A Non-Gaussian Airline Model for Seasonal Adjustment John A.D. Aston, US Census Bureau Siem Jan Koopman, Free University Amsterdam Contact-John Aston, SRD, US Census Bureau, Washington, DC, USA (jaston@niss.org) Key Words: Airline Model, ARIMA, Non-Gaussian Models, Seasonal Adjustment, Importance Sampling. Abstract: The Airline model, introduced by Box and Jenkins in their sem- inal book Time Series Analysis: Forecasting and Control, is rou- tinely used to model economic time series. This model is pa- rameterized by two factors, and gaussianity is usually assumed for the underlying noise component. Here, this model is gener- alised to include a non-Gaussian component to model outliers in the data. The model is examined using a state-space modelling approach, and importance sampling (see Durbin and Koopman). It utilises the decomposition method for ARIMA models devel- oped by Hillmer and Tiao. This is necessary in order to pre- serve the airline structure whilst allowing a flexibility to include non-Gaussian noise terms for different components in the model. Different forms for the generalisation of the noise term are in- vestigated. The models are interrogated through the use of a real series, the US Automobile Retail Series. The new models allow outliers to be accounted for, whilst keeping the underlying struc- tures that are currently used to aid reporting of economic data. 1. Introduction The aim of seasonal adjustment is straightforward: remove sea- sonal variations from time series observations. For many eco- nomic time series the seasonal cycle is clearly visible in a time series graph but it can not be observed. An important task of the U.S. Bureau of the Census is to extract the seasonal component in economic time series and to remove it from the series. Years of experience from experts and users have culminated in the cur- rent Census X-12-Arima procedure which is described and doc- umented in Findley, Monsell, Bell, Otto, and Chen (1998) and Ladiray and Quenneville (2001). The Census X-12 program has become the standard seasonal adjustment method for many sta- tistical agencies worldwide. A simple approach to estimating seasonal effects that are ad- ditive is to consider seasonal means by computing the sample mean of observations associated with a particular season (e.g. calender months or quarters). With regression and other statis- tical models, seasonal effects can be estimated by including so- called seasonal dummy variables which in effect allow different JA is a National Institute of Statistical Sciences Postdoctoral Fellow at US Census Bureau. Some of this work was carried out when SJK was an American Statistical Association visiting fellow at the US Census Bureau. This paper re- ports the results of research and analysis undertaken at the U.S. Census Bureau. It has undergone a Census Bureau review more limited in scope than that given to official Census Bureau publications. This report is released to inform interested parties of ongoing research and to encourage discussion of work in progress. means for different seasons. Alternatively, seasonal effects can be removed directly by considering the sum of s observations of the last year where s is the number of seasons in a year. When focus is on the growth rate of seasonally adjusted time series, this approach amounts to taking annual differences of the time series. For both approaches seasonal effects are determined by weighting the appropriate observations using zeros and ones as unnormalised weights. As a result the observation weights are not discounted when they lie further away. In the dummy case the weighting patterns are too wide for practical purposes while in the seasonal-sum case they are too short. More advanced meth- ods of seasonal adjustment aim at discounting weighting patterns which are not too wide and not too short. Weighting patterns for seasonally adjusted data can be determined explicitly by choos- ing a set of moving average filters or implicitly by estimating a seasonal time series model. One of the most commonly used sea- sonal models is the airline model introduced by Box and Jenkins (1976) who used it to study a time series of monthly number of US airline passengers. The airline model is given by (1 - B)(1 - B s )y t = (1 - θB)(1 - ΘB s )ξ t (1) where ξ t ∼N (0,σ 2 ),t =1,...,n with observation y t , back- shift operator B so that By t = y t−1 and B s y t = y t−s , seasonal length s (s =4 for quarterly data and s = 12 for monthly data) and white noise disturbance ξ t . The model requires non-seasonal, Δ(B)=1 - B, and seasonal, Δ s (B)=1 - B s , differencing and is based on a moving average polynomial of degree s+1. The dy- namic characteristics of the model depend on two parameters Θ and θ, which essentially describe the seasonal and non-seasonal structure of the data, respectively, although not completely inde- pendently of one another. The airline model falls within the class of seasonal autoregressive integrated moving average (ARIMA) models. For any seasonal adjustment procedure, a difficult problem to handle in practice is the treatment of outliers in a time series. The identification of outliers for a given time series and a given model specification leads to two particular problems. The first is that an outlier can only be identified with respect to a specific model. An observation may be an outlier for one model but not for another model while both models can be nested or even fall within the same class. The second problem is related to the sample choice. An observation common to two different samples may be iden- tified as an outlier in one sample while it is not an outlier for another sample. When an outlier is identified in a time series, it is not guaranteed that the observation will again be identified as an outlier when the time series is extended with more recent ob- servations even if the same model and the same outlier detection method are used. Such outlier detection practices lead to seasonal