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

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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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 isnt always better

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 8 Why Do Users Want Seasonally Adjusted Data? Seasonal movements can make features difficult or impossible to see

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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)

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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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

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

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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 27 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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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 32 Models Multiplicative model: Y t = S t ´ × T t × I t = S t ´ × N t where S t ´ = S t × TD t × H t Additive model: Y t = S t ´ + T t + I t = S t ´ + N t where S t ´ = S t + TD t + H t

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 33 Objectives Estimate N t (remove effects of S t ) for seasonal adjustment Estimate T t (remove effects of S t and I t ) for trend estimation

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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)

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

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RegARIMA Model log = ´ X t + Z t transformationsARIMA process X t = Regressor for trading day and holiday or calendar effects, additive outliers, temporary changes, level shifts, ramps, and user-defined effects D t = Leap-year adjustment, or subjective prior adjustment () YtYt DtDt Catherine Hood Consulting

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 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 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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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 38 Autocorrelation The major statistical tool for ARIMA models is the sample autocorrelation coefficient r k = ( Y t – Y ) 2 t=k+1 n t=1 n __ ( Y t-k – Y )( Y t – Y )

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 39 Autocorrelations r 1 indicates how successive values of Y relate to each other, r 2 indicates how Y values two periods apart relate to each other, and so on.

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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 Z t = w t – r w t-r where 1, 2, …, q are the MA parameters and w t is white noise r=1 q

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 43 Random Walk Constrained AR Model Z t = Z t-1 + w t with 1 = 1 First differenced model Z t = Z t-1 + w t Z t – Z t-1 = w t (1 – B) Z t = w t Seasonal difference model Z t – Z t-12 = w t

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 44 ARMA processes The autoregressive moving average process, ARMA(p,q) is defined by Z t – r Z t–r = r w t–r where again w t is white noise qp r=1r=0

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 45 Seasonal Processes A seasonal AR process Z t = r Z t-Sr + w t A seasonal MA process Z t = w t – Θ r w t-r where 1,..., P and Θ 1, …, Θ Q are fixed constants, {w t } 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 = ´ X t + Z t transformationsARIMA process X t = Regressor for trading day and holiday or calendar effects, additive outliers, temporary changes, level shifts, ramps, and user-defined effects D t = Leap-year adjustment, or subjective prior adjustment () YtYt DtDt Catherine Hood Consulting

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 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)

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 48 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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Seasonal Adjustment RegARIMA Models (Forecasts, Backcasts, and Preadjustments) Modeling and Model Comparison Diagnostics and Graphs Seasonal Adjustment Diagnostics and Graphs

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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)

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

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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 53 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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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 54 AMB Approach Fit RegARIMA model y t = x´ t + Z t Given an ARIMA model for series Z t, (B) (B) Z t = Θ (B) (B) w t and the model Y t = S t + N t, determine models for components S t and N t

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 55 Where... S t independent of T t independent of I t ( S t independent of N t ) S t, T t, I t follow ARIMA models consistent with the model for Z t (hence so does N t ) I t is white noise (or low order MA)

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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 )

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 73 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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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 76 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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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 77 Running in Windows TRAMO/SEATS for Windows Windows Interface to X-12-ARIMA

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 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

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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 79 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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© 2007 Catherine C.H. Hood Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007 Catherine Hood Consulting 80 Resources X-12-ARIMA website Seasonal adjustment papers pages TRAMO/SEATS website Papers and course information

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

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