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Seasonal Adjustment

Seasonal Adjustment . Topics. Motivation and theoretical background (Øyvind Langsrud) Seasonal adjustment step-by-step (László Sajtos) (A few) issues on seasonal adjustment (László Sajtos). Presented by. Øyvind Langsrud Statistics Norway.

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Seasonal Adjustment

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  1. Seasonal Adjustment

  2. Topics • Motivation and theoretical background (Øyvind Langsrud) • Seasonal adjustment step-by-step (László Sajtos) • (A few) issues on seasonal adjustment (László Sajtos)

  3. Presented by Øyvind Langsrud Statistics Norway

  4. Time series with seasonal and non-seasonal variation

  5. Removing the seasonal variation

  6. Removing also the non-seasonal variation

  7. Monthly time series example Trend and seasonality can be seen How to find it by computation?

  8. Quick and dirty calculation of trend by ordinary linear regression:y = a + b*time + e time = 2000.000, 2000.083, 2000.167, 2000.250, 2000.333, 2000.417, 2000.500, 2000.583, 2000.667, 2000.750, 2000.833, 2000.917, 2001.000, 2001.083, …... a = -6619.731 b = 3.351223

  9. Including seasonality in "the dirty model"y = a+ b*time + cmonth+ e

  10. Including seasonality in "the dirty model" y = a+ b*time + cmonth+ e Transforming to seasonal adjustment language a + b*time → Tt cmonth → St e → It a = -6468.505 b = 3.275956 c = mnd0 mnd2 mnd3 mnd4 mnd5 mnd6 -9.19620250 -16.59062737 -6.79790939 -8.51090569 -1.18890200 6.33881598 mnd7 mnd8 mnd9 mnd10 mnd11 mnd12 1.84439111 4.62139480 -2.56494236 -0.04409251 1.53598811 30.55299181 yt = Tt + St+ It

  11. Trend from "the dirty model" yt = Tt + St + It

  12. Seasonality from "the dirty model" yt = Tt + St+ It

  13. Seasonal adjustment by "the dirty model" yt = Tt + St+ It

  14. Question to the audience:What is wrong with this ordinary regression approach ?

  15. Irregular component by "the dirty model" yt = Tt + St + It

  16. In practise a multiplicative model is used:yt = Tt× St× It yt is not the original series but a series that is corrected for holiday and trading day effects (calendar adjusted) yt =Tt× St × It

  17. yt = Tt× St× It Note that the seasonal factors vary slightly along time

  18. yt = Tt× St×It This time the irregular component looks more as true noise Note that correlated neighbour values is allowed (autocorrelation)

  19. yt = Tt×St× It This is seasonally adjusted data as published by Statistics Norway

  20. Multiplicative model: yt = Tt× St× It Additive model: yt = Tt+ St+ ItHow to calculate Tt, St,and It from yt? • This is done by filtering techniques • One element of this methodology is how to calculate the trend from seasonally adjusted data • This is a question of smoothing a noisy series

  21. 2000-2014

  22. 2007-2012

  23. Smoothing by averaging Pt = (Yt-1+ Yt +Yt+1)/3

  24. Also called filtering Pt = (Yt-2+ Yt-1+ Yt +Yt+1 +Yt+2)/5 The filter is [1,1,1,1,1]/5

  25. Here the filter length is 9

  26. Filtering can be performed twice 3x3 filter 3-term moving average of a 3-term moving average The final filter is [1,2,3,2,1]/9 Pt = (Yt-2+ 2Yt-1+ 3Yt +2Yt+1 +Yt+2)/9 2x12 filter [1/2,1,1,1,1,1,1,1,1,1,1,1,1/2]/12 Also called a centred 12-term moving average Question to the audience: Why is this filter of special interest?

  27. Henderson filters Finding filters with good properties is an interesting topic … Hederson (1916) introduces the so-called Henderson filters X-12-ARIMA uses this type of filter to calculate the trend The filter length determines the degree of smoothing

  28. Question to the audience: Why does the filtered series stop in 2009?

  29. Non-available observations at the end: Two solutions Asymmetric filters Asymmetric variant of Henderson [-0.034,0.116,0.383,0.534,0,0,0] Can be used at the last observation Forecasts in place of the unobserved values The “starting series” for the X12-ARIMA decompositions is a calendar adjusted series which is based on reg-ARIMA modelling The reg-ARIMA modelling can also be used to produced forecasts X12-ARIMA uses these forecasts in trend calculations

  30. Finding the seasonal component by filtering From a series with the trend removed we make 12 series January-values, February-values, … Each of these series is smoothed by filtering Altogether these smoothed series are the seasonal component

  31. The X12-ARIMA algorithm The decomposition is made by several iterative steps Seasonal component from series with trend removed Trend from series with seasonal component removed Initial estimate of trend using the 2x12 moving average One element is downweighting of observations with an extreme irregular component

  32. X12-ARIMA or SEATS Both method can be viewed as filtering techniques X12-ARIMA A non-parametric method No model assumed SEATS The components are assumed to follow ARIMA models The filters are derived from modelling Possible to do inference and to make forecasts with confidence intervals • So why the name X12-ARIMA when this method is the one that is not based on ARIMA? • Answer on the next slide

  33. Calendar adjustment by reg-ARIMA modelling Seasonal ARIMA model Correlated errors (autocorrelation) Differencing the series makes the model quite good without explicit parameters for trend and seasonality Need to decide the type of ARIMA model: ARIMA(p,d,q)(P,D,Q) Regression parameters in the model Calendar effects: Trading day, Moving holyday, … Outliers and level shifts Here y can be a log-transformed and leap-year adjusted variant of the original data "The dirty model" mentioned earlier:

  34. This slide is “stolen” from https://www.scss.tcd.ie/Rozenn.Dahyot/ST7005/15SeasonalARIMA.pdf • Here B is the backshift operator: BYt =Yt-1 • ARIMA(0,1,1)(0,1,1) • Most common model • Airline model

  35. Example of regression variables in reg-ARIMA modelling Easter 2000 and 2001: Easter in April 2008: Easter in March 2002: 4 of 5 Norwegian Easter days in March Trading day Six parameters needed to model seven days Mon: Number of Mondays minus Number of Sundays Easter Mon Tue Wed Thu Fri Sat Jan 2000 0.0000000 0 -1 -1 -1 -1 0 Feb 2000 0.0000000 0 1 0 0 0 0 Mar 2000 -0.2571429 0 0 1 1 1 0 Apr 2000 0.2571429 -1 -1 -1 -1 -1 0 May 2000 0.0000000 1 1 1 0 0 0 Jun 2000 0.0000000 0 0 0 1 1 0 Jul 2000 0.0000000 0 -1 -1 -1 -1 0 Aug 2000 0.0000000 0 1 1 1 0 0 Sep 2000 0.0000000 0 0 0 0 1 1 Oct 2000 0.0000000 0 0 -1 -1 -1 -1 Nov 2000 0.0000000 0 0 1 1 0 0 Dec 2000 0.0000000 -1 -1 -1 -1 0 0 Jan 2001 0.0000000 1 1 1 0 0 0 Feb 2001 0.0000000 0 0 0 0 0 0 Mar 2001 -0.2571429 0 0 0 1 1 1 Apr 2001 0.2571429 0 -1 -1 -1 -1 -1 May 2001 0.0000000 0 1 1 1 0 0 Jun 2001 0.0000000 0 0 0 0 1 1 Jul 2001 0.0000000 0 0 -1 -1 -1 -1 Aug 2001 0.0000000 0 0 1 1 1 0 Sep 2001 0.0000000 -1 -1 -1 -1 -1 0 Oct 2001 0.0000000 1 1 1 0 0 0 Nov 2001 0.0000000 0 0 0 1 1 0 Dec 2001 0.0000000 0 -1 -1 -1 -1 0 Jan 2002 0.0000000 0 1 1 1 0 0 Feb 2002 0.0000000 0 0 0 0 0 0 Mar 2002 0.5428571 -1 -1 -1 -1 0 0 Apr 2002 -0.5428571 1 1 0 0 0 0 May 2002 0.0000000 0 0 1 1 1 0 : : : Mar 2008 0.7428571 0 -1 -1 -1 -1 0 Apr 2008 -0.7428571 0 1 1 0 0 0 May 2008 0.0000000 0 0 0 1 1 1 Jun 2008 0.0000000 0 -1 -1 -1 -1 -1 Jul 2008 0.0000000 0 1 1 1 0 0 Aug 2008 0.0000000 -1 -1 -1 -1 0 0 Sep 2008 0.0000000 1 1 0 0 0 0 Oct 2008 0.0000000 0 0 1 1 1 0 Nov 2008 0.0000000 -1 -1 -1 -1 -1 0 Dec 2008 0.0000000 1 1 1 0 0 0

  36. Trading day:Separate effect of each day or common effect of all weekdays? Question to the audience: Why exactly equal t-values? Regression Model -------------------------------------------------------------- Parameter Standard Variable Estimate Error t-value -------------------------------------------------------------- Trading Day Mon -0.0019 0.00193 -1.00 Tue 0.0064 0.00194 3.31 Wed 0.0018 0.00190 0.94 Thu -0.0016 0.00195 -0.81 Fri 0.0138 0.00188 7.37 Sat 0.0034 0.00193 1.73 *Sun (derived) -0.0219 0.00196 -11.16 Regression Model -------------------------------------------------------------- Parameter Standard Variable Estimate Error t-value -------------------------------------------------------------- Trading Day Weekday 0.0036 0.00053 6.87 **Sat/Sun (derived) -0.0090 0.00131 -6.87

  37. Outliers An extreme observation caused by a special event can be problematic Can influence the modelling in a negative way Parameter estimates Forecasts Decomposition Solution Include the outlier as a dummy variable in the reg-ARIMA modelling ….0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0…. The outlier is included in the irregular component after modelling The observation is still included in seasonally adjusted data But has no effect on the trend • Question to the audience: Examples of special events?

  38. Level shift is handled similar to outliers Regression variable: ….0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1…. Level shift is included in the trend

  39. Presented by • László Sajtos • HungarianCentralStatistical Office

  40. Topics • Seasonaladjustmentstep-by-step • (A few) issuesonseasonaladjustment

  41. Seasonaladjustmentstep-by-step

  42. Seasonaladjustmentstep-by-step: structure Input data STEPS withcheckpoints Notacceptableresults Preliminaryresults Ifresultsareacceptable Output data

  43. Time series analysis (STEP 0) • Basic conditions • Length of time series (enoughlongto be seasonallyadjusted?) • Monthlydatasets: atleast 3-year long • Quarterlydatasets: atleast 4-year long • Atleast 5-7-year longtime series is optimal! • Expertinformation • Collectingexpertdatafromthesectionsaboutdatasets (potentialoutliers, methodologicalchanges, changesinexteriorfactors (e.g. law), connectionstoothertime series and sectors)

  44. Graphicalanalysis, test forseasonality (STEP 1) • Graphicalanalysisviabasic and sophisticatedgraphs Plottedrawdataset Spectralanalysis: autocorrelogram and auto-regressivespectrum • Identifying and explainingmissingobservationsandoutliers • Correction of datafaults • Test forseasonality

  45. Graphicalanalysis, an example(2000-2013) Seasonality Seemsadditive Probablyoutliers Data:Hungarianmonthlyretailvolume index, food

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