An Introduction to Time Series Ginger Davis VIGRE Computational Finance Seminar Rice University November 26, 2003

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 Y t = μ t

White Noise Process Basic building block for time series processes

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

Autocovariance Covariance of Y t with its own lagged value Example: Calculate autocovariances for:

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

Stationarity, cont. 2 processes –1 covariance stationary, 1 not covariance stationary

Stationarity, cont. Covariance stationary processes –Covariance between Y t and Y t-j depends only on j (length of time separating the observations) and not on t (date of the observation)

Stationarity, cont. Strict stationarity –For any values of j 1, j 2, …, j n, the joint distribution of (Y t, Y t+j 1, Y t+j 2,..., Y t+j n ) depends only on the intervals separating the dates and not on the date itself

Gaussian Processes Gaussian process {Y t } –Joint density is Gaussian for any What can be said about a covariance stationary Gaussian process?

Ergodicity A covariance-stationary process is said to be ergodic for the mean if converges in probability to E(Y t ) as

Describing the dynamics of a Time Series Moving Average (MA) processes Autoregressive (AR) processes Autoregressive / Moving Average (ARMA) processes Autoregressive conditional heteroscedastic (ARCH) processes

Moving Average Processes MA(1): First Order MA process “moving average” –Y t is constructed from a weighted sum of the two most recent values of.

Properties of MA(1) for j>1

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

Autocorrelation Function for White Noise:

Autocorrelation Function for MA(1):

Moving Average Processes of higher order MA(q): q th order moving average process Properties of MA(q)

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

Properties of AR(1)

Properties of AR(1), cont.

Autocorrelation Function for AR(1):

Gaussian White Noise

AR(1),

Autoregressive Processes of higher order p th order autoregression: AR(p) Stationarity: We will assume that the roots of the following all lie outside the unit circle.

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

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

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

U. S. Federal Funds Rate

Modeling the Variance AR(p): ARCH(m) –Autoregressive conditional heteroscedastic process of order m –Square of u t follows an AR(m) process –w t is a new white noise process

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