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Table 4: The Performance of the Best Trading Rule in Other Five Sub-Samples
This table reports the performance of the best one in the universe of 5,162 technical trading rules in each sub-sample. Five sub-samples are created by the measure taken in Brock et al. (1992). The annualized return of the best rule is calculated with one-way transaction cost 0.05%. SPA (C) is the SPA p− value. * denotes significance at the 10% level, **denotes significance at the 5% level, and ***
denotes significance at the 1% level.
Sub-sample the Best Rule in Each Sub-sample Annualized Return SPA (C)
Sub-sample 1 MSV 29.12% 0.428
Sub-sample 2 VMA 10.79% 0.004 ***
Sub-sample 3 VMA 13.55% 0.000 ***
Sub-sample 4 VMA 12.39% 0.186
Sub-sample 5 MSV 13.08% 0.654
2.5 Market Timing Ability Test
We have shown that investors can get higher excess return by the VMA rule, and now we turn to finding the reason possibly explaining its higher profitability. If a technical trading rule can be profitable, the stock return must be predictable. Higher profitability of a trading rule might stem from that it possesses higher predictive ability for stock returns.
Tests of market timing ability are common ways to evaluate whether a forecast is to have value or not in financial literature. These tests are regularly used to study whether mutual fund managers, portfolio managers, investment newsletter recommendations or trading rules’ signals offer any market timing ability (Henriksson, 1984; Cumby and Modest, 1987; Lee and Rahman, 1990; Graham and Harvey, 1996; Kho, 1996; Kleiman et al., 1996; Daniel et al., 1997; Neely and Weller, 1999; Bollen and Busse, 2001). For any trading rule, timing implies that excess returns are positive after its recommended long positions and negative after its recommended short positions. In other words, the forecasting ability of a trading rule can be evaluated by these tests.
In this study, we apply the market timing ability tests by Cumby and Modest (1987) since it is often used for investigating whether a trading rule can predict the sign of a one-period-ahead
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excess return.2 This test is carried out by regressing excess returns on one forecast position measured by an indicator variable z, which is either + 1 or - 1, depending upon whether the trading rule signals a long or a short position. If a trading rule does possess forecasting ability, a significantly positive relation between the one-period-ahead excess return and the forecast position can be found.
We extend the test of Cumby and Modest (1987) by studying forecasting ability of long po-sitions and short popo-sitions separately and including the no-popo-sitions variable. By this extension, in addition to the market timing ability of long positions and short positions, we can also study the average direction of price changes when a trading rule recommends not to have any position.
Since the VMA rule is the variant of the Moving Average system, we focus on comparing the forecasting ability of the best VMA rule with that of the best MA rule. Besides, we also make comparisons between the market timing ability of the best VMA rule and that of the best MSV rule, because the MSV rule is the second best in our previous SPA tests.
In order to carry out a reliable hypothesis testing, in which the market timing ability of the best VMA rule is compared with that of the best MA and best MSV rule, we apply the Seemingly Unrelated Regressions (SUR). We consider the specification of a system of three equations, each with three explanatory variables and 20,496 observations.3 The SUR model is as follows:
2The assumption that relative risk premiums are constant over the sample period is made in this study.
3Here we delete the first 30 observations since all of these three best rules need at least 30 days to generate trading signals.
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4 log P , is the log return of the DJIA (multiplied by 100), and it can also be interpreted as the daily one-period-ahead excess return. The independent variables for each i are three forecast variables as follows, z1i,t, z2i,t and z3i,t. They are set to one when the ith rule recommends no position, a short position, and a long position respectively; otherwise, they are zero. The sum of these three dummy variables at time t is one. β1i, β2iand β3iare slope coefficients for the ith rule.If the ithrule does possess ability to detect future downward and upward trends, β2iand β3i
will be significantly negative and positive, respectively. A positive β2i (negative β3i) denotes that the ith rule predicts there will be a downward (upward) trend in the future, but the actual price rises (falls). Further, we can interpret the value of β2i(β3i) as the mean profit/loss per day over the period in which the ith rule signals short (long) positions. Then β1i can be explained as the mean price change as the ith rule recommends not to have any position. In Panel A of Table 5, we present the estimation results for the SUR model along with the Newey-West (1987) robust standard errors. First of all, there is strong evidence indicating that the best VMA rule does have predictive ability for both upward and downward price movements, while the
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best MA and best MSV rule are only capable of detecting upward price movements. We also compare the mean profit per day based on signals of long position and short-position issued by the best VMA rule, the best VMA rule, and the best MSV rule. The Wald test results in Table 6 show that the mean profit the best VMA rule gains from its long positions is significantly greater than those of the others. Its mean profit over the short-position periods is significantly greater than that of the best MSV rule, while there is no significant difference in the average short-position profit between the best VMA rule and the best MA rule.
In addition to mean profits over the short-position periods and the long-position periods for the three competing rules (based on the size and value of β2and β3), we also compute the mean profit per day (MPPD) as follows:
MPPDi = wli∗β3i− wsi∗β2i, (12)
where wli∗ and wsi∗ for each rule are the proportion of time spent on long positions and on short positions relative to the entire sample period, respectively. Essentially, the MPPD can be interpreted as an overall performance measure of a trading rule, which is weighted by the proportions of time spent on long positions and short positions. The results are reported in Panel B and C of Table 5. All of those three rules have significantly positive mean profit per day. Furthermore, the best VMA rule earns more than the others, as presented in Table 6.
To summarize, the fact that the best VMA can predict both upward and downward price movements, while other competing rules are only capable of detecting upward price movement give a strong evidence that higher profitability of the VMA rule might just stems from its better forecasting ability of stock returns. In addition, the results that the time the best VMA spent in the market is the least (roughly 63.6%), while the mean profit it gains in the market is the most (see Table 6), offer another piece of evidence that the best VMA rule does enjoy a better timing in generating profitable trading signals.
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Table 5: Cumby-Modest market timing tests for the best VMA, best MA and best MSV rule
We carry out Cumby-Modest market timing tests by applying Seemingly Unrelated Regressions (SUR) as follows:
4 log P = Xiβi+ ei, i = VMA, MA, MSV,
where Xi = (z1i, z2i, z3i) and βi = (β1i, β2i, β3i)0. For each equation i, the dependent variable 4 log P , is the log return of the DJIA (multiplied by 100). The independent variables, z1i,t, z2i,tand z3i,t, are three dummy variables. They are set to one when the ith rule recommends no position, a short position, and a long position respectively; otherwise, they are set to zero. β1i, β2iand β3iare slope coefficients for the ithrule. We use w∗l and w∗s, the proportion of the time spent long and short to the full sample period, to measure a trading rule’s overall performance, the mean profit per day (MPPD). Panel A reports the regression results for the best VMA, best MA and best MSV rule, respectively. Panel B presents their overall performances. The w∗l and ws∗of each rule is reported in Panel C. In the parentheses are the Newey-West (1987) robust standard errors. * denotes significance at the 10% level, ** denotes significance at the 5%
level, and *** denotes significance at the 1% level.
Position Coefficient i = VMA i = MA i = MSV
Panel A: Regression slopes
No Position β1i 0.020 0.008 -0.003
(0.01) (0.01) (0.06)
Short Position β2i -0.117 *** -0.057 -0.023
(0.02) (0.04) (0.02)
Long Position β3i 0.105 *** 0.041 *** 0.039 ***
(0.01) (0.01) (0.01)
Panel B: The overall trading performance
Mean profit per day (MPPDi) wli∗β3i− wsi∗β2i 0.070 *** 0.029 *** 0.033 ***
(0.01) (0.01) (0.01)
Panel C: Weights adopted in theMPPDi
The time spent long/Full sample size wli∗ 0.382 0.543 0.651
The time spent short/Full sample size w∗si 0.254 0.126 0.324
The total trading time/Full sample size wi 0.636 0.669 0.975
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Table 6: Comparisons in the trading performances of the best VMA, best MA and best MSV rule
This table reports comparisons between the trading performances of the best VMA rule and that of the best MA and best MSV rule. For the ithrule, the value of β2iand β3ican be interpreted as the mean profit/loss per day over its short-position periods and long-position periods, while β1iis the average excess return when there is no signal released by this rule. The MPPDimeasures its mean profit per day. These comparisons are implemented by the Wald test based on the Newey-West (1987) covariance matrix. In the parentheses are the Newey-West (1987) robust standard errors. * denotes significance at the 10% level, ** denotes significance at the 5% level, and ***
denotes significance at the 1% level.
2.6 Is the Profitability of the Trading Rule Asymmetric in Different