Unformatted text preview: Notes 5 Supplementary
Other transformations
We have seen how diﬀerencing can be a useful transformation for achieving stationarity.
However, the logarithm transformation, perhaps followed by diﬀerencing is also a useful
method in certain circumstances.
We sometimes encounter series where increased dispersion seems to be associated with
increased levels of the series  the larger the level of the series, the more variation there
is around that level and conversely.
Speciﬁcally, suppose that Zt > 0 for all t with
E (Zt ) = µt , V ar(Zt ) = µt σ By Taylor series expansion,
log (Zt ) ≈ log (µt ) + Zt − µt
+ ...
µt Therefore
E [log (Zt )] ≈ log (µt ), V ar[log (Zt )] ≈ σ 2
In other words, if the standard deviation of the series is proportional to the level of the
series, then transforming to logarithms will produce a series with approximately constant variance.
Percentage change
Suppose Zt tends to have relatively stable percentage changes from one time period to
the next.
Speciﬁcally, assume that
Zt = (1 + Xt )Zt−1
where 100Xt is the percentage change (possibly ve) from Zt−1 to Zt . Then
log (Zt ) − log (Zt−1 ) = log ( Zt
) = log (1 + Xt ).
Zt−1 If Xt is restricted to, say Xt  < 0.2, i.e. the percentage changes are at most ±20%,
then, to a good approximation,
log (1 + Xt ) ≈ Xt
Consequently,
[log (Zt )] ≈ Xt
will be relatively stable and perhaps modelled by a stationary model. 1 ...
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This note was uploaded on 11/02/2011 for the course STAT 4005 taught by Professor Wu,kaho during the Spring '08 term at CUHK.
 Spring '08
 WU,KaHo

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