Seasonal Adjustment Lecture JULY2012

7/25/2019 Seasonal Adjustment Lecture JULY2012

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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.


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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.


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Example of Actual and Its Seasonally Adjusted Series


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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.


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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)


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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.


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Example of Quarterly Data Showing Seasonality


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Example of Monthly Data Showing Seasonality


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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).


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The more complete models are,

for additive decomposition, and

for multiplicative decomposition.


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The seasonally adjusted series for the additive model is,

For the multiplicative model, the seasonally adjusted series is,


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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


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Decomposition ProcessX11 and X12


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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.


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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


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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.


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TRAMO-SEATS


TRAMO - Time Series Regression with ARIMA Noise,

Missing Observations, and Outliers

SEATS - Signal Extraction in ARIMA Time Series


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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


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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


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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.


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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.


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is the penalty parameter that controls for thesmoothness of the series. The default values for are:

That is, the HP filter chooses TRtto minimize:


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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.


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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;

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Seasonal Adjustment Lecture JULY2012

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