8.2 MA, AR, and ARMA
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moving average of n + 1 consecutive shocks. The notation in (8.2) makes it easy to
understand why the process is named “moving average.” In the literature, however, a
notation is commonly used where the first coefficient is set to unity and the remaining
coefficients are denoted by θ with their sign reversed. A general MA(q) process is
therefore characterized by
y i = ε i − θ 1 ε i−1 − · · · − θ q ε i−q
(8.3)
with the coefficients θ k related to the coefficients a j through a 0 = 1 and a j = −θ j
for j > 0.
We point out that in electrical engineering (8.2) is called a finite-impulse response
(FIR) filter, provided that the random shocks ε j are replaced by a sampled input signal
x j . The output signal y i is then given by the weighted average of the most recent input
samples x j , where the filter coefficients a j are the weights applied to the respective
samples. The name “finite-impulse response” describes the feature, that a particular
input samples x j requires n + 1 time steps to trickle through the filter. For example,
an input signal with a single non-zero value x k = 1 will produce the filter coefficients
as output signal y i .
A slightly more complex filter is an infinite-impulse response (IIR) filter, which
is based on feeding a fraction of the output signal back to the input. Its functional
dependence is given by
y i =
n
j=1
b j y i− j + a 0 x i ,
(8.4)
where the coefficients b k describe the weights of how much of the previous output
signals y i− j is fed back to the input. The name “infinite-impulse” describes the feature
that combinations of coefficients can cause a single non-zero input sample to result
in an output signal with infinite duration.
An example of an IIR filter is the exponentially weighted moving average
(EWMA) filter, which is given by
y i =
1
m + 1
(my i−1 + x i ) ,
(8.5)
related to the coefficients in (8.4) through b 1 = m/(m + 1) and a 0 = 1/(m + 1). The
interpretation is straightforward. The output consists of a heavily weighted previous
sample (with weight m/(m + 1)) and a weakly weighted new sample update (with
weight 1/(m + 1)) in case m is moderately large. This filter is also a low-pass filter,
because it smoothes fluctuations of the input signal x i over approximately m samples.
IIR and FIR filters are widely used in digital data-acquisition applications to clean
up noisy signals. But let us return to the discussion of statistical processes that are
excited by random shocks ε j .
In the time-series literature the processes, corresponding to IIR filters from electrical engineering, are called autoregressive (AR) processes, which describe the feeding
95
moving average of n + 1 consecutive shocks. The notation in (8.2) makes it easy to
understand why the process is named “moving average.” In the literature, however, a
notation is commonly used where the first coefficient is set to unity and the remaining
coefficients are denoted by θ with their sign reversed. A general MA(q) process is
therefore characterized by
y i = ε i − θ 1 ε i−1 − · · · − θ q ε i−q
(8.3)
with the coefficients θ k related to the coefficients a j through a 0 = 1 and a j = −θ j
for j > 0.
We point out that in electrical engineering (8.2) is called a finite-impulse response
(FIR) filter, provided that the random shocks ε j are replaced by a sampled input signal
x j . The output signal y i is then given by the weighted average of the most recent input
samples x j , where the filter coefficients a j are the weights applied to the respective
samples. The name “finite-impulse response” describes the feature, that a particular
input samples x j requires n + 1 time steps to trickle through the filter. For example,
an input signal with a single non-zero value x k = 1 will produce the filter coefficients
as output signal y i .
A slightly more complex filter is an infinite-impulse response (IIR) filter, which
is based on feeding a fraction of the output signal back to the input. Its functional
dependence is given by
y i =
n
j=1
b j y i− j + a 0 x i ,
(8.4)
where the coefficients b k describe the weights of how much of the previous output
signals y i− j is fed back to the input. The name “infinite-impulse” describes the feature
that combinations of coefficients can cause a single non-zero input sample to result
in an output signal with infinite duration.
An example of an IIR filter is the exponentially weighted moving average
(EWMA) filter, which is given by
y i =
1
m + 1
(my i−1 + x i ) ,
(8.5)
related to the coefficients in (8.4) through b 1 = m/(m + 1) and a 0 = 1/(m + 1). The
interpretation is straightforward. The output consists of a heavily weighted previous
sample (with weight m/(m + 1)) and a weakly weighted new sample update (with
weight 1/(m + 1)) in case m is moderately large. This filter is also a low-pass filter,
because it smoothes fluctuations of the input signal x i over approximately m samples.
IIR and FIR filters are widely used in digital data-acquisition applications to clean
up noisy signals. But let us return to the discussion of statistical processes that are
excited by random shocks ε j .
In the time-series literature the processes, corresponding to IIR filters from electrical engineering, are called autoregressive (AR) processes, which describe the feeding