An Econometric Analysis of ETF and ETF Futures in Financial and Energy Markets Using Generated Regressors *

Michael McAleer Department of Quantitative Finance National Tsing Hua University, Taiwan and Discipline of Business Analytics University of Sydney Business School, Australia and Econometric Institute, Erasmus School of Economics Erasmus University Rotterdam, The Netherlands and Department of Quantitative Economics Complutense University of Madrid, Spain and Institute of Advanced Sciences Yokohama National University, Japan


Introduction
The Global Financial Crisis (GFC) was not only unexpected and unpredicted, but also had a marked and sustained impact on the world economy, in general, and also on international financial markets. After the GFC had subsided, oil prices recovered and stabilized at a price between US$90 and US$110 per barrel. This period of relative stability lasted from January 2011 to June 2014. However, in mid-2014 oil prices nosedived from a high of US$107.95 per barrel to a low of US$26.19 per barrel on February 11, 2016. According to a World Bank Report (Baffes, Kose, Ohnsorge, and Stocker, 2015), the plunge in oil prices was mainly driven by supply factors, namely the growth of unconventional oil production, such as Canadian oil sand and US shale oil. In particular, spurred by the shale oil boom, the USA nearly doubled its 2011 daily production levels to over 11 million barrels in June 2014. This surge allowed the USA to surpass Saudi Arabia as the oil and natural gas liquids global production leader, as reported by the International Energy Agency (IEA) (Bloomberg, July 4, 2014).
Responding to the surge in unconventional oil production, at the 166 th OPEC meeting held on November 27, 2014, OPEC decided not to curtail daily production, choosing instead to maintain a stable production of 30 million barrels per day, a policy that was enacted on December 14, 2011. This decision represented abandonment of OPEC's price targeting policy, with the tradeoff of possibly maintaining their current market share. However, this course of action may well have led to persistently low oil prices.
Such low oil prices have major ramifications on the banking sector. In addition to being forced to increase reserves for losses in the oil and gas portfolio, banks have also tried to shrink the credit lines offered to energy companies, even as energy companies become more dependent on banking loans. This sentiment is echoed by Devi Aurora, a senior director at Standard & Poor's in New York, who was reported to have said (Financial Times, January 15, 2016):

"[Energy] Companies have a tendency to draw on bank lines once other options dry up."
Faced with the dual pressures of low oil prices and a compromised ability to generate cash flows, oil companies are increasingly in danger of defaulting on loans. As reported in the Wall Street Journal, "Coming to the Oil Patch: Bad Loans to Outnumber the Good", March 24, 2016: "Fifty-one North American oil-and-gas producers have already filed for bankruptcy since the start of 2015, cases totaling $17.4 billion in cumulative debt, according to law firm Haynes and Boone LLP. That trails the number from September 2008 to December 2009 during the global financial crisis, when there were 62 filings, but is expected to grow: About 175 companies are at high risk of not being able to meet loan covenants, according to Deloitte LLP." From recent data, it is clear that oil price collapses of greater than 50% are not unprecedented events. For example, in 1986, there was a similar supply glut, which also led to a plunge in oil prices. In particular, that year marked OPEC's decision to revert its production target back to 30 million barrels per day, ending a significant decline in oil production since the Iran-Iraq war in 1979. This reversion, combined with an influx of oil supply from Mexico and the North Sea, caused the price of oil to collapse from US$26.53 per barrel on January 6, 1985 to US$10.25 per barrel at its low point on March 31, 1986. Around this time, the US government attempted to stimulate the sluggish economy and guard against deflation through several monetary and fiscal policies such as interest rate cuts. In spite of these measures, low oil prices persisted, thereby contributing to a global economic slowdown and a major downward correction in global financial markets on October 19, 1987. This day, which came to be known as Black Monday, saw the S&P 500 drop 20.4%, falling from 282.7 to 225.06. Another significant plunge in oil prices, this time of the order of 40%, occurred between October 1997 and March 1998 amid the Asian Financial Crisis. This crisis was propelled primarily by an unexpected speculative attack on the Thai baht. The resulting drastic devaluation of the Thai currency not only wrought considerable damage to the East Asian economy, but also impacted global financial markets. The US Federal Reserve was forced to bail out a well-known hedge fund, Long Term Capital Management (LTCM), on September 23, 1998. During the economic slump, which lasted from 1997 to 1998, the global oil demand receded substantially, with oil prices reaching a low of US$10.82 per barrel on December 10, 1998.
The most dramatic example of a sudden oil price collapse occurred a decade later in the wake of the Global Financial Crisis (GFC). While there is no consensus on the exact starting and ending dates of the GFC, for the purposes of this paper, we consider the GFC to span the time period from October 9, 2007 to March 9, 2009, which corresponds to the S&P 500 dropping from a high of 1565.26 to a low of 672.88. Oil prices reached an historical high of US$145.31 per barrel on July 3, 2008, but tumbled to US$30.28 per barrel just 6 months later, on December 23, 2009. The GFC was spurred by a tsunami of financial chaos, including the housing bubble which, in turn, led to an epidemic of defaults in subprime mortgages. Subsequently, banks and insurance companies sold trillions of dollars of Credit Default Swaps (CDSs), which not only involved subprime mortgage loans, but also many other financial instruments and institutions. This resulted in Lehman Brothers going bankrupt on September 15, 2008, and the US Treasury being forced to bail out AIG in the same month. The GFC led to a dramatic diminution in the global oil demand and, in turn, tumbling energy prices (Van Vactor, January 1, 2009).
In light of the preceding discussion, it is clear that there is an intrinsic link between the financial and energy sectors. One way to unearth the link between two or more sectors is by analyzing their spillover effects, which are measures of how the shocks to returns in different assets affect each other's subsequent volatility in both spot and futures markets.
In conducting spillover effect analysis, an important consideration is the choice of indices used to represent the assets or sectors under comparison. One reasonable selection of measures to examine volatility spillovers between the energy and financial sectors is the Energy Select Sector index (Ticker: IXE) and the Financial Select Sector index (Ticker: IXM). Both of these are sub-indices of S&P500, reflecting the overall economic condition of their respective sectors. One shortcoming of using these indices, however, is the fact that they are not tradable, and hence may be of little practical use to investors.
One way to overcome this drawback is by employing derivatives of the IXE and IXM indices, as opposed to the indices themselves. Financial derivatives which are not only highly representative of the underlying indices but can also be traded on both the spot and futures markets, include Exchange Traded Funds (ETFs), otherwise known as implied tradable spot prices. Another financial derivative that has not yet been considered in practice, primarily as it typically does not exist in many financial markets, but may well have practical importance, is ETF futures.

Advances in Economics, Business and Management Research (AEBMR), volume 26
For the reasons specified above, in order to probe the relationship between the energy and financial sectors, we apply not only IXE and IXM, but also ETFs and ETF futures in conducting spillover effect analysis within and across these two sectors. In particular, for both the energy and financial sectors, we will select one index (namely, IXE or IXM), one ETF, and construct one ETF futures from which to analyze all 15 possible pairwise combinations of spillover effects. The list of indices, ETFs, and ETF futures that we will use in the empirical analysis is as follows: Financial Select Sector Index (IXM), Energy Select Sector Index (IXE), Financial Select Sector SPDR Fund (XLF), Energy Select Sector SPDR Fund (XLE), Financial ETF futures (XLFf), and Energy ETF futures (XLEf).
An important point to clarify is that, despite the delisting of ETF futures in March 1, 2011, due to low trading volume, our analysis will include up-to-date ETF futures data from each sector. This is made possible by the use of "generated regressors" to construct both Financial ETF futures and Energy ETF futures. More details on this methodological approach will be discussed in Section 3.
An Exchange Traded Fund (ETF) is a tradable spot index whose aim is to replicate the return of an underlying benchmark index. For instance, SPDR® S&P 500® ETF, issued by State Street Bank & Trust Company, tracks the performance of the S&P 500 Index. In contrast to investing in a single stock, ETFs invest in a basket of stocks or commodities, thereby diversifying the non-systematic risk and decreasing the levels of risk and volatility. Furthermore, unlike actively-managed mutual funds, most ETF managers take a passive management style and collect lower managing fees. Whereas mutual funds are limited to trades based on end-of-day prices, ETFs are traded like stocks. Besides the points listed above, ETFs have the following additional advantages over traditional mutual funds: (i) ETFs offer greater transparency compared with mutual funds in the sense that ETFs are required to reveal their holdings data on a daily basis, whereas mutual funds are mandated only to disclose holdings data on a quarterly basis.
(ii) ETFs are more flexible than mutual funds because investors can short sell them when they are bearish on the market. Although short selling may be considered risky compared with conventional investing, it can be a useful strategy if executed by savvy investors when the market is overvalued.
To recap, the purpose for this paper is to investigate spillover effects within and across the energy and financial sectors in terms of both the US spot and futures markets by applying indices, ETF, and ETF futures. For the empirical analysis, we select two indices and two ETFs, and generate two ETF futures from which to analyze all 15 possible pairwise combinations of spillover effects. Specifically, the list of variables we use is as follows: Financial Select Sector Index (IXM), Energy Select Sector Index (IXE), Financial Select Sector SPDR Fund (XLF), Energy Select Sector SPDR Fund (XLE), Financial ETF futures (XLFf), and Energy ETF futures (XLEf). In order to carry out this analysis, the techniques to be used are generated regressors and the multivariate conditional volatility Diagonal BEKK model. The empirical result will be discussed in greater detail in Section 5.
The remainder of the paper is organized as follows. In Section 2, the brief literature on the topic is reviewed. In Section 3, the empirical models are presented, and the data are discussed in Section 4. In Section 5, the empirical results are analyzed, and some concluding comments are given in Section 6.

Brief Literature Review
The literature on the use of ETFs and testing for co-volatility spillovers is rather sparse. Chang, Li, and McAleer (2015) conducted a comprehensive review of the literature related to co-volatility spillovers between energy markets and agricultural commodities. One of the major findings of their review paper was that most researchers fail to employ valid statistical techniques in testing for spillover effects. Multivariate conditional volatility models, namely BEKK and DCC, have typically been used to test for spillover effects between energy and agricultural markets. However, these models are either problematic in and of themselves (in the case of DCC), or have been used in erroneous manners (in the case of BEKK).
Specifically, the scalar DCC model lacks regularity conditions, while a serious technical deficiency related to estimating the full BEKK and scalar DCC models through Quasi-Maximum Likelihood Estimates (QMLE) is the absence of any asymptotic properties. In contrast, the multivariate diagonal BEKK conditional volatility model possesses both regularity conditions and asymptotic properties. For these reasons, Chang, Wang, and McAleer (2016) applied the multivariate diagonal BEKK conditional volatility model in testing the volatility spillovers for bio-ethanol, sugarcane and corn, while this paper also applies the multivariate diagonal BEKK conditional volatility model in testing the volatility spillover effects within and across the US financial and energy markets.
As described above, an exchange traded fund (ETF) is a tradable asset whose aim is to track an underlying index representing the economic condition of an entire sector. Thus, ETFs have great value to investors as they facilitate a systematic reduction in risk within a trading portfolio. Chang and Ke (2014) applied ETFs in the US energy sector to investigate the causality between flows and returns through the Vector-AutoRegressive (VAR) model to test four hypotheses, namely, the price pressure, information, feedback trading, and smoothing hypotheses. One noteworthy aspect of their methodology was the fact that they analyzed not just the entire sample period, but also divided the data into three sub-periods, namely, before, during, and after the Global Financial Crisis (GFC), a methodology also used by McAleer, Jimenez-Martin, and Perez-Amaral (2013). The use of the three sub-periods will also be used in the paper.
Chen and Huang (2010) used ETFs to examine volatility spillovers, albeit in a rudimentary manner, between an ETF and its underlying stock index in 9 different countries. They used the GARCH-ARMA and EGARCH-ARMA models, and found that there were volatility spillover effects for the stock index and ETF. Unfortunately, as in the case of estimating the full BEKK and scalar DCC models through Quasi-Maximum Likelihood Estimation (QMLE) methods, EGARCH has no known regularity conditions, and the statistical properties of the estimators of the parameters are not available under general conditions (see McAleer and Hafner, 2014).
One paper which used the diagonal BEKK model to examine ETFs was by Chang, Hsieh, and McAleer (2016). The authors investigated the causality and spillover effects between VIX, consisting of different moving average processes, and ETF returns by using vector autoregressive (VAR) models and diagonal BEKK models. The empirical results show that daily VIX returns have: (1) significant negative effects on European ETF returns in the short run; (2) stronger significant effects on single market ETF returns than on European ETF returns; and (3) lower impacts on the European ETF returns than on S&P500 returns.
In some financial research contexts, it may be necessary or advantageous to generate a new index representing a certain sector that may be of interest. One way in which this may be performed is through the use of generated variables. Chang (2015) applied generated variables to develop a daily Tourism Financial Conditions Index (TFCI), based on nominal exchange rates, interest rates, and a tourism industry stock index that is listed on the Taiwan Stock Exchange. The empirical results indicated that the generated TFCI was accurately estimated through the estimated conditional means of the tourism stock index returns. As described in the introduction, the paper is interested in the co-volatility spillover effects across and within the financial and energy sectors in both the spot and futures markets. While energy and financial indices and ETFs are already available to analyze spot markets, it is necessary to use generated variables to construct ETF futures to analyze futures markets.
The paper combines several of the elements reviewed above to create a novel methodology to test for spillover effects in a statistically valid and comprehensive way that can be of immense practical use to investors. In particular, we use the diagonal BEKK model which, as mentioned above, has valid asymptotic and regularity properties as compared with the full BEKK and scalar DCC models, in order to test for spillovers within and across the financial spot (indices and ETFs) and futures (ETF futures via generated regressors) markets. This analysis is conducted for four time periods namely, before-GFC, during-GFC, after-GFC, and the entire sample period.

Methodology
The primary purpose of this paper is to test volatility spillover effects among ETF and ETF futures in the financial and energy sectors. In the previous literature, a great deal of confusion has arisen about how spillover effects should be tested, with published academic papers often using dubious methodologies. Indeed, many so-called tests of spillovers are not, in fact, tests of spillovers at all. The following section presents three novel tests of spillovers, namely full volatility spillovers, full co-volatility spillovers, and partial co-volatility spillovers. For further details, see Chang, Li and McAleer (2015).
Tests of spillovers require estimation of a multivariate volatility model, with appropriate regularity conditions and asymptotic properties of the Quasi Maximum Likelihood Estimation (QMLE) of the associated parameters underlying the conditional mean and conditional variance. As the first step of the estimation of multivariate conditional volatility model is the estimation of multiple univariate conditional volatility models, an appropriate and widely-used univariate conditional volatility model will be discussed below.
This section is organized as follow: (1) A brief discussion of the most widely-used univariate conditional volatility model; (2) A definition of three novel spillover effects; (3) A discussion of the most widely-used multivariate model of conditional volatility.
In order to accommodate volatility spillover effects, alternative multivariate volatility models of the conditional covariances are available. Examples of such multivariate models include: (1)  The first step in estimating multivariate models is to obtain the standardized shocks from the conditional mean returns shocks. For this reason, the most widely used univariate conditional volatility model, namely GARCH, will be presented briefly, followed by the most widely estimated multivariate conditional covariance model, namely a specific version of BEKK.
Consider the conditional mean of financial returns as follows: (1) where the returns, , represent the log-difference in financial commodity or agricultural prices, , is the information set at time t-1, and is a conditionally heteroskedastic returns shock. In order to derive conditional volatility specifications, it is necessary to specify the stochastic processes underlying the returns shocks, .

Univariate Conditional Volatility Models
Alternative univariate conditional volatility models are of interest in single index models to describe individual financial assets and markets. Univariate conditional volatilities can also be used to standardize the conditional covariances in alternative multivariate conditional volatility models to estimate conditional correlations, which are particularly useful in developing dynamic hedging strategies.
The most popular univariate conditional volatility model is discussed below, together with the associated regularity conditions, and the conditions underlying the asymptotic properties of consistency and asymptotic normality.

Random Coefficient Autoregressive Process and GARCH
Consider the random coefficient autoregressive process of order one: (2) where and is the standardized residual. Tsay (1987) derived the ARCH(1) model of Engle (1982) from equation (2) as: ( 3) where is conditional volatility, and is the information set available at time t-1. The use of an infinite lag length for the random coefficient autoregressive process in equation (2), with appropriate geometric restrictions (or stability conditions) on the random coefficients, leads to the GARCH model of Bollerslev (1986). From the specification of equation (2), it is clear that both and should be positive as they are the unconditional variances of two separate stochastic processes.
The QMLE of the parameters of ARCH and GARCH have been shown to be consistent and asymptotically normal in several papers. For example, Ling and McAleer (2003) showed that the QMLE for GARCH(p,q) is consistent if the second moment is finite. Moreover, a weak sufficient log-moment condition for the QMLE of GARCH(1,1) to be consistent and asymptotically normal is given by: (4) which is not easy to check in practice as it involves two unknown parameters and a random variable. The more restrictive second moment condition, namely , is much easier to check in practice.
In general, the proofs of the asymptotic properties follow from the fact that ARCH and GARCH can be derived from a random coefficient autoregressive process (see McAleer et al. (2008) for a general proof of multivariate models that are based on proving that they satisfy the regularity conditions given in Jeantheau (1998) for consistency).

Multivariate Conditional Volatility Models
The multivariate extension of univariate GARCH is given as variations of the BEKK model in Baba et al. (1985) and Engle and Kroner (1995). In order to establish volatility spillovers in a multivariate framework, it is useful to define the multivariate extension of the relationship between the returns shocks and the standardized residuals, that is, . The multivariate extension of equation (1), namely , can remain unchanged by assuming that the three components are now, respectively, vectors, where is the number of financial assets.
The multivariate definition of the relationship between and is: where is a diagonal matrix comprising the univariate conditional volatilities. Define the conditional covariance matrix of as . As the vector, , is assumed to be independently and identically distributed (iid) for all elements, the conditional correlation matrix of , which is equivalent to the conditional correlation matrix of , is given by . Therefore, the conditional expectation of (4) is defined as: .
( 6) Equivalently, the conditional correlation matrix, , can be defined as: . (7) Equation (5) is useful if a model of is available for purposes of estimating , whereas equation (6) is useful if a model of is available for purposes of estimating .
Equation (5) is convenient for a discussion of volatility spillover effects, while both equations (5) and (6) are instructive for a discussion of asymptotic properties. As the elements of are consistent and asymptotically normal, the consistency of in (5) depends on consistent estimation of , whereas the consistency of in (6) depends on consistent estimation of . As both and are products of matrices, neither the QMLE of nor can be asymptotically normal, based on the definitions given in equations (5) and (6).

Full and Partial Volatility and Co-volatility Spillovers
Volatility spillovers are defined in Chang, Li and McAleer (2015) as the delayed effect of a returns shock in one asset on the subsequent volatility or co-volatility in another asset. Therefore, a model relating to returns shocks is essential, and this will be addressed in the following sub-section. Spillovers can be defined in terms of full volatility spillovers and full co-volatility spillovers, as well as partial co-volatility spillovers, as follows: (1) Full volatility spillovers: (2) Full co-volatility spillovers: (3) Partial co-volatility spillovers: where is returns shocks, and is the conditional covariance matrix of . Volatility spillovers in the spot and derivatives markets is crucial for purposes of dynamic hedging.
Full volatility spillovers occur when the returns shock from financial asset k affects the volatility of a different financial asset i. Full co-volatility spillovers occur when the returns shock from financial asset k affects the co-volatility between two different financial assets, i and j. Partial co-volatility spillovers occur when the returns shock from financial asset k affects the co-volatility between two financial assets, i and j, one of which can be asset k. When m = 2, only (1) and (3) are possible as full co-volatility spillovers depend on the existence of a third financial asset.
As mentioned above, spillovers require a model that relates the conditional volatility matrix, , to a matrix of delayed returns shocks. The most frequently used models of multivariate conditional covariance are alternative specifications of the BEKK model, with appropriate parametric restrictions, which will be considered below.

Diagonal and Scalar BEKK
The vector random coefficient autoregressive process of order one is the multivariate extension of equation (2), and is given as: (11) where and are vectors, is an matrix of random coefficients, and: , . Technically, a vectorization of a full (that is, non-diagonal or non-scalar) matrix A to vec A can have dimension as high as , whereas vectorization of a symmetric matrix A to vech A can have dimension as low as .
In a case where A is either a diagonal matrix or the special case of a scalar matrix, , McAleer et al. (2008) showed that the multivariate extension of GARCH(1,1) from equation (10), incorporating an infinite geometric lag in terms of the returns shocks, is given as the diagonal or scalar BEKK model, namely: (12) where A and B are both either diagonal or scalar matrices. The matrix A is crucial in the interpretation of symmetric and asymmetric weights attached to the returns shocks, as well as the subsequent analysis of spillover effects.
McAleer et al. (2008) showed that the QMLE of the parameters of the diagonal or scalar BEKK models were consistent and asymptotically normal, so that standard statistical inference on testing hypotheses is valid. Moreover, as in (11) can be estimated consistently, in equation (6) can also be estimated consistently.
In terms of volatility spillovers, as the off-diagonal terms in the second term on the right-hand side of equation (11), , have typical (i,j) elements , there are no full volatility or full co-volatility spillovers. However, partial co-volatility spillovers are not only possible, but they can also be tested using valid statistical procedures.

Triangular, Hadamard and full BEKK
Without actually deriving the model from an appropriate stochastic process, Baba et al. (1985) and Engle and Kroner (1995) considered the full BEKK model, as well as the special cases of triangular and Hadamard (element-by-element multiplication) BEKK models. The specification of the multivariate model is the same as the specification in equation (11), namely: (13) Advances in Economics, Business and Management Research (AEBMR), volume 26 except that A and B are full, Hadamard or triangular matrices, rather than diagonal or scalar matrices, as in (11).
It is possible to examine spillover effects using each of these models, but it is not possible to test or analyze spillover effects as the QMLE of the parameters in equation (12) have no known asymptotic properties. Although estimation of the full, Hadamard and triangular BEKK models is available in some standard econometric and statistical software packages, it is not clear how the likelihood functions might be determined. Moreover, the so-called "curse of dimensionality", whereby the number of parameters to be estimated is excessively large, makes convergence of any estimation algorithm somewhat problematic. This is in sharp contrast to a number of published papers in the literature, whereby volatility spillovers have been tested incorrectly based on the off-diagonal terms in the matrix A in equation (12).

Generated Regressors
One of the primary purposes of the paper is to investigate the spillover effects within and across the energy and financial sectors for both US spot and futures market by applying indices, ETF, and ETF futures. While energy and financial indices and ETFs are already available for spot markets, it is necessary to use generated variables to construct ETF futures for futures markets. The generated ETF futures proposed in the paper focus on economic activities related to the financial and energy industries, respectively. The three components of the Financial ETF futures (XLFf), each of which can be constructed from data downloaded from Bloomberg or Yahoo Finance, are as follows: (1) Financial Select Sector SPDR Fund (XLF); (2) Generic 1 st S&P 500 index futures (SP1); and (3) Generic 1 st FTSE 100 index futures (Z1).
The other three components of the Energy ETF futures (XLEf), each of which can be constructed from data downloaded from Bloomberg or Yahoo Finance, are as follows: (1) Energy Select Sector SPDR Fund (XLE); (2) Generic 1 st Crude Oil WTI futures (CL1); and (3) Generic 1 st Natural Gas futures (NG1).
The ETF futures discussed above are based on estimation of a regression model, which may be referred to as the generating model. The model-based weights for the components of Financial ETF futures and Energy ETF futures will be estimated by OLS. The traditional method of examining the statistical properties of generated variables, and more specifically generated regressors, use variables that are typically stationary. In empirical finance, the variables considered can be financial returns, in which the variables are typically stationary, or financial stock prices, where the variables are typically non-stationary.  (1992)). For purposes of determining whether the generated ETF futures are a reasonable construction of the latent variable, will be used as a statistical indicator.
The models to be estimated below are linear in the variables, with the appropriate weights to be estimated empirically. Accordingly, XLFf is defined as: (14) where c denotes the constant term, and denotes the shocks to XLFf, which need not be independently or identically distributed, especially for daily data. The parameters θ1, θ2 and θ3 are the weights attached to one-period lagged Financial ETF, Generic 1 st S&P 500 index futures, and Generic FTSE 100 index futures, respectively.
As XLFf is a latent variable, it is necessary to link XLFf to observable data. The latent variable is defined as being the conditional mean of an observable variable, namely the Financial Select Sector SPDR Fund (XLF), which is a tradable spot index, reflecting the financial select index that is listed on the NYSE, as follows: (15) where XLF is observed, XLFf is latent, and the measurement error in XLF is denoted by , which need not be independently or identically distributed, especially for daily data.
Given the zero mean assumption for , the means of XLF and XLFf will be identical, as will their estimates. Using equations (13) and (14), the empirical model for estimating the weights for XLF is given as: (16) where , which should be distinguished from the return shocks, , in equations (1) and (4) above, need not be independently or identically distributed, especially for daily data.
The parameters in equation (15) can be estimated by OLS or QMLE, depending on the specification of the conditional volatility of , to yield estimates of XLF, if SP1 and Z1 are stationary. As XLF is a non-stationary price, there is no reason to expect the combined error, , to be conditionally heteroskedastic. Alternatively, Instrumental Variables (IV) or Generalized Method of Moments (GMM) can be used to estimate the parameters in equation (15) to obtain an estimate of XLF, and hence also an estimate of the latent variable, XLFf, although finding suitable instruments can be problematic when daily data are used.
Cointegration could also be used to estimate the parameters in equation (15), but only if consistent estimates of the parameters are desired, and if statistical inference is intended for the estimates. As we are interested only in the fitted values of ETF to generate ETF futures, namely XLF to obtain XLFf, these alternative methods are eschewed in favour of the Ordinary Least Squares (OLS) estimates. In view of the definition in equation (14), the estimates of XLF will also provide estimates of the latent XLFf.
Similar logic to the above applies to the energy case. XLEf is defined as follows: (17) where XLE is observed, XLEf is latent, and the measurement error in XLE is denoted by , which need not be independently or identically distributed, especially for daily data.
Given the zero mean assumption for , the means of XLE and XLEf will be identical, as will their estimates. Using equations (13) and (14), the empirical model for estimating the weights for XLE is given as:

Advances in Economics, Business and Management Research (AEBMR), volume 26
where need not be independently or identically distributed, especially for daily data. As there would seem to be no known optimality properties for the OLS estimates of ETF futures, the OLS estimates of XLE will be used to estimate XLEf, though no optimality properties are claimed for the generated XLE futures.

Data and Variables
As shown in Table 1, we choose the following indices, ETFs, and ETF futures for the empirical analysis: Financial Select Sector Index (IXM), Energy Select Sector Index (IXE), Financial Select Sector SPDR Fund (XLF), Energy Select Sector SPDR Fund (XLE), Financial ETF futures (XLFf), and Energy ETF futures (XLEf). Estimates of XLFf using Generated Regressors via the software R is shown in equation (18): where XLEf is Energy ETF futures, XLE is Energy Select Sector SPDR® Fund, CL1 is Generic 1 st Crude Oil WTI futures, NG1 is Generic 1 st Natural Gas futures, and t-ratios are shown in parentheses. As stated previously, the t-ratios do not have the standard asymptotic normal distribution as the variables are non-stationary, but the extremely high value of suggests that the generated variable is a useful construction of the latent variable Daily data for the financial select sector index, energy select sector index, financial ETF, energy ETF, and the constituents of the Financial ETF futures and Energy ETF futures (namely, generic 1 st S&P 500 index futures, generic 1 st FTSE 500 index futures, generic 1 st Crude Oil futures, and generic 1 st Natural Gas futures), were downloaded from Bloomberg or Yahoo Finance. In the case of a national holiday, the missing value is replaced by the value of the previous day. ETF fund returns are calculated by taking the log difference of adjusted prices and multiplying by 100, that is, (logPt -logPt-1)*100. The relevant descriptive statistics are shown in Table 2, implying that the returns of all variables are not normal. The Augmented Dickey Fuller (ADF) and PP (Phillips-Perron) test for unit roots are shown in Table 3. The unit roots tests indicate that the returns of all variables are stationary. Note: The Jarque-Bera Lagrange Multiplier test is asymptotically chi-squared, and is based on testing skewness and kurtosis against the normal distribution.

Table 3: Unit Root Tests
The empirical analysis was conducted in its entirety and also subdivided into three sub-periods, namely (i) before-GFC, from December 22, 1998

Hypothesis Testing of Co-volatility Spillovers
This paper uses the Diagonal BEKK model, in which the co-volatility spillover effects are a function of the diagonal elements of matrix A and the returns shocks of asset i at time t-1. A rejection of the null hypothesis H0, as shown in the definition of the test of co-volatility spillover effects in Section 3, indicates significance of the co-volatility spillovers from the returns shocks of asset j at time t-1 to the co-volatility between assets i and j at time t.
In the empirical analysis, we selected two indices and two ETFs, and generated two ETF futures, from which to analyze all 15 possible pairwise combinations of spillover effects based on the multivariate diagonal BEKK model, specifically, the co-volatility spillovers for all cases in which the estimates of A in the Diagonal BEKK model are significant. The diagonal BEKK model shown in equation (11) was estimated by QMLE using the econometric software package EViews 8.  Table 4 shows the estimates of the diagonal elements of A in the Diagonal BEKK model for each pairwise comparison analyzed (as described below), while Table 5 shows the mean returns shocks for each asset, both for the entire time period and for each of the three sub-periods. Tables 6 shows the mean co-volatility spillovers, which are calculated by applying the definition of the co-volatility spillover effects discussed in Section 3.   Note: Co-volatility Spillover = ∂Qij,t / ∂εj,t−1= aii*ajj*εi,t−1 ; mean co-volatility spillovers use the mean return shocks from Table 5.

Calculating Average Co-volatility Spillovers
As can be seen in Table 6 and the explanation below, the data were separated into 5 groups, which will be described in detail below.
Group 1: Cross-sector spot-spot spillover effects, specifically, the spillover effects between each of the pairs: (a) financial index and energy index, (b) financial ETF and energy ETF, (c) financial index and energy ETF, and (d) energy index and financial ETF.
Group 2: Cross-sector futures-futures spillover effects, specifically, the spillover effects between (a) financial ETF futures and energy ETF futures. Group 4: Within-sector spot-spot spillover effects, specifically, the spillover effects between (a) financial index and financial ETF and (b) energy index and energy ETF.
Group 5: Within-sector spot-futures spillover effects, specifically, the spillover effects between each of the pairs: (a) financial index and financial ETF futures, (b) financial ETF and financial ETF futures, (c) energy index and energy ETF futures, and (d) energy ETF and energy ETF futures.
The following paragraphs describe the average co-volatility spillover effects for each of the 5 groups mentioned above, and also across each of the 4 time periods, namely "before-GFC", "during-GFC", "after-GFC", and "all".
In Group 1, before-GFC, namely, cross-sector spot-spot spillovers, it was found that in all cases, co-volatility spillovers were statistically significant and negative. For each of the four pairs, the magnitude of the spillovers of the financial spot asset, namely, IXM (Financial Select Sector Index) or XLF (Financial ETF), on subsequent co-volatility between itself and its corresponding energy spot asset, namely IXE (Energy Select Sector Index) or XLE (energy ETF), was numerically greater than the spillovers of the energy spot asset on the same subsequent co-volatility pair.
In Group 1, during-GFC, it was found that in all cases, co-volatility spillovers were again statistically significant and negative. For each pair, the magnitude of the spillovers of the financial spot asset, namely, IXM (Financial Select Sector Index) or XLF (Financial ETF), on subsequent co-volatility between itself and its corresponding energy spot asset, namely, IXE (Energy Select Sector Index) or XLE (energy ETF), was similar to as the spillover effect of the energy spot asset on the same subsequent co-volatility pair.
In Group 1, after-GFC, it was found that in all cases, co-volatility spillovers were statistically significant. For each pair, the spillovers of the financial spot asset, namely, IXM (Financial Select Sector Index) or XLF (Financial ETF) on subsequent co-volatility between itself and its corresponding energy spot asset, namely, IXE (Energy Select Sector Index) or XLE (energy ETF), was negative and greater than the positive spillovers of the energy spot asset on the same subsequent co-volatility pair.
In terms of the aggregation of the three periods for Group 1, it was found that in all cases, co-volatility spillovers were statistically significant and negative. For each pair, the magnitude of the spillovers of spillover of the futures asset, namely, XLFf (Financial ETF futures) or XLEf (Energy ETF futures), on subsequent co-volatility between itself and its corresponding cross-sector spot asset, namely, IXE (Energy Select Sector Index) or XLE (energy ETF) and IXM (Financial Select Sector Index) or XLF (financial ETF), respectively, was greater than the spillovers of the spot asset on the same subsequent co-volatility pair.
In Group 3, during-GFC, it was found that co-volatility spillovers between XLEf (energy ETF futures) and XLF (financial ETF) or IXM (financial index), namely, cases 3.a.1 to 3.b.2, were statistically significant and negative. For each pair, the magnitude of spillovers of XLEf (Energy ETF futures) on subsequent co-volatility between itself and its corresponding cross-sector spot asset, namely, IXM (Financial Select Sector Index) or XLF (financial ETF), was greater than the spillovers of the spot asset on the same subsequent co-volatility pair. However, in each of the cases involving the co-volatility between financial ETF futures and a spot energy asset (namely, energy ETF or energy index), specifically, cases 3.c.1 to 3.d.2, non-significant co-volatility effects were found.
In Group 3, after-GFC, it was found that in all cases, co-volatility spillovers were statistically significant. For each pair, the magnitude of the spillover effect of XLFf (Financial ETF futures) on subsequent co-volatility between itself and its corresponding cross-sector energy spot asset, namely, IXE (Energy Select Sector Index) or XLE (energy ETF), was greater than the spillovers of the energy spot asset on the same subsequent co-volatility pair. However, the spillovers of XLEf (energy ETF futures) on subsequent co-volatility between itself and its corresponding cross-sector financial spot asset, namely, IXM (financial Select Sector Index) or XLF (financial ETF), were positive and smaller than the negative spillovers of the financial spot asset on the same subsequent co-volatility pair.
In Group 3, combining all three periods, it was found that in all cases, co-volatility spillovers were statistically significant and negative. For each pair, the magnitude of spillovers of XLFf (Financial ETF futures) on subsequent co-volatility between itself and its corresponding cross-sector energy spot asset, namely, IXE (Energy Select Sector Index) or XLE (energy ETF), was the similar to the spillovers of the energy spot asset on the same subsequent co-volatility pair. However, the spillovers of XLEf (energy ETF futures) on subsequent co-volatility between itself and its corresponding cross-sector financial spot asset, namely, IXM (financial Select Sector Index) or XLF (financial ETF), were greater than the spillovers of the financial spot asset on the same subsequent co-volatility pair.
In Group 4, it was found that in all cases, co-volatility spillovers were statistically significant over the four time periods. In terms of the magnitude of within-sector spot-spot co-volatility effects, the spillovers of IXM (Financial Select Sector Index) on subsequent co-volatility between itself and XLF (Financial ETF), was the similar to the spillovers of XLF on the same subsequent co-volatility pair, 3. Asymmetric spillover effects were found in all cases of spot-futures ETF within sectors. Moreover, in all cases, spillover effects of ETF futures on its co-volatility with the corresponding ETF are stronger than in the reverse case (see group 5).
4. The co-volatility spillovers in all groups over all time periods are statistically significant, except for cases 3.c.1 to 3.d.2 During-GFC. 5. Additionally, with the exception of the insignificant cases, the co-volatility spillovers are stronger During-GFC than for the other time periods (see groups 1, 2, and 4). 6. In terms of the current relationship between the financial and energy sectors, the After-GFC spillovers are of greater relevance than the spillovers the three sub-periods are combined into a single sample.

Concluding Remarks
The primary purpose of the paper was to investigate the co-volatility spillovers within and across the US energy and financial sectors in both their spot (namely, IXE, IXM, XLF, and XLE) and futures (namely, XLFf and XLEf) markets, by using "generated regressors" and a multivariate conditional volatility model, namely Diagonal BEKK. The daily data used in the empirical analysis are from 1998/12/23 to 2016/4/22. The data set was analyzed in its entirety, and also subdivided into three time periods, namely "before-GFC", "during-GFC", "after-GFC".
In Group 1, before and after the Global Financial Crisis, the magnitude of the spillovers of the financial spot asset, namely, IXM (Financial Select Sector Index) or XLF (Financial ETF), on subsequent co-volatility between itself and its corresponding energy spot asset, namely, IXE (Energy Select Sector Index) or XLE (energy ETF), was greater than the spillovers of the energy spot asset on the same subsequent co-volatility pair.
However, during the GFC, the pattern changed dramatically. All of the spillovers were stronger, and the spillovers of the financial spot asset on the subsequent co-volatility between itself and its corresponding energy spot asset was the similar to the spillovers of the energy spot asset on the same subsequent co-volatility pair.
Other significant spillover patterns were also found between financial ETF index and energy ETF index in their spot-spot, spot-futures, and futures-futures co-volatility, namely, Groups 2 and 3, when combining all three periods. In terms of the within-sector spot-spot and spot-futures markets, namely, Groups 4 and 5, significant spillovers of ETF futures on subsequent co-volatility between ETF and ETF futures were also found.
It is apparent that there is an intrinsic relationship between the Financial ETF and Energy ETF, both in their spot and futures markets. The energy ETF and financial ETF have statistically significant co-volatility spillovers for all time periods. These empirical results suggest that financial and energy ETFs are suitable for constructing a financial portfolio from an optimal risk management perspective, and also for dynamic hedging purposes.