1. Methods, Diagnostics, and Practices for Seasonal Adjustment
Catherine C. H. Hood
Introductory Overview Lecture: Seasonal Adjustment

2. Acknowledgements Many thanks to
David Findley, Brian Monsell, Kathy McDonald-Johnson, Roxanne Feldpausch
Agustín Maravall
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

3. Outline Basic concepts
Software packages for seasonal adjustment production
Mechanics of X-12 and SEATS
Overview of current practices
Recent developments in research areas
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

4. Time Series A time series is a set of observations ordered in time
Usually most helpful if collected at regular intervals
In other words, a sequence of repeated measurements of the same concept over regular, consecutive time intervals
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

5. Time Series Data
Occurs in many areas: economics, finance, environment, medicine
Methods for time series are older than those for general stochastic processes and Markov Chains
The aims of time series analysis are to describe and summarize time series data, fit models, and make forecasts
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

6. Why are time series data different from other data?
Data are not independent
Much of the statistical theory relies on the data being independent and identically distributed
Large samples sizes are good, but long time series are not always the best
Series often change with time, so bigger isn’t always better
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

7. What Are Our Users Looking for in an Economic Time Series?
Important features of economic indicator series include
Direction
Turning points
In addition, we want to see if the series is increasing/decreasing more slowly than it was before
Consistency between indicators
Econometric models require seasonally adjusted data.
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

8. Why Do Users Want Seasonally Adjusted Data?
Seasonal movements can make features difficult or impossible to see
Econometric models require seasonally adjusted data.
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

9. Classical Decomposition
One method of describing a time series
Decompose the series into various components
Trend – long term movements in the level of the series
Seasonal effects – cyclical fluctuations reasonably stable in terms of annual timing (including moving holidays and working day effects)
Cycles – cyclical fluctuations longer than a year
Irregular – other random or short-term unpredictable fluctuations
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

10. Causes of Seasonal Effects
Possible causes are
Natural factors
Administrative or legal measures
Social/cultural/religious traditions (e.g., fixed holidays, timing of vacations)
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

11. Causes of Irregular Effects
Possible causes
Unseasonable weather/natural disasters
Strikes
Sampling error
Nonsampling error
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

12. Other Effects
Trading Day: The number of working or trading days in a period
Moving Holidays: Events which occur at regular intervals but not at exactly the same time each year
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

13. May 2007
S M T W T F S
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

14. June 2007
S M T W T F S
1 2
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

15. Moving Holiday Effects
Holidays not at exactly the same time each year
Easter
Labor Day
Thanksgiving
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

16. “Combined” Effects
Trading day and moving holiday effects are both persistent, predictable, calendar-related effects, so trading day and holiday effects often included with the seasonal effects
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

22. The Simple Case The time series would have
No growth or decline from year to year, only rather repetitive within-year movements about an unchanging level
No trading day or moving holidays
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

23. Change in Variations
What if the magnitude of seasonal fluctuations is proportional to level of series?
take logarithms
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

26. Log Transformations
Appropriate when the variability in a series increases as its level increases, and when all values of the series are positive
Change multiplicative relationships into additive relationships
Increases/decreases can be thought of in terms of percentages
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

27. Problem: Extreme Values
Solution:
These effects can be estimated also, but they can be difficult to estimate when seasonality and trend are present
How do you estimate extreme values (outliers) in the presence of seasonality and trend?
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

30. Which of these values are outliers (extreme values)?

31. Trading Day and Other Effects
What if trading day and/or other effects (holiday, outliers) are present?
X-11: TD, holiday regression on the irregular component, extreme value modifications
SEATS: RegARIMA models for a regression on the original series
X-12: Use X-11 methods or RegARIMA models
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

32. Models Multiplicative model: Yt = St´ × Tt × It = St´ × Nt where
St´ = St × TDt × Ht
Additive model:
Yt = St´ + Tt + It
= St´ + Nt
where
St´ = St + TDt + Ht
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

33. Objectives Estimate Nt (remove effects of St ) for seasonal adjustment
Estimate Tt (remove effects of St and It) for trend estimation
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

34. How Do We Estimate the Components?
Seasonal adjustment is normally done with off-the-shelf programs such as:
X-11 or X-12-ARIMA (Census Bureau),
X-11-ARIMA (Statistics Canada),
Decomp, SABL, STAMP,
TRAMO/SEATS (Bank of Spain)
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

35. (Forecasts, Backcasts, and Preadjustments)
RegARIMA Models
(Forecasts, Backcasts, and Preadjustments)
Modeling and Model Comparison Diagnostics
and Graphs
Seasonal Adjustment
Seasonal Adjustment
Diagnostics and Graphs

36. ( ) RegARIMA Model log = ´ Xt + Zt Yt Dt
transformations ARIMA process
Xt = Regressor for trading day and holiday or calendar effects, additive outliers, temporary
changes, level shifts, ramps, and
user-defined effects
Dt = Leap-year adjustment, or “subjective” prior adjustment ( Yt ) Dt
Catherine Hood Consulting

37. ARIMA Models and Forecasting
If we can describe the way the points in the series are related to each other (the autocorrelations), then we can describe the series using the relationships that we’ve found
AutoRegressive Integrated Moving Average Models (ARIMA) are mathematical models of the autocorrelation in a time series
One way to describe time series
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

38. Autocorrelation
The major statistical tool for ARIMA models is the sample autocorrelation coefficient n __ __ 
( Yt – Y )
( Yt-k – Y )
The autocorrelation is the correlation of the time series with itself, lagged by 1,2, or more periods.
which describes the relationship between various values of the time series that are lagged k periods apart.
rk =
t=k+1 n  __
( Yt – Y )2
t=1
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

39. Autocorrelations
r1 indicates how successive values of Y relate to each other,
r2 indicates how Y values two periods apart relate to each other,
and so on.
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

40. ACF
Together, the autocorrelations at lags 1, 2, 3, etc. make up the autocorrelation function or ACF and then we plot the autocorrelations by the lags
The ACF values reflect how strongly the series is related to its past values over time
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

41. Autoregressive Processes
The autoregressive process of order p is denoted AR(p), and defined by
Zt =  r Zt-r + wt
where 1 , , p are fixed constants and {wt} white noise, a sequence of independent (or uncorrelated) random variables with mean 0 and variance  2 p
r=1
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

42. Moving Average Processes
The moving average process of order q, denoted MA(q), includes lagged error terms t–1 to t–q, written as
Zt = wt –  r wt-r
where 1 , 2 , … , q are the MA parameters and wt is white noise q
r=1
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

43. Random Walk Constrained AR Model Zt = Zt-1 + wt with 1 = 1
First differenced model
Zt = Zt-1 + wt Zt – Zt-1 = wt
(1 – B) Zt = wt
Seasonal difference model
Zt – Zt-12 = wt
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

44. ARMA processes
The autoregressive moving average process, ARMA(p,q) is defined by
Zt –  r Zt–r =  r wt–r
where again wt is white noise p q
r=1
r=0
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

45. Seasonal Processes A seasonal AR process Zt =  r Zt-Sr + wt
A seasonal MA process
Zt = wt –  Θr wt-r
where 1 , , P and Θ1 , … , ΘQ are fixed constants, {wt} is white noise, and S is the frequency of the series (12 for monthly or 4 for quarterly) p
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

46. ( ) RegARIMA Model log = ´ Xt + Zt Yt Dt
transformations ARIMA process
Xt = Regressor for trading day and holiday or calendar effects, additive outliers, temporary
changes, level shifts, ramps, and
user-defined effects
Dt = Leap-year adjustment, or “subjective” prior adjustment ( Yt ) Dt
Catherine Hood Consulting

47. RegARIMA Model Uses
Extend the series with forecasts (or possibly backcasts)
Detect and adjust for outliers to improve the forecasts and seasonal adjustments
Estimate missing data
Detect and directly estimate trading day effects and other effects (e.g. moving holiday effects, user-defined effects)
Forecasting isn’t just for seasonal adjustment. FTD is using forecasts from X-12 for quality control.
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

48. Automatic Procedures
Both X-12-ARIMA and SEATS have procedures for the automatic identification of
ARIMA model
Outliers
Trading Day effects
Easter effects
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

49. (Forecasts, Backcasts, and Preadjustments)
RegARIMA Models
(Forecasts, Backcasts, and Preadjustments)
Modeling and Model Comparison Diagnostics
and Graphs
Seasonal Adjustment
Seasonal Adjustment
Diagnostics and Graphs

50. How are component estimates formed?
X-11, X-12: limited set of fixed filters
ARIMA Model-based (AMB):
Fit ARIMA model to series
This model, plus assumptions, determine component models
Signal extraction to produce component estimates and mean squared errors (MSE)
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

51. Example Trend Filter from X-12-ARIMA
A centered 12-term moving average
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

52. Example: 3x3 Filters 3 x 3 filter for Qtr 1, 1990 (or Jan 1990) 9
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

53. Example Seasonal Filter from X-12-ARIMA: 3x3 Filter
Recall that Y = TSI, so SI = Y/T, i.e., the detrended series
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

54. AMB Approach Fit RegARIMA model yt = x´t  + Zt
Given an ARIMA model for series Zt,
 (B)  (B) Zt = Θ (B)  (B) wt
and the model Yt = St + Nt , determine models for components St and Nt
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

55. Where . . .
St independent of Tt independent of It (  St independent of Nt )
St , Tt , It follow ARIMA models consistent with the model for Zt (hence so does Nt)
It is white noise (or low order MA)
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

56. Canonical Decomposition
Problem: There is more than one admissible decomposition
Solution: Use the canonical decomposition, the decomposition that corresponds to minimizing the white noise in the seasonal component
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

57. Properties of the Canonical Decomposition
Unique (and usually exists)
Minimizes innovation variances of seasonal and trend; maximizes irregular variance
Forecasts of St follow a fixed seasonal pattern
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

58. Advantages of AMB Seasonal Adjustment
Flexible approach with a wide range of models and parameter values
Model selection can be guided by accepted statistical principals
Filters are tailored to individual series through parameter estimation, and are “optimal” given
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

59. Advantages of AMB Seasonal Adjustment (2)
Signal extraction calculations provide error variances of component estimates with MSE based on the model
Approach easily extends (in principle) to accommodate a sampling error component
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

60. At the End of the Series
X-11: asymmetric filters (from ad-hoc modifications to symmetric filters)
X-11-ARIMA, X-12: one year (optionally longer) forecast extension
AMB: full forecast extension
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

61. Issues Relating to Current Practices
X-12 versus SEATS
Use of RegARIMA models, for forecasting, trading day, holidays, etc.
Diagnostics
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

62. Agreement in Current Practices
Compute the concurrent factors (running the seasonal adjustment software every month with the most recent data) instead of projected factors
Use regARIMA models whenever possible (ARIMA models required for SEATS)
Continue to publish the original series along with the seasonal adjustment
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

63. X-12 vs SEATS Eurostat recommends use of either program
US Census Bureau recommends use of X-12-ARIMA
According to research, X-12 is more accurate than SEATS for most series
X-12 works better for short series (4 to 7 years) and for longer series (over 15 years)
X-12 has better diagnostics
In practice, I’ve noticed with SEATS smoother adjustments with smaller revisions for some of our irregular series
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

64. Setting Options
To reduce revisions, best to set certain options for production
Most agencies let the software choose the options and then fix the settings for production
Problems come with SEATS because model used is not always the model specified, and model coefficients also are not always the ones specified
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

65. Trading Day and Moving Holiday Settings
In Europe, there has been a lot of work on “user-defined” variables that include trading days and moving holidays to incorporate country-specific holidays
Most agencies in the U.S. use built-in trading day and built-in moving holidays from X-12-ARIMA
Unfortunately, not all the built-in variables are useful for every situation
Some agencies avoid trading day altogether
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

66. Outlier Settings
At the Australia Bureau of Statistics, they have a very rigorous procedure of outlier identification, including meta data on certain unusual events
Most other agencies use the automatic outlier selection procedure
At the U.S. Census Bureau
Choose new outliers with every run
At annual review time, set outliers for current data and set a high critical value for the new data coming in
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

67. Direct/Indirect Definitions
If a time series is a sum (or other composite) of component series
Direct adjustment – a seasonal adjustment of the aggregate series obtained by seasonally adjusting the sum of the component series
Indirect adjustment – a seasonal adjustment of the aggregate series obtained from the sum of the seasonally adjusted component series
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

68. Example – Direct and Indirect Adjustment
US = NE + MW + SO + WE
Indirect seasonal adjustment of US:
SA(NE) + SA(MW) + SA(SO) + SA(WE)
Direct seasonal adjustment of US: SA( NE + MW + SO + WE )
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

69. Comment on Yearly Totals
When do yearly totals of the original series and the seasonally adjusted series coincide?
When the series has
An additive decomposition
A seasonal pattern that is fixed from one year to the next
No trading adjustments
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

70. Areas for Improvement in Current Practices
Concurrent adjustment
Use of regARIMA models
Moving holidays and other user-defined effects
Setting options (to reduce revisions) and checking the options regularly
Software to make it easier to check diagnostics regularly
Training in ARIMA modeling and diagnostics
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

71. Recent Developments and Research Areas
X-13 (X-13-SEATS)
Improved and new diagnostics (for both X-12 and SEATS)
New filters for X-12 and new, more flexible models for SEATS
Supplemental and utility software
Documentation and training
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

72. Newest X-12
Version 0.3 includes a new automatic ARIMA-modeling procedure based on the program TRAMO from the Bank of Spain
The next release (X-13) will include ARIMA-model-based seasonal adjustment options
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

73. Model-based Adjustment
SEATS, developed by Agustín Maravall at the Bank of Spain
REGCMPT, developed by Bill Bell at the Census Bureau
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

74. SEATS Disadvantages No diagnostics for the adjustment
No methods for series with different variability in different months
No user-defined regressors
Not very flexible ARIMA models
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

75. REGCMPT Advantages Still being tested
Methods for different variability in different months
Can build very flexible regARIMA models
Still being tested
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

76. X-13-SEATS Advantages Disadvantage
Would combine the model-based adjustments from SEATS with diagnostics from X-12, and keep the ability to use X-11-type adjustments also
Disadvantage
????
Possibly, within the next few years, besides all the other choices we have to make, we’ll also have to decide between X11 adjustments and SEATS adjustments for production work at the Bureau
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

77. Running in Windows TRAMO/SEATS for Windows
Windows Interface to X-12-ARIMA
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

78. Supplemental Software
X-12-Graph in SAS and in R
X-12-Data and X-12-Rvw
Programs to help write user-defined variables for custom trading day and moving holidays
Excel interfaces to run SEATS and X-12 from Excel
Interfaces to other software are available
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

79. Documentation and Training
“Getting Started” papers to use with the Windows version, written for novice users
Documentation on commonly used options for both X-12 and SEATS
Training
Advanced Diagnostics
RegARIMA Modeling
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

80. Resources X-12-ARIMA website www.census.gov/srd/www/x12a
Seasonal adjustment papers pages
TRAMO/SEATS website
Papers and course information
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

81. Contact Information
Catherine Hood
Catherine Hood Consulting
1090 Kennedy Creek Road
Auburntown, TN
Telephone: (615)
Web:
Catherine Hood Consulting
Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007