摘要
當用簡單線性回歸模型估計最佳避險比例的時候，估計誤差是一個一直沒有解決的問題。Herbst (1989) 指出Ederingtion (1979)用OLS回歸估計期貨對現貨的最佳避險比例時會產生有偏的估計量，因為殘差項有嚴重的序列相關，這違反了OLS回歸的基本假設。
同時，現貨和期貨的時間序列資料可能符合I（1）過程並且不共整合。這就產生了偽回歸的問題，用OLS估計出來的估計量是偽估計量。由於這個原因，近期的研究在將現貨資料和期貨資料進行差分後用OLS線性回歸來估計最佳避險比例。這些研究將非平穩的資料過程轉換成平穩的資料過程，只有這樣才能避免解決偽回歸的問題。因此，之前對避險比例的研究僅僅是建立在共整合或者穩態的基礎上的，這對於hedge ratio 的研究並不是必須的。即使將解釋變數和被解釋變數一階差分後得到穩態線性回歸方程，又會出現異方差的問題，從而導致另一個估計誤差。
本研究為在出現不同計量問題的時候選擇避險比例的估計方法，減少估計誤差提供了指導作用。首先，我們用蒙地卡羅模擬的方法模擬出違背OLS各種假設的共整合方程和偽回歸方程, 這些被違背的假設包括存在測量誤差，異方差以及序列相關。然後我們用八種估計方法估計這些回歸式。最後，我們找到在每一種情況下最優的估計方法。在實證研究中, 我們把CVAR (Gatarek and Johansen, 2014) 模型和八種估計方法結合起來計算避險的效果，我們發現這個模型的避險效果要好於在研究避險比例時盛行的VAR和EC模型。

Abstract
Estimation error has been a major problem when the simple linear regression model is used to estimate optimal hedge ratios. Herbst (1989) pointed out that Ederingtion’s (1979) method of using the OLS regression to estimate optimal hedge ratios between spot and futures rates of foreign currencies would yield biased estimators because the error terms are significantly serially correlated which violates stringent assumptions of the OLS regression.
Meanwhile, time series data of spot and futures may follow the I (1) process and is not cointegrated. Thus the OLS will yield spurious estimators. Due to this problem, more recent studies estimate the optimal hedge ratios based on the change of spots and the change of futures contracts. These studies transform the nonstationary process into the stationary process in order to avoid solving the possible spurious regression problem. Thus, they only study hedge ratio problem based on either the cointegration or stationary situation, which in fact is not necessary. Even if we get stationary equations by taking first difference of dependent and independent variables, heteroscedasticity usually appears, making another estimation error.
This paper provides a guideline on how to choose the estimation methods to solve the estimation error problem in different situations. We perform Monte Carlo simulations of cointegrated and spurious regressions with and without measurement errors and heteroscedasticity. Serial correlation is simulated in all the cases. Then eight estimation methods are used to estimate these regressions. Finally, we find the best estimation method in each case. In empirical study, we use CVAR (Gatarek and Johansen, 2014) model combined with our eight estimation methods to compute post-sample performances and we show that the present method produces better results than those of the VAR and EC models, which are prevalent in hedge ratio study.

References
Boscovich, R. J. (1757). De litteraria expeditione per pontificiam ditionem, et synopsis amplioris operis, ac habentur plura ejus ex exemplaria etiam sensorum impressa. Bononiensi Scientarum et Artum Instituto Atque Academia Commentarii, 4, 353-396.
Charnes, A., Cooper, W. W., & Ferguson, R. O. (1955). Optimal estimation of executive compensation by linear programming. Management science, 1(2), 138-151.
Chen, K., Ying, Z., Zhang, H., & Zhao, L. (2008). Analysis of least absolute deviation. Biometrika, 95(1), 107-122.
Choi, C. Y., Hu, L., & Ogaki, M. (2008). Robust estimation for structural spurious regressions and a Hausman-type cointegration test. Journal of Econometrics, 142(1), 327-351.
Chou, W. L., Denis, K. F., & Lee, C. F. (1997). Hedging with the Nikkei index futures: the conventional model versus the error correction model. The Quarterly Review of Economics and Finance, 36, 495–505.
Ederington, L. (1979). The hedging performance of the new futures markets. Journal of Finance, 34, 157-170.
Engle, R. F., & Granger, C. W. (1987). Co-integration and error correction: representation, es- timation, and testing. Econometrica, 55, 251-276.
Gatarek, L. T., & Johansen, S. (2014). Optimal hedging with the cointegrated vector autoregressive model. Available at SSRN 2509711.
Ghosh, A. (1993). Hedging with stock index futures: Estimation and forecasting with error correction model. The Journal of Futures Markets, 13, 743–752.
Granger, C. W., & Newbold, P. (1974). Spurious regressions in economics. Journal of Econometrics 4, 111-120.
Harris, T. E. (1950). Regression using minimum absolute deviations. American Statistician, 4(1), 14-15.
Herbst, A. F., Kare, D. D., & Caples, S. C. (1989). Hedging effectiveness and minimum risk hedge ratios in the presence of autocorrelation: foreign currency futures. Journal of Futures Markets, 9(3), 185-197.
Hong, Y. (1996). Testing for independence between two covariance stationary time series. Biometrika, 83(3), 615-625.
Lien, D. (1996). On the conventional definition of currency hedge ratio. The Journal of Futures Market, 16, 219-226.
Lien, D. (1996). The effect of the cointegration relationship on futures hedging: a note. The Journal of Futures Markets (1986-1998), 16(7), 773.
Lien, D., & Tse, Y. K. (1999). Fractional cointegration and futures hedging. Journal of Futures Markets, 19, 457-474.
Lien, D. (2004). Cointegration and the optimal hedge ratio: the general case. The Quarterly Review of Economics and Finance, 44, 654-658.
Lien, D. (2005). A note on the superiority of the OLS hedge ratio. The Journal of Futures Markets, 25, 1121-1126.
Lien, D. (2007). Statistical properties of post-sample hedging effectiveness. International Review of Financial Analysis, 16(3), 293-300.
Lien, D., & Shrestha, K. (2008). Hedging effectiveness comparisons: A note. International Review of Economics & Finance, 17(3), 391-396.
Lien, D. (2010). A note on the relationship between the variability of the hedge ratio and hedging performance. The Journal of Futures Markets, 30, 1100-1104.
Lien, D. (2010). Effects of omitting information variables on optimal hedge ratio estimation: a note. The Journal of Futures Markets, 30, 795-800.
Lien, D., & Shrestha, K. (2012). The effects of price dynamics on optimal futures hedging. Annals of Financial Economics, 7(02), 1250008.
Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.
Phillips, P. C. (1986). Understanding spurious regressions in econometrics. Journal of Econometrics, 33(3), 311-340.
Phillips, P. C., & Hansen, B. E. (1990). Statistical inference in instrumental variables regression with I (1) processes. The Review of Economic Studies, 99-125.
Phillips, P. C., & Ouliaris, S. (1988). Testing for cointegration using principal components methods. Journal of Economic Dynamics and Control, 12(2), 205-230.
Phillips, P. C., & Ouliaris, S. (1990). Asymptotic properties of residual based tests for cointegration. Econometrica, 165-193.
Pollard, D. (1991). Asymptotics for least absolute deviation regression estimators. Econometric Theory, 7(02), 186-199.
Wang, C. S. H., Rosa, C., & Hafner, C. M. (2014). A simple solution of the spurious regression problem. Journal of Time Series Econometrics, forthcoming.
Wang, C. S. H., Hsiao, C., & Xie, Y. M. (2015). On the Need of Estimating the Parameters of GARCH Models. Computational Statistics & Data Analysis, forthcoming.
Wright, S. (1928). Appendix B to Wright, P. The Tariff on Animal and Vegetable Oils, New.
Yule, G. U. (1926). Why do we sometimes get nonsense-correlations between Time-Series? A study in sampling and the nature of time-series. Journal of the Royal Statistical Society, 1-63.