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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.
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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 Yt = μt
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 Yt 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 Yt and Yt-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 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
Gaussian Processes • Gaussian process {Yt} • 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(Yt) 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” • Yt 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)
Moving Average Processesof higher order • MA(q): qth order moving average process • Properties of MA(q)
Autoregressive Processes • AR(1): First order autoregression • Stationarity: We will assume • Can represent as an MA
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.
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
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
References • Investopia.com • Economagic.com • Hamilton, J. D. (1994), Time Series Analysis, Princeton, New Jersey: Princeton University Press.