On the Stationarity of Futures Hedge Ratios

Stationarity of hedge ratios can be viewed as a first step for portfolio hedging since it represents that the sensitivity of spot and futures returns follow a process whose main characteristics do not depend on time. However, we provide evidence that the hedge ratios of the main European stock indices are better described as a combination of two different mean-reverting stationary processes, which depend on the state of the market. Also, when analysing the dynamics of hedge ratios at intraday level, results display a similar picture suggesting that intraday dynamics of the hedge between spot and futures are driven mainly by market participants with similar perspectives of the investment horizon.


Introduction
Stationarity of hedge ratios indicates a stable relationship between spot and futures prices. Since hedgers seek for reducing the risk of their investments, reliable dynamics of hedge ratios are expected. If not, futures markets may lose its usefulness to hedgers since the risk diversification can be hard to achieve. The property of stationarity motivates investors to use diversification strategies and can be utilised by policy makers to stabilize financial markets.
A hedge is a spread between a spot asset and a futures position that reduces risk 1 . Thus, the hedge ratio is defined as the number of futures contracts bought or sold divided by the number of spot contracts whose risk is being hedged. A considerable amount of research has focused on modelling the distribution of spot and futures prices and applies the results to estimate the optimal hedge ratio using various type of models (see Chen et al., 2003;Floros and Vougas, 2004;Salvador and Arago, 2014;Wang et al., 2014). Although most of the previous studies on optimal hedge ratios are successful in capturing the time-varying covariance-variances, almost all of them focus only on the estimate of the hedge ratios. The main purpose of this paper is to further examine and understand the stationarity of hedge ratios over time, as the literature provides limited information about it 2 .
Stationarity of hedge ratios indicates a stable relationship between spot and futures prices. Since hedgers seek for reducing the risk of their investments, reliable dynamics of hedge ratios are expected. If not, futures markets may lose its usefulness to hedgers since the risk diversification can be hard to achieve. The property of stationarity motivates investors to use diversification strategies and can be utilised by policy makers to stabilize financial markets.
The analysis of stationarity of hedge ratios can be viewed as a first step for portfolio hedging in any asset or market. Hedge ratios represent the sensitivity of spot prices to changes in futures prices, and measure how changes in the futures market affect spot markets. If this sensitivity is stationary, we are confident that the predictions we make lie into reasonable bounds. As a stationary process, the mean value of this sensitivity is constant and the variance is finite, so deviations of the predicted sensitivity to the observed sensitivity will be due to short-term corrections.
On the other hand, if the sensitivity of spot and futures returns is not stationary it will represent a burden to the successful implementation of a hedging strategy since accurate predictions of this sensitivity will be hard to achieve.
The potential deviations from stationarity for the sensitivity between spot and futures returns can be due to two main reasons. First, during crisis periods it is possible that the dynamics of the relationship between spot and futures markets follow different dynamics. This would lead to periods where the expected values for the sensitivities differ, casting doubts on the predictability of the hedge ratios. Second, the investment horizon considered might not capture completely the stationary characteristics of the relationship between spot and future returns. This can be the case when we look at this relationship at a very high frequency. If deviations from the expected hedge ratio take too long to revert (due to the presence of short-term traders), focusing the analysis at high frequency observations over short horizons may blur the stationary characteristics of this sensitivity between spot and futures markets.
Early studies, such as Ederington (1979) and Anderson and Danthine (1981), assume a constant optimal hedge ratio which can be obtained as the slope coefficient of an OLS regression. When the optimal hedge ratios depend on the conditional distributions of spot and futures price movements, then the hedge ratios vary over time as this distribution changes. Subsequent studies show the variability of hedge ratios over time, and support the hypothesis that optimal hedge ratios are time-varying and non-stationary (see Baillie and Myers, 1991). These studies report that hedge ratios contain a unit root and therefore behave much like a random walk. Grammatikos and Saunders (1983) were the first to examine the stability of hedge ratios. They concluded that hedge ratio stability (stationarity) in currencies could not be rejected. Furthermore, Malliaris and Urrutia (1991)  hedge ratio (hT) and the carry cost rate futures hedge ratio (hc) vary as predicted both within and across spot asset carry cost rate (c) regimes. They report that the BAM is not statistically significant different from low and high c periods.
The contribution of our article is to examine whether time-varying hedge ratios, calculated from a set of European stock indices (German DAX30, British FTSE100, French CAC40 and Spanish IBEX35), are stationary over time. The novelty of the paper lays on the analysis of the hedge ratio stationarity from a statedependent perspective. A hedger expects that her strategy allows her to reduce risk all the time, but especially during periods of financial distress. During periods of market instability, positions in the spot market are likely to lose value and hedgers rely on futures markets position to minimise losses. Thus, analysing hedge ratio stationarity in both states (low and high volatility) is a crucial topic which shows evidence on the usefulness of futures markets as hedging markets.
The studies we mentioned previously about HR stationarity do not make this distinction among states, and this can lead to misleading conclusions about the stationarity of hedge ratios series across volatility regimes. There are several papers from other areas of economics or financial economics that report mixed evidence on the stationarity hypothesis, depending on the volatility regime. See, for example, Holmes (2010) for the long-run purchasing power, Kanas and Genius (2005) for the US/UK real exchange rate, and Camacho (2011) for the US real GDP. Recently, Cotter and Salvador (2015) analyze the relationship between expected return and risk in the US market, and find that market volatility follows non-stationary dynamics during period of high instability. However, since the volatility process will eventually come back to the tranquil state, the whole process remains stationary.
Given the evidence reported in previous papers when analysing the stationarity properties of economic/financial time-series from a state-dependent perspective, we support that it is necessary to re-examine the stationarity properties of futures hedge ratios. The non-stationary volatility in spot markets during these high-volatility states may have a negative effect on the stationarity of the hedge ratios and on the hedging effectiveness of hedging ratios. Therefore, we test for state-dependent stationarity to verify the usefulness of futures markets as hedging markets in both states (especially during periods of financial instability).
The results in this paper provide empirical evidence that time-varying hedge ratios are stationary over time. Thus, we confirm the stable relationship between futures and spot returns across time. If we take a closer look at the evolution of spot and futures dynamics, and analyse stationarity across states, we find that hedge ratios are described better as a combination of two different mean-reverting stationary processes, which depend on the state of the market. Our state-dependent analysis confirms the existence of futures markets as hedging markets. In both states, the hedge ratios follow stable and predictable processes, which can be used to manage investors' risk. However, although correlations follow a stable stationary process in both states, during periods of financial turmoil the correlations between spot and futures are different from the ones during calm periods. This last result can provide an explanation to the controversy caused by the evidence of greater hedging effectiveness using static hedge ratios than using simple dynamics ones, and as why there have been several recent papers which both theoretically (Lien, 2012) and empirically (Alizadeh and Nomikos, 2008;Salvador and Arago, 2014) showed a greater effectiveness of regime-switching models. The intuition is that omitting the regime-switching specification leads to inefficient hedges compared not only to the ones considering state-dependence, but also to the static ones.
Finally, we also analyse the dynamics of optimal hedge ratios at intraday level (we extend the study by Lai and Sheu, 2010). Since executing an intraday hedging strategy would be very expensive, we focus on providing new insights about the dynamics of the spot and futures markets at ultra-high frequency.
Our results display a similar picture in the dynamics of spot-future returns at this intraday frequency. The evidence suggests that intraday movements in the hedge between spot asset and futures position are driven mainly by market participants, with similar perspectives of investment horizon. Even if there are different types of traders at this frequency, they do not have a significant impact on the hedge between spot and futures positions.
The rest of the paper is organised as follows. Section 2 provides a description of the database. Section 3 develops the model used to obtain the dynamic hedge ratios. Section 4 analyses time-series stationarity of estimated hedge ratios using both a standard perspective and the regime-switching framework. In Section 5, we take a closer look at stationarity properties of hedge ratios using intraday data and Section 6 concludes.

Data Description
The dataset contains daily data ( [Insert Table 1 about here] Table 1 presents the statistical properties of the price and returns series. The returns of spot and futures prices follow all stylized facts of financial time series such as leptokurtosis, volatility clustering, and leverage effects (see Bollerslev et al., 1994).
We estimate time-varying hedge ratios using GARCH models, which are very popular in the literature to capture the stylized facts of financial time series (see, for example, Degiannakis and Floros, 2010;Floros and Salvador, 2016). In the next section, we develop the empirical models to obtain dynamic hedge ratios and we describe their patterns.

Methodology
In more traditional hedge-ratio estimation methodology, the covariance matrix of spot and futures prices (and therefore the hedge ratio) is assumed constant through time. However, according to Lee (1999), given the time-varying nature of the covariance in financial markets, the OLS assumption is inappropriate when estimating optimal hedge ratios. There has been a large body of research that has applied the GARCH framework to infer time-varying hedge ratios (Cecchetti et al., 1988;Kroner and Sultan, 1993;Park and Switzer, 1995). In the GARCH model 5 , the conditional variance of a time-series depends on the squared residuals of the process (Bollerslev, 1986). It also captures the tendency for volatility clustering in financial data, and utilises the information in one market own history (univariate GARCH) or uses information from more than one markets history (multivariate GARCH). According to Conrad et al. (1991), multivariate GARCH models provide more precise estimates of the parameters because they utilise information in the entire variance-covariance matrix of the errors and allow the variance and covariance to depend on the information set in a vector of the ARMA manner (Engle and Kroner, 1995).Although GARCH models are useful for estimating time-varying optimal hedge ratios, a timevarying covariance matrix of spot and futures prices is not sufficient to establish that the optimal hedge ratio is time-varying 6 .
In this study we use a bivariate model with GARCH errors, the Diag-BEKK(p,q) model, to estimate the dynamic variance-covariance matrix of spot and futures log-returns. The Diag-BEKK(p,q) framework of log-spot (s) and log-futures where the dynamic hedge ratios are computed as the quotient between the conditional spot-futures covariance and the futures variance.
Recent studies on HR estimation include Lai and Lien (2017)

Empirical Results
The estimation of the model is conducted using conditional quasi maximum likelihood estimation 7 . Diagnostic tests and information criteria were employed to determine the lag orders, the validation of the assumptions concerning symmetry and diagonality. The results from the Diag-BEKK(1,1) model (Eq.1) are presented in Table 2. The coefficients are all statistically significant and imply volatility clustering.
Both spot and futures log-returns exhibit strong persistence in volatility, but it is the futures market that shows the strongest persistence.
[Insert Table 2 about here] [Insert Figure  1 about here] Figure 2 shows the plot of time-varying hedge ratios obtained from Eq.2. The DAX hedge ratios are quite volatile during the first part of the sample, but they seem to stabilise after 2005. Despite the evident peaks in volatilities in all countries, the hedge ratios follow a smooth pattern along the sample period where they seem to return always to a predetermined value. As from visual description of the hedge ratios, we cannot infer about their stationarity. Next section provides a formal study of the hedge ratio stationarity and the implications for optimal hedging.
[Insert Figure  2 about here]

Analysing the (Non) Stationarity of the Hedge Ratios
The purpose of the paper is to distinguish hedge ratios series that appear to have a unit root from those that appear to be stationary over time. For this purpose, we 7 The conditional log-likelihood function for a single observation can be written as , where  represents a vector of parameters and n is the sample size (for more details see Xekalaki and Degiannakis, 2010). employ two well-known unit root tests 8 , i.e. the Augmented Dickey Fuller (ADF) and Phillips and Perron (PP) tests, which test the hypotheses:

Ho: there is a unit root
Ha: there is stationarity

Unit Root Theory
The ADF test assumes that series t y follows an AR(p) process , and modifies the t-ratio of the a coefficient; hence, the serial correlation does not affect the asymptotic distribution of the test statistic 9 .

Regime-Switching ADF Test
Regime switching models are an important methodology to model nonlinear dynamics and widely applied to economic data including business cycles, bull and bear markets, interest rates and inflation. There are two common features of these models. First, past states can recur over time. Second, the number of states is finite and small (it is usually two and at most four). In contrast to the regime switching models, structural break models can capture dynamic instability by assuming an infinite or a much larger number of states at the cost of extra restrictions. If there is a change in the data dynamics, it will be captured by a new state. The restriction in these models is that the parameters in a new state are different from those in the previous ones. This condition is imposed for estimation tractability. However, it prevents the data divided by breakpoints from sharing the same model parameters, and could incur some loss in estimation precision.
Did hedge ratios have distinct dynamics or revert to a historical state with the same dynamics during the sample period analysed? Existing econometric models have difficulty answering such questions. In our paper we took the first assumption where past states can recur over time (in terms of bull and bear markets) instead of being considered new states. For more information about the regime switching models, see Samitas and Armenatzoglou (2014)   [Insert Table 3 about here]

Empirical Results
The implication of this result is that optimal hedges on stock indices tend to fluctuate around a mean-reverting value. This stable relationship, between correlations of spot and futures markets, can be exploited by hedgers to reduce risk of their investments. This result of stationarity in the hedge ratios can be viewed as good news, since it implies a reliable relationship between the spot and futures prices and a confirmation that futures markets are useful for hedgers.
Besides this first analysis, we also examine the stationarity of hedge ratios by looking at low and high volatile periods. The advantage of our approach is that we do not need to assume which periods correspond to low/high volatility states. The estimation procedure makes this classification (regime-switching methodology). reverting processes, one when the process is in low-volatility periods, and another one when the process is in high-volatility periods 11 . Within each state, hedge ratios tend to fluctuate around different state-dependant values, instead of just one common value independent of state. The good news for hedgers is that hedge ratios follow stationary processes in both states, which implies that they can use these markets to manage their risk at any time (even under the most needful times of market turmoil). 11 Note that state one is the low volatility state, while state two is the high volatility state. Figure 3 shows the probability of being in a state of low volatility and complements Figure 2 which shows, in shaded areas, the observations that correspond to high volatility periods when compared to the estimated hedge ratios.
[Insert Figure 3 about here] The hedge-ratios process changes continuously among regimes. Nevertheless, the hedge-ratios within each regime are stationary, and the dynamics of the correlation in the different regimes are not the same. Thus, if we are interested in shorter horizons hedges, not considering different states can be a cause of a worse hedging performance.
These differences among regimes can be observed more clearly in Figure 4.
For all markets there are obvious differences in the distributions of the hedge ratios during LV states and HV states. During HV states the optimal HRs are consistently higher than during LV states. Based on the way the HRs are computed, this suggests that the covariance between spots and futures markets is higher during these periods of uncertainty.
[Insert Figure 4 about here] This result can provide an explanation to very recent evidence, which shows, both theoretically and empirically, that hedge ratios obtained from regime switching models outperform the rest of strategies (both static and dynamic). Lien (2012) characterizes conditions under which the regime-switching hedge strategy performs better than the OLS hedge strategy and where the GARCH effects prevail. These conditions would allow the RS-GARCH hedge strategy to dominate both OLS and GARCH hedge strategies.
Recently, Alizadeh and Nomikos (2008), for commodities, and Salvador and Arago (2014), for stock indices, report a greater performance of (multiple-) regimeswitching strategies than those obtained through single-regime models. Our results about this state-dependent stationarity of hedge ratios support this previous evidence.
When analysing the performance of hedging strategies, we usually look at shorter horizons and we tend to follow the false dynamics. Neglecting the switching of HRs' regimes causes a worse hedging effectiveness. Given that both states are stationary, optimal hedging can be exploited in any volatility regime. However, we need to identify the proper dynamics for each one of the regimes.
[Insert Table 5 about here] In in the low volatility state is negative and significant providing evidence of stationarity of hedge ratios during this low volatility state. However, if we look at high volatility states it seems that the process followed by optimal hedge ratios is non-stationary.
This result highlights the importance of modelling the trend of the time series properly. Similar results apply when using standard unit-root tests. Wrong trend specification leads into wrong/incorrect conclusions about the stationarity of hedge ratios. We recommend the use of a state dependent drift when testing the stationarity of HRs.

Hedge Ratio Stationarity for Intraday Data
Dynamic hedging is usually expensive to implement since it involves transaction costs any time the hedged portfolio is re-balanced. Therefore, hedging is more rational at low frequencies. However, if investors conduct the hedging, the hedge dynamics will not differ across different sampling frequencies.
On the other hand, if both investors and short-term traders conduct the hedging (i.e. swap trading between futures and spot for speculation), then the hedge dynamics will differ across different sampling frequencies. In this section, we try to unmask this hypothesis by looking at the stationarity patterns of intraday hedge ratios.
The dataset is comprised by hourly observations of the DAX index and its corresponding future contract from 3 rd of January, 2000 to 30 th of December, 2010 (25138 observations) 12 . As in the previous datasets, we first compute the dynamic hedge ratios based on eqs.1 and 2. A plot of the estimated intraday hedge ratios is displayed in Figure 4. The hedge ratios seem to follow a smooth pattern although it is not possible to draw any conclusion about its stationarity from this figure. Therefore, we run the corresponding stationarity tests to provide new insights.
[Insert Figure  4 about here] Table 6 displays the unit-root tests when we consider the regime-switching approach and distinguish between high and low volatility regimes. The empirical results show that we do reject the unit root for both regimes, when a switching intercept is employed.
[Insert Table 6 about here] If we had found evidence in favour of unit root presence, then we should have obtained a more complex picture for the distributions of spot and futures returns at this intraday frequency. That would have implied that, looking at longer horizons, the spot-future correlations would have seemed to follow a stationary process, although when looking at intra-day horizons, the dynamics of the spreads between these two markets would have followed unpredictable dynamics. Nevertheless, there is no such discrepancy in our findings. There is no evidence that the dynamics of hedge ratios vary across different sampling frequencies. There is no evidence that the agents driving the spread of these markets at intraday level are mainly short-term traders.
Even if market participants have different perspectives of their investment horizon, this is not evident in regime-switching unit root testing. Our results suggest evidence of investors prevailing at both daily and intra-day frequencies. The impact of the short-term traders do not affect significantly the dynamics of the hedge between spot and futures at the intra-day frequency.
It is also unclear how transaction costs affect rebalancing the optimal hedge position, although it may discourage speculators in general. On the other hand, professional speculators may employ day trading or speculation in securities. In line with Tse and Williams (2013) we do support that spot-futures markets need to be fully examined using high frequency intraday data.

Conclusion
Static and dynamic models of various forms have been employed in the literature to calculate hedge ratios. However, there is to date no definite conclusion concerning stationarity of the dynamic hedge ratios. We focus on the characteristics of optimal hedge ratios for the DAX30 (Germany), FTSE100 (UK), CAC40 (France), and IBEX35 (Spain) indices over the period 2000-2013. We estimate dynamic hedge ratios by a bivariate diagonal multivariate GARCH-type model and we examine stationarity of hedge ratios by employing standard econometric methods of unit root tests and a new state-dependent approach following the RS-ADF test.
We find that dynamic hedge ratios are stationary over time when the entire sample is considered. This result implies a stable relationship in spot-futures correlations that can be used by hedgers to reduce the risk in their investments.
However, when we consider shorter horizons and distinguish between volatility states (i.e. high and low volatile periods), we show that the dynamic hedge ratios follow different stationary processes during periods of calm and periods of financial turmoil.
These results support evidence in previous studies that report a greater hedging performance of dynamic strategies using regime-switching models.
The different processes followed by the hedge ratios for volatile periods are associated with changes in the variances and the covariance between spot and future returns. This has important implications for hedgers. First, financial analysts and hedgers must determine the effect of this unexpected change in the risk on their position. Second, they should determine the factors causing this shifted stationarity.
The good news for investors is that futures markets can be seen as a hedging market at any time or state of the market (even for the most necessary periods of market turmoil).
The results for the dynamic hedge ratios at intraday level are also in line with the high frequency results. There is no clear evidence that the spreads are distorted by short-term market participants. Our conclusion is that the role of speculators in the determination of intra-day spot-futures stock dynamics is not as relevant as the one taken by hedgers. Further research should consider structural breaks tests in both spot and futures returns, and examine if hedging effectiveness change when HRs are stationary or not.

Tables
The Table shows summary statistics and stationarity tests for prices ( ) t t f s , and of the 4 European stock indices (German DAX30, the British FTSE100, the French CAC40 and the Spanish IBEX35) in the spot and futures markets. Panel A shows the descriptive statistics and the Jarque-Bera normality test for the log returns of spot and futures markets and Panel B shows the same information for the prices in both markets. *** , ** and * represents rejection of the null hypothesis at 1%, 5% and 10% levels of significance, respectively.   The Table shows the estimated parameters for the model in eq.1 for the logreturns on the spot and futures markets for the DAX30, FTSE100, CAC40 and IBEX35 indices. Standard errors are computed using Bollerslev-Wooldridge (1992) specification correcting for heteroskedasticity ( *** , ** and * represents statistical significance at 1%, 5% and 10% levels of significance, respectively).  The Table shows the estimated parameters for the RS-ADF test presented in eq.4. Dependent variables in each column represent the estimated HRs using eq.2 for the spots and futures returns on the DAX30, FTSE100, CAC40 and IBEX35 indices (sample period May 2000-November 2013). Standard errors are computed using Bollerslev-Wooldridge (1992) specification correcting for heteroskedasticity ( *** , ** and * represents statistical significance at 1%, 5% and 10% levels of significance, respectively). 1.16e-05 *** (9.3970)

Panel A.-Summary statistics for log-returns
The Table shows the estimated parameters for the RS-ADF test presented in eq.4 but omitting the drift component. Dependent variables in each column represent the estimated HRs using eq.2 for the spots and futures returns on the DAX30, FTSE100, CAC40 and IBEX35 indices (sample period May 2000-November 2013). Standard errors have been corrected for heteroskedasticity ( *** , ** and * represents statistical significance at 1%, 5% and 10% levels of significance, respectively).

Hedge ratios Intraday Germany
This figure plots the estimated HRs according to eq.2 using the intraday (hourly) returns on the spot and futures stock indices in Germany.