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Ec 178 –
ECON & BUS FORECASTING
LECTURE NOTES
Foster, UCSD
February 25, 2010
TOPIC 2  SMOOTHING & TRENDS
A.
Filters and Smoothing
1.
Filters:
a)
A filter is a function which transforms a raw data series into a new series.
1)
For raw data y
t
, let x
t
= f(y
t
) be the transformed series.
2)
x
t
is a linear filter if f(y
t
) is a linear function.
Example:
x
t
=
α
1
y
t+1
+
α
2
y
t
+
α
3
y
t1
.
3)
If
Σα
i
= 1, then x
t
is an averaging transformation.
b)
Some common filters.
•
Series mean
•
Simple and Exponentiallyweighted moving averages
•
HodrickPrescott and other nonlinear filters
•
Consecutive differencing
c)
Uses of filters.
[Fig. 1]
1)
Smooth data to reduce randomness
and reveal underlying patterns.
2)
Remove seasonal component to pro
duce seasonallyadjusted series.
3)
Generate simple, cheap forecasts.
4)
Create stationary ARMA series.
2.
Simple Moving Average of Length m:
a)
Let (m)
ℓ
t
denote a simple moving average, the average of the m
most recent values of y available at time t.
(Note that
ℓ
t
is the
smoothed level
of series y
t
.)
b)
Numerical example and notes.
1)
A simple MA is appropriate for smoothing out a data series y
t
that has no up or down trend and no recurring seasonal
pattern.
2)
A larger m gives a smoother result, but we always lose m1 values at the beginning
of the smoothed series (m)
ℓ
t
.
3)
Simple MA has given way to EWMA and nonlinear filters.
Simple MA
t
y
t
(2)
ℓ
t
(4)
ℓ
t
1
2
3
4
5
6
3
5
2
5
2
4
///
4.0
3.5
3.5
3.5
3.0
///
///
///
3.7
5
3.5
0
3.2
5
y
t
x
t
Fig. 1
)
2
{int
1
)
(
1
1
1
0
≥
∈
+
+
+
=
=
+



=

∑
m
m
y
y
y
y
m
m
m
t
t
t
m
s
s
t
t
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SMOOTHING/TRENDS
p. 2
3.
Centered Moving Averages of Length m:
a)
Let CMA(m)
t
be a centered moving average of length m, an average of m consecutive
values of series y centered on time period t.
b)
Numerical examples and notes.
1)
A CMA is appropriate for smoothing series with trend
or recurring seasonal variation (repeating every m
periods).
2)
Note that some observations are always lost at the
beginning and end of the smoothed series CMA(m)
t
.
3)
In the example at right:
•
CMA(3)
t
= [y
t1
+ y
t
+ y
t+1
]/3
•
CMA(6)
t
= [y
t3
+ 2(y
t2
+ y
t1
+ y
t
+ y
t+1
+ y
t+2
) + y
t+3
]/
12
4.
ExponentiallyWeighted Moving Averages:
a)
For smoothing parameter
, let ( )
α
ℓ α
t
or just
ℓ
t
denote an
exponentiallyweighted moving average of series y at
time t.
b)
An EWMA is a weighted average
of all past observations on y
t
going back forever, with
more weight given to recent observations than to those from the remote past.
1)
By backward substitution we obtain the result below:
2)
Since 0 < (1
) < 1, as s
−α
→
∞
, (1
)
s
→
0, so the weights are
declining to zero as we go back in time.
c)
For recursive equations like EWMA, we must provide initial values.
1)
One way to initialize is to set ( )
ℓ α
1
= y
1
.
This implies a “runin
period” as
ℓ
t
gradually forgets the arbitrary starting value and
settles in to the average level of y
t
.
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 Spring '10
 Foster

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