Evaluating the Predictability of ETF Volatility Indexes...

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





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.





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





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).





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.





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





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





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:





,,;)()()(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





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.





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.





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.





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.





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.

References

Adjei, F., 2009, Diversification, Performance, and Performance Persistence in Exchange-Traded Funds.

International Review of Applied Financial Issues and Economics 1(1), 4-19.





Badshah, I. U., Bart F. and Alireza, T. R., 2013, Contemporaneous Spill-Over among Equity, Gold, and

Exchange Rate Implied Volatility Indices. Journal of Futures Markets 33(6), 493-599.

Chaiyuth, Padungsaksawasdi and Daigler, R. T., 2013, The Return-Implied Volatility Relation for

Commodity ETFs. Journal of Futures Markets 34(3),261-281.

Chau, F., Deesomsak, R. and Lau, M., 2011, Investor Sentiment and Feedback Trading: Evidence from

the Exchange-traded Fund Markets. International Review of Financial Analysis 20(5), 292-305.

Chen, A. S. and Lin, J. W., 2004, Cointegration and DetectableLinear and Nonlinear Causality:

Analysis Using the London Metal Exchange Lead Contract. Applied Economics 36(11), 1157-1167.

Chen, J. H., Diaz, J. F. and J. W., Huang J. W., 2013, High Technology ETF Forecasting: Application

of Grey Relational Analysis and Artificial Neural Networks. Frontiers in Finance & Economics 10(2),

129-155.

Cheng, M. and Madhavan, A., 2009, The Dynamics of Leveraged and Inverse Exchange-Traded Funds.

Journal of Investment Management 7(4), 43-62.

Ciner, C., 2011, Commodity Prices and Inflation: Testing in the Frequency Domain. Research in

International Business and Finance 25(3), 229-237.

Dacco, R. and Satchell, S., 1999, Why do Regime Switching Models Forecast so Badly? Journal of

Forecasting 18(1),1-16.

Daigler, R. T., Hibbert, A. M. and Pavlova, I., 2014, Examining the Return–Volatility Relation for

Foreign Exchange: Evidence from the Euro VIX. Journal of Futures Markets 34(1), 74-92.

Elton, J. Edwin, Martin J. Gruber, George, Comer and Kai, L., 2002, Spiders: Where Are the Bugs?

Journal of Business 75(3), 453-472.

Engle, R. F. and Granger, C. W. J., 1987, Co-integration and Error-correction: Representation,

Estimation, and Testing. Econometrica 55, 251-276.

Granger, C. W. J. and Teräsvirta, T., 1993, Modelling Nonlinear Economic Relationship: Oxford

University Press, Oxford.

Hasbrouck, J., 2003, Intraday Price Formation in U.S. Equity Index Markets. Journal of Finance 58(6),

2375-2400.

Horst J., Nijiman, T., and Verbeek, M., 2001, Eliminating Look-ahead Bias in Evaluating Persistence

in Mutual Fund Performance. Journal of Empirical Finance 8(4), 345-373.

Hsu, K. C. and Chiang, H. C., 2011, Nonlinear Effects of Monetary Policy on Stock Returns in a

Smooth Transition Autoregressive Model. The Quarterly Review of Economics and Finance 51(4),

339-349.

Jubinski, D. and Lipton, A. F., 2013, VIX, Gold, Silver, and Oil: How Do Commodities React to

Financial Market Volatility? Journal of Accounting and Finance 13(1), 70-88.

Lin, C. J., and Teräsvirta T., 1994, Testing the Constancy of Regression Parameters against Continuous

Structural Change. Journal of Econometrics 62, 211-228.

Liu, M. L., Q., Ji, and Ying, F., 2013, How Does Oil Market Uncertainty Interact with Other Markets?

An Empirical Analysis of Implied Volatility Index. Energy 55, 860-868.

Low, C., 2004, The Fear and Exuberance from Implied Volatility of S&P 100 Index Options. Journal

of Business 77, 527-546.

McMillan, D. G., 2001, Nonlinear Predictability of Stock Market Returns: Evidence form

Nonparametric and Threshold Models. International Review of Economics and Finance 10(4), 353-

368.

Qadan, M. and Yagil, J., 2012, Fear Sentiments and Gold Price: Testing Causality In-mean and In-

variance. Applied Economics Letters 19(4), 363-366.

Sarantis, N., 1999, Modeling Non-linearities in Real Effective Exchange Rates. Journal of

International Money and Finance 18(1), 27-45.

Sari, R., Ugur, Soytas, Erk, Hacihasanoglu, 2011, Do Global Risk Perceptions Influence World Oil

Prices? Energy Economics 33(3), 515-524.





Sarwar, G., 2011, The VIX Market Volatility Index and U.S. Stock Index Returns. Journal of

International Business and Economics 11(4), 167-179.

Sarwar, G., 2012, Intertemporal Relations between the Market Volatility Index and Stock Index

Returns. Applied Financial Economics 22(11), 899-909.

Terasvirta, T., 1994, Specification, Estimation and Evaluation of Smooth Transition Autoregressive

Models. Journal of the American Statistical Association 89(425), 208-218.

Teräsvirta, T. and Anderson, H. M., 1992, Characterizing Nonlinearities In Business Cycles Using

Smooth Transition Autoregressive Models. Journal of Applied Econometrics 7, 119-136.

Vaneesha B. D., Hany, G., and Glenn, R. M., 2013, Did Intraday Trading by Leveraged and Inverse

Leveraged ETFs Create Excess Price Volatility? A Look at REITs and the Broad Market. Journal of

Real Estate Portfolio Management 19(1), 1-16.

Whaley, R. E., 2009, Understanding the VIX. Journal of Portfolio Management 35(3), 98-105.

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.





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


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


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





(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


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)





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)



Table 8: Parameter Estimates with lagged ETF-VIX as the transition variable-STARX


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) -





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)



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





(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)





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)





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)



Table 12: Estimation outcomes with lagged ETF-VIX as the transition variable-STARX


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) -





'

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)



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





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





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





Appendix A Table A1: Descriptive statistics of ETF volatility index and ETF

Panel A


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


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.

Evaluating the Predictability of ETF Volatility Indexes...

Documents

Transcript of Evaluating the Predictability of ETF Volatility Indexes...