Empirical Results for Density Forecast

Statistical Measures of Density Forecast Performance

Table 9 reports the Amisano and Giacomini (2007) (AG) test results for the predictive density comparison of individual models and a wide range of combination models. The benchmark model is the kernel density estimation based on historical unconditional quantile regression. Panel A displays that all of individual density fore-

casts under-perform the benchmark historical estimation without exception (AG test statistics are all negative), which indicates that all macro-financial variables fail to provide superior density forecast than the benchmark model in the term of AG test statistics. This fact is line with the empirical finding that most of macro-financial variables cannot predict the conditional mean of the equity premium (Goyal and Welch (2008)).

Panel B and C of Table 9, however, show the significant improvement of density forecast performance after utilizing both model and distributional information. All values of the AG test statistics become strongly positive and some of them are even statistically significant at the 5% level. More specifically, Panel B reports the AG test results for the estimation strategy of combining models of every quantile estimation in the first step, followed by kernel density estimation employing distributional informa- tion in the second step. In contrast, Panel C reverses the combination strategy, which estimates the individual kernel density first and then combines the different models of density forecasts in the second step. Table 9 shows that both the distributional and model information are useful to make forecasts in density estimation.

Economic Measures of Density Forecast Performance

Table 2.10 presents the economic evaluation of a wide ranges of density forecasts under relative risk aversion preference (CRRA). Again, the economic significance of density forecast are evaluated by two metrics: (1) Average Utility Gain ∆U (%) can be considered as the portfolio management fee that an investor under CRRA prefer- ence would be willing to pay to have access to the proposed conditional kernel density forecast relative to unconditional benchmark forecast. (2) CER refers to certainty equivalent rate of return that provides the same utility level as the risky portfolio. As Panel A of Table 10 shows, most conditional density forecasts based on macro-finance

variables fail to provide economic gain relative to historical benchmark kernel den- sity forecasts. Panel B reports the results of economic evaluation for the estimation strategy of model combination in the first step followed by kernel density estimation in the second step. As we can see, combining the information both across different models and multiple quantiles can lead to superior density forecast in term of utility metrics. All values of ∆U (%) and CER obtained from the proposed density forecast in Panel B are greater than those derived from benchmark kernel density estimation in which no predictive variables are used to make the forecast. In contrast, Panel C reverses the combination order in which the kernel density estimation is implemented for each macro-finance variable first, followed by combining different density forecasts in the second step. Again density forecasts utilizing both model and distributional information significantly outperform the benchmark kernel density estimation.

Figure 2.7 and 2.8 displays how the average utility gain ∆U (%) and CER under CRRA preference changes with relative risk aversion coefficient γ. I intend to show that the conclusion arrived in table 2.10 is robust to the choice of relative risk aversion coefficient (risk appetite of an investor).

The first two panels of Figure 2.7 show the utility gain based on density fore- cast which pools the distributional information in the first step, followed by models combination in the second step. The other panels of Figure 2.7 derive the value of ∆U (%) based on the density forecast in which the order of combination strategies reverse , namely, model combination in the quantile estimation followed by density kernel estimation. As we can see, pooling distributional information as well as model information can produce better density forecasts relative to the hisotrical benchmark forecast, which results in higher economic value in the measurement of average utility gain ∆U (%) and CER30_{. The first combination strategy perform slightly better than}

30_{∆U (%) is positive in Figure 2.7, indicating a better density forecast relative to the historical}

the second combination strategy, and the likelihood weight density forecast gener- ates a higher utility gain than the other density forecasts. Furthermore, the average utility gain ∆U (%) attains the greatest value when investors are mildly risk averse γ = 3, but decays slowly as the risk aversion coefficient γ increases. This fact is not surprising since the forecast of tail distribution become more and more important as investors become more and more risk averse, but the accuracy of forecasts in the tail of distribution deteriorates dramatically31. Therefore, the relative utility gain decreases as investors become more risk averse since the accuracy of forecast in the tails of the distribution decrease relative to historical benchmark forecast. This can explain why the value of ∆U (%) decays gradually as the risk aversion coefficient γ increases.

followed with kernel density estimation is also higher than that derived from the historical benchmark density forecast.