Before one can specify a model for a given data set, one must have an initial guess about the data generation process. The first step is always to plot the time series. In most cases such a plot gives first answers to questions like: ''Is the time series under consideration stationary?'' or ''Do the time series show a seasonal pattern?''
Figure 5.1 displays the quarterly unemployment rate for Germany
(West) from the first quarter of 1962 to the forth quarter of 1991. The data are
published by the OECD (Franses; 1998, Table DA.10).
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The solid line represents the original series and the dashed line shows the
seasonally adjusted series. It is easy to see, that this quarterly time series possesses
a distinct seasonal pattern with spikes recurring always in the first quarter of the
year.
After the inspection of the plot, one can use the sample autocorrelation function (ACF) and
the sample partial autocorrelation function (PACF) to specify the order of the ARMA part (see
acf
,
pacf
,
acfplot
and
pacfplot
). Another convenient tool for
first stage model specification is the extended autocorrelation function (EACF), because the
EACF does not require that the time series under consideration is stationary and it allows a
simultaneous specification of the autoregressive and moving average order. Unfortunately, the
EACF can not be applied to series that show a seasonal pattern. However, we will present the
EACF later in Section 5.4.5, where we use it for checking the residuals
resulting from the fitted models.
Figures 5.2, 5.3 and 5.4 display the sample ACF
of three different transformations of the unemployment rate () for Germany.
Using the difference--or backshift--operator , these kinds of transformations of the
unemployment rate can be written compactly as
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where operates as
and
denotes the seasonal period.
and
stand for nonseasonal and seasonal differencing. The superscripts
and
indicate that, in general, the differencing may be applied
and
times.
Figure 5.2 shows the sample ACF of the original data of the unemployment rate
(). The fact, that the time series is neither subjected to nonseasonal nor to seasonal
differencing, implies that
. Furthermore, we set
, since the unemployment rate is
recorded quarterly. The sample ACF of the unemployment rate declines very slowly, i.e. that
this time series is clearly nonstationary. But it is difficult to isolate any seasonal
pattern as all autocorrelations are dominated by the effect of the nonseasonal unit root.
Figure 5.3 displays the sample ACF of the first differences of the
unemployment rate (
) with
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Since this transformation is aimed at eliminating only the nonseasonal unit root, we set
and
. Again, we set
because of the frequency of the time series under
consideration.
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Taking the first differences produces a very clear pattern in the sample ACF. There are
very large positive autocorrelations at the seasonal frequencies (lag 4, 8, 12, etc.),
flanked by negative autocorrelations at the 'satellites', which are the autocorrelations
right before and after the seasonal lags. The slow decline of the seasonal
autocorrelations indicates seasonal instationarity. Analogous to the analysis of
nonseasonal nonstationarity, this may be dealt by seasonal differencing; i.e. by applying
the
operator in conjunction with the usual lag operator
(Mills; 1990, Chapter 10).
Eventually, Figure 5.4 displays the sample ACF of the unemployment rate that was subjected to the final transformation
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|
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Since this transformation is used to remove both the nonseasonal and the seasonal unit root,
we set . What the transformation
finally does is seasonally
differencing the first differences of the unemployment rate. By means of this transformation
we obtain a stationary time series that can be modeled by fitting an appropriate ARMA model.
After this illustrative introduction, we can now switch to theoretical considerations. As we
already saw in practice, a seasonal model for the time series
may take the
following form
where
and
indicate nonseasonal and seasonal
differencing and
gives the season.
represents a white noise innovation.
and
are the usual AR and MA lag operator polynomials for ARMA
models
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and
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Since the and
must account for seasonal autocorrelation, at least
one of them must be of minimum order
. This means that the identification of models of
the form (5.1) can lead to a large number of parameters that have to be
estimated and to a model specification that is rather difficult to interpret.
Box and Jenkins (1976) developed an argument for using a restricted version of equation (5.1), that should be adequate to fit many seasonal time series. Starting point for their approach was the fact, that in seasonal data there are two time intervals of importance. Suppose, that we still deal with a quarterly series, we expect the following to occur (Mills; 1990, Chapter 10):
Referring to Figure 5.1 that displays the quarterly unemployment rate for Germany, it is obvious that the seasonal effect implies that an observation in the first quarter of a given year is related to the observations of the first quarter for previous years. We can model this feature by means of a seasonal model
and
stand for a seasonal AR polynomial
of order
and a seasonal MA polynomial of order
respectively:
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and
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which satisfy the standard stationarity and invertibility conditions. denotes the error
series. The characteristics of this process are explained below.
It is obvious that the above given seasonal model (5.2) is simply a special case of the usual ARIMA model, since the autoregressive and moving average relationship is modeled for observations of the same seasonal time interval in different years. Using equation (5.2) relationships between observations for the same quarters in successive years can be modeled.
Furthermore, we assume a relationship between the observations for successive quarters of a
year, i.e. that the corresponding error series (
, etc.) may be
autocorrelated. These autocorrelations may be represented by a nonseasonal model
is ARIMA
with
representing a process of innovations (white noise
process).
Substituting (5.3) into (5.2) yields the general multiplicative seasonal model
In equation (5.4) we additionally include the constant term in
order to allow for a deterministic trend in the model (Shumway and Stoffer; 2000). In the
following we use the short-hand notation SARIMA (
) to
characterize a multiplicative seasonal ARIMA model like (5.4).
Before to start with the issues of identification and estimation
of a multiplicative SARIMA model a short example may be helpful,
that sheds some light on the connection between a multiplicative
SARIMA (
) and a simple ARMA (
) model
and reveals that the SARIMA methodology leads to parsimonious
models.
Polynomials in the lag operator are algebraically similar to simple polynomials . So
it is possible to calculate the product of two lag polynomials (Hamilton; 1994, Chapter 2).
Given that fact, every multiplicative SARIMA model can be telescoped out into an ordinary
ARMA() model in the variable
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For example, let us assume that the series
follows a SARIMA(
) process. In that case, we have
After some calculations one obtains
where
. Thus, the multiplicative SARIMA model has an ARMA(0,13)
representation where only the coefficients
![]() ![]() |
are not zero. All other coefficients of the MA polynomial are zero.
Thus, we are back in the well-known ARIMA() world. However, if we know that the
original model is a SARIMA(0,1,1)
(12,0,1,1), we have to estimate only the two
coefficients
and
. For the ARMA(0,13) we would estimate
instead the three coefficients
,
, and
. Thus it is
obvious that SARIMA models allow for a parsimonious model building.
In the following, a model specification like (5.6) is called an expanded model. In Section 5.4 it is shown, that this kind of specification is required for estimation purposes, since a multiplicative model like (5.5) cannot be estimated directly.