Comparing models and data
Suppose we would like to see whether output is more or less volatile than
investment
We want a measure of volatility that is invariant to scale
Simple variance or standard deviation is not scale invariant
Suppose for example that investment is a constant fraction
λ
of GDP
x
=
λy
Now
sd
(
x
) =
λsd
(
y
)
A simple remedy would be rescale both variables by their mean prior to
computing the standard deviation
sd
(
x
µ
x
) =
1
µ
x
sd
(
x
) =
1
λµ
y
λsd
(
y
) =
sd
(
y
µ
y
)
Now if we have variables that are non-stationary we need to apply some sort
of
fi
lter prior to computing measures of dispersion.
Suppose
x
gt
is the value of the growth component at
t,
and
x
ct
=
x
t
−
x
gt
is the value of cyclical component
Now rather than dividing by the mean, we should divide by the trend prior
to computing the standard deviation
The percentage standard deviation of the
fi
ltered series is given by
%
sd
(
x
ct
) = 100
×
sd
μ
x
ct
x
gt
¶
In our simple example, if our
fi
lter has the property that
x
gt
=
λy
gt
,

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