## 18.2 Conditional variance and scale

Similar to conditional expectations, as we have discussed in Sections 5.3 and 6.3, the conditional h steps ahead variance is in general available only for the pure additive models. While the conditional expectation might be required on its own to use as a point forecast, the conditional variance is typically needed to produce prediction intervals. However, it becomes useful only in cases of distributions that support convolution (addition of random variables), which limits its usefulness to pure additive model and to additive models applied to the data in logarithms. For example, if we deal with Inverse Gaussian distribution, then the h-steps-ahead values will not follow Inverse Gaussian distribution, and we would need to revert to simulations in order to obtain the proper statistics for it. Another situation would be a multiplicative error model that relies on Normal distribution - the product of Normal distributions is not a Normal distribution, so the statistics would need to be obtained using simulations again.

If we deal with pure additive model with either Normal, Laplace, S or Generalised Normal distributions, then the formulae derived in Section 5.3 can be used to produce h-steps-ahead conditional variance. Having obtained those values, we can then produce conditional h-steps-ahead scales for the distributions (which would be needed, for example, to generate quantiles from these distributions), using the relations between the variance and scale in those distributions (discussed in Section 5.5):

- Normal: scale is \(\sigma^2_h\);
- Laplace: \(s_h = \sigma_h \sqrt{\frac{1}{2}}\);
- S: \(s_h = \sqrt{\sigma_h}\sqrt[4]{\frac{1}{120}}\);
- Generalised Normal: \(s_h = \sigma_h \sqrt{\frac{\Gamma(1/\beta)}{\Gamma(3/\beta)}}\).

If the variance is needed for the other combinations of model/distributions, simulations would need to be done to produce multiple trajectories, similar to how it was done in Subsection 18.1.1. An alternative to this would be the calculation of in-sample multistep forecast errors (similar to how it was discussed in Sections 11.3 and 14.7.1). Having the matrix of forecast errors, we can then calculate variance for each horizon \(h\).

### 18.2.1 Scale model

In case of the scale model (Chapter 17), the situation becomes more complicated, because we no longer assume that the variance of the error term is constant (homoscedastic) – we now assume that it is a model on its own. In this case, we need to take a step back to the recursion (5.9) and when taking the variance, introduce the time varying variance \(\sigma_{t+h}^2\).

*Remark*. Note the difference between \(\sigma_{t+h}^2\) and \(\sigma_{h}^2\) in our notations - the former is the variance of the error term for the specific step \(t+h\), while the latter is the conditional variance \(h\) steps ahead, which is derived based on the assumption of homoscedasticity.

Making that substitution leads to the following analytical formula for the h-steps-ahead conditional variance in case of scale model: \[\begin{equation} \text{V}(y_{t+h}|t) = \sum_{i=1}^d \left(\mathbf{w}_{m_i}^\prime \sum_{j=1}^{\lceil\frac{h}{m_i}\rceil-1} \mathbf{F}_{m_i}^{j-1} \mathbf{g}_{m_i} \mathbf{g}^\prime_{m_i} (\mathbf{F}_{m_i}^\prime)^{j-1} \mathbf{w}_{m_i} \sigma_{t+h-j}^2 \right) + \sigma_{t+h}^2 . \tag{18.1} \end{equation}\] This variance can then be used, for example, to produce quantiles from the assumed distribution.

As mentioned above, in case of the not purely additive model or model with other distributions than Normal, Laplace, S or Generalised Normal, the conditional variance can be obtained using simulations. In case of the scale model, the principles will be the same, just assuming that each error term \(\epsilon_{t+h}\) has its own scale, obtained from the estimated scale model. The rest of the logic will be exactly the same as discussed in Subsection 18.1.1.