A non‐linear dynamic model for multiplicative seasonal‐trend decomposition

A non‐linear dynamic model is introduced for multiplicative seasonal time series that follows and extends the X‐11 paradigm where the observed time series is a product of trend, seasonal and irregular factors. A selection of standard seasonal and trend component models used in additive dynamic time series models are adapted for the multiplicative framework and a non‐linear filtering procedure is proposed. The results are illustrated and compared to X‐11 and log‐additive models using real data. In particular it is shown that the new procedures do not suffer from the trend bias present in log‐additive models. Copyright © 2002 John Wiley & Sons, Ltd.     

Non-Gaussian seasonal adjustment: X-12-ARIMA versus robust structural models

This study compares X-12-ARIMA and MING, two new seasonal adjustment methods designed to handle outliers and structural changes in a time series. X-12-ARIMA is a successor to the X-11-ARIMA seasonal adjustment method, and is being developed at the US Bureau of the Census. MING is a ‘Mixture based Non-Gaussian’ method for seasonal adjustment using time series structural models and is implemented as a function in the S-Plus language. The procedures are compared using 29 macroeconomic time series from the US Bureau of the Census. These series have both outliers and structural changes, providing a good testbed for comparing non-Gaussian methods. For the 29 series, the X-12-ARIMA decomposition consistently leads to smoother seasonal factors which are as or more ‘flexible’ than the MING seasonal component. On the other hand, MING is more stable, particularly in the way it handles outliers and level shifts. This study relies heavily on graphical tools for comparing seasonal adjustment methods.     

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.     

The Effects of Disaggregation on Forecasting Nonstationary Time Series

This paper focuses on the effects of disaggregation on forecast accuracy for nonstationary time series using dynamic factor models. We compare the forecasts obtained directly from the aggregated series based on its univariate model with the aggregation of the forecasts obtained for each component of the aggregate. Within this framework (first obtain the forecasts for the component series and then aggregate the forecasts), we try two different approaches: (i) generate forecasts from the multivariate dynamic factor model and (ii) generate the forecasts from univariate models for each component of the aggregate. In this regard, we provide analytical conditions for the equality of forecasts. The results are applied to quarterly gross domestic product (GDP) data of several European countries of the euro area and to their aggregated GDP. This will be compared to the prediction obtained directly from modeling and forecasting the aggregate GDP of these European countries. In particular, we would like to check whether long‐run relationships between the levels of the components are useful for improving the forecasting accuracy of the aggregate growth rate. We will make forecasts at the country level and then pool them to obtain the forecast of the aggregate. The empirical analysis suggests that forecasts built by aggregating the country‐specific models are more accurate than forecasts constructed using the aggregated data. Copyright © 2014 John Wiley & Sons, Ltd.     

A nonparametric method for asymmetrically extending signal extraction filters

Two important problems in the X‐11 seasonal adjustment methodology are the construction of standard errors and the handling of the boundaries. We adapt the ‘implied model approach’ of Kaiser and Maravall to achieve both objectives in a nonparametric fashion. The frequency response function of an X‐11 linear filter is used, together with the periodogram of the differenced data, to define spectral density estimates for signal and noise. These spectra are then used to define a matrix smoother, which in turn generates an estimate of the signal that is linear in the data. Estimates of the signal are provided at all time points in the sample, and the associated time‐varying signal extraction mean squared errors are a by‐product of the matrix smoother theory. After explaining our method, it is applied to popular nonparametric filters such as the Hodrick–Prescott (HP), the Henderson trend, and ideal low‐pass and band‐pass filters, as well as X‐11 seasonal adjustment, trend, and irregular filters. Finally, we illustrate the method on several time series and provide comparisons with X‐12‐ARIMA seasonal adjustments. Copyright © 2010 John Wiley & Sons, Ltd.     

Periodically integrated subset autoregressions for dutch industrial production and money stock

The univariate quarterly Dutch series of industrial production and money stock are both modelled with a periodically integrated subset autoregression (PISA). This model for a non-stationary series allows the lag orders, the values of the parameters and the cyclical patterns to vary over the seasons. The PISA models are found by applying a general-to-simple specification strategy, which deals with non-stationarity and periodicity simultaneously. It is found that the two series show a common asymmetric cyclical behaviour. This paper further proposes a test for periodicity in the errors, with which it is argued that a non-periodic model for the industrial production and money stock is misspecified and that seasonal adjustment does not remove periodicity in the autocorrelation function.     

Forecasting with latent structure time series models: an application to nominal interest rates

In this paper we develop a latent structure extension of a commonly used structural time series model and use the model as a basis for forecasting. Each unobserved regime has its own unique slope and variances to describe the process generating the data, and at any given time period the model predicts a priori which regime best characterizes the data. This is accomplished by using a multinomial logit model in which the primary explanatory variable is a measure of how consistent each regime has been with recent observations. The model is especially well suited to forecasting series which are subject to frequent and/or major shocks. An application to nominal interest rates shows that the behaviour of the three‐month US Treasury bill rate is adequately explained by three regimes. The forecasting accuracy is superior to that produced by a traditional single‐regime model and a standard ARIMA model with a conditionally heteroscedastic error. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

Backward‐in‐Time Selection of the Order of Dynamic Regression Prediction Model

We investigate the optimal structure of dynamic regression models used in multivariate time series prediction and propose a scheme to form the lagged variable structure called Backward‐in‐Time Selection (BTS), which takes into account feedback and multicollinearity, often present in multivariate time series. We compare BTS to other known methods, also in conjunction with regularization techniques used for the estimation of model parameters, namely principal components, partial least squares and ridge regression estimation. The predictive efficiency of the different models is assessed by means of Monte Carlo simulations for different settings of feedback and multicollinearity. The results show that BTS has consistently good prediction performance, while other popular methods have varying and often inferior performance. The prediction performance of BTS was also found the best when tested on human electroencephalograms of an epileptic seizure, and for the prediction of returns of indices of world financial markets.Copyright © 2013 John Wiley & Sons, Ltd.     

Short‐term forecasting of GDP using large datasets: a pseudo real‐time forecast evaluation exercise

This paper performs a large‐scale forecast evaluation exercise to assess the performance of different models for the short‐term forecasting of GDP, resorting to large datasets from ten European countries. Several versions of factor models are considered and cross‐country evidence is provided. The forecasting exercise is performed in a simulated real‐time context, which takes account of publication lags in the individual series. In general, we find that factor models perform best and models that exploit monthly information outperform models that use purely quarterly data. However, the improvement over the simpler, quarterly models remains contained. Copyright © 2009 John Wiley & Sons, Ltd.     

Factor forecasts for the UK

Data are now readily available for a very large number of macroeconomic variables that are potentially useful when forecasting. We argue that recent developments in the theory of dynamic factor models enable such large data sets to be summarized by relatively few estimated factors, which can then be used to improve forecast accuracy. In this paper we construct a large macroeconomic data set for the UK, with about 80 variables, model it using a dynamic factor model, and compare the resulting forecasts with those from a set of standard time‐series models. We find that just six factors are sufficient to explain 50% of the variability of all the variables in the data set. These factors, which can be shown to be related to key variables in the economy, and their use leads to considerable improvements upon standard time‐series benchmarks in terms of forecasting performance. Copyright © 2005 John Wiley & Sons, Ltd.     

Multivariate Bayesian regression applied to the problem of network security

An Erratum has been published for this article in Journal of Forecasting 23(6): 461 (2004) . This paper examines the problem of intrusion in computer systems that causes major breaches or allows unauthorized information manipulation. A new intrusion‐detection system using Bayesian multivariate regression is proposed to predict such unauthorized invasions before they occur and to take further action. We develop and use a multivariate dynamic linear model based on a unique approach leaving the unknown observational variance matrix distribution unspecified. The result is simultaneous forecasting free of the Wishart limitations that is proved faster and more reliable. Our proposed system uses software agent technology. The distributed software agent environment places an agent in each of the computer system workstations. The agent environment creates a user profile for each user. Every user has his or her profile monitored by the agent system and according to our statistical model prediction is possible. Implementation aspects are discussed using real data and an assessment of the model is provided. Copyright © 2002 John Wiley & Sons, Ltd.     

Robust estimation of level and trend

Numerical state space models are efficiently implemented for the estimation of the underlying level and trend of a time series. The model specification is chosen so that the estimation is insensitive to outliers yet adapts rapidly to step changes in level. An example illustrates, by means of projection plots, how at times of uncertainty in the evolution of the series the inferred distribution of level and trend may be multi-modal.     

Exponential smoothing: Estimation by maximum likelihood

In this paper several forecasting methods based on exponential smoothing with an underlying seasonal autoregressive-moving average (SARIMA) model are considered. The relations between the smoothing constants and the coefficients of the autoregressive and moving average polynomials are used. On that basis, a maximum likelihood procedure for parameter estimation is described. The approach rules out the need for initial smoothed values. Prediction intervals are also obtained as a by-product of the approach and a fast algorithm for implementing the method is outlined.     

Selection and estimation of component models for seasonal time series

We present a method for investigating the evolution of trend and seasonality in an observed time series. A general model is fitted to a residual spectrum, using components to represent the seasonality. We show graphically how well the fitted spectrum captures the evidence for evolving seasonality associated with the different seasonal frequencies. We apply the method to model two time series and illustrate the resulting forecasts and seasonal adjustment for one series. Copyright © 2000 John Wiley & Sons, Ltd.     

Unemployment variation over the business cycles: a comparison of forecasting models

Asymmetry has been well documented in the business cycle literature. The asymmetric business cycle suggests that major macroeconomic series, such as a country's unemployment rate, are non‐linear and, therefore, the use of linear models to explain their behaviour and forecast their future values may not be appropriate. Many researchers have focused on providing evidence for the non‐linearity in the unemployment series. Only recently have there been some developments in applying non‐linear models to estimate and forecast unemployment rates. A major concern of non‐linear modelling is the model specification problem; it is very hard to test all possible non‐linear specifications, and to select the most appropriate specification for a particular model. Artificial neural network (ANN) models provide a solution to the difficulty of forecasting unemployment over the asymmetric business cycle. ANN models are non‐linear, do not rely upon the classical regression assumptions, are capable of learning the structure of all kinds of patterns in a data set with a specified degree of accuracy, and can then use this structure to forecast future values of the data. In this paper, we apply two ANN models, a back‐propagation model and a generalized regression neural network model to estimate and forecast post‐war aggregate unemployment rates in the USA, Canada, UK, France and Japan. We compare the out‐of‐sample forecast results obtained by the ANN models with those obtained by several linear and non‐linear times series models currently used in the literature. It is shown that the artificial neural network models are able to forecast the unemployment series as well as, and in some cases better than, the other univariate econometrics time series models in our test. Copyright © 2004 John Wiley & Sons, Ltd.     

Dynamic Model Averaging and CPI Inflation Forecasts: A Comparison between the Euro Area and the United States

The paper forecasts consumer price inflation in the euro area (EA) and in the USA between 1980:Q1 and 2012:Q4 based on a large set of predictors, with dynamic model averaging (DMA) and dynamic model selection (DMS). DMA/DMS allows not solely for coefficients to change over time, but also for changes in the entire forecasting model over time. DMA/DMS provides on average the best inflation forecasts with regard to alternative approaches (such as the random walk). DMS outperforms DMA. These results are robust for different sample periods and for various forecast horizons. The paper highlights common features between the USA and the EA. First, two groups of predictors forecast inflation: temporary fundamentals that have a frequent impact on inflation but only for short time periods; and persistent fundamentals whose switches are less frequent over time. Second, the importance of some variables (particularly international food commodity prices, house prices and oil prices) as predictors for consumer price index inflation increases when such variables experience large shocks. The paper also shows that significant differences prevail in the forecasting models between the USA and the EA. Such differences can be explained by the structure of these respective economies. Copyright © 2015 John Wiley & Sons, Ltd.     

Exponential smoothing of seasonal data: a comparison

A parsimonious method of exponential smoothing is introduced for time series generated from a combination of local trends and local seasonal effects. It is compared with the additive version of the Holt–Winters method of forecasting on a standard collection of real time series. Copyright © 2001 John Wiley & Sons, Ltd.     

Real‐Time Forecasts of Inflation: The Role of Financial Variables

We present a mixed‐frequency model for daily forecasts of euro area inflation. The model combines a monthly index of core inflation with daily data from financial markets; estimates are carried out with the MIDAS regression approach. The forecasting ability of the model in real time is compared with that of standard VARs and of daily quotes of economic derivatives on euro area inflation. We find that the inclusion of daily variables helps to reduce forecast errors with respect to models that consider only monthly variables. The mixed‐frequency model also displays superior predictive performance with respect to forecasts solely based on economic derivatives. Copyright © 2012 John Wiley & Sons, Ltd.     

Forecasting key macroeconomic variables from a large number of predictors: a state space approach

We use state space methods to estimate a large dynamic factor model for the Norwegian economy involving 93 variables for 1978Q2–2005Q4. The model is used to obtain forecasts for 22 key variables that can be derived from the original variables by aggregation. To investigate the potential gain in using such a large information set, we compare the forecasting properties of the dynamic factor model with those of univariate benchmark models. We find that there is an overall gain in using the dynamic factor model, but that the gain is notable only for a few of the key variables. Copyright © 2009 John Wiley & Sons, Ltd.