# 14 Autoregressive Moving Average Processes

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```					14 AUTOREGRESSIVE MOVING AVERAGE PROCESSES

14 Autoregressive Moving Average Processes
We can now combine the MA and AR ideas to form a autoregressive moving average (ARMA) model. This is done by replacing the error term e t in the autoregressive model by an MA process, thus: y t = [φ0 + φ1 y t −1 + φ2 y t −2 + · · · + φp y t −p ] + [ψ0 + e t − ψ1 e t −1 − ψ2 e t −2 − · · · − ψq e t −q .] We combine the constants φ0 and ψ0 into one constant c, to get: y t = [c + φ1 y t −1 + φ2 y t −2 + · · · + φp y t −p ] + [e t − ψ1 e t −1 − ψ2 e t −2 − · · · − ψq e t −q ]. Using the backshift operator, we would have: (1 − φ1 B − φ2 B 2 − · · · − φp B p )y t = c + (1 − ψ1 B − ψ2 B 2 − · · · − ψq B q )e t . This is denoted ARMA(p, q). ARMA(p, q) means a model that has an autoregressive model of order p with error that is a moving average of order q. EXERCISE: write down the formula for an ARMA(1,2) model.

EXERCISE: look at the plot of an ARMA(3,2) process. Does it appear to be stationary?

EXERCISE: In the second column of the table below is a sequence of data values y 1 , . . . , y 5 . Use them to create values from the following ARMA(1,1) model: y t = 19 − 0.4y t −1 + e t − 0.5e t −1 , so that ˆ y t = 19 − 0.4y t −1 − 0.5e t −1 . Do this by ﬁrst completing the ﬁrst row in the fourth column, which is the error term of the process e t = ˆ ˆ y t − y t , and then using this to compute the y t in the next row. The ﬁrst value of the MA process cannot be ˆ calculated (there is no e 0 ), and so to start things off we assume that y 1 = 0.

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14 AUTOREGRESSIVE MOVING AVERAGE PROCESSES

Time t

Data y t

ˆ Fitted values y t

ˆ Error Process e t = y t − y t

1

9.5

0

2

13.7

3

8.7

4

16.1

5

15.3

14.1 Which ARMA model to choose?
The class of ARMA models is vast. Even ARMA models with the same p and q can have very different behaviour (see ﬁgure attached). Which ARMA model to pick to model a given time series is then a bewildering task. Fortunately, it turns out that many models with different p and q look more or less the same, so this choice may not be so crucial (that is to say, very similar predictions and goodness of ﬁt can often be obtained from ARMA models with different p and q). We will discuss this more in the next handout.

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18 FIGURES

18.15 Autoregressive Moving Average Processes
3 2 1
MA(3)

0 −1 −2 −3 0 10 20 30 40 50 t 60 70 80 90 100

3 2
ARMA(3,2)

1 0 −1 −2 −3 0 10 20 30 40 50 t 60 70 80 90 100

Figure 49: An MA process of order 3 and the ARMA(2,3) process formed from it (of equation y t = 0.1 + 0.3y t −1 − 0.3y t −1 + e t − 0.6e t −1 − 0.3e t −2 − 0.4e t −3 ).

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