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The set of autoregressive integrated moving average (ARIMA) models are a
complete set to model any kind of stationary set. These models have two main
components a moving average and an autoregressive component. Both can be present
simultaneous or not and have several terms. This model can be reduced to an
exponential model in some simpler cases [9].
We can see in Figure 5 the result of applying splinef() function. This function
adjusts 3-degree polynomials to the intervals between the different values, and
performs predictions using a linear regression using only the two most recent points
[8]. In this chart and also in all the following charts shown in Table 2, the forecast line
is shown in red, the black line corresponds to the historical time series, and the gray
bars correspond to the confidence intervals of 80% and 95% for the forecast value.
It can be seen in Figure 5, that from 2002 to 2009, there is very good agreement
between the historical time series and the forecasts. Only in 2010 and 2011, the
adjustment is worse due to an unexpected variation. However, as the forecast only
takes into account the last two years, all other points, however well adjusted, are
ignored in the calculation of the predicted value.
Fig. 5.
Graphic on Capital Transfers showing a forecast line by the splinef() function
In the first graph of Table 2, an application of exponential smoothing model, the
forecast is almost constant for the next 2 years, but with distinct confidence intervals
showing that two years prediction is more uncertainty than one year.
The ARIMA models, shown in the second graph of Table 2, show that there is
some similarity between the prediction line and the historical data line, but lagged one
year. In this sense, as expected, forecasts have a growing trend. The model on the 3th
chart in Table 3 has the form of a straight line regression. Clearly, there is an
increasing linear trend and seasonality that has not been identified.
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