BANDWIDTH SELECTION IN KERNEL SMOOTHING OF TIME SERIES

Abstract. The kernel smoothing method has been considered as a useful tool for identification and prediction in time series models. In practice this method is to be tuned by a smoothing parameter. For selection of the smoothing parameter, Härdle and Vieu (Kernel regression smoothing of time series. J. Time Ser. Anal. 13(1992), 209–32) considered a cross-validation rule and proved its asymptotic optimality. In this paper we strengthen their result for a wider use of the kernel smoothing of time series.     

FREQUENCY-DOMAIN ESTIMATION OF BILINEAR TIME SERIES MODELS

Abstract. Two frequency-domain methods of estimation of the parameters of linear time series models–one based on maximum likelihood, called the 'Whittle criterion', and the other based on least squares, called the 'Taniguchi criterion'–are discussed in this paper. A heuristic justification for their use in models such as bilinear models is given. The estimation theory and associated asymptotic theory of these methods are numerically illustrated for the bilinear model BL( p ,0, p , 1). For that purpose, an approach based on the calculus of Kronecker product matrices is used to obtain the derivatives of the spectral density function of the state-space form of the model.     

KERNEL REGRESSION SMOOTHING OF TIME SERIES

Abstract. A class of non-parametric regression smoothers for times series is defined by the kernel method. The kernel approach allows flexible modelling of a time series without reference to a specific parametric class. The technique is applicable to detection of non-linear dependences in time series and to prediction in smooth regression models with serially correlated observations.
In practice these estimators are to be tuned by a smoothing parameter. A data-driven selector for this smoothing parameter is presented that asymptotically minimizes a squared error measure. We prove asymptotic optimality of this selector. We illustrate the technique with a simulated example and by constructing a smooth prediction curve for the variation of gold prices. In both cases the non-parametric method proves to be useful in uncovering non-linear structure.     

PREDICTION ERROR OF MULTIVARIATE TIME SERIES WITH MIS-SPECIFIED MODELS

We consider the situation in which an incorrectly specified autoregressive moving-average model is used to predict future values of a stationary multivariate time series. The use of an incorrect model for prediction results in an increase in mean-square prediction error over that of the optimal predictor, and an expression for this increase is first given for fixed values of the parameters in the incorrect model. For the case in which the incorrect model is an autoregression, we also take into account parameter estimation error by first deriving the asymptotic distribution and limiting moment properties of the least-squares estimator of the parameters in the mis-specified model. An asymptotic approximation to the increase in mean-square prediction error is then obtained. Numerical examples are provided to demonstrate the accuracy of the asymptotic approximation in finite samples. Our results are consistent with those obtained in the univariate case, indicating that fitted autoregressions of high order can yield substantially sub-optimal forecasts.     

On Estimating Conditional Mean-Squared Prediction Error in Autoregressive Models

Abstract. Zhang and Shaman considered the problem of estimating the conditional mean-squared prediciton error (CMSPE) for a Gaussian autoregressive (AR) process. They used the final prediction error (FPE) of Akaike to estimate CMSPE and proposed that FPE's effectiveness be judged by its asymptotic correlation with CMSPE. However, as pointed out by Kabaila and He, the derivation of this correlation by Zhang and Shaman is incomplete, and the performance of FPE in estimating CMSPE is also poor in Kabaila and He's simulation study. Kabaila and He further proposed an alternative estimator of CMSPE, V , in the stationary AR(1) model. They reported that V has a larger normalized correlation with CMSPE through Monte Carlo simulation results. In this paper, we propose a generalization of V , V˜, in the higher-order AR model, and obtain the asymptotic correlation of FPE and V˜ with CMSPE. We show that the limit of the normalized correlation of V˜ with CMSPE is larger than that of FPE with CMSPE, and hence Kabaila and He's finding is justified theoretically. In addition, the performances of the above estimators of CMSPE are re-examined in terms of mean-squared errors (MSE). Our main conclusion is that from the MSE point of view, V˜ is the best choice among a family of asymptotically unbiased estimators of CMSPE including FPE and V˜ as its special cases.     

EFFICIENT METHOD OF MOMENTS ESTIMATORS FOR INTEGER TIME SERIES MODELS

The parameters of integer autoregressive models with Poisson, or negative binomial innovations can be estimated by maximum likelihood where the prediction error decomposition, together with convolution methods, is used to write down the likelihood function. When a moving average component is introduced this is not the case. To address this problem an efficient method of moment estimator is proposed where the estimated standard errors for the parameters are obtained using subsampling methods. The small sample properties of the estimator are investigated using Monte Carlo methods, while the approach is demonstrated using two well‐known examples from the time series literature.     

ASYMPTOTIC RELATIVE EFFICIENCY OF SOME TESTS OF FIT IN TIME SERIES MODELS

Abstract. Two frequency domain tests of fit for autoregressive moving average time series models are considered. The tests are slight generalizations of those introduced by Cameron (1978) and Milhøj (1981). It is shown that according to asymptotic relative efficiency the test by Milhøj outperforms the test by Cameron. However, if asymptotic relative efficiency is used as a standard of comparison, both of these tests are extremely poor as compared to the well-known time domain test of Box and Pierce (1970), for the asymptotic relative efficiency of the frequency domain tests as compared to the Box-Pierce test is zero.     

A STATE SPACE TIME SERIES MODELLING METHOD WITHOUT INDIVIDUAL DETRENDING

Abstract. A state space method for building time series models without detrending each component of data vectors individually is presented. The method uses the recent algorithm based on the singular-value decomposition of the Hankel matrix and a two-step sequential procedure suggested by the notion of dynamic aggregation. Some asymptotic properties of the estimators of the model parameter and error estimates are also presented.     

Multivariate time delay analysis based local KPCA fault prognosis approach for nonlinear processes☆

Currently, some fault prognosis technology occasionally has relatively unsatisfied performance especially for in-cipient faults in nonlinear processes duo to their large time delay and complex internal connection. To overcome this deficiency, multivariate time delay analysis is incorporated into the high sensitive local kernel principal com-ponent analysis. In this approach, mutual information estimation and Bayesian information criterion (BIC) are separately used to acquire the correlation degree and time delay of the process variables. Moreover, in order to achieve prediction, time series prediction by back propagation (BP) network is applied whose input is multivar-iate correlated time series other than the original time series. Then the multivariate time delayed series and future values obtained by time series prediction are combined to construct the input of local kernel principal component analysis (LKPCA) model for incipient fault prognosis. The new method has been exemplified in a sim-ple nonlinear process and the complicated Tennessee Eastman (TE) benchmark process. The results indicate that the new method has superiority in the fault prognosis sensitivity over other traditional fault prognosis methods. ? 2016 The Chemical Industry and Engineering Society of China, and Chemical Industry Press. Al rights reserved.     

NONPARAMETRIC ESTIMATORS FOR TIME SERIES

Abstract. Kernel multivariate probability density and regression estimators are applied to a univariate strictly stationary time series X _r We consider estimators of the joint probability density of X _t at different t -values, of conditional probability densities, and of the conditional expectation of functionals of X _v given past behaviour. The methods seem of particular relevance in light of recent interest in non-Gaussian time series models. Under a strong mixing condition multivariate central limit theorems for estimators at distinct points are established, the asymptotic distributions being of the same nature as those which would derive from independent multivariate observations.     

A NEW WAY TO ESTIMATE ORDERS IN TIME SERIES

Abstract. In this paper we propose the order determination quantity (ODQ) as a new way to solve order estimation problems in time series analysis. We estimate orders according to ODQ > 0 or ODQ < 0 instead of by minimizing. Theoretical analysis and simulation have shown that the ODQ has higher identifiability for unknown true orders, provides clear separation points and requires less computational effort than the existing order estimation criteria such as Akaike's information criterion (AIC), Bayes information criterion (BIC), φ and predictive least squares (PLS).     

RESIDUAL AUTOCOVARIANCES AND UNIT ROOT TESTS BASED ON INSTRUMENTAL VARIABLE ESTIMATORS FROM TIME SERIES REGRESSION MODELS

Abstract. Hall (Testing for a unit root in the presence of moving average errors. Biometrika 76 (1989), 49–56; Joint hypothesis tests for a random walk based on instrumental variable estimators. J. Time Ser. Anal. 13 (1992), 29–45), Pantula and Hall (Testing for unit roots in autoregressive moving average models:an instrumental variable approach. J. Econometrics 48 (1991), 325–53) and Lee and Schmidt (Unit root tests based on instrumental variable estimation. Int. Econ. Rev. 39 (1994), 449–62) proposed instrumental variable (IV) based tests for a unit root in an ARMA(p+ 1, q) time series. To perform the tests it is essentially necessary to know (p, q) but in many cases this information is unknown. In practice a natural solution to this problem is to estimate (p, q) from the data using a strategy based on the residual autocovariances from the IV regression. In this paper we examine the properties of these residual autocovariances under various assumptions about the true nature of the time series. This analysis allows us to propose a model selection procedure which has desirable asymptotic and finite sample properties whether the time series is stationary or possesses a unit root. A sideproduct of our analysis is that we extend Box and Pierce's (Distribution of residual autocorrelations in autoregressive integrated moving average time series models. J. Am. Statist. Assoc. 65 (1970), 1509–26) analysis of the least squares residual autocorrelations to the residual autocovariances from IV regressions.     

ASYMPTOTIC PROPERTIES OF SOME PRELIMINARY ESTIMATORS FOR AUTOREGRESSIVE MOVING AVERAGE TIME SERIES MODELS

Abstract. Some simple preliminary estimators for the coefficients of mixed autoregressive moving average time series models are considered. As the first step the estimators require the fitting of a long autoregression to the data. The first two methods of the paper are non-iterative and generally inefficient. The estimators are Yule-Walker type modifications of the least squares estimators of the coefficients in auxiliary linear regression models derived, respectively, for the coefficients of the long autoregression and for the coefficients of the corresponding long moving average approximation of the model. Both of these estimators are shown to be strongly consistent and their asymptotic distributions are derived. The asymptotic distributions are used in studying the loss in efficiency and in constructing the third estimator of the paper which is an asymptotically efficient two-step estimator. A numerical illustration of the third estimator with real data is given.     

A GENERALIZED BLOCK BOOTSTRAP FOR SEASONAL TIME SERIES

When time‐series data contain a periodic/seasonal component, the usual block bootstrap procedures are not directly applicable. We propose a modification of the block bootstrap – the generalized seasonal block bootstrap (GSBB) – and show its asymptotic consistency without undue restrictions on the relative size of the period and block size. Notably, it is exactly such restrictions that limit the applicability of other proposals of block bootstrap methods for time series with periodicities. The finite‐sample performance of the GSBB is also illustrated by means of a small simulation experiment.     

A TEST FOR NON-LINEARITY OF PREDICTION IN TIME SERIES

Abstract. A test for non-linearity of prediction in time series is suggested; the test makes use of the partial correlation between a series and its square, eliminating linear terms. The distribution of partial correlation estimates is considered both theoretically and by simulation. Extensions of the test to include partial correlations with other nonlinear functions of the observed data are suggested. The nature of optimum predictors for some particular non-linear time series is described, and the performance of the test when used with simulated non-linear series is investigated.     

A DIAGNOSTIC TEST FOR NONLINEAR SERIAL DEPENDENCE IN TIME SERIES FITTING ERRORS

Abstract. Time series analysts have begun to consider the applicability of nonlinear models. In order for nonlinear models to be accepted by practitioners, practicai tests must be avilable to test for the presence of nonlinearity in both raw time series and in the residuals from fitted models. A diagnostic test, based on the bispectrum, for the presence of nonlinear serial dependence in these time series is investigated here using artificial data. Detection of such nonlinear dependence is taken to indicate that nonlinear modelling methods are necessary. The theory behind the test is reviewed and simulations driven by pseudorandom numbers are presented for a variety of models and sample sizes. The simulations indicate that the test has substantial power for many models. In addition, theoretical and empirical results are presented which show that the bispectral diagnostic test is equally powerful for both the source series and for the fitting errors from a line& model. Thus, while the test is suitable for use as a diagnostic test on the fitting errors of linear time series models, prior linear modeling of the time series is not required.     

ESTIMATION FOR NON-LINEAR TIME SERIES MODELS USING ESTIMATING EQUATIONS

Godambe's (1985) theorem on optimal estimating equations for stochastic processes is applied to non-linear time series estimation problems. Examples are considered from the usual classes of non-linear time series models. A recursive estimation procedure based on optimal estimating equations is provided. It is also shown that pre-filtered estimates can be used to obtain the optimal estimate from a non-linear state-space model.     

Recursive Prediction and Likelihood Evaluation for Periodic ARMA Models

This paper explores recursive prediction and likelihood evaluation techniques for periodic autoregressive moving-average (PARMA) time series models. The innovations algorithm is used to develop a simple recursive scheme for computing one-step-ahead predictors and their mean squared errors. The asymptotic form of this recursion is explored. The prediction results are then used to develop an efficient (and exact) PARMA likelihood evaluation algorithm for Gaussian series. We then show how a multivariate autoregressive moving average (ARMA) likelihood can be evaluated by writing the multivariate ARMA model in PARMA form. Explicit calculations for PARMA(1, 1) models and periodic autoregressions are included.     

Prediction Interval for Autoregressive Time Series via Oracally Efficient Estimation of Multi‐Step‐Ahead Innovation Distribution Function

A kernel distribution estimator (KDE) is proposed for multi‐step‐ahead prediction error distribution of autoregressive time series, based on prediction residuals. Under general assumptions, the KDE is proved to be oracally efficient as the infeasible KDE and the empirical cumulative distribution function (cdf) based on unobserved prediction errors. Quantile estimator is obtained from the oracally efficient KDE, and prediction interval for multi‐step‐ahead future observation is constructed using the estimated quantiles and shown to achieve asymptotically the nominal confidence levels. Simulation examples corroborate the asymptotic theory.     

Inference on Multivariate Heteroscedastic Time Varying Random Coefficient Models

In this article, we introduce the general setting of a multivariate time series autoregressive model with stochastic time‐varying coefficients and time‐varying conditional variance of the error process. This allows modelling VAR dynamics for non‐stationary time series and estimation of time‐varying parameter processes by the well‐known rolling regression estimation techniques. We establish consistency, convergence rates, and asymptotic normality for kernel estimators of the paths of coefficient processes and provide pointwise valid standard errors. The method is applied to a popular seven‐variable dataset to analyse evidence of time variation in empirical objects of interest for the DSGE (dynamic stochastic general equilibrium) literature.