2.6 Is the Profitability of the Trading Rule Asymmetric in Different Market Condi-
2.6.1 Market Timing Ability in Bull and Bear Market
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lower variance; while bear markets are associated with lower average, but a more volatile stock returns. Specifically, we are interested in investigating whether the profitability of the VMA rule in bull markets is significantly different from that in bear markets? If the profitability of the VMA rule in bear markets is significantly higher, it may imply that the information of mar-ket volatility in bear marmar-kets is more crucial for technical trading rules to forecast future price movements.
In this study, the dating algorithm proposed by Pagan and Sossounov (2003) is used to identify bull markets and bear markets in the DJIA daily index.4,5 According to Pagan and Sossounov (2003), a bull market occurs during the period when the stock price rises from a trough point and ends in a peak point; while a bear market exists during the period as the stock price moves from a peak point to a trough point. With this dating algorithm, all possible peaks and troughs in the DJIA index can be found and subsequently be used to identify bull markets and bear markets. Table 7 presents a summary of all bull markets and bear markets so identified.
There are 23 and 22 mutually exclusive and exhaustive bull markets and bear markets. We can find that, on average, durations of bull markets tend to be longer than those of bear markets in the DJIA. Specifically, Bull markets have lasted for an average of 28 months (583 days), while bear markets have lasted for 15 months (324 days). Table 7 also shows that bull markets are associated with higher but more stable stock returns while bear markets are associated with low but volatile stock returns. These results are coincident with the findings in the literature.
2.6.1 Market Timing Ability in Bull and Bear Market
To ascertain whether the VMA rule has asymmetric, yet still dominating, performance over the equity cycles, the test of Cumby and Modest (1987) introduced in Section 2.5 is applied here again with the number of independent variables for each rule Xi extended from three to six as
4The dating algorithm of Pagan and Sossounov (2003) is presented in the Appendix.
5The original parameter settings in Pagan and Sossounov’s dating algorithm are suitable for monthly data. In order to apply this algorithm to daily data, we revise the parameters based on the premise that there is 21 trading days in one month.
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Table 7: Bull and bear markets in the DJIA daily price index
This table reports a summary of the bull and bear markets identified with Pagan and Sossounov (2003) algorithm over the full sample period. In the ”Days” and ”Duration” columns, we record the number of days for the whole bull and bear markets and the average days a bull or a bear market lasts.
Market Definition Number Days Duration log Return, Mean log Return, S.D.
Bull Trough to peak 23 13,407 583 0.094 1.012
Bear Peak to trough 22 7,119 324 -0.124 1.397
follows:
Xi = (zbl1i, z2ibl, zbl3i, z1ibr, z2ibr, z3ibr), i =VMA, MA, MSV, (13) where the superscript bl and br represent bull and bear markets in the DJIA; z1ibl, zbl2iand z3ibl are dummy variables of the ith rule in bull markets, which are set to one when the ith rule in bull markets recommends no position, a short position, and a long position respectively; otherwise, they are set to zero; similarly, z1ibr, zbr2i and zbr3i are the corresponding dummy variables for the ithrule in bear markets. Correspondingly, β1ibl, β2ibl, β3ibl, β1ibr, β2ibr and β3ibr are the associated slope coefficients for the ithrule. If the ithrule can forecast future price increases (decreases) in all market conditions, β3ibl and β3ibr (β2ibl and β2ibr) should be significantly positive (negative). We also useMPPDto measure the overall trading performance of the rule.
Table 8 reports the results of Cumby-Modest market timing tests for the best VMA, best MA and best MSV rule in bull markets and bear markets. According to the estimated results, the predictive ability of the best VMA rule for price movements is indeed asymmetric in bull and bear markets. It can forecast price increases in bull markets, but not in bear markets. Conversely, it is capable of detecting downward price actions in bear markets but not in bull markets. For the best MA and the best MSV rule, the asymmetry in their respective forecasting ability under different market conditions still exists, and is similar to that in the best VMA rule. Moreover, they incur losses significantly on average as they forecast price increases in bear markets and price decreases in bull markets.
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As reported in Table 9, the profitability, measured by theMPPD, of the best VMA rule in bull and bear markets are all significantly higher than those of the best MA and the best MSV rule.
This answers our second question. In other words, the higher profitability of the best VMA rule is not sensitive to different market conditions. In bull markets, the mean profit from the long positions signaled by the best VMA rule is significantly higher, and the mean loss entailed from its short-position signals is significantly less than the other two trading rules. In bear markets, although there is no significant difference between the mean profit in the downward forecasting of the best VMA rule and those of the others, the best VMA rule has significant less mean loss in the upward prediction.
It is noteworthy that, in bear markets, the mean profit of the best MA rule and the best MSV rule are not significantly different from 0, while the mean profit of the best VMA rule is significantly positive. as shown in Panel B of Table 8. This offers another piece of evidence that, when asset prices change rapidly or drastically on the market, technical trading rules which do not incorporate any market volatility information cannot respond enough to the changing market conditions. Therefore, our results suggest the information of market volatility in this kind of market should be seriously considered in technical trading rules. Another interesting result in Table 10 is that the trading performance of the best VMA rule in bear markets is better than that in bull markets. As mentioned before, better profitability of the VMA rule in bear markets may imply that the device for detecting price movements, the market volatility ratio, is particularly suitable for bear markets.
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Table 8: Cumby-Modest market timing tests for the best VMA, best MA and best MSV rule in bull and bear markets
The market timing test for the trading rules in bull and bear markets is carried out by Seemingly Unrelated Regressions (SUR) as follows:
4 log P = Xiβi+ ei, i = VMA, MA, MSV,
where Xi = (zbl1i, z2ibl, z3ibl, zbr1i, z2ibr, z3ibr) and βi = (βbl1i, βbl2i, βbl3i, βbr1i, β2ibr, βbr3i)0. For each equation i, the dependent variable, 4 log P , is the log return of the DJIA (multiplied by 100). The market conditions, bull and bear markets, are represented by the index bl and br. The independent variables, z1i,tbl , z2i,tbl and z3i,tbl , (z1i,tbr , z2i,tbr and z3i,tbr ), are dummy variables for bull (bear) markets. They are set to one when the ithrule in bull (bear) markets recommends no position, a short position, and a long position respectively; otherwise, they are set to zero. β1ibl, β2ibland β3ibl(β1ibr, βbr2i and βbr3i) are the associated slope coefficients for the ithrule in bull markets. Panel A reports the regression results and trading performances for these three best rules in bull markets while the corresponding results in bear markets are presented in Panel B. We use w∗l and ws∗to measure the trading performance of the rule, the mean profit per day (MPPD).
w∗bll (ws∗bl) is defined as the proportion of the time spent long (short) to all bull periods, while wl∗br(w∗brs ) is the ratio the trading rule spent for long (short) positions to all bear periods. The results are reported in Panel C on next page. In the parentheses are 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.
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Table 8 Continued
Measurement Weight i = VMA i = MA i = MSV
Panel C: Weights adopted in measuring a technical trading rule’s performance Bull market
The time spent long/All bull periods wli∗bl 0.511 0.658 0.709
The time spent short/All bull periods wsi∗bl 0.138 0.077 0.261
The total trading time/All bull periods wibl 0.649 0.735 0.970
Bear market
The time spent long/All bear periods wli∗br 0.141 0.327 0.544
The time spent short/All bear periods wsi∗br 0.473 0.217 0.443
The total trading time/All bear periods wibr 0.614 0.544 0.987
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Table 9: Comparisons in the trading performances of the trading rules in each market condition
This table reports comparisons between the trading performances of the best VMA rule and that of the best MA and best MSV rule in bull and bear markets, respectively. Bull and bear markets are represented by the index bl and br. For the ithrule, the value of β2ibland β3ibl (βbr2i and β3ibr) can be interpreted as the mean profit/loss over its short-position periods and long-position periods in bull (bear) markets, while β1ibl(βbr1i) is the average excess return over the no-position periods in bull (bear) markets. MPPDbl
i (MPPDbr
i ) measures the mean profit per day the ithrule gains in bull (bear) markets. These comparisons are implemented by the Wald test based on the Newey-West (1987) covariance matrix. The Newey-West (1987) robust standard errors are in the parentheses. * denotes significance at the 10% level,
** denotes significance at the 5% level, and *** denotes significance at the 1% level.
Position Comparison i = VMA, j = MA i = VMA, j = MSV
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Table 10: Comparisons in the trading performances of one trading rule in different market conditions
This table reports comparisons between the trading performances of one trading rule, including the best VMA, best MA, and best MSV rule, in bull markets and that in bear markets. Bull and bear markets are represented by the index bl and br respectively. For the ithrule, the value of βbl2iand β2ibr(βbl3iand β3ibr) can be explained as the mean profit/loss over its short-position (long-positions) periods in bull and bear markets, whileMPPDbl
i andMPPDbr
i measure its mean profit per day in these two market conditions. We use the Wald test to implement these comparisons, 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.
Rule (i) Position Comparison bl = Bull, br = Bear
VMA Short Position β2ibl− β2ibr= 0 0.221 ***