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STAT 497 LECTURE NOTES 3. STATIONARY TIME SERIES PROCESSES (ARMA PROCESSES OR BOX-JENKINS PROCESSES). AUTOREGRESSIVE PROCESSES. AR( p ) PROCESS: or where. AR(p) PROCESS. Because the process is always invertible .

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stat 497 lecture notes 3

STAT 497LECTURE NOTES 3

STATIONARY TIME SERIES PROCESSES

(ARMA PROCESSES OR BOX-JENKINS PROCESSES)

autoregressive processes
AUTOREGRESSIVE PROCESSES
  • AR(p) PROCESS:

or

where

ar p process
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.
ar 1 process
AR(1) PROCESS

where atWN(0, )

  • Always invertible.
  • To be stationary, the roots of (B)=1B=0

must lie outside the unit circle.

ar 1 process1
AR(1) PROCESS
  • OR using the characteristic equation, the roots of m=0 must lie inside the unit circle.

B=1 |B|<|1|

||<1 STATIONARITY CONDITION

ar 1 process2
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.
ar 1 process3
AR(1) PROCESS
  • PROCESS MEAN: 
ar 1 process4
AR(1) PROCESS
  • AUTOCOVARIANCE FUNCTION: K
ar 1 process6
AR(1) PROCESS

When ||<1, the process is stationary and the ACF decays exponentially.

ar 1 process7
AR(1) PROCESS
  • 0 <  < 1  All autocorrelations are positive.
  • 1 <  < 0  The sign of the autocorrelation shows an alternating pattern beginning a negative value.
ar 1 process8
AR(1) PROCESS
  • RSF: Using the geometric series
ar 1 process9
AR(1) PROCESS
  • RSF: By operator method _ We know that
ar 1 process10
AR(1) PROCESS
  • RSF: By recursion
the second order autoregressive process
THE SECOND ORDER AUTOREGRESSIVE PROCESS
  • AR(2) PROCESS: Consider the series satisfying

where atWN(0, ).

ar 2 process
AR(2) PROCESS
  • Always invertible.
  • Already in 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.

ar 2 process2
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)
ar 2 process3
AR(2) PROCESS
  • THE AUTOCOVARIANCE FUNCTION: Assuming stationarity and that atis independent of Yt-k, we have
ar 2 process8
AR(2) PROCESS

ACF: It is known as Yule-Walker Equations

ACF shows an exponential decay or sinusoidal behavior.

ar 2 process9
AR(2) PROCESS
  • PACF:

PACF cuts off after lag 2.

ar 2 process10
AR(2) PROCESS
  • RANDOM SHOCK FORM: Using the Operator Method
the p th order autoregressive process ar p process
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

ar p process1
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.
moving average processes
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 tosses=average pay-off on the last tosses:

MOVING AVERAGE PROCESS

moving average processes1
MOVING AVERAGE PROCESSES
  • Consider the process satisfying
moving average processes2
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.
ma 1 process
MA(1) PROCESS
  • From autocovariance generating function
ma 1 process1
MA(1) PROCESS
  • ACF

ACF cuts off after lag 1.

General property of MA(1) processes: 2|k|<1

ma 1 process3
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B=0 lie outside the unit circle or the root of the characteristic equation m=0 lie inside the unit circle.

 INVERTIBILITY CONDITION: ||<1.

ma 1 process4
MA(1) PROCESS
  • It is already in RSF.
  • IF:

1=

2=2

ma 1 process5
MA(1) PROCESS
  • IF: By operator method
the second order moving average process ma 2 process
THE SECOND ORDER MOVING AVERAGE PROCESS_MA(2) PROCESS
  • Consider the moving average process of order 2:
ma 2 process
MA(2) PROCESS
  • From autocovariance generating function
ma 2 process1
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.
ma 2 process2
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.

ma 2 process3
MA(2) PROCESS
  • Invertibility condition for MA(2) process
ma 2 process4
MA(2) PROCESS
  • It is already in RSF form.
  • IF: Using the operator method:
ma q process
MA(q) PROCESS
  • The autocovariance function:
  • ACF:
the autoregressive moving average processes arma p q processes
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.
arma p q processes
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:

arma p q processes1
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.
arma 1 1 processes
ARMA(1, 1) PROCESSES
  • The ARMA(1, 1) process can be written as
  • Stationary if ||<1.
  • Invertible if ||<1.
arma 1 1 processes1
ARMA(1, 1) PROCESSES
  • Autocovariance function:
arma 1 1 process
ARMA(1,1) PROCESS
  • The process variance
arma 1 1 process2
ARMA(1,1) PROCESS
  • Both ACF and PACF tails of after lag 1.