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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

An Introduction to Time Series

Ginger Davis

VIGRE Computational Finance Seminar Rice University

November 26, 2003

what is a time series
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
White Noise Process
  • Basic building block for time series processes
white noise processes cont
White Noise Processes, cont.
  • Independent White Noise Process
    • Slightly stronger condition that and are independent
  • Gaussian White Noise Process
  • Covariance of Yt with its own lagged value
  • Example: Calculate autocovariances for:
  • Covariance-stationary or weakly stationary process
    • Neither the mean nor the autocovariances depend on the date t
stationarity cont
Stationarity, cont.
  • 2 processes
    • 1 covariance stationary, 1 not covariance stationary
stationarity cont8
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 cont9
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 Processes
  • Gaussian process {Yt}
    • Joint density

is Gaussian for any

  • What can be said about a covariance stationary Gaussian process?
  • 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
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
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 .
  • Covariance stationary
    • Mean and autocovariances are not functions of time
  • Autocorrelation of a covariance-stationary process
  • MA(1)
moving average processes of higher order
Moving Average Processesof higher order
  • MA(q): qth order moving average process
  • Properties of MA(q)
autoregressive processes
Autoregressive Processes
  • AR(1): First order autoregression
  • Stationarity: We will assume
  • Can represent as an MA
autoregressive processes of higher order
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
Properties of AR(p)
  • Can solve for autocovariances / autocorrelations using Yule-Walker equations
mixed autoregressive moving average processes
Mixed Autoregressive Moving Average Processes
  • ARMA(p,q) includes both autoregressive and moving average terms
time series models for financial data
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
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
  • Hamilton, J. D. (1994), Time Series Analysis, Princeton, New Jersey: Princeton University Press.