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

Methods, Diagnostics, andPractices for Seasonal

Adjustment

Catherine C. H. Hood Introductory Overview Lecture: Seasonal Adjustment

Methods, Diagnostics, and Practices for Seasonal Adjustment---June 2007

Catherine Hood Consulting

2© 2007 Catherine C.H. Hood

Acknowledgements

Many thanks to ● David Findley, Brian Monsell, Kathy

McDonald-Johnson, Roxanne Feldpausch● Agustín Maravall




Outline

Basic concepts Software packages for seasonal

adjustment production● Mechanics of X-12 and SEATS

Overview of current practices Recent developments in research areas




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




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




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




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




Why Do Users Want Seasonally Adjusted Data?

Seasonal movements can make features difficult or impossible to see




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




Causes of Seasonal Effects

Possible causes are● Natural factors● Administrative or legal measures● Social/cultural/religious traditions

(e.g., fixed holidays, timing of vacations)




Causes of Irregular Effects

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




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




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




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




Moving Holiday Effects

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




“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




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




Change in Variations

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




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




Problem: Extreme Values

Solution:● These effects can be estimated also, but

they can be difficult to estimate when seasonality and trend are present

Which of these values are outliers (extreme values)?




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




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




Objectives

Estimate Nt (remove effects of St ) for seasonal adjustment

Estimate Tt (remove effects of St and It) for trend estimation




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)

Seasonal Adjustment

RegARIMA Models(Forecasts, Backcasts, and Preadjustments)

Modeling and Model Comparison Diagnostics and Graphs

Seasonal Adjustment Diagnostics and Graphs

RegARIMA Model

log = ´ Xt + Zt

transformations ARIMA process

Xt = Regressor for trading day and holiday or calendar effects, additive outliers,

temporarychanges, level shifts, ramps, and user-defined effects

Dt = Leap-year adjustment, or “subjective” prior adjustment

( )Yt

Dt





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




Autocorrelation The major statistical tool for ARIMA

models is the sample autocorrelation coefficient

rk =

( Yt – Y )2

t=k+1

n

t=1

n

__ __

__

( Yt-k – Y )( Yt – Y )




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.




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




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

r=1

p




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

r=1

q




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 modelZt – Zt-12 = wt




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

qp

r=1 r=0




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

RegARIMA Model

log = ´ Xt + Zt

transformations ARIMA process

Xt = Regressor for trading day and holiday or calendar effects, additive outliers,

temporarychanges, level shifts, ramps, and user-defined effects

Dt = Leap-year adjustment, or “subjective” prior adjustment

( )Yt

Dt





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)




Automatic Procedures

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

Seasonal Adjustment

RegARIMA Models(Forecasts, Backcasts, and Preadjustments)

Modeling and Model Comparison Diagnostics and Graphs

Seasonal Adjustment Diagnostics and Graphs




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)




Example Trend Filter from X-12-ARIMA

A centered 12-term moving average




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




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

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




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




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)




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




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




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




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




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




Issues Relating to Current Practices

X-12 versus SEATS Use of RegARIMA models, for

forecasting, trading day, holidays, etc. Diagnostics




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




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




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




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




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




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




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 )




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




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




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




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




Model-based Adjustment

SEATS, developed by Agustín Maravall at the Bank of Spain

REGCMPT, developed by Bill Bell at the Census Bureau




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




REGCMPT

Advantages● Methods for different variability in

different months● Can build very flexible regARIMA

models Still being tested




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● ????




Running in Windows

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




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




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




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

http://www.census.gov/srd/www/x12a

http://www.bde.es/english/

http://www.catherinechhood.net/




Contact Information

Catherine HoodCatherine Hood Consulting1090 Kennedy Creek RoadAuburntown, TN 37016-9614

Telephone: (615) 408-5021 Email: [email protected]

Web: www.catherinechhood.net

mailto:[email protected]

http://www.catherinechhood.net/

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