Puzzled by derivation of time series prediction based on its log

Question

From Introductory Time Series with R:

If the random variation is modelled by a multiplicative factor and the variable is positive, an additive decomposition model for $\log(x_t)$ [where $x_t$ is the observation at time $t$] can be used:

$\log(x_t) = m_t + s_t + z_t$

[where $m_t$, $s_t$ and $z_t$ are, respectively, terms for the trend, seasonal effect and an error term at time $t$]

Some care is required when the exponential function is applied to the predicted mean of $\log(x_t)$ to obtain a prediction for the mean value $x_t$, as the effect is usually to bias the predictions. If the random series $z_t$ are normally distributed with mean 0 and variance $\sigma^2$ , then the predicted mean value at time $t$ based on Equation (1.4) is given by

$\hat{x}_t = e^{m_t + s_t} e^{\frac{1}{2}σ^2}$

The discussion about the caution that is necessary with this sort of model that follows makes sense and, unless it is necessary, I will omit it. What is confusing me is why $e^{\frac{1}{2}\sigma^2}$ is used to represent the error term. There is apparently some point about mathematical statistics that I'm not aware of, but would like to know. Thank you.

Answer 1

the reasoning must (except for the seasonal term) be the same as for a log normal random variable. Let $X$ be $N(\mu,\sigma^2)$ and consider $Y = \exp(X)$ then the expectation of $Y$ is given by $$ E[Y] = \exp(\mu + \sigma^2/2). $$ Then the seasoal term just shifts the mean further and you arrive at the formula that you have up there.

You can see more details in the wikipedia article on the log-normal distribution.

Answer 2

The $e^\frac{\hat{\sigma}^2}{2}$ does not represent the error term. It is part of the point forecast, i.e., the mean of the future realizations. It comes from the fact that the lognormal distribution is asymmetric (see the formula for the mean of a lognormal distribution) and is thus a consequence of modeling on the log scale.