STAT 497 LECTURE NOTES 3

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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

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### STAT 497LECTURE 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.
• 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

where atWN(0, )

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

must lie outside the unit circle.

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) 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) PROCESS
• PROCESS MEAN: 
AR(1) PROCESS
• AUTOCOVARIANCE FUNCTION: K
AR(1) PROCESS

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

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) PROCESS
• RSF: Using the geometric series
AR(1) PROCESS
• RSF: By operator method _ We know that
AR(1) PROCESS
• RSF: By recursion
THE SECOND ORDER AUTOREGRESSIVE PROCESS
• AR(2) PROCESS: Consider the series satisfying

where atWN(0, ).

AR(2) PROCESS
• Always invertible.
• 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) 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) PROCESS
• THE AUTOCOVARIANCE FUNCTION: Assuming stationarity and that atis independent of Yt-k, we have
AR(2) PROCESS

ACF: It is known as Yule-Walker Equations

ACF shows an exponential decay or sinusoidal behavior.

AR(2) PROCESS
• PACF:

PACF cuts off after lag 2.

AR(2) PROCESS
• RANDOM SHOCK FORM: Using the Operator Method
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) 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
• 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 PROCESSES
• Consider the process satisfying
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
• From autocovariance generating function
MA(1) PROCESS
• ACF

ACF cuts off after lag 1.

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

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) PROCESS
• It is already in RSF.
• IF:

1=

2=2

MA(1) PROCESS
• IF: By operator method
THE SECOND ORDER MOVING AVERAGE PROCESS_MA(2) PROCESS
• Consider the moving average process of order 2:
MA(2) PROCESS
• From autocovariance generating function
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) 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) PROCESS
• Invertibility condition for MA(2) process
MA(2) PROCESS
• It is already in RSF form.
• IF: Using the operator method:
MA(q) PROCESS
• The autocovariance function:
• ACF:
• If we assume that the series is partly autoregressive and partly moving average, we obtain a mixed ARMA process.
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) 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
• The ARMA(1, 1) process can be written as
• Stationary if ||<1.
• Invertible if ||<1.
ARMA(1, 1) PROCESSES
• Autocovariance function:
ARMA(1,1) PROCESS
• The process variance
ARMA(1,1) PROCESS
• Both ACF and PACF tails of after lag 1.