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.     

A class of periodic trend models for seasonal time series

Trend and seasonality are the most prominent features of economic time series that are observed at the subannual frequency. Modeling these components serves a variety of analytical purposes, including seasonal adjustment and forecasting. In this paper we introduce unobserved components models for which both the trend and seasonal components arise from systematically sampling a multivariate transition equation, according to which each season evolves as a random walk with a drift. By modeling the disturbance covariance matrix we can encompass traditional models for seasonal time series, like the basic structural model, and can formulate more elaborate ones, dealing with season specific features, such as seasonal heterogeneity and correlation, along with the different role of the nonstationary cycles defined at the fundamental and the harmonic frequencies in determining the shape of the seasonal pattern.     

Unobserved components in ARCH models: An application to seasonal adjustment

The paper deals with unobserved components in ARIMA models with GARCH errors, in the context of an actual application, namely seasonal adjustment of the monthly Spanish money supply series. The series shows clear evidence of (moderate) non-linearity, which does not disappear with simple outlier correction. The GARCH structure explains reasonably well the non-linearity, and this explanation is robust with respect to the GARCH specification. We look at the time variation of the standard error of the adjusted series estimator and show how it can be measured. Next, we look at the implications this variation has on short-term monetary control. The non-linearity seems to have a small effect in practice. It is further seen that the conditional variance of the GARCH process may, in turn, be decomposed into components. In fact, the conditional variance of the money supply series is the sum of a weak linear trend, a strong non-linear seasonal component, and a moderate non-linear irregular component. This information has policy implications: for example, there are periods in the year when policy can be more assertive because information is more precise. Finally, looking at the non-linear components of the money supply it is seen how linear combinations of non-linear series can produce series that behave linearly.     

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.     

Linear signal extraction with intervention techniques in non-linear time series

Seasonal adjustment is performed in some data-producing agencies according to the ARIMA-model-based signal extraction theory. A stochastic linear process parametrized in terms of an ARIMA model is first fitted to the series, and from this model the models for the trend, cycle, seasonal, and irregular component can be derived. A spectrum is associated to every component model and is used to compute the optimal Wiener–Kolmogorov filter. Since the modelling is linear, prior linearization of the series with intervention techniques is performed. This paper discusses the performance of linear signal extraction with intervention techniques in non-linear processes. In particular, the following issues are discussed: (1) the ability of intervention techniques to linearize time series which present non-linearities; (2) the stability of the linear projection giving the components estimators under non-linear misspecifications; (3) the capacity of the WK filter to preserve the linearity in some components and the non-linearities in others. Copyright © 1998 John Wiley & Sons, Ltd.     

Recursive estimation and forecasting of non-stationary time series

The paper presents a unified, fully recursive approach to the modelling and forecasting of non-stationary time-series. The basic time-series model, which is based on the well-known ‘component’ or ‘structuraL’ form, is formulated in state-space terms. A novel spectral decomposition procedure, based on the exploitation of recursive smoothing algorithms, is then utilized to simplify the procedures of model identification and estimation. Finally, the fully recursive formulation allows for conventional or self-adaptive implementation of state-space forecasting and seasonal adjustment. Although the paper is restricted to the consideration of univariate time series, the basic approach can be extended to handle explanatory variables or full multivariable (vector) series.     

Analysis of many short time sequences: Forecast improvements achieved by shrinkage

Credibility models in actuarial science deal with multiple short time series where each series represents claim amounts of different insurance groups. Commonly used credibility models imply shrinkage of group-specific estimates towards their average. In this paper we model the claim size yu in group i and at time t as the sum of three independent components: y_it = μ_r + δ_i + ?_it. The first component, μt = μ_t?1 + m_t, represents time-varying levels that are common to all groups. The second component, δ_i, represents random group offsets that are the same in all periods, and the third component represents independent measurement errors. In this paper we show how to obtain forecasts from this model and we discuss the nature of the forecasts, with particular emphasis on shrinkage. We also assess the forecast improvements that can be expected from such a model. Finally, we discuss an extension of the above model which also allows the group offsets to change over time. We assume that the offsets for different groups follow independent random walks.     

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.     

Forecasting with missing data: application to coastal wave heights

This paper presents a comparative analysis of linear and mixed models for short‐term forecasting of a real data series with a high percentage of missing data. Data are the series of significant wave heights registered at regular periods of three hours by a buoy placed in the Bay of Biscay. The series is interpolated with a linear predictor which minimizes the forecast mean square error. The linear models are seasonal ARIMA models and the mixed models have a linear component and a non‐linear seasonal component. The non‐linear component is estimated by a non‐parametric regression of data versus time. Short‐term forecasts, no more than two days ahead, are of interest because they can be used by the port authorities to notify the fleet. Several models are fitted and compared by their forecasting behaviour. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

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.     

Modelling and forecasting time series sampled at different frequencies

This paper discusses how to specify an observable high‐frequency model for a vector of time series sampled at high and low frequencies. To this end we first study how aggregation over time affects both the dynamic components of a time series and their observability, in a multivariate linear framework. We find that the basic dynamic components remain unchanged but some of them, mainly those related to the seasonal structure, become unobservable. Building on these results, we propose a structured specification method built on the idea that the models relating the variables in high and low sampling frequencies should be mutually consistent. After specifying a consistent and observable high‐frequency model, standard state‐space techniques provide an adequate framework for estimation, diagnostic checking, data interpolation and forecasting. An example using national accounting data illustrates the practical application of this method. Copyright © 2008 John Wiley & Sons, Ltd.     

Exponential smoothing: The state of the art

This paper is a critical review of exponential smoothing since the original work by Brown and Holt in the 1950s. Exponential smoothing is based on a pragmatic approach to forecasting which is shared in this review. The aim is to develop state-of-the-art guidelines for application of the exponential smoothing methodology. The first part of the paper discusses the class of relatively simple models which rely on the Holt-Winters procedure for seasonal adjustment of the data. Next, we review general exponential smoothing (GES), which uses Fourier functions of time to model seasonality. The research is reviewed according to the following questions. What are the useful properties of these models? What parameters should be used? How should the models be initialized? After the review of model-building, we turn to problems in the maintenance of forecasting systems based on exponential smoothing. Topics in the maintenance area include the use of quality control models to detect bias in the forecast errors, adaptive parameters to improve the response to structural changes in the time series, and two-stage forecasting, whereby we use a model of the errors or some other model of the data to improve our initial forecasts. Some of the major conclusions: the parameter ranges and starting values typically used in practice are arbitrary and may detract from accuracy. The empirical evidence favours Holt's model for trends over that of Brown. A linear trend should be damped at long horizons. The empirical evidence favours the Holt-Winters approach to seasonal data over GES. It is difficult to justify GES in standard form–the equivalent ARIMA model is simpler and more efficient. The cumulative sum of the errors appears to be the most practical forecast monitoring device. There is no evidence that adaptive parameters improve forecast accuracy. In fact, the reverse may be true.     

Seasonal heteroscedasticity and trends

This paper discusses the possibility of accommodating features such as seasonal heteroscedasticity and trends in a seasonal model. The former takes place when one or more seasonal effects are more variable than others and it is quite pervasive in hydrology, although interesting examples are found in economics, where it has recently been shown to characterize the output of the manufacturing series; seasonal trends occur when the seasonal effect shows a systematic tendency to increase or decrease its amplitude over the years. We consider different models of seasonality available in the literature and we argue that the Harrison and Stevens seasonal model enhances the flexibility that is necessary to capture effects associated to particular seasons. The resulting seasonally heteroscedastic model provides an explanation for the periodicity in the series alternative to that provided by the literature on periodic integration. © 1998 John Wiley & Sons, Ltd.     

The impact of seasonal constants on forecasting seasonally cointegrated time series

In this paper we focus on the effect of (i) deleting, (ii) restricting or (iii) not restricting seasonal intercept terms on forecasting sets of seasonally cointegrated macroeconomic time series for Austria, Germany and the UK. A first empirical result is that the number of cointegrating vectors as well as the relevant estimated parameter values vary across the three models. A second result is that the quality of out-of-sample forecasts critically depends on the way seasonal constants are treated. In most cases, predictive performance can be improved by restricting the effects of seasonal constants. However, we find that the relative advantages and disadvantages of each of the three methods vary across the data sets and may depend on sample-specific features. © 1998 John Wiley & Sons, Ltd.     

Forecasting time series which are inherently discontinuous

Commonly used forecasting methods often produce meaningless forecasts when time series display abrupt changes in level. Measuring and accounting for the effect of discontinuities can have a significant impact on forecasting accuracy. In addition, if discontinuities are considered non-random and their cause is known, then adjustments can be made to more reliably represent the trend, seasonal and random component. This paper concerns a computational method used in forecasting inherently discontinuous time series. The method provides screening to determine the locations and types of discontinuities. The paper includes analyses of actual time series which are typical of certain types of inherently discontinuous processes.     

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.     

Prediction‐based adaptive compositional model for seasonal time series analysis

In this paper we propose a new class of seasonal time series models, based on a stable seasonal composition assumption. With the objective of forecasting the sum of the next ? observations, the concept of rolling season is adopted and a structure of rolling conditional distributions is formulated. The probabilistic properties, estimation and prediction procedures, and the forecasting performance of the model are studied and demonstrated with simulations and real examples.