A large number of prior studies have developed models for futures hedging. Constructed from high-frequency data and realized beta framework of Andersen et al. (2005, 2006), this chapter has analyzed the dynamics in the realized daily hedge ratio, which serves as an accurate esti-mate for the integrated daily hedge ratio. Moreover, a two-regime threshold autoregressive model is applied to detect the regime-switching feather of the ratio. Empirical studies on two equity index futures show that the hedge ratio behaves in a regime-dependent dynamics and tends to be more volatile in the low regime than in the high regime. The result then supports the argument of time-varying hedge ratio.
4.1. Research Problem and Objective
A large number of studies have documented that the conventional regression-based static ap-proach is inappropriate for futures hedge because the joint distribution of spot and futures prices is not constant through time.30 Henceforth, models for time-varying (or conditional) minimum-variance hedge ratios have emerged and been discussed. For example, the bivariate GARCH model of Baillie and Myers (1991) or the random coefficient autoregressive model of Bera et al. (1997) both lead estimates of dynamic hedge ratios. The latter directly estimates time-varying coefficients instead of conditional second moments so that different behavior may be appeared in the estimated hedge ratio. The insight into the hedge ratio behavior may further the development of dynamic hedge ratio models. Set against this background, we analysis the dynamic property in integrated daily hedge ratio via the realized beta framework of Andersen et al. (2005, 2006), which allows explicitly for approximating the integrated
30 See, for example, Baillie and Myers (1991), Kroner and Sultan (1993), and among many others for evidences of this.
hedge ratio from underlying covariance and variance components.
Empirical studies are conducted on the S&P 500 and the NASDAQ 100 futures contracts.
Using the intraday data for the spot and the futures over a six-year period from January 1, 1999 to December 31, 2004, their realized daily hedge ratios are firstly constructed. To ex-plore the possibility of nonlinear dynamics in the realized hedge ratio series, a two-regime Self-Exciting Threshold Autoregressive (SETAR) model is then considered in this paper. The two-regime SETAR is tackled with a linear Autoregressive (AR) model if the threshold effect is not significant. In testing the linear AR against the nonlinear SETAR models, a heteroske-dasticity-consistent LM-based statistic of Hansen (1996) is applied via a bootstrap procedure.
Empirical results conclude the realized daily hedge ratio is characterized as regime-dependent dynamics and is likely to be positively autocorrelated so that the usual assumption of constant hedge ratio seems inappropriate. In addition, the main effect of the regime demonstrates dif-ferent variation in the daily-realized hedge ratio. That’s, the hedge ratio tends to be more vola-tile in the low regime than in the high regime. The rest of the article is organized as follows.
Next, we demonstrate the SETAR model and then elaborate on the realized daily hedge ratio.
This is followed by providing the empirical evidence, and finally, the conclusion is presented in the last section.
4.2. A Two-Regime SETAR Model
To explore the dynamics in the integrated daily hedge ratio, consider the two-regime SETAR model with the form:
pa-rameter. The autoregressive order is m ≥1, and the parameters αj and βj are autoregres-sive slopes. The SETAR is composed of two regime-dependent piecewise linear models for which the regime-switching dynamics is controlled by a lagged dependent variable. Hansen (1996) provided a heteroskedasticity-consistent LM-based statistic for testing the linearity against the nonlinear SETAR. As the threshold parameter is not identified under the null, the asymptotic distribution of the statistic is not standard and may be approximated by using a bootstrap procedure. In estimating the SETAR model, the sequential conditional least squares or the sequential conditional quasi-maximum likelihood method may be applied. The thresh-old variable of Equation (4.1) is set as yt d− ≡yt−1 because it should provide relevant infor-mation for hedgers when making hedging decisions.
4.3. Realized Daily Hedge Ratios
We investigate the dynamics of the integrated daily hedge ratio in the context of hedging with two stock index futures, namely the S&P 500 and the NASDAQ 100 (traded in CME), over a six-year period from January 1, 1999 to December 31, 2004. Applying the realized beta framework of Andersen et al. (2005, 2006) to this one-period hedging problem, the estimates of the integrated daily hedge ratios for each of the two index pairs are given by
, ,
ˆt RCovsf t RVf t
y = →yt (4.2)
almost surely for all t as the time between sampling observations →0. In this chapter, both the realized daily hedge ratio series are constructed from the transactions prices based on the previous tick method.31 The price data sets from the Tick Data Inc. are equidistant in time, and 5-minute sampling is often used in practice, see, Andersen et al. (2001a) for discussion.
31 Several sampling methods have been proposed for constructing the intraday returns, such as the linear inter-polation method of Andersen and Bollerslev (1997). One of the discussions of these methods can be referred to Hansen and Lunde (2006).
Specifically, for the futures contracts, the nearest month contract is rolled to the next month when the daily volume of the current contract is exceeded. Moreover, to construct the realized covariance, the futures returns after 15:00 for each day are dropped because the futures mar-kets close fifteen minutes later than the spot marmar-kets. The 5-minute returns of the futures and the spot are used to calculate the realized daily hedge ratio for each product. Table 4.1 reports their descriptive statistics and dynamic dependence. The Phillips-Perron unit root tests con-clude that the two realized daily hedge ratios are stationary, and the Ljung-Box Q statistics conclude that they are strongly autocorrelated. The preliminary analysis suggests that the re-alized daily hedge ratio may be modeled as stationary I(0) processes although the realized variance and covariance may be well approximated by a nonlinear fractionally cointegration (Andersen et al., 2006).
Table 4.1 Statistics and Dynamic Dependences of Realized Daily Hedge Ratios
Index S&P 500 NASDAQ 100
Mean 0.801 0.804
Std. Dev. 0.094 0.105
Skewness -1.045 -1.268
Kurtosis 4.878 5.367
Q(5) 978.094 1525.645
PP-AR test -2.466 -2.564
PP-ARD test -25.470 -22.845
PP-TS test -32.298 -26.208
Observations 1508 1508
Notes: The Q(k) statistic (Ljung and Box, 1978) tests the null hypothesis of no autocorrelation up to order k. PP test is a nonparametric unit root test proposed by Phillips and Perron (1988); PP-AR test is based on zero drift AR(1) process; PP-ARD test is based on AR(1) model with drift; and PP-TS test is based on trend stationary AR(1) model. Statistics in bold indicate significance at the 5% level.
4.4. Empirical Results
The bootstrap-calculated asymptotic p-value for the S&P 500 (p =0.05) and the NASDAQ 100 (p =0.02) concludes that the null hypothesis of a single regime (no threshold effect) is
rejected at the 5% significance level.32 The least squares estimates of the threshold are ˆ 0.82
γ = and γ =ˆ 0.79 for the S&P 500 and the NASDAQ 100, respectively, where the optimal autoregressive orders for them is set by the Bayesian information criteria. The opti-mal order for our data is 4 and 8 for the S&P 500 and the NASDAQ 100, respectively. The threshold principle divides the linear regression into two regimes by piecewise linear AR function depending on whether the previous realized daily hedge ratio has been exceeding the threshold estimate. Table 4.2 reports the parameter estimates for the two-regime SETAR mod-els with heteroskedasticity consistent standard errors. Although the realized daily hedge ratio is likely to be positively autocorrelated, its variation in the two regimes behaves differently. It is observed that the realized hedge ratio tends to be more volatile in the low regime than in the high regime. As a result, the empirical results show the null of time-invariant hedge ratio hy-pothesis seems inappropriate and thus support the argument of time-varying hedge ratios.
Table 4.2 Two-regime SETAR estimates Notes: Standard errors are in brackets. Statistics in bold indicate significance at the 5% level.
32 The bootstrap replications 1,000 and the trimming percentage 15% are used, see Hansen (1996) for details.