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. 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.

||<1 STATIONARITY CONDITION 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 Keep this part as it is

AR(1) PROCESS

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. 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.

AR(2) PROCESS

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 at is independent of Ytk, we have

AR(2) PROCESS

AR(2) PROCESS

AR(2) PROCESS

AR(2) PROCESS

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 PROCESS Errors are the average of this period’s random error and last period’s 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.

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.

THE FIRST ORDER MOVING AVERAGE PROCESS_MA(1) PROCESS Consider the process satisfying

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 PACF:

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:

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

MA(q) PROCESS The autocovariance function: ACF:

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 For the process to be invertible, the roots of lie outside the unit circle. For the process to be stationary, the roots of 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

ARMA(1,1) PROCESS Both ACF and PACF tails of after lag 1.

ARMA(1,1) PROCESS IF:

ARMA(1,1) PROCESS RSF:

AR(1) PROCESS

AR(2) PROCESS

MA(1) PROCESS

MA(2) PROCESS

ARMA(1,1) PROCESS

ARMA(1,1) PROCESS (contd.)