Bootstrap inference in local polynomial regression of time series

In this paper we consider the inferential aspect of the nonparametric estimation of a conditional function , where X _t,m represents the vector containing the m conditioning lagged values of the series. Here is an arbitrary measurable function. The local polynomial estimator of order p is used for the estimation of the function g, and of its partial derivatives up to a total order p. We consider α-mixing processes, and we propose the use of a particular resampling method, the local polynomial bootstrap, for the approximation of the sampling distribution of the estimator. After analyzing the consistency of the proposed method, we present a simulation study which gives evidence of its finite sample behaviour.     

Robust recursive estimation in nonlinear time series

Aase (1983) has dealt with recursive estimation in nonlinear time series of autoregressive type including its asymptotic properties. This contribution modifies the results for the case of nonlinear time series with outliers using the principle of M-estimation from robust statistics. Strong consistency of the robust recursive estimates is preserved under corresponding assumptions. Several types of such estimates are compared by means of a numerical simulation.     

A recursive approach to parameter estimation in regression and time series models

In this paper we discuss the recursive (or on line) estimation in (i) regression and (ii) autoregressive integrated moving average (ARIMA) time series models. The adopted approach uses Kalman filtering techniques to calculate estimates recursively. This approach is used for the estimation of constant as well as time varying parameters. In the first section of the paper we consider the linear regression model. We discuss recursive estimation both for constant and time varying parameters. For constant parameters, Kalman filtering specializes to recursive least squares. In general, we allow the parameters to vary according to an autoregressive integrated moving average process and update the parameter estimates recursively. Since the stochastic model for the parameter changes will "be rarely known, simplifying assumptions have to be made. In particular we assume a random walk model for the time varying parameters and show how to determine whether the parameters are changing over time. This is illustrated with an example.     

Minimum distance estimation in linear regression with strong mixing errors

Abstract

Minimum distance estimation on the linear regression model with independent errors is known to yield an efficient and robust estimator. We extend the method to the model with strong mixing errors and obtain an estimator of the vector of the regression parameters. The goal of this article is to demonstrate the proposed estimator still retains efficiency and robustness. To that end, this article investigates asymptotic distributional properties of the proposed estimator and compares it with other estimators. The efficiency and the robustness of the proposed estimator are empirically shown, and its superiority over the other estimators is established.     

Quantile regression estimation of partially linear additive models

In this paper, we consider the estimation of partially linear additive quantile regression models where the conditional quantile function comprises a linear parametric component and a nonparametric additive component. We propose a two-step estimation approach: in the first step, we approximate the conditional quantile function using a series estimation method. In the second step, the nonparametric additive component is recovered using either a local polynomial estimator or a weighted Nadaraya–Watson estimator. Both consistency and asymptotic normality of the proposed estimators are established. Particularly, we show that the first-stage estimator for the finite-dimensional parameters attains the semiparametric efficiency bound under homoskedasticity, and that the second-stage estimators for the nonparametric additive component have an oracle efficiency property. Monte Carlo experiments are conducted to assess the finite sample performance of the proposed estimators. An application to a real data set is also illustrated.     

Uniform consistency rate of kNN regression estimation for functional time series data

In this paper, we investigate the k-nearest neighbours (kNN) estimation of nonparametric regression model for strong mixing functional time series data. More precisely, we establish the uniform almost complete convergence rate of the kNN estimator under some mild conditions. Furthermore, a simulation study and an empirical application to the real data analysis of sea surface temperature (SST) are carried out to illustrate the finite sample performances and the usefulness of the kNN approach.     

Predicting hospital census using time series regression methods

We first describe the time series modeling problem in a general way. Then some specific assumptions and observations which are pertinent to the application of these models are made. We next propose a specific approach to the modeling problem, one which yields efficient, easily calculated estimators of all parameters (under the stated assumptions). Finally, the technique is applied to the problem of modeling the census of a particular hospital.     

Convergence rates for uniform confidence intervals based on local polynomial regression estimators

We investigate the convergence rates of uniform bias-corrected confidence intervals for a smooth curve using local polynomial regression for both the interior and boundary region. We discuss the cases when the degree of the polynomial is odd and even. The uniform confidence intervals are based on the volume-of-tube formula modified for biased estimators. We empirically show that the proposed uniform confidence intervals attain, at least approximately, nominal coverage. Finally, we investigate the performance of the volume-of-tube based confidence intervals for independent non-Gaussian errors.     

Volatility estimation from short time series of stock prices

We consider estimation of the historical volatility of stock prices. It is assumed that the stock prices are represented as time series formed as samples of the solution of a stochastic differential equation with random and time-varying parameters; these parameters are not observable directly and have unknown evolution law. The price samples are available with limited frequency only. In this setting, the estimation has to be based on short time series, and the estimation error can be significant. We suggest some supplements to the existing nonparametric methods of volatility estimation. Two modifications of the standard summation formula for the volatility are derived. In addition, a linear transformation eliminating the appreciation rate and preserving the volatility is suggested.     

Nonparametric regression estimation in a null recurrent time series

We derive an asymptotic theory of nonparametric estimation for a time series regression model Z_t=f(X_t)+W_t, where {X_t} and {Z_t} are observed nonstationary processes, and {W_t} is an unobserved stationary process. The class of nonstationary processes allowed for {X_t} is a subclass of the class of null recurrent Markov chains. This subclass contains the random walk, unit root processes and nonlinear processes. The process {W_t} is assumed to be linear and stationary.     

More efficient local polynomial regression with random-effects panel data models

We propose a modification on the local polynomial estimation procedure to account for the “within-subject” correlation presented in panel data. The proposed procedure is rather simple to compute and has a closed-form expression. We study the asymptotic bias and variance of the proposed procedure and show that it outperforms the working independence estimator uniformly up to the first order. Simulation study shows that the gains in efficiency with the proposed method in the presence of “within-subject” correlation can be significant in small samples. For illustration purposes, the procedure is applied to explore the impact of market concentration on airfare.     

Wavelet estimation in time-varying coefficient time series models with measurement errors

The article studies a time-varying coefficient time series model in which some of the covariates are measured with additive errors. In order to overcome the bias of estimator of the coefficient functions when measurement errors are ignored, we propose a modified least squares estimator based on wavelet procedures. The advantage of the wavelet method is to avoid the restrictive smoothness requirement for varying-coefficient functions of the traditional smoothing approaches, such as kernel and local polynomial methods. The asymptotic properties of the proposed wavelet estimators are established under the α-mixing conditions and without specifying the error distribution. These results can be used to make asymptotically valid statistical inference.     

Automatic polynomial wavelet regression

In Oh, Naveau and Lee (2001) a simple method is proposed for reducing the bias at the boundaries for wavelet thresholding regression. The idea is to model the regression function as a sum of wavelet basis functions and a low-order polynomial. The latter is expected to account for the boundary problem. Practical implementation of this method requires the choice of the order of the low-order polynomial, as well as the wavelet thresholding value. This paper proposes two automatic methods for making such choices. Finite sample performances of these two methods are evaluated via numerical experiments.     

Adaptive wavelet series estimation in separable nonparametric regression models

It is well-known that multivariate curve estimation suffers from the curse of dimensionality. However, reasonable estimators are possible, even in several dimensions, under appropriate restrictions on the complexity of the curve. In the present paper we explore how much appropriate wavelet estimators can exploit a typical restriction on the curve such as additivity. We first propose an adaptive and simultaneous estimation procedure for all additive components in additive regression models and discuss rate of convergence results and data-dependent truncation rules for wavelet series estimators. To speed up computation we then introduce a wavelet version of functional ANOVA algorithm for additive regression models and propose a regularization algorithm which guarantees an adaptive solution to the multivariate estimation problem. Some simulations indicate that wavelets methods complement nicely the existing methodology for nonparametric multivariate curve estimation.     

Using difference-based methods for inference in nonparametric regression with time series errors

Summary. We show that difference-based methods can be used to construct simple and explicit estimators of error covariance and autoregressive parameters in nonparametric regression with time series errors. When the error process is Gaussian our estimators are efficient, but they are available well beyond the Gaussian case. As an illustration of their usefulness we show that difference-based estimators can be used to produce a simplified version of time series cross-validation. This new approach produces a bandwidth selector that is equivalent, to both first and second orders, to that given by the full time series cross-validation algorithm. Other applications of difference-based methods are to variance estimation and construction of confidence bands in nonparametric regression.     

Change detection in linear regression with time series errors

A new test for detecting a change in linear regression parameters assuming a general weakly dependent error structure is given. It extends earlier methods based on cumulative sums assuming independent errors. The novelty is in the new standardization method and in smoothing when the time series is dominated by high frequencies. Simulations show the excellent performance of the test. Examples are taken from environmental applications. The algorithm is easy to implement. Testing for multiple changes can be done by segmentation. The Canadian Journal of Statistics 38:65–79; 2010 © 2009 Statistical Society of Canada     

On multivariate variable-kernel density estimates for time series

Let X₁ be a strictly stationary multiple time series with values in R^d and with a common density f. Let X₁,.,.,X_n, be n consecutive observations of X₁. Let k = k_n, be a sequence of positive integers, and let H_ni be the distance from X_i to its kth nearest neighbour among X_j, j i. The multivariate variable-kernel estimate f_n, of f is defined by where K is a given density. The complete convergence of f_n, to f on compact sets is established for time series satisfying a dependence condition (referred to as the strong mixing condition in the locally transitive sense) weaker than the strong mixing condition. Appropriate choices of k are explicitly given. The results apply to autoregressive processes and bilinear time-series models.     

Testing for changes in autocovariances of nonparametric time series models

In the literature on change-point analysis, much attention has been paid to detecting changes in certain marginal characteristics, such as mean, variance, and marginal distribution. For time series data with nonparametric time trend, we study the change-point problem for the autocovariance structure of the unobservable error process. To derive the asymptotic distribution of the cumulative sum test statistic, we develop substantial theory for uniform convergence of weighted partial sums and weighted quadratic forms. Our asymptotic results improve upon existing works in several important aspects. The performance of the test statistic is examined through simulations and an application to interest rates data.