Flexible Least Squares
( Multicriteria Estimation)

Last Updated: 9 January 2012

Site Maintained By: Leigh Tesfatsion Professor of Econ, Math, and ECpE Heady Hall 375 Iowa State University FAX: (515) 294-0221 http://www.econ.iastate.edu/tesfatsi/ tesfatsi AT iastate.edu

Table of Contents: Overview FLS Research Summary Budget-Line Analogy for FLS Frontiers (K > 1) Publications Developing the Basic FLS Methodology FLS Software & Manual Availability Illustrative FLS Applications

Overview

FLS Research Summary:

The theoretical relationships postulated for economic processes typically fall into four conceptually distinct categories: cross-sectional (simultaneous); dynamic; measurement; and probabilistic. Any actual economic process will inevitably behave in a manner that is incompatible to some degree with each postulated theoretical relationship. "All models are wrong, but some are useful." (GEP Box, 1979)

Consequently, the estimation of a model (postulated theory) conditional on given data would seem, intrinsically, to be a multicriteria optimization problem. Associated with any set of estimated (fitted) theoretical relationships will be a set of discrepancy terms reflecting the extent to which the estimated relationships are incompatible with the data. An econometrician undertaking the estimation would presumably prefer each type of discrepancy to be small. However, beyond a particular point a further decrease in one type of discrepancy comes at the cost of an increase in another.

In a series of studies listed in the following publications section and summarized in Kalaba and Tesfatsion (CSDA,1996 pdf 1.6MB), Bob Kalaba and I develop a multicriteria Flexible Least Squares (FLS) approach to model estimation that encompasses a wide range of views regarding the appropriate interpretation and treatment of theory-data discrepancy terms.

At one end of the range, stressing minimal reliance on stochastic priors for discrepancy terms, we develop FLS for Time-Varying Linear Regression (FLS-TVLR). For any given data and any given linear model (theory) proposed to explain the data, each possible estimated model generates two conceptually-distinct types of discrepancy terms, dynamic and measurement. The dynamic discrepancy terms reflect time variation in successive coefficient vectors (relative to a null of constancy), and the measurement discrepancy terms reflect differences between actual observed outcomes and theoretically predicted outcomes based on the null of a linear regression model. The dynamic and measurement errors are separately aggregated into sums of residual squared errors RD and RM.

The basic FLS-TVLR objective is to determine the Residual Efficiency Frontier (REF), i.e., the family of all estimated models that are equally efficient with respect to achieving vector-minimal squared error sums (RD,RMB) for dynamic and measurement errors. By construction, given any estimated model M along the REF with residual squared error sums (RD,RMB), the residual squared error sums (RD',RM') corresponding to any other estimated model M' are at least as large as the error sums (RD,RMB) with either RD' > RD or RM' > RM.

We also develop Generalized FLS for Approximately Linear Systems (GFLS-ALS), an extension of FLS-TVLR to models for which the dynamic relationships and measurement relationships are both postulated to be general linear systems of equations. The concept of the REF is correspondingly generalized to a planar "cost-efficient frontier" in which costs are separately assessed for dynamic and measurement discrepancies.

As detailed in the software/manual section, below, our software implementations for FLS-TVLR and GFLS-ALS have been incorporated into the statistical packages SHAZAM and GAUSS.

More generally, suppose a model (theory) has been conjectured as a possible explanation for given data, and suppose the modeler has specified a K-dimensional vector of "incompatibility cost functions" for measuring the degree of incompatibility between theory and data in accordance with K different goodness-of-fit criteria. We develop a constructive FLS procedure for the determination of the corresponding Cost-Efficient Frontier (CEF), i.e., the family of all estimated models that are equally efficient with respect to achieving vector-minimalization of these K incompatibility cost functions. In particular, we derive a recurrence relation for updating the CEF at time t+1 as a function of the CEF at time t together with a K-dimensional vector of incremental incompatibility costs associated with new data obtained between time t and time t+1.

We show that this general K-dimensional CEF construction encompasses a number of other estimation methods. For example, for the time-varying linear regression (TVLR) problem, the REF determined via FLS-TVLR is a special case of the CEF with K=2. Moreover, in the special limiting case in which joint probability assessments can be used to achieve a complete amalgamation of all TVLR theory-data discrepancy terms into a single (K=1) real-valued incompatibility cost function taking the form of a posterior distribution, the CEF recurrence relation reduces to the standard Kalman Filter.

Finally, we also clarify the interesting relationship between FLS and the "general to specific" econometric methodology advocated by David Hendry and J.F. Richard, and between the FLS REF/CEF constructions and Ed Leamer's proposed information contract curve (global sensitivity analysis).

Budget-Line Analogy for FLS Frontiers (K > 1):

The question naturally arises whether an FLS residual efficiency frontier (REF) or cost-efficient frontier (CEF) of estimated models determined on the basis of K distinct goodness-of-fit criteria for K distinct types of discrepancy terms (K > 1) can be further reduced, ideally to a single "best" estimated model.

As will now be argued, this is akin to asking for the further reduction of budget lines (iso-cost curves) in standard economic analysis. Of course a single point can always be selected along a budget line by the introduction of a scalar-valued utility function inducing a specific preferential weighting across conceptually distinct types of goods and services. The key question, however, is whether this specific weighting can be done in a scientifically objective manner, assented to by others, or whether it simply represents the personal preferences of the researcher doing the weighting.

The argument can be made as follows:

• Any model estimation procedure conducted for a given postulated model conditional on a given data set is essentially a method for distributing the irreducible incompatibility between the model and data across conceptually distinct types of discrepancy terms.

• This irreducible incompatibility distributed across discrepancy terms can be compared to an "income" distributed across the purchase of goods and services. Deciding on the precise representation for discrepancy terms (with corresponding costs) is akin to deciding on the precise representation for goods and services (with corresponding prices) on which an income is to be spent. However these representation issues are resolved, they do not affect the objective existence of an irreducible theory-data incompatibility (income).

• The choice of a particular estimated model along a REF/CEF, conditional on a given model, given data, and a given representation of discrepancy terms (with corresponding costs), is analogous to the selection of a consumption bundle along a budget line conditional on a given income and a given representation for goods and services (with corresponding prices). Starting at any estimated model along the REF/CEF, a decision to lower the cost associated with any one type of discrepancy by moving to a different estimated model must necessarily result in an increase in the cost associated with at least one other type of discrepancy. Similarly, starting at any point along the budget line, a decision to increase expenditures on any one type of good or service must necessarily result in a decrease in expenditures on at least one other type of good or service.

• Objective agreement can reasonably be sought for the construction of a REF/CEF, just as it can reasonably be sought for the construction of a budget line.

• However, the selection of any one "best" estimated model along a REF/CEF is akin to a consumer bringing a utility function to bear in order to select one "best" consumption bundle along a budget line. Any such unique selection must necessarily involve the personal preferences of the selector for one particular cost-efficient choice over another on the basis of some additional personally-introduced criterion.

• To the extent that a group of researchers can agree on the relative importance of various types of discrepancies for an application at hand, they might be able to construct a "social welfare function" for selecting a unique estimated model from among the family of cost-efficient estimated models constituting a REF/CEF. But this does not change the fact that any such unique selection will rely on the subjective judgements of the researchers concerning which cost-efficient distribution of discrepancies is "best" for this application.

NOTE: It is with immense sadness I report that Robert E. Kalaba, a great scholar, mentor, colleague, and friend, died on September 29, 2004.

Publications Developing the Basic FLS Methodology (Chronological Order)

• Benjamin Hamidi, Bertrand Maillet, and Paul Merlin (2011), "A Robust Time-Varying Style Analysis for Hedge Funds based on Dynamic Quantiles" (pdf,563KB), CES/CNRS Working Paper, Paris, February.

  Abstract: Financial management styles are often active, time-varying, and dependent on market opportunities. This paper adapts the time-varying flexible least squares (FLS) approach to L-estimators based on quantile regression, resulting in a time-varying multi-quantile robust approach for financial management style analysis. The approach is used to study hedge fund strategies as defined in the HFR database, using monthly data from January 1995 to January 2010.

• Giovanni Montana, Kostas Triantafyllopoulos, and Theo Tsagaris (2009), "Flexible Least Squares for Temporal Data Mining and Statistical Arbitrage" (pdf,348KB), Expert Systems with Applications, Vol. 36, Issue 2, Part 2, pp. 2819-2830.

  Abstract: "A number of recent emerging applications call for studying data streams, potentially infinite flows of information updated in realtime. When multiple co-evolving data streams are observed, an important task is to determine how these streams depend on each other, accounting for dynamic dependence patterns without imposing any restrictive probabilistic law governing this dependence. In this paper we argue that flexible least squares (FLS), a penalized version of ordinary least squares that accommodates for time-varying regression coefficients, can be deployed successfully in this context. Our motivating application is statistical arbitrage, an investment strategy that exploits patterns detected in financial data streams. We demonstrate that FLS is algebraically equivalent to the well-known Kalman filter equations, and take advantage of this equivalence to gain a better understanding of FLS and suggest a more efficient algorithm. Promising experimental results obtained from a FLS-based algorithmic trading system for the S&P 500 Futures Index are reported."

• Claudio Morana (2009), "An Omnibus Noise Filter" (pdf,237KB), Computational Statistics, Vol. 24, 459-479.

  Abstract: "A new noise filtering approach, based on flexible least squares (FLS) estimation of an unobserved component local level model, is introduced. The proposed FLS filter has been found to perform well in Monte Carlo analysis, independently of the persistence properties of the data and the size of the signal to noise ratio, outperforming in general even the Wiener Kolmogorov filter, which, theoretically, is a minimum mean square estimator. Moreover, a key advantage of the proposed filter, relatively to available competitors, is that any persistence property of the data can be handled, without any pretesting, being computationally fast and not demanding, and easy to be implemented as well."

• Michael Markov, Vadim Mottl, and Ilya Muchnik (2004), "Principles of Nonstationary Regression Estimation: A New Approach to Dynamic Multi-Factor Models in Finance" (pdf,1.5MB), DIMACS Technical Report 2004-47, October.

  Abstract: "We review existing financial multi-factor models from the standpoint of their performance in detecting hidden investment portfolio dynamics. Using practical examples, we present and analyze the shortcomings of these models in detecting both gradual and rapid changes in investment portfolio structure. We then lay the groundwork for a new approach, which we call Dynamic Style Analysis (DSA), representing a true time-series multi-factor portfolio analysis model. At the core of the methodology, we present a new dynamic regression model, which we call Constrained Flexible Least Squares (CFLS). One of the most important features of the DSA model is that it is fully adaptive, i.e., all model parameters are determined from data. The major concepts of the new methodology are gradually introduced and applied to analyses of both model portfolios and well-known public US mutual funds."

• M.J. Manohar Rao (2000), "Estimating Time-Varying Parameters in Linear Regression Models using a Two-Part Decomposition of the Optimal Control Formulation" (pdf), Sãnkhya: The Indian Journal of Statistics 62, 433–467.

  Abstract: "This paper discusses an econometric technique based on optimal control theory which, by employing a variation of the near-neighbourhood search problem, is seen to be suitable for the type of research that requires estimating time-varying parameters for linear regression models. The methodology is based on the characterization of the time-varying parameter (TVP) problem as an optimal control problem, with an explicit allowance for welfare loss considerations, which leads to an algorithm capable of updating the values of the time-varying parameters as well as their covariance matrices. The technique adopts an instruments-targets approach, with the initial condition and the emphasis on parameter flexibility being the instruments; and the total welfare loss and the norm of the error vector being the targets. The methodology is a blend of the flexible least squares and Kalman filter techniques."

• Helmut Lütkepohl and Helmut Herwatz (1996), "Specification of Varying Coefficient Time Series Models via Generalized Flexible Least Squares," Journal of econometrics 70 (1), 261–290. Available from ScienceDirect.Com using doi:10.1016/0304-4076(94)01692-5

  Abstract: "Flexible Least Squares (FLS) is a method for recursively estimating the time paths of the coefficients of a regression model with time-varying coefficients. In its standard form the FLS solution is capable of capturing smooth changes of the coefficients over the sample period. For time series models erratic coefficient changes, for instance, due to seasonal variation, are possible. A generalization of FLS is proposed which can account for such phenomena. The method is applied to artificially generated as well as real economic data. Specifically, West German income and consumption time series are analyzed in some detail."

• Robert Kalaba and Leigh Tesfatsion (1996), "A Multicriteria Approach to Model Specification and Estimation" (pdf,1.6MB), Computational Statistics and Data Analysis, Vol. 21, pp. 193-214. The published article is also available from Science Direct.

  Abstract: This study considers why multicriteria techniques have not been widely adopted in econometrics to date. It then reviews a multicriteria Flexible Least Squares (FLS) approach to model estimation developed earlier by the authors for which the basic objective is state filtering, i.e., sequentially estimating the sequence of states through which a process has passed as time proceeds and new process data is obtained. Suppose a model (theory) for a given process has been proposed, and the modeler has specified a vector of theory-data incompatibility cost functions for measuring the degree of incompatibility between the given model and observed process data. The FLS approach then involves the constructive determination and sequential updating over time of the Cost-Efficient Frontier (CEF), the family of all estimated forms of the model that are equally efficient with respect to achieving minimal theory-data incompatibility costs conditional on the current set of observed process data. This FLS approach is shown to encompass a variety of previously proposed model estimation methods as special cases: for example, FLS for Time-Varying Linear Regression (FLS-TVLR); Generalized FLS for Approximately Linear Systems (GFLS-ALS); the Larsen-Peschon Filter; the Swerling Filter; the Viterbi Filter; and the Kalman Filter. Connections with Hendry and Richard's "general-to-specific" econometric methodology and Leamer's "information contract curve" (global sensitivity analysis) are also clarified.

• Robert Kalaba and Leigh Tesfatsion (1993), "A Multicriteria Approach to Dynamic Estimation," Chapter 18, pp. 288-300, in R. Day and P. Chen (eds.), Nonlinear Dynamics and Evolutionary Economics, Oxford University Press.

  Abstract: This chapter reviews work by the authors on the multicriteria Flexible Least Squares (FLS) approach to model estimation.

• Robert Kalaba and Leigh Tesfatsion (1991), "A Unified Approach to Dynamic Estimation" (pdf,710KB), Information Sciences Vols. 57-58 (September-December), pp. 159-169. The published article is available from Science Direct.

  Abstract: Discrepancies between assumed dynamical models and observations are often handled by making further probabilistic assumptions, a tactic which has both strengths and weaknesses. A re-examination of filtering and smoothing is conducted, and an alternative multicriteria Flexible Least Squares (FLS) approach is advanced that does not require a probabilistic interpretation for discrepancy terms. This approach involves vector minimization as a key ingredient, and it specializes to the well-known Kalman, Viterbi, Larson-Peschon, and Swerling filters.

• Robert Kalaba and Leigh Tesfatsion (1990a), "Flexible Least Squares for Approximately Linear Systems (pdf,1.2MB), IEEE Transactions on Systems, Man, and Cybernetics, Vol. 20, No. 5, pp. 978-989. The published article is also available from IEEE Xplore.

  Abstract: The problem of filtering and smoothing for a system described by approximately linear dynamic and measurement relations has been studied for many decades. Yet the potential problem of misspecified dynamics, which makes the usual probabilistic assumptions involving normality and independence questionable at best, has not received the attention it merits. This study proposes a probability-free filter that meets this misspecification problem head on, referred to as Generalized Flexible Least Squares for Approximately Linear Systems (GFLS-ALS). A Fortran program implementation is provided for GFLS-ALS, and references to simulation and empirical results are given. Although GFLS-ALS has close connections with the standard Kalman filter, it is concretely demonstrated that there are also important conceptual and computational distinctions. The Kalman filter provides a unique estimate for the state sequence, conditional on maintained probability assumptions for discrepancy terms. In contrast, the GFLS-ALS filter provides a family of state sequence estimates, each of which is vector-minimally incompatible with the prior dynamical and measurement specifications.

  NOTE: A program implementation for GFLS-ALS has been incorporated into the statistical package GAUSS. See the section on Software & Manual Availability below.

• Robert Kalaba and Leigh Tesfatsion (1990b), "A Further Note on Flexible Least Squares and Kalman Filtering" (pdf,174KB), Journal of Economic Dynamics and Control, Vol. 14, No. 1 (February), 183-185. The published article is available from Science Direct.

  Abstract: The logical relationship between the multicriteria Flexible Least Squares (FLS) method and Kalman filtering is clarified.

• Robert Kalaba and Leigh Tesfatsion (1990c), "An Organizing Principle for Dynamic Estimation," Journal of Optimization Theory and Applications, Vol. 64, No. 3 (March 1990), 445-470. The published article is available from SpringerLink.

  Abstract: This study develops a general multicriteria Flexible Least Squares (FLS) framework for the sequential estimation of process states. Three well-known state estimation algorithms (the Viterbi, Larson-Peschon, and Kalman filters) are derived as monocriterion specializations. The FLS framework is used to clarify both Bayesian and classical statistical procedures for treating potential model specification errors. Recently developed bicriteria specializations (FLS-TVLR, GFLS-ALS), explicitly designed to take model specification errors into account, are also reviewed. The latter specializations concretely demonstrate how the FLS framework can be used to construct estimation algorithms capable of handling disparate sources of information coherently and systematically, without forced scalarization.

• Leigh Tesfatsion and John Veitch (1990), "U.S. Money Demand Instability: A Flexible Least Squares Approach" (pdf,1.4MB), Journal of Economic Dynamics and Control, Vol. 14, No. 1 (February), pp. 151-173. The published article is also available from Science Direct.

  Abstract: This study uses the Flexible Least Squares method for Time-Varying Linear Regression (FLS-TVLR) to investigate coefficient stability for the Goldfeld U.S. money demand model over the volatile period 1959:Q2 to 1985:Q3. The only constraint imposed on coefficient variation over time is a smoothness prior. Nevertheless, the time paths traced out by the FLS-TVLR coefficient estimates exhibit systematic idiosyncratic time variations as well as simultaneous shift movements in 1974 during the time of the first oil price shock. Moreover, the FLS-TVLR estimates also indicate that the "unit root" nonstationarity problem reported by OLS money demand studies disappears if the coefficients are allowed to exhibit even small amounts of time variation.

• Robert Kalaba and Leigh Tesfatsion (1989a), "Time-Varying Linear Regression Via Flexible Least Squares" (pdf,2.4MB), Computers and Mathematics with Applications, Vol. 17, pp. 1215-1245. The published article is available from Science Direct.

  Abstract: This study develops in detail a multicriteria Flexible Least Squares method for Time-Varying Linear Regression (FLS-TVLR). The basic FLS-TVLR objective is to determine the Residual Efficiency Frontier (REF), that is, the set of all coefficient trajectory estimates that yield vector-minimal sums of squared residual measurement and dynamic errors conditional on a given set of observations.

  NOTE: A program implementation for FLS-TVLR has been incorporated into the statistical packages GAUSS and SHAZAM. See the section on Software & Manual Availability below.

• Robert Kalaba and Leigh Tesfatsion (1989b), "Sequential Nonlinear Estimation with Nonaugmented Priors," Journal of Optimization Theory and Applications, Vol. 60, No. 3 (March), pp. 421-438. The published article is available from SpringerLink.

  Abstract: A "flexible least cost method" is proposed for investigating the basic compatibility of theory and observations in the absence of valid or known stochastic characterizations for residual error terms.

• Robert Kalaba and Leigh Tesfatsion (1988a), "The Flexible Least Squares Approach to Time-Varying Linear Regression" (pdf,360KB), Journal of Economic Dynamics and Control, Vol. 12, No. 1 (March), 43-48. The published article is available from Science Direct.

  Abstract: This study proposes a Flexible Least Squares (FLS) method for state estimation when the dynamic equations are unknown but the process state evolves only slowly over time. A smoothness prior is introduced in place of an explicit specification for the unknown dynamic equations governing the evolution of the process state. Simulation experiments illustrating the method are presented.

• Robert Kalaba and Leigh Tesfatsion (1988b), "Exact Sequential Filtering, Smoothing, and Prediction for Nonlinear Systems" (pdf,1.4MB), Nonlinear Analysis, Vol. 12, Number 6 (June), 599-615. The published article is available from Science Direct.

  Abstract: This study develops two algorithms for the exact sequential updating of the optimal solution for a general discrete-time nonlinear least squares estimation problem as the process length increases and new observations are obtained. One algorithm proceeds via an imbedding on the process length and the final state vector. The second algorithm proceeds via an imbedding on the process length and the final observation vector. Each algorithm generates optimal (least cost) filtered and smoothed state estimates, together with optimal one-step-ahead state predictions.

• Robert Kalaba, Karl Spingarn, and Leigh Tesfatsion (1981), "A Sequential Method for Nonlinear Filtering: Numerical Implementation and Comparisons," Journal of Optimization Theory and Applications, Vol. 34, No. 4 (August), pp. 1144-1149. The published article is available from SpringerLink.

  Abstract: Exact equations are presented for sequentially updating the optimal solution for a discrete-time analog of the basic Sridhar nonlinear filtering problem as the process length increases and new observations are obtained. A tabular method is described for implementing numerically the sequential filtering equations. The accuracy and efficiency of the tabular method are illustrated by means of several numerical examples.

• Robert Kalaba and Leigh Tesfatsion (1981), "An Exact Sequential Solution Procedure for a Class of Discrete-Time Nonlinear Estimation Problems" (pdf,595KB), IEEE Transactions on Automatic control, Vol. AC-26, pp. 1144-1149. The published article is available from IEEE Xplore.

  Abstract: This study develops an exact procedure for sequentially updating the optimal solution for a general discrete-time nonlinear least squares estimation problem as the process length increases and new observations are obtained.

• Robert Kalaba and Leigh Tesfatsion (1980), "A Least-Squares Model Specification Test for a Class of Dynamic Nonlinear Economic Models with Systematically Varying Parameters," Journal of Optimization Theory and Applications, Vol. 32, No. 4 (December), pp. 537-567. The published article is available from SpringerLink.

  Abstract: This study develops a least-squares measure for simultaneously testing the basic compatibility of prior dynamical, observational, and distributional model specifications against actual data for a class of dynamic nonlinear economic models with parameters explicitly modeled as nonlinear functions of endogenous and exogenous variables. Using invariant imbedding techniques, an algorithm is derived for sequentially updating the optimal least-squares estimates for parameters, endogenous variables, and squared residual modeling error sums as the duration of the process increases and new observations are obtained.

FLS Software & Manual Availability

• FLS-TVLR Program:

  Flexible Least Squares for Time-Varying Linear Regression (FLS-TVLR) is implemented by the FLS-TVLR Program (html,34KB), a Fortran program developed by Kalaba and Tesfatsion (the copyright holders) which is released here as free open-source software under the terms of the Artistic License Agreement (html). The FLS-TVLR program is also included in the statistical packages SHAZAM and GAUSS - TSM (Time Series Methods).

• FLS-TVLR Manual:

  Robert Kalaba and Leigh Tesfatsion, "Time-Varying Linear Regression Via Flexible Least Squares" (pdf,2.4MB), Computers and Mathematics with Applications Vol. 17 (1989), pp. 1215-1245. The published article is available from Science Direct.

• GFLS-ALS Program:

  Generalized Flexible Least Squares for Approximately Linear Systems (GFLS-ALS) is implemented by the GFLS-ALS Program (html,46KB), a Fortran program developed by Kalaba and Tesfatsion (the copyright holders) which is released here as free open-source software under the terms of the Artistic License Agreement. A Help File (html) for the GFLS-ALS program is available. The GFLS-ALS program is also included in the statistical package GAUSS - TSM (Time Series Methods).

• GFLS-ALS Manual:

  Robert Kalaba and Leigh Tesfatsion, "Flexible Least Squares for Approximately Linear Systems" (pdf,1.2MB), IEEE Transactions on Systems, Man, and Cybernetics, Vol. 20, No. 5 (1990), pp. 978-989

Illustrative FLS Applications

• Carlos A. Arango A., Martha A. Misas A., and Juan Nicolández (2004), "La Demanda de Especies Monetarias en Colombia: Estructura Y Pronóstico" (pdf,538KB), Banco de la República, Borradores de Economia, No. 002964, September.

• Jon R. Bond, Richard Fleisher, and B. Dan Wood (2003), "The Marginal and Time-Varying Effect of Public Approval on Presidential Success in Congress" (html), The Journal of Politics 65 (1), 92–110.

• Adam Brinker, Joe Parcell, and Kevin Dhuyvetter (2007), “Cross-Hedging Distillers Dried Grains: Exploring Corn and Soybean Meal Futures Contracts" (pdf,274KB), Proceedings, NCCC-134 Conference on Applied Commodity Price Analysis, Forecasting, and Market Risk Management, Chicago, IL.

• Lars-Erik Cederman (2001), "Exploring the Dynamics of the Democratic Peace" (html), Journal of Conflict Resolution, Vol. 45, No. 6, 818-833.

• Jim Clayton and Greg MacKinnon (2001), "The Time-Varying Nature of the Link Between REIT, Real Estate and Financial Asset Returns" (pdf,6.3MB), Journal of Real Estate Portfolio Management, January-March Issue.

• Zsolt Darvas and Balázs Varga (2007), "Inflation Persistence in the New Members of the EU" (pdf,497KB), Working Paper, Corvinus University of Budapest, December 2007.

• Jeffrey H. Dorfman and Kenneth A. Foster (1991), "Estimating Productivity Changes with Flexible Coefficients," Western Journal of Agricultural Economics 16 (2), 280–90.

• Markus Ebner and Thorsten Neumann (2005), "Time-Varying Betas of German Stock Returns" (html), Financial Markets and Portfolio Management 19(1), 29-46.

• Markus Ebner and Thorsten Neumann (2008), "Time-Varying Factor Models for Equity Portfolio Construction" (html), The European Journal of Finance 14(5), July, 381-395.

• Akhter Faroque and William Veloce (2008), "Fundamentals versus the Leading Index: The Forecasting of Canada's Output Growth Since 1991, an Encompassing Approach", Applied Economics, November.

• Jason R. V. Franken and Joe L. Parcell (2003), "Market Integration: Case Studies of Structural Change" (pdf,311KB), NCR-134 Conference on Applied Commodity Price Analysis, Forecasting, and Market Risk Management, St. Louis, Missouri, April 21-22.

• Hayette Gatfaoui (2007), "Are Credit Default Swap Spreads Market Driven?" (pdf,18pp), 4th AFE-QASS Conference, Samos, Greece, July.

• Hayette Gatfoui (2006), "Credit Risk and Market Risk: Analyzing US Credit Spreads" (pdf,956KB), Working Paper, Rouen School of Management, Economics and Finance Department, Mont-Saint-Aignan Cedex, France, August.

• Eliana González, Miguel I. Gómez, Luiz F. Melo, and Joseé Luis Torres (2006), "Forecasting Food Inflation in Developing Countries with Inflation Targeting Regimes: The Colombian Case" (download site), Banco de la República, Borradores de Economia, No. 002735, October.

• Ling T. He (2001), "Time Variation Paths of International Transmission of Stock Volatility -- U.S. vs. Hong Kong and South Korea," Global Finance Journal 12 (1), 79–93.

• Ling T. He (2004), "Instability of Risk Loadings of Industrial Stocks" (pdf,217KB), Journal of Business and Economics Research, Vol. 2, No. 2.

• Ling T. He (2005), "Instability and Predictability of Factor Betas of Industrial Stocks: The Flexible Least Squares Solutions", Quarterly Review of Economics and Finance 45 (4-5), 619–640.

• Todd Hubbs, Todd H. Kuethe, and Timothy G. Baker (2009), “Evaluating the Dynamic Nature of Market Risk” (pdf,1.3MB), Proceedings, NCCC-134 Conference on Applied Commodity Price Analysis, Forecasting, and Market Risk Management, St. Louis, MO.

• Peter Kennedy (2003), A Guide to Econometrics: Fifth Edition (Publisher's Book Guide), MIT Press, 623 pp.

• Andreas Kladroba (2005), "Flexible Least Squares Estimation of State Space Models: An Alternative to Kalman Filtering?" (pdf,126KB), Working Paper 149, Universitat Duisburg-Essen.

• Todd H. Kuethe, Kenneth A. Foster, and Raymond J.G.M. Florax (2008), "A Spatial Hedonic Model with Time-Varying Parameters: A New Method Using Flexible Least Squares" (pdf,295KB), Selected Paper, American Agricultural Economics Association Annual Meeting, Orlando, Florida, July 27-29. (Slide Presentation,317K,pdf)

• Manish Kumar (2011), "A Time-Varying Parameter Vector Autoregression Model for Forecasting Emerging Market Exchange Rates" (pdf,364KB), International Journal of Economic Sciences and Applied Research, Vol. 3(2), 21-39.

• Andrew C. Lee and Man-Keun Kim (2005), "Time Varying Coefficients: An Application of Flexible Least Squares to Cattle Captive Supply," Selected Paper, American Agricultural Economics Association Annual Meeting, Providence, Rhode Island, July 24-27.

• Helmut Lütkepohl (1993), The Sources of the U.S. Money Demand Instability," Empirical Economics 18(4), 729-743.

• Helmut Lütkepohl and Helmut Herwatz (1996), "Specification of Varying Coefficient Time Series Models via Generalized Flexible Least Squares," Journal of econometrics 70 (1), 261–290.

• Vêlayoudom Marimoutou, Denis Peguin, and Anne Peguin-Feissole (2009), "The `Distance-Varying' Gravity Model in International Economics: Is the Distance an Obstable to Trade?" (pdf,333KB), Economics Bulletin, Vol. 29, No. 2, pp. 1157-1173.

• Michael Markov, Vadim Mottl, and Ilya Muchnik (2004), "Principles of Nonstationary Regression Estimation: A New Approach to Dynamic Multi-Factor Models in Finance" (pdf,1.5MB), DIMACS Technical Report 2004-47, October. (Constrained Flexible Least Squares)

• Giovanni Montana, Kostas Triantafyllopoulos, and Theo Tsagaris (2009), "Flexible Least Squares for Temporal Data Mining and Statistical Arbitrage" (pdf,348KB), Expert Systems with Applications, Vol. 36, Issue 2, Part 2, pp. 2819-2830.

• Claudio Morana (2009), "An Omnibus Noise Filter" (pdf,237KB), Computational Statistics, Vol. 24, 459-479.

• Claudio Morana (2009), "On the Macroeconomic Causes of Exchange Rate Volatility" (pdf,518KB), International Journal of Forecasting, Volume 25, Issue 2, April-June, pages 328-350. doi:10.1016/j.ijforecast.2009.01.013

• Claudio Morana (2009), "Realized Betas and the Cross-Section of Expected Returns", Applied Financial Economics, Vol. 19, Issue 17, pp. 1371-1381.

• Claudio Morana (2005), "The Japanese Deflation: Has it Had Any Real Effects? Could It Have Been Avoided?" (pdf), Applied Economics 37, 1337-1352.

• Christopher John (C.J.) O'Donnell (2006), "Some Econometric Options for Dealing with Unknown Functional Form" (pdf), Working Paper, School of Economics, University of Queensland, Bridbane QLD 4072.

• Joe Parcell (2003), "An Empirical Analysis of the Demand for Wholesale Pork Primals: Seasonality and Structural Change,” Journal of Agricultural and Resource Economics 28(2), 335-48.

• Michael C. Poray, Kenneth A. Foster, and Jeffrey H. Dorfman (2001), "Measuring an Almost Ideal Demand System with Generalized Flexible Least Squares," (pdf), Working Paper 01-01, Department of Agricultural Economics, Purdue University.

• M.J. Manohar Rao (2000), "Estimating Time-Varying Parameters in Linear Regression Models using a Two-Part Decomposition of the Optimal Control Formulation" (pdf), Sãnkhya: The Indian Journal of Statistics 62, 433–467.

• A. Ramachandra Rao (1995), "Mulivariate Flexible Least Squares Analysis of Hydrological Time Series" (pdf,381KB), Modelling and Management of Sustainable Basin-Scale Water Resource Systems, Proceedings of a Boulder Symposium, July 1995, IAHS Publication No. 231, pp. 359-366.

• Stâle Størdal and Darius M. Adams (2005), "Testing for Variation in the Western Oregon Softwood Log Price Structure," Canadian Journal of Forestry Research 35(3), 713–723. doi:10.1139/x05-008

• Leigh Tesfatsion and John Veitch (1990), "U.S. Money Demand Instability: A Flexible Least Squares Approach" (pdf,1.4MB), Journal of Economic Dynamics and Control 14(1), 151-173.

• K. Triantafyllopoulos and G. Montana (2011), "Dynamic modeling of mean-reverting spreads for statistical arbitrage", Computational Management Science, Springer, vol. 8(1), pages 23-49, April.

• Benu Varman and Wolfgang Schneider (1989), "Measuring Capital Flight: A Time-Varying Regression Analysis," Universitat Kiel Discussion Paper No. 82/89, December.

• Curt Wells (1996), The Kalman Filter in Finance (Chapter 3:FLS), Series: Advanced Studies in Theoretical and Applied Econometrics, Vol. 32, 192 pp., Hardcover, ISBN: 978-0-7923-3771-3.

• B. Dan Wood (2000), "Weak Theories and Parameter Instability: Using Flexible Least Squares to Take Time-Varying Relationships Seriously," American Journal of Political Science 44(3), 603-618.

• Chiang Yat-Hung, So Chun-Kei Joinkey, and Tang Bo-Sin (2008), "Time-varying performance of four Asia-Pacific REITs" (html), Journal of Property Investment and Finance 26(3), 210-231. DOI: 10.1108/14635780810871605