A Benchmarking Approach to Forecast Combination

This article is concerned with the development of a statistical model-based approach to optimally combine forecasts derived from an extrapolative model, such as an autoregressive integrated moving average (ARIMA) time series model, with forecasts of a particular characteristic of the same series obtained from independent sources. The methods derived combine the strengths of all forecasting approaches considered in the combination scheme. The implications of the general theory are investigated in the context of some commonly encountered seasonal ARIMA models. An empirical example to illustrate the method is included.     

A recursive approach to parameter estimation in regression and time series models

In this paper we discuss the recursive (or on line) estimation in (i) regression and (ii) autoregressive integrated moving average (ARIMA) time series models. The adopted approach uses Kalman filtering techniques to calculate estimates recursively. This approach is used for the estimation of constant as well as time varying parameters. In the first section of the paper we consider the linear regression model. We discuss recursive estimation both for constant and time varying parameters. For constant parameters, Kalman filtering specializes to recursive least squares. In general, we allow the parameters to vary according to an autoregressive integrated moving average process and update the parameter estimates recursively. Since the stochastic model for the parameter changes will "be rarely known, simplifying assumptions have to be made. In particular we assume a random walk model for the time varying parameters and show how to determine whether the parameters are changing over time. This is illustrated with an example.     

Quantile Regression for Location‐Scale Time Series Models with Conditional Heteroscedasticity

This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean‐variance models are reinterpreted as conditional location‐scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location‐scale time series models under mild conditions. In the application of this result to ARMA‐generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.     

Effect of systematic sampling on arima models

Given a general homogeneous non-stationary autoregressive integrated moving average process ARIMA(p,d,q), the corresponding model for the subseries obtained by a systematic sampling is derived. The article then shows that the sampled subseries approaches approximately to an integrated moving average process IMA(d,l), l≤(d-l), regardless of the autoregressive and moving average structures in the original series. In particular, the sampled subseries from an ARIMA (p,l,q) process approaches approximately to a simple random walk model.     

Threshold Vector Arma Models

In this article, we propose the threshold vector autoregressive moving average model (TVARMA). It is a multivariate nonlinear time series model characterized by two or more regimes that follow a vector ARMA structure and where the switching among them is regulated by a latent variable. The TVARMA model represents a generalization of some nonlinear models proposed in the literature and shows interesting features that are explored. The condition for the strong and weak stationarity of the TVARMA model are presented and the moments up to order two of the process are derived.     

Sampled autocovariance and autocorrelation results for linear time processes

We derive an exact formula for the covariance between the sampled autocovariances at any two lags for a finite time series realisation from a general stationary autoregressive moving average process. We indicate, through one particular example, how this result can be used to deduce analogous formulae for any nonstationary model of the ARUMA class, a generalisation of the ARIMA models. Such formulae then allow us to obtain approximate expressions for the convariances between all pairs of serial correlations for finite realisations from the ARUMA model. We also note that, in the limit as the series length n → ∞, our results for the ARMA class retrieve those of Bartlett (1946). Finally, we investigate an improvement to the approximation that is obtained by applying Bartlett's general asymptotic formula to finite series realisations. That such an improvement should exist can immediately be seen by consideration of out results for the simplest case of a white noise process. However, we deduce the final improved approapproximation, for general models, in two ways - from (corrected) results due to Davies and Newbold (1980), and by an alternative approach to theirs.     

Sampling the Future: A Bayesian Approach to Forecasting From Univariate Time Series Models

The Box–Jenkins methodology for modeling and forecasting from univariate time series models has long been considered a standard to which other forecasting techniques have been compared. To a Bayesian statistician, however, the method lacks an important facet—a provision for modeling uncertainty about parameter estimates. We present a technique called sampling the future for including this feature in both the estimation and forecasting stages. Although it is relatively easy to use Bayesian methods to estimate the parameters in an autoregressive integrated moving average (ARIMA) model, there are severe difficulties in producing forecasts from such a model. The multiperiod predictive density does not have a convenient closed form, so approximations are needed. In this article, exact Bayesian forecasting is approximated by simulating the joint predictive distribution. First, parameter sets are randomly generated from the joint posterior distribution. These are then used to simulate future paths of the time series. This bundle of many possible realizations is used to project the future in several ways. Highest probability forecast regions are formed and portrayed with computer graphics. The predictive density's shape is explored. Finally, we discuss a method that allows the analyst to subjectively modify the posterior distribution on the parameters and produce alternate forecasts.     

An optimal k of kth MA-ARIMA models under AR(p) models

In this article, we discuss finding the optimal k of (i) kth simple moving average, (ii) kth weighted moving average, and (iii) kth exponential weighted moving average based on simulated autoregressive AR(p) model. We run a simulation using the three above examining method under specific conditions. The main finding is that the optimal k = 4 and then k = 3. Especially, the fourth WMA ARIMA model, fourth EWMA ARIMA model, and third EWMA ARIMA model are the best forecasting models among others, respectively. For all the six real data reveal the similar results of simulation study.     

Bayesian inference and forecasting in the stationary bilinear model

A stationary bilinear (SB) model can be used to describe processes with a time-varying degree of persistence that depends on past shocks. This study develops methods for Bayesian inference, model comparison, and forecasting in the SB model. Using monthly U.K. inflation data, we find that the SB model outperforms the random walk, first-order autoregressive AR(1), and autoregressive moving average ARMA(1,1) models in terms of root mean squared forecast errors. In addition, the SB model is superior to these three models in terms of predictive likelihood for the majority of forecast observations.     

Forecasting Study of Shanghai’s and Shenzhen’s Stock Markets Using a Hybrid Forecast Method

In this article, a novel hybrid method to forecast stock price is proposed. This hybrid method is based on wavelet transform, wavelet denoising, linear models (autoregressive integrated moving average (ARIMA) model and exponential smoothing (ES) model), and nonlinear models (BP Neural Network and RBF Neural Network). The wavelet transform provides a set of better-behaved constitutive series than stock series for prediction. Wavelet denoising is used to eliminate some slight random fluctuations of stock series. ARIMA model and ES model are used to forecast the linear component of denoised stock series, and then BP Neural Network and RBF Neural Network are developed as tools for nonlinear pattern recognition to correct the estimation error of the prediction of linear models. The proposed method is examined in the stock market of Shanghai and Shenzhen and the results are compared with some of the most recent stock price forecast methods. The results show that the proposed hybrid method can provide a considerable improvement for the forecasting accuracy. Meanwhile, this proposed method can also be applied to analysis and forecast reliability of products or systems and improve the accuracy of reliability engineering.     

Semiparametric Time Series Models with Log‐concave Innovations: Maximum Likelihood Estimation and its Consistency

We study semiparametric time series models with innovations following a log‐concave distribution. We propose a general maximum likelihood framework that allows us to estimate simultaneously the parameters of the model and the density of the innovations. This framework can be easily adapted to many well‐known models, including autoregressive moving average (ARMA), generalized autoregressive conditionally heteroscedastic (GARCH), and ARMA‐GARCH models. Furthermore, we show that the estimator under our new framework is consistent in both ARMA and ARMA‐GARCH settings. We demonstrate its finite sample performance via a thorough simulation study and apply it to model the daily log‐return of the FTSE 100 index.     

Short-term forecasting with a computationally efficient nonparametric transfer function model

In this paper a semi-parametric approach is developed to model non-linear relationships in time series data using polynomial splines. Polynomial splines require very little assumption about the functional form of the underlying relationship, so they are very flexible and can be used to model highly non-linear relationships. Polynomial splines are also computationally very efficient. The serial correlation in the data is accounted for by modelling the noise as an autoregressive integrated moving average (ARIMA) process, by doing so, the efficiency in nonparametric estimation is improved and correct inferences can be obtained. The explicit structure of the ARIMA model allows the correlation information to be used to improve forecasting performance. An algorithm is developed to automatically select and estimate the polynomial spline model and the ARIMA model through backfitting. This method is applied on a real-life data set to forecast hourly electricity usage. The non-linear effect of temperature on hourly electricity usage is allowed to be different at different hours of the day and days of the week. The forecasting performance of the developed method is evaluated in post-sample forecasting and compared with several well-accepted models. The results show the performance of the proposed model is comparable with a long short-term memory deep learning model.     

Forecasting Economic Time Series With Structural and Box-Jenkins Models: A Case Study

The basic structural model is a univariate time series model consisting of a slowly changing trend component, a slowly changing seasonal component, and a random irregular component. It is part of a class of models that have a number of advantages over the seasonal ARIMA models adopted by Box and Jenkins (1976). This article reports the results of an exercise in which the basic structural model was estimated for six U.K. macroeconomic time series and the forecasting performance compared with that of ARIMA models previously fitted by Prothero and Wallis (1976).     

Parameter Estimation Robust to Low-Frequency Contamination

We provide methods to robustly estimate the parameters of stationary ergodic short-memory time series models in the potential presence of additive low-frequency contamination. The types of contamination covered include level shifts (changes in mean) and monotone or smooth time trends, both of which have been shown to bias parameter estimates toward regions of persistence in a variety of contexts. The estimators presented here minimize trimmed frequency domain quasi-maximum likelihood (FDQML) objective functions without requiring specification of the low-frequency contaminating component. When proper sample size-dependent trimmings are used, the FDQML estimators are consistent and asymptotically normal, asymptotically eliminating the presence of any spurious persistence. These asymptotic results also hold in the absence of additive low-frequency contamination, enabling the practitioner to robustly estimate model parameters without prior knowledge of whether contamination is present. Popular time series models that fit into the framework of this article include autoregressive moving average (ARMA), stochastic volatility, generalized autoregressive conditional heteroscedasticity (GARCH), and autoregressive conditional heteroscedasticity (ARCH) models. We explore the finite sample properties of the trimmed FDQML estimators of the parameters of some of these models, providing practical guidance on trimming choice. Empirical estimation results suggest that a large portion of the apparent persistence in certain volatility time series may indeed be spurious. Supplementary materials for this article are available online.     

Predictive Accuracy Gain From Disaggregate Sampling in ARIMA Models

We compare the forecast accuracy of autoregressive integrated moving average (ARIMA) models based on data observed with high and low frequency, respectively. We discuss how, for instance, a quarterly model can be used to predict one quarter ahead even if only annual data are available, and we compare the variance of the prediction error in this case with the variance if quarterly observations were indeed available. Results on the expected information gain are presented for a number of ARIMA models including models that describe the seasonally adjusted gross national product (GNP) series in the Netherlands. Disaggregation from annual to quarterly GNP data has reduced the variance of short-run forecast errors considerably, but further disaggregation from quarterly to monthly data is found to hardly improve the accuracy of monthly forecasts.     

A prediction approach for precise marketing based on ARIMA-ARCH Model: A case of China Mobile

The autoregressive integrated moving average (ARIMA) model presents improved performance in forecasting short-term trends because it considers the dependence of time series and the interference of stochastic volatility. Thus, in this study, we establish ARIMA(0, 2, 1) based on the historical data of large-scale online marketing promotions to realize precise marketing of China Mobile's Ling Xi Voice app in the communication market. We eliminate the auto-regression effect of residual series by establishing the ARIMA model combined with the autoregressive conditional heteroskedasticity (ARCH) model denoted as ARIMA(0, 2, 1) ? ARCH(1), the ARIMA model combined with the generalized ARCH (GARCH) model denoted as ARIMA(0, 2, 1) ? GARCH(1, 1), and the ARIMA model combined with the threshold GARCH model denoted as ARIMA(0, 2, 1) ? TGARCH(2, 1). The performance of the aforementioned models is then compared for validation. Considering the characteristics of the communication markets and the attractive statistical properties of ARIMA, we apply ARIMA(0, 2, 1) to forecast the cumulative number of Ling Xi Voice app users for precise marketing that offers reliable agreement for China Mobile to further advertise and study the market demand. Our analysis contributes toward the development of the current knowledge on forecasting the number of app users in the communication market and provides a new idea to increase the market share for communication operators.     

Multivariate time series prediction using a hybridization of VARMA models and Bayesian networks

In this paper, a new hybrid model of vector autoregressive moving average (VARMA) models and Bayesian networks is proposed to improve the forecasting performance of multivariate time series. In the proposed model, the VARMA model, which is a popular linear model in time series forecasting, is specified to capture the linear characteristics. Then the errors of the VARMA model are clustered into some trends by K-means algorithm with Krzanowski–Lai cluster validity index determining the number of trends, and a Bayesian network is built to learn the relationship between the data and the trend of its corresponding VARMA error. Finally, the estimated values of the VARMA model are compensated by the probabilities of their corresponding VARMA errors belonging to each trend, which are obtained from the Bayesian network. Compared with VARMA models, the experimental results with a simulation study and two multivariate real-world data sets indicate that the proposed model can effectively improve the prediction performance.     

Bootstrapping moving average models

Summary In recent years, the bootstrap method has been extended to time series analysis where the observations are serially correlated. Contributions have focused on the autoregressive model producing alternative resampling procedures. In contrast, apart from some empirical applications, very little attention has been paid to the possibility of extending the use of the bootstrap method to pure moving average (MA) or mixed ARMA models. In this paper, we present a new bootstrap procedure which can be applied to assess the distributional properties of the moving average parameters estimates obtained by a least square approach. We discuss the methodology and the limits of its usage. Finally, the performance of the bootstrap approach is compared with that of the competing alternative given by the Monte Carlo simulation. Research partially supported by CNR and MURST.     

Bridge Estimation for Linear Regression Models with Mixing Properties

Penalized regression methods have for quite some time been a popular choice for addressing challenges in high dimensional data analysis. Despite their popularity, their application to time series data has been limited. This paper concerns bridge penalized methods in a linear regression time series model. We first prove consistency, sparsity and asymptotic normality of bridge estimators under a general mixing model. Next, as a special case of mixing errors, we consider bridge regression with autoregressive and moving average (ARMA) error models and develop a computational algorithm that can simultaneously select important predictors and the orders of ARMA models. Simulated and real data examples demonstrate the effective performance of the proposed algorithm and the improvement over ordinary bridge regression.     

Premia in Forward Foreign Exchange as Unobserved Components: A Note

We reconsider the signal-extraction approach to measuring premia in the pricing of forward foreign exchange, put forward by Wolff, in which the difference between the forward rate and the associated future spot rate is modeled as an autoregressive moving average (ARMA) model for the risk premium buried in a white-noise forecast error. We point out that an ARMA model for the risk premium is not always identifiable from information on the difference between the forward rate and the future spot rate only. We present solutions to the problem of identification and show how the model for the risk premium can be estimated in a direct way, provided that the identification problem is solved. For reason of comparison, we use the series analyzed by Wolff to estimate the models for risk premia. The results confirm the earlier finding that premia in forward exchange exhibit a certain degree of persistence over time.