Advances in Time Series Methods and Applications

Ian McLeod’s contributions to time series are both broad and influential. His work has put Canada and the University of Western Ontario on the map in the time series community. This article strives to give a partial picture of McLeod’s diverse contributions and their impact by reviewing the development of portmanteau statistics, long memory (persistence) models, the concept of duality in McLeod’s work, and his contributions to intervention analysis.

The LASSO (Tibshirani, J R Stat Soc Ser B 58(1):267–288, 1996, [30]) and the adaptive LASSO (Zou, J Am Stat Assoc 101:1418–1429, 2006, [37]) are popular in regression analysis for their advantage of simultaneous variable selection and parameter estimation, and also have been applied to autoregressive time series models. We propose the doubly adaptive LASSO (daLASSO), or PLAC-weighted adaptive LASSO, for modelling stationary vector autoregressive processes. The procedure is doubly adaptive in the sense that its adaptive weights are formulated as functions of the norms of the partial lag autocorrelation matrix function (Heyse, 1985, [17]) and Yule–Walker or ordinary least squares estimates of a vector time series. The existing papers ignore the partial lag autocorrelation information inherent in a VAR process. The procedure shows promising results for VAR models. The procedure excels in terms of VAR lag order identification.

There has been an increasing interest recently in the analysis of financial data that arrives at irregular intervals. An important class of models is the autoregressive Conditional Duration (ACD) model introduced by Engle and Russell (Econometrica 66:1127–1162, 1998, [22]) and its various generalizations. These models have been used to describe duration clustering for financial data such as the arrival times of trades and price changes. However, relatively few evaluation procedures for the adequacy of ACD models are currently available in the literature. Given its simplicity, a commonly used diagnostic test is the Box-Pierce/Ljung-Box statistic adapted to the estimated standardized residuals of ACD models, but its asymptotic distribution is not the standard one due to parameter estimation uncertainty. In this paper we propose a test for duration clustering and a test for the adequacy of ACD models using wavelet methods. The first test exploits the one-sided nature of duration clustering. An ACD process is positively autocorrelated at all lags, resulting in a spectral mode at frequency zero. In particular, it has a spectral peak at zero when duration clustering is persistent or when duration clustering is small at each individual lag but carries over a long distributional lag. As a joint time-frequency decomposition method, wavelets can effectively capture spectral peaks and thus are expected to be powerful. Our second test checks the adequacy of an ACD model by using a wavelet-based spectral density of the estimated standardized residuals over the whole frequency. Unlike the Box-Pierce/Ljung-Box tests, the proposed diagnostic test has a convenient asymptotic “nuisance parameter-free” property—parameter estimation uncertainty has no impact on the asymptotic distribution of the test statistic. Moreover, it can check a wide range of alternatives and is powerful when the spectrum of the standardized duration residuals is nonsmooth, as can arise from neglected persistent duration clustering, seasonality, calender effects and business cycles. For each of the two new tests, we propose and justify a suitable data-driven method to choose the finest scale—the smoothing parameter in wavelet estimation. This makes the methods fully operational in practice. We present a simulation study, illustrating the merits of the wavelet-based procedures. An application with tick-by-tick trading data of Alcoa stock is presented.

We derive the asymptotic distribution of residual autocorrelations for the Weibull autoregressive conditional duration (ACD) model, and this leads to a portmanteau test for the adequacy of the fitted Weibull ACD model. The finite-sample performance of this test is evaluated by simulation experiments and a real data example is also reported.

This paper studies the residual autocorrelation functions (ACFs) of partially nonstationary multivariate autoregressive moving-average (ARMA) models. The limiting distributions of the full rank estimators and the Gaussian reduced rank estimators are derived. Using these results, we derive the limiting distributions of the residual ACFs under full rank and reduce rank estimations. Based on these limiting distributions, we construct the portmanteau statistics for model checking. It is shown that these statistics asymptotically follow \(\chi ^2\)-distributions. Simulations are carried out to assess their performances in finite samples and two real examples are given.

In this article, we investigate the portmanteau tests and the Lagrange multiplier (LM) test for goodness of fit in autoregressive and moving average models with uncorrelated errors. Under the assumption that the error is not independent, the classical portmanteau tests and LM test are asymptotically distributed as a weighted sum of chi-squared random variables that can be far from the chi-squared distribution. To conduct the tests, we must estimate these weights using nonparametric methods. Therefore, by employing the method of Kiefer et al. (Econometrica, 68:695–714, 2000, [11]), we propose new test statistics for the portmanteau tests and the LM test. The asymptotic null distribution of these test statistics is not standard, but can be tabulated by means of simulations. In finite-sample simulations, we demonstrate that our proposed test has a good ability to control the type I error, and that the loss of power is not substantial.

We propose generalized \(C(\alpha )\) tests for testing linear and nonlinear parameter restrictions in models specified by estimating functions. The proposed procedures allow for general forms of serial dependence and heteroskedasticity, and can be implemented using any root-n consistent restricted estimator. The asymptotic distribution of the proposed statistic is established under weak regularity conditions. We show that earlier \(C(\alpha )\)-type statistics are included as special cases. The problem of testing hypotheses fixing a subvector of the complete parameter vector is discussed in detail as another special case. We also show that such tests provide a simple general solution to the problem of accounting for estimated parameters in the context of two-step procedures where a subvector of model parameters is estimated in a first step and then treated as fixed.

Regression analysis for multinomial/categorical time series is not adequately discussed in the literature. Furthermore, when categories of a multinomial response at a given time are ordinal, the regression analysis for such ordinal categorical time series becomes more complex. In this paper, we first develop a lag 1 transitional logit probabilities based correlation model for the multinomial responses recorded over time. This model is referred to as a multinomial dynamic logits (MDL) model. To accommodate the ordinal nature of the responses we then compute the binary distributions for the cumulative transitional responses with cumulative logits as the binary probabilities. These binary distributions are next used to construct a pseudo likelihood function for inferences for the repeated ordinal multinomial data. More specifically, for the purpose of model fitting, the likelihood estimation is developed for the regression and dynamic dependence parameters involved in the MDL model.

Due to the lack of a suitable modeling procedure and the difficulty to identify the threshold variable and estimate the threshold values, the threshold autoregressive moving average (TARMA) model with multi-regime has not attracted much attention in application. Therefore, the chief goal of our paper is to propose a simple and yet widely applicable modeling procedure for multi-regime TARMA models. Under no threshold case, we utilize extended least squares estimate (ELSE) and linear arranged regression to obtain a test statistic \(\hat{F}\), which is proved to follow an approximate F distribution. And then, based on the statistic \(\hat{F}\), we employ some scatter plots to identify the number and locations of the potential thresholds. Finally, the procedures are considered to build a TARMA model by these statistics and the Akaike information criterion (AIC). Simulation experiments and the application to a real data example demonstrate that both the power of the test statistic and the model-building can work very well in the case of TARMA models.

Nonparametric statistical procedures are commonly used in analyzing for trend in water resources time series (Chapter 23, Hipel and McLeod in Time series modelling of water resources and environmental systems. Elsevier, New York, 2005 [10]). One popular procedure is the seasonal Mann–Kendall tau test for detecting monotonic trend in seasonal time series data with serial dependence (Hirsch and Slack in Water Resour Res 20(6):727–732, 1984 [12]). However there is little rigorous discussion in the literature about its validity and alternatives. In this paper, the asymptotic normality of a seasonal Mann–Kendall test is determined for a large family of absolutely regular processes, a bootstrap sampling version of this test is proposed and its performance is studied through simulation. These simulations compare the performance of the traditional test, the bootstrapped version referred to above, as well as a bootstrapped version of Spearman’s rho partial correlation. The simulation results indicate that both bootstrap tests perform comparably to the traditional test when the seasonal effect is deterministic, but the traditional test can fail to converge to the nominal levels when the seasonal effect is stochastic. Both bootstrapped tests perform similarly to each other in terms of accuracy and power.

A brief and accessible derivation of the asymptotic distribution of Pearson’s goodness-of-fit statistic is proposed. Additionally, a shifted gamma distribution is introduced as an accurate approximation to be utilized when the chi-squared distribution proves to be inadequate. It is also explained that the exact probability mass function of this test statistic can be readily determined from its moment-generating function via symbolic computations. Two illustrative numerical examples are included.

Many power crises have occurred in developing and developed countries such as through disruptions in transmission lines, excessive demand during heat waves, and regulatory failures. The 2011 Great Japan Earthquake caused one of most severe power crises ever recorded. This study measures the industry’s ability to conserve power without critically reducing production (“power saving rate”) as one of the indicator of resilience as a lesson of disaster. The quantification of the power saving rate leads to grasping the potential power reduction of industrial sector or production losses caused by the future incidents in many regions or countries. Using time series data sets of monthly industrial production and power consumption, this study investigates the power saving rate of Japanese industries during power shortages after the great earthquake. The results demonstrates the size of power saving rate right after the disaster, during the first severe peak demand season, as well as long-term continuous efforts of power saving in different business.

There is now considerable scientific consensus that the acknowledged increase in global temperatures is due to the increasing levels of atmospheric carbon dioxide arising from the burning of fossil fuels. Large scale global circulation models support this consensus and there have also been statistical studies which relate the trend in temperatures to the carbon dioxide increase. However, causal dependence of one trending series upon another cannot be readily proved using statistical means. In this paper we model the trend corrected series by times series methods which provide a plausible representation of their dependence. A consequence of trend correction and our use of relatively short series is that our model is unable to give precise long-term predictions, but it does illuminate the relationships and interaction between the series.

Although clean and sustainable wind energy has long been recognized as one of the most attractive electric power sources, generation of wind power is still much easier than its integration into liberalized electricity markets. One of the key obstacles on the way of wider implementation of wind energy is its highly volatile and intermittent nature. This has boosted an interest in developing a fully probabilistic forecast of wind speed, aiming to assess a variety of related uncertainties. Nonetheless, most of the available methodology for constructing a future predictive density for wind speed are based on parametric distributional assumptions on the observed wind data, and such conditions are often too restrictive and infeasible in practice. In this paper we propose a new nonparametric data-driven approach to probabilistic wind speed forecasting, adaptively combining sieve bootstrap and regime switching models. Our new bootstrapped regime switching (BRS) model delivers highly competitive, sharp and calibrated ensembles of wind speed forecasts, governed by various states of wind direction, and imposes minimal requirements on the observed wind data. The proposed methodology is illustrated by developing probabilistic wind speed forecasts for a site in the Washington State, USA.