evaluating the predictability of etf volatility indexes...
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Proceedings of the Fifth Asia-Pacific Conference on Global Business, Economics, Finance
and Social Sciences (AP16Mauritius Conference) ISBN - 978-1-943579-38-9
Ebene-Mauritius, 21-23 January, 2016. Paper ID: M620
1 www.globalbizresearch.org
Evaluating the Predictability of ETF Volatility Indexes
Using the STARX Models
Jo-Hui, Chen,
Dept. of Finance,
Chung Yuan Christian University, Taiwan.
E-mail: [email protected]
Yu-Fang, Huang,
Fu Jen Catholic University, Taiwan.
E-mail: [email protected]
___________________________________________________________________________
Abstract
Metal and energy prices are crucial for economic growth and have significant impacts on the
financial performance of investment decisions. Some commercial exchange traded funds
(ETF) volatility indexes (VIX), such as the oil ETF volatility index, energy sector ETF
volatility index, gold ETF volatility index, silver ETF volatility index and gold miners ETF
volatility index, have been published by the Chicago Board of Options Exchange. The current
study applies smooth transition autoregressive (STAR(X)) models to examine whether ETF-
VIX have forecasting abilities. This work also investigates whether ETF-VIX can play the
same role as that of the market index for ETFs. The results indicate that the nonlinear STARX
models can predict the changes in ETF-VIX more accurately than the linear autoregressive
(AR) and multiple regression models. Specifically, the results in this study use commodity
ETF-VIX as transition variables for forecasting the changes and comparing them to the
precision transition variable to examine the return-implied volatility relation. The results
indicated a weak explanation for the return-volatility relation for commercial ETF but not for
the traditional price series in equity market.
___________________________________________________________________________
Key Words: ETF Volatility Index, ETF, STARX Model JEL Classification: G 1
Proceedings of the Fifth Asia-Pacific Conference on Global Business, Economics, Finance
and Social Sciences (AP16Mauritius Conference) ISBN - 978-1-943579-38-9
Ebene-Mauritius, 21-23 January, 2016. Paper ID: M620
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1. Introduction
Exchange traded funds (ETFs) have been gaining popularity among investors for the past
decade because of their unique features, such as lower costs, trading flexibility, tax efficiency,
and exposure to a variety of markets. The Investment Company Institute (ICI) reported that at
the end of November 2013, approximately 8,097 ETFs have been in existence, all of which
have a combined market capitalization of close to $8.7 trillion worldwide. ETF assets have
increased at remarkable speed in recent years, making them important financial vehicles.
ETFs have also become investment vehicles offering investors with broad equity market
indexes, industry sectors, and other asset classes.
Some financial institutes1 create and manage all kinds of ETFs to be listed and traded on
the American Stock Exchange (AMEX). Hence, some studies have focused on the returns and
risk management of ETFs. For instance, Adjei (2009) explored 150 ETFs listed in the US
equity market as well as the degree of diversification and the persistence performance of
ETFs, reporting that few ETFs have not diversified well. Moreover, the performance of ETFs
is the same as that of the market, although ETFs have higher reward-to-risk ratios than the
market. However, a significant limitation of previous studies is that applying past data cannot
predict future performance. Horst, Nijman, and Verbeek (2001) investigated the performance
persistence of mutual funds and found that historical price cannot effectively predict future
price. Cheerfully, the Chicago Board Options Exchange (CBOE) has introduced new implied
volatility indexes in ETFs that reflect expected volatility for options on ETFs.
Some studies on the VIX have been published, but none on ETF-VIX. Sarwar (2011)
indicated that the VIX serves as a powerful predictive indicator of the developed derivative
markets. Sari et al. (2011) investigated the relationships among oil, silver, gold, euro as well
as the VIX and employed Vector Autoregressive (VAR) model. They also demonstrated that
the VIX, oil, gold and silver seem to appear as the long run driving variables of oil prices.
This implies when a common stochastic shock impacts on the finicial market, the VIX will
change first comparing with other finicial products. Furthermore, Jubinski and Lipton (2013)
examined how the VIX influences the series of gold, silver, and oil futures returns. They
found that gold and silver have a positive relationship with the VIX and that oil has a negative
relationship with the VIX.
The study focuses on the new implied volatility indexes of ETF introduced recently by the
CBOE, namely, the investigation of the switching behavior among oil ETF volatility index
(OVX), energy sector ETF volatility index (VXXLE), gold ETF volatility index (GVZ), silver
ETF volatility index (VXSLV), and gold miners ETF volatility index (VXGDX). Since
energy and precious metals are the important natural resources and are key factors of
1 The financial institutes include SPDRs, WEBs, iShares, Street TRACKs, and other entities.
Proceedings of the Fifth Asia-Pacific Conference on Global Business, Economics, Finance
and Social Sciences (AP16Mauritius Conference) ISBN - 978-1-943579-38-9
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economic growth. Their prices had increased in recent years, so that those who demand for
resources can be hedging or investing. Therefore, the motivation of the present paper based
on the correlation between VIX and stock index to test whether the ETF-VIX and ETF can
provide some information.
The study employs smooth transition autoregressive (STAR) models to capture the trend
in ETF prices. To the best of our knowledge, the present study is the first to examine the
differences among the ETF volatility indexes of oil, energy, silver, and gold. The current
study aims to forecast ETF-VIX prices by using the non-linear STARX model, as proposed
by McMillan (2001). The main advantage of using the STAR model is that it allows for a
more general transition function, so the transition processes between switches are smooth.
The study attempts to depict the relation between ETF-VIX and ETF and detect the ETF-VIX
changes on financial and economic variables.
2. Literature Review
ETFs generally track an investment product or a particular index. ETF prices change along
with the supply and demand of investment and are transparent throughout the trading day.
Since the generation of the S&P 500 Depositary-Receipts (SPDR; symbol SPY) in 1993 by
the American Stock Exchange, these have been used widely since 1993. The number of
academic studies on ETFs reflects the popularity of ETFs in investment leads. Most studies
examined ETFs in relation to premiums and discounts, pricing effectiveness, and relationships
of ETFs with related derivative markets. For example, Elton et al. (2002) found that the
SPDR underperforms both its underlying index and traditional S&P 500 index funds, proving
that even the net asset value (NAV) of first generation fully duplicate ETFs and track the
index very closely. Furthermore, some papers have discussed price efficiency.
In addition, several authors have called into question the relationships among ETFs and
related derivative markets. The derivative-based ETFs have been divided into two types,
namely, leveraged and inversed ETFs. Cheng and Madhavan (2009) reported on the impact of
daily leveraged ETF (LETF) rebalancing on the underlying market volatility through the
constituent stocks toward the closing of each trading day. They also derived the relation
between an LETF return and an underlying index return. In this context, Vaneesha et al.
(2013) reviewed previous studies to examine the influence of LETF and inverse leveraged
ETF (ILETF) trading activities on the intraday returns of real estate investment trusts (REITs)
and the Dow Jones Index. They found that statistically significant results of LETFs and
ILETFs influenced the price changes of REITs more than they influenced the Dow Jones
Index (Vaneesha et al., 2013).
Another significant issue is whether investor sentiment influences financial asset prices,
especially after the dramatic rise and fall of the stock markets (Chau et al., 2011). In that
Proceedings of the Fifth Asia-Pacific Conference on Global Business, Economics, Finance
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study, the authors verified the viewpoint that feedback trading activity is highly motived by
the appearance of sentiment-driven noise trading. Meanwhile, many studies have explored
this area using different methodologies and data sets, offering various conclusions on these.
The VIX serves as an alarm for the market to gauge fears. When the stock market is expected
to rise, investors are expected to sell the S&P 500 options for portfolio insurance (Whaley,
2009). Sarwar (2012) analyzed the connection between the VIX and returns of the S&P 100,
500, and 600 indexes from 1992–2011, proving that significantly negative synchronic
relations exist among daily innovations in the VIX and S&P 100, 500, and 600 returns. In
particular, the relationship is more significant when the VIX is both high and volatile, thus
confirming that the VIX is more of a measure of investor fear and portfolio insurance stock
price than of positive investor sentiment. Another, Qadan and Yagil (2012) employed the
Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) model to examine the
causal relationship between the VIX and the price of gold futures and confirmed that changes
in the VIX directly induced the returns of gold futures. They also proved that the message
flow among the gold market and the investors’ fears influence not only the mean movements
but also the volatility. Furthermore, they suggested that the VIX, as a measure of investor
sentiment or risk aversion, could be used to perform the cross-market uncertainty linkages
and to uncover more information than the historical price series (Qadan and Yagil, 2012).
In recent years, interest in the performance of ETFs on oil, energy, gold, silver, and gold
miners have grown. Chen and Lin (2004) investigated commercial product prices by
employing linear and nonlinear Granger causality approaches, and detected the price change
among the quarterly inventory, UK Treasury bill interest rates, futures prices, and spot prices.
They found that spot prices are affected by inventory and interest rates on futures prices.
Ciner (2011) applied Granger causality model to examine the relations among commodity
prices and economic and financial variables such as industrial production, federal funds rate,
stock prices, and dollar exchange value. They confirmed that commodity prices have
significant relations with inflation, industrial production, interest rate, S&P 500, and exchange
rate.
Focusing on the commodities and foreign currency markets, Sari et al. (2011) used the
VIX as an indicator of global risk to estimate the relationship among oil, silver, gold, and the
dollar/euro exchange rate. They discovered that the VIX has an impact on returns in the metal
and oil markets. Daigler et al. (2014) investigated the effects between equity implied volatility
indexes2 and the euro‐currency ETF (FXE). These researchers also examined the effects
among equity implied volatility indexes (VIX), exchange rate volatility index (EVZ), and the
euro‐currency exchange‐traded fund (FXE), thus confirming that returns and implied
2 There are two indexes, namely the EVZ (euro) and VIX (S&P500).
Proceedings of the Fifth Asia-Pacific Conference on Global Business, Economics, Finance
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volatility exist for the euro and that the volatility of the US and Europe stock markets
influences volatility in the USD/euro market.
To address these concerns, some financial innovation productions have been published by
the CBOE, particularly relative VIXxes. On May 2007, the reflection of expected volatility
for options on ETFs was introduced by the CBOE. Badshah, Bart, and Alireza (2013)
employed a structural VAR (SVAR) approach to investigate the spillover effects among the
implied VIX for VIX, GVZ (gold ETF volitility index), and EVZ (euro/dollar exchange rate
volatility index). They found that the VIX increases the spillover effects of the GVZ and
EVZ, whereas the GVZ and EVZ produce bi-directional spillover effects. Liu et al. (2013)
applied unit root and Granger causality test to measure the nexus with OVX, VIX, EVZ and
GVZ, and found that the VIX strongly causes OVX (oil ETF volatility index). This finding
implies that investors interested in trading OVX futures can readjust investment strategies
based on the changes in the VIX.
Various approaches have been proposed to solve the ETF price change. One study applied
Vector Error Correction Model (VECM) to estimate ETFs and the US equity market
(Hasbrouck, 2003), whereas some studies applied GARCH method to evaluate the relation
among the VIX and ETFs (Qadan and Yagil, 2012). McMillan (2001) used the STAR model
to estimate the variation of stock market returns, eventually proving that the model
outperformed the linear model in both in-sample and out-of-sample forecasts. The current
study contributes to the literature by applying the new implied volatility indexes recently
introduced by the CBOE and by employing the STAR model to examine the asymmetric
effects of ETF price trends. One of the most prominent regime-switching models is the
smooth transition regression (STR) model introduced by Teräsvirta and Anderson (1992) and
Granger and Teräsvirta (1993) in the field of macro-econometrics. Teräsvirta and Anderson
(1992) used the STAR model to capture the different dynamics of business cycles in OECD
industrial output series, whereas Hsu and Chiang (2011) examined the relation among the
Federal funds rate, industrial production, and stock returns using STAR models, and found
that (1) monetary policy and excess returns on stock prices are positive and nonlinear, and (2)
policy has a larger impact on stock returns in a bear-market (high) regime than in a typical
regime.
3. Data and Methodology
The current study applied ETF-VIX published by the CBOE from 16 March 2011 to 31
December 2013, with the sample period of 1 January 2014 to 31 March 2014. The sample
ETF data came from Yahoo Finance, and the exogenous variables data came from Yahoo
Finance and Bloomberg. ETF-VIX and ETFs included oil, energy, gold, silver, and gold
industries, as shown in Table 1.
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A variety of financial and macroeconomic series were employed, namely, West Texas
crude oil-spot price, gold spot price (KITCO), silver spot price (KITCO), London inter-bank
offered rate (one-month) and US dollar index (USDX) (Jubinski and Lipton, 2013) and
Commodity Research Bureau Futures Price Index (CRB) (Chen et al., 2013), and were
assumed to influence ETF volatility indexes returns as shown in Table 2. As mentioned earlier
literature, US dollar index might also co-drive both oil and the precious metals
contemporaneously because they are denominated in the dollar currency. Furthermore, CRB
index had informational content on predicting the direction of inflation. Interest rates helped
to predict inflation and expected the future path of monetary policy. The present study applied
ETF-VIX as a transition variable.
3.1 The Non-Linear STAR(X) Model
The STARX model, based on the smooth transition regression (STR) and autoregressive
smooth transition regression (STAR) models, has been previously estimated (Teräsvirta and
Anderson,1992; Teräsvirta, 1994). The present study used the STAR and STARX models
proposed by Teräsvirta and Anderson (1992) and McMillan (2001), respectively. The STARX
model uses exogenous variables as explanatory variables, and has an autoregressive transition
variable.
3.1.1 The STAR Model
Consider the following two-regime STAR model of order p:
tdtttt cyFwwy ,;220110 , (1)
where dty is the transition variable, d is the number of periods and cyF dt ,; is the
transition function bounded by zero and unity. In the expression given by ),0( 2 Nt ,
,,...,1
jpjj j=1,2, ,,...,1
pttt yyw c is the threshold and provides the location
of the transition function, whereas defines the speed of the transition. Equation (1) changes
with values of the transition variable and is composed of two different transition functions,
namely, the LSTAR and the ESTAR. The logistic function is given by
1 exp1,;
cycyF dtdt , .0 (2)
The logistic function is monotonically increasing in ,dty as
cy dt with 0,; cyF dt , or ccyF dt ,; , and the model is situated at
upper regime; and as cy dt with 1,; cyF dt , or ccyF dt ,; , in
which the model is situated at lower regime. When cyF dt ,;, changes into:
,0,; cyF dt cy dt , 1,; cyF dt , cy dt , Equation (1) associated with
Proceedings of the Fifth Asia-Pacific Conference on Global Business, Economics, Finance
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Equation (2) becomes a TAR (p) model. On the contrary, when ,0 Equation (1) becomes
a linear AR(p) model. The LSTAR model is an asymmetric realization. Figure 1 shows that
the transition function increases monotonically from zero to unity with yt-d.
The exponential function is given by
2exp1,; cycyF dtdt , .0 (3)
When 1,;, cyF dt , Equation (1) becomes linear and 0 . Based on the
boundary, one regime obtains a probability of one, whereas the other shows a probability of
zero. Hence, c is the local parameter (threshold value) and the transition function is
symmetric around c. The local dynamics are the same for both high and low values of dty ,
but the mid-range behavior is different. Figure 2 shows that the minimum value of the
transition function equals zero with yt-d.
The STAR model used in this study consisted of four stages. The first stage involved the
specification of a linear AR model, which was essential in estimating the lag length of the
autoregressive process. The Akaike information criteria (AIC) was used as the criterion for
selecting lag length (i.e., lag length with the minimum AIC). The second step involved the
selection of possible candidates for the transition variable and testing for the appropriateness
of linearity. Following Teräsvirta and Anderson (1992), the current study employed an
auxiliary equation to test a linear AR model against a nonlinear STAR model. The auxiliary
equation can be expressed as:
.1
3
4
1
2
3
1
210
p
j
tdtjtj
p
j
dtjtj
p
j
dtjtjtt yyyyyywy (4)
The hypothesis to be tested is .0: 4320 H The third step involved the
selection of the appropriate type of smooth transition model, which was either an LSTAR or
an ESTAR model. Following Teräsvirta and Anderson (1992), the present study tested a
sequence of null hypothesis, as follows:
.,...,1,0: 404 pjH j (5)
.,...,1,00: 4303 pjH jj (6)
.,...,1,00: 43202 pjH jjj (7)
If H04 is rejected, then the tested model fits the LSTAR model. If H04 is accepted but H03 is
rejected, the model fits the ESTAR model. However, if both H04 and H03 are accepted but H02
is rejected, the model fits the LSTAR model. Finally, to estimate the STAR model, the
threshold value c and transition variable were determined using a two-dimensional grid
search over data points of the transition variable y and different values of , in which the
Proceedings of the Fifth Asia-Pacific Conference on Global Business, Economics, Finance
and Social Sciences (AP16Mauritius Conference) ISBN - 978-1-943579-38-9
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minimum sum of squared errors served as the optimal estimates.
3.2. The STARX Model
The STARX model can be written as:
tdt
p
i
tii
p
i
tit cZFxxy
,;1
1,
''
0
1
10 , (8)
where cyF dt ,; is the transition function bounded by zero and unity. As described above,
the STARX model was divided into two transition functions. The logistic STARX (LSTARX)
model is given by
1 exp1
cxxF dtdt .0 (9)
The present study permits a smooth transition among the different dynamics of positive and
negative returns, where d is the lag parameter, is the smoothing parameter, and c is the
transition parameter. The LSTARX function enables the parameter to alter monotonically with
dtx . As dtxF , becomes a Heaviside function,
,,1,,0 cxxFcxxF dtdtdtdt the STARX model is reduced to a TARX (p)
model, and when ,0 the model becomes a linear model of order p.
The exponential STARX (ESTARX) model is given by
2 exp1 cxxF dtdt ; .0 (10)
The ESTARX function permits the parameter to change c symmetrically with dtx . If
or 0 , the ESTARX model becomes a linear model. The dynamics of the middle
ground differ from those of the larger returns. The ESTAR model distinguishes among those
resulting from larger and smaller trades, and can also capture the effects of transition costs on
trader behavior or market depth.
Next, the estimation of STAR models, particularly the smoothing of parameter , may
have a problem without converging very slowly. A solution suggested by Terasvirta and
Anderson (1992) and Terasvirta (1994) is to scale the smoothing parameter using the
standard deviation of the transition variable, and analogously in the ESTAR model, to scale
using the variance of the transition variable using Equations (11) and (12), respectively, as
shown below:
1/ exp1
dtdtdt xcxxF , (11)
dtdtdt xcxxF 22/ exp1 (12)
3.3 Empirical Model
The STAR and STARX models were employed in the current study, as shown below:
Proceedings of the Fifth Asia-Pacific Conference on Global Business, Economics, Finance
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,,;)()()(1
''
0
1
0 tdtjt
p
j
jjt
p
j
jt cZFETFsVIXETFsVIXETFsVIX
(13)
iti
p
i
p
i
itiiti
p
i
itit CRBINTUSDPETFsVIX 4
1 1
32
1
10)(
tdt
p
i
p
i
itiiti
p
i
itiit
p
i
i cZFCRBINTUSDP
,;1 1
'
4
'
3
1
'
2
1
'
1
'
0 (14)
where VIX(ETFs) is ETF-VIX, P is relative spot price (i.e., oil, gold, and silver), USD is the
exogenous variables of US dollar index, INT is London inter-bank offered rate (one-month),
CRB is CRB index, and dtZ is the lag of ETFs-VIX. The present study further examined the
forecasting of ETFs, using ETFs and ETF-VIX as transition variables, and compared their
performance, as shown in Equations (15) and (16) below:
tdtti
p
i
iiti
p
i
it cETFsFxxETFs
,;)( 1,
1
''
0,
1
0 , (15)
tdt
p
i
tii
p
i
tiit cETFsVIXFxxETFs
,;)(1
1,
''
0
1
1,0 . (16)
3.4 Sample Forecast Comparisons
The added value of the nonlinear features of the models was determined, in order to
compare the forecasts from the STAR and STARX models with those from the benchmark
linear models, The forecasts were then evaluated based on three criteria, namely, the root
mean square error (RMSE), mean absolute error (MAE) values, and Theil’s inequality
coefficient (Theil’ U).
4. Results and Discussion
The present study applied STARX models to examine ETF-VIX and ETF and therefore
selected the best forecasting model.
4.1 Data Description
The basic statistics shows that the OVX, VXXLE, GVZ, VXSLV, LIBOR, and CRB
indexes exhibit leptokurtic distribution, whereas VXGDX, OIL, GOLD, SILVER, USDX,
and the five ETFs display platykurtic distribution. All variables do not show normal
distribution. The descriptive statistics of variables are displayed in Tables A1 and A2.
In order to avoid the problem of spurious regression, it is important to test the procdure of
unit roots. This study used the Dickey-Fuller test to examine whether there is the presence of
stationary series. The results show that the unit root test fails to reject the null of the unit root
for USDX, LIBOR, CRB, XLE, GLD, and GDX, suggesting that time-series variables are
non-stationary. In addition, the tests of first difference reject the null hypothesis; hence, the
time-series variables appear stationary after the first differencing. Evidence shows that as with
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the cointegration time series, the time series integrated of order one may have stationary
linear combinations without differences (Engle and Granger, 1987). The results show one
cointegrating vector, at most, with the use of the trace statistic. This study confirmed that all
variables could achieve the stationarity condition after the first differencing and they also
could display Johansen effect.3
4.2 Results of Linearity Estimation
Table 3 depicts the parameter estimation results of the AR model using the AIC minimum
standard. All coefficients of the ETF-VIX lag periods are significantly different from zero.
The Q test indicates that the AR model of ETF-VIX accepts the null hypothesis for residual
non-autocorrelation, whereas the ARCH test indicates that ETF-VIX rejects the null
hypothesis for the residual non-heterogeneous variation and that normal distribution is not
met.
For the multiple regression models, the current study first screens the optimal economic
variables for the ETF-VIX of all samples through stepwise regression. Then, the significant
lag period of the macroeconomic variable with a p-value is identified using the estimation
linear model proposed by McMillan (2001). The purpose is to determine the optimal multiple
regression model of the ETF-VIX on the basis of the AIC minimum standard. The optimal lag
period of the economic variable is also shown in Table 4. The coefficients in the ETF-VIX lag
periods are significantly different from zero. However, the adjusted R2 of GVZ was only
0.1798 reflecting that the optimal lagged economic variables does not contain CRB index,
since the gold price fluctuations and inflation rates are the same. The Q test results indicate
that all ETF-VIXs reject the null hypothesis for residual non-autocorrelation. The ARCH test
results also show that ETF-VIXs reject the null hypothesis for residual non-heterogeneous
variation. The JB test results reveal that all ETF-VIX residuals estimated by the multiple
regression models do not satisfy the normal distribution.
4.3 Results of Linearity Test and Determination of Lag Order for Transition Variable
The first phase in specifying the STAR model involves selecting the linear AR model4.
Similar to the approach of McMillian (2001), an AR (p) was considered in selecting the
optimum number of lags using the maximum F-statistic or the minimum p-value (Granger
and Teräsvirta, 1993). Panel A of Table 5 shows that the linearity hypothesis is rejected, with
the results of the STAR model indicating the following: d=1 for OVX, d=6 for VXXLE, d=2
for GVZ, d=4 for VXGDX, and d=6 for VXSLV. In addition, panel B of Table 5 shows that
the result of the STARX model exhibits d=1 for all ETF-VIXs. These findings imply that
3 The result is available from the authors upon request. 4 The term d is the lagged period of the transition variable. The values in the table and those in parentheses are the F-statistics and p-values, respectively.
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linearity is rejected in favor of STAR(X) nonlinearity. Hence, ETF-VIX was selected as the
transition variable in this study.
4.4 Results of Parameter Constancy Test and Determining the Type of Models
The findings of the initial procedure identified the appropriate STAR(X) models. This
section identifies a set of specification tests to determine between LSTAR(X) and ESTAR(X).
Table 6 indicates that the appropriate forms of the transition function are ESTAR and
LSTARX. Following Sarantis (1999), 03H has the largest F-value in panel A of Table 6, thus
fitting the ESTAR for all ETF-VIX. In panel B, 02H has the largest F-value, thus fitting the
LSTAR.
4.5 Results of Parameter Estimate
After confirming the transition variables and transition functions of the non-linear models,
the nonlinear least squares (NLS) method was used to estimate the STAR and STARX
models. Teräsvirta (1994) reported that the parameters in a test model must be tested after
they are estimated. For example, the threshold coefficient (c) must be significantly different
from zero. Threshold (c) is a structural transition point of time series data and should be
estimated within the range of the sample observation value, that is, the transition speed should
be more than zero (γ > 0). The threshold (c) and the transition speed (γ) of the model are
repeatedly tested in different starting values, and the reasonability of the threshold estimation
should be compared. Model convergence and the maximum likelihood value serve as the
bases in selecting the STAR and STARX models.
Table 7 shows the estimation results on the convergence of ETF-VIX with the STAR
model. The threshold values of ETF-VIX are significantly different from zero in terms of
transition speed γ. Except for VXGDX and VXSLV; the remaining ETF-VIXs represent a
significant case5. A large γ indicates a sharp transition between different conditions, whereas
a small γ indicates a smooth transition. For the ESTAR model, the γ coefficient of VXXLE is
the largest, and that of OVX is the smallest in Figure 3. The residual test results indicate that
the residuals of ETF-VIX meet the non-autocorrelation hypothesis. With the exception of
those of GVZ, the other residuals of ETF-VIX reject the null hypothesis for homogeneous
variation; hence, all residuals of ETF-VIX do not meet the normal distribution.
Table 8 shows the parameter estimation results of the sample ETF-VIX of the STARX
model. As can be seen, the threshold estimation results of the ETF-VIXs are significantly
different from zero. In terms of transition speed, the ETF-VIXs of all samples reject the null
hypothesis with zero estimation parameters. Therefore, LSTARX models show a slower
transition speed than the STARX model. Specifically, the fastest one is OVX, followed by
5 Lin and Teräsvirta (1994) emphasized that the non-significant conversion speed can be due to the
different conversion speed levels that do not exhibit nonlinear changes in the same conversion function.
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VXXLE. The transition conditions of all ETF-VIXs between different states are shown in
Figure 4. The residual test indicates that except for the residuals of VXGDX and VXSLV, the
rest of the residuals meet non-autocorrelation. The ETF-VIX residuals of all samples meet the
homogeneous variation hypothesis but not the normal distribution. Moreover, the adjusted R2
of ETF-VIX using the STAR or STARX model shows an upward tendency, indicating the
acceptable predictive capacity of ETF-VIX.
4.6 Results of ETF Testing
The present study also examined ETF and assessed whether using ETF as a transition
variable under the STARX model was better than using ETF-VIX.
4.6.1 Results of Linearity Test
As in the previous procedure, panel A of Table 9, in which ETF is the transition variable,
shows that the linearity hypothesis is rejected. The result of the STARX model indicates that
d=1 for ETFs. Therefore, panel B of Table 9, in which ETF-VIX is the transition variable,
shows the results of the STARX model: d=1 for USO, GDX, and SLV; d=3 for XLE; and
d=2 for GLD. These findings indicate that linearity is rejected in favor of STARX
nonlinearity based on the selected transition variable.
4.6.2 Results of Parameter Constancy Test and Determining the Type of Models
The initial procedure aimed to identify the suitable STARX. Therefore, the next procedure
aimed to identify a set of specification tests for the determination between LSTARX and
ESTARX. Table 10 indicates that the appropriate forms of the transition functions are
ESTARX and LSTARX. In this context, the maximum F-statistic was used to select a model.
(Sarantis, 1999)
4.6.3 Results of Parameter Estimate
After confirming the transition function as shown in Table 11, the NLS approach was used
to estimate the STARX model. The judgment standard and process of this approach were
based on Teräsvirta’s (1994) study, in which the transition speed is more than zero (γ > 0).
The threshold coefficient (c) must be significantly different from zero, and the value should
be in the range of the observation values. Table 11 also shows the estimation results under the
STARX model using ETF as the transition variable. As can be seen, all ETF threshold values
are significantly different from zero. In terms of transition speed γ, all ETFs also represent a
significant case. A large γ indicates a sharp transition between different conditions, whereas a
small γ indicates a smooth transition. Therefore, the transition speed is smooth under the
LSTAR model. The residual test result indicates that except for the residual of XLE, the
residuals of ETF meet the non-autocorrelation hypothesis, and all residuals of ETF accept the
null hypothesis for non-heterogeneous variation. However, all residuals of ETF do not meet
the normal distribution.
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Table 12 shows the parameter estimation results of ETF under the STARX model using
ETF-VIX as the transition variable. The threshold estimation results of ETF are significantly
different from zero. In terms of transition speed, all ETFs also reject the null hypothesis with
an estimation parameter of zero. Meanwhile, most of the STARX models show a slow
transition speed. The slowest transition speed is that of USO, followed by XLE and GLD. On
the contrary, most of the LSTARX models show a fast transition speed. The fastest transition
speed is that of GDX, followed by SLV. The residual test results indicate that the residuals of
all sample ETFs reject non-autocorrelation. In addition, the residuals of USU, GLD, and SLV
meet the non-heterogeneous hypothesis, although heterogeneous variables exist between XLE
and GDX exist. All ETFs do not meet the normal distribution.
4.7 Results of Out-of-Sample Forecast Evaluation
The preceding tests demonstrate that the residual test and the adjusted R2 of the STAR and
STARX models can provide better adaptability than the linear model for intra-sample
estimation. However, this result does not mean that the predictive model of ETF-VIX is also
optimal for extra-sample estimation. Generally, extra-sample predictive capacity does not
only inspect the predictive capacity of a model, but also assists investors or fund managers in
analyzing investment decisions to reduce uncertainty and risk in the process. Therefore, the
current study used three different predictive indicators, namely, RMSE, MAE and Theil’U, as
the standards to evaluate the predictive capacity of the model. The extra-sample predictive
period was from January 1, 2014 to March 31, 2014. Here, a small indicating value is taken to
mean that the predictive capacity of the model is good.
Table 13 shows the poor extra-sample predictive results of all models. As can be seen
from the RMSE and Theil'U performance indexes, the LSTARX model is suitable for OVX
and VXXLE, the linear AR model can apply for GVZ and VXGDX, and the ESTAR model
is appropriate for VXSLV. However, upon using the MAE performance index to measure the
inference in OVX, the ESTAR model is found to be most suitable. The current study shows
that the nonlinear STAR and STARX models can provide good ETF-VIX adaptability, and
that these two models are better than the linear model in evaluating extra-sample ETF-VIX.
This conclusion is consistent with that of Dacco and Satchell (1999). In particular, a status
transition model can increase the intra-sample adaptability of a linear model, but a better
extra-sample predictive capacity than that of the linear model cannot be guaranteed.
The present research further employed the STARX model to predict ETF tendency by
using ETF and ETF-VIX as transition variables. Table 14 shows that if ETF is used as the
transition variable, a good extra-sample predictive capacity can be achieved. This conclusion
is consistent with that of Chaiyuth and Daigler (2013) who reported that correlation between
commodity ETF-VIX and ETF is very weak.
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5. Conclusions and Recommendations
VIX is an important indicator in evaluating the volatility of equity. Investors first estimate
the fluctuation of investment targets, and then calculate the value of equities using proper
models. This process plays an important role in risk management. Therefore, a correct
variation of equity estimation for warrant evaluation is critical. Existing studies have mostly
investigated the relationship between stock markets and VIX, whereas the relationship
between ETF and ETF-VIX has been rarely examined. Many previous studies have used the
linear model to predict ETF or its GARCH effects. However, the ETFs of commodities,
including petroleum, gold and silver, which are important international commodities, affect
hedge and international policies. When these commodities are affected by the policies of
various countries and by systematic risk, the ETF change in these commodities may be
converted smoothly. In this context, the nonlinear model may be suitable in describing the
ETF change. Therefore, the present study predicts ETF-VIX using linear AR, linear multiple
regression models, and nonlinear STAR and STARX models to evaluate ETF as well as
determine the optimal predictive model.
The linear test results using the AR model and the multiple regression models, which are
used to estimate ETF-VIX, show that all ETF-VIX have the form of STAR and STARX
models, namely, changes in ETF-VIX indicating nonlinear routine and symmetric or
asymmetric linear conversion situations. For intra-sample estimation, the residual test and R2
of the STAR and STARX models can provide better adaptability than the linear AR and
multiple regression models. These results indicate the nonlinear relationship between ETF-
VIX and previous ETF-VIX or economic variables, such as oil price, gold price, silver price,
London inter-bank offered rate, US dollar index, and CRB index. Therefore, the nonlinear
STAR or STARX model can describe the changes in ETF-VIX more efficiently than the
linear AR and multiple regression models. However, the extra-sample predictive results
depict that the STAR(X) model is suitable in predicting 60% of the samples.
There are three contributes in the study of ETFs-VIX. First, ETFs-VIX does not fully
respond to ETFs price changes, unlike the VIX and stock indices. Next, the present study
demonstrated the effectiveness of predictability of a nonlinear STARX model for ETF-VIXs.
Since the relationship between return and implied volatility is nonlinear and asymmetric
(Low, 2004). Finally, the work extend limited previous research on the explanation of the
return-volatility relation to investigate the relation of implied volatilities between ETFs-VIX
and ETFs. Therefore, the results also present an important discovery, that is, using ETF as a
transition variable can better predict changes in ETF than ETF-VIX.
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Table 1: ETF volatility index and ETFs
Panel A: ETF volatility index
Symbol Meaning Underlying Start Date
OVX Oil ETF Volatility Index United States Oil Fund (USO) 2007.5.10
VXXLE Energy Sector ETF Volatility Index SPDR XLE ETF (XLE) 2011.3.16
GVZ Gold ETF Volatility Index SPDR Gold Shares (GLD) 2008.6.3
VXSLV Silver ETF Volatility Index iShares Silver Trust (SLV) 2011.3.16
VXGDX Gold Miners ETF Volatility Index NYSE Arca Gold Miners Index (GDX) 2011.3.16
Panel B: ETFs
USO United States Oil Fund WTI6 light 2006.4.10
XLE Energy Select Sector SPDR Fund S&P Energy Select Sector Index 1998.12.16
GLD SPDR Gold Shares Gold Price 2004.11.18
SLV iShares Silver Trust London Silver Fix Price 2011.6.13
GDX NYSE Arca Gold Miners Index Gold Price 2006.5.16
Source: Yahoo Finance.
Table 2: Exogenous Variables
ETFs-
VIX OVX VXXLE GVZ VXGDX VXSLV
Variable USO XLE GLD GDX SLV
Index WTI oil price WTI oil price Gold spot price Gold spot price Silver spot price
Exchange
Rate US Dollar Index (USDX)
Interest
rate 1-Month London Interbank Offered Rate (LIBOR), based on U.S. Dollar
CRB
Index Commodity Research Bureau Futures Price Index (CRB)
Source: Yahoo finance and Bloomberg.
Table 3: Parameter Estimates of ETF-VIX - AR model
ETF-VIX OVX VXXLE GVZ VXGDX VXSLV
0 0.4729 (0.1230) 0.3110
(0.1256)
0.6301**
(0.0040)
1.0541**
(0.0112)
1.0047***
(0.0034)
1 0.7972***
(0.0000)
0.9388***
(0.0000)
0.9066***
(0.0000)
0.7938***
(0.0000)
0.9732***
(0.0000)
2 0.0903**
(0.0246) - -
0.1130***
(0.0051) -
6 West Texas Intermediate.
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3 - -0.08738*
(0.0517) - - -
4 - 0.1173**
(0.0211)
0.0631**
(0.0067) - -
5 - -0.1467***
(0.0039) - - -
6 0.0959***
(0.0000)
0.1645***
(0.0000) -
0.0653***
(0.0034) -
Q-statistic 4.2925 (0.637) 1.0855
(0.982) 9.2607 (0.159) 6.0714 (0.415)
1.3931
(0.966)
ARCH 36.5331***
(0.0000)
34.2880***
(0.0000)
8.5787***
(0.0000)
14.7425***
(0.0000)
12.7299***
(0.0000)
JB 56865.89***(0.00
00)
2491.92***
(0.0000)
9062.79***
(0.0000)
1861.23***
(0.0000)
18619.60***
(0.0000) 2R 0.9409 0.9592 0.9295 0.9217 0.9476
Notes: P-values are in parentheses. The figures are the standard regression coefficients. * significant at
10% level; **significant at 5% level; ***significant at 1% level.
Table 4: Parameter Estimates of ETF-VIX --- Multiple Regression model
ETF-VIX OVX VXXLE GVZ VXGDX VXSLV
0 143.7732***
(0.0000)
209.0893***
(0.0000)
147.5482***
(0.0000)
225.2161***
(0.0000)
232.9209***
(0.0000)
11 -0.4155***
(0.0000)
-0.4010
(0.0000)
-0.0120***
(0.0000)
-0.0236***
(0.0000)
-0.5315***
(0.0000)
26 -1.4898***
(0.0000)
-1.9626***
(0.0000)
-1.1201***
(0.0000)
-1.2568***
(0.0000)
-2.5866***
(0.0000)
31 74.6569***
(0.0000)
65.0385***
(0.0006)
67.9127**
(0.0017)
70.2684***
(0.0000) -
32 70.03912***
(0.0000)
47.0191**
(0.0134)
43.1632**
(0.0487) - -
36 - - -50.1924**
(0.0161)
-45.3817*
(0.0523) -
41 - -0.0504***
(0.0000)
-0.0649***
(0.0000)
-0.0500**
(0.0285) -
42 - - - -0.0300*
(0.0523) -
43 - - - -0.0339**
(0.0285) -
46 0.0292***
(0.0007)
0.0220*
(0.0728) - -
0.0544***
(0.0023)
Q-statistic 2187.90***
(0.0000)
2702.50***
(0.0000)
2911.10***
(0.0000)
2748.00***
(0.0000)
3126.50***
(0.0000)
ARCH 23.51504***
(0.0000)
223.0693***
(0.0000)
414.8907***
(0.0000)
294.8404***
(0.0000)
574.8837***
(0.0000)
JB 442.63***
(0.0000)
103.86***
(0.0000)
180.96***
(0.0000)
75.27***
(0.0000)
307.80***
(0.0000)
2R 0.7396 0.6590 0.1798 0.3500 0.3649
Notes: p-values are in parentheses. The figures are the standard regression coefficients. * significant at
10% level; **significant at 5% level; ***significant at 1% level.
Table 5: Linearity tests and determination of lag order for transition variable
Panel A ETF-VIX---AR
lag OVX VXXLE GVZ VXGDX VXSLV
1 16.4772 # 2.4801 0.3559 1.4348 0.5728
(0.0000) (0.0015) (0.9066) (0.1691) (0.6330)
2 4.3648 1.6412 2.2118# 1.5003 0.8372
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(0.0000) (0.0584) (0.0402) (0.1437) (0.4737)
3 2.8504 2.2152 1.6063 1.2414 1.8746
(0.0000) (0.0051) (0.1426) (0.2663) (0.1325)
4 4.7886 2.4289 1.2727 1.8174# 2.8523
(0.0000) (0.0019) (0.2676) (0.0619) (0.0366)
5 5.6012 2.3954 0.7374 1.3406 3.1307
(0.0000) (0.0022) (0.6196) (0.2121) (0.0251)
6 4.9469 2.9601# 0.7673 0.9836 3.3799#
(0.0000) (0.0001) (0.5958) (0.4521) (0.0180)
Panel B ETF-VIX--- Multiple Regression
lag OVX VXXLE GVZ VXGDX VXSLV
1 198.3198# 295.9103# 425.8525# 242.4701# 903.3877#
(0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
2 113.7245 164.7047 229.4508 146.4880 438.9559
(0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
3 80.6335 112.7283 162.6935 103.2420 291.7842
(0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
4 61.3179 97.6794 138.1974 80.3104 224.2976
(0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
5 52.5169 89.9131 119.3465 65.5407 181.3503
(0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
6 48.1092 79.5220 105.3909 58.8687 152.2321
(0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
Notes: # denotes the optimal lagged period with the maximum F-statistic (or minimum p-value) reject the linearity hypothesis.
Table 6: Tests of functional form---STAR(X)
Panel A: STAR
ETF-VIX d 0: 404 jH 0: 40303 jjH 0: 430202 jjjH Model
OVX 1 10.6015 (0.0000) 18.4775 (0.0000) 17.0277 (0.0000) ESTAR
VXXLE 6 4.5106 (0.0005) 5.1545 (0.0001) 1.3039 (0.2603) ESTAR
GVZ 2 1.3678 (0.2554) 3.2086 (0.0410) 2.0405 (0.1307) ESTAR
VXGDX 4 1.1078 (0.3452) 2.5895 (0.0519) 1.7408 (0.1573) ESTAR
VXSLV 6 0.4336 (0.5104) 9.5518 (0.0021) 0.1602 (0.6891) ESTAR
Panel B: STARX
ETF-VIX d 0: 404 jH 0: 40303 jjH 0: 430202 jjjH Model
OVX 1 9.8663 (0.0000) 17.5081 (0.0000) 474.8257 (0.0000) LSTARX
VXXLE 1 3.3630 (0.0028) 5.6414 (0.0000) 827.0156 (0.0000) LSTARX
GVZ 1 8.3551 (0.0000) 3.6365 (0.0015) 620.5323 (0.0000) LSTARX
VXGDX 1 3.6165 (0.0008) 1.4691 (0.1752) 699.8661 (0.0000) LSTARX
VXSLV 1 0.5492 (0.6488) 0.1943 (0.9003) 2724.2860 (0.0000) LSTARX
Note:d is the optimal lag length for the transition variable. p-values are in parentheses.
Table 7: Parameter Estimates with lagged ETF-VIX as the transition variable-STAR
ETF-VIX OVX VXXLE GVZ VXGDX VXSLV
0 1.3060
(0.5610)
-267.0241
(0.1637)
7.1940
(0.0350)
12.2534
(0.0743)
161.4047
(0.3699)
1 1.0669
(0.0000)
2.1382
(0.0000)
0.3670
(0.0289)
0.2486
(0.0501)
-2.9284
(0.5382)
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2 -0.1226
(0.0219) - -
0.3515
(0.0099) -
3 - -0.9738
(0.0670) - - -
4 - -0.0392
(0.9560)
0.2781
(0.0206) - -
5 - 2.1964
(0.0064) - - -
6 0.0344
(0.2346)
6.5692
(0.2795) -
0.1136
(0.2251) -
'
0 1974.1840
(0.9985)
267.4264
(0.1631)
-6.6238
(0.0528)
-11.2391
(0.1031)
-160.4564
(0.3727)
'
1 -454.1135
(0.9985)
-1.2520
(0.0000)
0.5702
(0.0007)
0.6420
(0.0000)
3.9027
(0.4121)
'
2 320.9948
(0.9985) - -
-0.3074
(0.0391) -
'
3 - 0.9294
(0.0824) - - -
'
4 - 0.1182
(0.8686)
-0.2433
(0.0490) - -
'
5 - -2.3013
(0.0044) - - -
'
6 0.6981
(0.9985)
-6.4038
(0.2918) -
-0.0762
(0.4392) -
0.0002
(0.9985)
643.1420
(0.2779)
30.9781
(0.1036)
30.3768**
(0.0474)
223.5330**
(0.0145)
c 42.9506***
(0.0000)
30.1508***
(0.0000)
19.8180***
(0.0000)
41.6823***
(0.0000)
36.8355***
(0.0000) 2R 0.9487 0.9631 0.9308 0.9248 0.9493
Q-statistic 10.8720
(0.1440)
6.9878
(0.3220)
8.0982
(0.1510)
4.8298
(0.5660)
2.1564
(0.9050)
ARCH 0.4421
(0.9961)
1.0758
(0.2830)
8.0954
(0.0000)
0.9759
(0.5631)
0.9273
(0.6318)
JB 46051.27
(0.0000)
1295.90
(0.0000)
8895.65
(0.0000)
1605.15
(0.0000)
20310.32
(0.0000)
Notes: p-values are in parentheses. The figures are the standard regression coefficients. * significant at
10% level; **significant at 5% level; ***significant at 1% level.
Table 8: Parameter Estimates with lagged ETF-VIX as the transition variable-STARX
ETF-VIX OVX VXXLE GVZ VXGDX VXSLV
0 -45.4106
(0.1009)
-20.6089
(0.3390)
-47.3107
(0.5122)
-13.0245
(0.8384)
155.1834
(0.9553)
11 0.2346
(0.0020)
0.0703
(0.1028)
-0.0101
(0.1142)
0.0030
(0.6849)
-8.6638
(0.9640)
26 0.3401
(0.1769)
0.0814
(0.6578)
0.6427
(0.3299)
0.1220
(0.7891)
-5.0488
(0.9638)
31 185.4738
(0.4301)
38.5335
(0.7489)
228.6764
(0.5650)
-195.9086
(0.3436) -
32 183.2387
(0.4472)
-25.7944
(0.8305)
-246.1395
(0.6096) - -
36 - - 39.3634
(0.8471)
188.5849
(0.3505) -
41 - 0.0618
(0.1522)
-0.0027
(0.9234)
-0.2047
(0.3459) -
42 - - - 0.2561
(0.3567) -
43 - - - -0.0556
(0.7922) -
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46 0.0014
(0.9356)
-0.0468
(0.2535) - -
0.2316
(0.9610)
'
0 154.6879
(0.0002)
130.6739
(0.0028)
134.0489
(0.2050)
94.0019
(0.3650)
-276.2683
(0.9634)
'
11 -0.4979
(0.0000)
-0.2041
(0.0426)
0.0163
(0.1252)
-0.0070
(0.5870)
18.4648
(0.9647)
'
26 -0.8374
(0.0441)
-0.4485
(0.2769)
-1.2569
(0.1932)
-0.2372
(0.7574)
10.8427
(0.9642)
'
31 -324.6975
(0.4437)
-93.7633
(0.7350)
-391.7049
(0.5588)
364.6466
(0.3009) -
'
32 345.7154
(0.4251)
70.6030
(0.7973)
424.9671
(0.5965) - -
'
36 - - -74.1021
(0.8287)
-356.3281
(0.3055) -
'
41 - -0.1227
(0.0711)
-3.52E-05
(0.9994)
0.3178
(0.3488) -
'
42 - - - -0.3792
(0.3673) -
'
43 - - - 0.0695
(0.8315) -
'
46 0.0065
(0.7896)
0.0865
(0.1460) - -
-0.4560
(0.9646)
0.7192
(0.0000)
0.6334
(0.0000)
0.4084
(0.0075)
0.3876
(0.0241)
0.0413
(0.9646)
c 29.4988***
(0.0000)
29.1394***
(0.0000)
16.4198***
(0.0065)
32.5037***
(0.0000)
68.4822**
(0.0418)
2R 0.9482 0.9599 0.8723 0.9193 0.9040
Q-statistic 7.4531
(0.2810)
0.3481
(0.8400)
7.7486
(0.2570)
19.5210
(0.0030)
173.42
(0.0000)
ARCH 0.8803
(0.5866)
0.8949
(0.7971)
0.8983
(0.7314)
1.1801
(0.1046)
1.1674
(0.1557)
JB 59947.98
(0.0000)
1871.10
(0.0000)
7856.27
(0.0000)
1752.04
(0.0000)
22462.72
(0.0000)
Notes: p-values are in parentheses. The figures are the standard regression coefficients. * significant at
10% level; **significant at 5% level; ***significant at 1% level.
Table 9: Linearity tests and determination of lag order for transition variable---STARX
Panel A: ETF as the transition variable
lag USO XLE GLD GDX SLV
1 15.2028# 199.1036# 41.2414# 85.6012# 18.4155#
(0.0000) (0.0015) (0.9066) (0.1691) (0.6330)
2 9.3950 106.6133 3.0521 44.5134 4.3343
(0.0000) (0.0000) (0.0003) (0.0000) (0.0000)
3 7.6069 72.0953 1.8225 28.9750 4.1105
(0.0000) (0.0000) (0.0412) (0.0000) (0.0000)
4 13.3542 66.4355 2.3545 22.5749 3.4775
(0.0000) (0.0000) (0.0057) (0.0000) (0.0000)
5 6.2257 62.6350 2.3205 18.1682 3.2875
(0.0000) (0.0000) (0.0066) (0.0000) (0.0000)
6 6.2700 67.2705 2.7286 15.3334 3.9647
(0.0000) (0.0000) (0.0013) (0.0000) (0.0000)
Panel B: ETF-VIX as the transition variable
lag USO XLE GLD GDX SLV
1 4.9784# 23.96298 3.2042 8.2848# 4.9454#
(0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
2 4.7259 14.7802 3.6157# 7.5544 4.1358
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(0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
3 4.7293 65.7129# 2.3512 7.7384 4.0049
(0.0000) (0.0000) (0.0000) (0.0000) (0.0000)
4 4.4716 14.3577 2.5810 6.8252 3.6342
(0.0000) (0.0000) (0.0001) (0.0000) (0.0000)
5 4.5923 14.2411 1.9561 6.4375 3.1752
(0.0000) (0.0000) (0.0066) (0.0000) (0.0000)
6 4.4343 12.8823 1.7584 7.1044 2.7881
(0.0000) (0.0000) (0.0195) (0.0000) (0.0000) Notes: # denotes the optimal lagged period with the maximum F-statistic (or minimum p-value) reject the linearity hypothesis.
Table 10: Test of functional form---STARX
Panel A: ETF as the transition variable
ETF d 0: 404 jH 0: 40303 jjH
0: 430202 jjjH
Model
USO 1 1.9290 (0.0531) 3.9693 (0.0001) 37.9106 (0.0000) LSTARX
XLE 1 0.9138 (0.5125) 1.40209 (0.1829) 592.5200 (0.0000) LSTARX
GLD 1 2.7898 (0.0256) 7.6329 (0.0000) 107.9183 (0.0000) LSTARX
SLV 1 1.6015 (0.0738) 1.8018 (0.0348) 246.0330 (0.0000) LSTARX
GDX 1 3.3171 (0.0003) 3.2948 (0.0003) 45.3537 (0.0000) LSTARX
Panel B: ETF-VIX as the transition variable
ETF d 0: 404 jH 0: 40303 jjH
0: 430202 jjjH
Model
USO 1 3.2153 (0.0000) 6.2741 (0.0000) 4.2283 (0.0000) ESTRAX
XLE 3 10.9044 (0.0000) 17.0606 (0.0000) 11.5530 (0.0000) ESTRAX
GLD 2 2.0740 (0.0422) 5.2202 (0.0000) 3.3141 (0.0018) ESTRAX
SLV 1 1.9172 (0.0191) 4.8713 (0.0000) 16.1830 (0.0000) LSTRAX
GDX 1 1.9021 (0.0271) 3.9415 (0.0000) 8.2934 (0.0000) LSTRAX
Note:d is the optimal lag length for the transition variable. p-values are in parentheses.
Table 11: Estimation outcomes with lagged ETF as the transition variable-STARX
ETF USO XLE GLD GDX SLV
0 87.1045 (0.1055) 37.4532* (0.0595) 36.6874* (0.0265) -27.2306
(0.1032) 12.2225 (0.2508)
21 -0.1657 (0.5789) 0.0225 (0.7875) 0.0130 (0.4152) -0.0007
(0.8814) -0.0792 (0.7332)
22 - - - - 0.3600 (0.0343)
24 -0.0889 (0.3527) - - - -
25 - - - - 0.1252 (0.2437)
26 - -0.0276 (0.6779) - - -
31 0.2216 (0.4222) - 0.9295 (0.3036) 0.7100
(0.1623) 0.2478 (0.4487)
32 - - -0.6517 (0.4511) -0.6794
(0.2061) -0.3248 (0.3296)
36 -0.4270 (0.1504) -0.0806 (0.6451) - 0.4741
(0.0699) -
41 - 19.9293 (0.9013) - -96.1811
(0.3093) -
42 - -318.2916 (0.1983) - - 100.7972 (0.1437)
43 -34.9184 (0.3304) - 57.6067** (0.0016) -216.8758
(0.1953) 12.8404 (0.8897)
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44 - 161.0332 (0.4499) - 328.3716*
(0.0543) -
45 - 163.4531 (0.3627) - - -3.7105 (0.9574)
46 - - - -12.8433
(0.9009) 103.3454 (0.1175)
51 0.0370 (0.4629) - - 0.0074
(0.9343) -
52 - - - -0.1134
(0.3538) -
53 -0.0733 (0.3833) - - 0.0451
(0.6964) -
54 -0.0085 (0.8595) - - -0.0058
(0.9627) -0.0278** (0.0414)
55 - -0.0851 (0.2503) - -0.0382
(0.7645) -
56 - 0.0783 (0.3139) - 0.0628
(0.5143) -
'
0 -83.1938 (0.1518) 68.1720* (0.0690) 279.0009*** (0.0002) 118.0441***
(0.0000) 55.7374 (0.3005)
'
21 0.2837 (0.4191) -0.0518 (0.7563) -0.0344 (0.3771) 0.0030
(0.6654) -0.0549 (0.9403)
'
22 - - - - -0.8288 (0.1733)
'
24 0.1314 (0.2619) - - - -
'
25 - - - - -0.5122 (0.1583)
'
26 - 0.0674 (0.6271) - - -
'
31 -0.2547 (0.4788) -2.3380 (0.1823) -1.1678
(0.1401) -1.2352 (0.2344)
'
32 - 1.5407 (0.3776) 1.1739
(0.1657) 1.5218 (0.1543)
'
36 0.5564 (0.1140) 0.1884 (0.6024) - -0.8912**
(0.0279) -
'
41 - -21.0822 (0.9435) - 139.7227
(0.2791) -
'
42 - 624.7018 (0.1893) - - 258.2755 (0.1398)
'
43 52.2942 (0.1688) - -112.8828*** (0.0048) 315.3174
(0.1747) 24.5694 (0.9187)
'
44 - -326.2338 (0.4517) - -468.7643*
(0.0517) -
'
45 - -336.4014 (0.3608) - - -25.7302 (0.8983)
'
46 - - - 21.6042
(0.8823) -290.1819 (0.1758)
'
51 - - - -0.0052
(0.9670) -
'
52 -0.0562 (0.4343) - - 0.1594
(0.3572) -
'
53 0.1135 (0.3037) - - -0.0473
(0.7723) -
'
54 0.0127 (0.8661) - - 0.0118
(0.9484) 0.1070** (0.0106)
'
55 - 0.2031 (0.2382) - 0.0540
(0.7756) -
'
56 - -0.1909 (0.2837) - -0.1013
(0.4735) -
0.5229*** (0.0000) 0.3641*** (0.0003) 0.4948*** (0.0000) 0.5791***
(0.0000) 0.3685*** (0.0000)
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c 30.0707***
(0.0000)
71.8414***
(0.0000)
159.7010***
(0.0000)
36.5714***
(0.0000) 46.0583***(0.0002)
2R 0.9674 0.9824 0.9870 0.9940 0.9873
Q 10.5700 (0.1030) 23.4200*** (0.0001) 1.0868 (0.9820) 1.7196
(0.9440) 6.3318 (0.3870)
ARCH 1.2065 (0.0721) 1.1748 (0.1047) 1.2530 (0.1033) 0.2156
(0.6426) 1.0999 (0.2311)
JB 87.6076*** (0.0000) 271.1260*** (0.0000) 809.2688*** (0.0000) 35.9675***
(0.0000) 1382.8230***(0.0000)
Notes: p-values are in parentheses. The figures are the standard regression coefficients. * significant at
10% level; **significant at 5% level; ***significant at 1% level.
Table 12: Estimation outcomes with lagged ETF-VIX as the transition variable-STARX
ETF USO XLE GLD GDX SLV
0 -15.6496 (0.0005) -4.4167 (0.6405) 6.4105 (0.2227) -143.8947
(0.0000) -2.9719 (0.3182)
11 -0.0068 (0.7961) -0.2787 (0.000) -0.1964 (0.0134) 0.1816 (0.0653) -0.0450 (0.0167)
12 - - 0.1490 (0.0706) - 0.0500 (0.0058)
13 0.0548 (0.0171) - - - -
16 - - - 0.3173 (0.0000) -
21 0.3172 (0.0000) 0.3138 (0.0000) 0.1038 (0.0000) 0.0477 (0.0000) 0.7289 (0.0000)
22 0.0652 (0.0341) - -0.0073 (0.2356) - 0.2296 (0.0000)
24 0.0128 (0.6963) - - - -
25 0.0140 (0.6812) - - - -0.0102 (0.7672)
26 -0.0511 (0.0590) - - - -
31 -0.0224 (0.7596) - -1.2918 (0.0000) -0.3988 (0.3567) -0.3743 (0.0001)
32 - - 1.2012 (0.0001) 0.5067 (0.2639) 0.3564 (0.0001)
34 0.1252 (0.2522) - - - -
36 -0.1866 (0.0312) - - -0.5797 (0.0115) -
41 - -44.7609 (0.1734) - - -
42 - - - - -22.0161 (0.4265)
43 22.8239 (0.0000) - 11.6256 (0.0197) -201.8195
(0.0118) -37.2745 (0.3132)
44 77.6576 (0.0325) 129.3607 (0.0376) - 285.9546 (0.0000) -
45 -74.3225 (0.0373) -43.8120 (0.3725) - - 33.9968 (0.3751)
46 - - - -23.5439 (0.6113) 28.0486 (0.3372)
51 -0.0178 (0.2200) - - 0.1667 (0.0221) -0.0024 (0.8177)
52 - - - 0.0434 (0.6703) -
53 0.0047 (0.8310) - - 0.0056 (0.9549) -
54 0.0248 (0.3466) 0.2183 (0.0066) - -0.0018 (0.9852) 0.0111 (0.2956)
55 0.0219 (0.2608) -0.0128 (0.8039) - 0.0261 (0.7922) -
56 - -0.1255 (0.0654) - 0.0042 (0.9510) - '
0 28.6354 (0.0000) 136.0818 (0.0000) 602.1840 (0.6667) 172.9657 (0.0000) 57.8624 (0.0000) '
11 0.0349 (0.2853) 0.1248 (0.0553) -2.2177 (0.6708) -0.1115 (0.2241) -0.1006 (0.0207)
'
12 - - 1.7973 (0.6739) - 0.0656 (0.0796)
'
13 -0.0263 (0.3835) - - - -
'
16 - - - -0.2299 (0.0018) -
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'
21 0.0165 (0.7366) -0.0588 (0.3588) -0.3182 (0.6537) -0.0124 (0.0030) -0.6452 (0.0000)
'
22 -0.0861 (0.0967) - 0.2148 (0.6487) - 0.1713 (0.0869)
'
24 0.0588 (0.2803) - - - -
'
25 -0.1088 (0.0642) - - - -0.0305 (0.6280)
'
26 0.1333 (0.0037) - - - -
'
31 -0.0347 (0.7799) - 4.6339 (0.6628) -1.0071 (0.0845) -0.2664 (0.3124)
'
32 - - -10.6144 (0.6601) 0.2497 (0.6869) -0.2817 (0.2796)
'
34 -0.0517 (0.7785) - - - -
'
36 -0.0046 (0.9755) - - -0.0976 (0.7512) -
'41 - 55.3855 (0.3328) - - - '
42 - - - - 30.9780 (0.2699)
'
43 -6.7202 (0.4869) - 221.2312 (0.6703) 382.0468 (0.0032) 67.9595 (0.0688)
'
44 -95.1380 (0.0311) -432.3791 (0.0220) - -342.3409
(0.0000) -
'
45 81.5004 (0.0840) 149.4637 (0.3803) - - -41.6030 (0.2819)
'
46 - - - 61.9865 (0.5039) -46.7294 (0.1149)
'
51 0.0177 (0.2582) - - -0.1698 (0.0296) 0.0085 (0.5548)
'
52 - - - -0.0397 (0.7049) -
'
53 -0.0035 (0.8769) - - 0.0084 (0.9349) -
'
54 -0.0263 (0.3231) -0.2871 (0.0120) - 0.0174 (0.8604) -0.0078 (0.4766)
'
55 -0.0259 (0.1914) -0.0042 (0.9594) - -0.0160 (0.8723) -
'
56 - 0.1320 (0.1166) - 0.0008 (0.9906) -
1.0181*** (0.0000) 0.9358*** (0.0000) 0.0369*** (0.6951) 3.1080*** (0.0000) 13.7711***
(0.0113)
c 31.2623*** (0.0000) 31.6877*** (0.0000) 20.2401*** (0.0000) 35.6034*** 0.0000) 54.3883***
(0.0000) 2R 0.9628 0.9117 0.9797 0.9744 0.9842
Q 93.9810*** (0.0000) 1724.600*** (0.0000) 103.300*** (0.0000) 1164.10***(0.0000) 93.7660***(0.0000)
ARCH 2.4643 (0.1169) 13.2477*** (0.0000) 1.0244 (0.4294) 2.4182*** (0.0000) 0.8908 (0.7819)
JB 122.3221*** (0.0000) 2.4059 (0.3003) 980.7379*** (0.0000) 1.9598 (0.3754) 415.6381***
(0.0000)
Notes: p-values are in parentheses. The figures are the standard regression coefficients. * significant at
10% level; **significant at 5% level; ***significant at 1% level.
Table 13: Forecast Errors of the Respective Models for ETF-VIX
RMSE AR STAR Multiple linear STARX
OVX 0.7575 0.7605 4.2292 0.7272
VXXLE 0.8547 0.8611 2.7670 0.8203
GVZ 0.6797 0.7147 4.0973 0.6949
VXGDX 0.8676 0.8834 8.0083 1.0399
VXSLV 1.0494 1.0435 10.0252 1.9428
Continue
MAE AR STAR Multiple linear STARX
OVX 0.5564 0.5451 3.4135 0.5737
VXXLE 0.6550 0.6602 2.5038 0.6416
GVZ 0.5547 0.5743 3.9160 0.5605
VXGDX 0.6868 0.6885 7.7208 0.8954
VXSLV 0.8031 0.7993 9.7188 1.6268
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Theil'U AR STAR Multiple linear STARX
OVX 0.0193 0.0194 0.0949 0.0185
VXXLE 0.0252 0.0254 0.0839 0.0241
GVZ 0.0197 0.0207 0.1072 0.0201
VXGDX 0.0119 0.0121 0.0995 0.0142
VXSLV 0.0175 0.0174 0.1447 0.0318
Table 14: Forecast Errors of the Respective Models for ETF
RMSE USO XLE GLD GDX SLV
STARX(ETF) 0.3923 0.7682 1.4248 0.7892 0.3411
STARX(ETFVIX) 2.0917 1.9791 1.3474 4.2760 0.3885
MAE
STARX(ETF) 0.3161 0.6047 1.1334 0.6705 0.2579
STARX(ETFVIX) 1.6739 1.6762 1.1059 3.6736 0.3053
Theil'U
STARX(ETF) 0.0056 0.0045 0.0057 0.0162 0.0087
STARX(ETFVIX) 0.0302 0.0115 0.0054 0.0927 0.0099
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Figure 1: Graph of the logistic function Figure 2: Graph of the exponential function
Figure 3: Transition functions in the STAR for ETF-VIX
0.0
0.2
0.4
0.6
0.8
1.0
10 20 30 40 50 60 70
OVX
0.0
0.2
0.4
0.6
0.8
1.0
10 20 30 40 50 60
VXXLE
0.0
0.2
0.4
0.6
0.8
1.0
10 15 20 25 30 35 40 45
GVZ
0.0
0.2
0.4
0.6
0.8
1.0
24 28 32 36 40 44 48 52 56 60
VXGDX
0.9984
0.9986
0.9988
0.9990
0.9992
0.9994
0.9996
0.9998
1.0000
20 30 40 50 60 70 80 90
VXSLV
Figure 4: Transition functions in the STARX (ETF-VIX) for ETF-VIX
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
10 20 30 40 50 60 70
OVX
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
10 20 30 40 50 60
VXXLE
.4
.5
.6
.7
.8
.9
10 15 20 25 30 35 40 45
GVZ
.3
.4
.5
.6
.7
.8
24 28 32 36 40 44 48 52 56 60
VXGDX
0.0
0.2
0.4
0.6
0.8
1.0
20 30 40 50 60 70 80 90
VXSLV
Proceedings of the Fifth Asia-Pacific Conference on Global Business, Economics, Finance
and Social Sciences (AP16Mauritius Conference) ISBN - 978-1-943579-38-9
Ebene-Mauritius, 21-23 January, 2016. Paper ID: M620
27 www.globalbizresearch.org
Appendix A Table A1: Descriptive statistics of ETF volatility index and ETF
Panel A
ETF-VIX OVX VXXLE GVZ VXGDX VXSLV
Mean 31.0703 24.5548 20.5785 37.1091 38.1254
Maximum 69.1200 57.4700 39.9500 56.5000 80.6400
Minimum 15.2000 14.4400 11.9700 24.8400 20.8600
Std. Dev. 8.6725 8.1764 5.3206 6.8112 10.3776
Skewness 0.7780 1.5142 0.9263 0.4560 1.1272
Kurtosis 3.6697 4.8590 3.8211 2.2113 4.7963
Jarque-Bera 84.1740*** 370.4086*** 120.4612*** 42.6409*** 243.7359***
Panel B
ETF USO XLE GLD GDX SLV
Mean 35.9023 72.3294 151.5376 43.7391 29.1262
Maximum 45.1500 88.0800 184.5900 65.1500 47.2600
Minimum 29.4600 53.9900 114.8200 20.3100 17.8900
Std. Dev. 3.1332 7.1146 16.2034 12.4160 6.1890
Skewness 0.5112 0.2304 -0.5390 -0.4272 0.0749
Kurtosis 2.7712 2.3988 2.2766 1.9031 2.5270
Jarque-Bera 32.2009*** 16.8282*** 49.4346*** 56.7088*** 7.2219**
Notes: *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively.
Table A2: Descriptive statistics of relative variables
Variables OIL GOLD SILVER USDX LIBOR CRB
Mean 96.0215 151.5376 30.0037 79.5981 0.215748 495.2199
Median 96.0300 155.3650 30.8000 80.0425 0.21 484.47
Maximum 113.8100 184.5900 48.7000 84.5800 0.2963 579.68
Minimum 75.5100 114.8200 18.6100 72.9330 0.02955 207.74
Std. Dev. 7.3536 16.2034 6.2556 2.7232 0.031291 33.51334
Skewness -0.0430 -0.5390 0.0467 -0.6701 0.109027 0.170414
Kurtosis 2.5319 2.2766 2.4680 2.5666 4.137927 10.4434
Jarque-Bera 6.643514** 49.43456*** 8.545265** 58.1914*** 39.32191*** 1628.598***
Notes: *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively.