ERRORS-IN-VARIABLES MODELS: A GENERALIZED FUNCTIONS APPROACH

Identification in errors-in-variables regression models was recently extended to wide models classes by S. Schennach (Econometrica, 2007) (S) via use of generalized functions. In this paper the problems of non- and semi- parametric identification in such models are re-examined. Nonparametric identification holds under weaker assumptions than in (S); the proof here does not rely on decomposition of generalized functions into ordinary and singular parts, which may not hold. Conditions for continuity of the identification mapping are provided and a consistent nonparametric plug-in estimator for regression functions in the L₁ space constructed. Semiparametric identification via a finite set of moments is shown to hold for classes of functions that are explicitly characterized; unlike (S) existence of a moment generating function for the measurement

NON AND SEMI-PARAMETRIC ESTIMATION IN MODELS WITH UNKNOWN SMOOTHNESS

Many asymptotic results for kernel-based estimators were established under some smoothness assumption on density. For cases where smoothness assumptions that are used to derive unbiasedness or asymptotic rate may not hold we propose a combined estimator that could lead to the best available rate without knowledge of density smoothness. A Monte Carlo example confirms good performance of the combined estimator.

Consequences of lack of smoothness in nonparametric estimation (in Russian)

Nonparametric estimation is widely used in statistics and econometrics with many asymptotic results relying on smoothness of the underlying distribution, however, there are cases where such assumptions may not hold in practice. Lack of smoothness may have undesirable consequences such as an incorrect choice of window width, large estimation biases and incorrect inference. Optimal combinations of estimators based on different kernel/bandwidth can achieve automatically the best unknown rate of convergence. The combined estimator was successfully applied in density estimation, estimation of average derivatives and for smoothed maximum score in a binary choice model. In the extreme case when density does not exist the estimator "estimates" a non-existent function; nevertheless its limit process can be described in terms of generalized (in terms of generalized functions) Gaussian processes. Inference about existence of density and about its smoothness is not yet well developed; some preliminary results are discussed.

REDUCED-DIMENSION CONTROL REGRESSION

A model to investigate the relationship between one variable and another usually requires controls for numerous other effects which are not constant across the sample; where the model omits some elements of the true process, estimates of parameters of interest will typically be inconsistent. Here we investigate conditions under which, with a set of potential controls which is large (possibly infinite), orthogonal transformations of a subset of potential controls can nonetheless be used in a parsimonious regression involving a reduced number of orthogonal components (the ‘reduced-dimension control regression’), to produce consistent (and asymptotically normal, given further restrictions) estimates of a parameter of interest, in a general setting. We examine selection of the particular orthogonal directions, using a new criterion which takes into account both the magnitude of the eigenvalue and the correlation of the eigenvector with the variable of interest. Simulation experiments show good finite-sample performance of the method.

Properties of Estimates of Daily GARCH Parameters Basaed on Intra-Day Observations

We consider estimates of the parameters of GARCH models of daily financial returns, obtained using intra-day (high-frequency) returns data to estimate the daily conditional volatility.Two potential bases for estimation are considered. One uses aggregation of high-frequency Quasi- ML estimates, using aggregation results of Drost and Nijman (1993). The other uses the integrated volatility of Andersen and Bollerslev (1998), and obtains coefficients from a model estimated by LAD or OLS, in the former case providing consistency and asymptotic normality in some cases where moments of the volatility estimation error may not exist. In particular, we consider estimation in this way of an ARCH approximation, and obtain GARCH parameters by a method related to that of Galbraith and Zinde-Walsh (1997) for ARMA processes. We offer some simulation evidence on small-sample performance, and characterize the gains relative to standard quasi-ML estimates based on daily data alone. Nous considérons les estimés des paramètres des modèles GARCH pour les rendements financiers journaliers, qui sont obtenus à l'aide des données intra-jour (haute fréquence) pour estimer la volatilité journalière. Deux bases potentielles sont evaluées. La première est fondée sur l'aggrégation des estimés quasi-vraisemblance-maximale, en profitant des résultats de Drost et Nijman (1993). L'autre utilise la volatilité integrée de Andersen et Bollerslev (1998), et obtient les coefficients d'un modèle estimé par LAD ou MCO; la première méthode résiste mieux à la possibilité de non-existence des moments de l'erreur en estimation de volatilité. En particulier, nous considérons l'estimation par approximation ARCH, et nous obtenons les paramètres par une méthode liée à celle de Galbraith et Zinde-Walsh (1997) pour les processus ARMA. Nous offrons des résultats provenant des simulations sur la performance des méthodes en échantillons finis, et nous décrivons les atouts relatifs à l'estimation standard de quasi-VM basée uniquement sur les données journalières.GARCH, high frequency data, integrated volatility, LAD, GARCH, données haute fréquence, volatilité intégrée, LAD

ROBUST KERNEL ESTIMATOR FOR DENSITIES OF UNKNOWN

Results on nonparametric kernel estimators of density differ according to the assumed degree of density smoothness; it is often assumed that the density function is at least twice differentiable. However, there are cases where non-smooth density functions may be of interest. We provide asymptotic results for kernel estimation of a continuous density for an arbitrary bandwidth/kernel pair. We also derive the limit joint distribution of kernel density estimators coresponding to different bandwidths and kernel functions. Using these reults, we construct an estimator that combines several estimators for different bandwidth/kernel pairs to protect against the negative consequences of errors in assumptions about order of smoothness. The results of a Monte Carlo experiment confirm the usefulness of the combined estimator. We demonstrate that while in the standard normal case the combined estimator has a relatively higher mean squared error than the standard kernel estimator, both estimators are highly accurate. On the other hand, for a non-smooth density where the MSE gets very large, the combined estimator provides uniformly better results than the standard estimator.