Estimation and prediction under structural instability: The case of the U.S. pulp and paper market

The objectives of this paper are: first, to show empirically the relevance of using adaptive estimation techniques over more traditional estimation approaches when economic systems are believed to be structurally unstable over time; and secondly, to compare in an empirical framework two adaptive estimation techniques: Kalman filtering and the Carbone–Longini filter. For that purpose, an econometric model for the U.S. pulp and paper market is examined under the assumption of structural instability and, hence, constitutes the basis for comparing forecasting performances and estimation accuracy achieved by each technique. A version of Kalman filtering, modified in line with the basic idea of ‘tracking’ characterizing the Carbone–Longini filter, is also presented and applied. The analysis of the results shows that it may be worth using adapative estimation methods to estimate structurally unstable models, even if there is no prior knowledge about the patterns of variation of the parameters. Also, it shows the Carbone–Longini filter and Kalman filtering as being complementary estimation techniques. An estimation/forecasting methodology involving a sequential application mode of these two techniques is suggested.     

Updating the forecast function of ARIMA models and the link with DLMs

This paper shows that the whole forecast function of ARIMA time series models, and not just the eventual forecast function, may be updated each time an observation is received. The paper also shows that the coefficients in the updating equations for the forecast function may be expressed in exactly the same form as the Kalman filter updating equations for canonical time series DLMs. Moreover, the adaptive factors in the updating equations are shown to be a simple function of the ARIMA model parameters. Copyright © 1999 John Wiley & Sons, Ltd.     

Seasonal adjustment and kalman filtering: Extension to periodic variances

This paper reviews the relations between the methods of seasonal adjustment used by official statistical agencies and the ‘model-based’ methods that postulate explicit stochastic models for the unobserved components of a time series and apply optimal signal extraction theory to obtain a seasonally adjusted series. The Kalman filter implementation of the model-based methods is described and some recent results on its properties are reviewed. The model-based methods employ homogeneous or time-invariant models that assume in particular that the autocovariance structure does not vary with the season. Relaxing this leads to the class of models known as periodic models, and an example of a seasonally heterosceclastic unobserved-components ARIMA (SHUCARIMA) model is presented. The calculation of the standard error of a seasonally adjusted series via the Kalman filter is extended to this periodic model and illustrated for a monthly rainfall series.     

Convergence and the Constant Dynamic Linear Model

It is well known that, as calculated using the Kalman filter recurrence relationships, the posterior parameter variance and the adaptive vector of observable constant dynamic linear models converge to limiting values. However, most proofs are tortuous, some have subtle errors and some relate only to specific cases. An elegant probabilistic convergence proof demonstrates that the limit is independent of the initial parametric prior. The result is extended to a class of multivariate dynamic linear models. Finally the proof is shown to apply to many non-observable constant DLMs. © 1997 John Wiley & Sons, Ltd.     

A non‐Gaussian generalization of the Airline model for robust seasonal adjustment

In their seminal book Time Series Analysis: Forecasting and Control, Box and Jenkins (1976) introduce the Airline model, which is still routinely used for the modelling of economic seasonal time series. The Airline model is for a differenced time series (in levels and seasons) and constitutes a linear moving average of lagged Gaussian disturbances which depends on two coefficients and a fixed variance. In this paper a novel approach to seasonal adjustment is developed that is based on the Airline model and that accounts for outliers and breaks in time series. For this purpose we consider the canonical representation of the Airline model. It takes the model as a sum of trend, seasonal and irregular (unobserved) components which are uniquely identified as a result of the canonical decomposition. The resulting unobserved components time series model is extended by components that allow for outliers and breaks. When all components depend on Gaussian disturbances, the model can be cast in state space form and the Kalman filter can compute the exact log‐likelihood function. Related filtering and smoothing algorithms can be used to compute minimum mean squared error estimates of the unobserved components. However, the outlier and break components typically rely on heavy‐tailed densities such as the t or the mixture of normals. For this class of non‐Gaussian models, Monte Carlo simulation techniques will be used for estimation, signal extraction and seasonal adjustment. This robust approach to seasonal adjustment allows outliers to be accounted for, while keeping the underlying structures that are currently used to aid reporting of economic time series data. Copyright © 2006 John Wiley & Sons, Ltd.     

Forecasting ability of GARCH vs Kalman filter method: evidence from daily UK time-varying beta

This paper investigates the forecasting ability of four different GARCH models and the Kalman filter method. The four GARCH models applied are the bivariate GARCH, BEKK GARCH, GARCH-GJR and the GARCH-X model. The paper also compares the forecasting ability of the non-GARCH model: the Kalman method. Forecast errors based on 20 UK company daily stock return (based on estimated time-varying beta) forecasts are employed to evaluate out-of-sample forecasting ability of both GARCH models and Kalman method. Measures of forecast errors overwhelmingly support the Kalman filter approach. Among the GARCH models the GJR model appears to provide somewhat more accurate forecasts than the other bivariate GARCH models. Copyright © 2008 John Wiley & Sons, Ltd.     

Estimation and forecasting of long-memory processes with missing values

This paper addresses the issues of maximum likelihood estimation and forecasting of a long-memory time series with missing values. A state-space representation of the underlying long-memory process is proposed. By incorporating this representation with the Kalman filter, the proposed method allows not only for an efficient estimation of an ARFIMA model but also for the estimation of future values under the presence of missing data. This procedure is illustrated through an analysis of a foreign exchange data set. An investment scheme is developed which demonstrates the usefulness of the proposed approach. © 1997 John Wiley & Sons, Ltd.     

Monitoring for outliers and level shifts in kalman filter implementations of exponential smoothing

This paper presents a new application of a Kalman filter implementation of exponential smoothing with monitoring for outliers and level shifts. The assumption is that each observation comes from one of three models: steady, outlier, or level shift. This concept was introduced as a multiprocess model by Harrison and Stevens (1976). However, their handling of the models is different. In this paper four different model-selection criteria are introduced and compared by applying them to data. The new features of the application include the four model-selection criteria and the estimation of the required parameters by maximum likelihood.     

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.     

Nonlinearities in the CAPM: Evidence from Developed and Emerging Markets

This paper examines the forecasting ability of the nonlinear specifications of the market model. We propose a conditional two‐moment market model with a time‐varying systematic covariance (beta) risk in the form of a mean reverting process of the state‐space model via the Kalman filter algorithm. In addition, we account for the systematic component of co‐skewness and co‐kurtosis by considering higher moments. The analysis is implemented using data from the stock indices of several developed and emerging stock markets. The empirical findings favour the time‐varying market model approaches, which outperform linear model specifications both in terms of model fit and predictability. Precisely, higher moments are necessary for datasets that involve structural changes and/or market inefficiencies which are common in most of the emerging stock markets. Copyright © 2016 John Wiley & Sons, Ltd.     

Estimation under exact linear time-varying constraints,with an application to population projections

Estimation problems sometimes have inherent constraints which, when used, increase efficiency. When these constraints vary over time, the Kalman filter provides a convenient method of imposing them. This paper applies the Kalman filter to the problem of estimating state (provincial) populations given annual national population and national net arrivals, together with actual state populations in census years. An advantage with this approach is that the resulting projections can be evaluated by the provision of standard errors and the quality of one step ahead predictions. With our data the method seems to perform well for population projections, but poorly for net arrivals.     

Variance intervention

Variance intervention is a simple state-space approach to handling sharp discontinuities of level or slope in the states or parameters of models for non-stationary time-series. It derives from earlier procedures used in the 1960s for the design of self-adaptive, state variable feedback control systems. In the alternative state-space forecasting context considered in the present paper, it is particularly useful when applied to structural time series models. The paper compares the variance intervention procedure with the related ‘subjective intervention’ approach proposed by West and Harrison in a recent issue of the Journal of Forecasting, and demonstrates it efficacy by application to various time-series data, including those used by West and Harrison.     

A state space approach to forecasting the final vintage of revised data with an application to the index of industrial production

There is considerable interest in the index of industrial production (IIP) as an indicator of the state of the UK's industrial base and, more generally, as a leading economic indicator. However, this index, in common with a number of key macroeconomic time series, is subject to revision as more information becomes available. This raises the problem of forecasting the final vintage of data on IIP. We construct a state space model to solve this problem which incorporates bias adjustments, a model of the measurement error process, and a dynamic model for the final vintage of IIP. Application of the Kalman filter produces an optimal forecast of the final vintage of data.     

Forecasting Longevity Gains Using a Seemingly Unrelated Time Series Model

In this paper a multivariate time series model using the seemingly unrelated time series equation (SUTSE) framework is proposed to forecast longevity gains. The proposed model is represented in state space form and uses Kalman filtering to estimate the unobservable components and fixed parameters. We apply the model both to male mortality rates in Portugal and the USA. Our results compare favorably, in terms of mean absolute percentage error, in‐sample and out‐of‐sample, to those obtained by the Lee–Carter method and some of its extensions. Copyright © 2015 John Wiley & Sons, Ltd.     

Minimum variance prediction of bilinear time series: Direct and adaptive versions

Bilinear models of time series are considered. Minimum variance predictor for bilinear time series, homogeneous in the input and output, is proposed. Results of minimum variance prediction of bilinear time series are included. They are compared to the results of linear prediction of bilinear time series. A minimum variance prediction algorithm for bilinear time series of the general form is developed and an adaptive version of minimum variance algorithm is derived.     

Forecasting the Daily Time‐Varying Beta of European Banks During the Crisis Period: Comparison Between GARCH Models and the Kalman Filter

This intention of this paper is to empirically forecast the daily betas of a few European banks by means of four generalized autoregressive conditional heteroscedasticity (GARCH) models and the Kalman filter method during the pre‐global financial crisis period and the crisis period. The four GARCH models employed are BEKK GARCH, DCC GARCH, DCC‐MIDAS GARCH and Gaussian‐copula GARCH. The data consist of daily stock prices from 2001 to 2013 from two large banks each from Austria, Belgium, Greece, Holland, Ireland, Italy, Portugal and Spain. We apply the rolling forecasting method and the model confidence sets (MCS) to compare the daily forecasting ability of the five models during one month of the pre‐crisis (January 2007) and the crisis (January 2013) periods. Based on the MCS results, the BEKK proves the best model in the January 2007 period, and the Kalman filter overly outperforms the other models during the January 2013 period. Results have implications regarding the choice of model during different periods by practitioners and academics. Copyright © 2016 John Wiley & Sons, Ltd.     

A large Bayesian VAR with a block-specific shrinkage: A forecasting application for Italian industrial production

This paper proposes a Bayesian vector autoregression (BVAR) model with the Kalman filter to forecast the Italian industrial production index in a pseudo real-time experiment. Minnesota priors are adopted as a general framework, but a different shrinkage pattern is imposed for both the VAR coefficients and the Kalman gain, depending on the informative contribution of each variable investigated at frequency level. Both a time-varying and a constant selection for the shrinkage are proposed. Overall, the new BVAR models significantly improve the forecasting performance in comparison with the more traditional versions based on standard Minnesota priors with a single shrinkage, equal for all the variables, and selected on the basis of some optimal criteria. Very promising results come out in terms of density forecasting as well.     

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.     

Bayes Inference for Technological Substitution Data with Data-based Transformation

Bayesian inference via Gibbs sampling is studied for forecasting technological substitutions. The Box–Cox transformation is applied to the time series AR(1) data to enhance the linear model fit. We compute Bayes point and interval estimates for each of the parameters from the Gibbs sampler. The unknown parameters are the regression coefficients, the power in the Box–Cox transformation, the serial correlation coefficient, and the variance of the disturbance terms. In addition, we forecast the future technological substitution rate and its interval. Model validation and model choice issues are also addressed. Two numerical examples with real data sets are given.©1997 John Wiley & Sons, Ltd.     

A Bayesian analysis of periodic integration

Recent empirical research into the seasonal and trend properties of macroeconomic time series using periodic models has resulted in strong evidence in favour of periodic integration (PI). PI implies that the differencing filter necessary to remove a stochastic trend varies across seasons and, hence, that seasonal fluctuations are related to the stochastic trend. Previous studies finding evidence of PI have used classical econometric techniques. In this paper, we investigate the possible sensitivity of this empirical result by using Bayesian techniques. An application of posterior odds analysis and highest posterior density interval tests to several quarterly UK macroeconomic series suggests strong evidence for PI, even when we allow for structural breaks in the deterministic seasonals. A predictive exercise indicates that PI usually outperforms other competing models in terms of out-of-sample forecasting. © 1997 John Wiley & Sons, Ltd.