Inverse Problems in Econometrics: 

An Annotated Bibliography

Marine Carrasco
marine.carrasco [at] umontreal.ca
Home page at the Université de Montréal

Contents



The author thankfully acknowledges financial support from the NSF. She also wishes to thank Kausik Gangopadhyay and Josef Perktold for helping with this web page. [Last update: March 30, 2006]


Introduction

Inverse problems can be described as functional equations where the value of the function is known or easily estimable but the argument is unknown. Many problems in econometrics can be stated in the form of inverse problems where the argument itself is a function. For example, consider a nonlinear regression where the functional form is the object of interest. One can readily estimate the conditional expectation of the dependent variable given a vector of instruments. From this estimate, one would like to recover the unknown functional form.

This webpage provides a list of references relative to the estimation of the solution to inverse problems. It focuses mainly on integral equations of the first kind. Solving these equations is particularly challenging as the solution does not necessarily exist, may not be unique, and is not continuous. As a result, a regularized (or smoothed) solution needs to be implemented. Various regularization methods can be used: Tikhonov, spectral cut-off, and Landweber-Fridman. Integral equations of the first kind appear, for example, in the generalized method of moments when the number of moment conditions is infinite, and in the nonparametric estimation of instrumental variable regressions. Integral equations of the second kind are also of interest, their solutions may not be unique but are continuous. Such equations arise when additive models and measurement error models are estimated nonparametrically.

While the theory of inverse problems is part of mathematics, it has found applications in many fields in particular in physics and geophysics. It is only recently that it was applied in econometrics. In this field, the parameter of interest is defined as a solution of a functional equation depending on the data distribution. Hence, the operator in the underlying inverse problem is usually unknown. When studying the asymptotic properties of the solution, we have to take into account the error due to the estimation of the operator.

First, we will give a list of the references relative to the integral equations of the first kind and the second kind. Then, we give a list of applications of the integral equations of the first kind to econometrics and statistics. Finally, we give the references providing the mathematical tools necessary to deal with inverse problems.

This overview is mainly based on the following article :
" Linear Inverse Problems and Structural Econometrics: Estimation Based on Spectral Decomposition and Regularization ", by Marine Carrasco, Jean-Pierre Florens, and Eric Renault, to appear in the Handbook of Econometrics, Vol. 6B, edited by J. Heckman and E. Leamer, 2007.

This web page is still under construction and will be updated on a regular basis. Please email me with your comments or if you wish to have your paper included or updated.


Integral Equations of the First Kind and Regularization

Let H and E be two Hilbert spaces considered only over the real scalars for the sake of notational simplicity. Let K be a linear operator on Domain of K as a Subset of H into E . Typically, K is an compact integral operator. An integral equation (also called Fredholm equations) of the first kind is one of the form :

K time phi equal to r


Integral Equations of the Second Kind

An Integral Equation of the Second Kind (also called Fredholm equation of the second type) is defined by :

(I-K) phi = r

where phi is an element of a Hilbert space H and K is a compact operator from H to H and r is an element of H . K and r are known functions of a data generating process characterized by a measure Pi , and the functional parameter of interest is the function phi . In most cases, H is a functional space and K is an integral operator defined by its kernel k .

Many examples of this problem can be found.

Rational Expectations Asset Pricing Models

Partially Nonparametric Forecasting Model

Backfitting Estimation in Additive Models

An additive model is defined by,

Equation

The function phi is the solution of the equation :

phi - E [E( phi (Z)|W)|Z] = E(Y |Z) - E [E(Y |W)|Z]

and psi is the solution of an equation of the same nature obtained by a permutation of W and Z. This is an example of the Integral Equation of the Second Kind as the unknown function phi is an element of H . The operator K is defined by K phi = E [E( phi (Z)|W)|Z]. And, the function r is equal to E(Y |Z) - E [E(Y |W)|Z]. 

The backfitting algorithm of Breiman and Friedman (1985), and Hastie and Tibshirani (1990) is widely used to estimate phi and psi in the equation above. Mammen, Linton, and Nielsen (1999) derive the asymptotic distribution of the backfitting procedure. Alternatively, Newey (1994), Tjostheim and Auestad (1994), and Linton and Nielsen (1995) propose to estimate phi (respectively psi ) by integrating an estimator of E [Y |Z = z,W = w] with respect to w (respectively z)

Estimation of the Bias Function in a Measurement Error Equation

Equation

where eta , U's are random unknown elements and Y1 and Y2 are two measurements of eta contaminated by a bias term depending on observable elements Z1 and Z2 . The unobservable component eta is eliminated by differentiation to obtain

Y = phi ( Z1 ) - phi ( Z2 ) + U

when Y = Y1 - Y2 and E (Y | Z1 , Z2 ) = phi ( Z1 ) - phi ( Z2 ). We assume that i.i.d. observations of (Y, Z1 , Z2 ) are available. Moreover, the order of measurements is arbitrary or equivalently ( Y1 , Y2 , Z1 , Z2 ) is distributed identically to ( Y2 , Y1 , Z2 , Z1 ). This implies that (Y, Z1 , Z2 ) and (-Y, Z2 , Z1 ) have the same distribution.

This equation introduces an overidentification property because it constrains the Conditional expectation of Y given Z1 and Z2 . In order to define for any F (and in particular for the estimated one), the parameter phi is now defined as the solution of the minimization problem:

Equation

Or, equivalently,

Equation

This can be viewed as an Integral Equation of the Second Kind, when K is the conditional expectation operator. An application can be found in the following paper:


Applications

Estimation of the Density

Let f be an unknown density and F the corresponding cumulative distribution function. While F can be estimated at a parametric rate, the estimator of f will have a slower rate because it is solution of an inverse problem :

Equation

Ridge Regression

The Ridge estimator can be reinterpreted as a Tikhonov regularized solution to an inverse problem y = X θ. For a review of the Ridge estimator, see:

Factor Models

Factor models are frequently adopted in macroeconomics and finance where there are many potential explanatory variables. There is a strong link between the prediction given by a factor model and that of the spectral cut-off regularized solution to the equation y = X θ. This link is outlined in Section 5 of the Handbook chapter. For references on factor models, see:

Functional Linear Model

In this model, the explained variable is a scalar and the explanatory variable is a function, while the parameter to estimate is also a function. Obviously, some regularization is necessary.

Deconvolution

Assume we observe iid realizations of a random variable Y with unknown pdf h, where Y satisfies

Y = X + Z

where X and Z are independent random variables with pdf phi and g respectively. The aim is to get an estimator of phi assuming g is known.

The most common approach to solving this equation is to use a deconvolution kernel estimator, this method was pioneered by Carroll and Hall (1988) and Stefanski and Carroll (1990). It is essentially equivalent to inverting this equation by means of the continuous spectrum of K, see Carroll, Van Rooij, and Ruymgaart (1991) and Subsection 5.4.2 of the Handbook Chapter.

Generalized Method of Moments

First introduced by Hansen (1982),  Generalized Method of Moments (GMM) became the cornerstone of modern structural econometrics. In Hansen (1982), the number of moment conditions is supposed to be finite. However in practice, there is often an infinity of moment conditions that arises from an infinity of potential instruments. It occurs also that there is a continuum of moment conditions. These different cases can be encompassed in the same framework, where the moment functions take their values in finite or infinite dimensional Hilbert spaces. To implement the optimal method of moments, it is necessary to invert a covariance operator, in other words, to solve an integral equation of the first kind. It is trivial in the finite dimensional case (inversion of a matrix) but it is much more challenging in infinite dimension.

Instrumental variables

An economic relationship between a response variable Y and a vector Z of explanatory variables is often represented by an equation:

Y = phi (Z)+ U

where the function phi defines the parameter of interest while U is an error term. This relationship does not characterize the function phi if the residual term is not constrained. This difficulty is solved if it is assumed that E[U | Z] = 0. However in numerous structural econometric models, this condition does not hold because Z is endogenous. 

There are three possible ways to add assumptions to the above equation in order to characterize phi

The First Approach consists in replacing E[U | Z] = 0 by E[U | W] = 0. Thus, W is a vector of instrumental variables. 

The Second Approach is now called control function approach and was systematized by:

A Third Definition follows from the literature on treatment models. This Approach has been discussed in the following references.

Game Theoretic Model

This example and the next one present economic models formalized by nonlinear inverse problems.The analysis of nonlinear functional equations raises numerous questions: uniqueness and existence of the solution, asymptotic properties of the estimator, implementation of the estimation procedure and numerical computation of the solution. These questions are usually solved locally by a linear approximation of the nonlinear problem deduced from a suitable concept of derivative. A strong concept of derivation (typically Frechet derivative) is needed to deal with the implicit form of the model, which requires the use of the Implicit Function theorem.

Solution of a Differential Equation

In several models like the analysis of the consumer surplus, the function of interest is the solution of a differential equation depending on the data generating process. A theoretical treatment is given in:



Mathematical Tools

A good knowledge of (functional) analysis is needed to grasp the difficulties linked with inverse problems.

Hilbert Spaces and Operators

Hilbert Schmidt Operators are discussed in Dunford and Schwartz (1988,p. 1009), Dautray and Lyons (1988, Vol 5, p.41, chapter VIII). These two books (collections of books I should say as they are several volumes) are very complete and more advanced than Debnath and Mikusinski for instance, but are also more difficult to read.

Reproducing Kernel Hilbert Space

Models based on reproducing kernels are the foundation for penalized likelihood estimation and splines (see e.g. Berlinet and Thomas-Agnan, 2004). However, it has been little used in econometrics so far. The theory of reproducing kernels becomes very useful when the econometrician has an infinite number of moment conditions and wants to exploit all of them in an efficient way. 

Law of Large Numbers and Functional Central Limit Theorem in Hilbert Spaces

Estimation of Operators

For the estimation of the conditional expectation operator and its adjoint, there are two possible approaches. One uses the spectral decomposition of the operator.  The second approach consists in using a nonparametric estimator of the density. The papers below are mainly concerned with the estimation of diffusions. When the data are the discrete observations of a diffusion process, the nonparametric estimations of a single eigenvalue-eigenfunction pair and of the marginal distribution are enough to recover a nonparametric estimate of the diffusion coefficients. 


Related Web Pages

Inverse Problems at the University of Alabama
A repository for information of interest to researchers in inverse problems. Included are conference/seminar announcements, new books, and some selected articles.

Inverse problem from Wikipedia
Wikipedia is a online encyclopedia that anyone can edit. The page on inverse problems has a general introduction and includes references to applications in geophysics, ocean and athmospheric sciences, and medical imaging.

Ill-posed Inverse Problems webpage of Rama Cont
Rama Cont is professor at Ecole Polytechnique (France) and works on inverse problems in finance among other things.

Journals:

The papers in these journals are written with physics applications in mind.

Others :

Inverse Problems Network
The IPNet is a free network for researchers working the area of Inverse and/or Ill-Posed Problems. The goal is to promote communication between scientists working in these areas, to provide a newsletter 'IPNet Digest' for notices and scientific queries of general interest, and to provide a central database of up-to-date e-mail addresses, institutional affiliations, and websites of members.

Finnish Inverse Problems Society
The mission of the society is to promote research activities within the field of inverse problems in Finland and maintain and improve the connections between the various research groups working in this area. The society is interdisciplinary in nature, bringing together the expertise in mathematical, physical and engineering sciences.

Mathworld definition of inverse problem

Workshop on Inverse Problems, Toulouse, 2005

Matlab programs used in  Carrasco, M., M. Chernov, J.-P. Florens, and E. Ghysels (2006) “ Efficient estimation of general dynamic models with a continuum of moment conditions ”, forthcoming in Journal of Econometrics can be downloaded from [zipped files]

List of Authors and Affiliations


Alphabetical List of References

This is the list of references of the Handbook chapter. This list will be updated with new references regularly. (The link at the end of a reference points to the section in which the reference was mentioned or discussed)
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