Seasonal adjustment methods - TRAMO-SEATS and X-13ARIMA-SEATS

This section describes selected aspects of the seasonal adjustment methods and technical issues including descriptions of the theoretical models used by X-12-ARIMA and TRAMO-SEATS. Information provided here are vital for a good understanding of the results produced by the software.

TRAMO-SEATS is a model-based seasonal adjustment method developed by Victor Gómez (Ministerio de Hacienda), and Agustin Maravall (Banco de España). It consists of two linked programs: TRAMO and SEATS. TRAMO (Time Series Regression with ARIMA Noise, Missing Observations, and Outliers) performs estimation, forecasting, and interpolation of regression models with missing observations and ARIMA errors, in the presence of possibly several types of outlier. SEATS (Signal Extraction in ARIMA Time Series) performs an ARIMA-based decomposition of an observed time series into unobserved components. Information about the TRAMO-SEATS method available in this section derives directly from papers by Victor Gómez and Agustin Maravall; the most important ones are: GÓMEZ, V., and MARAVALL, A. (1996), GÓMEZ, V., and MARAVALL, A. (2001a, b) and MARAVALL, A. (2009). More information about the TRAMO-SEATS method, TRAMO-SEATS software (DOS version and TSW+ – Tramo Seats Windows software and several interfaces) and its documentation as well as papers on methodology and application of the programs, can be found in the dedicated section of the Banco de España website.

X-13ARIMA-SEATS is a seasonal adjustment program developed and supported by the U.S. Census Bureau. It is based on the U.S. Census Bureau's earlier X-11 program, the X-11-ARIMA program developed at Statistics Canada, the X-12-ARIMA program developed by the U.S. Census Bureau, and the SEATS program developed at the Banco de España. The program is now used by the U.S. Census Bureau for a seasonal adjustment of time series. Users can download the X-13ARIMA-SEATS application, which is a Windows interface for the X-13ARIMA-SEATS program. Detailed information on X-13ARIMA-SEATS can be found at a dedicated U.S. Census Bureau webpage.

In contrast to the earlier product (X-12-ARIMA), X-13ARIMA-SEATS includes not only the enhanced X-11 seasonal adjustment procedure but also the capability to generate ARIMA model-based seasonal adjustment using a version of the SEATS procedure originally developed by Victor Gómez and Agustín Maravall at the Banco de España. The program also includes a variety of new tools to overcome adjustment problems and thereby enlarge the range of economic time series that can be adequately seasonally adjusted.

In general, X-13ARIMA-SEATS can perform seasonal adjustment in two ways: either using ARIMA model-based seasonal adjustment as in SEATS or by means of an enhanced X-11 method.

The seasonal adjustment methods available in JDemetra+ aim to decompose a time series into components and remove seasonal fluctuations from the observed time series. The X-11 method considers monthly and quarterly series while SEATS is able to decompose series with 2, 3, 4, 6 and 12 observations per year. The main components, each representing the impact of certain types of phenomena on the time series ($X_{t}$), are:

• The trend ($T_{t}$) that captures long-term and medium-term behaviour;

• The seasonal component ($S_{t}$) representing intra-year fluctuations, monthly or quarterly, that are repeated more or less regularly year after year;

• The irregular component ($I_{t}$) combining all the other more or less erratic fluctuations not covered by the previous components.

In general, the trend consists of 2 sub-components:

• The long-term evolution of the series;

• The cycle, that represents the smooth, almost periodic movement around the long-term evolution of the series. It reveals a succession of phases of growth and recession.

For seasonal adjustment purposes both TRAMO-SEATS and X-13ARIMA-SEATS do not separate the long-term trend from the cycle as these two components are usually too short to perform their reliable estimation. Consequently, hereafter TRAMO-SEATS and X-13ARIMA-SEATS estimate the trend component. However, the original TRAMO-SEATS may separate the long-term trend from the cycle through the Hodrick-Precsott filter using the output of the standard decomposition. It should be remembered that JDemetra+ refers to the trend-cycle as trend ($T_{t}$), and consequently this convention is used in this document.

TRAMO-SEATS considers two decomposition models:

• The additive model: $X_{t} = T_{t} + S_{t} + I_{t}$;

• The log additive model: $log(X_{t}) = log(T_{t}) + log(S_{t}) + log(I_{t})$.

Apart from these two decomposition types X-13ARIMA-SEATS allows the user to apply also the multiplicative model: $X_{t} = T_{t} \times S_{t} \times I_{t}$.

A time series $x_{t}$, which is a subject to a decomposition, is assumed to be a realisation of a discrete-time stochastic, covariance-stationary linear process, which is a collection of random variables $x_{t}$, where $t$ denotes time. It can be shown that any stochastic, covariance-stationary process can be presented in the form:

$x_{t} = \mu_{t} + {\widetilde{x}}_{t}$, [1]

where $\mu_{t}$ is a linearly deterministic component and ${\widetilde{x}}_{t}$ is a linearly interderministic component, such as:

${\widetilde{x}}_{t} = {\sum_{j = 0}^{\infty}\psi_{j}a}_{t - j}$, [2]

where $\sum_{j = 0}^{\infty}\psi_{i}^{2} < \infty$ (coefficients $\psi_{j}$ are absolutely summable), $\psi_{0} = 1$ and $a_{t}$ is the white noise error with zero mean and constant variance $V_{a}$. The error term $a_{t}$ represents the one-period ahead forecast error of $x_{t}$, that is:

$a_{t} = {\widetilde{x}}_{t} - {\widehat{x}}_{t|t - 1}$, [3]

where ${\widehat{x}}_{t|t - 1}$ is the forecast of ${\widetilde{x}}_{t}$ made at period $t - 1$. As $a_{t}$ represents what is new in ${\widetilde{x}}_{t}$ in point $t$, i.e., not contained in the past values of ${\widetilde{x}}_{t}$, it is also called innovation of the process. From [3] ${\widetilde{x}}_{t}$ can be viewed as a linear filter applied to the innovations.

The equation 7.1 is called a Wold representation. It presents a process as a sum of linearly deterministic component $\mu_{t}$ and linearly interderministic component $\sum_{j = 0}^{\infty}\psi_{j}a_{t - j}$, the first one is perfectly predictable once the history of the process $x_{t - 1}$ is known and the second one is impossible to predict perfectly. This explains why the stochastic process cannot be perfectly predicted.

Under suitable conditions ${\widetilde{x}}_{t}$ can be presented as a weighted sum of its past values and $a_{t}$, i.e.:

${ {\widetilde{x}}_{t} = \sum_{j = 0}^{\infty}\pi_{j}{\widetilde{x}}_{t - j} + a}_{t}$, [4]

In general, for the observed time series, the assumptions concerning the nature of the process [1] do not hold for various reasons. Firstly, most observed time series display a mean that cannot be assumed to be constant due to the presence of a trend and the seasonal movements. Secondly, the variance of the time series may vary in time. Finally, the observed time series usually contain outliers, calendar effects and regression effects, which are treated as deterministic. Therefore, in practice a prior transformation and an adjustment need to be applied to the time series. The constant variance is usually achieved through taking a logarithmic transformation and the correction for the deterministic effects, while stationarity of the mean is achieved by applying regular and seasonal differencing. These processes, jointly referred to as preadjustment or linearization, can be performed with the TRAMO or RegARIMA models. Besides the linearisation, forecasts and backcasts of stochastic time series are estimated with the ARIMA model, allowing for later application of linear filters at both ends of time series. The estimation performed with these models delivers the stochastic part of the time series, called the linearised series, which is assumed to be an output of a linear stochastic process.¹ The deterministic effects are removed from the time series and used to form the final components.

In the next step the linearised series is decomposed into its components. There is a fundamental difference in how this process is performed in TRAMO-SEATS and X-13ARIMA-SEATS. In TRAMO-SEATS the decomposition is performed by the SEATS procedure, which follows a so called ARIMA model based approach. In principle, it aims to derive the components with statistical models. More information is given in the SEATS section. X-13ARIMA-SEATS offers two algorithms for decomposition: SEATS and X-11. The X-11 algorithm, which is described in the X-11 section section, decomposes a series by means of linear filters. Finally, in both methods the final components are derived by the assignment of the deterministic effects to the stochastic components. Consequently, the role of the ARIMA models is different in each method. TRAMO-SEATS applies the ARIMA models both in the preadjustment step and in the decomposition procedure. On the contrary, when the X-11 algorithm is used for decomposition, X-13ARIMA-SEATS uses the ARIMA model only in the preadjustment step. In summary, the decomposition procedure that results in an estimation of the seasonal component requires prior identification of the deterministic effects and their removal from the time series. This is achieved through the linearisation process performed by the TRAMO and the RegARIMA models, shortly discussed in the Linearisation with the TRAMO and RegARIMA models section.The linearised series is then decomposed into the stochastic components with SEATS or X-11 algorithms.