5 supp - Notes 5 Supplementary Other transformations We...

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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.

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