STAT 497 LECTURE NOTES 3

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STAT 497LECTURE NOTES 3

• STATIONARY TIME SERIES PROCESSES
• (ARMA PROCESSES OR BOX-JENKINS PROCESSES)

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AUTOREGRESSIVE PROCESSES

• AR(p) PROCESS
• or
• where

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AR(p) PROCESS

• Because the
  process is always invertible.
• To be stationary, the roots of ?p(B)0 must lie
  outside the unit circle.
• The AR process is useful in describing situations
  in which the present value of a time series
  depends on its preceding values plus a random
  shock.

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AR(1) PROCESS

• where at?WN(0, )
• Always invertible.
• To be stationary, the roots of ?(B)1??B0
• must lie outside the unit circle.

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AR(1) PROCESS

• OR using the characteristic equation, the roots
  of m??0 must lie inside the unit circle.
• ?B??1 ? Blt??1
• ??lt1 ?STATIONARITY CONDITION

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AR(1) PROCESS

• This process sometimes called as the Markov
  process because the distribution of Yt given
  Yt?1,Yt?2, is exactly the same as the
  distribution of Yt given Yt?1.

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AR(1) PROCESS

• PROCESS MEAN ?

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AR(1) PROCESS

• AUTOCOVARIANCE FUNCTION ?k

Keep this part as it is
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AR(1) PROCESS

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AR(1) PROCESS
When ?lt1, the process is stationary and the ACF
decays exponentially.
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AR(1) PROCESS

• 0 lt ? lt 1 ? All autocorrelations are positive.
• ?1 lt ? lt 0 ? The sign of the autocorrelation
  shows an alternating pattern beginning a negative
  value.

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AR(1) PROCESS

• RSF Using the geometric series

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AR(1) PROCESS

• RSF By operator method _ We know that

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AR(1) PROCESS

• RSF By recursion

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THE SECOND ORDER AUTOREGRESSIVE PROCESS

• AR(2) PROCESS Consider the series satisfying

where at?WN(0, ).
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AR(2) PROCESS

• Always invertible.
• Already in the Inverted Form.
• To be stationary, the roots of
• must lie outside the unit circle.
• OR the roots of the characteristic equation
• must lie inside the unit circle.

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AR(2) PROCESS
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AR(2) PROCESS

• Considering both real and complex roots, we have
  the following stationary conditions for AR(2)
  process (see page 84 for the proof)

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AR(2) PROCESS

• THE AUTOCOVARIANCE FUNCTION Assuming
  stationarity and that at is independent of Yt?k,
  we have

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AR(2) PROCESS

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AR(2) PROCESS

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AR(2) PROCESS

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AR(2) PROCESS

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AR(2) PROCESS

• ACF It is known as Yule-Walker Equations

ACF shows an exponential decay or sinusoidal
behavior.
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AR(2) PROCESS

• PACF

PACF cuts off after lag 2.
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AR(2) PROCESS

• RANDOM SHOCK FORM Using the Operator Method

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The p-th ORDER AUTOREGRESSIVE PROCESS AR(p)
PROCESS

• Consider the process satisfying

where at?WN(0, ).
provided that roots of all lie outside the unit
circle
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AR(p) PROCESS

• ACF Yule-Walker Equations
• ACF tails of as a mixture of exponential decay
  or damped sine wave (if some roots are complex).
• PACF cuts off after lag p.

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MOVING AVERAGE PROCESSES

• Suppose you win 1 TL if a fair coin shows a head
  and lose 1 TL if it shows tail. Denote the
  outcome on toss t by at.
• The average winning on the last 4 tossesaverage
  pay-off on the last tosses

MOVING AVERAGE PROCESS
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MOVING AVERAGE PROCESS

• Errors are the average of this periods random
  error and last periods random error.
• No memory of past levels.
• The impact of shock to the series takes exactly
  1-period to vanish for MA(1) process. In MA(2)
  process, the shock takes 2-periods and then fade
  away.
• In MA(1) process, the correlation would last only
  one period.

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MOVING AVERAGE PROCESSES

• Consider the process satisfying

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MOVING AVERAGE PROCESSES

• Because , MA
  processes are always stationary.
• Invertible if the roots of ?q(B)0 all lie
  outside the unit circle.
• It is a useful process to describe events
  producing an immediate effects that lasts for
  short period of time.

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THE FIRST ORDER MOVING AVERAGE PROCESS_MA(1)
PROCESS

• Consider the process satisfying

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MA(1) PROCESS

• From autocovariance generating function

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MA(1) PROCESS

• ACF

ACF cuts off after lag 1.
General property of MA(1) processes 2?klt1
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MA(1) PROCESS

• PACF

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MA(1) PROCESS

• Basic characteristic of MA(1) Process
• ACF cuts off after lag 1.
• PACF tails of exponentially depending on the sign
  of ?.
• Always stationary.
• Invertible if the root of 1??B0 lie outside the
  unit circle or the root of the characteristic
  equation m??0 lie inside the unit circle.
• ? INVERTIBILITY CONDITION ?lt1.

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MA(1) PROCESS

• It is already in RSF.
• IF

?1??
?2??2
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MA(1) PROCESS

• IF By operator method

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THE SECOND ORDER MOVING AVERAGE PROCESS_MA(2)
PROCESS

• Consider the moving average process of order 2

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MA(2) PROCESS

• From autocovariance generating function

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MA(2) PROCESS

• ACF
• ACF cuts off after lag 2.
• PACF tails of exponentially or a damped sine
  waves depending on a sign and magnitude of
  parameters.

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MA(2) PROCESS

• Always stationary.
• Invertible if the roots of
• all lie outside the unit circle.
• OR if the roots of
• all lie inside the unit circle.

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MA(2) PROCESS

• Invertibility condition for MA(2) process

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MA(2) PROCESS

• It is already in RSF form.
• IF Using the operator method

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The q-th ORDER MOVING PROCESS_ MA( q) PROCESS
Consider the MA(q) process
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MA(q) PROCESS

• The autocovariance function
• ACF

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THE AUTOREGRESSIVE MOVING AVERAGE
PROCESSES_ARMA(p, q) PROCESSES

• If we assume that the series is partly
  autoregressive and partly moving average, we
  obtain a mixed ARMA process.

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ARMA(p, q) PROCESSES

• For the process to be invertible, the roots of
• lie outside the unit
  circle.
• For the process to be stationary, the roots of
• lie outside the unit
  circle.
• Assuming that and
  share no common roots,
• Pure AR Representation
• Pure MA Representation

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ARMA(p, q) PROCESSES

• Autocovariance function
• ACF
• Like AR(p) process, it tails of after lag q.
• PACF Like MA(q), it tails of after lag p.

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ARMA(1, 1) PROCESSES

• The ARMA(1, 1) process can be written as

• Stationary if ?lt1.

• Invertible if ?lt1.

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ARMA(1, 1) PROCESSES

• Autocovariance function

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ARMA(1,1) PROCESS

• The process variance

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ARMA(1,1) PROCESS

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ARMA(1,1) PROCESS

• Both ACF and PACF tails of after lag 1.

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ARMA(1,1) PROCESS

• IF

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ARMA(1,1) PROCESS

• RSF

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AR(1) PROCESS
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AR(2) PROCESS
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MA(1) PROCESS
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MA(2) PROCESS
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ARMA(1,1) PROCESS
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ARMA(1,1) PROCESS (contd.)