Title: An Introduction to Time Series
1An Introduction to Time Series
- Ginger Davis
- VIGRE Computational Finance Seminar Rice
University - November 26, 2003
2What 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
3White Noise Process
- Basic building block for time series processes
4White Noise Processes, cont.
- Independent White Noise Process
- Slightly stronger condition that and
are independent - Gaussian White Noise Process
5Autocovariance
- Covariance of Yt with its own lagged value
- Example Calculate autocovariances for
6Stationarity
- Covariance-stationary or weakly stationary
process - Neither the mean nor the autocovariances depend
on the date t
7Stationarity, cont.
- 2 processes
- 1 covariance stationary, 1 not covariance
stationary
8Stationarity, 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)
9Stationarity, cont.
- Strict stationarity
- For any values of j1, j2, , jn, the joint
distribution of (Yt, Ytj1, Ytj2, ..., Ytjn)
depends only on the intervals separating the
dates and not on the date itself
10Gaussian Processes
- Gaussian process Yt
- Joint density
-
- is Gaussian for any
- What can be said about a covariance stationary
Gaussian process?
11Ergodicity
- A covariance-stationary process is said to be
ergodic for the mean if - converges in probability to E(Yt) as
12Describing the dynamics of a Time Series
- Moving Average (MA) processes
- Autoregressive (AR) processes
- Autoregressive / Moving Average (ARMA) processes
- Autoregressive conditional heteroscedastic (ARCH)
processes
13Moving Average Processes
- MA(1) First Order MA process
- moving average
- Yt is constructed from a weighted sum of the two
most recent values of .
14Properties of MA(1)
for jgt1
15MA(1)
- Covariance stationary
- Mean and autocovariances are not functions of
time - Autocorrelation of a covariance-stationary
process - MA(1)
16Autocorrelation Function for White Noise
17Autocorrelation Function for MA(1)
18Moving Average Processesof higher order
- MA(q) qth order moving average process
- Properties of MA(q)
19Autoregressive Processes
- AR(1) First order autoregression
- Stationarity We will assume
- Can represent as an MA
20Properties of AR(1)
21Properties of AR(1), cont.
22Autocorrelation Function for AR(1)
23Autocorrelation Function for AR(1)
24Gaussian White Noise
25AR(1),
26AR(1),
27AR(1),
28Autoregressive 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.
29Properties of AR(p)
- Can solve for autocovariances / autocorrelations
using Yule-Walker equations
30Mixed Autoregressive Moving Average Processes
- ARMA(p,q) includes both autoregressive and moving
average terms
31Time 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
32U. S. Federal Funds Rate
33Modeling 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
34References
- Investopia.com
- Economagic.com
- Hamilton, J. D. (1994), Time Series Analysis,
Princeton, New Jersey Princeton University
Press.