8.8 Zoo of Models
107
Fig. 8.8 The accumulated returns for Apple stocks as they relay happened (black) and simulated
data using the algorithm, discussed in the text, for m = 1 (blue) and m = 4 (red)
data from Mauna Loa. There, we also found that differencing the data points, as
suggested by the Box-Jenkins methodology, made the time series “more stationary.”
Thus, by adding one or more differencing steps
z i = y i − y i−1 ,
(8.39)
we create a new time series z i that is closer to being stationary than the original
series y i . This is then called an ARIMA( p, d = 1, q)-model, if only a single differencing step is prepended, else d can also assume larger values. Since the inverse of
differentiation is integration, and, at the end, the differentiation needs to be undone,
the model is called an auto-regressive integrated moving average, or ARIMA-model.
Note that the differentiation stage can be described through the lag-operator ˆ
L from
(8.26), which permits us to write
z i = (1 − ˆ
L)y i
(8.40)
and the second derivative becomes
u i = z i − z i−1 = (1 − ˆ
L)z i = (1 − ˆ
L)
2 y i = y i − 2y i−1 + y i−2
(8.41)
where we observe that the last expression is indeed the second derivative, because it
can be written as u i = (y i − y i−1 ) − (y i−1 − y i−2 ).
A wonderful example of using ARIMA models is discussed in [9, 10], where the
authors analyze the impact of terrorist attacks on the revenue generated by tourism
in Italy in the period from 1971 until 1988. They base their analysis on the data
set ITALY.XLS, available from [11]. The file contains three columns with values
107
Fig. 8.8 The accumulated returns for Apple stocks as they relay happened (black) and simulated
data using the algorithm, discussed in the text, for m = 1 (blue) and m = 4 (red)
data from Mauna Loa. There, we also found that differencing the data points, as
suggested by the Box-Jenkins methodology, made the time series “more stationary.”
Thus, by adding one or more differencing steps
z i = y i − y i−1 ,
(8.39)
we create a new time series z i that is closer to being stationary than the original
series y i . This is then called an ARIMA( p, d = 1, q)-model, if only a single differencing step is prepended, else d can also assume larger values. Since the inverse of
differentiation is integration, and, at the end, the differentiation needs to be undone,
the model is called an auto-regressive integrated moving average, or ARIMA-model.
Note that the differentiation stage can be described through the lag-operator ˆ
L from
(8.26), which permits us to write
z i = (1 − ˆ
L)y i
(8.40)
and the second derivative becomes
u i = z i − z i−1 = (1 − ˆ
L)z i = (1 − ˆ
L)
2 y i = y i − 2y i−1 + y i−2
(8.41)
where we observe that the last expression is indeed the second derivative, because it
can be written as u i = (y i − y i−1 ) − (y i−1 − y i−2 ).
A wonderful example of using ARIMA models is discussed in [9, 10], where the
authors analyze the impact of terrorist attacks on the revenue generated by tourism
in Italy in the period from 1971 until 1988. They base their analysis on the data
set ITALY.XLS, available from [11]. The file contains three columns with values