Essays On Recursive Nonparametric Kernel Estimation Of Regression

In the practice of economics it is common that the data are observed not as a sample of fixed size, but rather as an ongoing sequence of a time series. It could be computationally advantageous if the estimate of the unknown function could be updated for each newly arriving data point. On some occasions, there is also a need to update the existing estimate with the newly realized observations. Kalman filter and Bayesian estimation are the commonly encountered techniques to handle these problems in the paradigm of linear parametric estimation. However, few procedures are available for nonlinear models, especially in the nonparametric setting. This thesis attempts to formulate such an estimator using the recursive version of the Nadaraya-Watson estimator.;The recursive estimator for the conditional mean of a nonparametric regression model with independent observations was thoroughly explored in the late 1970\u27s and early 1980\u27s by authors such as Greblicki and Pawlak (1987). The first chapter of this thesis summarizes the constructs and methods of analysis developed in connection with such estimators for independent observations and briefly demonstrates some of their asymptotic properties under the chosen conditions. However, economic time series are generated as economic agents engage in intertemporal optimization and are usually heterogeneous, correlated and unlikely to be linear. There is an incentive for us to extend the study of this recursive nonparametric regression estimator to the case where the observations are correlated. This investigation forms the content of Chapter Two. In Chapter Three, we propose a recursive version of nonparametric kernel estimator of the derivative of a regression function and establish the conditions to ensure that it is consistent and has an asymptotically normal distribution. In Chapter Four, we show an implementation of the recursive estimator and examine its finite sample properties

Does the Box-Cox transformation help in forecasting macroeconomic time series?

The paper investigates whether transforming a time series leads to an improvement in forecasting accuracy. The class of transformations that is considered is the Box-Cox power transformation, which applies to series measured on a ratio scale. We propose a nonparametric approach for estimating the optimal transformation parameter based on the frequency domain estimation of the prediction error variance, and also conduct an extensive recursive forecast experiment on a large set of seasonal monthly macroeconomic time series related to industrial production and retail turnover. In about one fifth of the series considered the Box-Cox transformation produces forecasts significantly better than the untransformed data at one-step-ahead horizon; in most of the cases the logarithmic transformation is the relevant one. As the forecast horizon increases, the evidence in favour of a transformation becomes less strong. Typically, the na¨ıve predictor that just reverses the transformation leads to a lower mean square error than the optimal predictor at short forecast leads. We also discuss whether the preliminary in-sample frequency domain assessment conducted provides a reliable guidance which series should be transformed for improving significantly the predictive performance.Forecasts comparisons; Multi-step forecasting; Rolling forecasts; Nonparametric estimation of prediction error variance.

Nonparametric specification analysis of dynamic parametric models

Time series parametric models generally cater to a particular objective, such as forecasting, and it is therefore desirable to judge such models solely on the basis of their performance in the fullfillment of that objective. We propose a specification testing procedure which concentrates power on the parametric model's ability to estimate a set of characteristics of the finite dimensional distributions of the process. It is based on the comparison between a nonparametric estimate of the said characteristic and its parametric boot-strap analogue. Applications of this principle are proposed for the assessment of recursive dynamic models in the estimation of conditional means and conditional quantiles for mixing processes and for the estimation of dependence in long memory processes

Consistency of the recursive nonparametric regression estimation for dependent functional data

We consider the recursive estimation of a regression functional where the explanatory variables take values in some functional space. We prove the almost sure convergence of such estimates for dependent functional data. Also we derive the mean quadratic error of the considered class of estimators. Our results are established with rates and asymptotic appear bounds, under strong mixing condition.Comment: 12 page

Consistency of a recursive estimate of mixing distributions

Mixture models have received considerable attention recently and Newton [Sankhy\={a} Ser. A 64 (2002) 306--322] proposed a fast recursive algorithm for estimating a mixing distribution. We prove almost sure consistency of this recursive estimate in the weak topology under mild conditions on the family of densities being mixed. This recursive estimate depends on the data ordering and a permutation-invariant modification is proposed, which is an average of the original over permutations of the data sequence. A Rao--Blackwell argument is used to prove consistency in probability of this alternative estimate. Several simulations are presented, comparing the finite-sample performance of the recursive estimate and a Monte Carlo approximation to the permutation-invariant alternative along with that of the nonparametric maximum likelihood estimate and a nonparametric Bayes estimate.Comment: Published in at http://dx.doi.org/10.1214/08-AOS639 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On nonparametric estimation of a mixing density via the predictive recursion algorithm

Nonparametric estimation of a mixing density based on observations from the corresponding mixture is a challenging statistical problem. This paper surveys the literature on a fast, recursive estimator based on the predictive recursion algorithm. After introducing the algorithm and giving a few examples, I summarize the available asymptotic convergence theory, describe an important semiparametric extension, and highlight two interesting applications. I conclude with a discussion of several recent developments in this area and some open problems.Comment: 22 pages, 5 figures. Comments welcome at https://www.researchers.one/article/2018-12-

Real Time Changes in Monetary Policy

This paper investigates potential changes in monetary policy over the last decades using a nonparametric vector autoregression model. In the proposed model, the conditional mean and variance are time-dependent and estimated using a nonparametric local linear method, which allows for different forms of nonlinearity, conditional heteroskedasticity, and non-normality. Our results suggest that there have been gradual and abrupt changes in the variances of shocks, in the monetary transmission mechanism, and in the Fed’s reaction function. The response of output was strongest during Volcker’s disinflationary period and has since been slowly decreasing over time. There have been some abrupt changes in the response of inflation, especially in the early 1980s, but we can not conclude that it is weaker now than in previous periods. Finally, we find significant evidence that policy was passive during some parts of Burn’s period, and active during Volcker’s disinflationary period and Greenspan’s period. However, we find that the uncovered behavior of the parameters is more complex than general conclusions suggest, since they display considerable nonlinearities over time. A particular appeal of the recursive estimation of the proposed VAR-ARCH is the detection of discrete local deviations as well as more gradual ones, without smoothing the timing or magnitude of the changes.Monetary Policy, Taylor Rule, Local Estimation, Nonlinearity, Nonparametric, Monetary Policy; Taylor Rule; Local Estimation; Nonlinearity; Nonparametric; Structural Vector Autoregression; Autoregressive Conditional Heteroskedasticity;

Asymptotic normality of recursive estimators under strong mixing conditions

The main purpose of this paper is to estimate the regression function by using a recursive nonparametric kernel approach. We derive the asymptotic normality for a general class of recursive kernel estimate of the regression function, under strong mixing conditions. Our purpose is to extend the work of Roussas and Tran [17] concerning the Devroye-Wagner estimate

Bias Reduction by Recursive Mean Adjustment in Dynamic Panel Data Models

Accurate estimation of the dominant root of a stationary but persistent time series are required to determine the speed at which economic time series, such as real exchange rates or interest rates, adjust towards their mean values. In practice, accuracy is hampered by downward small- sample bias. Recursive mean adjustment has been found to be a useful bias reduction strategy in the regression context. In this paper, we study recursive mean adjustment in dynamic panel data models. When there exists cross-sectional heterogeneity in the dominant root, the recursive mean adjusted SUR estimator is appropriate. When homogeneity restrictions can be imposed, a pooled recursive mean adjusted GLS estimator with fixed e¤ects is the desired estimator. Application of these techniques to a small panel of five eurocurrency rates finds that these interest rates are unit root nonstationary as the bias-corrected autoregressive coefficient exceeds 1.Small sample bias, Recursive mean adjustment, Panel Data, Cross-sectional dependence, Interest rate dynamics