00 000.000 000.004 00
Average Fraction > XrateSimulation meanXrate mean
0.004 000.000 890.002 41
0.630 000.014 220.013 94
0.684 000.597 010.564 39
0.996 000.000 870.001 00
0.422 000.013 640.013 79
0.002 000.000 020.003 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 regime-shift models. These models are generated from estimatedparameters and computer generated normal random numbers.
what the trading rules are picking up for any of the series. For the DM andBP series, the buy±sell dierences are actually negative. For all the series, thep-values for the buy±sell dierences are all close to zero.This should not rule out this model in general, but at these relatively highfrequencies (weekly) it does not seem to capture what is going on. There maybe some numerical problems in estimation as the probabilities, p and q, areclose to one at this time horizon. In Engle and Hamilton (1990), theconditional mean estimates are signi®cant and larger than those foundhere. This may be due to the use of quarterly data. It remains to be seenwhether other estimation techniques can help repair these results for theregime shift model.
22
Advanced Trading Rules
1.4.5 Interest rate differentialsThe use of the previous simple processes for foreign exchange movementsignores much of the information available in world ®nancial markets. Thissection incorporates some of this information into further simulations.The relation that will be used here is uncovered interest parity. This relationcan be written as:Et st1 
st  i t
i t
where i and i  are the domestic and foreign interest rates and st is the log ofthe exchange rate. In a risk-neutral world, the interest rate dierential overthe appropriate horizon should be equal to the expected drift of the exchangerate.Although uncovered parity, and theories closely related to it, have beenrejected by several studies, it is important to see if this long-range persistentdrift could be causing what the trading rules are picking up. For this test amodel of the form:st1  st  it
i t  "t
where "t is independently identically distributed noise will be used. One majorproblem is getting the interest rates and their timing correct. This is extremelydicult. For the weekly exchange rates used here, weekly eurorates would bethe most useful series to have. This study is constrained by what is availableon the EHRA tapes. For the dollar, weekly eurorates are available at dailyfrequency and will be used as the risk-free dollar rate for each week beginningat the close on Wednesday. Unfortunately, the other currencies do not havesuch rates available. The weekly rates are constructed from daily ex postovernight rates from Wednesday to the following Tuesday. Assuming theexpectations hypothesis holds at the very short end of the term structure:it;7 
6X
it;7 
6X
or
Et it1;1
i0
i0
iti;1  et
where Et et   0. The expected drift term it i t is therefore it i~  et wherei~ is the ex post rate constructed from the overnight rates. Therefore:st1
st  it
i~t  t
where Et t   0.The time period studied is shortened due to data availability. For the BPand DM, the series now starts in January 1975. For the JY, the series begins
Technical trading rules and regime shifts in foreign exchange
23
in October 1977. The lengths of the BP, DM and JY series are 832, 832 and690 weeks respectively.Rather than immediately adjusting these series for the interest dierential,a slightly dierent approach is taken at ®rst. Representative series of theform:st1  st  t  "twill be simulated. The drift, t , is obtained from the appropriate interestdierential. An estimate of the residual series "^t is obtained by removing thedrift from the actual exchange rate changes. This is then scrambled withreplacement, and a new series is generated using the original drift series andthe scrambled residuals. This gives us representative exchange rate seriesre¯ecting the appropriate information from the interest rates.These simulations are then run through the same trading rule tests run inprevious sections. Results of these tests are presented in Table 1.11. Theresults are comparable to those found for the random walk simulations inTables 1.2, 1.3 and 1.4. For all three series, none of the rules generate buy±selldierences as large as those generated from the original series. The adjustment for the interest dierential appears to have had little eect on thetrading-rule results.Table 1.12 repeats some of the earlier GARCH simulations accounting forinterest dierentials. In this case, the more traditional approach of subtracting the expected drift from the exchange rate returns is done. A GARCHmodel is then ®tted to these `zero drift' terms and simulated back usingscrambled, standardized residuals as in section 1.4.3. Comparing Table 1.12with Table 1.8 shows very few dierences. Adjusting the exchange rate seriesusing the expected drift has very little impact on the GARCH simulations.The large (small) returns during buy (sell) are still not replicated well by thesimulated null model.1.4.6 Simulated method of moments estimatesThe previous tests have not incorporated the trading-rule diagnostic tests intothe estimation procedure. This section presents a method where the two canbe brought together in one combined procedure.One problem with the trading-rule measures is that it is dicult to deriveanalytic results for these measures. One technique for estimating parametersusing conditions which can only be simulated is simulated method ofmoments. This technique was developed for cross-section data by McFadden(1989) and Pakes and Pollard (1989). It is extended to time series cases inDue and Singleton (1993) and Ingram and Lee (1991).
24
Advanced Trading RulesTable 1.11 Interest rate drift
Result
Buy
b
FractionBuySell
s
Fraction > XrateFraction > XrateFraction > Xrate
0.012 000.010 000.006 00
0.312 000.616 000.714 00
0.406 000.284 000.328 00
0.982 000.992 000.994 00
0.650 000.296 000.152 00
0.002 000.000 000.000 00
Average Fraction > XrateSimulation meanXrate mean
0.006 000.000 140.001 55
0.552 000.014 680.014 57
0.320 000.476 600.507 31
0.994 000.000 110.001 65
0.330 000.014 610.014 91
0.000 000.000 020.003 20
Fraction > XrateFraction > XrateFraction > Xrate
0.032 000.110 000.046 00
0.462 000.756 000.822 00
0.544 000.540 000.750 00
0.990 000.960 000.972 00
0.606 000.212 000.166 00
0.004 000.024 000.008 00
Average Fraction > XrateSimulation meanXrate mean
0.040 000.000 280.001 57
0.704 000.015 410.015 05
0.630 000.538 980.510 50
0.982 000.000 480.001 35
0.294 000.015 440.015 82
0.004 000.000 200.002 92
Fraction > XrateFraction > XrateFraction > Xrate
0.022 000.002 000.124 00
0.354 000.350 000.576 00
0.762 000.866 000.812 00
0.996 000.996 000.978 00
0.638 000.664 000.594 00
0.002 000.000 000.012 00
Average Fraction > XrateSimulation meanXrate mean
0.024 000.000 770.002 26
0.424 000.015 210.015 31
0.814 000.596 420.527 17
0.992 000.000 980.001 21
0.642 000.015 330.015 02
0.000 000.000 210.003 48
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 replications of series generated with conditional drift equal togiven interest rate dierentials. rt  t  "t where t corresponds to the interest rate dierential attime t.
We will follow the procedure of ®tting a linear process to the data using aset of moment conditions which include the trading rules. The trading rulesmust ®rst be modi®ed to ®t into a moment condition framework. De®ne rt asthe returns series of interest. Also, let pt be the price at time t, where:rt  log pt 
log pt 1 
Again, use the moving average of length L at time t:mat L  1=L
L 1Xi0
pt i
Technical trading rules and regime shifts in foreign exchange
25
Table 1.12 GARCH zero drift
Result
Buy
b
FractionBuySell
s
Fraction > XrateFraction > XrateFraction > Xrate
0.064 000.030 000.014 00
0.368 000.514 000.548 00
0.362 000.278 000.342 00
0.902 000.934 000.978 00
0.660 000.420 000.332 00
0.036 000.006 000.004 00
Average Fraction > XrateSimulation meanXrate mean
0.024 000.000 510.001 97
0.480 000.014 800.014 62
0.318 000.512 140.541 96
0.944 000.000 350.001 49
0.444 000.015 190.015 02
0.006 000.000 850.003 46
Fraction > XrateFraction > XrateFraction > Xrate
0.156 000.138 000.034 00
0.698 000.792 000.788 00
0.270 000.182 000.388 00
0.916 000.894 000.950 00
0.608 000.486 000.466 00
0.050 000.054 000.008 00
Average Fraction > XrateSimulation meanXrate mean
0.092 000.000 520.001 59
0.762 000.015 760.013 85
0.298 000.477 580.519 42
0.924 000.000 690.001 76
0.536 000.015 130.014 62
0.026 000.001 210.003 35
Fraction > XrateFraction > XrateFraction > Xrate
0.024 000.016 000.194 00
0.360 000.436 000.352 00
0.568 000.556 000.598 00
0.962 000.970 000.932 00
0.528 000.478 000.636 00
0.006 000.002 000.040 00
Average Fraction > XrateSimulation meanXrate mean
0.050 000.000 600.002 01
0.372 000.015 220.015 43
0.576 000.492 500.473 34
0.970 000.000 520.001 93
0.546 000.015 130.014 99
0.000 000.001 120.003 94
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. Models are estimated andsimulated using foreign exchange returns series with interest rate dierentials removed.
One ®rst guess for trading rule related moment might be:  pt 1E Srmat 1 twhere S x  1 if x  1 and S x  1 if x < 1. This will not do forsimulated method of moments, since the ®rst derivatives of this momentwill not necessarily be continuous in the parameters of the process rt . Thecondition must be replaced with a `smooth substitute'. The hyperbolictangent does a good job of being just such a function. 7 Replace the abovecondition with:
26
Advanced Trading Rules
E tanh

pt 11=mat 1
 1 rt
This condition can now be added to a more standard set of momentconditions. 8The estimation procedure will attempt to ®t an AR(2) to each of theexchange rate series. When using any method of moments estimator,choosing the moment conditions to use is not always a trivial procedure.Here, the choice of moments will follow the goal of trying to see whether alinear model does a good job of replicating some properties of the data(autocovariances) as well as the trading-rule results. This goal does not intendto get the tightest estimates of the parameters on the model. For this reason,the set of moment conditions will be rather small relative to other studies. Theactual data will be aligned to simulated data using the mean variance, the ®rstthree lagged autocovariances and one trading-rule moment. This gives a totalof six moment conditions. For the trading-rule moment condition, the thirtyweek moving average is used. The results are generally similar across theother rules. 9There are two ®nal details left for estimation. The variance±covariancematrix is estimated using the Newey±West (1987) weighting using ten lags.The lag length has been moved from ®ve to ®fty and the results have notchanged greatly. This is important in this case, since the moving average maygenerate very long-range dependence in the estimated moments. Finally, thenumber of simulations is set to ®fty times the sample size. For most of theseseries, this gives simulation samples in the range of 40 000 to 45 000.Results of the estimation are given in Table 1.13. This table shows theestimated parameters and the chi-squared goodness-of-®t estimate for theAR(2). For the BP series, the results show a weak, but insigni®cant AR(1)parameter combined with a rejection of the moment conditions as indicatedby the  2 test. The AR(2) is not able to match up with both the covariancesand the trading-rule results. The next rows present results for similarestimation removing the trading-rule moment condition. Similar parameterestimates are obtained, but now the goodness-of-®t statistic is only signi®cantat the 18 per cent level. The trading-rule condition has clearly added animportant restriction for this time series.The row labelled `DM' repeats this procedure for the DM series. In thiscase, the model estimates two larger AR coecients and the goodness-of-®ttest is only signi®cant at the 13 per cent level. The AR(2) is not stronglyrejected here. Part of the reason for this can be seen in Table 1.1. There issome correlation in this series at the ®rst two lags which allows the estimatedAR coecients to be larger. When the rule is removed the chi-square statisticstill remains small with a signi®cance level of 73 per cent.
Technical trading rules and regime shifts in foreign exchange
27
Table 1.13 SMM estimationrt    1 rt 1
  2 rt 2
"t  N 0; 1
  "t
Series
Condition


1
2
2
BP
Rule
0.046(0.054)
1.407(0.074)
0.043(0.035)
0.017(0.041)
8.261(0.016)
BP
No rule
0.023(0.057)
1.450(0.073)
0.031(0.034)
0.029(0.039)
1.793(0.181)
DM
Rule
0.099(0.052)
1.410(0.061)
0.051(0.031)
0.042(0.043)
3.989(0.136)
DM
No rule
0.071(0.054)
1.427(0.062)
0.052(0.031)
0.045(0.043)
0.120(0.729)
JY
Rule
0.152(0.055)
1.364(0.062)
0.104(0.039)
0.103(0.042)
6.819(0.033)
JY
No rule
0.125(0.059)
1.405(0.062)
0.100(0.039)
0.088(0.040)
2.066(0.150)
BPZD
Rule
0.075(0.059)
1.463(0.078)
0.063(0.035)
0.017(0.041)
8.540(0.014)
BPZD
No rule
0.043(0.063)
1.492(0.078)
0.043(0.035)
0.022(0.040)
1.773(0.183)
DMZD
Rule
0.021(0.055)
1.404(0.063)
0.063(0.031)
0.047(0.045)
3.035(0.219)
DMZD
No rule
0.011(0.058)
1.428(0.063)
0.063(0.031)
0.048(0.045)
0.009(0.924)
ZYZD
Rule
0.061(0.070)
1.501(0.062)
0.098(0.040)
0.113(0.043)
5.734(0.057)
JYZD
No rule
0.006(0.073)
1.509(0.064)
0.100(0.042)
0.083(0.044)
1.230(0.541)
Parameters estimated by simulated method of moments. Numbers in parentheses are asymptoticstandard errors for the parameters and the p-value for the chi-squared goodness-of-®t test.Moments used are the mean, variance, 3 autocovariances and the 30-day moving-average tradingrule. The chi-squared statistic has 6 4  2 degrees of freedom when the trading rule is used and5 4  1 degrees of freedom when it is not used. The variance±covariance matrix is estimatedusing the Newey±West (1987) technique with 10 lags.
28
Advanced Trading Rules
For the JY series, the AR(2) speci®cation is rejected at the 3 per cent level. Inthis case, the model is estimating the largest AR parameters of the threeseries. However, these appear to not be enough to match the trading-rulecondition. This is again demonstrated by removing this condition. After thisis done, the chi-squared statistic drops to 2.1, which has a signi®cance level of0.5. The next rows in the table repeat these results for the zero drift series.These series generate results similar to those for the original series.The simulated method of moments procedure has added to the earlierresults. The procedure rejected the simple linear speci®cation for two of thethree foreign exchange series. This rejection followed from a procedure thatcombined standard autocovariance moments with conditions based on thetrading rules.1.5 ECONOMIC SIGNIFICANCE OF TRADING-RULE PROFITSThe tests run in the previous section have shown the moving-average tradingrules to be able to detect periods of high and low returns. These returns arestatistically large when compared with several dierent stochastic processesfor the exchange rate series. These results are interesting in attempting tomodel the exact dynamics in the foreign exchange market, but they do notgive us the economic signi®cance of these rules. This section will attempt tomeasure the trading-rule results. Transaction costs and interest rates will beaccounted for and some attempts will be made to measure the riskiness of thestrategies relative to other assets.The moving-average trading rules will be implemented as suggested by theprevious tests. When the current price is above the long moving-average buyis indicated and when it is below a sell is indicated. The implementation testsperformed here will concentrate on the thirty-day moving average alone.When a buy is indicated in a currency, the trader takes a long position in thatcurrency and deposits this in foreign bonds. In the rules used here, the traderalso will borrow dollars and invest these funds in the foreign currency. Thetrader will take a 50 per cent leveraged position. 10 This generally follows theprocedure used in Dooley and Shafer (1983). Sweeney (1986) takes a slightlymore cautious route of never borrowing and moving only from domesticbonds to foreign bonds conditional on the signal. This strategy leaves thetrader exposed to foreign exchange risk only part of the time. There isobviously a continuous range of adjusting the leverage parameter whichmoves the outcome of the strategy both in terms of risk and return. In thisstudy, the 50 per cent leverage strategy will be used for comparability withother studies and for the purpose of risk comparisons with the stock market.Trading is done once a week. When the rule signals a change in position, a
Technical trading rules and regime shifts in foreign exchange
29
Table 1.14 Rule implementation summary
Series
Trades
BPDMJYCRSP VW$ RF
364326
Return/year
Return/week
Std/week
16.712.620.115.29.5
0.350.260.410.320.18
2.252.192.172.180.05
0.070.080.03
BH
BH/week
BH(std)
9.96.28.4
0.200.130.17
1.571.541.52
This table summarizes the results of the trading rules over the full sample. is the estimatedCAPM beta for the trading strategy estimated using weekly data. t(return-CRSP) is a t-statisticfor equality of the returns for the strategy and CRSP. BH stands for the buy and hold strategy inthe foreign currency holding foreign bonds.
trade is made. Transactions costs are assumed to be 0.1 per cent of the size ofthe trade. This appears to be a reasonable estimate and is used in Dooley andShafer (1983). Some studies are slightly above this number (Sweeney, 1986,uses 1/8 per cent), while others claim that this is a maximum for foreignexchange trading. The weekly eurodollar rate series and daily call moneyovernight rates are used again with compounding occurring at dailyfrequencies for the daily series. An interest rate dierential of 3 per centper year is used to estimate the borrowing rates from the lending rates fromthe tape. This is probably an upper bound on the borrowing and lendingspread and is estimated from the current prime rate CD spread. Results willbe compared with those from buying and holding stocks in the US market.The CRSP value weighted index, including dividends, will be used torepresent this asset. All tests begin in October 1977 and end in December1989.Table 1.14 presents some summary statistics comparing the results for thevarious assets. The row labelled `BP' gives the trading strategy for the pound.The table shows that the strategy executed thirty-six trades and yielded anaverage return of 16.7 per cent per year continuously compounded. It had aweekly standard deviation over the period of 2.25 per cent. The columnlabelled ` ' estimates the CAPM beta for the dynamic strategy using theCRSP portfolio as the market proxy. Although a static CAPM based only ondomestic securities is probably not a good representation of risk, it is stillinteresting to observe how correlated the strategy is with the stock marketand how much potential there is for diversi®cation. For all currencystrategies, the is negative and very close to zero. The last three columnspresent results for a buy and hold strategy in the foreign currency and bonds.For the pound, this is 9.9 per cent with a weekly standard deviation of 1.57per cent. This should be compared with the return to only holding dollar
30
Advanced Trading Rules
bonds (reported in the last row) of 9.5 per cent with a weekly standarddeviation of 0.05 per cent.The next three rows present results for the DM, JY and CRSP seriesrespectively. All the series have similar standard deviation and beta riskcharacteristics. The DM underperforms CRSP by about 2.6 per cent and theJY exceeds the CRSP series by about 5 per cent. In each case, the strategiesdramatically dominate the buy and hold portfolios.Two currencies give returns in excess of the CRSP return. The importanteconomic question is whether these dynamic strategies oer an important newsecurity in terms of risk and return. This is a dicult question to answerwithout an appropriate model for risk or the exact stochastic process foreither foreign exchange or stocks. A fairly straightforward technique will beused to try to get some initial answers to this question. Returns will bemeasured over ®xed horizons chosen at random out of the entire sample. Inother words ®x the horizon at one year and estimate returns at randomlychosen one-year periods during the sample. This will generate a jointdistribution of stock and exchange rate returns which can be compared. 11Results for 500 simulations at the 1-year horizon are presented in Table1.15. For the BP series, the simulations gave an average annual return of 19.5per cent with a standard deviation of 13.7 per cent. This compares with areturn of 16.2 per cent with a standard deviation of 18 per cent for the CRSPseries. The table also presents some other risk measures. The ®rst,prob < RF 5%, reports the estimated probability of getting a return ofless than 5 per cent below the risk-free rate. This number attempts to capturesome aspect of `drawdown' risk. For the BP series, this happens in 15 per centof the simulations as compared with 29 per cent for the CRSP series