Forecasting functional time series

We propose forecasting functional time series using weighted functional principal component regression and weighted functional partial least squares regression. These approaches allow for smooth functions, assign higher weights to more recent data, and provide a modeling scheme that is easily adapted to allow for constraints and other information. We illustrate our approaches using age-specific French female mortality rates from 1816 to 2006 and age-specific Australian fertility rates from 1921 to 2006, and show that these weighted methods improve forecast accuracy in comparison to their unweighted counterparts. We also propose two new bootstrap methods to construct prediction intervals, and evaluate and compare their empirical coverage probabilities.     

Bootstrapping time series models

This paper surveys recent development in bootstrap methods and the modifications needed for their applicability in time series models. The paper discusses some guidelines for empirical researchers in econometric analysis of time series. Different sampling schemes for bootstrap data generation and different forms of bootstrap test statistics are discussed. The paper also discusses the applicability of direct bootstrapping of data in dynamic models and cointegrating regression models. It is argued that bootstrapping residuals is the preferable approach. The bootstrap procedures covered include the recursive bootstrap, the moving block bootstrap and the stationary bootstrap.     

Bootstrapping time series models

This paper surveys recent development in bootstrap methods and the modifications needed for their applicability in time series models. The paper discusses some guidelines for empirical researchers in econometric analysis of time series. Different sampling schemes for bootstrap data generation and different forms of bootstrap test statistics are discussed. The paper also discusses the applicability of direct bootstrapping of data in dynamic models and cointegrating regression models. It is argued that bootstrapping residuals is the preferable approach. The bootstrap procedures covered include the recursive bootstrap, the moving block bootstrap and the stationary bootstrap.     

Forecasting power-transformed time series data

When there is an interest in forecasting the growth rates as well as the levels of a single macro-economic time series, a practitioner faces the question of whether a forecasting model should be constructed for growth rates, for levels, or for both. In this paper, we investigate this issue for 10 US (un-)employment series, where we evaluate the forecasts from a non-linear time series model for power-transformed data. Our main finding is that models for growth rates (levels) do not automatically result in the most accurate forecasts of growth rates (levels).     

On count time series prediction

We consider the problem of assessing prediction for count time series based on either the Poisson distribution or the negative binomial distribution. By a suitable parametrization we employ both distributions with the same mean. We regress the mean on its past values and the values of the response and after obtaining consistent estimators of the regression parameters, regardless of the response distribution, we employ different criteria to study the prediction problem. We show by simulation and data examples that scoring rules and diagnostic graphs that have been proposed for independent but not identically distributed data can be adapted in the setting of count dependent data.     

Forecasting spring flow time series

Summary The paper deals with a statistical analysis, carried out to define the underlying reason of some of the damage observed in many buildings of a southern Italian town. Engineering considerations, substantiated by specific measurements, attributed them to the lowering of the groundwater table in the area below the building locations. Due to two coinciding events which occurred in the preceding years, i.e. a persistent drought and the start up of a system of wells, it was not possible to define the cause of the former phenomenon. As shutting down the wells could generate additional problems, an accurate picture of the whole situation was necessary, before taking any action. By taking advantage of some fragmentary data belonging to the flow of a spring located in the area and on the basis of the knowledge of the rainfall data recorded in the Italian hydrographic service directory, two models have been developed which reproduce the spring flow time series in relation to the rainfall recorded in the surrounding area. By comparing the spring flow predictions with the actual data it has been possible to highlight the main role played by the wells.     

Polynomial trends in time series

The J^* test which was previously proposed by the present author for the detection of a trend in a time series does not depend on any quantitative assumptions, but in the case of a polynomial trend it depends on its degree; if this degree is too high, the test cannot be applied. The author finds a bound of the significance level at which the test can be applied when the sample size, as well as a bound of the degree of the trend, are given. Asymptotic results are used only when we trust the asymptotic distribution of J^* under the null hypothesis.     

Missing observations in time series

A method based on forecasting techniques is proposed to estimate missing observations in time series. Using mean squares, this method is compared to the minimum mean square estimate.     

Assessing influence in time series

This paper studies influential observations on the spectrum of a stationary stochastic process. We introduce a leave-one-out procedure in spectral density estimation to identify influential points. A simulated envelope is proposed to assess the magnitude of influence when the data follow an autoregressive integrated moving average model. Practical illustrations are discussed in two examples.     

Forecasting contemporal time series aggregates

Forecast of a contemporal aggregate of several time series can be obtained from ‘1’ an aggregate series, ‘2’ individual component processes, or ‘3’ a joint multiple forecasting model. Through general Hilbert space theory and some illustrative examples, this paper establishes the relative efficiencies among the three methods     

Data tilting for time series

Summary. We develop a general methodology for tilting time series data. Attention is focused on a large class of regression problems, where errors are expressed through autoregressive processes. The class has a range of important applications and in the context of our work may be used to illustrate the application of tilting methods to interval estimation in regression, robust statistical inference and estimation subject to constraints. The method can be viewed as 'empirical likelihood with nuisance parameters'.     

Resampling-based bias-corrected time series prediction

In this paper, we consider estimation of the mean squared prediction error (MSPE) of the best linear predictor of (possibly) nonlinear functions of finitely many future observations in a stationary time series. We develop a resampling methodology for estimating the MSPE when the unknown parameters in the best linear predictor are estimated. Further, we propose a bias corrected MSPE estimator based on the bootstrap and establish its second order accuracy. Finite sample properties of the method are investigated through a simulation study.     

Smoothing for discrete-valued time series

We deal with smoothed estimators for conditional probability functions of discrete-valued time series { Y_t } under two different settings. When the conditional distribution of Y_t given its lagged values falls in a parametric family and depends on exogenous random variables, a smoothed maximum (partial) likelihood estimator for the unknown parameter is proposed. While there is no prior information on the distribution, various nonparametric estimation methods have been compared and the adjusted Nadaraya–Watson estimator stands out as it shares the advantages of both Nadaraya–Watson and local linear regression estimators. The asymptotic normality of the estimators proposed has been established in the manner of sparse asymptotics, which shows that the smoothed methods proposed outperform their conventional, unsmoothed, parametric counterparts under very mild conditions. Simulation results lend further support to this assertion. Finally, the new method is illustrated via a real data set concerning the relationship between the number of daily hospital admissions and the levels of pollutants in Hong Kong in 1994–1995. An ad hoc model selection procedure based on a local Akaike information criterion is proposed to select the significant pollutant indices.     

Clustering time series by linear dependency

Statistics and Computing - We present a new way to find clusters in large vectors of time series by using a measure of similarity between two time series, the generalized cross correlation. This...     

Quantile smoothing in financial time series

Various parametric models have been designed to analyze volatility in time series of financial market data. For maximum likelihood estimation these parametric methods require the assumption of a known conditional distribution. In this paper we examine the conditional distribution of daily DAX returns with the help of nonparametric methods. We use kernel estimators for conditional quantiles resulting from a kernel estimation of conditional distributions. This work was financially supported by the Deutsche Forschungsgemeinschaft