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ADVANCED TRADING RULES

Elsevier Science

Chapter One

BLAKE LEBARON

1.1 INTRODUCTION

Techniques for using past prices to forecast future prices have a long and colourful history. Since the introduction of floating rates in 1973, the foreign exchange market has become another potential target for 'technical' analysts who attempt to predict potential trends in pricing using a vast repertoire of tools with colourful names such as channels, tumbles, steps and stumbles. These market technicians have generally been discredited in the academic literature since their methods are sometimes difficult to put to rigorous tests. This chapter attempts to settle some of these discrepancies through the use of bootstrapping techniques.

For stock returns, many early studies generally showed technical analysis to be useless, while for foreign exchange rates there is no early study showing the techniques to be of no use. Dooley and Shafer (1983) found interesting results using a simple filter rule on several daily foreign exchange rate series. In later work, Sweeney (1986) documents the profitability of a similar rule on the German mark. In an extensive study, Schulmeister (1987) repeats these results for several different types of rules. Also, Taylor (1992) finds that technical trading rules do about as well as some of his more sophisticated trend-detecting methods.

While these tests were proceeding, other researchers were trying to use more traditional economic models to forecast exchange rates with much less success. The most important of these was Meese and Rogo (1983). These results showed the random walk to be the best out-of-sample exchange rate forecasting model. Recently, results using nonlinear techniques have been mixed. Hsieh (1989) finds most of the evidence for nonlinearities in daily exchange rates is coming from changing conditional variances. Diebold and Nason (1990) and Meese and Rose (1990) found no improvements using nonparametric techniques in out-of-sample forecasting. However, LeBaron (1992) and Kim (1989) show small out-of-sample forecast improvements. During some periods, LeBaron (1992) found forecast improvements of over 5 per cent in mean squared error for the German mark. Both of these papers relied on some results connecting volatility with conditional serial correlations of the series.

This chapter breaks off from the traditional time series approaches and uses a technical trading rule methodology. With the bootstrap techniques of Efron (1979), some of the technical rules can be put to a more thorough test. This is done for stock returns in Brock, Lakonishok and LeBaron (1992). This chapter will use similar methods to study exchange rates. These allow not only the testing of simple random walk models, but the testing of any reasonable null model that can be simulated on the computer. In this sense, the trading rule moves from being a profit-making tool to a new kind of specification test. The trading rules will also be used as moment conditions in a simulated method of moments framework for estimating linear models.

Finally, the economic significance of these results will be explored. Returns from the trading rules applied to the actual series will be tested. Distributions of returns from the exchange rate series will be compared with those from risk-free assets and stock returns. These tests are important in determining the actual economic magnitude of the deviations from random walk behaviour that are observed.

Section 1.2 introduces the simple rules used. Section 1.3 describes the null models used. Section 1.4 presents results for the various specification tests. Section 1.5 implements the trading rules and compares return distributions and section 1.6 summarizes and concludes.

1.2 TECHNICAL TRADING RULES

This section outlines the technical rules used in this chapter. The rules are closely related to those used by actual traders. All the rules used here are of the moving average or oscillator type. Here, signals are generated based on the relative levels of the price series and a moving average of past prices:

ma_t = (1/L)[L - 1.summation over (i=0)] p_t-i

For actual traders, this rule generates a buy signal when the current price level is above the moving average and a sell signal when it is below the moving average. This chapter will use these signals to study various conditional moments of the series during buy and sell periods. Estimates of these conditional moments are obtained from foreign exchange time series, and these estimates are then compared with those from simulated stochastic processes. Section 1.4 of this chapter differs from most trading rule studies which look at actual trading profits from a rule. Actual trading profits will be explored in section 1.5.

1.3 NULL MODELS FOR FOREIGN EXCHANGE MOVEMENTS

This section describes some of the null models which will be used for comparison with the actual exchange rate series. These models will be run through the same trading rule systems as the actual data and then compared with those series. Several of these models will be bootstrapped in the spirit of Efron (1979) using resampled residuals from the estimated null model. This closely follows some of the methods used in Brock, Lakonishok and LeBaron (1992) for the Dow Jones stock price series.

The first comparison model used is the random walk:

log(p_t) = log(p_t-1) + ε_t

Log differences of the actual series are used as the distribution for ε_t and resampled or scrambled with replacement to generate a new random walk series. The new returns series will have all the same unconditional properties as the original series, but any conditional dependence will be lost.

The second model used is the GARCH model (Engle, 1982; Bollerslev, 1986). This model attempts to capture some of the conditional heteroskedasticity in foreign exchange rates. The model estimated here is of the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This model allows for an AR(2) process in returns. The specification was identified using the Schwartz (1978) criterion. Only the Japanese yen series required the two lags, but for better comparisons across exchange rates the same model is used. Estimation of this model is done using maximum likelihood.

Simulations of this model follow those for the random walk. Standardized residuals of the GARCH model are estimated as:

ε_t/√_t

These residuals are scrambled and the scrambled residuals are then used to rebuild a GARCH representation for the data series. Using the actual residuals for the simulations allows the residual distribution to differ from normality. Bollerslev and Wooldridge (1990) have shown that the previous parameter estimates will be consistent under certain deviations from normality. Therefore, the estimated residuals will also be consistent.

The third model has been proposed for foreign exchange markets in a paper by Engle and Hamilton (1990). It suggests that exchange rates follow long persistent swings following a two-state Markov chain. It is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This model allows both the mean and variance for exchange rate returns to move between two different states. Since this model is capable of generating persistent trends, it presents a strong possibility for generating the results seen using the trading rules. Estimation is done using maximum likelihood. For this model, the simulations will use normally distributed random numbers from a computer random number generator.

1.4 EMPIRICAL RESULTS

1.4.1 Data summary

The data used in this chapter are all from the EHRA macro data tape from the Federal Reserve Bank. Weekly exchange rates for the British pound (BP), the German mark (DM) and the Japanese yen (JY) are sampled every Wednesday from January 1974 to February 1991 at 12:00 pm EST.

Returns are created using log first differences of these weekly exchange rates quoted in dollars/fx. Table 1.1 presents some summary statistics for these return series. All three series show little evidence of skewness and are slightly leptokurtic. These properties are common for many high frequency asset returns series. The first ten autocorrelations are given in the rows labelled ρ_n. The Bartlett asymptotic standard error for these series is 0.033. The BP shows little evidence of any autocorrelation except for lags four and eight, while the DM shows some weak evidence of correlation, and the JY shows strong evidence for some autocorrelation. The Ljung–Box–Pierce statistics are shown on the last row. These are calculated for ten lags and are distributed [chi square] 10 under the null of independently identically distributed. The p-values are included for each in parentheses. The BP and JY series reject independence, whereas the DM series does not.

The interest rate series used are also from the EHRA macro data tape. For the dollar, the weekly eurodollar rate is used. For the pound, the international money market call money rate is used. For the mark, the Frankfurt interbank call money rate is used, and for the yen, the Tokyo unconditional lender rate. Weekly rates are constructed ex post from the compounded rates from Wednesday to Tuesday. These rates can only be viewed as proxies for the desirable situation of having a set of interest rates from the same offshore market at the same maturity. At this time that is not available.

1.4.2 Random walk comparisons

In this section, simulations are performed comparing conditional moments from the technical trading rules with a bootstrapped random walk generated from the actual returns time series scrambled with replacement. Three moving-average rules will be used, the twenty-week, thirty-week and fifty-week moving averages. These are fairly common lengths used by traders. We will see that the results are not very sensitive to the lengths used. The moving-average rules force us to start the study after a certain number of weeks have passed. For this chapter, all tests for all the rules begin after week fifty. This gives the same number of weekly observations for all three rules.

Table 1.2 presents the results comparing the actual series for the BP with 500 simulated random walks. Six comparison statistics are computed in this table. First, the column labelled 'Buy' refers to the conditional mean during buy periods. This is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where N_b is the number of buy signals in the sample and I^b_t is an indicator variable for a buy signal at time t. The second column, labelled σ_b', looks at the standard deviation of this same set of returns. This is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This gives a simple idea of how risky the buy or sell periods might be and tells us something about what is happening to conditional variance. The third column, labelled 'Fraction buy', is just the fraction of buy weeks N_b/N. The next two columns, 'Sell' and 'σ_s' repeat the previous descriptions for the sell periods. Let m_s be the mean during the sell periods. The final column, 'Buy–Sell', refers to the difference between the buy and sell means, m_b - m_s.

(Continues...)

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