Since stock index futures were introduced to the market in the late 1970s, they have become popular hedging instruments. Due to growing fluctuations in stock markets, it is increasingly important to choose the optimal hedging strategy. Early studies used traditional ordinary least squares (OLS) regressions to estimate optimal hedge ratios (HRs) and hedging effectiveness (HE) (Johnson 1960; Ederington 1979). Although the OLS method identifies the relationship between a stock index and its futures, it has two major drawbacks. First, the stock index and its futures series exhibit nonstationary and conditional volatilities that are often neglected by the OLS model. When the nonstationary, conditional volatilities of a stock index portfolio and its futures are not taken into consideration, the HR estimated by the OLS method can lead to a suboptimal hedging decision. Second, the OLS approach involves a one-time hedging decision and does not allow for any adjustment in the HR once this decision has been made. When new information becomes available, investors using the OLS hedging model cannot immediately adjust their hedging decisions.

To represent the dynamic nature of HRs, most recent studies have employed autoregressive conditional heteroscedasticity (ARCH), generalized autoregressive conditional heteroscedasticity (GARCH), cointegration or state-space models (SSMs).¹¹ 1 Chen et al (2003) and Lien and Tse (2002) provide excellent reviews of HR research. ARCH-based models consider time-varying volatility and are widely used in estimating optimal HRs.²² 2 Holmes (1996), Baillie and Myers (1991), Kroner and Sulten (1993) and Hsu et al (2008) use different ARCH and GARCH models to estimate time-varying HRs under conditional heteroscedasticity. Although the conditional heteroscedasticity hedging estimation method allows the HR to change over time, Miffre (2004) found that ARCH and GARCH models provide limited improvement to HE. In contrast, Ghosh (1993a,b) proposed an error-correction hedging model that considers the nonstationary nature of financial time series. Ghosh (1993a,b) uses the Engle–Granger two-step method to estimate the cointegrating relationship between stock portfolios and futures. When a cointegrating relationship exists, the optimal HR can be estimated through an error-correction regression. The advantage of the cointegration method is that, when the stock price index and futures possess unit roots, it can avoid spurious regression (Ferson et al 2003). However, even if the HRs in the error-correction regression are easy to compute, this model cannot outperform other hedging models. The disadvantage of the error-correction hedging model is that information cannot be updated during the estimation process. This leads to an inaccurate calculation of the long-term relationship between a stock index and its futures: a serious drawback considering that accurate identification of the long-run relationship is critical to HE.

The third approach to estimating the HR is an SSM. Hatemi-J and Roca (2006) introduced the Kalman filter (KF) model for estimating a time-varying HR. Lee et al (2006) estimated HRs with a random-coefficient autoregressive regime-switching model. Although SSMs demonstrate better efficiency, the drawbacks of traditional state-space hedging models are twofold. First, traditional time-varying SSMs (see, for example, Hatemi-J and Roca 2006; Lee et al 2006) use the stock index as the regressand and its futures as the regressor; thus, they extract the trend between the stock index and its futures. When the stock index and futures follow different trends, traditional SSMs cannot obtain the single best common trend. Second, although SSMs exhibit better efficiency in the autoregressive moving average (ARMA) process, the assumption of stationarity of the regressand and regressors is crucial to the estimation of HRs (Harvey 1990; Kim and Nelson 1999). The violation of this stationarity by the regressand and regressors can cause inconsistency in the estimation of the HR, resulting in the utilization of a suboptimal HR.³³ 3 Please refer to Chang et al (2009) for further discussion of the KF.

Many empirical studies have compared methods of estimating HRs. Benet (1992) used the OLS method to estimate the HR for the exchange rate futures of eight countries and observed a significant decrease in hedging performance from in-sample to out-of-sample estimation. Park and Switzer (1995) compared the HE of a conventional OLS model, an OLS model with cointegration and a bivariate GARCH model. They did so by examining Standard & Poor’s 500 (S&P 500) and Toronto 35 index futures. Of the three models, the bivariate GARCH model had the best HE. Holmes (1996) assessed estimation methods for the optimal HR of the Financial Times Stock Exchange 100 (FTSE 100) stock futures using OLS, cointegration and GARCH models. The optimal HR estimated by OLS outperformed those estimated via the more advanced econometric techniques of cointegration and GARCH. Ghosh (1993b) studied S&P 500 futures with the cointegration model and concluded that the traditional OLS approach could not be applied to the stock index and futures data because they could be nonstationary. When unit roots exist in the stock index and its futures, an OLS method may underestimate the HR. Hatemi-J and Roca (2006) examined the HE of Australian equity and futures using the time-varying parameter Kalman filter (TVPKF). The TVPKF outperformed the conventional OLS model. According to these empirical results, a single dominant model does not exist for optimizing dynamic hedging strategies.

This paper proposes a new method of determining the optimal HR: the Kalman filter error-correction model (KF–ECM). When we construct a hedging model involving a stock index and its futures, the Kalman filter state-space model (KF–SSM) (that is, the new econometric methodology proposed by Chang et al (2009)) is used to extract the single best latent common trend between the stock index and its futures. The difference between the long-run relationships estimated by the traditional maximum likelihood error-correction method (ML–ECM) and the KF–SSM is that the KF–SSM procedure allows for information on the stock index and its futures to be updated. After the best common trend has been obtained, we substitute it into an ECM to estimate the HR. Since the KF–ECM combines the merits of error-correction and state-space hedge models, we expect it to exhibit better HE than other, more traditional methods (see, for example, Baillie and Myers 1991; Ederington 1979; Ghosh 1993a,b; Holmes 1996).

In practice, the HR depends on the conditional information about the spot returns and their futures returns; therefore, when the conditional information changes over time, the HR also changes. Hedging strategies that employ an invariant HR are inadequate for multiperiod hedging (Benet 1992; Lien and Luo 1993). To compare the HE under time-varying circumstances, this study employs a rolling regression (moving window) to calculate the HE index (HEI), as proposed by Park and Switzer (1995). Rolling regression is a common forecasting method in time series analysis. Each time we perform a rolling regression, we add a period of data from the prediction sample. After the estimation is complete, we compare the predicted value with the actual value. We repeat this step until all data from the prediction samples has been added to the estimate. We then use the rolling regression to compare the out-of-sample performance of all examined hedging models.⁴⁴ 4 The hedging models include the OLS, GARCH, traditional TVPKF and ECMs.

This study contributes to the literature in two ways. First, we provide evidence that in high-dimensional applications, where some series are more important than others in driving the trend, the KF–ECM method provides a superior way of extracting common trends. When we use the KF–ECM to extract latent common trends, the KF method can set different weights according to the importance of the latent trends and extract the single best trend. By contrast, the ML–ECM assigns the same weight to all possible latent common trends and extracts the best trends; no restriction is placed on the number of trends. In high-dimensional financial applications and when a single best trend is required, the KF–ECM may be a more effective approach than the ECM. Second, the KF–ECM uses maximum likelihood estimation (MLE) to obtain the parameters. Even if the distributions of the data have thick tails, reasonable parameter estimates can be obtained. By contrast, ARCH and GARCH hedge models may be challenged by order selection with conditional correlations. HRs can be biased when the ARCH and GARCH models are incorrectly specified. For these reasons, investors may adopt the KF–ECM as a more accurate method for the estimation of the optimal HR.

The remainder of this paper is organized as follows. Section 2 describes the characteristics and estimation methods of the KF–ECM hedging model. Section 3 presents the empirical results for the KF–ECM, compares the common trends extracted using MLE and KF–SSM, and examines the effectiveness of the different hedging models. Finally, Section 4 presents the conclusions of this study.

Kalman (1960) proposed an SSM known as the KF for computing the optimal parameters of an unobserved trend at time $t$ based on the available information at $t$. The KF has been used to analyze dynamic time series in such fields as aeronautical and electrical engineering.⁵⁵ 5 Refer to Harvey (1990), Hamilton (1994) and Kim and Nelson (1999) for their excellent surveys. Although the KF allows the state variable estimate to be continually updated as new observations become available, the latent trend must be a stationary series. To mitigate this drawback, Chang et al (2009) considered an SSM with integrated time series,

where $x_t$ is a scalar latent variable with fixed initial value $x_0$; $y_t$ is an $m\times 1$ observable time series; and $u_t$ and ${\upsilon }_t$ are sequences of independent and identically distributed (iid) errors with means of zero and variances of $\mathrm{\Lambda }$ and $1$, respectively.⁶⁶ 6 The variance of ${\upsilon }_t$ is set to be unity for the identification of parameter $\beta$. The other conditions imposed in the KF model are standard. For convenience, we may assume that $y_t=[\begin{array}{c}\hfill S_t\hfill \\ \hfill F_t\hfill \end{array}]$, where $S_t$ is the stock index and $F_t$ represents the stock index futures. $\beta =[\begin{array}{c}\hfill {\beta }_s\hfill \\ \hfill {\beta }_f\hfill \end{array}]$ and $u_t=[\begin{array}{c}\hfill u_{st}\hfill \\ \hfill u_{ft}\hfill \end{array}]$ are coefficients and errors, respectively. Equations (2.1) and (2.2) are referred to as the measurement and the state-space equations, respectively. Let $y_t$ be the observed variable vector, which can be represented by an unobserved variable $x_t$. The KF–SSM proposed by Chang et al (2009) was used to extract the best common stochastic trends between $y_t$ and $x_t$.

The following two stages are proposed to estimate the Kalman state space with the KF–ECM.

1. (1)

   Extract the common trend using the KF–SSM.

   Unlike traditional cointegration estimation procedures, the KF requires three steps: prediction, estimation and updating.⁷⁷ 7 Estimation using the Engle–Granger or Johansen procedures does not involve updating steps. In the prediction step, we use the linear conditional expectations of the mean and variance,

   	$x_{t|t-1}$	$=E(x_t\mid {\mathrm{\Theta }}_{t-1})=x_{t-1|t-1},$		(2.3)
   	$y_{t|t-1}$	$=E(y_t\mid {\mathrm{\Theta }}_{t-1})=\beta x_{t|t-1},$		(2.4)

   where ${\mathrm{\Theta }}_t$ is a sigma field generated by $y_1,\mathrm{\dots },y_t$. The mean square error (MSE) of prediction using (2.3) and (2.4) is defined below:

   	${\omega }_{t|t-1}$	$=E[(x_t-x_{t|t-1}){(x_t-x_{t|t-1})}^{\prime }]$
   		$=E[(x_{t-1}-x_{t-1|t-1}){(x_{t-1}-x_{t-1|t-1})}^{\prime }]+E[{\upsilon }_t{\upsilon }_t^{\prime }]$
   		$={\omega }_{t-1|t-1}+1,$		(2.5)
   	${\mathrm{\Sigma }}_{t|t-1}$	$=E[(y_t-y_{t|t-1}){(y_t-y_{t|t-1})}^{\prime }]$
   		$=E[(x_t-x_{t|t-1}){(x_t-x_{t|t-1})}^{\prime }\beta {\beta }^{\prime }]+E[u_t{u}_t^{\prime }]$
   		$={\omega }_{t|t-1}\beta {\beta }^{\prime }+\mathrm{\Lambda }.$		(2.6)

   When the prediction value $\beta x_{t|t-1}$ is obtained, the cointegrated SSM is decomposed into permanent and transitory components (as in Chang et al 2009):

   	$$y_t=y_t^{\mathrm{P}}+y_t^{\mathrm{T}},$$
   	$$y_t^{\mathrm{P}}=\beta x_{t|t-1}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{y}_t^{\mathrm{T}}=y_t-\beta x_{t|t-1},$$

   where $y_t^{\mathrm{P}}$ and $y_t^{\mathrm{T}}$ are permanent and transitory components, respectively. Because $y_t^{\mathrm{P}}$ is a function of $x_{t|t-1}$ and $y_t$ is a series with an I(1) stochastic common trend, the common trend can be extracted using MLE:

   	$\ln L(\theta )$	$=-{\displaystyle \frac{n}{2}}\log det\mathrm{\Sigma }-\frac{1}{2}\text{tr}{\mathrm{\Sigma }}^{-1}{\displaystyle \sum _{t=1}^n}{\epsilon }_t{\epsilon }_t^{\prime },$		(2.7)
   	${\epsilon }_t$	$=y_t-y_{t|t-1},$		(2.8)

   where $\mathrm{\Sigma }$ is the steady-state value of ${\mathrm{\Sigma }}_{t|t-1}$. When the prediction error ${\epsilon }_t$ is obtained through MLE, it contains new information about $\beta$ beyond what is found in information set ${\mathrm{\Theta }}_{t-1}$. The estimation of $\beta$ is updated using the updating equation:

   	$x_{t|t}$	$=x_{t|t-1}+{\omega }_{t|t-1}{\beta }^{\prime }{\mathrm{\Sigma }}_{t|t-1}^{-1}(y_t-y_{t|t-1})$		(2.9)
   and
   	${\omega }_{t|t}$	$={\omega }_{t|t-1}-{\omega }_{t|t-1}^2{\beta }^{\prime }{\mathrm{\Sigma }}_{t|t-1}^{-1}\beta .$		(2.10)

   The authors repeat these three steps for $t=1,2,\mathrm{\dots },T$. Any given values of $\beta$ and $\mathrm{\Lambda }$ correspond with steady-state values of ${\omega }_{t|t}$ and ${\mathrm{\Sigma }}_{t|t}$ that are denoted by $\omega$ and $\mathrm{\Sigma }$. The unobserved state-space parameter $\beta$ can be extracted iteratively.

   After the common trend has been estimated, the measurement equation (2.1) can be rewritten as

   	$S_t$	$={\widehat{\beta }}_s{x}_t+u_{st},$		(2.11)
   	$F_t$	$={\widehat{\beta }}_f{x}_t+u_{ft}.$		(2.12)

   Because $x_t$ is a latent variable, we rewrite $x_t$ as a function of (2.12),

   $$x_t=\frac{F_t}{{\widehat{\beta }}_f}-\frac{u_{ft}}{{\widehat{\beta }}_f}.$$

   (2.13)

   Substituting (2.13) into (2.11), the stochastic common trend between the stock index and futures can be expressed as

   $$S_t-\frac{{\widehat{\beta }}_s}{{\widehat{\beta }}_f}{F}_t=u_{st}-\frac{{\widehat{\beta }}_s}{{\widehat{\beta }}_f}{u}_{ft}.$$

   (2.14)

2. (2)

   Substitute the common trend in (2.14) into the ECM

   $$s_t={\alpha }_0+{\alpha }_1\left(u_{st-1}-\frac{{\widehat{\beta }}_s}{{\widehat{\beta }}_f}{u}_{ft-1}\right)+\theta f_t+\sum _{j=1}^q{\delta }_j{f}_{t-j}+\sum _{i=1}^q{\xi }_i{s}_{t-i}+e_t,$$

   (2.15)

   where $f_t=F_t-F_{t-1}$ and $s_t=S_t-S_{t-1}$ are daily changes in the futures price $F_t$ and spot price $S_t$, and $\theta$ is the HR. The new two-stage method for estimating the HR is called the KF–ECM estimator.

This paper examines the stock indexes of the nineteen industries included in the Taiwan Stock Exchange (TWSE). The sample comprises daily data from July 21, 1998 to March 27, 2019, a period that includes both the global financial crisis and the European debt crisis.⁸⁸ 8 Weekends and holidays have been deleted from the series. Table 1 shows the percentage distribution of the industries in the Taiwan Capitalization Weighted Stock Index (TAIEX).

Table 6 presents the augmented Dickey–Fuller (ADF) unit test results. For the null hypothesis of a unit root, the level series cannot reject the null hypothesis. When we apply the ADF test for the first difference series, the ADF $t$-statistic indicates the rejection of the null hypothesis of a unit root. Therefore, unit roots exist in all nineteen industrial series.

This paper proposes a hedging model that combines the KF with error correction, namely the KF–ECM. We compare the HE of the KF–ECM with that of four other models: the dynamic OLS, the GARCH(1,1), the TVPKF and an error-correction hedging model. The KF–ECM exhibits greater HE than any other model, according to both in-sample and out-of-sample evidence. Compared with the other models, the improvement in HE offered by the KF–ECM is small; even so, it can still be regarded as a robust method of estimating the HR. In contrast with the ML–ECM, the KF–ECM method does not require linearity and normality assumptions. Even if the distributions of the data have thick tails, reasonable parameter estimates can be obtained using the KF–ECM. The KF–ECM can be a valuable tool for investors to use in dynamic hedging.

Although the KF–ECM boasts a highly dynamic HE, this method has two limitations. First, numerical optimization is required to calculate the parameters. Because much complexity and necessary computing time result from a large number of parameters, we must set restrictions on the parameters when estimating. Second, when estimating the HR, the KF–ECM does not consider possible transaction costs. Chen et al (2003) indicated that transaction costs do affect the optimal HR.

In future research, two research directions can be pursued. First, transaction costs can be factored into the KF–ECM to explore their effect on the optimal HR. Second, because the KF–ECM can extract the single best trend from multiple portfolios, researchers can explore the HE of the KF–ECM when expanding hedging instruments from single-index futures to multiple-asset portfolios.

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.