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I found in Section 3.1 of this paper (see this screen shot), the author used log(y) due to "the context of a concentration time series". A book (Hastie and Tibshirani, 1990) is cited, however I cannot get this book. So I asked this question: why we should log transform the y, when it is a time series?

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  • $\begingroup$ I can confirm what the quotation implies: Hastie and Tibshirani are being cited here because of the use of a generalized additive model with smooth functions $s_j()$ and an error term $\varepsilon_i$. $\endgroup$ – Nick Cox Apr 23 at 16:18
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The reason is the "concentration" part of the "concentration time series": log transformations are often useful in analysis of chemical concentration data.

As the concentrations of the pollutants at issue in the linked paper are necessarily positive, may have inherent measurement errors that are proportional to the measured values rather than independent of measured values, and the covariates being considered might be better related to percentage changes in those concentrations rather than to linear changes, a log transformation of the concentration values could let the model fit better while meeting the underlying assumption (stated in the end of the sentence you copy here) that the error terms $\varepsilon_i$ are normally distributed with variance constant over the range of values.

More discussion about use of log transforms specifically in time-series analysis is on this page.

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