1. An Introduction to Time Series
Ginger Davis
VIGRE Computational Finance Seminar
Rice University
November 26, 2003
2. What is a Time Series?
• Time Series
– Collection of observations
indexed by the date of each
observation
• Lag Operator
– Represented by the symbol L
• Mean of Yt = μt
T
y
y
y ,
,
, 2
1
1
t
t x
Lx
3. White Noise Process
• Basic building block for time series processes
0
0
2
2
t
t
t
t
t
E
E
E
4. White Noise Processes, cont.
• Independent White Noise Process
– Slightly stronger condition that and are
independent
• Gaussian White Noise Process
2
,
0
~
N
t
t
5. Autocovariance
• Covariance of Yt with its own lagged value
• Example: Calculate autocovariances for:
j
t
j
t
t
t
jt Y
Y
E
j
t
t
j
t
t
jt
t
t
E
Y
Y
E
Y
6. Stationarity
• Covariance-stationary or weakly stationary
process
– Neither the mean nor the autocovariances depend on
the date t
j
j
t
t
t
Y
Y
E
Y
E
7. Stationarity, cont.
• 2 processes
– 1 covariance stationary, 1 not covariance
stationary
t
t
t
t
t
Y
Y
8. Stationarity, cont.
• Covariance stationary processes
– Covariance between Yt and Yt-j depends only on
j (length of time separating the observations) and
not on t (date of the observation)
j
j
9. Stationarity, cont.
• Strict stationarity
– For any values of j1, j2, …, jn, the joint
distribution of (Yt, Yt+j1
, Yt+j2
, ..., Yt+jn
) depends
only on the intervals separating the dates and
not on the date itself
10. Gaussian Processes
• Gaussian process {Yt}
– Joint density
is Gaussian for any
• What can be said about a covariance stationary
Gaussian process?
n
n
j
t
j
t j
t
j
t
t
Y
Y
Y y
y
y
f
,
,
, 1
1
1 ,
,
,
n
j
j
j ,
,
, 2
1
11. Ergodicity
• A covariance-stationary process is said to be
ergodic for the mean if
converges in probability to E(Yt) as
T
t
t
y
T
y
1
1
T
12. Describing the dynamics
of a Time Series
• Moving Average (MA) processes
• Autoregressive (AR) processes
• Autoregressive / Moving Average (ARMA)
processes
• Autoregressive conditional heteroscedastic
(ARCH) processes
13. Moving Average Processes
• MA(1): First Order MA process
• “moving average”
– Yt is constructed from a weighted sum of the two
most recent values of .
1
t
t
t
Y
14. Properties of MA(1)
0
1
2
2
2
1
2
2
2
1
1
2
1
1
1
2
2
2
1
2
1
2
2
1
2
j
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
Y
Y
E
E
E
Y
Y
E
E
E
Y
E
Y
E
for j>1
15. MA(1)
• Covariance stationary
– Mean and autocovariances are not functions of time
• Autocorrelation of a covariance-stationary
process
• MA(1)
0
j
j
2
2
2
2
1
1
1
17. Autocorrelation Function for MA(1):
1
8
.
0
t
t
t
Y
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20
Lag
Autocorrelation
18. Moving Average Processes
of higher order
• MA(q): qth order moving average process
• Properties of MA(q)
q
t
q
t
t
t
t
Y
2
2
1
1
q
j
q
j
j
j
q
q
j
j
j
j
q
,
0
,
,
2
,
1
,
1
2
2
2
1
1
2
2
2
2
2
1
0
19. Autoregressive Processes
• AR(1): First order autoregression
• Stationarity: We will assume
• Can represent as an MA
t
t
t Y
c
Y
1
1
2
2
1
2
2
1
1
t
t
t
t
t
t
t
c
c
c
c
Y
:
)
(
20. Properties of AR(1)
2
2
2
4
2
2
2
2
1
2
0
1
1
1
t
t
t
t
E
Y
E
c
21. Properties of AR(1), cont.
j
j
j
j
j
j
j
j
j
t
j
t
j
t
j
t
j
t
t
t
j
t
t
j
E
Y
Y
E
0
2
2
2
4
2
2
4
2
2
2
1
2
2
1
1
1
22. Autocorrelation Function for AR(1):
t
t
t Y
Y
1
8
.
0
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20
Lag
Autocorrelation
23. Autocorrelation Function for AR(1):
t
t
t Y
Y
1
8
.
0
-0.5
0.0
0.5
1.0
0 5 10 15 20
Lag
Autocorrelation
28. Autoregressive Processes
of higher order
• pth order autoregression: AR(p)
• Stationarity: We will assume that the roots of
the following all lie outside the unit circle.
t
p
t
p
t
t
t Y
Y
Y
c
Y
2
2
1
1
0
1 2
2
1
p
p z
z
z
29. Properties of AR(p)
• Can solve for autocovariances /
autocorrelations using Yule-Walker
equations
p
c
2
1
1
30. Mixed Autoregressive Moving
Average Processes
• ARMA(p,q) includes both autoregressive and
moving average terms
q
t
q
t
t
t
p
t
p
t
t
t Y
Y
Y
c
Y
2
2
1
1
2
2
1
1
31. Time Series Models
for Financial Data
• A Motivating Example
– Federal Funds rate
– We are interested in forecasting not only the
level of the series, but also its variance.
– Variance is not constant over time
32. U. S. Federal Funds Rate
Time
1955 1960 1965 1970 1975
2
4
6
8
10
12
33. Modeling the Variance
• AR(p):
• ARCH(m)
– Autoregressive conditional heteroscedastic process
of order m
– Square of ut follows an AR(m) process
– wt is a new white noise process
t
p
t
p
t
t
t u
y
y
y
c
y
2
2
1
1
t
m
t
m
t
t
t w
u
u
u
u
2
2
2
2
2
1
1
2
