marine.carrasco [at] umontreal.ca

Home page at the Université de Montréal

- Introduction
- Integral Equations of the First Kind and Regularization
- Integral Equations of the Second Kind
- Applications
- Mathematical Tools
- Related Web Pages
- List of Authors and Affiliations
- Alphabetical List of References

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]

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.

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

- Engl, H. W., M. Hanke, and A. Neubauer (1996)
*Regularization of Inverse Problems*, Kluwer Academic Publishers. - Groetsch, C. (1993)
*Inverse Problems in Mathematical Sciences*, Vieweg Mathematics for Scientists and Engineers, Wiesbaden.- This book offers a good introduction to inverse problems and is very accessible.

- Kress, R. (1999),
*Linear Integral Equations*, Springer.

- This book reviews various regularization methods: Tikhonov, Spectral cut-off, and Landweber-Fridman.

- Loubes, J.M. and A. Vanhems (2003), “Saturation Spaces for Regularization Methods in Inverse Problems”, Discussion Paper, GREMAQ, University of Toulouse, presented at ESEM 2003, Stockholm.
- Nashed, N. Z. and G. Wahba (1974) “Generalized inverses in reproducing
kernel spaces: An approach to regularization of linear operator equations”,
*SIAM Journal of Mathematical Analysis*, 5, 974-987. - Natterer, F.(1984) “Error bounds for Tikhonov regularization in
Hilbert scales”,
*Applicable Analysis*, 18, 29-37. - Tautenhahn, U. (1996) “Error estimates for regularization methods
in Hilbert scales”,
*SIAM Journal of Numerical Analysis*, 33, 2120-2130. - Van Rooij, A., F. Ruymgaart, and W. Van Zwet (2000) “Asymptotic
Efficiency of Inverse Estimators”,
*Theory of Probability and its Applications*, 44, 4, 722-738. - Vapnik V. (1998),
*Statistical Learning Theory*, Wiley, New York. - Wahba, G. (1973) “Convergence Rates of Certain Approximate Solutions
to Fredholm Integral Equations of the First Kind”,
*Journal of Approximation Theory*, 7, 167-185.

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

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

Many examples of this problem can be found.

- Bosq, D. (1998)
*Nonparametric Statistics for Stochastic Processes. Estimation and Prediction*, Lecture Notes in Statistics, 110. Springer-Verlag, NewYork. - Lucas, R. (1978) “Asset Prices in an Exchange Economy”,
*Econometrica*, 46, 1429-1446.

- Bollerslev, T. (1986), “Generalized Autoregressive Conditional
Heteroskedasticity”,
*Journal of Econometrics*31, 307-327. - Engle R.F., (1990), “Discussion: Stock Market Volatility and the
Crash of ’87”,
*Review of Financial Studies*3, 103-106. - Engle, R.F., and V.K. Ng (1993),“Measuring and Testing the Impact
of News on Volatility”,
*The Journal of Finance*XLVIII, 1749-1778. - Linton, O. and E. Mammen (2005),“Estimating Semiparametric ARCH(1)
models by kernel smoothing methods”,
*Econometrica*, 73, 771-836.

The function is the solution of the equation :

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

and 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 is an element of . The operator K is defined by K = E [E( (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 and 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 (respectively ) by integrating an estimator of E [Y |Z = z,W = w] with respect to w (respectively z)

- Breiman, L. and J.H. Friedman (1985) “Estimating Optimal Transformations
for Multiple Regression and Correlation”,
*Journal of American Statistical Association*, 80, 580-619. - Florens, J.-P., M. Mouchart, and J.-M. Rolin (1990)
*Elements of Bayesian Statistics*, Dekker, New York. - Hastie, T.J. and R.J. Tibshirani (1990), Generalized Additive Models, Chapman and Hall, London.
- Linton, O. and J.P. Nielsen (1995) “A Kernel Method of Estimating
Structured Nonparametric regression Based on Marginal Integration”,
*Biometrika*, 82, 93-100. - Mammen, E., O. Linton, and J. Nielsen (1999) “The existence and
asymptotic properties of a backfitting projection algorithm under weak conditions”,
*The Annals of Statistics*, 27, 1443-1490. - Newey, W. (1994) “Kernel Estimation of Partial Means”,
*Econometric Theory*, 10, 233-253 - Tjostheim, D. and B. Auestad (1994) “Nonparametric Identification
of Nonlinear Time Series Projections”,
*Journal of American Statistical Association*, 89, 1398- 1409.

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

Y = ( ) - ( ) + U

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

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

Or, equivalently,

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:

- Gaspar, P. and J.-P. Florens, (1998), “Estimation of the Sea State
Biais in Radar Altimeter Measurements of Sea Level: Results from a Nonparametric
Method”,
*Journal of Geophysical Research*, 103 (15), 803-814.

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 :

- Hardle, W. and O. Linton (1994) “Applied Nonparametric Methods”,
*Handbook of Econometrics*, Vol. IV, edited by R.F. Engle and D.L. McFadden, North Holland, Amsterdam.

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:

- Judge, G., W. Griffiths, R. C. Hill, H. Lutkepohl, and T-C. Lee
(1980)
*The Theory and Practice of Econometrics*, John Wiley and Sons, New York.

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:

- Forni, M., M. Hallin, M., Lippi, and L. Reichlin (2000) “The generalized
dynamic factor model: identification and estimation”,
*Review of Economic and Statistics*, 82, 4, 540-552. - Forni, M. and L. Reichlin (1998) “Let’s Get Real: A Factor Analytical
Approach to Disaggregated Business Cycle Dynamics”,
*Review of Economic Studies*, 65, 453-473. - Stock, J. and M. Watson (1998) “Diffusion Indexes”, NBER working paper 6702.
- Stock, J. and M. Watson (2002) “Macroeconomic Forecasting Using
Diffusion Indexes”,
*Journal of Business and Economic Statistics*, 20, 147-162. - Ross, S. (1976) “The Arbitrage Theory of Capital Asset Pricing”,
*Journal of Finance*, 13, 341-360.

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.

- Cardot, H., F. Ferraty, and P. Sarda (2003) “Spline Estimators
for the Functional Linear Model",
*Statistica Sinica*, 13, 571-591. - Hall, P. and J. Horowitz (2005b) " Methodology
and convergence rates for functional linear regression '', mimeo,
Northwestern University.
- The authors show that the Ridge estimator is more robust than the spectral cut-off estimator when the eigenvalues are close to each other.

- Ramsay, J. O. and B.W. Silverman (1997)
*Functional Data Analysis*. Springer, New York.- This book provides an overview of models that involve curves.

- Van Rooij, A., F. Ruymgaart, and W. Van Zwet (2000) “Asymptotic
Efficiency of Inverse Estimators”,
*Theory of Probability and its Applications*, 44, 4, 722-738.

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 and g respectively. The aim is to get an estimator of 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.

- Carrasco, M. and J.-P. Florens (2002) “ Spectral
method for deconvolving a density ”, mimeo, University of Rochester.

- This paper recasts the deconvolution problem into that of solving a Fredholm equation of the first kind. By an adequate choice of the spaces of reference, the operator to invert is compact. The regularization method employed here is Tikhonov.

- Carroll, R. and P. Hall (1988) “Optimal Rates of Convergence for
Deconvolving a Density”,
*Journal of American Statistical Association*, 83, No.404, 1184-1186. - Carroll, R., A. Van Rooij, and F. Ruymgaart (1991) “Theoretical
Aspects of Illposed Problems in Statistics”,
*Acta Applicandae Mathematicae*, 24, 113-140. - Stefanski, L. and R. Carroll (1990) “Deconvoluting Kernel Density
Estimators”,
*Statistics*, 2, 169-184. - Van Rooij, A. and F. Ruymgaart (1991) “Regularized Deconvolution
on the Circle and the Sphere”, in
*Nonparametric Functional Estimation and Related Topics*, edited by G. Roussas, 679-690, Kluwer Academic Publishers, the Netherlands.

- They propose a regularized inverse to a deconvolution problem where g has the circle for support. They invert the operator K using its continuous spectrum.

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.

- Arellano, M., L. Hansen, and
E. Sentana (2005) “ Underidentification?
”, mimeo, CEMFI.
- The authors show that when the parameter, θ₀, is locally unidentified by a moment condition then it results in a continuum of moment conditions indexed by θ. They apply the test for overidentifying restrictions proposed by Carrasco and Florens (2000) to test for the null of underidentification.

- 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*.- This paper extends the results of Carrasco and Florens (2000) to a time-series context. The main application of the results of this paper is in finance. Indeed, financial data are often assumed to be the discrete observations of a diffusion process. While the likelihood does not have a closed-form expression in this case, the conditional characteristic function (CF) is known or can be easily estimated via simulations. The paper explains how to use the conditional CF to obtain an efficient estimator.

- Carrasco, M. and J.-P. Florens (2000) “Generalization of GMM to
a continuum of moment conditions”,
*Econometric Theory*, 16, 797-834.- The authors extend the "classical" Method of Moments to the case where there is a continuum of moment conditions. In particular they show how to invert the covariance operator using a regularized inverse. The regularization used here and in the other papers by Carrasco and Florens is Tikhnonov regularization. Assumptotic properties of the estimator are derived assuming i.i.d. data.

- Carrasco, M. and J.-P. Florens (2001) “ Efficient
GMM Estimation Using the Empirical Characteristic Function ”, mimeo,
University of Rochester.
- The CF provides a continuum of moment conditions. Traditionally, statisticians exploited only a few of these moments. The authors show how GMM can handle the full continuum of moment conditions, resulting in an estimator which is as efficient as MLE. This paper assumes i.i.d. data.

- Carrasco, M. and J.-P. Florens (2004) “ On
the Asymptotic Efficiency of GMM ”, mimeo, University of Rochester.
- This paper uses results on RKHS to establish under which conditions GMM is efficient. It also provides new results on semi-parametric efficiency bound for Markov models.

- Hansen, L.P., (1982), “Large Sample Properties of Generalized
Method of Moments Estimators”,
*Econometrica*, 50, 1029-1054.- This paper defines and studies the properties of the GMM estimator in time series. Here, the number of moment conditions is assumed to be finite.

- Kutoyants, Yu. (1984),
*Parameter estimation for stochastic processes*, Heldermann Verlag, Berlin.

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

Y = (Z)+ U

where the function defines the parameter of interest while U is an error term. This relationship does not characterize the function 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 .

- Florens, J.-P., J. Heckman, C. Meghir and E. Vytlacil (2003),
“ Instrumental
Variables, Local Instrumental Variables and Control Functions ”,
IDEI working paper n. 249, University of Toulouse.

- The authors give a general comparison between these three concepts and their extensions to more general treatment models.

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

- Blundell, R., X. Chen, and D. Kristensen (2003) “
Semi-Nonparametric
IV Estimation of Shape-Invariant Engel Curves ”, Cemmap working
paper CWP 15/03, University College London.

- This paper proposes a sieves estimator of the Engel curve.

- Blundell, R. and J. Powell (2003) “Endogeneity in Nonparametric
and Semiparametric Regression Models”, in
*Advances in Economics and Econometrics*, Vol. 2, eds by M. Dewatripont, L.P. Hansen and S.J. Turnovsky, Cambridge University Press, 312-357. - Darolles, S., J.-P. Florens, and E. Renault (2002), “ Nonparametric
Instrumental Regression ”, Working paper 05-2002, CRDE.

- They prove both the consistency and the asymptotic distribution of a kernel estimator of .

- Florens, J.-P. (2003) “Inverse Problems in Structural Econometrics:
The Example of Instrumental Variables”, in
*Advances in Economics and Econometrics*, Vol. 2, eds by M. Dewatripont, L.P. Hansen and S.J. Turnovsky, Cambridge University Press, 284-311. - Hall, P. and J. Horowitz (2005a) ``Nonparametric Methods for Inference
in the Presence of Instrumental Variables'',
*Annals of Statistics*, Vol 33, n. 6.

- They give the optimal rate of convergence of the kernel estimator under conditions which differ from those of Darolles, Florens, and Renault (2002).

- Newey, W., and J. Powell (2003), “Instrumental Variables for Nonparametric
Models”,
*Econometrica*, 71, 1565-1578.

- They were the first to identify the problem above as an ill-posed problem. They prove the consistency of a series estimator of .

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

- Newey, W., Powell, J., and F. Vella (1999),“Nonparametric Estimation
of Triangular Simultaneous Equations Models”,
*Econometrica*, 67, 565-604.

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

- Das, M. (2005) “Instrumental variables estimators of nonparametric
models with discrete endogenous regressors”,
*Journal of Econometrics*, 124, 335-361. - Florens, J.-P. and L. Malavolti (2002) “ Instrumental
Regression with Discrete Variables ”, mimeo University of Toulouse,
presented at ESEM 2002, Venice.

- Here, the explanatory variable is binary. Hence the operator has a finite range and no regularization is needed.

- Heckman, J., H. Ichimura, J. Smith, and P. Todd (1998), “Characterizing
Selection Bias Using Experimental Data”,
*Econometrica*, 66, 1017-1098. - Heckman, J., and E. Vytlacil (2000), “Local Instrumental Variables”,
in
*Nonlinear Statistical Modeling: Proceedings of the Thirteenth International Symposium in Economic Theory and Econometrics: Essays in Honor of Takeshi Amemiya*, ed. by C. Hsiao, K. Morimune, and J. Powells. Cambridge: Cambridge University Press, 1-46. - Imbens, G., and J. Angrist (1994), “Identification and Estimation
of Local Average Treatment Effects”,
*Econometrica*, 62, 467-476.

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.

- Florens, J.-P., C. Protopopescu, and J.F. Richard, (1997), “Identification
and Estimation of a Class of Game Theoretic Models”, GREMAQ, University of
Toulouse.

- This paper (but not the next one) uses the framework of inverse problems.

- Guerre, E., I. Perrigne, and Q. Vuong, (2000), “Optimal Nonparametric
Estimation of First-Price Auctions”,
*Econometrica*, 68 (3), 525-574.

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:

- Loubes, J.M. and A. Vanhems (2001) “Differential Equation and Endogeneity”, Discussion Paper, GREMAQ, University of Toulouse, presented at ESEM 2002, Venice.
- Vanhems, A. (2006) "Nonparametric Study of Solutions of Differential Equations", Econometric Theory, 22, 127-157.

- Debnath, L. and P. Mikusinski (1999)
*Introduction to Hilbert Spaces with Applications*, Academic Press. San Diego.- This book provides all the basic definitions and properties of Hilbert Spaces.

- Ait-Sahalia, Y., L.P. Hansen, and J.A. Scheinkman (2004) “
Operator
Methods for Continuous-Time Markov Processes ”, forthcoming in
the
*Handbook of Financial Econometrics*, edited by L.P. Hansen and Y. Ait-Sahalia, North Holland.- This chapter provides an excellent survey of operator methods for the purpose of Financial Econometrics.

** 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.

- Dunford, N. and J. Schwartz (1988)
*Linear Operators, Part II: Spectral Theory*, Wiley, New York. - Dautray, R. and J.-L. Lions (1988)
*Analyse mathematique et calcul numerique pour les sciences et les techniques*. Vol. 5. Spectre des operateurs, Masson, Paris.

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.

- Aronszajn, N. (1950) “Theory of Reproducing Kernels”,
*Transactions of the American Mathematical Society*, Vol. 68, 3, 337-404. - Berlinet, A. and C. Thomas-Agnan (2004)
*Reproducing Kernel Hilbert Spaces in Probability and Statistics*, Kluwer Academic Publishers, Boston.

- This excellent book is an up to date and accessible account on RKHS. It contains theory, many examples of RKHS, and numerous applications. A full chapter is devoted to nonparametric curve estimation (mainly spline density estimation).

- Kailath, T. (1971) "RKHS Approach to Detection and Estimation
Problems - Part I: Deterministic Signals in Gaussian Noise", IEEE Trans. Inform. Theory ,
IT-17, 530-549.

- This very interesting paper builds on Parzen's work. It gives several examples of RKHS of processes, including processes defined over spaces of dimension n.

- Nashed, N. Z. and G. Wahba (1974) “Generalized inverses in reproducing
kernel spaces: An approach to regularization of linear operator equations”,
*SIAM Journal of Mathematical Analysis*, 5, 974-987. - Parzen, E. (1959) “Statistical Inference on time series by Hilbert
Space Methods,I.”, Technical Report No.23, Applied Mathematics and Statistics
Laboratory, Stanford. Reprinted in (1967)
*Time series analysis papers*, Holden-Day, San Francisco.

- This paper was the first to show the use of RKHS for statistical inference in continous-time models. Most results of this paper and the next one are discussed in details in Berlinet and Thomas-Agnan (2004).

- Parzen, E. (1970) “Statistical Inference on time series by RKHS
methods”, Proc. 12th Biennal Canadian Mathematical Seminar, R. Pyke, ed.
American Mathematical Society, Providence.

- This paper is the sequel of the previous one. It gives several examples of kernels for which the norms in the associated RKHS have a closed-form expression.

- Saitoh, S. (1997)
*Integral transforms, reproducing kernels and their applications*, Longman.

- This book is more technical than Berlinet and Thomas-Agnan (2004).

- Chen, X., and H. White (1992), “Central Limit and Functional Central
Limit Theorems for Hilbert Space-Valued Dependent Processes”, Working Paper,
University of San Diego.
- This is the working paper version of the ET (1998) paper. It contains extra results on the limit of a random function premultiplied by an operator.

- Chen, X. and H. White (1996) “Law of Large Numbers for Hilbert
Space-Valued mixingales with Applications”,
*Econometric Theory*, 12, 284-304.- The paper gives weak and strong laws of large numbers for near epoch dependent (NED) processes.

- Chen, X. and H. White (1998) “Central Limit and Functional Central
Limit Theorems for Hilbert Space-Valued Dependent Processes”,
*Econometric Theory*, 14, 260-284.- This paper provides weak convergence theorems for NED processes that might have trending mean (hence are not covariance stationary).

- Davidson, J. (1994)
*Stochastic Limit Theory*, Oxford University Press, Oxford.- This is a good reference book for the definitions of various temporal dependences (mixing, NED) and limiting properties (law of large numbers and central limit theorems). Attention: the first edition has typos (for instance the definition of beta-mixing is wrong).

- Politis, D. and J. Romano (1994) “Limit theorems for weakly dependent
Hilbert space valued random variables with application to the stationary
bootstrap”,
*Statistica Sinica*, 4, 451-476.- This paper provides functional central limit theorems for mixing processes.

- van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes
. Springer, New York.
- This book focuses on i.i.d. processes.

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.

- Chen, X., L.P. Hansen and J. Scheinkman (1998) “Shape-preserving
Estimation of Diffusions”, mimeo, University of Chicago.

- This paper estimates the conditional expectation operator by B-spline wavelets.

- Darolles, S., J.-P. Florens, and E. Renault (1998), “Nonlinear
Principal Components and Inference on a Conditional Expectation Operator
with Applications to Markov Processes”, presented in Paris-Berlin conference
1998, Garchy, France.

- First, they estimate the spectrum of the operator. Then, they recover the operator from its spectral decomposition using Mercer's theorem.

- Darolles, S., J.-P. Florens, and C. Gourieroux (2004) “Kernel
Based Nonlinear Canonical Analysis and Time Reversibility”,
*Journal of Econometrics*, 119, 323- 353.

- This paper estimates the conditional expectation operator by kernel smoothing.

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.

- Journal of Inverse and Ill-posed problems
- Inverse Problems
- Inverse Problems in Science and Engineering

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

- Chunrong Ai, Department of Economics, University of Florida
- Yacine Ait-Sahalia, Bendheim Center for Finance , Princeton University.
- Joshua Angrist, Department of economics, Massachusetts Institute of Technology
- Manuel Arellano, CEMFI, Casado del Alisal 5 - 28014, Madrid, Spain
- Nachman Aronszajn
- Bjørn Auestad, Det teknisk- naturvitenskapelige fakultet
- Jushan Bai, Department of Economics, Department of Economics New York University
- Robert L. Basmann, Department of Economics, Binghamton University (SUNY)
- Alain Berlinet, Institut de Mathématiques et de Modélisation de Montpellier (I3M) - UMR 5149
- Richard Blundell, University College London
- Tim Bollerslev, Department of Economics, Duke University
- Denis Bosq, Laboratoire de Statistique Théorique et Appliquée, Universite Pierre et Marie Curie, Paris 6
- Leo Breiman, Professor Emeritus, Department of Statistics, University of California at Berkeley
- Herve Cardot, INRA, Dijon, France
- Marine Carrasco, Département de sciences économiques, Université de Montréal
- Raymond J. Carroll, Department of Statistics, Texas A&M University
- George Chacko, Harvard Business School
- Xiaohong Chen, Department of Economics, New York University
- Mikhail Chernov, Graduate School of Business, Columbia University
- Serge Darolles, Société Générale Asset Management
- Mitali Das, Department of Economics, Columbia University
- Robert Dautray
- James Davidson, School of Business and Economics, University of Exeter
- Lokenath Debnath, Department of Mathematics, University of Central Florida
- Nelson Dunford
- Heinz W. Engl
- Robert F. Engle, Stern School of Business, New York University
- Jianqing Fan, Department of Operations Research & Financial Engineering, Princeton University
- Frederic Ferraty, Laboratoire de Statistique et Probabilités, Université Paul Sabatier ( Toulouse III )
- Andrey Feuerverger, Department of Statistics, University of Toronto
- Jean-Pierre Florens, University of Toulouse I, France
- Jerome H. Friedman, Department of Statistics, Stanford University
- Mario Forni, Faculty of Economics, University of Modena and Reggio Emilia, Italy
- A. Ronald Gallant, Fuqua School of Business, Duke University
- Philippe Gaspar
- Eric Ghysels, Department of Economics, University of North Carolina - Chapel Hill
- Christian Gourieroux, Department of Economics, University of Toronto, and ENSAE, France
- William Griffiths, Department of Economics, University of Melbourne
- Charles W. Groetsch, Department of Mathematics, University of Cincinnati
- Emmanuel Guerre, Laboratoire de Statistique Théorique et Appliquée, Universite Pierre et Marie Curie, Paris 6
- Peter Hall, Mathematical Sciences Institute, Australian National University
- Marc Hallin, Université libre de Bruxelles, Belgium
- Martin Hanke-Bourgeois, Arbeitsgruppe Numerische Mathematik, Johannes Gutenberg-Universität, Germany
- Lars Peter Hansen, Department of Economics, University of Chicago
- Wolfgang Hardle, Department of Statistics, Humboldt University
- Trevor J. Hastie, Department of Statistics, Stanford University
- Jerry Hausman, Department of Economics, MIT
- James Heckman, Department of Economics, University of Chicago
- David F. Hendry, Professor of Economics, University of Oxford & Fellow Nuffield College, Oxford
- R. Carter Hill, Department of Economics, Louisiana State University
- Arthur E. Hoerl
- Joel Horowitz, Department of Economics, Northwestern University
- Robert Hussey, Department of Economics, Georgetown University
- Hidehiko Ichimura, Graduate School of Public Policy, The University of Tokyo
- Guido Imbens, Department of Economics, University of California at Berkeley
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- Vladislav Kargin, Cornerstone Research
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- Rainer Kress, Institut für Numerische und Angewandte Mathematik (NAM), Georg-August-Universität Göttingen, Germany
- Denis Kristensen, Department of Economics, University of Wisconsin-Madison
- Yury A. Kutoyants, Université du Maine, Faculté des Sciences, Laboratoire de Statistique et Processus
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- Tsoung-Chao Lee, Professor Emeritus, Agriculture and Resource Economics, University of Connecticut
- Oliver Linton, London School of Economics
- Jacques-Louis Lions
- Marco Lippi, Dipartimento di Scienze Economiche, Universita' di Roma "La Sapienza", Italy
- Jonathan R. Long
- Jean-Michel Loubes, Département de mathématiques, Université de Montpellier 2, France
- Robert E. Lucas, Jr., Department of Economics, University of Chicago
- David G. Luenberger, Department of Management Science and Engineering, Stanford University
- Helmut Lütkepohl, Economics Department, European University Institute, Italy
- Laëtitia Malavolti, GREMAQ - Université de Toulouse I, FRANCE
- Edmond Malinvaud
- Enno Mammen, Department of Economics, University of Mannheim
- Alexander Vladimirovich Manzhirov, Department of Higher Mathematics, Moscow State Academy of Engineering and Computer Science, Russia
- Philip McDunnough, Department of Statistics, University of Toronto
- Costas Meghir, Department of Economics, University College London
- Piotr Mikusinski, Department of Mathematics, University of Central Florida
- Michel Mouchart
- M. Zuhair Nashed, Department of Mathematics, University of Central Florida
- Frank Natterer, Fachbereich Mathematik, Institut für Numerische und instrumentelle Mathematik, Germany
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- Whitney K. Newey, Department of Economics, MIT
- Serena Ng, Department of Economics, University of Michigan
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- Alexei Onatski, Department of Economics, Columbia University
- Art B. Owen, Department of Statistics, Stanford University
- Adrian Pagan, Department of economics, Australian National University
- Emanuel Parzen, Department of Statistics, Texas A&M University
- Isabelle Perrigne, Department of Economics, Penn State University
- Dimitris N. Politis, Department of Mathematics, University of California at San Diego
- Andrei Dmitrievich Polyanin, Institute for Problems in Mechanics, Russian Academy of Sciences
- Costin Protopopescu, Grequam, Universite d'Aix-Marseille, France
- James Powell, Department of Economics, University of California at Berkeley
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- Jean-Marie Rolin, Institut de Statistique, Universite Catholique de Louvain
- Joseph P. Romano, Department of Statistics, Stanford University
- Sheldon M. Ross, Department of Industrial Engineering and Operations Research, University of California at Berkeley
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- Grace Wahba, Department of Statistics, University of Wisconsin-Madison
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- Halbert White, Department of Economics, University of California at San Diego
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