On estimation and prediction procedures for ar(1) models with power transformation

The power transformation of Box and Cox (1964) has been shown to be quite useful in short-term forecasting for the linear regression model with AR(1) dependence structure (see, for example, Lee and Lu, 1987, 1989). It is crucial to have good estimates of the power transformation and serial. correlation parameters, because they form the basis for estimating other parameters and predicting future observations. The prediction of future observations is the main focus of this paper. We propose to estimate these two parameters by minimizing the mean squared prediction errors. These estimates and the corresponding predictions compare favourably, via revs and simulated data, with those obtained by the maximum likelihood method. Similar results are also demonstrated in the repeated measurements setting.     

Deletion diagnostics for transformations of time series

Deletion diagnostics are derived for the effect of individual observations on the estimated transformation of a time series. The paper uses the modified power transformation of Box and Cox to provide a parametric family of transformations. Inference about the transformation parameter is made through regression on a constructed variable. The effect of deletion of observations on residuals and on the estimate of the regression parameter are obtained. Index plots of the diagnostic quantities are shown to be highly informative. Structural time series modelling is used, so that the results readily extend to inference about regression on other explanatory variables.     

Bayesian analysis of concurrent time series with application to regional IBM revenue data

Business data frequently arise in the form of concurrent time series. We present a general framework for simultaneous modeling and fitting of such series using the class of Box—Jenkins models. This framework is an exchangeable hierarchical Bayesian model incorporating dependence among the series. Our motivating data set consists of regional IBM revenue available monthly for several geographic regions. Stationary seasonal autoregressive models are simultaneously fit to the regional data series using various error covariance specifications for the strong interregional dependence. A modified Gibbs sampling algorithm is used to carry out the fitting and to enable all subsequent inference. Graphical techniques using predictive distributions are employed to assess model adequacy and to select among models. Outlier estimation and prediction under the chosen model are used for planning and to measure the effect of special promotional events.     

Identifying the time‐effect factors of multiple time series

The Peña–Box model is considered for finding the time‐effect factors of a multiple time series. This paper first establishes the connection between the Peña–Box model and the vector ARMA model. According to the Peña–Box model, some series can be ignored while modelling the vector ARMA model. A consistent estimator is then proposed to identify the model for nonlinear and nonstationary time series. Finally, the finite‐sample behaviour of the estimator is illustrated via simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

On Selecting a Power Transformation in Time-Series Analysis

The primary aim of this paper is to select an appropriate power transformation when we use ARMA models for a given time series. We propose a Bayesian procedure for estimating the power transformation as well as other parameters in time series models. The posterior distributions of interest are obtained utilizing the Gibbs sampler, a Markov Chain Monte Carlo (MCMC) method. The proposed methodology is illustrated with two real data sets. The performance of the proposed procedure is compared with other competing procedures. © 1997 John Wiley & Sons, Ltd.     

ARMA Models and the Box–Jenkins Methodology

The purpose of this paper is to apply the Box–Jenkins methodology to ARIMA models and determine the reasons why in empirical tests it is found that the post-sample forecasting the accuracy of such models is generally worse than much simpler time series methods. The paper concludes that the major problem is the way of making the series stationary in its mean (i.e. the method of differencing) that has been proposed by Box and Jenkins. If alternative approaches are utilized to remove and extrapolate the trend in the data, ARMA models outperform the models selected through Box–Jenkins methodology. In addition, it is shown that using ARMA models to seasonally adjusted data slightly improves post-sample accuracies while simplifying the use of ARMA models. It is also confirmed that transformations slightly improve post-sample forecasting accuracy, particularly for long forecasting horizons. Finally, it is demonstrated that AR(1), AR(2) and ARMA(1,1) models can produce more accurate post-sample forecasts than those found through the application of Box–Jenkins methodology.© 1997 John Wiley & Sons, Ltd.     

Forecasting output growth rates and median output growth rates: a hierarchical Bayesian approach

This paper describes procedures for forecasting countries' output growth rates and medians of a set of output growth rates using Hierarchical Bayesian (HB) models. The purpose of this paper is to show how the γ‐shrinkage forecast of Zellner and Hong ( 1989 ) emerges from a hierarchical Bayesian model and to describe how the Gibbs sampler can be used to fit this model to yield possibly improved output growth rate and median output growth rate forecasts. The procedures described in this paper offer two primary methodological contributions to previous work on this topic: (1) the weights associated with widely‐used shrinkage forecasts are determined endogenously, and (2) the posterior predictive density of the future median output growth rate is obtained numerically from which optimal point and interval forecasts are calculated. Using IMF data, we find that the HB median output growth rate forecasts outperform forecasts obtained from variety of benchmark models. Copyright © 2001 John Wiley & Sons, Ltd.     

A comparative arima analysis of the 111 series of the makridakis competition

The 111 series of the Makridakis competition are used to address a number of questions pertaining to use of the Box–Jenkins technique. The ARIMA models developed are compared to the ARIMA models developed independently by Andersen for the Makridakis competition. The time required to perform the analysis for each series is discussed in terms of model complexity. Forecast accuracy, measured as the MAPE for the one step ahead forecast, is discussed for different series lengths.     

Assessing inefficiency in the s&p 500 futures market

We analyse the price movement of the S&P 500 futures market for violations of the efficient market hypothesis on a short-term basis. To assess market inefficiency we construct a model and find that the returns, i.e. the difference in the logarithm of closing prices on consecutive days, exhibit the usual conditional heteroscedasticity behaviour typical of long series of financial data. To account for this non-linear behaviour we scale the returns by a volatility factor which depends on the daily high, low, and closing price. The rescaled series, which may be interpreted as the trend-countertrend component of the time series, is modelled using Box and Jenkins techniques. The resulting model is an ARMA(1,1). The scale factors are assumed to form a time series and are modelled using a semi-non-parametric method which avoids the restrictive assumptions of most ARCH or GARCH models. Using the combined model we perform 1000 simulations of market data, each simulation comprising 250 days (approximately one year). We then formulate a naive trading strategy which is based on the ratio of the one-day-ahead expected return to its one-day-ahead expected conditional standard deviation. The trading strategy has four adjustable parameters which are set to maximize profits for the simulation data. Next, we apply the trading strategy to one year of recent out-of-sample data. Our conclusion is that the S&P 500 futures market exhibits only slight inefficiencies, but that there exist, in principle, better trading strategies which take account of risk than the benchmark strategy of buy-and-hold. We have also constructed a linear model for the return series. Using the linear model, we have simulated returns and determined the optimum values for the adjustable parameters of the trading strategy. In this case, the optimum trading strategy is the same as the benchmark strategy, buy-and-hold. Finally, we have compared the profitability of the optimized trading strategy, based on the non-linear model, to three ad hoc trading strategies using the out-of-sample data. The three ad hoc strategies are more profitable than the optimized strategy.     

Combination of Forecasts across Estimation Windows: An Application to Air Travel Demand

This paper applies combining forecasts of air travel demand generated from the same model but over different estimation windows. The combination approach used resorts to Pesaran and Pick (Journal of Business Economics and Statistics 2011; 29 : 307–318), but the empirical application is extended in several ways. The forecasts are based on a seasonal Box–Jenkins model (SARIMA), which is adequate to forecast monthly air travel demand with distinct seasonal patterns at the largest German airport: Frankfurt am Main. Furthermore, forecasts with forecast horizons from 1 to 12 months ahead, which are based on different average estimation windows, expanding windows and single rolling windows, are compared with baseline forecasts based on an expanding window of the observations after a structural break. The forecast exercise shows that the average window forecasts mostly outperform the alternative single window forecasts. Copyright © 2016 John Wiley & Sons, Ltd.     

The accuracy of extrapolation methods; an automatic box–jenkins package sift

Hill and Woodworth (1980) proposed an algorithm suitable for identifying Box–Jenkins models automatically without reliance on the investigator. This paper first reviews the method. It is then used on the 111 series analysed by Anderson in the Makridakis forecasting competition. The results show that the automatic method of Hill and Woodworth is comparable in terms of accuracy to the full Box–Jenkins identification procedure.     

Forecasting high‐frequency financial data with the ARFIMA–ARCH model

Financial data series are often described as exhibiting two non‐standard time series features. First, variance often changes over time, with alternating phases of high and low volatility. Such behaviour is well captured by ARCH models. Second, long memory may cause a slower decay of the autocorrelation function than would be implied by ARMA models. Fractionally integrated models have been offered as explanations. Recently, the ARFIMA–ARCH model class has been suggested as a way of coping with both phenomena simultaneously. For estimation we implement the bias correction of Cox and Reid ( 1987 ). For daily data on the Swiss 1‐month Euromarket interest rate during the period 1986–1989, the ARFIMA–ARCH (5,d,2/4) model with non‐integer d is selected by AIC. Model‐based out‐of‐sample forecasts for the mean are better than predictions based on conditionally homoscedastic white noise only for longer horizons (τ > 40). Regarding volatility forecasts, however, the selected ARFIMA–ARCH models dominate. Copyright © 2001 John Wiley & Sons, Ltd.     

Forecasting with unbalanced panel data

This paper derives the best linear unbiased prediction (BLUP) for an unbalanced panel data model. Starting with a simple error component regression model with unbalanced panel data and random effects, it generalizes the BLUP derived by Taub (Journal of Econometrics, 1979, 10, 103–108) to unbalanced panels. Next it derives the BLUP for an unequally spaced panel data model with serial correlation of the AR(1) type in the remainder disturbances considered by Baltagi and Wu (Econometric Theory, 1999, 15, 814–823). This in turn extends the BLUP for a panel data model with AR(1) type remainder disturbances derived by Baltagi and Li (Journal of Forecasting, 1992, 11, 561–567) from the balanced to the unequally spaced panel data case. The derivations are easily implemented and reduce to tractable expressions using an extension of the Fuller and Battese (Journal of Econometrics, 1974, 2, 67–78) transformation from the balanced to the unbalanced panel data case.     

Transformations and dynamic linear models

The dynamic linear model (DLM) with additive Gaussian errors provides a useful statistical tool that is easily implemented because of the simplicity of updating a normal model that has a natural conjugate prior. If the model is not linear or if it does not have additive Gaussian errors, then numerical methods are usually required to update the distributions of the unknown parameters. If the dimension of the parameter space is small, numerical methods are feasible. However, as the number of unknown parameters increases, the numerial methods rapidly grow in complexity and cost. This article addresses the situation where a state dependent transformation of the observations follows the DLM, but a priori the appropriate transformation is not known. The Box-Cox family, which is indexed by a single parameter, illustrates the methodology. A prior distribution is constructed over a grid of points for the transformation parameter. For each value of the grid the relevant parameter esitmates and forecasts are obtained for the transformed series. These quantities are then integrated by the current distribution of the transformation parameter. When a new observation becomes available, parallel Kalman filters are used to update the distributions of the unknown parameters and to compute the likelihood of the transformation parameter at each grid point. The distribution of the transformation parameter is then updated.     

Temporal aggregation and systematic sampling in structural time-series models

Given a structural time-series model specified at a basic time interval, this paper deals with the problems of forecasting efficiency and estimation accuracy generated when the data are collected at a timing interval which is a multiple of the time unit chosen to build the basic model. Results are presented for the simplest structural models, the trend plus error models, under the assumption that the parameters of the model are known. It is shown that the gains in forecasting efficiency and estimation accuracy for having data at finer intervals are considerable for both stock and flow variables with only one exception. No gain in forecasting efficiency is achieved in the case of a stock series that follows a random walk.     

A modified sequential Monte Carlo procedure for the efficient recursive estimation of extreme quantiles

Many applications in science involve finding estimates of unobserved variables from observed data, by combining model predictions with observations. The sequential Monte Carlo (SMC) is a well‐established technique for estimating the distribution of unobserved variables that are conditional on current observations. While the SMC is very successful at estimating the first central moments, estimating the extreme quantiles of a distribution via the current SMC methods is computationally very expensive. The purpose of this paper is to develop a new framework using probability distortion. We use an SMC with distorted weights in order to make computationally efficient inferences about tail probabilities of future interest rates using the Cox–Ingersoll–Ross (CIR) model, as well as with an observed yield curve. We show that the proposed method yields acceptable estimates about tail quantiles at a fraction of the computational cost of the full Monte Carlo.     

Forecasting exchange rates using cointegration models and intra‐day data

We present a cointegration analysis on the triangle (USD–DEM, USD–JPY, DEM–JPY) of foreign exchange rates using intra‐day data. A vector autoregressive model is estimated and evaluated in terms of out‐of‐sample forecast accuracy measures. Its economic value is measured on the basis of trading strategies that account for transaction costs. We show that the typical seasonal volatility in high‐frequency data can be accounted for by transforming the underlying time scale. Results are presented for the original and the modified time scales. We find that utilizing the cointegration relation among the exchange rates and the time scale transformation improves forecasting results. Copyright © 2002 John Wiley & Sons, Ltd.     

A unified view of statistical forecasting procedures

A large number of statistical forecasting procedures for univariate time series have been proposed in the literature. These range from simple methods, such as the exponentially weighted moving average, to more complex procedures such as Box–Jenkins ARIMA modelling and Harrison–Stevens Bayesian forecasting. This paper sets out to show the relationship between these various procedures by adopting a framework in which a time series model is viewed in terms of trend, seasonal and irregular components. The framework is then extended to cover models with explanatory variables. From the technical point of view the Kalman filter plays an important role in allowing an integrated treatment of these topics.     

A semiparametric method for predicting bankruptcy

Bankruptcy prediction methods based on a semiparametric logit model are proposed for simple random (prospective) and case–control (choice‐based; retrospective) data. The unknown parameters and prediction probabilities in the model are estimated by the local likelihood approach, and the resulting estimators are analyzed through their asymptotic biases and variances. The semiparametric bankruptcy prediction methods using these two types of data are shown to be essentially equivalent. Thus our proposed prediction model can be directly applied to data sampled from the two important designs. One real data example and simulations confirm that our prediction method is more powerful than alternatives, in the sense of yielding smaller out‐of‐sample error rates. Copyright © 2007 John Wiley & Sons, Ltd.     

Prediction in the Random Effects Model with MA (q) Remainder Disturbances

This paper considers the problem of forecasting in a panel data model with random individual effects and MA (q) remainder disturbances. It utilizes a recursive transformation for the MA (q) process derived by Baltagi and Li (Econometric Theory 1994; 10 : 396–408) which yields a simple generalized least‐squares estimator for this model. This recursive transformation is used in conjunction with Goldberger's result (Journal of the American Statistical Association 1962; 57 : 369–375) to derive an analytic expression for the best linear unbiased predictor, for the ith cross‐sectional unit, s periods ahead. Copyright © 2011 John Wiley & Sons, Ltd.