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
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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
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
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
10 www.globalbizresearch.org
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.
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
11 www.globalbizresearch.org
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.
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
12 www.globalbizresearch.org
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.
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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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.
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
14 www.globalbizresearch.org
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.
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
15 www.globalbizresearch.org
Badshah, I. U., Bart F. and Alireza, T. R., 2013, Contemporaneous Spill-Over among Equity, Gold, and
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)
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
19 www.globalbizresearch.org
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) -
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
20 www.globalbizresearch.org
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
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
21 www.globalbizresearch.org
(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