Stochastic dominance in stock market 2

Chapter Size and Power of Some Stochastic Dominance Tests 2.1 Introduction Consumers choose from among uncertain alternatives to maximize their utility. One of the necessary conditions for optimizing behavior in economic analysis is that an optimal portfolio cannot be inferior to another feasible portfolio for any increasing utility function. This can be characterized in terms of stochastic dominance (SD) between distributions of returns. The theory of SD provides a systematic framework for analyzing economic behavior under uncertainty. Although the SD methodology which focuses on economic decisions theory has been developed for more than three decades, while powerful SD tests have been only available recently. In the literature, Levy and Sarnat (1970, 1972), Joy and Porter (1974), Wingender and Groff (1989), Seyhun (1993) implement the SD rules empirically, but they have not discussed the testing procedure for SD and no statistical tests have been done. There are two broad classes of SD tests. One is minimum/maximum statistic while the other is based on distribution values computed on a set of grid points. McFadden (1989) first develops a SD test using the minimum/maximum statistic. Later, Klecan, McFadden and McFadden (1991, KMM), Kaur, Rao and Singh (1994, KRS) also propose some tests using the minimum/maximum statistic. On the other hand, Anderson (1996), and Davidson and Duclos (2000, DD) are the commonly used SD tests that compare the underlying distributions at a finite number of grid points. Although there have been differences in SD test methods in prior studies, the literature is rather silent on the performance of SD tests. Recently, Tse and Zhang (2004) present Monte Carlo studies to examine the size and power of some SD tests when the underlying distributions are independent. However, little is known about the size- and power-performance of SD tests when the underlying distributions are correlated. Zheng and Cushing (2001) point out that the conventional inference procedure usually requires samples to be independently drawn. Nevertheless, most of the economics or finance issues involve studies of dependent data, for example, the momentum puzzle discussed by Jegadeesh and Titman (1993, 2001) and Fong, Lean and Wong (2004) where the portfolio of winners is highly correlated with the portfolio of losers. Without confirmation of the size and power of the SD tests in the dependent situation, it will be difficult for academicians and practitioners to apply the SD tests to most of the empirical finance and economics studies. As the previous literature that study the performance of different SD tests overlook the dependent situation, this study attempts to fill the gap to examine the size- and powerperformance of several commonly used SD tests when the underlying distributions are correlated. Findings of this study will enable the academicians and practitioners to apply the SD tests efficiently in the dependent situations as required in most of the empirical issues. Monte Carlo simulations are used to compare the size- and powerperformance of three commonly used SD tests, including DD, Anderson and KRS. Specifically, this study attempts to determine which SD test is the most appropriate to be used for empirical research when the underlying distributions are correlated. In addition, the influence of heteroskedasticity on the size and power of SD tests as well as the issue of choosing the number of grids used for SD tests are included in this study. Tse and Zhang (2004) show that the DD test has superior sizeand power-performance compared to both Anderson and KRS tests when the underlying distributions are independent. The simulation results in this study show that the DD test also has good performance in both the power and size when the underlying distributions are correlated. In addition, the results show that heteroskedasticity significantly reduces the power of all SD tests. But, the DD test still has a good power-performance for large sample. The choice of the optimal number of grid points is an important issue for SD tests that compute the test statistics using different grid points. Too few grid points may not able to reveal more information from the distributions while too many grid points will violate the independence assumption of the grid statistics (Stoline and Ury 1979). Tse and Zhang (2004) propose to use 10-15 equal-distance major grids to satisfy the independence assumption for different grids. However, this may result in missing the detection of SD behavior for the distributions between any two consecutive major grids as the distance of these grids could be quite big due to the small number of grid points being used. This study follows the suggestion by Tse and Zhang (2004) using 10-15 equal-distance major grids. Moreover, it is suggested to use at least 10 equal-distance minor grids between any two consecutive major grids to ensure non-omission of important information between the major grids. The critical values for 100-150 grids which will violate the independence assumption of different grids cannot be used in this situation. Thus, it is suggested to use the critical values based on 10-15 major grids for all the major and minor grids to satisfy the independence assumption of grids. The simulation results support the suggestion of adding minor grid points. Furthermore, the results also show that the selection of the critical values based on 10-15 grids is appropriate for both major and minor grids. This chapter is organized as follows. The next section discusses expected utility theory. Section introduces SD theoretical framework and section describes some commonly used SD tests to be examined in this study. Section explains the procedure of Monte Carlo simulations while section exhibits and discusses the findings. The conclusion is in section 7. 2.2 Expected Utility Theory Since SD theory is developed on the foundation of expected utility paradigm, it will be beneficial for us to have some knowledge about expected utility theory beforehand. Investor's utility function determines combination of the optimum portfolio held by the investor and how much he will invest. The utility value of a prospect is a real number that represents the relative desirability of the prospect. If L is a set of prospects each defined over a fixed set of consequences and R is the set of real numbers, then u is a function mapping L into R. Let Li be the prospect associated with act Ai in a decision problem, then the relative desirability of the act is measured by the utility value of its prospect, u(Li). Following the definition of utility, the best act is the one for which u(Li) is a maximum. This is the most general criterion of choice in decision theory and is called utility maximization. Understanding of the properties of utility function can enlighten the process of rational choice and reduce the chances of making bad decisions. 10 2.2.1 Properties of Utility Function The first property of utility function is non-satiation or more is preferred to less. It means that the utility of x + dollar is always higher than the utility of x dollars. If utility increases as wealth increases, then the first derivative of utility, with respect to wealth, is positive. Thus, the first restriction placed on the utility function is a positive first derivative denoted as U ' (W ) > where U (W ) is the utility function. The second property of utility function is an assumption about investor's taste for risk. Three assumptions of risk are risk aversion, risk neutrality, and risk seeking. Risk aversion means that an investor will reject a fair gamble because the disutility of the loss is greater than the utility of an equivalent gain. Risk aversion implies that the second derivative of utility with respect to wealth is negative denoted as U " (W ) < . A function where an additional unit increase is less valuable than the last unit increase is a function with a negative second derivative. Risk neutrality means that an investor is indifferent to accept or reject a fair gamble. For the investor to be indifferent between investing and not investing, the expected utility of investing is same as the expected utility of not investing. Risk neutrality implies a zero second derivative denoted as U " (W ) = . Risk seeking means that an investor will select a fair gamble. Risk-seeking investors have utility functions with positive second derivatives denoted as U " (W ) > . The expected utility of investment must be higher than the expected utility of not investing. 11 The third property of utility function is an assumption about how the investor's preferences change with a change in wealth. If the investor increases the amount invested in risky assets as wealth increases, then the investor is said to exhibit decreasing absolute risk aversion. If the investor's investment in risky assets is unchanged as wealth changes, then the investor is said to exhibit constant absolute risk aversion. Finally, if the investor invests fewer dollars in risky assets as wealth increases, then the investor is said to exhibit increasing absolute risk aversion. A(W ) = − U " (W ) can be used to measure an investor’s absolute risk aversion. The U ' (W ) derivative of A(W) with respect to wealth, A'(W), is an appropriate measure of how absolute risk aversion behaves with respect to changes in wealth. Relative risk aversion, R(W), can be used as well to measure the changes in percentage instead of absolute amount. 2.3 Stochastic Dominance Theoretical Framework SD theorem has become an important tool in the analysis of choice under uncertainty. Dominance principles have important role in understanding and solving the portfolio problem in economics and finance. The SD selection rules for risk averters have been well studied in the literature, see for example, Markowitz (1952), Tobin (1958), Fishburn (1964), Hadar and Russell (1969), Hanoch and Levy (1969), Whitmore (1970), Tesfatsion (1976), Meyer (1977), Wong and Li (1999) and Li and Wong (1999). There are three basic SD rules as stated in the following definitions. 12 Definition 1: Suppose that an asset X has a cumulative distribution function (CDF) F(x), and an asset Y has a CDF G(y); both with the support [a,b] with a < b. Let Fi +1 (x ) = ∫ Fi (t ) dt , and x a Gi +1 ( y ) = ∫ Gi (t ) dt with a ≤ x, y ≤ b, for i = 1, 2. y (2.1) a 1) The asset X stochastically dominates the asset Y at first order, denoted by X f 1Y , or F f 1G if and only if F1 ( x ) ≤ G1 (x ) for all possible returns, x; 2) the asset X stochastically dominates the asset Y at second order, denoted by X f 2Y , or F f G if and only if F2 ( x ) ≤ G (x ) for all possible returns, x; and 3) the asset X stochastically dominates the asset Y at third order, denoted by X f 3Y , or F f G if and only if µ (F ) ≥ µ (G ) and F3 (x ) ≤ G3 ( x ) for all possible returns x b b a a where µ ( F ) = E ( X ) = ∫ xdF ( x) and µ (G ) = E (Y ) = ∫ xdG ( x) . Definition 2: Let { U s : s = 1, 2, 3} denotes a set of utility functions, where { } U s = u : (− 1) u (i ) ≥ 0, i = 1, ., s and i +1 u (i ) (x ) is the ith derivative of u(x). The extended sets of utility functions are: U 1E = {u : u is increasing}, U 2E = {u : u is increasing and concave}, { } U 3E = u ∈ U 2E : u ' is convex . The following theorem is the main result describes the basic relation between utility functions and distribution functions. 13 Theorem (Li and Wong 1999): Let X and Y be the random variables with CDF, F and G respectively. Suppose u is a utility function. For s = 1, and 3, X f s Y , if and only if u (F ) ≥ u (G ) for any u in U such that U s ⊆ U ⊆ U sE where u (H ) = ∫ u ( x ) dH ( x ), H = F , G. b a One may refer to Hanoch and Levy (1969), Rothschild and Stiglitz (1970), Hadar and Russell (1969, 1971), Bawa (1975), Tesfatsion (1976), Meyer (1977), Fishburn and Vickson (1978) and Li and Wong (1999) for the proof and the detail explanation of the theorem. SD test enables the application of SD within the theoretical framework circumscribed by the definitions and theorem laid down above. Application of SD in the comparison of different assets is important, because the above theorem shows that SD is equivalent to the choice of assets by utility maximization. SD also has a wide range of application in economics and finance, such as the study of market efficiency discussed in Liao and Chou (1995), Post (2003), and Post and Vliet (2004), the event study discussed in Larsen and Resnick (1999), the study of mutual fund performance by Kjetsaa and Kieff (2003), and the study of momentum strategy performance by Post and Levy (2004) and Fong, Lean and Wong (2004). 2.4 Various Stochastic Dominance Tests Varian (1983) and Green and Srivastava (1986) are the earliest work to establish statistical tests for the hypothesis of utility maximization when the distributions of returns are known. McFadden (1989) introduces a statistical test on empirical 14 distributions using SD statistical methodology by assuming that the paired observations are drawn from two independent distributions with equal sample sizes, and that the observations are independent over time. The hypothesis is set up as X dominates Y against the alternative hypothesis that X does not dominate Y. McFadden (1989) points out that even if the variables are not statistically independent, SD is a well-defined relationship between the marginal distributions. However, he indicates that the distribution of the testing statistic for SD depends on the joint distribution of paired observations, and that the use of this test would be inappropriate in application when the underlying distributions are dependent. Thereafter, different SD tests are introduced in the literature and they will be discussed in the following sub-sections with the assumption that the SD is studied between two dependent random variables, X and Y, where F and G are their corresponding CDF. KMM extend McFadden test by relaxing the assumption of independent underlying distributions. This extension is important to studies in the area of finance because most financial variables are dependent. The first- and second-order KMM test statistics are, respectively, ) ( = (max[F (x ) − G ( x )], max[G (x ) − F (x )]). d * = max[F1 (x ) − G1 (x )], max[G1 (x ) − F1 ( x )] , and x s* x x x Under the null hypothesis that X stochastically dominates Y at first (second) order, ( ) d * s * should not be greater than 0. KMM show that the test statistics converge in distribution to a maximum of a Gaussian process with a covariance function of ρ, 15 where the maximum is taken over a set of domain which makes the two distributions equivalent. However, Shin (1994) points out that the KMM test is not robust to tails. He claims that the KMM test may produce result accepted in the first-order null hypothesis but rejected in the second-order null hypothesis, especially when the minmax of CDF difference (or integrated CDF difference) is attained in the left tail of distribution. This is because a few negative extreme observations may cause the minmax of the CDF difference small and the min-max of the integrated CDF difference large. Recently, Linton, Maasoumi and Whang (2003) find that sub-sampling bootstrap technique is better than a traditional bootstrap method when computing the critical values of the KMM test. On the other hand, Barrett and Donald (2003) extend McFadden test by developing a Kolmogorov-Smirnov (KS) test applied to two independent samples with possibly unequal sample sizes. They state the hypotheses as H 0s : Fs ( z ) ≤ G s ( z ) for all z ∈ [0, z ], H 1s : Fs ( z ) > G s ( z ) for some z ∈ [0, z ]. This hypothesis is formed similar as in McFadden (1989). The KS test statistic is ⎛ N2 ⎞ ⎟⎟ Kˆ s = ⎜⎜ N ⎝ ⎠ 12 where Fˆs (z ) = [ ] sup Fˆs ( z ) − Gˆ s ( z ) , z N N 1 s −1 ˆ (z ) = ( ) (z − yi )+s −1 ; , and z − x G ∑ ∑ i + s N (s − 1)! i =1 N (s − 1)! i =1 N is the sample size, s = 1, 2, and z + = max(0, z ) . 16 The empirical size is estimated by calculating the percentage of rejections of the null hypothesis (i.e. the simulated statistic is greater than the critical value6) when the null hypothesis is true. The simulation results show that all the three SD tests are conservative as their empirical sizes are less than 0.05 except the case for the DD test with s = 1, k = and N = 100, 500 and 1000. From the table, it is noted that the empirical sizes of the DD test are closer to the critical values, while the empirical sizes of both the KRS and Anderson tests are close to zero. These results are consistent with the results in Tse and Zhang (2004) for independent distributions. In addition, the size of the DD test decreases as k increases from to 15, and the size of DD test decreases from first-order to third-order SD (except for sample size 50 at k = 15). Overall, the DD test’s biggest size is 0.1021 and the smallest size is 0.0009 in this simulation. The empirical size of the DD test is closest to the nominal size of 5% with N = 1000, k = 10 and s = 1. On the other hand, the effect of sample size on the size-performance of DD test is inconsistent. For example, size increases as samples become larger for the first-order SD. However, size decreases and then increases as samples become larger for the second-order SD. For the KRS test, its biggest size is in sample size of 1000 and its smallest size is in sample size of 50. In contrast to the DD test, the sizes of KRS test increase from first- to third-order SD. With the exception of case of s = and N = 50 which sees the sizes of the Anderson test increase as k increases, all other sizes of Anderson test in the simulations are either zero or very close to zero, and therefore, no further comparison can be made. The DD test follows the SMM distribution while both the Anderson and KRS tests follow the standardized normal distribution. At the nominal size of 5%, the SMM critical values of k for 6, 10 and 15 are 2.928, 3.254 and 3.487 respectively while the Z0.025 is 1.96 for KRS tests. 31 To examine the power-performance of SD tests, two series that generated with α i = 0.2, α j = -0.2 and β i = β j = 1.0 are investigated. In this case, ri stochastically dominates rj at all orders but not vice-versa. In the case of another two series that generated with α i = α j = 0.0 and β i = 0.0, β j = 2.0, there is no first-order SD between ri and r j , but ri stochastically dominates rj at the second and third orders. The empirical power is estimated by the probability of failing to reject null hypothesis when dominance exists in the population. Table 2.2a: Empirical Power of the DD and Anderson Tests for Homoskedasticity Process s=1 s=2 s=3 DD A DD A DD A 0.6931 0.0246 0.9536 0.0007 0.8776 0.0003 Parameters 50 0.5171 0.0047 0.9094 0.0000 0.8047 0.0000 of 0.3518 0.0040 0.8643 0.0000 0.7372 0.0000 Distributions 0.9887 0.1259 0.9999 0.0103 0.9984 0.0059 100 0.9455 0.0374 0.9990 0.0013 0.9956 0.0003 α i = 0.2 0.8962 0.0138 0.9977 0.0002 0.9931 0.0002 α j = -0.2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 500 1.0000 0.9990 1.0000 0.9998 1.0000 0.9993 β i =1.0 1.0000 0.9910 1.0000 0.9983 1.0000 0.9937 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 β j =1.0 1000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Average 0.8660 0.5167 0.9770 0.5009 0.9506 0.5000 Notes: The figures are the empirical relative frequency of accepting HA1 for s = 1, 2, 3. There are SD between X and Y for s = 1, 2, and thus HA1 is true for s = 1, 2, 3. N is the sample sizes, k is the number of major grid points; s is the order of SD. N k 10 15 10 15 10 15 10 15 32 Table 2.2b: Empirical Power of the DD and Anderson Tests for Homoskedasticity Process s=1 s=2 s=3 DD A DD A DD A Parameters 0.2011 0.2515 0.6465 0.4220 0.4308 0.2279 of 50 0.2590 0.3367 0.4471 0.2084 0.2345 0.0798 Distributions 0.2843 0.2612 0.3180 0.1043 0.1435 0.0269 0.0086 0.0209 0.9854 0.9589 0.9475 0.8807 100 0.0274 0.0622 0.9517 0.8830 0.8519 0.7098 α i = 0.0 0.0478 0.1046 0.9128 0.7862 0.7469 0.5430 α j = 0.0 0.0000 0.0000 1.0000 1.0000 1.0000 1.0000 500 0.0000 0.0000 1.0000 1.0000 1.0000 1.0000 β i =0.0 0.0000 0.0000 1.0000 1.0000 1.0000 1.0000 β j =2.0 0.0000 0.0000 1.0000 1.0000 1.0000 1.0000 1000 0.0000 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 0.0000 1.0000 Average 0.0690 0.0864 0.8551 0.7802 0.7796 0.7057 Notes: The figures are the empirical relative frequency of accepting HA1 for s = and 3. There are SD between X and Y for s = and and thus HA1 is true for s = and 3. N is the sample sizes, k is the number of major grid points; s is the order of SD. N k 10 15 10 15 10 15 10 15 Table 2.3: Empirical Power of the KRS Tests for Homoskedasticity Process A α i = 0.2 α j = -0.2 β i = β j = 1.0 s=1 N H A1 50 100 500 1000 0.0001 0.0048 0.7894 0.9925 0.4467 Average B α i = α j = 0.0 β i = 0.0 β j = 2.0 Average s=2 H A2 H A1 1.0000 1.0000 1.0000 1.0000 1.0000 0.1040 0.4003 0.9673 0.9996 0.6178 s=1 s=3 H A1 H A0 1.0000 1.0000 1.0000 1.0000 1.0000 0.0398 0.2147 0.8238 0.9704 0.5122 1.0000 1.0000 1.0000 1.0000 1.0000 H A2 s=2 s=3 N H A1 H A0 H A1 H A0 H A1 H A0 50 100 500 1000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0800 0.1552 0.3437 0.5103 0.2723 1.0000 1.0000 1.0000 1.0000 1.0000 0.1303 0.6719 1.0000 1.0000 0.7006 1.0000 1.0000 1.0000 1.0000 1.0000 Notes: The figures in Panel A are the empirical relative frequency of accepting H A1 or H A for s = 1, 2, 3. The figures in Panel B are the empirical relative frequency of accepting H A1 or H A for s = and 3. N is the sample sizes. 33 Tables 2.2a and 2.2b summarize the empirical powers of DD and Anderson tests while Table 2.3 shows the power of KRS test. When SD exists in the series for all three orders, the DD test has the highest power among the three SD tests being studied. As expected, the power is low for small sample sizes like 50 and 100, and it increases as the sample size increases for all the tests. In addition, as the sample size increases, the power of the DD test increases faster than those of the Anderson and KRS tests. For the situation when the first-order SD does not exist, while the second- and third-order SD exist (Table 2.2b), the DD test has higher power compared to both the Anderson and KRS tests in small sample sizes of 50 and 100. All the tests have power of one in sample sizes of 500 and 1000. For both situations, as expected, the empirical power increases significantly as the sample size increases. However, it is interesting to note from Table 2.2b that the power of both the DD and Anderson tests decrease as k increases i.e. the power is actually higher for smaller k. These results are possibly due to violation of the independence assumption of grids for larger k. In addition, the simulation results show that smaller order yields bigger power. Overall, the simulation results conclude that for the homoskedasticity processes, the DD test is superior to both the KRS and Anderson tests, as the DD test possess the highest power and its size is closer to the critical values while both KRS and Anderson tests have much smaller power and their sizes are close to zero. 34 2.6.2 Heteroskedasticity Process The experiments are repeated for heteroskedasticity process to check the robustness of the simulation results i.e. to check whether the size and power change when homoskedasticity is violated. Table 2.4: Empirical Sizes of the DD, Anderson and KRS Tests for Heteroskedasticity Process Parameters of Distributions: α i = 0.0, α j = 0.0, β i = 1.0, β j = 1.0 N k 50 10 15 100 10 15 500 10 15 1000 10 15 Average DD 0.0356 0.0108 0.0041 0.0845 0.0270 0.0142 0.1505 0.0850 0.0551 0.2376 0.1449 0.0938 0.0786 s=1 A 0.0002 0.0008 0.0030 0.0002 0.0000 0.0000 0.0004 0.0001 0.0001 0.0004 0.0000 0.0000 0.0004 KRS 0.0000 0.0000 0.0000 0.0000 0.0000 DD 0.0150 0.0046 0.0023 0.0174 0.0057 0.0028 0.0354 0.0119 0.0054 0.0683 0.0275 0.0112 0.0173 s=2 A 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 KRS 0.0000 0.0016 0.0022 0.0017 0.0014 DD 0.0058 0.0022 0.0005 0.0062 0.0020 0.0009 0.0113 0.0032 0.0009 0.0202 0.0059 0.0027 0.0052 s=3 A 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 KRS 0.0000 0.0009 0.0101 0.0095 0.0051 Notes: The figures are the empirical sizes of the DD, Anderson (A) and KRS tests. N is the sample sizes, k is the number of major grid points; s is the order of SD. Table 2.4 presents empirical sizes of the DD, Anderson and KRS tests for the heteroskedasticity process. Similar to the results for the homoskedasticity process, the simulation results show that both the Anderson and KRS tests in all orders and the DD test in the second and third orders are conservative, as their empirical sizes are less than 0.05 under all the situations. The empirical sizes of the Anderson test are zero for the second- and third-order SD and close to zero for the first-order SD while the sizes of the KRS test are zero for the first-order SD and close to zero for the second- and third-order SD test. The DD test is not conservative for the first-order SD 35 as its empirical sizes are greater than 0.05 in seven out of twelve situations, and the average size for the first-order SD is 0.0786. The size of the DD test decreases as k increases and as the order ascends from first to third. Compared with the homoskedasticity results in Table 2.1, the simulation results in Table 2.4 show that heteroskedasticity has no impact on the Anderson test for all orders, and has no impact on the KRS test for the first-order SD as their sizes are zero or very close to zero. However, it is found that heteroskedasticity causes the empirical sizes to be large for the DD test at all orders, and similarly for the KRS test at the second and third orders, especially for large sample sizes of 500 and 1000. Nevertheless, this effect is not serious and does not affect the performance of the tests as their sizes are still much less than the significant value. Heteroskedasticity creates the problem of under-rejection for H in the first-order SD of the DD test, especially for the case with and 10 grids in the large sample as the empirical size becomes as large as 0.2376 for N = 1000 and k = 6. Table 2.4 shows that a choice of grids of either 10 or 15 yields better size in the simulation. 36 Table 2.5a: Empirical Power of the DD and Anderson Tests for Heteroskedasticity Process s=1 s=2 s=3 DD A DD A DD A 0.5588 0.0190 0.7714 0.0009 0.5858 0.0004 Parameters 50 0.3830 0.0054 0.6708 0.0002 0.4745 0.0000 of 0.2717 0.0041 0.5873 0.0000 0.3924 0.0000 Distributions 0.9254 0.0820 0.9711 0.0116 0.9068 0.0052 100 0.8280 0.0282 0.9441 0.0016 0.8523 0.0008 α i = 0.2 0.7410 0.0084 0.9176 0.0003 0.7990 0.0000 1.0000 0.9959 1.0000 0.9872 1.0000 0.9159 α j = -0.2 500 1.0000 0.9749 1.0000 0.9173 1.0000 0.7663 β i = 1.0 1.0000 0.9307 1.0000 0.7958 1.0000 0.5907 1.0000 1.0000 1.0000 1.0000 1.0000 0.9994 β j = 1.0 1000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9986 1.0000 1.0000 1.0000 1.0000 1.0000 0.9986 Average 0.8090 0.5041 0.9052 0.4762 0.8342 0.4397 Notes: The figures are the empirical relative frequency of accepting HA1 for s = 1, 2, 3. There are SD between X and Y for s = 1, 2, and thus HA1 is true for s = 1, 2, 3. N is the sample sizes, k is the number of major grid points; s is the order of SD. N k 10 15 10 15 10 15 10 15 Table 2.5b: Empirical Power of the DD and Anderson Tests for Heteroskedasticity Process s=1 s=2 s=3 DD A DD A DD A 0.2453 0.2527 0.4940 0.2086 0.2872 0.0949 Parameters 50 0.2620 0.2182 0.3309 0.0952 0.1585 0.0307 of 0.2647 0.1546 0.2349 0.0492 0.0886 0.0103 Distributions 0.0579 0.1451 0.8889 0.6428 0.7299 0.4389 100 0.1051 0.2111 0.8018 0.4744 0.5880 0.2693 α i = 0.0 0.1442 0.2444 0.7248 0.3560 0.4621 0.1692 α j = 0.0 0.0000 0.0000 1.0000 1.0000 0.9997 0.9875 500 0.0000 0.0000 1.0000 1.0000 0.9993 0.9818 β i = 0.0 0.0000 0.0000 1.0000 1.0000 0.9990 0.9750 0.0000 0.0000 1.0000 1.0000 1.0000 0.9991 β j = 2.0 1000 0.0000 0.0000 1.0000 1.0000 1.0000 0.9982 0.0000 0.0000 1.0000 1.0000 0.9999 0.9985 Average 0.0899 0.1022 0.7896 0.6522 0.6927 0.5795 Notes: The figures are the empirical relative frequency of accepting HA1 for s = and 3. There are SD between X and Y for s = and and thus HA1 is true for s = and 3. N is the sample sizes, k is the number of major grid points; s is the order of SD. N k 10 15 10 15 10 15 10 15 37 Table 2.6: Empirical Power of the KRS Tests for Heteroskedasticity Process A α i = 0.2 α j = -0.2 β i = β j = 1.0 s=1 N H A1 50 100 500 1000 0.0002 0.0012 0.2506 0.7111 0.2408 Average B α i = α j = 0.0 β i = 0.0 β j = 2.0 Average s=2 H A2 H A1 1.0000 1.0000 1.0000 1.0000 1.0000 0.0543 0.1512 0.4808 0.7333 0.3549 s=1 s=3 H A1 H A0 1.0000 1.0000 1.0000 1.0000 1.0000 0.0178 0.0711 0.1193 0.1332 0.0854 1.0000 1.0000 1.0000 1.0000 1.0000 H A2 s=2 s=3 N H A1 H A0 H A1 H A0 H A1 H A0 50 100 500 1000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0455 0.0933 0.2223 0.3378 0.1747 1.0000 1.0000 1.0000 1.0000 1.0000 0.0704 0.3431 0.8013 0.9034 0.5296 1.0000 1.0000 1.0000 1.0000 1.0000 Notes: The figures in Panel A are the empirical relative frequency of accepting H A1 or H A for s = 1, 2, 3. The figures in Panel B are the empirical relative frequency of accepting H A1 or H A for s = and 3. N is the sample sizes. Table 2.5 summarizes the empirical powers of the DD and Anderson tests while Table 2.6 exhibits the power of the KRS test. The simulation results show that the DD test has the highest power among the three SD tests under all the situations being studied in this chapter. As such, it can be claimed that the DD test is superior to both the Anderson and KRS tests in the power performance. When first-, second- and third-order SD exist, the Anderson test has the lowest power, which may be ignored in small sample sizes like 50 and 100. On the other hand, in the absence of first-order SD, while second- and third-order SD exist, the KRS test has the lowest power. Hence, the Anderson test and the KRS test not dominate each other in term of power performance. In contrast, the DD test performs very well at all orders for the sample size above 100, especially when the sample size increases to 500 or 1000 that sees its power approaching one. As expected, the power for the three SD tests increases significantly as the sample size increases. For the DD and Anderson tests, their power 38 decreases as k increases. This result possibly attributed to the violation of the independence assumption of grids for larger k. In addition, the power is bigger for smaller order. Compared with the homoskedasticity results in Table 2.2, the simulation results for heteroskedasticity in Table 2.5 show that the power for the DD test decreases significantly under heteroskedasticity null when the first three orders SD exists, as well as when first-order SD is absent, but second- and third-order SD exist. Figures 2.3 and 2.4 below provide the evidence that heteroskedasticity reduces the power of the DD tests. Nevertheless, the DD test still has good power to detect the dominance alternative for sample size in excess of 500. On the other hand, the power for the Anderson test is inconsistent especially for small sample sizes when SD exists in all three orders. If comparing Table 2.3 and 2.6, the power of the KRS test reduces significantly under heteroskedasticity process. Therefore, it can be concluded that heteroskedasticity also reduces the power of the KRS test. 39 Figure 2.3: Power Graph for the DD Test under Homoskedasticity and Heteroskedasticity Process 0.6 0.5 power 0.4 0.3 0.2 0.1 -0.2 -0.133 -0.067 homoskedasticity 0.067 0.133 0.2 heteroskedasticity Figure 2.4: Power Graph for the DD Test under Homoskedasticity and Heteroskedasticity Process 1.2 power 0.8 0.6 0.4 0.2 0 0.33 0.67 homoskedasticity 1.33 1.67 heteroskedasticity 40 One may argue that the high power in big sample sizes of 500 and 1000 may be suffer from size distortion. A more appropriate power is by examining the sizeadjusted power as used in most of the literature (see for example, Harris and Tzavalis 1999 and Coe and Nason 2004). It is believe that using the size-adjusted power will enables us to better judge the relative performance of the three SD tests. The sizeadjusted power is the power calculated using the actual 5% critical value of the empirical distribution of the test statistics under H0 instead of using the standard critical value of SMM or normal. As the results are similar and for simplicity, the size-adjusted power with k = 10 for the homoskedasticity processes are reported here. Table 2.7: Size-adjusted Power of the DD and Anderson Tests for Homoskedasticity Process N 50 100 500 1000 Average s=1 DD A A. Parameters of Distributions: DD α i = 0.2, s=2 A α j = -0.2, β i = 1.0, s=3 DD β j = 1.0 0.8418 0.7879 0.4666 0.5050 0.5799 0.5834 0.5917 0.5932 0.6200 0.6174 B. Parameters of Distributions: 0.9934 0.9963 0.9961 0.9971 0.9957 α i = 0.0, 0.7850 0.7926 0.8045 0.8173 0.7999 α j = 0.0, β i = 0.0, 0.9888 0.9994 0.9992 0.9988 0.9966 β j = 2.0 50 0.0001 0.3406 0.9428 0.8191 0.9714 100 0.0000 0.0000 0.9462 0.8069 0.9991 500 0.0000 0.0000 0.9380 0.7997 1.0000 1000 0.0000 0.0000 0.9377 0.8027 1.0000 Average 0.0000 0.0852 0.9412 0.8071 0.9926 Notes: The figures in Panel A are the empirical relative frequency of accepting HA1 for s = 1, 2, 3. There are SD between X and Y for s = 1, 2, and thus HA1 is true for s = 1, 2, 3. The figures in Panel B are the empirical relative frequency of accepting HA1 for s = and 3. There are SD between X and Y for s = and and thus HA1 is true for s = and 3. N is the sample sizes, s is the order of SD. A 0.7851 0.7903 0.8040 0.8110 0.7976 0.9844 0.9950 0.9999 1.0000 0.9948 41 Table 2.8: Size-adjusted Power of the KRS Tests for Homoskedasticity Process A s=1 H A1 N 50 100 500 1000 Average N 50 100 500 1000 Average Notes: s=2 H A0 H A1 s=3 H A0 H A1 A. Parameters of Distributions: α i = 0.2, α j = -0.2, β i 0.8634 1.0000 0.9058 1.0000 0.9999 1.0000 1.0000 1.0000 0.9423 1.0000 B. Parameters of Distributions: 0.5669 1.0000 0.7356 1.0000 0.9966 1.0000 1.0000 1.0000 0.8248 1.0000 α i = 0.0, α j = 0.0, β i = 0.0, = 1.0, βj H A0 = 1.0 0.3512 0.5132 0.8988 0.9862 0.6874 β j = 2.0 1.0000 1.0000 1.0000 1.0000 1.0000 H A1 H A0 H A1 H A0 H A1 H A0 0.0013 0.0000 0.0000 0.0000 0.0003 0.9990 1.0000 1.0000 1.0000 1.0000 0.3053 0.3626 0.6050 0.7623 0.5088 1.0000 1.0000 1.0000 1.0000 1.0000 0.6407 0.9204 1.0000 1.0000 0.8903 1.0000 1.0000 1.0000 1.0000 1.0000 The figures in Panel A are the empirical relative frequency of accepting H A1 or H A for s = 1, 2, 3. The figures in Panel B are the empirical relative frequency of accepting H A1 or H A for s = and 3. N is the sample sizes. Table 2.7 summarizes the size-adjusted powers of the DD and Anderson tests while Table 2.8 exhibits the size-adjusted power of the KRS test. The simulation results show that the DD test still has a better power than both the Anderson and KRS tests. Although the power of Anderson and KRS tests increased significantly after adjusted by the empirical size, it does not affect our findings qualitatively. This study also tests different values of α and β for their effects on power. By fixing β i = β j = and α i = 0, the values of α j are changing from -0.2 to 0.2. The relative values of α i and α j will affect the two distributions in terms of first-order SD. The wider the difference between α i and α j , the bigger the gap of SD is. Power decreases and then increases as α j goes from -0.2 to 0.2. Figure 2.5 shows clearly 42 that the DD test has the best power for all values of α j . In a separate exercise, by fixing α i = α j = and β j = 1, and vary the values of β i from to 2. The relative values of β i and β j will alter the second-order SD (and hence altering the third-order SD) relationship between the distributions. The wider the difference between β i and β j , the bigger the respective gap of second-order SD (and third-order SD) is. Power increases by a little, decreases and increases again as β i goes from to 2. Figure 2.6 shows clearly that the DD test has the best power for all the values of β i . 43 Figure 2.5: Power Graph for the DD, Anderson, KRS Tests with Different Values of 0.6 0.5 power 0.4 0.3 0.2 0.1 -0.2 -0.133 -0.067 DD 0.067 Ads 0.133 0.2 KRS Figure 2.6: Power Graph for the DD, Anderson, KRS Tests with Different Values of 1.2 power 0.8 0.6 0.4 0.2 0 0.33 0.67 DD 1.33 Ads 1.67 KRS 44 It is noted that the number of grid points will affect the size- and powerperformance of SD tests. Increasing the number of grid points will reduce the size and power of the SD tests. This phenomenon may be due to the violation of the independence assumption of grids for larger k. However, less grid points may possibly miss out some important information between any two consecutive grids in the underlying distributions. In this study, Tse and Zhang’s (2004) recommendation with k major intervals, where k = 6, 10 and 15 is used. In addition, it is suggested that each major interval be partitioned into 10 (or more) equal-distance minor intervals to ensure non-omission of important information between any consecutive major grids. Nevertheless, the critical values for 10k grids, which will violate the independence assumption of different grids, cannot be used here. Thus, it is suggested to use the critical values in Stoline and Ury (1979) based on k major grids for all the major and minor grids to satisfy the independence assumption of grids. Based on the computing experience which is unreported here, distributions without minor intervals have bigger size and power than the distributions with minor intervals.7 This shows that without the imposition of minor grids, information within the minor intervals could be omitted which may in turn alter the decision. 2.7 Conclusion In empirical studies, returns of different assets are usually correlated. It is important to investigate statistical properties of SD tests by relaxing the independence assumption. This study investigates the size- and power-performance of three commonly used SD tests, namely the DD, Anderson and KRS, when the underlying distributions are correlated, and either homoskedastic or heteroskedastic. Although all the three tests The results are available on request. 45 are conservative, the DD test is found to be the best among the three SD tests. It is also found that the DD test has good power for correlated distributions. While heteroskedasticity significantly reduces the power of SD tests, the power of DD test is still reasonably good for large samples. Larger sample size and sufficient large number but independent grid points are needed to have better power. The Monte Carlo results suggest that when the sample size is at least 500 and that the number of major grid points is around 10 to 15, all three SD tests possess reasonably good power, especially so for the DD test. Furthermore, the simulation results also support the suggestion that each major interval be partitioned into 10 equal-distance minor intervals to ensure non-omission of important information in the underlying distributions. Although the independent assumption is critical for the Anderson and KRS tests, they are tested in this study for comparison purpose. The findings of this study will enable academicians and practitioners to apply the SD tests efficiently when dependence is in place. Empirical comparisons of new tests form interesting topics for future research within the growing literature on SD testing. 46 [...]... Tests for Heteroskedasticity Process s=1 s =2 s=3 DD A DD A DD A 0 .24 53 0 .25 27 0.4940 0 .20 86 0 .28 72 0.0949 Parameters 50 0 .26 20 0 .21 82 0.3309 0.09 52 0.1585 0.0307 of 0 .26 47 0.1546 0 .23 49 0.04 92 0.0886 0.0103 Distributions 0.0579 0.1451 0.8889 0.6 428 0. 729 9 0.4389 100 0.1051 0 .21 11 0.8018 0.4744 0.5880 0 .26 93 α i = 0.0 0.14 42 0 .24 44 0. 724 8 0.3560 0.4 621 0.16 92 α j = 0.0 0.0000 0.0000 1.0000 1.0000 0.9997... results for the independent situation with heteroskedasticity processes are similar with the dependent situation, they are not reported in this chapter 24 Figures 2. 1 and 2. 2 depict the CDFs for different values of α and β and their corresponding second- and third-order integrals as defined in (2. 1) 25 26 27 Figure 2. 1a to Figure 2. 1c depict the CDFs and their corresponding secondand third-order integrals... accepting HA1 for s = 1, 2, 3 There are SD between X and Y for s = 1, 2, 3 and thus HA1 is true for s = 1, 2, 3 N is the sample sizes, k is the number of major grid points; s is the order of SD N k 6 10 15 6 10 15 6 10 15 6 10 15 32 Table 2. 2b: Empirical Power of the DD and Anderson Tests for Homoskedasticity Process s=1 s =2 s=3 DD A DD A DD A Parameters 0 .20 11 0 .25 15 0.6465 0. 422 0 0.4308 0 .22 79 of... 0.0800 0.15 52 0.3437 0.5103 0 .27 23 1.0000 1.0000 1.0000 1.0000 1.0000 0.1303 0.6719 1.0000 1.0000 0.7006 1.0000 1.0000 1.0000 1.0000 1.0000 Notes: 0 The figures in Panel A are the empirical relative frequency of accepting H A1 or H A 2 for s = 1, 2, 3 0 The figures in Panel B are the empirical relative frequency of accepting H A1 or H A 2 for s = 2 and 3 N is the sample sizes 33 Tables 2. 2a and 2. 2b summarize... being used Ideally, we should choose sufficient large number of grids to reveal more information without compromising the grid point’s independence property In view of this, Tse and Zhang’s (20 04) recommendation of making k major intervals where k = 6, 10 and 15 is used in this study In addition, it is suggested that each major interval is then be partitioned into 10 (or more) equal-distance minor intervals... 100 500 1000 0.0001 0.0048 0.7894 0.9 925 0.4467 Average B α i = α j = 0.0 β i = 0.0 β j = 2. 0 Average s =2 H 0 A2 0 H A2 0.0398 0 .21 47 0. 823 8 0.9704 0.5 122 1.0000 1.0000 1.0000 1.0000 1.0000 H 0.1040 0.4003 0.9673 0.9996 0.6178 s=1 H A1 1.0000 1.0000 1.0000 1.0000 1.0000 H A1 1.0000 1.0000 1.0000 1.0000 1.0000 s=3 0 A2 s =2 s=3 N H A1 0 H A2 H A1 0 H A2 H A1 0 H A2 50 100 500 1000 0.0000 0.0000 0.0000... intervals to ensure nonomission of important information between any consecutive major grid points In the simulations, the two generated series are combined and sorted in ascending order This combined sample is used to determine the grid points The generated series are divided into k major intervals and each major interval is then partitioned to ten minor intervals Nevertheless, the critical values... 50 0 .25 90 0.3367 0.4471 0 .20 84 0 .23 45 0.0798 Distributions 0 .28 43 0 .26 12 0.3180 0.1043 0.1435 0. 026 9 0.0086 0. 020 9 0.9854 0.9589 0.9475 0.8807 100 0. 027 4 0.0 622 0.9517 0.8830 0.8519 0.7098 α i = 0.0 0.0478 0.1046 0.9 128 0.78 62 0.7469 0.5430 α j = 0.0 0.0000 0.0000 1.0000 1.0000 1.0000 1.0000 500 0.0000 0.0000 1.0000 1.0000 1.0000 1.0000 β i =0.0 0.0000 0.0000 1.0000 1.0000 1.0000 1.0000 β j =2. 0 0.0000... A2 s =2 s=3 N H A1 0 H A2 H A1 0 H A2 H A1 0 H A2 50 100 500 1000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0455 0.0933 0 .22 23 0.3378 0.1747 1.0000 1.0000 1.0000 1.0000 1.0000 0.0704 0.3431 0.8013 0.9034 0. 529 6 1.0000 1.0000 1.0000 1.0000 1.0000 Notes: 0 The figures in Panel A are the empirical relative frequency of accepting H A1 or H A 2 for s = 1, 2, 3 0 The figures in. .. 0.7714 0.0009 0.5858 0.0004 Parameters 50 0.3830 0.0054 0.6708 0.00 02 0.4745 0.0000 of 0 .27 17 0.0041 0.5873 0.0000 0.3 924 0.0000 Distributions 0. 925 4 0.0 820 0.9711 0.0116 0.9068 0.00 52 100 0. 828 0 0. 028 2 0.9441 0.0016 0.8 523 0.0008 α i = 0 .2 0.7410 0.0084 0.9176 0.0003 0.7990 0.0000 1.0000 0.9959 1.0000 0.98 72 1.0000 0.9159 α j = -0 .2 500 1.0000 0.9749 1.0000 0.9173 1.0000 0.7663 β i = 1.0 1.0000 0.9307 . 11. 0 011 0 001 Define 43 322 1 3 322 1 22 1 1 , where d j is the length of the j th interval. The Anderson test is used to examine the following hypotheses: () . : ,: ,0: , 0)(: : 12 11 0 XYH YXH ppIH ppIH SDorderFirst A A YXfA YXf f f ≠− =− − . reported in this chapter. 25 Figures 2. 1 and 2. 2 depict the CDFs for different values of α and β and their corresponding second- and third-order integrals as defined in (2. 1). 26 . () xS n xS n xGxF xZ GnFn n 2 , 2 , 22 11 ˆ ˆ + − = , and () () () [] .,;,, 1 2 1 2 2 , GFHyzhxHhx n xS n i iHn ==−−= ∑ = + KRS define () 0=xZ n when x is less than the minimum observation of the combined sample.