AVONEX 30 micrograms/0.5ml solution for injection in pre-filled pen (interferon beta-1a)
4. Possible side effects (Notes information)
Firstly, a VAR model without trading volume will be constructed, and the op-timal lag length will be determined. In this study, the Hannan-Quinn method was used to choose the optimal lag length, which will be used in the VAR model. In table 4.6 below, the optimal lag length for each country is given.
Table 4.6: Optimal lag length Country Optimal lag lengths
Canada 2
Australia 1
Russia 1
South Africa 2
Source: Thomson Reuters Datastream and Eviews
Next, a stability test will be done to determine whether the VAR is stable when the above-mentioned lags are included. This will be done by looking at the AR roots of the VAR for each country and is illustrated below.
Figure 4.9: AR roots Canada
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Inverse Roots of AR Characteristic Polynomial
Source: Thomson Reuters Datastream and Eviews
Figure 4.10: AR roots Australia
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Inverse Roots of AR Characteristic Polynomial
Source: Thomson Reuters Datastream and Eviews
Figure 4.11: AR roots Russia
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Inverse Roots of AR Characteristic Polynomial
Source: Thomson Reuters Datastream and Eviews
Figure 4.12: AR roots SA
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Inverse Roots of AR Characteristic Polynomial
Source: Thomson Reuters Datastream and Eviews
The results obtained from the stability tests show that all the roots of all the countries lie within the unit circle. This suggests that each VAR model is stable when estimated using the optimal lag length for each country.
The number of cointegrating equations is determined by making use of the Johansen cointegration method. According to Asteriou and Hall (2015), when testing for cointegration, if g variables are included in the model, the maximum number of cointegrating equations is g − 1. The trace test and maximum eigen-value statistics indicate that the maximum number of cointegrating equations should be included in the VECM; therefore, three cointegrating equations are included for each country. The results of the test statistics can be seen in the table below.
Table 4.7: Cointegration rank test Country Trace test Max-Eigen
Canada 3,1165 3,1165
Australia 8,1891 8,1891
Russia 0,0534 0,0534
South Africa 2,7913 2,7913
Source: Thomson Reuters Datastream and Eviews
The results of the VECM models of each country are illustrated below.
Table 4.8: Canada long run equations Dependent variable Coefficients
Closet 0, 0054 + 0, 9994Opent Hight 0, 0668 + 0, 9935Opent Lowt −0, 0537 + 1, 0051Opent Source: Thomson Reuters Datastream and Eviews
Table 4.9: Australia long run equations Dependent variable Coefficients
Closet 0, 0004 + 0, 9999Opent Hight 0, 1217 + 0, 9864Opent Lowt −0, 1369 + 1, 0153Opent
Source: Thomson Reuters Datastream and Eviews Table 4.10: Russia long run equations Dependent variable Coefficients
Closet −0, 0113 + 1, 0028Opent Hight 0, 0272 + 0, 9973Opent Lowt −0, 0289 + 1, 0032Opent Source: Thomson Reuters Datastream and Eviews
Table 4.11: SA long run equations Dependent variable Coefficients
Closet -0,0068 + 1,0006 Open
Hight 0,0203 + 0,9986 Open
Lowt -0,0401 + 1,0032 Open
Source: Thomson Reuters Datastream and Eviews
The results presented in tables 4.8 to 4.11 above, are in line with those obtained by Sreedharan (2004) and Oberholzer and Venter (2015a). The co-efficient of the opening price is positive and close to one in each case which is consistent with futures spot parity. Futures spot parity implies that the ratio of the futures price to the current spot price gives an estimate of the expecta-tion of the risk-free rate, which applies to the futures period (Los, 2000). The time period between the closing price and opening price (one trading day) is short, and, therefore, the coefficients are close to one.
According to Gujarati (2003), the relationship between variables can be expressed as an error correction mechanism if the variables are cointegrated.
The error correction mechanisms and their significance are illustrated below.
Table 4.12: Canada error correction coefficients
Equation Close High Low Open
Cointegrating equation 1 -0,1585 1,0875* 1,1766* 1,1156*
Cointegrating equation 2 0,2329 -0,6634* -0,1601 0,0552 Cointegrating equation 3 -0,1766 -0,4876* -0,7853* -0,1522*
Adjusted R2 0,0097 0,3843 0,3094 0,8664
* Denotes statistical significance at a 5% level
Source: Thomson Reuters Datastream and Eviews
Table 4.13: Australia error correction coefficients
Equation Close High Low Open
Cointegrating equation 1 -0,0344 1,1501* 0,9551* 1,1065*
Cointegrating equation 2 0,1761 -0,7907* -0,2179* 0,0091 Cointegrating equation 3 0,0026 -0,2079* -0,7438* 0,0125 Adjusted R2 -0,0039 0,4442 0,3623 0,9970
* Denotes statistical significance at a 5% level
Source: Thomson Reuters Datastream and Eviews
Table 4.14: Russia error correction coefficients
Equation Close High Low Open
Cointegrating equation 1 0,0762 1,0058* 0,9489* 0,7974*
Cointegrating equation 2 0,0601 -0,6491* -0,0722 0,1080*
Cointegrating equation 3 -0,1116 -0,2963* -0,7175* 0,0433
Adjusted R2 0,0096 0,3414 0,3826 0,7964
* Denotes statistical significance at a 5% level
Source: Thomson Reuters Datastream and Eviews
Table 4.15: SA error correction coefficients
Equation Close High Low Open
Cointegrating equation 1 -0,0723 0,9388* 0,9032* 0,8491*
Cointegrating equation 2 0,2740 -0,5685* -0,0569 0,1992 Cointegrating equation 3 -0,1587 -0,3037* -0,7536* 0,1167
Adjusted R2 0,0097 0,3281 0,2842 0,6743
* Denotes statistical significance at a 5% level
Source: Thomson Reuters Datastream and Eviews
When looking at the above results of the VECMs, given in tables 4.12 to 4.15, if the coefficients are positive and significant, it indicates that the vari-able is non-responsive and will take a long time to move back to equilibrium (Koop, 2006). However, if the coefficients are significant and the correct sign (negative), it indicates that when there is a deviation from the long-run equi-librium, the variable will move back to equilibrium. When the coefficient is closer to -1 than to 0, it means that the variable will move back to equilibrium rapidly. If cointegrating equation one for SA is considered, it can be seen that the high, low and opening prices are all significant but of the incorrect sign.
Therefore, they are non-responsive and will take a long time to move back to equilibrium. However, if cointegrating equation three for SA is considered, it is clear that both the high and low prices are significant and of the correct sign (negative), which means they will move back to equilibrium. Furthermore, the high price is closer to zero, which means it will move to equilibrium at a slower pace than the low price, as the low price is closer to -1 and will move back to equilibrium rapidly when there is a deviation from the long-run equilibrium.
An impulse response function shows what will happen to a variable if a shock is introduced in the model (Mitchell, 2000). In other words, if one variable increases rapidly (by one standard deviation), the impulse response function illustrates how the variables in the model react to the increase. The
impulse response functions for the intraday prices (highest, lowest, open and closing) of each country are illustrated below.
Figure 4.13: Canada impulse response
.004 Response of LNCANADA_CLOSE to Generalized One
S.D. Innovations Response of LNCANADA_HIGH to Generalized One
S.D. Innovations Response of LNCANADA_LOW to Generalized One
S.D. Innovations Response of LNCANADA_OPEN to Generalized One
S.D. Innovations
Source: Thomson Reuters Datastream and Eviews
Figure 4.14: Australia impulse response
Response of LNAUS_CLOSE to Generalized One S.D. Innovations
Response of LNAUS_HIGH to Generalized One S.D. Innovations
Response of LNAUS_LOW to Generalized One S.D. Innovations
Response of LNAUS_OPEN to Generalized One S.D. Innovations
Source: Thomson Reuters Datastream and Eviews
Figure 4.15: Russia impulse response
Response of LNRUS_CLOSE to Generalized One S.D. Innovations
Response of LNRUS_HIGH to Generalized One S.D. Innovations
Response of LNRUS_LOW to Generalized One S.D. Innovations
Response of LNRUS_OPEN to Generalized One S.D. Innovations
Source: Thomson Reuters Datastream and Eviews
Figure 4.16: SA impulse response
Response of LNSA_CLOSE to Generalized One S.D. Innovations
Response of LNSA_HIGH to Generalized One S.D. Innovations
Response of LNSA_LOW to Generalized One S.D. Innovations
Response of LNSA_OPEN to Generalized One S.D. Innovations
Source: Thomson Reuters Datastream and Eviews
Figures 4.13 to 4.16 above, show how each variable responds to a shock in any of the other variables (intra-day prices). In most cases, the variables respond by first increasing then decreasing slowly to reach a new equilibrium.
However, in some cases, a shock does not lead to a significant impact. One such case is the response of the logged intra-day low price of Australia to a shock in the intra-day opening price of Australia. It is evident that the shock in the intra-day opening price of Australia does not have a significant impact on the intra-day low price.
When looking at the response of the logged high price of SA, it is evident that a one standard deviation shock to the closing price causes the logged high price to rapidly increase and reach a peak and then slowly decrease afterwards
to reach a new equilibrium.
When trading volume is included as an exogenous variable in the model, similar results are obtained for the optimal lag length, cointegration tests and error correction coefficients. The trading volume coefficients and the adjusted R2 values are reported below.
Table 4.16: Canada error correction (including trading volume)
Equation Close High Low Open
Volume -0,0020* 0,0013* -0,0038* -0,0006 Adjusted R2 0,0147 0,3866 0,3295 0,8668
* Denotes statistical significance at a 5% level Source: Thomson Reuters Datastream and Eviews
Table 4.17: Australia error correction (including trading volume)
Equation Close High Low Open
Volume -0,0012 0,0021* -0,0037* 0 Adjusted R2 -0,0022 0,4539 0,3771 0,9970
* Denotes statistical significance at a 5% level Source: Thomson Reuters Datastream and Eviews
Table 4.18: Russia error correction (including trading volume)
Equation Close High Low Open
Volume 0,0007 0,0031* -0,0037* -0,0010*
Adjusted R2 0,0086 0,3616 0,4006 0,7975
* Denotes statistical significance at a 5% level Source: Thomson Reuters Datastream and Eviews
Table 4.19: SA error correction (including trading volume)
Equation Close High Low Open
Volume -0,0024* 0,0002 -0,0032* -0,0005 Adjusted R2 0,0097 0,3281 0,2842 0,6743
* Denotes statistical significance at a 5% level Source: Thomson Reuters Datastream and Eviews
When looking at the adjusted R2 values above, it can be seen that the explanatory power of the model does improve slightly. This is the case for both
developed and emerging markets. This implies that when investors consider the interrelationship between market prices of equity indices, including trading volume (new information) as an exogenous variable improves the explanatory power of the model, regardless of whether it is a developed or emerging market.
The informational efficiency of volatility modelling is considered in the next section.