The generalized hyperbolic distribution introduced in Barndorff-Nielsen (1977) has often been used to model return distributions in financial markets since its application to DAX stock returns by Eberlein and Keller (1995). Variance gamma distribution by Madan and Seneta (1990), the skewed Student’s
Normal inverse Gaussian (NIG) distribution which is a subclass of the generalized hyperbolic class of distributions has been successfully used in financial literature. Rydberg (1997) examined the performance of the NIG distribution using German and Danish stocks and Eriksson
Some research has been done on the parameter estimation of NIG distribution; for instance, Figueroa-López
The remainder of the paper is organized as follows. In Section 2, we define the NIG distribution and its feasible domain. Section 3 explains the adjusted estimation methods to be compared in later sections. In Section 4, simulation and real data application results using S&P 500 return data, U.S. indemnity loss data, and Danish fire loss data are provided. Section 5 finally concludes the paper.
The probability density function of the NIG distribution is as follows. For −
where
With the NIG distribution, the population moments are expressed in terms of
where
From (2.2), we can rewrite
As can be seen in (2.3), these parameters are not defined when 3
When the underlying distribution is NIG distribution, (
To impose the conditions on the estimators of NIG parameters such as −
where
By (2.2), we can easily see that the following relationship holds:
Thus, it is obvious that
This subsection chooses the value of
Among the considered sets of 250 S&P500 daily log return data above, we select three sets with
In Figure 2, RMSE of
Here, we compare the performance of the ordinary MLE, the ordinary MME,
For each scenario, we generate 1,000 observations to calculate estimates and repeat this process 1,000 times. We consider two sets of initial values in order to also see the effect of the initial value in maximum likelihood estimation. In Case 1, the maximization procedure starts at
Tables 3 and 4 show the effect of initial values on estimation. In particular, the performance of the ordinary MLE improves noticeably from Case 1 to Case 2 as seen with bold-faced numbers in Table 3.
We now compare the difference between the true density and the fitted density as indicated in Figures 3 and 4. As in the previous simulation, we generate 1,000 observations each from scenarios 1 and 2 to estimate the parameters. For 1,000 equidistant points between −0.1 and 0.1 such as
where
Tables 5 and 6 are RMSEs and the means and the standard deviations of the estimators in Scenario 2, respectively. In this scenario, MME is calculated for only 544 out of 1,000 simulations since
Figure 4 provides the averages of squared differences between the true density and the estimated density in Scenario 2. The density estimated with exp-MLE is closer to the true density than those of ordinary MLE and
There are other parameter estimation methods such as the EM algorithm and MCMC method. Karlis (2002) dealt with EM algorithm for NIG distribution and Karlis and Lillestöl (2004) proposed estimation for the parameters of the NIG distribution through an MCMC scheme based on the Gibbs sampler. We also considered EM algorithm and MCMC method; however, we did not include them because the results of EM were not as good as all the estimators mentioned above and the MCMC method took so much time compared to other estimation methods.
In a real data application, we estimate the parameters of NIG distribution for two data sets of a S&P500 daily log return and two data sets of insurance loss. The first S&P500 data set denoted as S&P500 I is collected from March 28, 2003 to April 24, 2004 and provides a negative value of
Table 7 provides the descriptive statistics for four data sets used.
Figure 5 compares a kernel density estimator with the estimated NIG density with
We run Kolmogorov-Smirnov (KS) test and Anderson-Darling (AD) test to examine if the NIG distribution is appropriate in describing the real datasets. Tables 8 and 9 show
This study reviews adjusted parameter estimation methods of
From the simulation studies, the ordinary MLE was greatly influenced by the initial value and its performance improved significantly by setting the
In the future, we would like to investigate other cases of generalized hyperbolic distribution in terms of the feasible domain since we only deal with NIG distribution that is a special case of the generalized hyperbolic distribution. We will also study estimation methods such as EM algorithm and the MCMC method with generalized hyperbolic distribution.
Parameter values used in simulation for choosing
Set 1 | −0.0011 | 0.0002 | 0.4747 | 0.3776 | 16731.2 | 16509.2 | 0.0143 | −0.0881 |
Set 2 | 0.0004 | 0.0002 | 0.1338 | 0.0341 | 2906.1 | 2219.5 | 0.1563 | −0.1845 |
Set 3 | −0.0006 | 0.0002 | 0.3524 | 0.2166 | 2874.5 | 2589.2 | 0.0471 | −0.0982 |
Parameter values used in simulations: 3
Scenario 1 | 77.361 | 0.259 | 0.0077 | 0.0002 | 0.0003 | 0.0001 | 0.013 | 5.013 |
Scenario 2 | 1539.2 | 102.21 | 0.1529 | −0.0098 | 0.0003 | 0.0001 | 0.013 | 0.013 |
RMSE of estimators in Scenario 1: Initial values are
MME | 8.7150 | 0.5804 | 6.53 × 10^{−4} | 5.02 × 10^{−5} | |
Case 1 | MLE | 2.75 × 10^{−3} | 6.94 × 10^{−4} | ||
exp-MLE | 0.2990 | 0.2136 | 5.81 × 10^{−5} | 1.45 × 10^{−5} | |
8.6675 | 0.6431 | 4.72 × 10^{−4} | 3.79 × 10^{−5} | ||
Case 2 | MLE | 5.42 × 10^{−5} | 3.81 × 10^{−5} | ||
exp-MLE | 1.5984 | 0.0966 | 4.12 × 10^{−6} | 6.48 × 10^{−6} | |
1.6592 | 0.1482 | 4.46 × 10^{−5} | 7.63 × 10^{−6} |
RMSE = root mean square error; MME = method of moments estimator; MLE = maximum likelihood estimator.
Means and standard deviations of estimators in Scenario 1: initial values are
True value | 77.3610 | 0.2590 | 0.0077 | 0.0002 | |
MME | 86.0768 (17.9973) | −0.3211 (8.1433) | 0.0083 (0.0014) | 0.0003 (0.0007) | |
Case 1 | MLE | 18.1338 (7.0908) | −13.1002 (5.4771) | 0.0049 (0.0002) | 0.0009 (0.0004) |
exp-MLE | 77.6608 (14.6287) | 0.4729 (5.4163) | 0.0077 (0.0007) | 0.0002 (0.0004) | |
86.0293 (9.0611) | −0.3837 (5.4339) | 0.0082 (0.0005) | 0.0003 (0.0004) | ||
Case 2 | MLE | 79.2738 (10.7438) | −0.1905 (6.1651) | 0.0077 (0.0006) | 0.0003 (0.0005) |
exp-MLE | 78.9602 (10.6734) | 0.1626 (5.2191) | 0.0077 (0.0006) | 0.0002 (0.0003) | |
79.0210 (10.5841) | 0.1110 (5.1661) | 0.0077 (0.0006) | 0.0002 (0.0003) |
Standard deviations are in parentheses. MME = method of moments estimator; MLE = maximum likelihood estimator.
RMSE of estimators in Scenario 2: Initial values are
MME | 696.228 | 76.786 | 0.0843 | 0.0084 | |
76.186 | 41.656 | 0.0359 | 0.0003 | ||
Case 1 | MLE | 359389 | 280447.7 | 11.7650 | 14.7400 |
exp-MLE | 61.896 | 55.029 | 0.0065 | 0.0104 | |
1025.828 | 61.173 | 0.1144 | 0.0078 | ||
Case 2 | MLE | 2248.810 | 61.465 | 0.1934 | 0.0013 |
exp-MLE | 45.489 | 41.594 | 0.0331 | 0.0004 | |
94.484 | 48.328 | 0.0403 | 0.0003 |
RMSE = root mean square error; MME = method of moments estimator; MLE = maximum likelihood estimator.
Means and standard deviations of estimators in Scenario 2: Initial values are
True value | 1539.2 | 102.21 | 0.1529 | −0.0098 | |
MME | 843.008 (1231.938) | 25.428 (935.797) | 0.0685 (0.0627) | −0.0014 (0.0403) | |
1463.049 (777.206) | 143.871 (682.236) | 0.1169 (0.1577) | −0.0095 (0.0464) | ||
Case 1 | MLE | 360928.2 (2017268) | −280345 (1560723) | 11.9188 (67.3032) | 14.7307 (82.4483) |
exp-MLE | 1601.132 (743.039) | −2.814 (682) | 0.1594 (0.1037) | 0.0005 (0.0311) | |
513.408 (817.254) | 41.040 (728.617) | 0.0384 (0.0583) | −0.0020 (0.0474) | ||
Case 2 | MLE | 3788.046 (14199.790) | 163.680 (784.071) | 0.3464 (1.4012) | −0.0112 (0.0587) |
exp-MLE | 1493.746 (589.716) | 143.809 (11.660) | 0.1197 (0.0505) | −0.0094 (0.0265) | |
1444.751 (816.193) | 150.543 (530.975) | 0.1125 (0.0645) | −0.0095 (0.0460) |
Standard deviations are in parentheses. MME = method of moments estimator; MLE = maximum likelihood estimator.
Descriptive statistics for S&P500_I, S&P500_II, U.S. indemnity loss, and Danish fire loss
S&P500_I | 0.0009 | 0.00007 | 0.0345 | −0.0256 | −0.0831 |
S&P500_II | 0.0002 | 0.00004 | −0.4899 | 3.9521 | 10.6565 |
U.S. indemnity loss | 8.5219 | 2.0446 | −0.4920 | 1.0581 | 1.9637 |
Danish fire loss | 0.7869 | 0.5136 | 1.7610 | 4.1790 | −2.9701 |
S&P500_I | S&P500_II | U.S. indemnity loss | Danish fire loss | ||||||
---|---|---|---|---|---|---|---|---|---|
KS | AD | KS | AD | KS | AD | KS | AD | ||
Case 1 | MME | NA | NA | 0.2993 | 0.8476 | 0.4602 | 0.5398 | NA | NA |
0.9267 | 0.9756 | 0.2993 | 0.8476 | 0.4602 | 0.5398 | 0.0854 | 0.0592 | ||
MLE | 0.1362 | ||||||||
exp-MLE | 0.9321 | 0.9786 | 0.5584 | 0.9231 | 0.9145 | 0.8972 | 0.0649 | 0.0541 | |
0.7506 | 0.6519 | 0.5386 | 0.9025 | 0.3907 | 0.3508 | 0.0817 | 0.0502 |
KS = Kolmogorov Smirnov-test; AD = Anderson-Darling test; NIG = normal inverse Gaussian; MME = method of moments estimator; MLE = maximum likelihood estimator.
S&P500_I | S&P500_II | U.S. indemnity loss | Danish fire loss | ||||||
---|---|---|---|---|---|---|---|---|---|
KS | AD | KS | AD | KS | AD | KS | AD | ||
Case 2 | MME | NA | NA | 0.2993 | 0.8476 | 0.4602 | 0.5398 | NA | NA |
0.9267 | 0.9756 | 0.2993 | 0.8476 | 0.4602 | 0.5398 | 0.0854 | 0.0592 | ||
MLE | 0.9309 | 0.9761 | 0.3323 | 0.8477 | 0.9088 | 0.9550 | 0.1251 | 0.0818 | |
exp-MLE | 0.9366 | 0.9761 | 0.5631 | 0.9232 | 0.9143 | 0.9549 | 0.1887 | 0.1803 | |
0.9313 | 0.9761 | 0.5487 | 0.9231 | 0.9142 | 0.9551 | 0.1682 | 0.1187 |
KS = Kolmogorov Smirnov-test; AD = Anderson-Darling test; NIG = normal inverse Gaussian; MME = method of moments estimator; MLE = maximum likelihood estimator.