ARIMA stands for Autoregressive Integrated
Moving Average models. Univariate (single vector) ARIMA is a forecasting
technique that projects the future values of a series based entirely
on its own inertia. Its main application is in the area
of short term forecasting requiring at least 40 historical data
points. It works best when your data exhibits a stable or consistent
pattern over time with a minimum amount of outliers.
Sometimes called BoxJenkins (after the original authors), ARIMA
is usually superior to exponential smoothing techniques when the
data is reasonably long and the correlation between past observations
is stable. If the data is short or highly volatile, then some
smoothing method may perform better. If you do not have at least
38 data points, you should consider some other method than ARIMA.
Basic Concepts:
The first step in applying ARIMA methodology is to check for
stationarity. "Stationarity" implies that the series remains at
a fairly constant level over time. If a trend exists, as in most
economic or business applications, then your data is NOT stationary.
The data should also show a constant variance in its fluctuations
over time. This is easily seen with a series that is heavily seasonal
and growing at a faster rate. In such a case, the ups and downs
in the seasonality will become more dramatic over time. Without
these stationarity conditions being met, many of the calculations
associated with the process cannot be computed.
Differencing:
If a graphical plot of the data indicates nonstationarity, then
you should "difference" the series. Differencing is an excellent
way of transforming a nonstationary series to a stationary one.
This is done by subtracting the observation in the current period
from the previous one. If this transformation is done only once
to a series, you say that the data has been "first differenced".
This process essentially eliminates the trend if your series is
growing at a fairly constant rate. If it is growing at an increasing
rate, you can apply the same procedure and difference the data
again. Your data would then be "second differenced".
Autocorrelations:
"Autocorrelations" are numerical values that indicate how a
data series is related to itself over time. More precisely, it
measures how strongly data values at a specified number of periods
apart are correlated to each other over time. The number of periods
apart is usually called the "lag". For example, an autocorrelation
at lag 1 measures how values 1 period apart are correlated to
one another throughout the series. An autocorrelation at lag 2
measures how the data two periods apart are correlated throughout
the series. Autocorrelations may range from +1 to 1. A value
close to +1 indicates a high positive correlation while a value
close to 1 implies a high negative correlation. These measures
are most often evaluated through graphical plots called "correlagrams".
A correlagram plots the auto correlation values for a given series
at different lags. This is referred to as the "autocorrelation
function" and is very important in the ARIMA method.
Autoregressive Models:
ARIMA methodology attempts to describe the movements in a stationary
time series as a function of what are called "autoregressive and
moving average" parameters. These are referred to as AR parameters
(autoregessive) and MA parameters (moving averages). An AR model
with only 1 parameter may be written as...
X(t) = A(1) * X(t1) + E(t)
where X(t) = time series under investigation
A(1) = the autoregressive parameter of order 1
X(t1) = the time series lagged 1 period
E(t) = the error term of the model
This simply means that any given value X(t) can be explained
by some function of its previous value, X(t1), plus some unexplainable
random error, E(t). If the estimated value of A(1) was .30, then
the current value of the series would be related to 30% of its
value 1 period ago. Of course, the series could be related to
more than just one past value. For example,
X(t) = A(1) * X(t1) + A(2) * X(t2) + E(t)
This indicates that the current value of the series is a combination
of the two immediately preceding values, X(t1) and X(t2), plus
some random error E(t). Our model is now an autoregressive model
of order 2.
Moving Average Models:
A second type of BoxJenkins model is called a "moving average"
model. Although these models look very similar to the AR model,
the concept behind them is quite different. Moving average parameters
relate what happens in period t only to the random errors that
occurred in past time periods, i.e. E(t1), E(t2), etc. rather
than to X(t1), X(t2), (Xt3) as in the autoregressive approaches.
A moving average model with one MA term may be written as follows...
X(t) = B(1) * E(t1) + E(t)
The term B(1) is called an MA of order 1. The negative sign
in front of the parameter is used for convention only and is usually
printed out auto matically by most computer programs. The above
model simply says that any given value of X(t) is directly related
only to the random error in the previous period, E(t1), and to
the current error term, E(t). As in the case of autoregressive
models, the moving average models can be extended to higher order
structures covering different combinations and moving average
lengths.
Mixed Models:
ARIMA methodology also allows models to be built that incorporate
both autoregressive and moving average parameters together. These
models are often referred to as "mixed models". Although this
makes for a more complicated forecasting tool, the structure may
indeed simulate the series better and produce a more accurate
forecast. Pure models imply that the structure consists only of
AR or MA parameters  not both.
The models developed by this approach are usually called ARIMA
models because they use a combination of autoregressive (AR),
integration (I)  referring to the reverse process of differencing
to produce the forecast, and moving average (MA) operations. An
ARIMA model is usually stated as ARIMA(p,d,q). This represents
the order of the autoregressive components (p), the number of
differencing operators (d), and the highest order of the moving
average term. For example, ARIMA(2,1,1) means that you have a
second order autoregressive model with a first order moving average
component whose series has been differenced once to induce stationarity.
Picking the Right Specification:
The main problem in classical BoxJenkins is trying to decide
which ARIMA specification to use i.e. how many AR and / or MA
parameters to include. This is what much of BoxJenkings [1976]
was devoted to the "identification process. It depended upon graphical
and numerical eval uation of the sample autocorrelation and partial
autocorrelation functions. Well, for your basic models, the task
is not too difficult. Each have autocorrelation functions that
look a certain way. However, when you go up in complexity, the
patterns are not so easily detected. To make matters more difficult,
your data represents only a sample of the underlying process.
This means that sampling errors (outliers, measurement error,
etc.) may distort the theoretical identification process. That
is why traditional ARIMA modeling is an art rather than a science.
