.000 120.003 76
(1, 50)
Fraction > XrateSimulation meanXrate mean
0.010 000.000 860.002 13
0.400 000.013 930.014 02
0.740 000.648 140.594 31
0.980 000.001 050.000 71
0.450 000.013 920.013 97
0.000 000.000 180.002 84
Average Fraction > XrateSimulation meanXrate mean
0.000 000.000 870.002 41
0.470 000.013 950.013 94
0.720 000.617 610.564 39
1.000 000.001 020.001 00
0.520 000.013 900.013 79
0.000 000.000 140.003 41
Buy±Sell
Technical trading rules and regime shifts in foreign exchange
15
Table 1.5 Skewness kurtosisRule
Result
Buy skew
Buy kurt.
Sell skew
Sell kurt.
(1, 20)(1, 30)(1, 50)
Fraction > XrateFraction > XrateFraction > Xrate
0.572 000.504 000.618 00
0.266 000.306 000.330 00
0.336 000.422 000.266 00
0.480 000.488 000.480 00
Average
Fraction > XrateSimulation meanXrate mean
0.564 000.198 990.134 46
0.296 005.416 805.941 69
0.326 000.217 490.335 41
0.480 005.459 275.511 24
(1, 20)(1, 30)(1, 50)
Fraction > XrateFraction > XrateFraction > Xrate
0.328 000.300 000.306 00
0.202 000.168 000.076 00
0.446 000.354 000.346 00
0.618 000.666 000.798 00
Average
Fraction > XrateSimulation meanXrate mean
0.312 000.341 520.443 44
0.136 004.282 225.553 64
0.350 000.353 440.394 10
0.724 004.279 913.437 40
(1, 20)(1, 30)(1, 50)
Fraction > XrateFraction > XrateFraction > Xrate
0.340 000.364 000.354 00
0.528 000.524 000.542 00
0.314 000.334 000.392 00
0.620 000.628 000.708 00
Average
Fraction > XrateSimulation meanXrate mean
0.330 000.383 540.490 69
0.530 005.089 464.964 34
0.346 000.407 570.541 88
0.668 005.052 354.631 71
BP
DM
JY
volatile than average and the sells were more volatile than average. Theresults are fairly weak for the average rule, but checking the individual rulesstronger rejections are found for the thirty- and ®fty-week moving averagesindividually. This result moves counter to a simple mean variance connectionfor the exchange rate from a dollar perspective. The higher conditionalreturns from the buy period should be compensating for more risk, butthese results show that for the DM the risk (in terms of own standarddeviation) is lower. Although this is puzzling, measuring the riskiness of aforeign exchange series is more complicated than estimating the standarddeviation, so strong conclusions about risk premia require more adequatemodelling of the exact risk±return trade o.Another check for changes in the conditional distributions of returns isperformed in Table 1.5. In this table, skewness and kurtosis are estimated forthe returns during the buy and sell periods. It is possible that these higher
16
Advanced Trading RulesTable 1.6 Subsamples: random walk
Buy
b
FractionBuySell
s
Fraction > XrateFraction > XrateFraction > Xrate
0.022 000.012 000.010 00
0.976 000.946 000.888 00
0.160 000.106 000.124 00
0.998 000.998 001.000 00
0.090 000.058 000.052 00
0.002 000.000 000.000 00
Average Fraction > XrateSimulation meanXrate mean
0.006 000.000 800.001 03
0.952 000.011 680.010 25
0.118 000.396 140.510 48
1.000 000.000 590.002 67
0.064 000.011 720.012 88
0.000 000.000 220.003 70
Fraction > XrateFraction > XrateFraction > Xrate
0.206 000.070 000.074 00
0.200 000.244 000.366 00
0.612 000.462 000.498 00
0.694 000.834 000.804 00
0.694 000.648 000.510 00
0.184 000.050 000.044 00
Average Fraction > XrateSimulation meanXrate mean
0.094 000.000 110.001 48
0.246 000.016 470.017 19
0.518 000.524 700.519 00
0.788 000.000 500.000 47
0.630 000.016 520.016 09
0.052 000.000 380.001 95
Fraction > XrateFraction > XrateFraction > Xrate
0.108 000.196 000.134 00
0.960 000.996 000.996 00
0.180 000.122 000.206 00
0.992 000.966 000.994 00
0.058 000.008 000.002 00
0.008 000.020 000.006 00
Average Fraction > XrateSimulation meanXrate mean
0.134 000.000 080.001 01
0.994 000.012 590.010 67
0.174 000.521 430.619 35
0.992 000.000 260.001 64
0.010 000.012 650.014 76
0.004 000.000 180.002 64
Fraction > XrateFraction > XrateFraction > Xrate
0.104 000.046 000.040 00
0.254 000.392 000.400 00
0.450 000.454 000.734 00
0.950 000.996 000.948 00
0.698 000.600 000.416 00
0.030 000.002 000.006 00
Average Fraction > XrateSimulation meanXrate mean
0.048 000.001 030.002 55
0.330 000.015 690.015 99
0.562 000.626 560.607 92
0.986 000.001 410.000 84
0.566 000.015 770.015 49
0.000 000.000 370.003 39
Fraction > XrateFraction > XrateFraction > Xrate
0.002 000.000 000.012 00
0.754 000.782 000.792 00
0.528 000.536 000.410 00
0.998 000.998 000.982 00
0.516 000.414 000.210 00
0.002 000.000 000.000 00
Average Fraction > XrateSimulation meanXrate mean
0.000 000.000 070.002 17
0.792 000.012 460.011 78
0.488 000.528 960.535 48
0.998 000.000 390.001 85
0.330 000.012 430.012 74
0.000 000.000 320.004 02
Rule
Result
Buy±Sell
BP ®rst half(1, 20)(1, 30)(1, 50)
BP second half(1, 20)(1, 30)(1, 50)
DM ®rst half(1, 20)(1, 30)(1, 50)
DM second half(1, 20)(1, 30)(1, 50)
JY ®rst half(1, 20)(1, 30)(1, 50)
Technical trading rules and regime shifts in foreign exchange
17
Table 1.6 (continued)
Buy
b
FractionBuySell
s
Fraction > XrateFraction > XrateFraction > Xrate
0.100 000.114 000.082 00
0.432 000.464 000.260 00
0.690 000.798 000.802 00
0.958 000.940 000.964 00
0.908 000.914 000.942 00
0.016 000.032 000.014 00
Average Fraction > XrateSimulation meanXrate mean
0.088 000.001 390.002 62
0.386 000.015 210.015 50
0.772 000.674 450.604 69
0.972 000.001 850.000 33
0.950 000.015 320.013 55
0.012 000.000 460.002 95
Rule
Result
Buy±Sell
JY second half(1, 20)(1, 30)(1, 50)
moments might give a better indication of the riskiness of returns during eachof the periods. This table combines the results for the three series into onetable. The individual tests are summarized with a single row entry giving theirsimulated p-value and the averages are presented in three rows for eachexchange rate. This table shows little dierence in the higher momentsfrom the actual series buy and sell periods and their simulation counterparts.Table 1.6 considers the stability of these results over various subsamples. Itis quite possible that these rules may be picking up certain nonstationarities inthe data series. The rules themselves are probably very good at checking forchanges in regime. If these regime changes are relatively infrequent, thensplitting the sample into two and repeating the tests makes it less likely that therules will detect any dierences between the buy and sell periods. Table 1.6presents results from such an experiment, where each series is broken in halfand the previous random walk simulations are repeated for each subsample.For the BP, the results are basically unchanged across the subsamples.However, the trading rule results look slightly less signi®cant in the secondsubsample. The simulated p-value for the average buy±sell dierence movesfrom 0 to 0.052. Also, the average buy±sell dierence falls from 0.37 per centto 0.195 per cent. The DM series shows similar results for the buy and sellmeans in the two dierent subsamples. The p-value for the average buy±selldierence moves from 0.004 in the ®rst subsample to 0 in the secondsubsample. The average buy±sell dierence increases from 0.26 per cent to0.34 per cent. For the standard deviations, the results look dierent. For thestandard deviations, the small volatility during buy periods is coming entirelyfrom the ®rst subperiod. For the average standard deviations, the p-value forb is 0.994 for the ®rst subsample and 0.330 for the second subsample. Theresults on s also are much stronger during the ®rst subsample with a p-valueof 0.01 during the ®rst subsample and 0.566 during the second subsample.
18
Advanced Trading RulesTable 1.7 GARCH(1, 1) parameter estimatesxt  a  b1 xt 1  b2 xt 2  "tht 
Xrate
0
0 
21"t 1 
1=2
"t  h t zt
ht 1
zt  N 0; 11
b1
b2
a
BP
2.2940(0.3504)
0.7287(0.0363)
0.1680(0.0303)
0.0832(0.0391)
0.0324(0.0393)
3.4473(4.5733)
DM
1.4131(0.3480)
0.7539(0.0319)
0.1889(0.0289)
0.0604(0.0378)
0.0935(0.0349)
6.8368(4.2092)
JY
1.3460(0.2403)
0.7610(0.0321)
0.1875(0.0308)
0.1179(0.0380)
0.0832(0.0389)
6.3114(4.1427)
Estimation is by maximum likelihood. Numbers in parentheses are asymptotic standard errors.
The results for the JY series change very little from the ®rst to the secondsubsample. The mean buy±sell dierence falls from 0.4 per cent to 0.3 percent. The p-value for this number goes from 0 to 0.012.
1.4.3 GARCH comparisonsTable 1.7 shows the parameter estimates for GARCH(1,1)-AR(2) model foreach of the three exchange rate series. The estimates show very similarestimates for the variance parameters, and 1 , for the three exchange rateseries. The AR(2) parameters show some signi®cant persistence in exchangerate movements for all three series, but the yen and the mark both show asomewhat larger amount of persistence with both the AR(1) and AR(2)parameters signi®cant.Standardized residuals from this model are run back through the samemodel to generate simulated time series for the three exchange rate series.Results of these simulations are presented in Table 1.8. This table shows thatthe GARCH model combined with the AR(2) causes some increase in themean buys and some decrease in the mean sells. Most of this is probablycoming from the persistence in the AR(2). However, the magnitude of thesedierences is not as great as that for the actual series.For the BP, the average buy±sell dierence for the three tests is 0.07 percent which compares with 0.29 per cent for the actual series. The simulatedp-value here is 0.01. For the BP, the GARCH model leaves the previous
Technical trading rules and regime shifts in foreign exchange
19
Table 1.8 GARCH bootstrap
Result
Buy
b
FractionBuySell
s
Fraction > XrateFraction > XrateFraction > Xrate
0.176 000.048 000.020 00
0.552 000.574 000.588 00
0.394 000.276 000.304 00
0.818 000.958 000.970 00
0.530 000.504 000.424 00
0.094 000.008 000.008 00
Average Fraction > XrateSimulation meanXrate mean
0.056 000.000 080.001 24
0.570 000.014 740.014 19
0.316 000.465 590.494 96
0.944 000.000 580.001 66
0.486 000.014 920.014 60
0.010 000.000 660.002 90
Fraction > XrateFraction > XrateFraction > Xrate
0.284 000.258 000.232 00
0.782 000.858 000.878 00
0.404 000.322 000.556 00
0.928 000.942 000.934 00
0.646 000.494 000.422 00
0.076 000.070 000.050 00
Average Fraction > XrateSimulation meanXrate mean
0.250 000.001 220.001 70
0.850 000.015 820.013 60
0.426 000.589 330.600 85
0.944 000.000 210.001 06
0.522 000.015 580.015 12
0.054 000.001 010.002 76
Fraction > XrateFraction > XrateFraction > Xrate
0.134 000.086 000.132 00
0.734 000.718 000.670 00
0.682 000.738 000.634 00
0.920 000.944 000.896 00
0.642 000.606 000.568 00
0.046 000.026 000.038 00
Average Fraction > XrateSimulation meanXrate mean
0.114 000.001 460.002 41
0.710 000.016 110.013 94
0.692 000.596 720.564 39
0.932 000.000 160.001 00
0.602 000.015 190.013 79
0.028 000.001 300.003 41
Rule
Buy±Sell
BP(1, 20)(1, 30)(1, 50)
DM(1, 20)(1, 30)(1, 50)
JY(1, 20)(1, 30)(1, 50)
Results from simulations of 500 GARCH models. These models are generated from estimatedparameters and standardized residuals from maximum likelihood.
results unchanged. Also, there are no eects on volatility as previouslymentioned.For the DM and JY series, the GARCH model has a slightly strongereect. The simulations generate average buy±sell dierences of 0.10 and 0.13per cent respectively. The p-values for these dierences are now 0.054 and0.028 respectively. The added persistence of the AR(2) has caused a largebuy±sell dierence for these series. Although this does have a small impact onthe results from the simulations, the dierences remain small relative to thebuy±sell dierence for the actual series.
20
Advanced Trading Rules
1.4.4 Regime shift bootstrapSome of the results for the GARCH model suggest that although this modelis moving in the right direction, the persistence generated is not strong enoughto generate the trading rule results that are seen in the data. The rules usedcontinue to generate buy or sell signals after the price has cut through themoving average, not just in the neighbourhood of the moving average.Long-range persistence could be generated using the regime shifting modelused by Engle and Hamilton (1990). In this model, conditional means andvariances follow a two-state Markov process. The parameter estimates forthis model are given in Table 1.9. For only one of the three series, the JY, areboth the conditional mean parameters signi®cantly dierent from zero. Forthe BP series, they are both insigni®cantly dierent from zero. There is also asign pattern reversal on the JY series. For this series, high variance periodsare high mean periods. For the other two series, this result is reversed.It seems doubtful that the magnitudes of the regime shift parameters will belarge enough to generate the conditional mean dierences. For example, forthe BP series, the conditional mean for St  0 is 0.05 per cent and for theSt  1 period it is 0.02 per cent. It is dicult to see how this will generate abuy±sell spread of 0.29 per cent. This is con®rmed in Table 1.10 which showsthe results for simulations of this model using a normal random numbergenerator to generate errors. There is little evidence of this model capturing
Table 1.9 Regime shift parameter estimatesxt  0  1 St  
0 
1 St zt
P St  1 j St 1  1  p
P St  0 j St 1  1  1
p
P St  1 j St 1  0  1
q
P St  0 j St 1  0  qzt  N 0; 1
Xrate
0
 1000
1
 1000
0  1000
1  1000
p
q
BP
2.7811(0.2447)
12.2139(0.3578)
0.4923(0.3851)
0.7119(0.6374)
0.9933(0.0033)
0.9260(0.0342)
DM
6.7422(0.4188)
9.0407(0.5350)
1.1889(0.5313)
0.6388(0.7815)
0.9940(0.0034)
0.9738(0.0136)
JY
4.8973(0.2892)
11.8531(0.5093)
0.7646(0.3632)
2.4587(0.7655)
0.9387(0.0157)
0.8773(0.0266)
Estimation is by maximum likelihood. Numbers in parentheses are asymptotic standard errors.
Technical trading rules and regime shifts in foreign exchange
21
Table 1.10 Regime shift bootstrap
Result
Buy
b
FractionBuySell
s
Fraction > XrateFraction > XrateFraction > Xrate
0.048 000.004 000.004 00
0.482 000.512 000.620 00
0.538 000.394 000.424 00
0.972 000.994 000.998 00
0.566 000.494 000.270 00
0.020 000.002 000.000 00
Average Fraction > XrateSimulation meanXrate mean
0.008 000.000 250.001 24
0.530 000.014 200.014 19
0.430 000.484 420.494 96
0.994 000.000 070.001 66
0.442 000.014 490.014 60
0.000 000.000 180.002 90
Fraction > XrateFraction > XrateFraction > Xrate
0.036 000.042 000.056 00
0.532 000.700 000.780 00
0.538 000.434 000.644 00
0.996 000.996 000.982 00
0.600 000.326 000.230 00
0.000 000.006 000.008 00
Average Fraction > XrateSimulation meanXrate mean
0.028 000.000 640.001 70
0.684 000.014 050.013 60
0.552 000.608 950.600 85
0.994 000.000 800.001 06
0.370 000.014 810.015 12
0.000 000.000 160.002 76
Fraction > XrateFraction > XrateFraction > Xrate
0.008 000.004 000.024 00
0.734 000.650 000.572 00
0.642 000.732 000.666 00
1.000 000.998 000.978 00
0.428 000.452 000.412 00
0.0