An Introduction to Time Series

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

3. White Noise Process • Basic building block for time series processes

4. White Noise Processes, cont. • Independent White Noise Process • Slightly stronger condition that and are independent • Gaussian White Noise Process

5. Autocovariance • Covariance of Yt with its own lagged value • Example: Calculate autocovariances for:

6. Stationarity • Covariance-stationary or weakly stationary process • Neither the mean nor the autocovariances depend on the date t

7. Stationarity, cont. • 2 processes • 1 covariance stationary, 1 not covariance stationary

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)

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?

11. Ergodicity • A covariance-stationary process is said to be ergodic for the mean if converges in probability to E(Yt) as

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 .

14. Properties of MA(1) for j>1

15. MA(1) • Covariance stationary • Mean and autocovariances are not functions of time • Autocorrelation of a covariance-stationary process • MA(1)

16. Autocorrelation Function for White Noise:

17. Autocorrelation Function for MA(1):

18. Moving Average Processesof higher order • MA(q): qth order moving average process • Properties of MA(q)

19. Autoregressive Processes • AR(1): First order autoregression • Stationarity: We will assume • Can represent as an MA

20. Properties of AR(1)

21. Properties of AR(1), cont.

22. Autocorrelation Function for AR(1):

23. Autocorrelation Function for AR(1):

24. Gaussian White Noise

25. AR(1),

26. AR(1),

27. AR(1),

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.

29. Properties of AR(p) • Can solve for autocovariances / autocorrelations using Yule-Walker equations

30. Mixed Autoregressive Moving Average Processes • ARMA(p,q) includes both autoregressive and moving average terms

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

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

34. References • Investopia.com • Economagic.com • Hamilton, J. D. (1994), Time Series Analysis, Princeton, New Jersey: Princeton University Press.