Testing Residuals for White Noise in Time Series

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Testing Residuals for White Noise in Time Series

• The Portmanteau Tests

Deborah Diamante Fall 2004
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• On a Measure of Lack of Fit in Time Series
  Models GM Ljung GEP Box Biometrika 1970
• Distribution of Residual Autocorrelations in
  ARIMA Time Series Models GEP Box DA Pierce
  JAMA 1978

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Overview

• Example

• Introduction to ARIMA in time series

• Distribution of B-P Test Statistic

• A better Test Statistic

• Return to Example

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What is the question?

• Does the model we fit to our data yield
  uncorrelated errors (residuals)?
• Hypotheses

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Example Dow Jones Utilities Index(Aug. 28
Dec. 18, 1972)

• What ARIMA model yields uncorrelated errors?

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Introduction to ARIMA Models in Time Series

• Usual definition, denoted ARIMA(p,d,q)
• Where B is the backshift operator defined as
• And

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• To simplify, call Yt the differenced time series
  so that
• The Yt can be written as a linear function of
  previous observations, previous white noise and
  current white noise
• In practice we must choose an appropriate model
  (p,d, and q) and estimate the model parameters.
  After fitting a model to some sample series, we
  wish to consider the stochastic properties of the
  residuals

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• The ACF (Autocorrelation function) at lag k
• The autocorrelations are uncorrelated with
  variances
• So that the statistics
• What if we replace the ACF with the sample ACF?

Anderson (1942)
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Distribution of B-P Test Statistic

• For the AR(p) process
• Can be rewritten as an MA(infinity) process
• Along with orthogonality constraints we have

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• Using the first order Taylor expansion about
  it can be shown that for k 1 m
• In matrix notation this is
• Thus, for QX(XX)-1X we have

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• It follows that the test statistic has
  Chi-squared distribution

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• Therefore, if we have Yt AR(p) then
• Similarly Box and Pierce show that if we have any
  ARIMA(p,d,q) then
• Requires n large relative to m

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A Better Test Statistic

• Ljung and Box make simple modification yielding
  substantially improved approximation!

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Return to Examplen 78, choose m 24, alpha
0.05
proc arima datadowj identify varDOWJ
nlag24 identify varDOWJ(1) nlag24 / Syntax
to fit AR(1) model to (1-B)DOWJ using
ML/ estimate p1 methodml run / Syntax to
fit MA(1) model to (1-B)DOWJ using ML/ estimate
q1 methodml run / Syntax to fit ARMA(1,1)
model to (1-B)DOWJ using ML/ estimate p1 q1
methodml run

• Fit an ARIMA(1,1,0)
• LB 38.88, p 0.0205
• Reject the null hypothesis

• Fit an ARIMA(0,1,1)
• LB 40.22, p 0.0145
• Reject the null hypothesis

• Fit an ARIMA(1,1,1)
• LB 33.50, p 0.0551
• Do not reject the null hypothesis

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Other Tests Exist

• McLeod-Li portmanteau test (1983)
• Turning Point Test
• Difference-Sign Test

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Thank You!

• Dr. Ravishanker
• Dr. Dey

Questions?