2.2 Kernel Density Estimates for Heavy-tailed Data
2.2.2 Experiments with Financial Data
The aim of this chapter is to construct models for heavy-tailed financial time series in which the density of the innovations is described in a non-parametric manner. Because the proposed approach is based on performing the estimation of the density in a transformed space, it is important to determine the impact of selecting a particular transformation function in the quality of the final estimator. Therefore, we investigate the performance of back-transformed kernel methods based on NIG, GHYP and stable transformations using actual financial data. The data used in these experiments consist of time series of 4000 consecutive daily returns from 59 assets included in the Dow Jones Composite Index1. The returns are computed using the daily closing
prices adjusted for dividends and splits, as published in http://www.finance.yahoo.com. The time period considered is from June 3, 1992 to March 31, 2008. Each time series of returns {Yt}4000t=1 is assumed to be generated by a stationary lag-one autoregressive process, in which the
volatility is assumed to follow an asymmetric GARCH process (Ding et al.,1993) Yt = φ0+ φ1Yt−1+ σtet
σt = κ + α(|σt−1et−1| − γσt−1et−1) + βσt−1, (2.6) 1AA, AEP, AES, AIG, ALEX, AMR, AXP, BA, BNI, C, CAT, CNP, CNW, CSX, D, DD, DIS, DUK, ED, EIX,
EXC, EXPD, FDX, FPL, GE, GM, GMT, HD, HON, HPQ, IBM, INTC, JBHT, JNJ, JPM, KO, LUV, MCD, MMM, MO, MRK, MSFT, NI, NSC, OSG, PCG, PEG, PFE, PG, R, SO, T, UNP, UTX, VZ, WMB, WMT, XOM and YRCW.
where κ > 0, α ≥ 0, β ≥ 0, −1 < γ < 1, −1 < φ1< 1. The innovations {et} are iidrv’s sampled
form a density f which has zero-mean and unit standard deviation. The dependence of σt on
the absolute value of the lagged innovation reflects the fact that, in empirical financial data, the absolute value of the returns frequently exhibit higher autocorrelations than higher powers of the returns (Ding et al., 1993;Taylor,1986). The dependence of σt on the lagged innovation
reflects the empirical observation that volatility has an asymmetric response to past positive and negative shocks (Cont,2001).
The parameters of the model are selected by maximizing the logarithm of the conditional likelihood (Brockwell and Davis,1996), with the assumtion that f is standard Gaussian, that is, et ∼
N
(0, 1). The assumption of Gaussian innovations is violated in practice. However,this estimation method (often referred to as quasi-maximum likelihood estimation) is generally consistent (Bollerslev and Wooldbridge,1992). In the estimation process, σ0is assumed to be
equal to the sample standard deviation of the series (denoted by ˆσ), e0is assumed to be 0 and
finally, φ0+ φ1Y0is taken to be equal to the sample mean of the series of returns (denoted by ˆµ).
Let ˆθ = ( ˆφ0, ˆφ1, ˆκ, ˆα, ˆγ, ˆβ) be the estimate of the model parameters obtained after calibration of (2.6) on the series {Yt}t4000=1. Then, the series of residuals of the process {rt( ˆθ)}4000t=1 is given by
rt( ˆθ) =
(
Yt− ˆφ1Yt−1− ˆφ0 t≥ 2
Yt− ˆµ t= 1
. (2.7)
These residuals should not present significant autocorrelations at lag 1. However, they may still be heteroskedastic. To eliminate the heteroskedasticity in the series of residuals, each rt( ˆθ) is
scaled by the corresponding estimate of the volatility ˆσt( ˆθ), where
ˆ σt( ˆθ) =
(
ˆκ + ˆα(|rt−1( ˆθ)| − ˆγrt−1( ˆθ)) + ˆβ ˆσt−1( ˆθ) t≥ 2
ˆκ + ˆβ ˆσ t= 1 . (2.8)
In this manner, we obtain the series of scaled residuals {ut( ˆθ)}t4000=1 where ut( ˆθ) = rt( ˆθ)/ ˆσt( ˆθ) .
These scaled residuals should be an accurate approximation of the actual innovations in the series of returns.
The plot on the top left of Figure2.2displays the series of 4000 returns for the stock AES. The middle-left plot in this figure indicates that this series presents a very small, but significant, autocorrelation at lag 1. The autocorrelations for the absolute values of the returns are larger and remain positive for longer times, as displayed in the bottom-left plot of Figure2.2. These strong autocorrelations are originated by the time-dependent structure of the volatility in the series of returns. The plot on the top right of Figure2.2displays the series of 4000 scaled residuals obtained after calibrating the time-series model (2.6) on the returns of AES. This series of scaled residuals exhibits no significant autocorrelations, as displayed by the middle-right plot in this figure. Finally, the autocorrelations for the absolute values of the series of scaled residuals are also very small (see the bottom-right plot in Figure 2.2). These properties confirm that the heteroskedasticity of the original series has been successfully eliminated and that the resulting scaled residuals are approximately independent.
Once the model described above has been calibrated for a particular asset, the corresponding 4000 scaled residuals are obtained and split on two consecutive blocks of 2000 elements. Then,
Chapter2. Semi-parametric Models for Financial Time-series 19 0 1000 2000 3000 4000 −60 −40 −20 0 20 AES Returns Day Retur n 0 1000 2000 3000 4000 −10 −5 0 5
AES Scaled Residuals
Day Retur n 0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 Lag A CF
ACF for AES Returns
0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 Lag A CF
ACF for AES Scaled Residuals
0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 Lag A CF
ACF for AES Absolute Returns
0 5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 Lag A CF
ACF for AES Abs. Scaled Residuals
Figure 2.2: Top left, middle left and bottom left: AES returns, empirical autocorrelations for AES returns and empirical autocorrelations for the absolute value of AES returns, respectively. Top right, middle right and bottom right: corresponding plots for the scaled residuals obtained after filtering the returns of AES stocks with the time-series model given by (2.6).
Table 2.2: Average log-likelihood obtained by each back-transformed method on the 59 assets.
NIG GHYP Stable -2777.543 -2776.339 -2762.829
1 2 3
NIG StableGH
CD
Figure 2.3: All to all comparison of the different back-transformed kernel density estimators by the Nemenyi test. The horizontal axis corresponds the average rank of each method on the 59 problems. Methods whose average ranks are not significantly different at the level α = 0.05 appear connected in the figure.
the back-transformed kernel density estimate is constructed using the scaled residuals in the first block. The log-likelihood of this estimate is evaluated in the second block of data. This out- of-sample measure of performance should be unbiased. Table2.2displays the average value of the log-likelihood obtained by each estimation technique on the 59 financial assets. According to this performance measure, the best method is the kernel estimator in which the data are transformed using the stable distribution.
To determine whether the differences in performance are statistically significant, we follow the methodology proposed by Demˇsar (2006). This framework is designed to compare the predictive performance of different methods in a collection of problems: The different methods are ranked according to their performance in the collection of problems considered. Statistical tests are then used to determine whether the differences in average ranks are significant. In our case, a Friedman rank sum test rejects the hypothesis that all the methods have an equivalent performance in the 59 problems that have been analyzed (p-value = 5.2 · 10−7). The average ranks of the different estimators with a Nemenyi test at a 95% confidence level are shown in Figure2.3. The differences in performance between methods whose average ranks differ less than a critical distance (CD) are not significant at this confidence level. The methods whose average ranks are not significantly different appear connected in the figure. These results confirm that the stable transformation is significantly better than the NIG or the GHYP transformations for approximating the conditional density of financial returns.
The only member of the stable family with finite variance is the Gaussian distribution. Other types of stable distributions have infinite variance. However, there is empirical evidence that the actual distributions of financial returns have finite second moments (Cont,2001). This has often been used to discard the stable family as realistic model for the unconditional distribution of financial returns. Nevertheless, even though the stable density used in the transformation may have infinite variance, the resulting back-transformed kernel approximation has in most cases finite second moment. This result is proved in AppendixA.2.
Chapter2. Semi-parametric Models for Financial Time-series 21