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

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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

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White Noise Process

• Basic building block for time series processes

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White Noise Processes, cont.

• Independent White Noise Process
• Slightly stronger condition that and
  are independent
• Gaussian White Noise Process

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Autocovariance

• Covariance of Yt with its own lagged value
• Example Calculate autocovariances for

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Stationarity

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

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Stationarity, cont.

• 2 processes
• 1 covariance stationary, 1 not covariance
  stationary

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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)

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Stationarity, cont.

• Strict stationarity
• For any values of j1, j2, , jn, the joint
  distribution of (Yt, Ytj1, Ytj2, ..., Ytjn)
  depends only on the intervals separating the
  dates and not on the date itself

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Gaussian Processes

• Gaussian process Yt
• Joint density
• is Gaussian for any
• What can be said about a covariance stationary
  Gaussian process?

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Ergodicity

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

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Describing the dynamics of a Time Series

• Moving Average (MA) processes
• Autoregressive (AR) processes
• Autoregressive / Moving Average (ARMA) processes
• Autoregressive conditional heteroscedastic (ARCH)
  processes

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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 .

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Properties of MA(1)
for jgt1
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MA(1)

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

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Autocorrelation Function for White Noise
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Autocorrelation Function for MA(1)
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Moving Average Processesof higher order

• MA(q) qth order moving average process
• Properties of MA(q)

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Autoregressive Processes

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

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Properties of AR(1)
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Properties of AR(1), cont.
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Autocorrelation Function for AR(1)
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Autocorrelation Function for AR(1)
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Gaussian White Noise
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AR(1),
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AR(1),
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AR(1),
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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.

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Properties of AR(p)

• Can solve for autocovariances / autocorrelations
  using Yule-Walker equations

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Mixed Autoregressive Moving Average Processes

• ARMA(p,q) includes both autoregressive and moving
  average terms

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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

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U. S. Federal Funds Rate
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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

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References

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