Financial Data Analysis
Professor S. Kou
Department of IEOR, Columbia University
Lecture 6. Seasonality, Unit Root Test, and Tests of Stationary
1S
e
a
s
o
n
a
l
i
t
y
Last time we learned MA, ARMA, and ARIMA models, which works for
stationary time series without any seasonality. However, many time series
data display seasonality.
For example, we can look at the quarterly earnings per share of Johnson
and Johnson from the
f
rst quarter of 1960 to the last quarter of 1980.
Although the e
f
ects of earnings announcement to stock prices are not very
clear, earnings may a
f
ect the short term prices, partly due to selfful
f
lling
prophecies. In other words, if many analysts focus on earnings, then the
stock prices will be a
f
ected by earnings in the short term.
#First, we will create a time series object.
jjEarning$date
 timeSeq(from="1/1/1960", to="10/1/1980",
by="quarters", format="%Y:%Q")
jjEarning.ts
 timeSeries (data=jjEarning$jj, pos=jjEarning$date)
#Then we plot the data
plot(jjEarning.ts)
The seasonality is very strong. In fact, it is immediately clear that the
period of the seasonality is about 4 (which means one year).
Now just like we take log of the stock prices to get returns, we can look
at the log transform of the earnings.
#Do the log transform
logJJ
 log(jjEarning.ts)
plot(logJJ)
We shall try to
f
t a seasonal ARIMA model to the log earnings.
A standard way to get ride of seasonality is by looking at the di
f
erence.
In our example, since we are quite certain the seasonality period is 4, we
can simply take the transform
1
−
=4
1
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is the backshifting operator. In other words, we shall only compare
the current quarter earning with the earnings in the same quarter of the
previous years.
After the transform, then we can
f
tanARIMAmode
l
.
1.1 Multiplicative Seasonal Model (Airline Model)
A popular way to generalize the above idea is to
f
t the Multiplicative Sea
sonal Model (also called the “airline model”):
(1
−
)(1
−
)
=(1
−
)(1
−
Θ
)
(1)
where
is the period.
Note that
(1
−
)(1
−
Θ
)
=(
1
−
−
Θ
+
Θ
+1
)
=
−
−
1
−
Θ
−
+
Θ
−
−
1
and
(1
−
)(1
−
)
=(
1
−
−
+
+1
)
=
−
−
1
−
−
+
−
−
1
Therefore, the airline model is
−
−
1
−
−
+
−
−
1
=
−
−
1
−
Θ
−
+
Θ
−
−
1
(2)
The motivation of the airline model is as follows. First, we expect
the value
is related to
−
. In the airline model, this is modeled as
a ARMA(0,1,1) with period
,i
.e
.aMA(1)mode
lforthe
f
rst order di
f
er
ence, except with period
. More precisely,
−
−
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 Fall '10
 StevenKou
 Financial Engineering, Null hypothesis, Statistical hypothesis testing, Statistical tests, Time series analysis, Airline Model

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