2.3 Transformations

A plot may suggest transforming the series, for example by taking logarithms or square roots. There are three main reasons:

1.: to stabilise the variance;
2.: to make the seasonal effect additive;
3.: to make the data closer to normally distributed.

The logarithm does the first two jobs at once when the series is multiplicative, as the next result shows.

Proposition 2.3.1.

(i): If \(x_t = m_t\,s_t\,\varepsilon _t\) with all factors positive, then \(\log x_t = \log m_t + \log s_t + \log \varepsilon _t\), an additive model.
(ii): If the standard deviation of \(x_t\) is proportional to its mean \(\mu _t\), say \(\operatorname {sd}(x_t) = c\,\mu _t\), then approximately \(\Var (\log x_t) \approx c^2\), which does not depend on \(t\).

Proof. (i) is the rule \(\log (abc) = \log a + \log b + \log c\). For (ii), by Taylor’s theorem \(\log x_t \approx \log \mu _t + (x_t - \mu _t)/\mu _t\) when \(x_t\) is close to \(\mu _t\). The right-hand side has variance \(\Var (x_t)/\mu _t^2 = c^2\mu _t^2/\mu _t^2 = c^2\). □

So when the seasonal swings, or the scatter, of a series grow with its level, take logarithms first.

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