# Technical Analysis and the London Stock Exchange: Testing Trading Rules Using the FT30

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6 324 T. C. Mills Table 4. Results for MA rules: sample, 1955± , 50, (1.92) (2.23) (3.59) 1, 50, (1.60) (2.69) (3.63) 1, 150, (1.69) (1.94) (3.14) 1, 150, (1.64) (1.96) (3.07) 5, 150, (1.54) (1.77) (2.86) 5, 150, (1.60) (1.79) (2.88) 1, 200, (1.80) (2.05) (3.34) 1, 200, (1.83) (2.08) (3.35) 2, 200, (1.66) (1.91) (3.09) 2, 200, (1.67) (1.97) (3.11) Average Rules are identi ed as (n, m, b), where b is the band, either 0 or `N(Buy)' and `N(Sell)' are the number of buy and sell signals generated by the rule. `Buy > 0' and `Sell > 0' are the fraction of buy and sell returns greater than zero. Numbers in parentheses are standard t-statistics testing the difference between the mean buy return and the unconditional mean return, between the mean sell return and the unconditional mean return and between buy sell and zero. The last row reports averages across all ten rules. The mean returns are scaled by a factor of 1000 for easier interpretation. with q s de ned analogously. The conditional standard deviations may then be de ned as s b ˆ E r h t q b 2 jb t Š 1=2 and, analogously, s s. These conditional expectations are estimated using appropriate sample means calculated from the FT30 returns and compared to expectations estimated in a similar way from simulated models. A decision has to be made as to what model(s) should be used to simulate the comparison series. Two features of the autocorrelations reported in table 1 inform our decision. First, it is clear that the correlation structure of returns is not constant across the whole sample, so it would be inappropriate to t a single model to the entire data set. Second, the returns exhibit autocorrelation in both levels and squares, so that ARMA models with GARCH disturbances are the natural class to focus attention upon. Using the three subperiods used above, we tted AR±ARCH models to the returns, the estimated models being shown in table 10. For all three samples an ARCH(3) process was required to t the conditional variance h t. For the two earlier periods the conditional mean return was adequately tted by AR(2) processes, albeit with rather different estimates, whereas the nal period required an AR(1) with intercept, a re ection of the pronounced (stochastic) trend in the index during these years. Extensions such as GARCH conditional variance equations and including h t as a regressor in the the conditional mean equation (GARCH-in-mean) were investigated, but were unsuccessful. The models presented in table 10 do, nevertheless, provide satisfactory statistical ts. The bootstrap simulations are performed by resampling, with replacement, the residuals from the tted models and, using these scrambled residuals and the estimated parameters, generating new return series from the models, from which the # 1997 John Wiley & Sons, Ltd.

9 Technical Analysis and London Stock Exchange 327 Table 9. Results for TRB Rules: sample, 1975± , (1.10) (0.06) (0.70) 50, (1.01) (0.15) (0.74) 150, (0.50) (0.66) (0.38) 150, (0.61) (0.24) (0.53) 200, (0.66) (0.63) (0.31) 200, (0.54) (0.10) (0.18) Average Cumulative returns are reported for xed 10-day periods after signals. Rules are identi ed as (m, b), where b is the band, either 0 or `N(Buy)' and `N(Sell)'" are the number of buy and sell signals generated by the rule. `Buy > 0' and `Sell > 0' are the fraction of buy and sell returns greater than zero. Numbers in parentheses are standard t-statistics testing the difference between the mean buy return and the unconditional 1-day mean return, between the mean sell return and the unconditional mean return and between buy sell and zero. The last row reports averages across all six rules. The mean returns are scaled by a factor of 1000 for easier interpretation. mean sell return, while none of the simulated series generated mean buy sell differences larger than the mean difference for the FT30. These results are consistent with the traditional tests presented in the corresponding table 3. The s b and s s entries show that every simulated buy conditional standard deviation exceeded that of the analogous FT30 standard deviation, whereas none of the simulated sell standard deviations was larger than the corresponding value from the observed series. The buy signals therefore pick out periods where higher conditional means are accompanied by lower volatilities, in contrast to sell periods where the conditional return is lower and volatility is Table 10. AR±ARCH models tted to sub-periods. 1935±1954: AR(2)±ARCH(3) r t ˆ 0:4225 r t 1 0:0626 r t 2 e t e t ˆ h 1=2 t z t z t N 0; 1 0:0769 0:0160 h t ˆ 0: :3529 e 2 t 1 0:2909 e 2 t 2 0:2339 e 2 t 3 0: :0229 0:0259 0: ±1974: AR(2)±ARCH(3) r t ˆ 0:2702 r t 1 0:1015 r t 2 e t e t ˆ h 1=2 t z t z t N 0; 1 0:0170 0:0157 h t ˆ 0: :3631 e 2 t 1 0:1676 e 2 t 2 0:1037 e 2 t 3 0: :0281 0:0212 0: ±1994: AR(1) with intercept±arch(3) r t ˆ 0:0006 0:0682 r t 1 e t e t ˆ h 1=2 t z t z t N 0; 1 0:0001 0:0156 h t ˆ 0: :1734 e 2 t 1 0:2098 e 2 t 2 0:1414 e 2 t 3 0: :0208 0:0222 0:0195

11 Technical Analysis and London Stock Exchange 329 Figure 2. FT30 trading pro ts 1935±1954. Figure 3. FT30 trading pro ts 1975±1994. gure 3 that the performance of the two strategies was very similar in the early years of the period when the market was growing relatively modestly. It is only when the market trend begins to dominate that the trading rule performs poorly: in particular, it misses out on the acceleration of prices in the early 1980s. The exponential growth observed in the FT30 in this period might suggest that a moving average rule based on the logarithms of x t (or, equivalently, a moving geometric average of x t itself) would be more appropriate. This was, in fact, tried, but gave an almost identical set of signals to the conventional moving (arithmetic) average.

12 330 T. C. Mills Table 11. Simulation tests from AR±ARCH bootstraps: sample, 1935±1954. Rule Result Buy s b Sell s s Buy Sell MA , 50, 0 % > FT30 TRB MA , 50, 0.01 % > FT30 TRB MA , 150, 0 % > FT30 TRB MA , 150, 0.1 % > FT30 TRB MA , 200, 0 % > FT30 TRB MA , 200, 0.01 % > FT30 TRB Returns are simulated 500 times using the estimated parameters and resampled, with replacement, residuals. % > FT30 denotes the fraction of simulations generating a mean or standard deviation larger than those from the actual FT30 series. Table 12. Simulation tests from AR±ARCH bootstraps: sample, 1955±1974. Rule Result Buy s b Sell s s Buy Sell MA , 50, 0 % > FT30 TRB MA , 50, 0.01 % > FT30 TRB MA , 150, 0 % > FT30 TRB MA , 150, 0.1 % > FT30 TRB MA , 200, 0 % > FT30 TRB MA , 200, 0.01 % > FT30 TRB Returns are simulated 500 times using the estimated parameters and resampled, with replacement, residuals. % > FT30 denotes the fraction of simulations generating a mean or standard deviation larger than those from the actual FT30 series. # 1997 John Wiley & Sons, Ltd.

13 Technical Analysis and London Stock Exchange 331 Table 13. Simulation tests from AR±ARCH bootstraps: sample, 1975±1994. Rule Result Buy s b Sell s s Buy Sell MA , 50, 0 % > FT30 TRB MA , 50, 0.01 % > FT30 TRB MA , 150, 0 % > FT30 TRB MA , 150, 0.1 % > FT30 TRB MA , 200, 0 % > FT30 TRB MA , 200, 0.01 % > FT30 TRB Returns are simulated 500 times using the estimated parameters and resampled, with replacement, residuals. % > FT30 denotes the fraction of simulations generating a mean or standard deviation larger than those from the actual FT30 series. It would therefore seem to be the case that trading rules `worked' when the market was driftless (see the return models tted in table 10, which did not have signi cant constants for the two earlier periods). These were years when, it might be argued, the market was less ef cient, in the sense of being more predictable, both linearly and nonlinearly: the R 2 of the AR±ARCH model tted to the 1935±1954 data is 0.145, while that of the model tted to the 1975±1994 data is only The results for the rst forty years of our sample are thus consistent, in almost every respect, with those of Brock et al. (1992) for the Dow in New York. It is in the last of our sub-samples that the performance of the trading rules deteriorates badly. Indeed, it could be argued from gure 3 that it is only from the early 1980s that a buy-and- hold strategy begins clearly to dominate. This, in fact, might not be inconsistent with the behaviour of the Dow. The sample of Brock et al. ended in 1986 and their nal subperiod began in 1962, so they were not in a position to be able to isolate any `structural shift' that might have taken place around An analysis of the recent behaviour of the New York Stock Exchange along these lines would thus be an interesting exercise to undertake. REFERENCES Allen, H. and Taylor, M. P. (1990) `Charts, noise and fundamentals in the London Foreign Exchange Market'. Economic Journal, 100 (Conference), 49±59. Brock, W., Lakonishok, J. and LeBaron, B. (1992) `Simple technical trading rules and the stochastic properties of stock returns'. Journal of Finance, 47, 1731±64. Efron, B. and Tibshirani, R. J. (1993) An Introduction to the Bootstrap. New York: Chapman and Hall. Malkiel, B. (1981) A Random Walk Down Wall Street, 2nd edn. New York: Norton. Maravall, A. (1983) `An application of nonlinear time series forecasting'. Journal of Business and Economic Statistics, 1, 66±74. NeftcËi, S. N. (1991) `Naive trading rules in nancial markets and Weiner±Kolmogorov prediction theory: A study of ``technical analysis'' '. Journal of Business, 64, 549±71. Sweeney, R. (1988) `Some new lter rule tests: Methods and results'. Journal of Financial and Quantitative Analysis, 23, 285±300. Taylor, M. P. and Allen, H. (1992) `The use of technical analysis in the foreign exchange market'. Journal of International Money and Finance, 11, 304±14.

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