1.5 Empirical analysis of S&P 100 equity returns
1.5.2 Out-of-sample model selection
The previous section revealed that the proposed models significantly beat benchmark models in in-sample model comparison tests. However, since it is essentially a fore- casting model for daily return distributions, it has to be investigated whether it has superior out-of-sample (OOS) forecasting performance. In this section, we consider a multivariate density forecasting based on out-of-sample log (composite) likelihoods to compare models.
We use the period from January 2006 to December 2010 pR “ 1259q as the in- sample period, and January 2011 to December 2012 pP “ 502q as the out-of-sample period. Giacomini and White (2006) test described in Section 1.3.2 requires rolling window or fixed window estimation scheme rather than expanding window one. To incorporate structural changes, we employ a rolling window rather than a fixed win- dow. We estimate the whole model using the data in the interval rt ´ R ` 1, ts and evaluate the model using the data at t ` 1 with those estimates at each time in out-of-sample period. We iterate 502 times for these estimations and evaluations. Out-of-sample density forecast comparisons
Table 1.12 presents t -statistics from pair-wise OOS model comparison tests. Similar to the results from in-sample tests, we discover three finding again. Copula models are significantly better than the multivariate t distribution, and jointly symmetric copula models significantly outperform the independence copula model. Also models using information of high frequency data significantly beat models using information of daily data.
These results reveal that three major components of the proposed model, sepa- rately specifying marginal distribution and dependence, capturing nonlinear depen- dence and exploiting information of high frequency data lead to improved forecasting performance.
Out-of-sample portfolio decision making
To investigate a economic gain of the proposed model, we consider asset allocation problems in an out-of-sample setting proposed by Patton (2004). The basic idea is simple: a better forecasting model should lead to a better portfolio decision.
We introduce a hypothetical portfolio of 104 stocks listed in Table 1.5 and assume that an investor maximizes his expected utility by choosing optimal portfolio weights on 104 stocks. The utility functions for the investor are the class of CRRA (constant relative risk averse) utility functions:
U pW q “
" W1´ρ
1´ρ if ρ ‰ 1
log pW q if ρ “ 1
where ρ is a relative risk aversion parameter and W is wealth. Optimal portfolio weights are determined by maximizing the expected utility under the multivariate predictive density for rt`1
ω˚t`1“ arg max
ωPWEtrU pW0p1 ` ω 1
rt`1qqs (1.22)
where ω is N ˆ1 portfolio weights, W0is initial wealth and W “ !
ω P r0, 1sN : 11ω ď 1). For more realistic settings, we only consider an investor with short-sale constraint. Since the conditional expectation above is taken with respect to conditional dis- tributions of next period returns rt`1, we may expect a better forecasting model (conditional distribution) for rt`1 to give better portfolio weights which generate higher average utilities. By comparing those average utilities, we may pick up better forecasting models. However, utility is not intuitively interpretable, so we convert the average utility to a “management fee”, which is a fixed amount that could be charged (or paid) each period making the investor indifferent between portfolio A
and portfolio B. The management fee C is the solution to the following equation: 1 P R`P ÿ t“R`1 U`1 ` ω˚1 A,t`1rt`1 ˘ “ 1 P R`P ÿ t“R`1 U`1 ` ω˚1 B,t`1rt`1´ C ˘
where initial wealth W0 sets to be 1, R is the length of the in-sample period, and P is the length of the out-of-sample period.
We keep the same R and P as in the previous section, and RRA parameter ρ sets to be 7. We obtain the conditional expectation in equation (1.22) through Monte Carlo integrals using simulated data from estimated models in the previous section. Table 1.13 presents the estimated management fee C in annualized percent be- tween any two models of twelve competing models. A positive number indicates that the model above outperforms the model to the left, and a negative one indicates the opposite. We compare copula models and the multivariate t distribution to see whether separately specifying marginals and dependence is influential. Portfolio de- cisions based on the multivariate t distribution yields smaller economic gains than those from copula based models except one based on Clayton copula. The gains by changing models from non-copula models to copula models are from 0.48% to 2.42%. Second, we find that models that use high frequency data come up with higher economic gains than models that do not use high frequency data. The gains range from 0.4% to 6.3%. This confirms the superiority of models capable of employ- ing high frequency data to models incapable of using high frequency data. Lastly, to see how important nonlinear dependence is, we compare copula models with the independence copula. The copula model based on t copula beats the independence copula whereas the other copula models do not. This suggests properly capturing nonlinear dependence generates higher economic gains. Overall, models based on t copula with high frequency data outperform all other models. As aforementioned, the model based on t copula substantially differs from the benchmark model, the
multivariate t distribution in that the latter does not separately specify marginal distributions and dependence whereas the former does.
In sum, we find the strong evidence of usefulness of high frequency data and copula approaches and mild evidence of importance of nonlinear dependence under the portfolio decision problems. Through the realistic portfolio decision problems, the proposed model proves to have an excellent forecasting capability which in turn generates large economic gains.