Unformatted text preview: Notes 5 Supplementary
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
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.
Suppose Zt tends to have relatively stable percentage changes from one time period to
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
[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