8.8 Zoo of Models
109
The authors of [9] then use the area under the curve to estimate the total financial
damage that a single terrorist act caused.
But we will move on to other time-series models.
ARFIMA
If the differencing parameter d is a fraction rather than an integer, the model is called
autoregressive fractionally integrated moving average, or ARFIMA( p, d, q)-model.
These models are particularly useful if the underlying process has very long memory,
which can be easily seen by writing the differencing steps in terms of the lag operator
ˆ
L and formally using the binomial expansion
(1 − L)
d
=
d
i=0
d
k
ˆ
L
k
= 1 − d L +
d(d − 1)
2!
ˆ
L
2
− · · ·
(8.43)
and we find that the series does not terminate for fractional values of d. Therefore,
information from much earlier samples contributes to the present sample.
EWMA
We already encountered the exponentially weighted moving average method in (8.5).
It is essentially an IIR filter that outputs the weighted average of the most-recent
output and a new value. The equation is reproduced in the following equation on the
left-hand side
y i =
1
m + 1
(my i−1 + x i )
or
σ
2
i =
1
m + 1
mσ
2
i−1 + u
2
i
,
(8.44)
where we average m times the previous output value y i−1 with the new value x i . In
this way, old values are “forgotten” on a time scale of m samples and continuously
updated by new values x i . The equation on the right-hand side is constructed in the
same way, but calculates continuously updated values of the volatility σ i , whose
day-to-day variation of the relative return u j is given by
u j =
S j − S j−1
S j−1
.
(8.45)
Here S j is, for example, the fluctuating stock value. We illustrate this in Fig. 8.10,
which shows the daily returns u (blue) and the corresponding value of σ , averaged
with m = 20 (red), for Apple Inc. (top) and for Coca-Cola (bottom) from March
2018 until March 2019, downloaded from https://finance.yahoo.com. We find the
high-tech stock from Apple to be more volatile than the rather stable stock from
Coca-Cola. The latter only shows a one-day glitch near trading day 220 that causes
σ to increase before returning towards the average value at a rate, determined by
m. Apparently, the thirst for soft-drinks is less volatile than the thirst for high-tech
products.
Instead of continuously updating the volatility alone, we can determine variations
of the relative covariance σ
XY between two sampled variables X i and Y i at sample
time i. The covariance matrix for two stocks X and Y that appears when calculating
109
The authors of [9] then use the area under the curve to estimate the total financial
damage that a single terrorist act caused.
But we will move on to other time-series models.
ARFIMA
If the differencing parameter d is a fraction rather than an integer, the model is called
autoregressive fractionally integrated moving average, or ARFIMA( p, d, q)-model.
These models are particularly useful if the underlying process has very long memory,
which can be easily seen by writing the differencing steps in terms of the lag operator
ˆ
L and formally using the binomial expansion
(1 − L)
d
=
d
i=0
d
k
ˆ
L
k
= 1 − d L +
d(d − 1)
2!
ˆ
L
2
− · · ·
(8.43)
and we find that the series does not terminate for fractional values of d. Therefore,
information from much earlier samples contributes to the present sample.
EWMA
We already encountered the exponentially weighted moving average method in (8.5).
It is essentially an IIR filter that outputs the weighted average of the most-recent
output and a new value. The equation is reproduced in the following equation on the
left-hand side
y i =
1
m + 1
(my i−1 + x i )
or
σ
2
i =
1
m + 1
mσ
2
i−1 + u
2
i
,
(8.44)
where we average m times the previous output value y i−1 with the new value x i . In
this way, old values are “forgotten” on a time scale of m samples and continuously
updated by new values x i . The equation on the right-hand side is constructed in the
same way, but calculates continuously updated values of the volatility σ i , whose
day-to-day variation of the relative return u j is given by
u j =
S j − S j−1
S j−1
.
(8.45)
Here S j is, for example, the fluctuating stock value. We illustrate this in Fig. 8.10,
which shows the daily returns u (blue) and the corresponding value of σ , averaged
with m = 20 (red), for Apple Inc. (top) and for Coca-Cola (bottom) from March
2018 until March 2019, downloaded from https://finance.yahoo.com. We find the
high-tech stock from Apple to be more volatile than the rather stable stock from
Coca-Cola. The latter only shows a one-day glitch near trading day 220 that causes
σ to increase before returning towards the average value at a rate, determined by
m. Apparently, the thirst for soft-drinks is less volatile than the thirst for high-tech
products.
Instead of continuously updating the volatility alone, we can determine variations
of the relative covariance σ
XY between two sampled variables X i and Y i at sample
time i. The covariance matrix for two stocks X and Y that appears when calculating