Random Series / White Noise

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Random Series / White Noise
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Notation

• WN (white noise) uncorrelated
• iid
• independent and identically distributed
• Yt iid N(m, s) Random Series
• et iid N(0, s) White Noise

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Data Generation

• Independent observations at every t from
• the normal distribution (m, s)

Yt
Yt
t
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Identification of WN Process

• How to determine if data are from WN process?

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Tests of Randomness - 1

• Timeplot of the Data
• Check trend
• Check heteroscedasticity
• Check seasonality

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Generating a Random Series Using Eviews

• Command nrnd generates a RND N(0, 1)

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Test of Randomness - 2 Correlogram
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Scatterplot and Correlation Coefficient - Review
Y
X
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Autocorrelation Coefficient

• Definition
• The correlation coefficient between Yt and
  Y(t-k) is called the autocorrelation coefficient
  at lag k and is denoted as rk . By definition,
  r0 1.
• Autocorrelation of a Random Series
• If the series is random, rk 0 for k 1,...

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Process Correlogram
rk
1
0
Lag, k
-1
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Sample Autocorrelation Coefficient

• Sample Autocorrelation at lag k.

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Standard Error of the Sample Autocorrelation
Coefficient

• Standard Error of the sample autocorrelation
• if the Series is Random.

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Z- Test of H0 rk 0
Reject H0 if Z lt -1.96 or Z gt 1.96
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Box-Ljung Q Statistic

• Definition

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Sampling Distribution of QBL(m) H0

• H0 r1r2rk 0
• QBL(m) H0 follows a c2 (DFm) distribution
• Reject H0 if QBL gt c2(95tile)

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Test of Normality - 1Graphical Test

• Normal Probability Plot of the Data
• Check the shape straight, convex,
• S-shaped

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Construction of a Normal Probability Plot

• Alternative estimates of the cumulative relative
  frequency of an observation
• pi (i - 0.5)/ n
• pi i / (n1)
• pi (i - 0.375) / (n0.25)
• Estimate of the percentile Normal
• Standardized Q(pi) NORMSINV(pi)
• Q(pi) NORMINV(pi, mean, stand. dev.)

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Non-Normal Populations

• Flat Skewed

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Test of Normality - 2Test Statistics

• Stand. Dev.
• Skewness
• Kurtosis

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The Jarque-Bera Test

• If the population is normal and the data are
  random, then
• follows approximately c2 with the 0f
  degrees of freedom 2.
• Reject H0 if JB gt 6

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Forecasting Random Series

• Given the data Y1,...,Yn, the one step ahead
  forecast Y(n1) is
• or Approx.

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Forecasting a Random Series

• If it is determined that Yt is RND N(m, s)
• a) The best point forecast of Yt E(Yt) m
• b) A 95 interval forecast of Yt
• (m 1.96 s, m1.96 s)
• for all t (one important long run implication of
  a stationary series.)

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The Sampling Distribution of the von-Neumann
Ratio

• The vN Ratio H0 follows an approximate normal
  with
• Expected Value of v E(v) 2
• Standard Error of v

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AppendixThe von Neumann Ratio

• Definition

The non Neumann Ratio of the regression residual
is the Durbin - Watson Statistic
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