7/25/2019 Seasonal Adjustment Lecture JULY2012
1/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Seasonal Adjustment
Seasonal adjustment of time series is mainly the
isolation of seasonal fluctuations. It consists of the
identification, estimation and removal of seasonal
variations and effect of trading days and moving
holidays (if present) from a time series.
After removal of seasonal variations, the resulting
series is referred to as seasonally adjusted series or
deseasonalized series.
7/25/2019 Seasonal Adjustment Lecture JULY2012
2/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Why is seasonal adjustment done?Seasonal adjustment is done to simplify data so that
they may be more easily interpreted by statistically
unsophisticated users without a significant loss ofinformation. (Bell and Hellmer, 1992)
Seasonal adjustment is mainly carried out for policy
makers or advisers who wish to be able, at a glance,
to read the trend of an economic time series without
being hampered by seasonal movements.
In the study of business cycles, seasonal adjustment is
essential when we want to estimate the trend-cycle
component.
7/25/2019 Seasonal Adjustment Lecture JULY2012
3/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Example of Actual and Its Seasonally Adjusted Series
7/25/2019 Seasonal Adjustment Lecture JULY2012
4/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Tests for Seasonality
Seasonality can be detected graphically, using multiple
line charts. However, in cases where presence ofseasonality is not clearly seen through visual
inspection, there are two commonly used statistical
tests for detecting presence of seasonality: the
Kruskal-Wallis test and the F-test based on the
analysis of variance using a linear regression model.
7/25/2019 Seasonal Adjustment Lecture JULY2012
5/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Decomposition of Time Series
An observed time series, yt, can be decomposed into four
components namely: trend (ytt), cycle (yt
c), seasonality (yts), and
irregularity (yti). For short series, it is difficult to disaggregate
the cycle from the trend and the two components are combinedinto the trend-cycle (yt
tc) component. Two decomposition
models are commonly used in relating the observed value with
its four components.
a. Additive Model: yt= yttc + yt
s + yti
b. Multiplicative Model: yt= yttc x yt
s x yti
Two other available decompositions are the log additive and thepseudo-additive decompositions, with the latter defined as,
yt= yttc x (yt
s + yti 1)
7/25/2019 Seasonal Adjustment Lecture JULY2012
6/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Multiplicative vs. Additive Decomposition
When the parameters describing the time series are not
changing over time, the time series can be modeled
adequately by the additive decomposition method. An
example is the unemployment rate.
When the time series exhibits increasing seasonal variation,
then the appropriate model is the multiplicative model. An
example is the number of tourist arrivals.
The bulk of economic time series handled by the U.S. Bureau
of Census and the U.S. Bureau of Labor Statistics are adjusted
using multiplicative decomposition. The Federal Reserve usesthe additive version more frequently because of the nature of
the time series it treats.
7/25/2019 Seasonal Adjustment Lecture JULY2012
7/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Example of Quarterly Data Showing Seasonality
7/25/2019 Seasonal Adjustment Lecture JULY2012
8/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Example of Monthly Data Showing Seasonality
7/25/2019 Seasonal Adjustment Lecture JULY2012
9/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Decomposition of Time Series
Since the values of the components of the observed series are
not known, these are estimated.
A series of less than 30 years of data is usually consideredshort when the purpose is to estimate the cycle.
To do seasonal adjustment, it is suggested that 5 to 15 years of
data points be used. This is to ensure that sufficient data isavailable to estimate the seasonal component.
A more complete decomposition includes trading day
variations (yttd) and Easter or moving holiday effects (yt
E) and
with yti partitioned into well-behaved noise (yt
i) and extreme
values (et).
7/25/2019 Seasonal Adjustment Lecture JULY2012
10/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Decomposition of Time Series
The more complete models are,
for additive decomposition, and
for multiplicative decomposition.
7/25/2019 Seasonal Adjustment Lecture JULY2012
11/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Decomposition of Time Series
The seasonally adjusted series for the additive model is,
For the multiplicative model, the seasonally adjusted series is,
7/25/2019 Seasonal Adjustment Lecture JULY2012
12/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
For series which exhibits much irregularity, and
consequently with et dominating it, an alternative
series to ytadj is the trend-cycle component, yttc.
The trend-cycle component, yttc, will show the trends
without being hampered not just by seasonality butalso by the high irregularity.
For short term indicators, most analysts prefer the
trend-cycle estimates than seasonally adjustedestimates.
Trend-Cycle Component or Seasonally Adjusted
7/25/2019 Seasonal Adjustment Lecture JULY2012
13/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Decomposition ProcessX11 and X12
7/25/2019 Seasonal Adjustment Lecture JULY2012
14/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Some Common Procedures for
Seasonal Adjustment
The majority of seasonal adjustment procedures being used
are based on univariate techniques and estimation of thecomponents of a time series is done in a simple automatic
manner.
Two broad classifications of seasonal adjustment methodsare:
a) those based on regression and linear estimation
theory; and
b) those based on the application of linear smoothing
filters or moving averages.
7/25/2019 Seasonal Adjustment Lecture JULY2012
15/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Most statistical agencies use methods based on
moving averages for seasonal adjustment.
The two most commonly used are the U.S. Bureau
of Census X11-Method II Variant and Statistics
Canadas X11 ARIMA.
These two methods follow an iterative estimation
procedure involving the major steps in the
decomposition of a time series.
Seasonal Adjustment Procedures
7/25/2019 Seasonal Adjustment Lecture JULY2012
16/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Main Steps in X11 ARIMA (Version 2000)
X11 ARIMA uses the Census X11 procedure on augmented data -
the time series plus one year of monthly or quarterly forecasts and
one year of backcasts from an ARIMA model. The X11 ARIMA
basically consists of:
a)modeling the original series using an ARIMA or Box-Jenkins Model;
b)forecasting one year of unadjusted data at each end of the series from
ARIMA models that fit and project the original series well; and
c)seasonally adjusting the augmented series using X11-Method II
variant.
The Easter and trading-day adjustments are applied even before a)
is done if one asks for it.
7/25/2019 Seasonal Adjustment Lecture JULY2012
17/24
TRAMO-SEATS
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
TRAMO - Time Series Regression with ARIMA Noise,
Missing Observations, and Outliers
SEATS - Signal Extraction in ARIMA Time Series
7/25/2019 Seasonal Adjustment Lecture JULY2012
18/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
A program for estimation and forecasting regression
models with possibly non-stationary (ARIMA) errors
and any sequence of missing values.The program interpolates these values, identifies and
corrects for several types of outliers, and estimates
special effects such as Trading Day and Easter andintervention variable type of effects.
Fully automatic model identification and outlier
correction procedures are available.
The program can pre-test for the level v. log
specification.
TRAMO
7/25/2019 Seasonal Adjustment Lecture JULY2012
19/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
SEATS
A program for estimation of unobserved components in
time series following the Auto-Regressive Integrated
Moving Average model based method.
The Trend, Seasonal, Irregular, and cyclical
components are estimated and forecasted with signal
extraction techniques (Kalman Filter) applied toARIMA models.
In Seasonal Adjustment, TRAMO pre-adjusts the series
to be adjusted by SEATS.
TRAMO-SEATS Program is due to Victor Gomez and
Agustin Maravall
7/25/2019 Seasonal Adjustment Lecture JULY2012
20/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
X12 and TRAMO/SEATS
X12 and TRAMO/SEATS are seasonal adjustment
procedures based on extracting components from a givenseries.
X12 uses a non-parametric moving average based method
to extract its components. TRAMO/SEATS bases itsdecomposition on an estimated parametric ARIMA model.
The main difference between the two methods is that X12
does not allow for missing values while TRAMO/SEATSwill interpolate the missing values based on an estimated
ARIMA model.
7/25/2019 Seasonal Adjustment Lecture JULY2012
21/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Hodrick-Prescott Filter - Permanent Component
This is a smoothing method that is widely used among
macroeconomists to obtain a smooth estimate of the long-
term trend component of a series. The method was firstused in a working paper (circulated in the early 1980's and
published in 1997) by Hodrick and Prescott to analyze
postwar U.S. business cycles.
The Hodrick-Prescott (HP) filter computes the permanent
component (TRt) of a series ytby minimizing the variance
of yt around TRt, subject to a penalty that constrains the
second difference of TRt.
7/25/2019 Seasonal Adjustment Lecture JULY2012
22/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Hodrick-Prescott Filter - Permanent Component
is the penalty parameter that controls for thesmoothness of the series. The default values for are:
That is, the HP filter chooses TRtto minimize:
7/25/2019 Seasonal Adjustment Lecture JULY2012
23/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Hodrick-Prescott Filter - Permanent Component
The parameter controls for the smoothness of the
series, by controlling the ratio of the variance of thecyclical component and the variance of the series.
The larger the , the smoother the TRt approaches
the linear trend.
King and Rabelo (1993) showed that the HP filter
can render stationary any integrated process of up to
the fourth order.
7/25/2019 Seasonal Adjustment Lecture JULY2012
24/24
SCHOOL OF STATISTICS, UNIVERSITY OF THE PHILIPPINES
Hodrick-Prescott Filter - Permanent Component
The HP has some disadvantages. Harvey and Jaeger (1993)
showed that the use of HP filter can lead to identification of
spurious cyclical behavior.
Moreover, users of HP filter should not be interested in data
points near the beginning or the end of the sample. Baxter and
King (1995) recommended that three years of data be droppedat both ends of the time series when the HP filter is applied
for quarterly or annual data.
Other extraction procedures: Baxter and King; Christiano andFitzgerald;