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X-12 ARIMA Eurostat, Luxembourg Seasonal Adjustment Introduction to X-12 Arima was developed by US Census Bureau and runs through the following steps: 1. The series is modified by any user defined prior adjustments. 2. the program fits a regARIMA model to the series in order to detect and adjust for outliers and other distorting effects to improve forecasts and seasonal adjustment. 3. It detect and estimates additional component (e.g. calendar effects) and extrapolate forward (forecast) and backwards (backcast). Eurostat, Luxembourg Seasonal Adjustment Introduction to X-12 Arima (2) 4. 5. The program then uses a series of moving averages to decompose a time series into three components. It does this in three iterations, getting successively better estimates of the components. During these iterations extreme values are identified and replaced. In the last step a wider range of diagnostic statistics are produced, describing the final seasonal adjustment, and giving pointers to possible improvements which could be made. Eurostat, Luxembourg Seasonal Adjustment What is a regARIMA model? A regARIMA model is a regression model with ARIMA errors. ARIMA stands for AutoRegressive Integrated Moving Average. When we use regression models to estimate some of the components in a time series, the errors from the regression model are correlated, and we use ARIMA models to model the correlation in the errors. ARIMA models are one way to describe the relationships between points in a time series. Besides using regARIMA models to estimate regression effects (such as outliers, trading day, and moving holidays), we also use ARIMA models to forecast the series. Research has shown that using forecasted values gives smaller revisions at the end of the series. Eurostat, Luxembourg Seasonal Adjustment What are seasonal filters? A filter is a weighted average where the weights sum to 1. Seasonal filters are the filters used to estimate the seasonal component. Ideally, seasonal filters are computed using values from the same month or quarter, for example, an estimate for January would come from a weighted average of the surrounding Januaries. The seasonal filters available in X-12-ARIMA consist of seasonal moving averages of consecutive values within a given month or quarter. An n x m moving average is an m-term simple average taken over n consecutive sequential spans. Eurostat, Luxembourg Seasonal Adjustment What are seasonal filters? (2) An example of a 3x3 filter (5 terms) for January 2003 (or Quarter 1, 2003) is: 2001.1 + 2002.1 + 2003.1 + 2002.1 + 2003.1 + 2004.1 + 2003.1 + 2004.1 + 2005.1 9 An example of a 3x5 filter for January 2003 (or Quarter 1, 2003) is: 2000.1 + 2001.1 + 2002.1 + 2003.1 + 2004.1 + 2001.1 + 2002.1 + 2003.1 + 2004.1 + 2005.1 + 2002.1 + 2003.1 + 2004.1 + 2005.1 + 2006.1 15 Eurostat, Luxembourg Seasonal Adjustment What are trend filters? Trend filters are weighted averages of consecutive months or quarters used to estimate the trend component. An example of a 2x4 filter (5-terms) for First Quarter 2005: 2004.3 + 2004.4 + 2005.1 + 2005.2 2004.4 + 2005.1 + 2005.2 + 2005.3 _______________________________________ 8 Notice that we are using the closest points, not just the closest points within the First Quarter like with the seasonal filters above. Notice also that every quarter has a weight of 1/4, though the Third Quarter uses values in both 2004 and 2005. Eurostat, Luxembourg Seasonal Adjustment Why do the seasonal factors change when new data are added? Keep in mind that the data used in the seasonal and the trend filters can go back several years. Let's look at an example using X-12-ARIMA's seasonal moving average filters. For example, if the last point in your series is January 2006, and you're using 3x5 seasonal filters, the value at January 2006 will effect the estimates for Januaries in 2004, and 2005. You can see the value for January 2003 in the equations below. The 3x5 filter for January 2004: 2001.1+2002.1+2003.1+2004.1+2005.1 2002.1+2003.1+2004.1+2005.1+2006.1 2003.1+2004.1+2005.1+2006.1+2007.1 _____________________________________________ 15 Eurostat, Luxembourg Seasonal Adjustment Why do the seasonal factors change when new data are added?(2) The 3x5 filter for January 2005: 2002.1+2003.1+2004.1+2005.1+2006.1 2003.1+2004.1+2005.1+2006.1+2007.1 2004.1+2005.1+2006.1+2007.1+2008.1 ________________________________________________ 15 You can see in the above equations that the new point at January 2006 will affect the estimates for the other Januaries. There is a similar effect for the trend filters, with new data effecting estimates for half the filter length. Eurostat, Luxembourg Seasonal Adjustment X12-ARIMA Stage 1. Initial Estimates Step 1.1: Initial Trend Estimate Compute a centered 12-term (13-term) moving average as a first estimate of the trend: Tt (1) 1 1 1 1 1 Yt 6 Yt 5 Yt Yt 5 Yt 6 24 12 12 12 24 The first trend estimation is always 2x12 or 2x4 Step 1.2: Initial Seasonal-Irregular component or “SI Ratio” The ratio of the original series to the estimated trend is the first estimate of the detrended series: SI (1) t Yt (1) Tt Eurostat, Luxembourg SI (1) t Yt Tt (1) Seasonal Adjustment X12-ARIMA Stage 1. Initial Estimates (cont..) Step 1.3: Initial Preliminary Seasonal Factor 5-term weighted moving average (3x3) is calculated for each month of the seasonal-irregular ratios (SI) to obtain preliminary estimates of the seasonal factors: 1 2 3 2 1 ) ) Sˆ t(1) SI t(1)24 SI t(112 SI t(1) SI t(112 SI t(1)24 9 9 9 9 9 Step 1.4: Initial Seasonal Factor Crude “unbiased” seasonal from step 1.3 via centering: S (1) t Sˆ t(1) 1 24 Sˆ t(1)6 121 Sˆt(1)5 121 Sˆ t(1)5 1 24 Sˆt(1)6 Sˆ t(1)6 Sˆ t(1)5 Sˆt(1)5 Sˆ t(1)6 (1) ˆ S St 24 12 12 24 1 t Step 1.5: Initial Seasonal Adjustment At(1) Yt S t(1) Eurostat, Luxembourg At(1) Yt St(1) Seasonal Adjustment X12-ARIMA Stage 2. Seasonal Factors and Seasonal Adjustment Step 2.1: Intermediate Trend Calculate an intermediate trend (Henderson) of length 2H+1 for datadetermined H. Tt where ( 2) H h j H ( 2 H 1) j At(1)j h (j 2 H 1) H j H , h j h j are the (2H+1)-term Henderson weights. Step 2.2: Final SI Ratios Calculate the detrended series from Henderson trend: SI ( 2) t Yt ( 2) Tt Eurostat, Luxembourg SI t(2) Yt Tt ( 2) Seasonal Adjustment X12-ARIMA Stage 2. Seasonal Factors and Seasonal Adjustment Step 2.3: Preliminary Seasonal Factor Calculate final “biased” seasonal factors via a “3x5” seasonal moving average: ˆS ( 2) 1 SI ( 2) 2 SI ( 2) 3 SI ( 2) 3 SI ( 2) 3 SI ( 2) 2 SI ( 2) 1 SI ( 2) t t 36 t 24 t 12 t t 12 t 24 t 36 15 15 15 15 15 15 15 Step 2.4: Seasonal Factor Calculate final “unbiased” seasonal factors via centering: S ( 2) t Sˆt( 2 ) 1 24 Sˆ t(26) 121 Sˆ t(25) 121 Sˆ t(25) 1 24 Sˆ t(26) S ( 2) t ˆ ( 2) Sˆ ( 2) Sˆt(26) Sˆ t(25) S ( 2) ˆ St t 5 t 6 24 12 12 24 Step 2.5: Seasonal Adjustment At( 2 ) Yt S t( 2) Eurostat, Luxembourg At(2) Yt St(2) Seasonal Adjustment Stage 3. Final Henderson Trend and Final Irregular Step 3.1: Final Trend Final trend from a Henderson trend filter determined from data: Tt ( 3) H h j H ( 2 H 1) j At(2)j Step 3.2: Final Irregular Final irregular factors as ratios between the seasonally adjusted series from Stage 2 and the final trend from Step 3.1: I ( 3) t At( 2) ( 3) Tt Eurostat, Luxembourg I t(3) At(2) Tt (3) Seasonal Adjustment Estimated Decomposition: Yt Tt (3) S ( 2) t I (3) t Yt Tt S I (3) Eurostat, Luxembourg ( 2) (3) t t Seasonal Adjustment Trend Filter Choices The first trend estimation (in Stage 1) is always 2x12 or 2x4. The Henderson trend filter choices (Step 2.1, 3.1) are based on noise-to-signal ratios, the size of the irregular variations relative to those of the trend and labelled I / C in the X-12 output. If I / C<1, the 9-term Henderson filter (H=4) is used. Otherwise, in Stage 2, the 13-term filter (H=6) is used. In Stage 3, the 13-term filter is used when 1<I / C <3.5, but the 23-term Henderson filter (H=11) is used when I / C≥3.5. Eurostat, Luxembourg Seasonal Adjustment Seasonal Filter Choices The criterion for selection of the seasonal moving average is based on the global I / S, which measures the relative size of irregular movements and seasonal movements averaged over all months or quarters. It is used to determine what seasonal moving average is applied using the following criteria: I/S Seasonal Moving Average 0<I / S<2.5 3x3 3.5<I / S<5.5 3x5 6.5<I / S 3x9 The global I / S ratio is calculated using data that ends in the last full calendar year available. If 2.5<I / S<3.5 or 5.5<I / S<6.5 then the I / S ratio will be calculated using one year less of data to see if the I / S ratio than falls into one of the ranges given above. The year removing is repeated either until the I / S ratio falls into one of the ranges or after five years a 3x5 moving average will be used. Eurostat, Luxembourg Seasonal Adjustment