Jan G. De Gooijer

Some recent developments in non-linear time series modelling, testing, and forecasting

Abstract Most of the recent work in time series analysis has been done on the assumption that the structure of the series can be described by linear time series models. However, there are occasions when subject-matter, theory or data suggest that linear models are unsatisfactory. In those cases it is desirable to look at non-linear alternatives. This paper gives an overview of the most recent developments in this area. Particular attention is paid to the strengths and weaknesses (advantages and disadvantages) of a large number of models and tests for non-linearity, focusing on ‘ready-to-use’ issues rather than discussing technical details. Various problems in forecasting from non-linear models are discussed. Some guidelines for practical non-linear time series modelling and forecasting are also included.

Dynamic factor analysis of nonstationary multivariate time series

A dynamic factor model is proposed for the analysis of multivariate nonstationary time series in the time domain. The nonstationarity in the series is represented by a linear time dependent mean function. This mild form of nonstationarity is often relevant in analyzing socio-economic time series met in practice. Through the use of an extended version of Molenaars stationary dynamic factor analysis method, the effect of nonstationarity on the latent factor series is incorporated in the dynamic nonstationary factor model (DNFM). It is shown that the estimation of the unknown parameters in this model can be easily carried out by reformulating the DNFM as a covariance structure model and adopting the ML algorithm proposed by Jöreskog. Furthermore, an empirical example is given to demonstrate the usefulness of the proposed DNFM and the analysis.

On Additive Conditional Quantiles With High-Dimensional Covariates

We investigate the estimation of the conditional quantile when many covariates are involved. In particular, we model the conditional quantile of a response as a nonlinear additive function of relevant covariates. For this setup, we propose a nonparametric smoother to estimate the unknown functions. The estimator provides direct computation of the nonlinear functions. Because it does not require any iteration, the estimator allows fast and routine data analysis. On the theoretical front, we also show asymptotic properties of the estimator, including mean squared error and limiting distribution. The theory confirms that for moderate dimension of the covariates, the estimator escapes the “curse of dimensionality” problem. Both simulated and real data examples are provided to illustrate the methodology.

On threshold moving-average models

In this paper the class of discrete self-exciting threshold moving-average (SETMA) models is studied in some detail. In particular, we consider various problems associated with the identification, estimation and testing of these models. A simple method for distinguishing between low order moving average (MA) and low order SETMA models is presented. Some simulation results illustrate the performance of the proposed method. We also derive a Lagrange multiplier (LM) test statistic for testing a linear MA model against a SETMA model. The small sample performance of the LM test is evaluated in a Monte Carlo study. A real example is used to illustrate the results.

25 Years of IIF Time Series Forecasting: A Selective Review

We review the past 25 years of time series research that has been published in journals managed by the International Institute of Forecasters (Journal of Forecasting 1982-1985; International Journal of Forecasting 1985-2005). During this period, over one third of all papers published in these journals concerned time series forecasting. We also review highly influential works on time series forecasting that have been published elsewhere during this period. Enormous progress has been made in many areas, but we find that there are a large number of topics in need of further development. We conclude with comments on possible future research directions in this field.

On forecasting SETAR processes

Suppose a time series {Yt} is generated by a first-order stationary self-exciting threshold autoregressive (SETAR) model with Gaussian innovations. The minimum mean squared error h-step ahead forecast for h> 2 involves a sequence of complicated numerical integrations and closed-form expressions are very difficult or even impossible to obtain. In this paper we derive explicit approximate expressions for E[Yt+hYs; s [less-than-or-equals, slant] t] and Var[Yt+hYs; s [less-than-or-equals, slant] t] (h> 2) for various SETAR models. The approximations are reasonably accurate as compared with alternative methods based on numerical integration and Monte Carlo experiments.

Exact moments of the sample autocorrelations from series generated by general arima processes of order (p, d, q), d=0 or 1

Abstract Formulae for the numerical computation of the first four exact moments of the sample autocorrelations, given a time series realisation from a general autoregressive moving average process of order (p, d, q) with d=0 or 1, are presented. The exact mean and variance of the sample autocorrelations are computed for various sample sizes and several time series models. The evaluated results are compared with those obtained from approximate formulae for the mean and variance of the sample autocorrelations. A specification of the numerical accuracy of the first two exact moments is included.

TARSO MODELING OF WATER TABLE DEPTHS

Threshold autoregressive self-exciting open-loop (TARSO) models are fitted to six time series of water table depths with precipitation excess as input variable. Basically, these models are nonlinear in structure because they incorporate several regimes which are separated by so-called thresholds. For each well a subset TARSO ((SS)TARSO) model is selected using a Bayes information criterion (BIC). (SS)TARSO models are used to simulate realizations of water table depths with lengths of 30 years, from which characteristics such as durations of exceedance are computed. The simulation performance of the fitted (SS)TARSO models is compared with results obtained from transfer function noise (TFN) models, dynamic regression (DR) models, and with a physical descriptive model, called SWATRE, extended with additional autoregressive moving average (ARMA) processes for the noise (SWATRE+ARMA). As compared to the linear TFN and DR models the (SS)TARSO models perform better because they incorporate several regimes. These regimes are the result of different soil layers or drainage levels. Furthermore, it is interesting that (SS)TARSO models show a good relative performance as compared to the SWATRE+ARMA models. A possible reason may be that inputs of SWATRE are uncertain.

Nonparametric conditional predictive regions for time series

Abstract Several nonparametric predictors based on the Nadaraya–Watson kernel regression estimator have been proposed in the literature. They include the conditional mean, the conditional median, and the conditional mode. In this paper, we consider three types of predictive regions for these predictors — the conditional percentile interval (CPI), the shortest conditional modal interval (SCMI), and the maximum conditional density region (MCDR). Further, we introduce a data-driven method for the choice of the optimal bandwidth. This method is based on the minimization of a cross-validation criterion given three different types of predictors. When the underlying conditional distribution is multi-modal, we show that the MCDR is much shorter in length than the CPI or SCMI irrespective of the type of predictor used. This point is illustrated using both a simulated and a real data set.

Forecasting exchange rates using TSMARS

In this article we use the Time Series Multivariate Adaptive Regression Splines (TSMARS) methodology to estimate and forecast non-linear structure in weekly exchange rates for four major currencies during the 1980s. The methodology is applied in three steps. First, univariate models are fitted to the data and the residuals are checked for outliers. Since significant outliers are spotted in all four currencies, the TSMARS methodology is reapplied in the second step with dummy variables representing the outliers. The empirical residuals of the models obtained in the second step pass the standard diagnostic tests for non-linearity, Gaussianity and randomness. Moreover, the estimated models can be sensibly interpreted from an economic standpoint. The out-of-sample forecasts generated by the TSMARS models are compared with those obtained from a pure random walk. We find that for two of the currencies, the models obtained using TSMARS provide forecasts which are superior to those of a random walk at all forecast horizons.