Methods, Diagnostics, and Practices for Seasonal Adjustment

1
Methods, Diagnostics, andPractices for
SeasonalAdjustment

• 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

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

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

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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

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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 isnt
  always better

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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

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Why Do Users Want Seasonally Adjusted Data?

• Seasonal movements can make features difficult
  or impossible to see

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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

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Causes of Seasonal Effects

• Possible causes are
• Natural factors
• Administrative or legal measures
• Social/cultural/religious traditions (e.g., fixed
  holidays, timing of vacations)

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Causes of Irregular Effects

• Possible causes
• Unseasonable weather/natural disasters
• Strikes
• Sampling error
• Nonsampling error

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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

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May 2007

• S M T W T F S
• 1 2 3 4 5
• 6 7 8 9 10 11 12
• 13 14 15 16 17 18 19
• 20 21 22 23 24 25 26
• 27 28 29 30 31

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June 2007

• S M T W T F S
• 1 2
• 3 4 5 6 7 8 9
• 10 11 12 13 14 15 16
• 17 18 19 20 21 22 23
• 24 25 26 27 28 29 30

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Moving Holiday Effects

• Holidays not at exactly the same time each year
• Easter
• Labor Day
• Thanksgiving

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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

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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

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Change in Variations

• What if the magnitude of seasonal fluctuations is
  proportional to level of series?
• take logarithms

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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

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Problem Extreme Values

• Solution
• These effects can be estimated also, but they can
  be difficult to estimate when seasonality and
  trend are present

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• Which of these values are outliers (extreme
  values)?

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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

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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

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Objectives

• Estimate Nt (remove effects of St ) for seasonal
  adjustment
• Estimate Tt (remove effects of St and It) for
  trend estimation

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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)

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

• log ? Xt Zt
• 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
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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 weve found
• AutoRegressive Integrated Moving Average Models
  (ARIMA) are mathematical models of the
  autocorrelation in a time series
• One way to describe time series

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Autocorrelation

• The major statistical tool for ARIMA models is
  the sample autocorrelation coefficient

__
__
n
?
( Yt-k Y )
( Yt Y )
rk
tk1
n
?
__
( Yt Y )2
t1
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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.

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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

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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
r1
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Moving Average Processes

• The moving average process of order q, denoted
  MA(q), includes lagged error terms t1 to tq,
  written as
• Zt wt ? ?r wt-r
• where ?1 , ?2 , , ?q are the MA parameters and
  wt is white noise

q
r1
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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

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ARMA processes

• The autoregressive moving average process,
  ARMA(p,q) is defined by
• Zt ? ?r Ztr ? ?r wtr
• where again wt is white noise

q
p
r1
r0
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Seasonal Processes

• A seasonal AR process
• Zt ? ?r Zt-Sr wt
• A seasonal MA process
• Zt wt ? Tr wt-r
• where ?1 , . . . , ?P and T1 , , TQ are fixed
  constants, wt is white noise, and S is the
  frequency of the series (12 for monthly or 4 for
  quarterly)

p
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RegARIMA Model

• log ? Xt Zt
• 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
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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)

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Automatic Procedures

• Both X-12-ARIMA and SEATS have procedures for the
  automatic identification of
• ARIMA model
• Outliers
• Trading Day effects
• Easter effects

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

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Example Trend Filter from X-12-ARIMA

• A centered 12-term moving average

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Example 3x3 Filters

• 3 x 3 filter for Qtr 1, 1990 (or Jan 1990)
• 1988.1 1989.1 1990.1
• 1989.1 1990.1 1991.1
• 1990.1 1991.1 1992.1
• 9

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Example Seasonal Filter from X-12-ARIMA 3x3
Filter

• Recall that Y TSI, so SI Y/T, i.e., the
  detrended series

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AMB Approach

• Fit RegARIMA model yt xt ? Zt
• Given an ARIMA model for series Zt,
• ? (B) ? (B) Zt T (B) ? (B) wt
• and the model Yt St Nt , determine models
  for components St and Nt

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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)

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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

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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

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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

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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

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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

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Issues Relating to Current Practices

• X-12 versus SEATS
• Use of RegARIMA models, for forecasting, trading
  day, holidays, etc.
• Diagnostics

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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

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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

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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

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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

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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

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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

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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 )

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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

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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

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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

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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

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Model-based Adjustment

• SEATS, developed by Agustín Maravall at the Bank
  of Spain
• REGCMPT, developed by Bill Bell at the Census
  Bureau

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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

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REGCMPT

• Advantages
• Methods for different variability in different
  months
• Can build very flexible regARIMA models
• Still being tested

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X-13-SEATS

• Advantages
• 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
• ????

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Running in Windows

• TRAMO/SEATS for Windows
• Windows Interface to X-12-ARIMA

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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

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Documentation and Training

• Documentation
• 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

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Resources

• X-12-ARIMA website www.census.gov/srd/www
  /x12a
• Seasonal adjustment papers pages
• TRAMO/SEATS website
• www.bde.es/english/
• Papers and course information www.catherinechhood
  .net

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Contact Information

• Catherine Hood
• Catherine Hood Consulting
• 1090 Kennedy Creek Road
• Auburntown, TN 37016-9614
• Telephone [phone]
• Email cath_at_catherinechhood.net
• Web www.catherinechhood.net

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