13.4.1 Parametric Density Models

13.4.1 Parametric Density Models

We next cover some popular parametric density models that have found widespread use particularly in finance applications, including the student-t, two-piece normal, and mixtures of normals.

13.4.1.1 Non-Gaussian Density Models

Even after accounting for volatility dynamics, the standardized residuals, ${\eta }_{t+1|t},$ are often found not to be normally distributed. To account for this, a variety of alternative parametric density models have been proposed. These typically seek to capture “stylized features” of the forecast errors such as skew or fat tails by means of flexible yet parsimonious densities that depend on only a few judiciously chosen parameters. We briefly describe some of the parametric methods that have been found to be useful in generating forecasts of economic and financial time series.

Hansen (1994) proposed the skewed-t distribution with zero mean and unit variance which takes the form

$$\begin{array}{r}p({\eta }_{t+1}|\gamma ,\lambda )=\{\begin{array}{ll}bc{\left(1+\frac{1}{\gamma -2}{\left(\frac{b{\eta }_{t+1}+a}{1-\lambda }\right)}^2\right)}^{-(\gamma +1)/2},& {\eta }_{t+1}<-a/b,\\ bc{\left(1+\frac{1}{\gamma -2}{\left(\frac{b{\eta }_{t+1}+a}{1+\lambda }\right)}^2\right)}^{-(\gamma +1)/2},& {\eta }_{t+1}\ge -a/b.\end{array}\end{array}\text{\hspace{1em}(13.28)}$$

The skewness $(\lambda )$ and kurtosis $(\gamma )$ parameters are subject to the restrictions $2<\gamma <$ $\infty ,-1<\lambda <1$ , while the scalar constants a, b, c are functions of λ and $\gamma :$

$$\begin{array}{l}a=4\lambda c\left(\frac{\gamma -2}{\gamma -1}\right),\\ b^2=1+3{\lambda }^2-a^2,\\ c=\frac{\Gamma ((\gamma +1)/2)}{\sqrt{\pi (\gamma -2)\Gamma (\gamma /2)}}.\end{array}$$

Here  is the gamma function. The parameters in (13.28) can be generalized to depend on lagged innovations, e.g., $\lambda ({\epsilon }_{t-1}),\gamma ({\epsilon }_{t-1})$ , which would produce an autoregressive conditional density model.

The two-piece normal distribution assumes that the predicted variable has a density function

$$\begin{array}{r}{p}_Y(y_{t+1}|Z_t)=\{\begin{array}{ll}\frac{\exp (-(y_{t+1}-{\mu }_{t+1|t}{)}^2/2{\sigma }_1^2)}{\sqrt{2\pi }({\sigma }_1+{\sigma }_2)/2}& \mathrm{for}{y}_{t+1}\le {\mu }_{t+1|t},\\ \frac{\exp (-(y_{t+1}-{\mu }_{t+1|t}{)}^2/2{\sigma }_2^2)}{\sqrt{2\pi }({\sigma }_1+{\sigma }_2)/2}& \mathrm{for}{y}_{t+1}>{\mu }_{t+1|t}.\end{array}\end{array}\text{\hspace{1em}(13.29)}$$

The mean and variance of this distribution is

$$E_t[y_{t+1}]={\mu }_{t+1|t}+\sqrt{\frac{2}{\pi }}({\sigma }_2-{\sigma }_1),\text{\hspace{1em}(13.30)}$$

$${\mathrm{Var}}_t(y_{t+1})=\left(1-\frac{2}{\pi }\right)({\sigma }_2-{\sigma }_1{)}^2+{\sigma }_1{\sigma }_2.\text{\hspace{1em}(13.31)}$$

If ${\sigma }_2>{\sigma }_1$ , the distribution is positively skewed and large positive values become more likely than large negative values. The distribution has fat tails provided that ${\sigma }_1\ne {\sigma }_2$ . Only one extra parameter is introduced relative to the standard normal distribution whose critical values can still be used subject to a scaling factor. This approach has been used by the Bank of England to generate the so-called fan charts which communicate the degree of uncertainty surrounding the Bank’s macroeconomic forecasts.

For this family of distributions the so-called balance of risk, i.e., the probability of experiencing an outcome below the mean, is given by $\mathrm{Pr}(y_{t+1}\le {\mu }_{t+1\mid t})={\sigma }_1/({\sigma }_1+$ ${\sigma }_2)$ . This equals 0.5 for ${\sigma }_1={\sigma }_2$ , but grows bigger if the volatility of outcomes below the mean is higher than that of outcomes above the mean.

Figure 13.4: Two-piece exponential density with $\mu =0$ and ${\sigma }_1=2.$ ${\sigma }_2=1$

As an illustration of this model, figure 13.4 compares the two-piece exponential density with $\mu =0$ and ${\sigma }_1=2,{\sigma }_2=1$ to a standard normal distribution. The two densities are very different with the two-piece density displaying a notably slower decay for negative values and a faster decay for positive values relative to the symmetric normal distribution. This type of two-piece density can thus be used to capture a pronounced left skew in the distribution.