An Introduction to Time Series

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# An Introduction to Time Series - PowerPoint PPT Presentation

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

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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 .
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.