Fitting a Pareto-Normal-Pareto distribution to the residuals of financial data

Summary

The Pareto-Normal-Pareto (PNP) distribution assumes that, for log returns of financial series, the innovations are normally distributed between two threshold values with Pareto tails below and above the respective thresholds. These threshold values can be estimated by maximum likelihood estimation (MLE). Monte Carlo simulations of normal, as well as heavy tailed error distributions, are used to compare the methods using this distribution with other methods to calculate Value-at-Risk (VaR) and Expected Shortfall (ESf). It is also applied to South African stock exchange data.

Appendix A. Estimation of PNP parameters

For the innovations of AR-GARCH models to have a zero expectation, it must hold that

$$E\left[ {Z|Z \le {u_1}} \right] + E\left[ {Z|{u_1} < Z < {u_2}} \right] + E\left[ {Z|Z \ge {u_2}} \right] = 0$$

or$\mu = - {\alpha _1}{z_1}\sigma - {\alpha _2}{z_2}\sigma - \sigma \left( {\phi \left( {{z_1}} \right) - \phi \left( {{z_2}} \right)} \right) - {{{\alpha _2}{\beta _2}} \over {1 - {\xi _2}}} + {{{\alpha _1}{\beta _1}} \over {1 - {\xi _1}}}.$

For the AR-GARCH restriction of unit variance, it must hold that

$$\matrix{ {1 + E{{\left[ Z \right]}^2} = {\sigma ^2}V = {\sigma ^2}\left\{ {{\alpha _1}\left[ {{{2\beta _1^{*2}} \over {\left( {1 - {\xi _1}} \right)\left( {1 - 2{\xi _1}} \right)}} + u_1^{*2} - {{2\beta _1^*u_1^*} \over {\left( {1 - {\xi _1}} \right)}}} \right]} \right. + \left( {1 - {\alpha _1} - {\alpha _2}} \right)\left( {1 + {\mu ^{*2}}} \right)} \\ { + \left[ {{z_1}\phi \left( {{z_1}} \right) - {z_2}\phi \left( {{z_2}} \right)} \right] + 2{\mu ^*}\left[ {\phi \left( {{z_1}} \right) - \phi \left( {{z_2}} \right)} \right]\left. { + {\alpha _2}\left[ {{{2\beta _2^{*2}} \over {\left( {1 - {\xi _2}} \right)\left( {1 - 2{\xi _2}} \right)}} + u_2^{*2} - {{2\beta _2^*u_2^*} \over {\left( {1 - {\xi _2}} \right)}}} \right]} \right\},} \\ {{\rm{with}}\,\,\beta _1^* = {{{\beta _1}} \over \sigma },\beta _2^* = {{{\beta _2}} \over \sigma },\quad \mu _1^* = {{{\mu _1}} \over \sigma },\,\mu _2^* = {{{\mu _2}} \over \sigma }\,{\rm{and}}\,{\mu ^*} = {\mu \over \sigma }.} \\ } $$

Parameters a₁, a₂, ξ₁, ξ₂, μ, σ are estimated by ML and from it ${\widehat{z}_1}$, ${\widehat{z}_2}$,,$\widehat\beta _1^*$,$\widehat\beta _2^*$,$\widehat{u}_1^*$,$\widehat{u}_2^*$and the factor $\widehat{V}$is calculated. Take ${\widehat\sigma ^2} = {1 \over {\widehat{V}}}$ and now calculate ${\widehat\mu }$, ${\widehat\beta _1}$, ${\widehat\beta _2}$, ${\widehat{u}_1}$ and ${\widehat{u}_2}$