Adaptive Pointwise Estimation in Time-Inhomogeneous Time-Series Models

This paper offers a new method for estimation and forecasting of the linear and nonlinear time series when the stationarity assumption is violated. Our general local parametric approach particularly applies to general varying-coefficient parametric models, such as AR or GARCH, whose coefficients may arbitrarily vary with time. Global parametric, smooth transition, and changepoint models are special cases. The method is based on an adaptive pointwise selection of the largest interval of homogeneity with a given right-end point by a local change-point analysis. We construct locally adaptive estimates that can perform this task and investigate them both from the theoretical point of view and by Monte Carlo simulations. In the particular case of GARCH estimation, the proposed method is applied to stock-index series and is shown to outperform the standard parametric GARCH model.adaptive pointwise estimation;autoregressive models;conditional heteroscedasticity models;local time-homogeneity

When did the 2001 recession really start?

The paper develops a non-parametric, non-stationary framework for business-cycle dating based on an innovative statistical methodology known as Adaptive Weights Smoothing (AWS). The methodology is used both for the study of the individual macroeconomic time series relevant to the dating of the business cycle as well as for the estimation of their joint dynamic. Since the business cycle is defined as the common dynamic of some set of macroeconomic indicators, its estimation depends fundamentally on the group of series monitored. We apply our dating approach to two sets of US economic indicators including the monthly series of industrial production, nonfarm payroll employment, real income, wholesale-retail trade and gross domestic product (GDP). We find evidence of a change in the methodology of the NBER's Business- Cycle Dating Committee: an extended set of five monthly macroeconomic indicators replaced in the dating of the last recession the set of indicators emphasized by the NBER's Business-Cycle Dating Committee in recent decades. This change seems to seriously affect the continuity in the outcome of the dating of business cycle. Had the dating been done on the traditional set of indicators, the last recession would have lasted one year and a half longer. We find that, independent of the set of coincident indicators monitored, the last economic contraction began in November 2000, four months before the date of the NBER's Business-Cycle Dating Committee.business cycle, non-parametric smoothing, non-stationarity

Adaptive hypothesis testing using wavelets

Let a function f be observed with a noise. We wish to test the null hypothesis that the function is identically zero, against a composite nonparametric alternative: functions from the alternative set are separated away from zero in an integral Ž e.g., L. 2 norm and also possess some smoothness properties. The minimax rate of testing for this problem was evaluated in earlier papers by Ingster and by Lepski and Spokoiny under different kinds of smoothness assumptions. It was shown that both the optimal rate of testing and the structure of optimal Ž in rate. tests depend on smoothness parameters which are usually unknown in practical applications. In this paper the problem of adaptive Ž assumption free. testing is considered. It is shown that adaptive testing without loss of efficiency is impossible. An extra log log-factor is inessential but unavoidable payment for the adaptation. A simple adaptive test based on wavelet technique is constructed which is nearly minimax for a wide range of Besov classes. 1. Introduction. Suppos

Optimal Pointwise Adaptive Methods in Nonparametric Estimation

The problem of optimal adaptive estimation of a function at a given point from noisy data is considered.Two procedures are proved to be asymptotically optimal for different settings.First we study the problem of bandwidth selection for nonparametric pointwise kernel estimation with a given kernel.We propose a bandwidth selection procedure and prove its optimality in the asymptotic sense. Moreover, this optimality is stated not only among kernel estimators with a variable kernel. The resulting estimator is optimal among all feasible estimators.The important feature of this procedure is that no prior information is used about smoothness properties of the estimated function i.e. the procedure is completely adaptive and "works" for the class of all functions. With it the attainable accuracy of estimation depends on the function itself and it is expressed in terms of "ideal" bandwidth corresponding to this function.The second procedure can be considered as a specification of the first one under the qualitative assumption that the function to be estimated belongs to some Hölder class Σ(β,L) with unknown parameters β, L.This assumption allows to choose a family of kernels in an optimal way and the resulting procedure appears to be asymptotically optimal in the adaptive sense

On estimation of non-smooth functionals

Let a function ƒ be observed with noise. In the present paper we concern the problem of nonparametric estimation of some non-smooth functionals of ƒ, more precisely, Lr -norm ∥ƒ∥r of ƒ. Existing in the literature results on estimation of functionals deal mostly with two extreme cases: estimation of a smooth (differentiable in L2) functional or estimation of a singular functional like the value of ƒ at a certain point or the maximum of ƒ. In the first case, the rate of estimation is typically n-1/2 , n being the number of observations. In the second case, the rate of functional estimation coincides with the nonparametric rate of estimation of the whole function ƒ in the corresponding norm. We show that the case of estimation of ∥ƒ∥r is in some sense intermediate between the above extreme two. The optimal rate of estimation is worse than n-1/2 but better than the usual nonparametric rate. The results depend on the value of r . For r even integer, the rate occurs to be n-β/(2β+1-1/r) where β is the degree of smoothness. If r is not even integer, then the nonparametric rate n -β/(2β+1) can be improved only by some logarithmic factor

Local adaptivity to inhomogeneous smoothness. 1. Resolution level

The problem of nonparametric estimation of functions of inhomogeneous smoothness is considered. The goal is to define the notion of local smoothness of a function ƒ(·), to evaluate the optimal rate of convergence of estimators (depending on this local smoothness) and to construct an asymptotically efficient locally adaptive estimator. We treat local (or δ - local) smoothness properties of a function ƒ( extperiodcentered) at a point t as the corresponding characteristics of this function on the interval [t-δ,t+δ]. The value δ measures the ``locality'' of our procedure. The smaller this value is taken the more precise is our resolution analysis. But this value can not be taken arbitrary small since we should ce able to restore local smoothness properties of a function from the noisy data. The main result of the paper describes just the maximal rate of convergence of this parameter δ to zero as the noise level ε goes to zero. We call this value the resolution level. The value of this level strongly depends on the upper considered smoothness β* what we wish to attain. If κ*ε is the bandwidth corresponding to this smoothness β* then the resolution level δ*ε can not be chosen less (in order) than κ*ε. In particular, this yields that it is impossible to improve at the same time the accuracy of our procedure (which is measured by the upper smoothness β*) and its local adaptive properties. If we improve the accuracy of estimation at subintervals where a function is of high smoothness then we will have a low accuracy in a larger vicinity near a point with small smoothness. The main results claim that if the parameter of locality δ is taken less (in order) than the resolution level, then the corresponding risk is (asymptotically) infinite. After that we construct estimators with a finite asymptotic risk for the case of δ coinciding with the resolution level

Regression Methods in Pricing American and Bermudan Options Using Consumption Processes

Numerical algorithms for the efficient pricing of multidimensional discrete-time American and Bermudan options are constructed using regression methods and a new approach for computing upper bounds of the options' price. Using the sample space with payoffs at optimal stopping times, we propose sequential estimates for continuation values, values of the consumption process, and stopping times on the sample paths. The approach allows the constructing of both lower and upper bounds for the price by Monte Carlo simulations. The algorithms are tested by pricing Bermudan max-calls and swaptions in the Libor market model.D.B. gratefully acknowledges the partial support of DFG through SFB 649. This work was completed while G.M. was a visitor at the Weierstrass-Institute für Angewandte Analysis und Stochastik (WIAS), Berlin, thanks to financial support from this institute and DFG (grant No. 436 RUS 17/137/05 and 436 RUS 17/24/07), which are gratefully acknowledged