set\/} of efficient models rather than to a uniquely
determined specification.

     In summary, although the MCDM literature has apparently not had much of
an impact on econometric and statistical procedures to date, some preliminary
steps toward a full-blown multicriteria approach have been taken.  Leamer
considers the trade-offs between a prior and a data-based conception of a
best estimate.  Hendry and Richard formalize a set of potentially conflicting
model design criteria which they argue will be objectively meaningful to
other researchers.  In terms of the general multicriteria framework outlined
in Section 5, above, the differences separating these two approaches reduce
to a different dimension $K$ for the basic cost vector $c$, a different idea
concerning which model criteria should be included in $c$, and a different
degree of recognition that conflicting model criteria result in {\it
set\/}-valued inferences in the form of a nondegenerate cost-efficient {\it
frontier\/} of alternative models.

     Our work on multicriteria estimation has its roots in ``Sridhar
filtering.''  In a series of studies initiated in the mid-nineteen sixties
focusing on continuous-time rigid-body dynamics (see, e.g., refs.\
\cite{bel,det}), R.\ Sridhar and other associates explored the idea of
forming a cost-of-estimation function as a weighted sum of squared
dynamic and measurement discrepancy terms.  In refs.\ \cite{kt0,kt00} we
extend this previous work by considering a broader class of models and by
deriving exact filtering equations for the determination of the
cost-minimizing solutions.  In a related study, Kohn and Ansley \cite{ka1}
discuss the relation between the use of Bayesian smoothness priors for
state-space smoothing and the use of a Sridhar-type penalized least squares
criterion function with quadratically specified dynamic and measurement costs
to achieve optimal function smoothing.  However, as in the earlier Sridhar
studies, the cost-of-estimation functions in these studies are still
formulated with uniquely specified penalty weights.

     The basic FLS approach, introduced in \cite{kt000}, instead focuses
attention on a cost {\it vector\/} $(c_D,c_M)$ incorporating separate penalty
costs for dynamic and measurement discrepancy terms.  This permits the
construction of a ``cost-efficient frontier,'' a curve in a two-dimensional
cost plane that provides an explicit way to determine the efficient
trade-offs between dynamic and measurement discrepancy terms.  Since the
costs indicate the relative {\it undesirability\/} of various discrepancy
term patterns rather than any intrinsic properties of the discrepancy terms
per se, quadratic cost specifications---while useful for tractability---are
in no sense required.  As indicated in Section 5 of this paper, we now view
the original FLS formulation as a special case of a more general
multicriteria estimation framework in which the cost vector $c$ can
incorporate whatever modelling criteria are deemed relevant for the problem
at hand.

.................................................................
% Here is the LaTeX file mc7.tex for Section 7
\vspace*{2mm}
\section{Concluding Remarks}

\indent
\indent
     This paper suggests that multicriteria methods such as FLS provide a
systematic way to approach the estimation of processes whose descriptions
embody potentially significant specification errors.  The heart of the FLS
approach is the recognition that conflicting model criteria result in {\it
set\/}-valued inferences in the form of a nondegenerate cost-efficient {\it
frontier\/} of alternative model specifications.  The power and elegance
achieved by the usual scalarization through the introduction of probabilistic
assumptions is impressive; but when doubt exists concerning the
appropriateness of these assumptions, FLS offers a contending conceptual
alternative.

.................................................................
% Here is the LaTeX file for the reference section mcref.tex
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