13.2.1 Location–Scale Models of Density Forecasts

13.2.1 Location–Scale Models of Density Forecasts

The most popular approach for generating density forecasts assumes that the predicted variable is generated by a conditional location–scale process of the form

seen in Example 13.1.1

$$y_{t+1}={\mu }_{t+1|t}+{\sigma }_{t+1|t}{\eta }_{t+1},{\eta }_{t+1}\sim (0,1),\text{\hspace{1em}(13.3)}$$

where $E\left[{\eta }_{t+1}|Z_t\right]=0$ and $E\left[{\eta }_{t+1}^2|Z_t\right]=1$ . Hence ${\mu }_{t+1|t}$ is the conditional mean of $y_{t+1}$ given conditioning information, $Z_t$ , while ${\sigma }_{t+1|t}$ is the conditional standard deviation or volatility of $y_{t+1}$ , again given $Z_t$ . Both the conditional mean and volatility can be time varying and will depend on a finite-dimensional vector of parameters, θ , which we suppress in the notation here. The distribution function of η, $P_{\eta },$ is usually assumed to be constant (time invariant), although the approach can easily be extended to allow for time-varying conditional skew, kurtosis, or tail risks.

Assuming that ${\eta }_{t+1}|Z_t$ has a time-invariant distribution function, $P_{\eta }$ , we can derive the conditional distribution function for $y_{t+1}$ given $Z_t$ as follows:

$$\begin{array}{l}P(y_{t+1}\le y|Z_t)=P\left(\frac{y_{t+1}-{\mu }_{t+h|t}}{{\sigma }_{t+1|t}}\le \frac{y-{\mu }_{t+1|t}}{{\sigma }_{t+1|t}}\right)\\ \equiv P_{\eta }\left(\frac{y-{\mu }_{t+1|t}}{{\sigma }_{t+1|t}}\right).\end{array}$$

This can readily be computed if a parametric model for $P_{\eta }$ is available. For example, if the density of η is a standard normal, we have for ${\eta }_{t+1}=(y_{t+1}-{\mu }_{t+1|t})/{\sigma }_{t+1|t}$

$$P_{\eta }({\eta }_{t+1})=(\sqrt{2\pi }{)}^{-1}{\int }_{-\infty }^{{\eta }_{t+1}}\exp (-0.5x^2)dx.$$

Semiparametric methods, on the other hand, do not make distributional assumptions about $\eta ,$ instead assuming that this can be approximated by the empirical distribution function of ${\hat{\eta }}_{t+1|t}=(y_{t+1}-{\hat{\mu }}_{t+1|t})/{\hat{\sigma }}_{t+1|t}$

Construction of an estimate for the conditional mean ${\mu }_{t+1|t}$ is standard and has already been covered. What is new in (13.3) is the presence of a time-varying conditional volatility, ${\sigma }_{t+1|t}$ . The parameters of the volatility process are often estimated using maximum likelihood or QMLE methods.

Example 13.2.1 (Estimation of parameters of location–scale model). Suppose the mean and variance functions and the density of the standardized innovations, ${\eta }_{t+1},$ are specified up to a finite-dimensional vector of unknown coefficients, θ, and consider a volatility model whose normalized innovations ${\eta }_{t+1}$ are assumed to be independent N(0, 1). For a sample of T observations $\{y_1,...,y_T\}$ , the log-likelihood function is

$$\mathrm{LL}(\theta |y_1,y_2,\dots ,y_T)=-\frac{1}{2}\sum _{t=1}^{T-1}\left(\ln (2\pi {\sigma }_{t+1|t}^2)+\frac{(y_{t+1}-{\mu }_{t+1|t}(\theta ){)}^2}{{\sigma }_{t+1|t}^2(\theta )}\right).\text{\hspace{1em}(13.4)}$$

This gives rise to a set of nonlinear first-order conditions. Maximization of the loglikelihood function therefore requires using numerical methods. Notice that even if one is only interested in estimating the parameters of the conditional mean, OLS methods are not efficient in the presence of volatility dynamics.

In the following we describe some simple and popular models for dynamics in the conditional volatility. With these in place, we then consider ways to generate distribution forecasts using various density forecasting models for the “normalized innovation,” ${\dot{\eta }}_{t+1}$