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The famous Hiemstra-Jones test (HJ test) developed by Hiemstra and Jones plays a significant role in studying nonlinear causality. In the last two decades there are numerous applications and theoretical extensions based on this pioneering work. However several works pointed out that counter-intuitive results are obtained from the HJ test and some researchers found that the HJ test is seriously over rejecting through simulation study. In this paper, we re-investigate HJ’s creative work in 1994 and found that their proposed estimators of the probabilities over different time intervals were not consistent to the target ones proposed in their criterion. To test the HJ’s novel hypothesis on Granger causality, we shall propose new estimators of the probabilities defined in HJ’s paper and reestablish the asymptotic properties which will induce new tests similar to HJ’s ones. Some simulation work will also be presented to support our new findings.

Chow, et al. (2016) use the theory of ascending stochastic dominance (ASD), descending stochastic dominance (DSD) to develop stochastic dominance (SD) tests for richness and poorness. In this paper, we extend their work by applying Markowitz stochastic dominance (MSD) and prospect stochastic dominance (PSD) to develop SD tests for the poor (test for poorness), the rich (test for richness), and for the middle class (we call it test for middle class) to achieve a more robust analysis of relative welfare levels in the study of income distributions. The findings of these tests can be used to draw preference for different types of increasing social welfare functions; namely, concave social welfare functions, convex social welfare functions, and S-shaped and reverse S-shaped social welfare functions. The feasibility of the proposed approach and the new tests of richness, poorness, AND the middle class developed for it are demonstrated using distributions of incomes in Hong Kong.

In this paper, we develop a new nonlinear Granger causality test to take into consideration different lags of variables and different values of the bound for panel data.

Bai, et al. (2011d) have developed the mean-variance-ratio (MVR) statistic to test the performance among assets for small samples. They have provided theoretical reasoning to use MVR and proved that their proposed statistic is uniformly most powerful unbiased. In this paper, we illustrate the superiority of the MVR test over the traditional Sharpe ratio (SR) test by applying both tests to analyze the performance of the S&P 500 index and the NASDAQ 100 index after the bursting of the Internet bubble in 2000s. Our findings show that while the traditional SR test concludes the two indices being analyzed to be indistinguishable in their performance, the MVR test statistic shows that the NASDAQ 100 index underperformed the S&P 500 index, which is the real situation after the bursting of the Internet bubble in 2000s. This shows the superiority of the MVR test statistic in revealing short term performance and, in turn, enables the investors to make better decisions in their investments.

Asset performance evaluation is one of the most important areas in investment analysis. In order to compare the performance among assets, several statistics have been developed; and among them, the Sharpe-ratio statistic is the most prevalent. However, the major limitation of the Sharpe-ratio statistic is that its distribution is only valid asymptotically, but not valid for small samples. Nevertheless, it is important in finance to test the performance among assets for small samples. To serve this purpose, we first provide theoretical reasoning and then develop both one-sided and two-sided mean-variance-ratio statistics to evaluate the performance among the assets for small samples. In this paper we further prove that our proposed statistics provide uniformly most powerful unbiased tests. We illustrate the superiority of our proposed test over the traditional Sharpe-ratio test by applying both tests to analyze the performance of funds from Commodity Trading Advisors. Our findings show that while the traditional Sharpe-ratio test concludes most of the CTA funds being analyzed as being indistinguishable in their performance, our proposed statistics show that some funds outperform the others. On the other hand, when we apply the Sharpe-ratio statistic on some other funds, we find that the statistic indicates that one fund is significantly outperforming another fund even though the difference between the two funds is insignificantly different from zero and/or even changes directions. However, when our proposed mean-variance-ratio statistic is applied, we could detect the change in the difference. This shows the superiority of our proposed statistic in revealing short term performance and in return, enables investors to make better decisions about their investments.

To circumvent the limitations of the tests for coefficients of variation and Sharpe ratio, we develop the mean-variance-ratio statistic to test for the equality of the mean-variance ratios. We prove that our proposed statistic is uniformly most powerful unbiased. In addition, we provide the reasonings of using our proposed test statistic.

The traditional linear Granger causality test has been widely used to examine the linear causality among several time series in bivariate settings as well as multivariate settings. Hiemstra and Jones (1994) develop a nonlinear Granger causality test in a bivariate setting to investigate the nonlinear causality between stock prices and trading volume. In this paper, we first discuss linear causality tests in multivariate settings and thereafter develop a nonlinear causality test in multivariate settings. A Monte Carlo simulation is conducted to demonstrate the superiority of our proposed multivariate test over its bivariate counterpart. In addition, we illustrate the applicability of our proposed test to analyze the relationships among different Chinese stock market indices.

The traditional estimated return for the Markowitz mean-variance optimization has been demonstrated to seriously depart from its theoretic optimal return. We prove that this phenomenon is natural and the estimated optimal return is always $\sqrt{\gamma}$ times larger than its theoretic counterpart where $\gamma = \frac 1{1-y}$ with $y$ as the ratio of the dimension to sample size. Thereafter, we develop new bootstrap-corrected estimations for the optimal return and its asset allocation and prove that these bootstrap-corrected estimates are proportionally consistent with their theoretic counterparts. Our theoretical results are further confirmed by our simulations, which show that the essence of the portfolio analysis problem could be adequately captured by our proposed approach. This greatly enhances the practical uses of the Markowitz mean-variance optimization procedure.

To circumvent the limitations of the Sharpe-ratio statistic on testing small samples, we develop the mean-variance-ratio (MVR) statistic to test the performance among assets for small samples. We provide theoretical reasoning to use MVR and prove that our proposed statistic is uniformly most powerful unbiased.

Davidson and Duclos (DD, 2000) develop the stochastic dominance statistics, T_j(x)(j=1,2,3), to test the hypothesis on statistically significant differences between any two cumulative density functions F and G for assets Y and Z, respectively. The DD test compares distributions at only a finite number of grid points and, under the null of equality of F and G, Davidson and Duclos show that T_j is asymptotically distributed as the studentized maximum modulus (SMM) distribution. Various studies examine the choice of grid points. One may argue that too many grids will violate the independence assumption required by the SMM distribution. We note that the limiting distributions of T_j(x) are not independent even for two points partition. More importantly, the limiting correlations would depend on the null distribution F=G. To overcome the difficulty, in this paper we improve the DD statistics by deriving the limiting process of the DD test statistic T_j(x) for the situations in which assets Y and Z which could be dependent and independent, and hence, the modified DD test can be performed by max_x|T_j(x)| and to take care of the dependency of the partitions. In addition, we propose a Bootstrap method to decide the critical point of the modified DD test.

The traditional linear Granger test has been widely used to examine the linear causality among several time series in bivariate settings as well as multivariate settings. Hiemstra and Jones (1994) develop a nonlinear Granger causality test in a bivariate setting to investigate the nonlinear causality between stock prices and trading volume. In this paper, we first discuss linear causality tests in multivariate settings and thereafter develop a non-linear causality test in multivariate settings.

Levy and Levy (2002, 2004) extend the stochastic dominance (SD) theory for risk averters and risk seekers by developing the prospect SD (PSD) and Markowitz SD (MSD) theory for investors with S‐shaped and reverse S‐shaped (RS‐shaped) utility functions, respectively. Davidson and Duclos (2000) develop SD tests for risk averters whereas Sriboonchitra et al. (2009) modify their statistics to obtain SD tests for risk seekers. In this paper, we extend their work by developing new statistics for both PSD and MSD of the first three orders. These statistics provide a tool to examine the preferences of investors with S‐shaped utility functions proposed by Kahneman and Tversky (1979) in their prospect theory and investors with RS‐shaped investors proposed by Markowitz (1952a). We also derive the limiting distributions of the test statistics to be stochastic processes. In addition, we propose a bootstrap method to decide the critical points of the tests and prove the consistency of the bootstrap tests. To illustrate the applicability of our proposed statistics, we apply them to study the preferences of investors with the corresponding S‐shaped and RS‐shaped utility functions vis‐a‐vis returns on iShares and vis‐a‐vis returns of traditional stocks and Internet stocks before and after the Internet bubble.

This paper extends the work of Korkie and Turtle (2002) by first proving that the traditional estimate for the optimal return of self-financing portfolios always over-estimates from its theoretic value. To circumvent the problem, we develop a Bootstrap estimate for the optimal return of self-financing portfolios and prove that this estimate is consistent with its counterpart parameter. We further demonstrate the superiority of our proposed estimate over the traditional estimate by simulation.