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Acknowledgements
The authors gratefully acknowledge financial support from the National Natural Science Foundation of China under grants 71001008 and 70733005, the Humanities and Social Science Research Foundation of the Ministry of Education of China under grant 09YJC630011 and the Specialized Research Fund for the Doctoral Program of Higher Education, Ministry of Education of China under grant 20101101120041. We would also like to thank Prof. Jean-Philippe Bouchaud, Prof. Michael Dempster, Samantha Hutton and Collette Teasdale, as well as the anonymous referees, for their helpful suggestions and corrections on an earlier draft of the paper that greatly improved the content. Naturally, all remaining errors are ours.
Notes
‡An up (down) stock market means that the excess stock market return is positive (negative). This also applies to the oil market.
§Otherwise nobody is willing to invest in the stock markets.
†Otherwise, if the realized returns are always larger than the risk-free returns, then nobody would be willing to invest in assets with risk-free returns, such as Treasury bonds, and if the realized returns are always less than the risk-free returns, then accordingly nobody would be willing to invest in the stock market.
†Here we adopt the WTI weekly average excess oil return series, which is calculated by subtracting the risk-free returns (weekly returns of the three-month US T-bills) from the original oil price returns.
‡Due to the role of crude oil in enterprise production, an oil price change may result in changes in production costs and profits and thus affect the performance of the stock price. Hence the vast majority of published CAPM/APT research investigating the relationship between oil price movements and stock prices uses the oil price as an independent variable. Here, in order to estimate the influence of stock market risk on oil market returns and avoid the endogeneity issue and biased parameter estimate as much as possible, we take the excess stock market return at time t–1 as the independent variable, and the excess oil market return at time t as the dependent variable. It should be noted that the lagged stock market return included is not of interest per se, but is just an instrument to address the endogeneity issue.
§In 1960, R.E. Kalman published his famous paper describing a recursive solution to the discrete-data linear filtering problem. Since that time, due in large part to the advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. Detailed information on the Kalman filtering approach can be obtained from Gao (Citation2006, chapter 11).
†Considering the interaction between the stock market and the oil market takes on a more linear feature, which can be seen from the empirical results below, therefore here we just test the symmetry effect for a linear influence.
†We follow the Chow test approach for a structural break test regarding the beta series, and find that every stock market beta series has a break point or interval. In this case, we calculated the influence of the beta series on the oil market in two separate sampling periods, and the results indicate that, in both sampling periods, the direction of influence of different advanced stock markets on the oil market, the nonlinear properties of their influence and the difference between the two kinds of stock markets are fairly similar to the present results. Therefore, basically, the break point or interval does not greatly impact the results.
†In fact, using the curve regression approach, we also estimate the relationship between American down stock market risk and the oil market using other nonlinear functions, such as inverse, cubic, logarithmic functions, etc., and we find that neither of the common nonlinear functions shows a significant result.